side the rectiline angle MAB either of which is equall to the rectiline angle geuen CDE which was required to be done An other construction also and demonstration after Pelitarâ—us And if the perpendiculer line chaunce to fall without the angle geuen namely if the angle geuen be an acute angle the selfe same manner of demonstration will serue but onely that in stede of the second common sentence must be vsed the 3. common sentence Appollonius putteth another construction demonstration of this propositioÌ„ which though the demonstration thereof depende of propositions put in the third booke yet for that the construction is very good for him that wil redely and mechanically without demonstration describe vpon a line geuen and to a point in it geuen a rectiline angle equall to a rectiline angle geuen I thought not amisse here to place it And it is thus Oenopides was the first inuenter of this proposition as witnesseth Eudemius The 15. Theoreme The 24. Proposition If two triangles haue two sides of the one equall to two sides of the other ech to his correspondent side and if the angle coÌ„tained vnder the equall sides of the one be greater then the angle contayned vnder the equall sides of the other the base also of the same shal be greater then the base of the other SVppose that there be two triangles ABC and DEF hauing two sides of the one that is AB and AC equall to two sides of the other that is to DE and DF ech to his correspondent side that is the side AB to the side DE and the side AC to the side DF and suppose that the angle BAC be greater then the angle EDF Then I saye that the base BC is greater then the base EF. For forasmuch as the angle BAC is greater then the angle EDF make by the 23. proposition vpon the right line DE and to the point in it geueÌ„ D an angle EDG equall to the angle geuen BAC And to one of these lines that is either to AC or DF put an equall line DG And by the first peticioÌ„ draw a right line from the point G to the point E and an other from the point F to the point G. And forasmuch as the line AB is equall to the line DE and the line AC to the line DG the one to the other and the angle BAC is by construction equall to the angle EDG therefore by the 4. proposition the base BC is equall to the base EG Agayne for as much as the line DG is equall to the line DF therâ— by the 5. proposition the angle DGF is equall to the angle DFG VVherefore the angle DFG is greater then the angle EGF VVherefore the angle EFG is much greater then the angle EGF And forasmuch as EFG is a triangle hauing the angle EFG greater then the angle EGF and by the 18. prâ—position vnder the greater angle is subtended the greater side therefore the side EG is greater then the side EF. But the side EG is equall to the side BC wherefore the side BC is greater then the side EF. If therefore two triangles haue two sides of the one equall to two sides of the other eche to his correspondent side and if the angle contayned vnder the equall sides of the one be greater then the angle contayned vnder the equall sides of the other the base also of the same shal be greater then the base of the other which was required to be proued In this Theoreme may be three cases For the angle EDG being put equall to the angle BAC and the line DG being put equall to the line AC and a line being drawen from E to G the line EG shall either fall aboue the line GF or vpon it or vnder it Euclides demonstration serueth when the line GE falleth aboue the line GF as we haue before manifestly seene But now let the line EG fall vnder the line E F as in the figure here put And forasmuch as these two lines AB and AC are equall to these two lines DE and DG the one to the other and they contayne equall angles therefore by the 4. proposition the base BC is equal to the base EG And forasmuch as within the triangle DEG the two linnes DF and FE are set vpon the side DE therfore by the 21. proposition the lines DF and Fâ— are lesse then the outward lines DG and GE but the line DG is equal to the line DF. Wherfore the line GE is greater then the line FE But GE is equall to BC. Wherefore the line BC is greater the the line EF. Which was required to be proued It may peraduenture semeâ— that Euclide should here in this proposition haue proued that not onely the bases of the triangles are vnequall but also that the areas of the same are vnequall for so in the fourth proposition after he had proued the base to be equall he proued also the areas to be equall But hereto may be answered that in equall angles and bases and vnequall angles and bases the consideration is not like For the angles and bases being equall the triangles also shall of necessitie be equall but the angles and bases being vnequall the areas shall not of necessitie be equall For the triangles may both be equall and vnequall and that may be the greater whiche hathe the greater angle and the greater base and it may also be the lesse And for that cause Euclide made no mencion of the comparison of the triangles VVhereof this also mought be a cause for that to the demonstration thereof are required certayne Propositions concerning parallel lines which we are not as yet come vnto Howbeit after the 37â— proposition of his booke you shal find the comparison of the areas of triangles which haue their sides equall and their bases and angles at the toppe vnequall The 16. Theoreme The 25. Proposition If two triangles haue two sides of the one equall to two sydes of the other eche to his correspondent syde and if the base of the one be greater then the base of the other the angle also of the same coÌ„tayned vnder the equall right linesâ— shall be greater then the angle of the other SVppose that there be two triangles A B C and DEF hauing two sides of tb'one that is AB and AC equall to two sides of the other that is to DE and DF ech to his correspondent side namely the side AB to the side DF and the side AC to the syde DF. But let the base BC be greater then the base EF. TheÌ„ I say thay the angle BAC is greater then the angle EDF For if not then is it either equall vnto it or lesse then it But the angle BAC is not equall to the angle EDF for if it were equall the base also BC should by the 4. proposition be equal to the base EF but by supposition it is not VVherfore
triangle vnto the section deuideth the angle of the triangle into two equall partes This construction is the halfe part of that Gnomical figure described in the 43. proposition of the first booke which Gnomical figure is of great vse in a maner in all Geometrical demonstrations The 4. Theoreme The 4. Proposition In equiangle triangles the sides which coÌ„taine the equall angles are proportionall and the sides which are subtended vnder the equall angles are of like proportion SVppose that there be two equiangle triangles ABC and DCE and let the angle ABC of the one triangle be equall vnto the angle DCE of the other triangle and the angle BAC equall vnto the angle CDE and moreouer the angle ACB equall vnto the angle DEC Then I say that those sides of the triangles ABC DCE which include the equall angles are proportionall and the side which are subtended vnder the equall angles are of like proportion For let two sides of the sayd triangles namely two of those sides which are subtended vnder equall angles as for example the sides BC and CE be so set that they both make one right line And because the angles ABC ACB are lesse then two right angles by the 17. of the first but the angle ACB is equall vnto the angle DEC therfore the angles ABC DEC are lesse theÌ„ two right angles Wherefore the lines BA ED being produced will at the length meete together Let them meete and ioyne together in the poynt F. And because by supposition the angle DCE is equall vnto the angle ABC therfore the line BF is by the 28. of the first a parallell vnto tâ—e line CD And forasmuch as by supposition the angle ACB is equall vnto the angle DEC therefore againe by the 28. of the first the line AC is a parallell vnto the line FE Wherefore FADC is a parallelogramme Wherfore the side FA is equall vnto the side DC and the side AC vnto the side FD by the 34. of the first And because vnto one of the sides of the triangle BFE namely to FE is drawen a parallell line AC therefore as BA is to AF so is BC to CE by the 2. of the sixt But AF is equall vnto CD Wherfore by the 11. of the fift as BA is to CD so is BC to CE which are sides subtended vnder equall angles Wherefore alternately by the 16. of the fift as AB is to BC so is DC to CE. Againe forasmuch as CD is a parallell vnto BF therefore againe by the 2. of the sixt as BC is to CE so is FD to DE. But FD is equall vnto AC Wherefore as BC is to CE so is AC to DE which are also sides subtended vnder equall angles Wherfore alternately by the 16. of the fift â—s BC is to CA so is CE to EDâ— Wherfore forasmuch as it hath bene demonstrated that as AB is vnto BCâ— so is DC vnto CEâ— but as DC is vnto CA so is CE vnto EDâ— it followeth of equalitie by the 22. of the fift that â—s BA is vnto AC so is CD vnto DEâ— Wherfore in eqâ—iangle triangleâ— the sides which include the equall angles are proportionall and the sides which are subtâ—nded vnder the equall angles are of like proportion â—hich was required to be demonstrated The 5. Theoreme The 5. Proposition If two triangles haue their sides proportionall the triangâ—â—s are equiangle and those angles in theÌ„ are equall vnder which are subtended sides of like proportion SVppose that there be two triangles ABC DEF hauing their sides proportionall as AB is to BC so let DE be to EF as BC is to AC so let EF be to DF and moreouer as BA is to AC so let ED be to DF. Then I say that the triangle ABC is equiangle vnto the triangle DEF and those angles in them are equall vnder which are subtended sides of like proportion that is the angle ABC is equall vnto the angle DEF and the angle BCA vnto the angle EFD and moreouer the angle BAC to the angle EDF Vpon the right line EF and vnto the pointes in it E F describe by the 23. of the first angles equall vnto the angles ABC ACB which let be FEG and EFG namely let the angle FEG be equall vnto the angle ABC and let the angle EFG be equall to the angle ACB And forasmuch as the angles ABC and ACB are lesse then two right angles by the 17. of the first therefore also the angles FEG and EFG are lesse then two right angles Wherefore by the 5. petition of the first the right lines EG FG shall at the length concurre Let theÌ„ concurre in the poynt G. Wherefore EFG is a triangle Wherefore the angle remayning BAC is equall vnto the angle remayning EGF by the first Corollary of the 32. of the first Wherfore the triangle ABC is equiangle vnto the triangle GEF Wherefore in the triangles ABC and EGF the sides which include the equall angles by the 4. of the sixt are proportionall and the sides which are subtended vnder the equall angles are of like proportion Wherefore as AB is to BC so is GE to EF. But as AB is to BC so by supposition is DE to EF. Wherefore as DE is to EF so is GE to EF by the 11. of the fift Wherefore either of these DE and EG haue to EF one and the same proportion Wherefore by the 9. of the fift DE is equall vnto EG And by the same reason also DF is equall vnto FG. Now forasmuch as DE is equall to EG and EF is common vnto them both therefore these two sides DE EF are equall vnto these two sides GE and EF and the base DF is equall vnto the base FG. Wherefore the angle DEF by the 8. of the first is equall vnto the angle GEF and the triangle DEF by the 4. of the first is equall vnto the triangle GEF and the rest of the angles of the one triangle are equall vnto the rest of the angles of the other triangle the one to the other vnder which are subtended equall sides Wherefore the angle DFE is equall vnto the angle GFE and the angle EDF vnto the angle EGF And becauseâ— the angle FED is equall vnto the angle GEF but the angle GEF is equall vnto the angle ABC therefore the angle ABC is also equall vnto the angle FED And by the same reason the angle ACB is equall vnto the angle DFEâ— and moreouer the angle BAC vnto the angle EDF Wherefore the triangle ABC is equiangle vnto the triangle DEF If two triangles therefore haue their sides proportionall the triangles shall be equiangle those angles in them shall be equall vnder which are subtended sides of like proportion which was required to be demonstrated The 6. Theoreme The 6. Proposition If there be two triangles wherof the one hath one angle equall to one angle of
â—ignifieth Last of all a Dodecahedron for that it is made of Pâ—ntagoâ— whose angles are more ample and large then the angles of the other bodies and by that â—eaâ—â—â— draw more â—â— rounâ—nes 〈◊〉 to the forme and nature of a sphere they assigned to a sphere namely 〈…〉 Who so will 〈…〉 in his Tineus shall â—ead of these figures and of their mutuall proportionâ—â—â—raunge maâ—terâ— which hâ—re are not to be entreated of this which is sayd shall be sufficient for the 〈◊〉 of them and for thâ— declaration of their diffinitions After all these diffinitions here set of Euclide Flussas hath added an other diffinition which 〈◊〉 of a Parallelipipedon which bicause it hath not hitherto of Euclide in any place bene defined and because it is very good and necessary to be had I thought good not to omitte it thus it is A parallelipipedon is a solide figure comprehended vnder foure playne quadrangle figures of which those which are opposite are parallels Because these fiue regular bodies here defined are not by these figures here set so fully and liuely expressed that the studious beholder can throughly according to their definitions conceyue them I haue here geuen of them other descriptions drawn in a playne by which ye may easily attayne to the knowledge of them For if ye draw the like formes in matter that wil bow and geue place as most aptly ye may do in fine pasted paper such as pastwiues make womeÌ„s pastes of theÌ„ with a knife cut euery line finely not through but halfe way only if theÌ„ ye bow and bende them accordingly ye shall most plainly and manifestly see the formes and shapes of these bodies euen as their definitions shew And it shall be very necessary for you to hadâ—â—tore of that pasted paper by you for so shal yoâ— vpon it 〈…〉 the formes of other bodies as Prismes and Parallelipopedons 〈…〉 set forth in these fiue bookes following and see the very 〈◊〉 of thâ—se bodies there meÌ„cioned which will make these bokes concerning bodies as easy vnto you as were the other bookes whose figures you might plainly see vpon a playne superficies Describe thiâ— figurâ— which consistâ—th of twâ—luâ—â—quilâ—â—â—â— and â—quianglâ— Pâ—ntâ—â—â—â—â— vpoâ— the foresaid mattâ—r and finely cut as before was â—â—ught tâ—â—â—lâ—uâ—n lines containâ—d within thâ— figurâ— and bow and folde the Penâ—â—gonâ— accordingly And they will so close toâ—ethâ—â— thaâ— thâ—y will â—â—kâ— thâ— very forme of a Dodecahedron If ye describe this figure which consisteth of twentie equilater and equiangle triangles vpon the foresaid matter and finely cut as before was shewed the ninâ—tâ—ne lines which are contayned within the figure and then bowe and folde them accordingly they will in such sort close together that therâ— will be made a perfecte forme of an Icosahedron Because in these fiue bookes there are sometimes required other bodies besides the foresaid fiue regular bodies as Pyramises of diuers formes Prismes and others I haue here set forth three figures of three sundry Pyramises one hauing to his base a triangle an other a quadrangle figure the other â— Pentagonâ— which if ye describe vpon the foresaid matter finely cut as it was before taught the lines contained within ech figure namely in the first three lines in the second fower lines and in the third fiue lines and so bend and folde them accordingly they will so close together at the toppes that they will â—ake Pyramids of that forme that their bases are of And if ye conceaue well the describing of these ye may most easily describe the body of a Pyramis of what forme so euer ye will. Because these fiue bookes following are somewhat hard for young beginners by reason they must in the figures described in a plaine imagine lines and superficieces to be eleuated and erected the one to the other and also conceaue solides or bodies which for that they haue not hitherto bene acquainted with will at the first sight be somwhat sâ—raunge vnto theÌ„ I haue for their more â—ase in this eleuenth booke at the end of the demonstration of euery Proposition either set new figures if they concerne the eleuating or erecting of lines or superficieces or els if they concerne bodies I haue shewed how they shall describe bodies to be compared with the constructions and demonstrations of the Propositions to them belonging And if they diligently weigh the maner obserued in this eleuenth booke touching the description of new figures agreing with the figures described in the plaine it shall not be hard for them of them selues to do the like in the other bookes following when they come to a Proposition which concerneth either the eleuating or erecting of lines and superficieces or any kindes of bodies to be imagined ¶ The 1. Theoreme The 1. Proposition That part of a right line should be in a ground playne superficies part eleuated vpward is impossible FOr if it be possible let part of the right line ABC namely the part AB be in a ground playne superficies and the other part therof namely BC be eleuated vpwarde And produce directly vpoÌ„ the ground playne superficies the right line AB beyond the point B vnto the point D. Wherfore vnto two right lines geuen ABC and ABD the line AB is a common section or part which is impossible For a right line can not touche a right line in 〈◊〉 pointes then one vâ—lesse those right be exactly agreing and laid the one vpon the other Wherfore that part of a right line should be in a ground plaine superficies and part eleuated vpward is impossible which was required to be proued This figure more plainly setteth forth the foresaid demonstratioÌ„ if ye eleuate the superficies wheriâ— the line BC. An other demonstration after Flâ—sâ—â—s If it be possible let there be a right line ABG whose part AB let be in the ground playne superficies AED and let the rest therof BG be eleuated on high that is without the playne AED Then I say that ABG is not one right line For forasmuch as AED is a plaine superficies produce directly equally vpon the sayd playne AED the right lyne AB towardes D which by the 4. definition of the first shall be a right line And from some one point of the right line ABD namely from C draâ— vnto the point G a right lyne CG Wherefore in the triangle 〈…〉 the outward angâ—â— ABâ— is eqâ—â—ll to the two inward and opposite angles by the 32. of the first and therfore it is lesse then two right angles by the 17. of the same Wherfore the lyne ABG forasmuch as it maketh an angle is not â— right line Whâ—refore that part of a right line should be in a ground playne superficies and part eleuated vpward is impossible If ye marke well the figure before added for the playâ—er declaration of Euclides demonstration iâ— will not be hard for you to coâ—â—â—â—e this figure which â—lussâ—s putteth for his demonstâ—â—tion â— wherein
same superficies Wherefore these right lines AB BD and DC are in one and the selfe same superficies and either of these angles ABD and BDC is a right angle by supposition Wherefore by the 28. of the first the line AB is a parallel to the line CD If therefore two right lines be erected perpendicularly to one and the selfe same playne superficies those right lines are parallels the one to the other which was required to be proued Here for the better vnderstanding of this 6. proposition I haue described an other figure as touching which if ye erect the superficies ABD perpendicularly to the superficies BDE and imagine only a line to be drawne from the poynt A to the poynt E if ye will ye may extend a thred from the saide poynt A to the poynt E and so compare it with the demonstration it will make both the proposition and also the demonstration most cleare vnto you ¶ An other demonstration of the sixth proposition by M. Dee Suppose that the two right lines AB CD be perpendicularly erected to one the same playne superficies namely the playne superficies OP Then I say that â—â— and CD are parallels Let the end points of the right lines AB and CD which touch the plaine supâ—â—â—â—cies Oâ— be the poyntes â— and D froÌ„â— to D let a straight line be drawne by the first petition and by the second petition let the straight line â—D be exteÌ„ded as to the poynts M N. Now forasmuch as the right line AB from the poynt â— produced doth cutte the line MN by construction Therefore by the second proposition of this eleuenth booke the right lines AB MN are in one plainâ— superficies Which let be QR cutting the superficies OP in the right line MN By the same meanes may we conclude the right line CD to be in one playne superficies with the right line MN But the right line MN by supposition is in the plaine superficies QR wherefore CD is in the plaine superficies QR And Aâ— the right line was proued to be in the same plaine superficies QR Therfore AB and CD are in one playne superficieâ— namely QR And forasmuch as the lines Aâ— and CD by supposition are perpendicular vpon the playne superficies OP therefore by the second definition of this booke with all the right lines drawne in the superficies OP and touching AB and CD the same perpeÌ„diculars Aâ— and CD do make right angles But by construction MN being drawne in the plaine superficies OP toucheth the perpendiculars AB and CD at the poyntes â— and D. Therefore the perpendiculars Aâ— and CD make with the right line MN two right angles namely ABN and CDM and MN the right line is proued to be in the one and the same playne superficies with the right lines AB CD namely in the playne superficies QR Whâ—refore by the second part of the 28. proposition of the first booke the right lineâ— AB and CD are parallelâ— If therefore two right lines be erected perpendicularly to one and the selfe same playne superficies those right lines are parallels the one to the other which was required to be demonstrated A Corollary added by M. Dee Hereby it is euident that any two right lines perpendicularly erected to one and the selfe same playne superficies are also them selues in one and the same playne superficies which is likewisâ— perpendicularly erected to the same playne superficies vnto which the two right lines are perpendicular The first part hereof is proued by the former construction and demonstration that the right lines AB and CD are in one and the same playne superficies Qâ— The second part is also manifest that is that the playne superficies QR is perpendicularly erected vpon the playne superficies OP for that Aâ— and CD being in the playne superficies QR are by supposition perpendicular to the playne superficies OP wherefore by the third definition of this booke QR is perpendicularly erected to or vpon OP which was required to be proued Io. d ee his aduise vpon the Assumpt of the 6. As concerning the making of the line DE equall to the right line AB verely the second of the first without some farther consideration is not properly enough alledged And no wonder it is for that in the former bookeâ— whatsoeâ—â—â—â—aâ—h of lines bene spoken the same hath alwayâ—s bene imagined to be in one onely playne superficies considered or executed But here the perpendicular line AB is not in the same playnâ— superficies that the right line DB is Therfore some other helpe must be put into the handes of young beginners how to bring this probleme to execution which is this most playne and briefe Vnderstand that BD the right line is the common section of the playne superficies wherein the perpendiculars AB and CD are of the other playne superficies to which they are perpendiculars The first of these in my former demonstration of the 6 â— I noted by the playne superficies QR and the other I noted by the plaine superficies OP Wherfore BD being a right line common to both the playne supâ—rficieces QR OP therby the ponits B and D are coÌ„mon to the playnes QR and OP Now from BD sufficiently extended cutte a right line equall to AB which suppose to be BF by the third of the first and orderly to BF make DE equall by the 3. oâ— the first if DE be greater then BF Which alwayes you may cause so to be by producing of DE sufficiently Now forasmuch as BF by construction is cutte equall to AB and DE also by construction put equâ—ll to BF therefore by the 1. common sentence DE is put equall to AB which was required to be done In like sort if DE were a line geuen to whome AB were to be cutte and made equall first out of the line DB suâ—â—iciently produced cutting of DG equall to DE by the third of the first and by the same 3. cutting from BA sufficiently produced BA equall to DG then is it euideÌ„t that to the right line DE the perpeÌ„dicular line AB is put equall And though this be easy to conceaue yet I haue designed the figure accordingly wherby you may instruct your imagination Many such helpes are in this booke requisite as well to enforme the young studentes therewith as also to master the froward gaynesayer of our conclusion or interrupter of our demonstrations course ¶ The 7. Theoreme The 7. Proposition If there be two parallel right lines and in either of them be taken a point at all aduentures a right line drawen by the said pointes is in the self same superficies with the parallel right lines SVppose that these two right lines AB and CD be parallels and in either of theÌ„ take a point at all aduentures namely E and F. Then I say that a right line drawen from the point E to the point F is in the selfe same plaine superficies that the
the whole line MG to the whole line EA by the 18. of the fifth Wherefore as MG the side of the cube is to EA the semidiameter so is the line FGHIM to the Octohedron ABKDLC inscribed in one the selfe same Sphere If therefore a cube and an Octohedron be contained in one and the selfe same Sphere they shall be in proportion the one to the other as the side of the cube is to the semidiameter of the Sphere which was required to be demonstrated A Corollary Distinctly to notefie the powers of the sides of the fiue solides by the power of the diameter of the sphere The sides of the tetrahedron and of the cube doo cut the power of the diameter of the sphere into two squares which are in proportion double the one to the other The octohedron cutteth the power of the diameter into two equall squares The Icosahedron into two squares whose proportion is duple to the proportion of a line diuided by an extreame and meane proportion whose lesse segmeÌ„t is the side of the Icosahedron And the dodecahedron into two squares whose proportion is quadruple to the proportion of a line diuided by an extreame and meane proportion whose lesse segment is the side of the dodecahedron For AD the diameter of the sphere contayneth in power AB the side of the tetrahedron and BD the side of the cube which BD is in power halfe of the side AB The diameter also of the sphere contayneth in power AC and CD two equall sides of the octohedron But the diameter contayneth in power the whole line AE and the greater segment thereof ED which is the side of the Icosahedron by the 15. of this booke Wheâ—fore their powers being in duple proportioÌ„ of that in which the sides are by the first corollary of the 20. of the sixth haue their proportion duple to the proportion of an extreame meane proportioÌ„ Farther the diameter coÌ„tayneth in power the whole line AF and his lesse segment FD which is the side of the dodecahedron by the same 15. of this booke Wherefore the whole hauing to the lesse â— double proportion of that which the extreame hath to the meane namely of the whole to the greater segment by the 10. diffinition of the fifth it followeth that the proportion of the power is double to the doubled proportion of the sides by the same first corollary of the 20. of the sixth that is is quadruple to the proportion of the extreame and of the meane by the diffinition of the sixth An aduertisment added by Flussas By this meanes therefore the diameter of a sphere being geuen there shall be geuen the side of euery one of the bodies inscribed And forasmuch as three of those bodies haue their sides commensurable in power onely and not in length vnto the diameter geuen for their powers are in the proportion of a square number to a number not square wherefore they haue not the proportion of a square number to a square number by the corollary of the 25. of the eight wherefore also their sides are incommensurabe in length by the 9. of the tenth therefore it is sufficient to compare the powers and not the lengths of those sides the one to the otherâ— which powers are contained in the power of the diameter namely from the power of the diameter let there ble taken away the power of the cube and there shall remayne the power of the Tetrahedron and taking away the power of the Tetrahedron there remayneth the power of the cube and taking away from the power of the diameter halfe the power thereof there shall be left the power of the side of the octohedron But forasmuch as the sides of the dodecahedron and of the Icosahedron are proued to be irrationall for the side of the Icosahedron is a lesse line by the 16. of the thirtenth and the side of the dedocahedron is a residuall line by the 17. of the same therfore those sides are vnto the diameter which is a rationall line set incommensurable both in length and in power Wherefore their comparison can not be diffined or described by any proportion expressed by numbers by the 8. of the tenth neither can they be compared the one to the other for irrational lines of diuers kindes are incoÌ„meÌ„surable the one to the other for if they should be commensurable they should be of one and the selfe same kinde by the 103. and 105. of the tenth which is impossible Wherefore we seking to compare them to the power of the diameter thought they could not be more aptly expressed then by such proportions which cutte that rationall power of the diameter according to their sides namely diuiding the power of the diameter by lines which haue that proportioÌ„ that the greater segment hath to the lesse to put the lesse segment to be the side of the Icosahedron deuiding the sayd power of the diameter by lines hauing the proportion of the whole to the lesse segment to expresse the side of the dodecahedron by the lesse segment which thing may well be done betwene magnitudes incommensurable The ende of the fourtenth Booke of Euclides Elementes after Flussas ¶ The fiftenth booke of Euclides Elementes THis finetenth and last booke of Euclide or rather the second boke of Appollonius or Hypsicles teacheth the inscription and circumscriptioÌ„ of the fiue regular bodies one within and about an other a thing vndoutedly plesant and delectable in minde to contemplate and also profitable and necessary in act to practise For without practise in act it is very hard to se and conceiue the constructions and demonstrations of the propositions of this booke vnles a man haue a very depe sharpe fine imagination Wherfore I would wish the diligent studeÌ„t in this booke to make the study thereof more pleasant vnto him to haue presently before his eyes the bodyes formed framed of pasted paper as I taught after the diffinitions of the eleuenth booke And then to drawe and describe the lines and diuisions and superficieces according to the constructions of the propositions In which descriptions if he be wary and diligent he shall finde all things in these solide matters as clere and as manifest vnto the eye as were things before taught only in plaine or superficial figures And although I haue before in the twelfth boke admonished the reader hereof yet bicause in this boke chiefly that thing is required I thought it should not be irkesome vnto him againe to be put in minde thereof Farther this is to be noted that in the Greke exemplars are found in this 15. booke only 5. propositions which 5. are also only touched and set forthe by Hypsicies vnto which Campane addeth 8. and so maketh vp the number of 13. Campane vndoubtedly although he were very well lerned and that generally in all kinds of learning yet assuredly being brought vp in a time of rudenes when all good letters were darkned barberousnes had
proposition after Prâ—â—lus A Corollary taken out of Flussates Demonstration leading to 〈◊〉 absurdiâ—â—e An addition oâ— Pelitarius Demonstration Three cases in this proposition The first case Construction Demonstration Three cases in this proposition The first case Euery case may happen seuen diuers wayes The like variety in ech of the other two cases Euclides construction and demostration serueth in all these cases and in their varities also Construction Demonstration How triangles are sayde to be in the selfe same parallel lines Comparison of two triangles whose sides being equal their bases and angles at the toppe are vnequall When they are lesse then two right angles Construction Demonstration Thre cases in this proposition Ech of these cases also may be diuersly Note An other addition of Pelitarius Construction Demonstration This Theoreme the conuerse of the 37. proposition An addition of Flâ—ssases An addition of Campanus Construction Demonstration leading to an absurditie This proposition is the conuerse of the 38. propositioÌ„s Demonstration Two cases in this proposition A corollary The selfe same demonstration will serue if the triangle the parallelogramme be vpon equall bases The conuerse of this proposition An other conuerse of the same propositioÌ„ Comparison of a triangle and a trapesium being vpon one the selfe same base and in the selfe same parallel lines Construction Demonstration Supplements Complementes Three cases in this Theoreme The first case This proposition called Gnomical and mistical The conuerse of this proposition Construction Demonstration Applications of spaces with excesses or wants an auncient inuention of Pithagoras How a figure is sayde to be applied to a line Three thinges geuen in this proposition The conuerse of this proposition Construction Demonstration An addition of Pelitarius To describe a square mechanically An addition of Procâ—â—â— The conuerse thereof Construction Demonstration Pithagoras the first inuenter of this proposition An addition of Pâ—lâ—tariâ—â— An other aditioÌ„ of Pelitarius An other addition of Pelitarius An other aditioÌ„ of Pelitarius A Corrollary This proposition is the conueâ—se of the former The argument of the second booke What is the power of a line Many compeÌ„dious rules of reckoning gathered one of this booke and also many rules of Algebra Two wonderfull propositions in this booke First definition What a parallelogramme is Fower kindes of parallelogrammes Second definiâ—ion A proposition added by Campane after the last proposition of the first booke Construction DemonstratioÌ„ Barlaam Barlaam Construction DemonstratioÌ„ Barlaam Construction DemonstratioÌ„ Barlaam Construction DemonstratioÌ„ A Corollary Barlaam Construction DemonstratioÌ„ Constrâ—ction Demonstration Construction Demonstration Construction DemonstratioÌ„ Many and singuler vses of this proposition This proposition can not be reduced vnto numbers Demonstration DemonstratioÌ„ A Corollary This Proposition true in all kindes of triangles Construction DemonstratioÌ„ The argument of this booke The first definition Definition of vnequall circles Second definition A contigent line Third definiâ—ion The touch of circles is 〈◊〉 in one poâ—â—â— onely Circles may touch togeâ—her two maâ—â—â— of wayes Fourth definition Fift definition Sixt definition Mixt angles Arkes Chordes Seuenth definition Difference of an angle of a Section and of an angle in a Section Eight definition Ninth definition Tenth definition Two definitions First Second Why Euclide defineth not equall Sections Constuction Demonstration leading to an impossibilitie Correlary DemonstratioÌ„ leading to an impossibilitie The first para of this Proposition Construction Demonstration The second part conuerst of the first Demonstration Demonstration leading to an impossibilitie Two cases in this Proposition Construction DemonstratioÌ„ leading to an impossibilitie Demonstratâ—on leading to an impossibilitie Two caseâ— in thys Proposition Construction The first part of this Proposition Demonstration Second part Third part This demonâ—â—rated by an argument leading to an impossibilie An other demonstration of the latter part of the Proposition leading also to an impossibilitie A Corollary Third part An other demonstration of the latter part leading also to an impossibility This Proposion is commoÌ„ly called Caâ—dâ— Panonis A Corollary Construction Demonstration An other demonstration of the same leading also to an impossibilitie Demonstration leading to an impossibilitie An other demonstration of the same leading also to an impossibilitie Construction Demonstration leading to an impossibilitie An other demonstration of the same leading also to an impossibilitie The same â—gaine demonstrated by an â—rgument leading to an absurdititie Demonstratiâ— leading to an impossibiliâ—ie An other demonstration after Pelitarius leading also to an absurditie Of circles which touch the one the other inwardly Of circles which touch the one the other outwardly An other demonstration after Pelitarius Flussates of circles which tooch the one the other outwardly Of circles which tooch the one the other inwardly The first part of this Theoreme Construction Demonstration Demonstration The second part which is the conuerse of the first An other demonstration of the first part after Campane Construction Demonstration An other demonstration after Campane The first part of this Theoreme Demonstration leading to an absurditie Second part Third part Construction Demonstration An addition of Pelitarius This Probleme commodious for the inscribing and circumscribing of figures in or abouâ— circles Demonstration leading to an impossibilitie An other deâ—onstration after Orontius Demonstration leading to an impossibilitie Two cases in thys Proposition the one when the angle set at the circumference includeth the center Demonstration The other wheÌ„ the same angle set at the circumference includeth not the center Construction Demonstration Three cases in this Proposition The first case The second case The third case Construction Demonstration Demonsâ—ration leading to an impossibilitie An adâ—ition of Campane dâ—moâ—strated by Pelitariâ—s Demonstration leading to an impossibilitie An other demonstration Construction Three cases in this Proposition The first case Demonstratioâ— The second case The third case An addition Construction Demonstration Demonstration leading to an impossibilitie Construction Demonstration The conuerse of the former Proposition Construction Demonstration Construction Demonsâ—ratioâ— Second part Thirâ— part The fift and last part An other Demonstration to proue that the angâ—e in a semicircle is a right angle A Corollary An addition of Pâ—litarius Demonstration leaâ—ing to an absurditâ—â— An addition of Campane Construction Demonstration Two cases in this Proposition Three cases in this Proposition The first case Construction Demonstratioâ— The second case Construction Demonstration The third case Construction Demonstration Consâ—â—uction Demonstration Two cases in this Proposition First case Demonstration The second câ—se Construction Demonstration Three cases in this â—roposition Construction Two cases in this Proposition The first case Demonstration The second case Construction Demonstration First Corollary Second Corollary Third Corollary This proposition is the coÌ„uerse of the former Construction Demonstration An other demonstration after Pelitarius The argument of this booke First definition Second definition The inscriptition and circumscription of rectiline â—igâ—res pertaiâ—eth only to regular figures The third definition The fourth definition The fift definition The sixt deuition Seuenth definition Construction Two cases in this Proposition First
caseâ— Second case Demonstration Construction Demonstration Construction Demonstration An other way after Peliâ—arius Construction Demonstration Construction Demonstration Demonstration leading to an impossibilitie Three cases in this Propositiâ—n The third case Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration A Proposition added by Petarilius Note Construction Demonstration An other way also after Pelitarius Construction Demonstration An other way to do the samâ— after Pelitarius Demonstration Demonstrâ—tion Demonstration leading to an absurditie A Corollary Construction Demonstration Demonstration An â—ther way to do the same after OroÌ„tius An other way after Pelitarius Construction Demonstration An addition of Flussates * A Poligonon figure is a figure consisting of many sides The argument of this fift booke The first aâ—thor of this booke Eudoxus The first definition A part taken two maner of wayes The fiâ—st way The second way How a lesse quantity is sayd to measure a greater In what significatioÌ„ Euclide here taketh a part Parâ— metienâ— or mensuranâ— Pars multiplicatiâ—a Pars aliquota This kinde of part coÌ„monly vsed in Arithmetique The other kinde of part Pars constitâ—ens or componens Pars aliquanta The second definition Numbers very necessary for the vnderstanding of this booke and the other bookes following The tâ—ird definition Rational proportion deuided â—â—to two kindes Proportion of equality Proportion of inequality Proportioâ— of the greater to the lesse Multiplex Duple proportion Triple quadruple Quintuple Superperticular Sesquialtera Sesquitertia Sesquiquarta Superpartiens Superbipartiens Supertripartiens Superquadripartiens Superquintipartiens Multiplex superperticular Dupla Sesquialtera Dupla sesquitertia Tripla sesquialtera Multiplex superpartiens Dupla superbipartiens Dupla supertripartiens Tripla superbipartiens Tripla superquadâ—ipartiens How to knoâ— the denomination of any proportion Proportion of the lesse in the greater Submultiplex Subsuperparticular Subsuperpertient c. The fourth definition Example of this definition in magnitudes Example thereof in numbers Note The fifth definition An example of this â—efinition in magnitudes Why Euclide in defining of Proportion vsed multiplication The sixth definition An example of this definitiin in magnitudes An example in numbers An other example in numbers An other example in numbers Note this particle according to any multiplication An example where the equimultiplices of the first and third exceedâ— the equimultiplices of the second and fourth and yet the quantities geuen are not in one and the selfe same proportion A rule to produce equimultiplices of the first and third equall to the equimultiplices of the secondâ— and fâ—urth Example thereof The seuenth definition 9 12 3 4 Proportionality of two sortes contiâ—uall and discontinuall An example of continuall proportionalitie in numbers 16.8.4.2.1 In coutinnall proportionalitie the quantities cannot be of one kinde Discontinuall propâ—rtionalitie Example of discontinual proportionality in numbers In discoÌ„tinual proportionalitie the proportions may be of diuers kindes The eight definition An example of this definition in magnitudes An example in numbers Note The ninth definition An example of this definition in magnitudes Example â—n numbers The tenth definition A rule to adde proportions to proportions 8. 4. 2. 1. 2 2 2 1 1 1 The eleuenth definition Example of this definition in magnitudâ— Example in numbers The twelfâ—h definition Example of this deâ—inition in magnituds Example in numbers The thirtenth definition Example of this definition in magnituds Example in numbers The fourtenth definition Example of this definition in magnituds Example in numbers The fiâ—tâ—ne definition This is the coÌ„uerse of the former definition Example in magnitudes Example in numbers The sixtene definition An example of this definition in magnitudes An example in numbers The seueÌ„tenth definition An example of this definition in magnitudes An example in numbers Note The eighttenth definition An example of this definition in magnitudes Example in numbers The nintenth definition An example of this definition in magnitudes Example in numbers The 20. definition The 2â— defiâ—ition These two last definitions not found in the greeke examplers Construction Demonstration Demonstrationâ— Construction Demonstration Construction Demonstration ALemmae or an assumpt A Corollary Conuerse proportion Construction Demonstration Two cases in this Propotion The second The second part demonstrated The first part of this Proposition demonstrated The second part of the proposition demonstrated First differâ—câ— of the first part Demonstratiâ— of tâ—e same first differeÌ„ce Second diffeâ—ence Third diâ—â—erence The second part â—f this proposition The first parâ— of this Proposition demonstrated The second part proued The first part of this proposition proued The second part demonstrated Construction Demonstrationâ— Constrâ—ction Demonstrationâ— Construction Demonstration An addition of Campane Demonstration Construction Demonstration DemoÌ„stration of alternate proportion Construction Demonstration DemonstratioÌ„ of proportion by diuision Constrâ—ctionâ— Demonsâ—ration Demonstration of proportion by composition This proposition is the conuerse of the former DemonstratioÌ„â—eâ—aing to an â—mpossibility That which the fift of this booke proued only touching multiplices this proueth generally of all magnitudes ALemma A Corollary Conuersion of proportion This proposition pertaineth to Proportion of equality inordinate proportionality The second difference The third difference Thâ—r proposition pertaineth to Proportion of equality in perturbate proportionality The third difference Proportion of equality in ordinate proportionality Construction Demonstration When there are more then three magnitudes in either order Aâ—CDEâ—GH Proportion of equality in perturbate proprotionalitie Construction Demonstration Note That which the second propositioÌ„ of this booke proued only touching multiplices is here proued generally touching magnitudes An other demonstration of the same affirmatiuely An other demonstration of the same affirmatiuely An other demonstration of the same DemonstratioÌ„ leading to an impossibilitie An other demonstration of the same affirmatiuely Demonstration Demonstrationâ— Demonstration The argument of this sixth booke This booke necessary for the vse of instrumentes of Geometry The first definition The second deâ—inition Reciprocall figures called mutuall figures The third definition The fourth definition The fifth definition An other example Of substraction of proportion The sixth definition Demonstration of the first part Demonstâ—ation of the second part A Corollary added by Flussates The first part of this Theoreme Demonstration of the second part A Corollary added by Flussates Construction Demonstration of the first part Demonstratiâ—n of the second part which is the conuerse of the first Construction Demonstration This is the conuerse of the former proposition Construction Demonstration Constructioâ— The first part of this proposition Demonstration leading to an impossibilitie The second part of this proposition Construction Demonstration Construction Demonstration Construction Demonstration A Corollary out of Flussates By this and the former propoâ—ition may a right line be deuided into what partes soeuer you will. Construction Demonstration An other way after Pelitarius An otâ—eâ— way after Pelitarius Construction Demonstration An other way after Campane Construction Demonstratioâ— A proposition added by Pelitarius The
first part of this proposition Demonstration of the of the same The second part which is the conuerse of the first The first parâ— of this proposition Demonstration of the same The second part which is the conuerse of the first Demonstration of the first part The second part which is the conuerse of the first The first part of thâ—â— Theoreme The second part which is the conuerse of the first A Coâ—ollary Description of the rectiline figure râ—quired Demonstration Demonstration A Corollary The first parâ— of this Theoreme The second part demonstrated The third part The first Corollary The second Corollary Demonstration The first part of this proposition The second part which is the conuerse of the first * Note that this is proued in the assumpt following An Assumpt An other demoÌ„stration of the second part after Flussates An other demonstration after flussates Demonstration of this propositioÌ„ wherein is first proued that the parallegramme EG is like to the whole parallelograÌ„me ABCD. That the parallelograÌ„me KH is like to the whole parallelogramme ABCD That the parallelogrammes EG and KH are like the one to the other An other Demonstration after Flussates An addition of Pelitarius Another addition of Pelitarius Construction Demonstration Demonstration * By the dimetieÌ„t is vnderstand here the dimetient which is â—rawen from the angle which is common to them both to the opposite angle Demonstration leading to an absurditie An other way after Flussates In this propositioÌ„ are two cases in the first the parallelogramme compared to the parallelograÌ„me described of the halfe line is described vpon a line greater theÌ„ the halfe line In the second vpoÌ„ a line lesse The first case where the parellelogramme compared namely AF is described vpon the line AK which is greater then the halfe line AC Demonstration of this case The second case where the parallelogramme compared namely AE is described vpon the line AD which is lesse then the line AC Demonstration of the second case Construction Two cases in this Proposition The first case The second case A Corollary added by Flussates and is put of Theon as an assumpt beâ—ore the 17. proposition of the teÌ„th booke which â—or that it followeth of this proposition I thought it not amisse here to place Construction Demonstration Construction Demoâ—straâ—ion An other way Construction Demonstration The conuerse of the former proposition Demonstration That the angles at thâ— ceÌ„ter are in proportioÌ„ the one to the other as the circumferences wheron they are That the angles at the circumferences are so also That the sectors are so also Construction of the Probleme Demonstartion of the same The first Corollary The second Corollary The third Corollary Demonstration of this proposition Demonstration of this propositions Demonstration of this proposition Demonstration of the first part of this proposition Demonstration of the second part Why Euclide in the middest of his workes was compelled to adde these three bookes of numbers Arithmetike of more excellency then Geometry Things intellectuall of more worthines theâ— things sensible Arithmetike ministreth prinâ—ciples and groundes in a maner to all sciences Boetius Cap. 2. Lib. prim Arithmeti Timaus The argument of the seuenth booke The first definition Without vnity should be confusion of thinges â—oetius in his booke dâ— vnitate vno An other desinition of vnity The second definition Differenâ—e betwene a point and vnity Boetius An other desinition of number Iordane An other definition of numbers Vnity hath in it the vertue and power of all numbers Number considered three maner of wayâ— The third definition The fourth definition The fifth definition The sixth definition Boetius An other definition of an eueÌ„ number Note Pithagoraâ— An other definition An other definition An other definition The seuenth definition An other definition of an od number An other definition The eight definition Campane An other deâ—inition of an eueÌ„ly euen number Flussates An other definition Boetius An other definition The ninth definition Campane An other definition Flussates An other definition An other definition The tenth definition This definition not found in the Greeks An other definition Boeâ—ius defâ—nition of a number euenly eueÌ„ and euenly â—d The eleuenth definition Flusâ—ates An other definition The twelfth definition Prime numbers called incomposed numbers The thirtenth definition The fourtenth definiâ—ion The fiftenth definition The sixtenth definition Two numbers required in multiplication The seuententh definition Why they are called superficiall numbers The eightenth definition Why they are called solid numbers The ninetenth definition Why it is called a square number The twenteth definition Why it is called a cube number The twenty one definition Why the definition of proportionall magnitudes is vnlike to the definitio of proportionall numbers The twenty two definition The twenty three definitioÌ„ Perfect numbers rare of great vse in magike in secret philosophy In what respect a number is perfect Two kinds of imperfect numbers A â—â—mber wanâ—â—ngâ— Common sentences â—irst common seâ—tence â—â—cond â—ommon sentence Third common sentence Fâ—urth common sentence â—iâ—th common sentence Sixth common sentence Seuenth comâ—mon sentence Constrâ—ctioâ— Demonstratiâ—â— leading to an absurditie The conuerse of â—his proposition How to â—now whether two numbers geuen be prime the one to the other Two cases in this probleme The first case The second case DemonstratioÌ„ of the second case That CF is a common measure to the numbers AB and CD That CF is the greatest common measure to AB and CD The second case Two cases in this Proposition The first case The second case This propositioÌ„ and the 6. proposition in discrete quantitie answer to the first of the fifth in continual quantitie Demonstration Construction Demonstration Thiâ— proposition and the next following in discret quaÌ„titie answereth to the fifth propositioÌ„ of the fifth boke in continuall quaÌ„tity Construction Demonstration Constuâ—ction Demonstration An other demonstration after Flussates Construction Demonstration Construction Demonstration This proposition iâ— discret quaÌ„titie answereth to the ninth propâ—sitioÌ„ of the fifth boke in continual quaÌ„titie Demonstration This in discret quaÌ„tity answereth to the twelfe proposition of the fifth in continual quaÌ„tity Demonstration This in discrete quantiâ—y answereth to the sixtenth proposition of the fifth booke in continuall quantitie Note This in discrete quantity anâ—wereth to tâ—â— tweÌ„ty one proposition oâ— the fifth booke in continuall quantitie Demonstration Certaine additions of â—aâ—pane The second case Propâ—rtionality deuided Prâ—portionaliâ—y composed Euerse proportionality The conuersâ— of the same prâ—position Demonstration A Corollary followiâ—g thâ—se propositions adâ—ed by Campaâ—e Coâ—strâ—ctioâ— Demonstration Demonsâ—raâ—ion â—emonstraâ—ion A Corollary added by Flussâ—tes Demonstration This proposition and the former may be extended to numbers how many soeuer The second part of this proposition which is the conuerse of the first Demonstration An assumpt added by Campane This proposition in numbers demonstrateth that which the 17. of the sixth demonstrateth in lines Demonstration The second part which is the conuerse of the
first Demonstration Demonstration leading to an impossibility This proposition in discret quaÌ„titie answereth to the 23. propositioÌ„ of the fifth boke in continual quaÌ„titie This and the eleuen propositions following declare the pâ—ssions and properties ofâ— prime nuÌ„bers Demonstration leading to an impossibility This is the coÌ„uerse of the former proposition Demonstrâ—tion leading to an absurditie Demonstration leading to an absurditie Demonstration leading to an absurditie Demonstration Demonstration Deâ—onstration Demonstration of the first part leading to an absurditie Demonstration of the second part which is the conâ—câ—se of the first leanâ—ng also to an absurditiâ— Demonstrasion leading to an absurditie Demonstrasion A Corollary â—â—ded by Campaue Demonstration lâ—ading to an impossibilitie An other demonstration Demonstration Two cases in this Proposition The first case The second case Demonstration Demonstration leading to an absurditie A Corollary added by Campaâ—e Two cases in this propositioÌ„ The first case Demonstration leading to an absurditie The second caseâ— Demonstration leading to an absurditie Demonstration leading to an impossibâ—â—â—â—â— Two cases in this propositioÌ„ The first case Demonstration leaâ—iâ—g â—o an absurâ—â—â—â—e The second case Demonstration leading to an absurditie A Corollary Demonstration The coâ—â—erse of the former proposition Demonstration Construction DemonstratioÌ„ leâ—ding to an â—bsuâ—diâ—ie A Corollary adâ—ed by Campane How to â—inde out the seconde least number and the third and so â—orth â—nâ—â—nitly How to siâ—â— out the least â—â—mâ— a conâ—ayâ—â—g â—â—e paâ—â—s of parts The Arguâ—â—â—â— of the eight books Demonstration leading to an absurdâ—â—ie Construction Demonstration This proposition is the â—â—uerse of the first Demonstrationâ— Two cases in this propositioÌ„ The first case Demonstration leading to an absurditie The second case Demonstration This proposition in numbers answereth to the of the sixth touching parellelogrammes Construction Demonstration An other demonstratioÌ„ after Campane Demonstration Demonstration leading to an impossibilitie Demonstration A Corollary added by Flussates Construction Demonstration This proposition is the conuerse of the former Construction Demonstration The first part of this proposition demonstrated The second part demonstrated Construction The first part of this prâ—position deâ—â—â—strated The second part demonstrated Construction Demonstration The first part of this proposition The second part is the conuerse of the first The first part of this proposition The second part is the conuerse of the first A negatâ—ue proportion The first part of this proposition The second part is the coÌ„uerse of the first A negatiue proposition The first part of this proposition The second part is the coÌ„uerse of the first Demonstration of the fiâ—st part of this proposition Demonstration of the second part Demonstration of the first part of this proposition The second part This proposition is the conuerse of the 18. proposition Construction Demonstration This proposition is the conuerse of the 19. proposition Construction Demonstration Demonstration Demonstration Demonstration Demonstration A Corollary added by Flussates Construction Construction Demonstration A Corollary added by Flussates Another Corollary added by Flussates The ArgumeÌ„t of the niâ—th booke Demonstration This proposition is the conuâ—rse oâ— tâ—e formâ—â— Demonstration A Corollary aâ—ded by Campane Demonstration Demonstration Demonstration A Corollary added by Campane Demonstration Demonstration Demonstration of the first part The second part demonstrated DemostratioÌ„ of the third part Demostration of the first part of this proposition The second pâ—rt demonstrated Demonstration of the first part leauing to an absuâ—â—itie Demonstration of the â—â—cond pâ—â—â— leading alâ—o to an absurditie Demonstration Demonstration leading to an absurditie An other demonstratioÌ„ aâ—ter Flussates Demonstration leading to an absurditie An other demonstratioÌ„ after Campane Demoâ—stration leading to an absurditie A propositioâ— added by Campane Construcâ—ion Demonstration Demonstration to proue that the numbers A and C are prime to B. Demonstratiou This proposition is the coÌ„uerse of the former Demonstration This answereth to the 2. of the second Demonstration This answereth to the 3. of the thirds Demonstration This answerâ—th to thâ— 4. of the second Demonstration This answereth to the 5. of the second Demonstration This answereth to the 6. of the second Demonstration This answereth to the 7. of the second Demonstration This answereth to the 8. of the second Demonstratition This answereth to thâ— 9. of the second Demonstration This answereth to the 10. oâ— the second Demonstration A negatiue propositiâ—n Demonstration leaâ—ing to an impossibilitie Demonstration leading to an absurditie Demonstration leading to an abjurditie Three cases in this proposition The first case The second case The third case Diuert cases â—n this proposition The first case Two cases in this Proposition The first case The second case Demonstration Demonstration Demonstration Demonstration Demonstration Demonstration Demonstration Demonstration Demonstration A proposition added by Campaâ—e An other added by him Demonstration leading to an absurditie Demonstration Demonstration Demonstration leading to an absurditie An other demonstration Demonstration Demonstration This proposition teachâ—th how to finde out a perfect number Construction Demonstration Demonstration leading to an absurditie The ArgumeÌ„t of the tenth booke Difference betwene number and magnitude A line is not made of points as number is made of vnities This booke the hardest to vnderstand of all the bookes of Euclide In this booke is entreated of a straunger maner of matter then in the former Many euen of the well learned haue thought that this booke can not well be vnderstanded without Algebra The nine former bookes the principles of this â—ooke well vnderstoode this booke will not be hard to vnderstand The fâ—rst definition The second definition Contraryes made manifest by the comparing of the one to the other The thirde definition What the power of a line is The fourth definition Vnto the supposed line first set may be compared infinite lines Why some mislike that the line first set should be called a rational line Flussates calleth this line a line certaine This rational line the grouÌ„d in a maner of all the propositions in this tenth booke Note The line Rationall of purpose The sixth deâ—inition Campâ—nus â—ath caused much oâ—scuritie in this tenth booke The seuenth definition Flussates in steede of this word irrationall vseth this word vncertayne Why they are called irrationall lines The cause of the obscurity and confusednes in this booke The eighth definition The ninth definitâ—on The tenth deâ—inition The eleuenth deâ—inition Construction Demonstration A Corollary Construction Demonstration This proposition teacheth that incontinuall quantitie which the first of the seuenth taught in discrete quantity Construction Demonstration leading to an abâ—urditie Two cases in this propositioÌ„ The first case This proposition teacheth that in continual quantity which the 2. of the sâ—â—ith taught in numbers The second case Demonstration leading to an absurditie A Corollary This Probleme reduced to a Theoreme This proposition teacheth that in continual quantity which the 3. of the second taught in numbers Construction Two cases in this Proposition The first case Demonstration leading to an absurditie The second case A Leâ—ma necesâ—ary
to be prâ—â—â—d beâ—oâ—e 〈◊〉 â—all to the demoÌ„â—â—ration Construction Demonstration leading to an absurditie A Corollary This Probleme reduced to a Theoreme Construction Demonstration How magnitudes are sayd to be in proportion the onâ— to the other as number is to number This proâ—osition is the conuerse of formâ—r Conâ—truction Demonstration A Corollary Construction Demonstration Construction Demonstration Demonâ—tration leading to an abâ—urdiâ—ie This is the 〈…〉 demonsâ—â—ation The first part demonstratâ—d An other demonstration of the first part An othâ—r demonâ—â—raâ—ion oâ— the same first part after Montaureus Demonstration of the seconde part which is the coâ—â—erse of the former An other demonstration of the second part This Assumpâ— followeth as a Corollary of the 25 but so as it might also be here in Methode placed you shall â—inde it after the 53. of this booke absolutely demonstrated for there it serueth to the 54. his demonsâ—ration DemoÌ„stratioÌ„ of the third part DemoÌ„stratioÌ„ of the fourth part which is the coÌ„uerse of the â— Conclusion of the whole proposition A Corâ—llary Proâ—e of the first part of the Corollary Profe of the second part Profe of the third pâ—rt Proâ—e oâ— the fourth part Certayne annotations â—ut of Montauâ—â—us Rules to know whether two superficiall numbers be like or no. This assumpt is the conuerse of the 26. of the eight Demonsâ—raâ—ioÌ„ oâ— the first part Demonstration of the second partâ— A Corollary To finde out the first line incommensurable in length onely to the line geuen To finde out the second line incommensurable both in length and in power to the line geuen Construction Demonstration Tâ—is is wiâ—h Zambert an Aâ—â—â—mpt but vâ—â—eâ—ly improperly â—lâ—ssateâ— maâ—eth iâ— a Corollary but the Greeâ—e and Montaureus maâ—e it a proposition but euery way an â—nfallible truth 〈…〉 Demonstration leading to an absurditie Demonstration leading to an absurditâ—e A Corollary A Corollary Demonstration An other way to proue that the lines A E C F are proportionall Demonstration of the first part Demonstration of the second paâ—t which is the conuerse of the first A Corollary Demonstration of the first part by an argument leadindg to an absurditie Demonstration of the second paâ—t leading also to an impossibilitie And this second part is the conuerse of the first Demonstration of the second part which is the conuerse of the first How to deuide the line BC redely in such sort as iâ— required in the propositioÌ„ Demonstrâ—tion of the second part which is the conuerse of tâ—e former An other demonstrationâ—y an argumeÌ„t leading to an absurditie An Assumpt A Corollary added by Montaureuâ— Cause Cause of increasing the difficulty of this booke Note Construction Demonstration Diuers caâ—es in this proposition The second case The first kind of rationall lines commensurable in length This particle in the proposition according to any of the foresayde wayes was not in vayne put The second kinde of rationall lines coÌ„mensurable in lengâ—h The third case The third kinde of rationall lines commensurable in length The fourth case This proposition is the conuerse of the former proposition Construction Demonstration An Assumpt Constâ—uction Demonstration Diffinition of a mediall line A Corollary This assumpt is nothing els but a part of the first proposition of the sixt booke 〈◊〉 How a square is sayde to be applied vppon a line Construction Demonstration Construction Demonstration Note A Corollary Construction Demonstration leading to an absurditie Construction Demonstâ—ation Construction Demonstration * A Corollary To finde out two square nâ—mbers exceeding the one the other by a square â—umber An Assumpt Construction Demonstration Montaureus maketh this an Assumpt as the Grecke text seemeth to do likewise but without a cause Construction Demonstration This Assumpt setteth foâ—th nothing â—ls but that which the first oâ— the sâ—â—t â—etteth â—orth and therefore in sâ—me examplars it is not founde Construction Demonstration Construction Demonstration Construction Demonstration A Corollary I. Dee * The second Corollary * Therefore if you deuide the square of the side AC by the side BC the portion DC will be the product c. as in the former Corollâ—ry I. Dâ—e * The thirde Corollary * Therfore if the parallelogramme of BA and AC be deuided by BC the product will geue the pâ—â—pâ—ndicular D A. These three Corollaryes in practise Logisticall and Geometricall are profitable An other demonstration of this fourth part of the determinatioÌ„ An Assumpt Construction Deâ—onstration The first part of the dâ—termination concluded The second part coÌ„cluded The totall conclusion Construction Demonstration The first part of the determination concluded The second part coÌ„cluded The totall conclusion Construction Demonstration The first part concluded The second part coÌ„cluded The third part coÌ„cluded The totall conclusion The first Senary by composition Diffinition of a binomiall line Sixe kindes of binomiall lines Demonstration Diffinition of a first bimediall line Construction Demonstration Diffinition of a second bimediall line Demonstration Diffinition of a greater line Diffinition of a line whose power is rationall and mediall Diffinition of a liâ—e containing in power two medials An Assumpt The second Senary by composition Demonstration leading to an impossibilitie A Corollary Demonstration leading to an impossibilâ—â—e DemonstratioÌ„ leading to an impossibiliâ—ie Demonstration leading to an impâ—ssibilitie Demonstration leading to an impossibiâ—â—â—e Demonstration leading to an impossibilitie Construction Demonstration leading to an absurditie Sixe kindes of binomiall lines A binomiall line coâ—â—â—steâ—h of two paâ—tâ—s Firsâ— dâ—â—â—initiâ—n Seconâ— diffinition Third â—â—â—â—â—â—ition Fourth diffinition Fifth difâ—inition Sixth diffinition The third Senary by composition Construction Demonstratiâ—n Construction Demonstration Construction Demonstration Constâ—uction Demonstration Construction Demonstration Construction Demonstration A Corollary added by Flussates M. d ee his booke called Tyâ—â—câ—niâ—m Mathematicum This Assumpt as was before noted fâ—llâ—weth most â—riâ—fly without farther demonstration of the 25. of this booke Demonstration An Assumpt The fourth Senary by composition Construction Demonstration The first part of this demonstration concluded The secoÌ„d part of the demonstration concluded The third part coÌ„cluded The totall conclusion Demoâ—stratioâ— The first part of this demonstration concluded The third part coÌ„cluded The fourth part coÌ„cludeâ— The fift part concluded The totalâ— conclusion Demonstration Construction Demonstration Demonstration Demonstration * Looke after the Assumpt concluded at this marke for plainer opening of this place The vse of this Assumpt is in the next proposition other following The fift Senary by composition Construction Demonstration Concluded that DG is a binomiall line Construction Demonstratiâ—n Concluded that DG is a binomiall line Construction Demonstration †* ‡ DG concluded a binomiall line A Corollary added by M. Dee Construction Demonstration Construction Demonstration Construction Demonstration The sixt Senary Construction Demonstration Construction Demonstration A Corollâ—ry addâ—d by Flussetes Note Construction Demonstration An other demonstration after P. Montaureus An other demonstration after Campane Construction Demonstration An other demonstratioÌ„ afâ—â—r Campane Construction Demonstration An Assumpt An other demonstration after Campanâ— Note
Second part of the first case The second case First part of the secoÌ„d case Second part of the secoÌ„d case Construction Two cases in this Proposition The first case The first part of the first case 〈◊〉 second 〈◊〉 of the 〈◊〉 case The second case A Corollary The first Senary by substraction Demonstration An other demonstration after Campane Diffinition of the eight irrationall line Diffinition of 〈◊〉 â—inth irrationall line An other demonstratioÌ„ after Campane Construction Demonstration Diffinition of the tenth irâ—ationall line Diffinition of the eleueÌ„th irrationall line â—â—â—â—iâ—ition of the twelueth irraâ—ionall line Diffinition of the thirtenth and last irrationall line An Assumpt of Campane I. Dee Though Campanes lemma be true yeâ— the maner of demonstrating it narrowly considered is not artificiall Second Senary Demonstration leading to an impossibilitie Demonstration leading to an absurditie Construction Demonstration leading to an absurditie Demonstration leading to an absurditie DemonstratioÌ„ leading to an impossibilitie Construction Demonstration 〈◊〉 an abjurdâ—tâ—â—â— Sixe kindes of reâ—iduall lines First diffinition Second diffinition Third diffinition Fourth diffinition Fifth diffinition Sixth diffinition Third Senary Construction Demonstratioâ— Construction Demoâ—stratiâ—â— Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstratioâ— An other more redie way to finde out the sixe residuall lines Fourth Senary The â—irst parâ— of the Construction The first part of the demonstration Note AI and FK concluded rational parallelogramme Note DH and FK parallelogrammes mediall Second part of the construction Second part of the demonstration LN is the onely liâ—e â—hat we sought consider First part of the construction The first part of thâ— demonstration AI and FK concluded parallelograÌ„mes mediall DH EK rationall The second part of the construction The second part of the demonstration * Analytically the proâ—e hereof followeth amoÌ„gâ— other thinges The line LN found which is the principall drift of all the former discourse The first part of the Construction The fiâ—st part of the demonstration Note AI and FK mediall Note DH and EK mediall Note AI incommensurable to EK Second part of the ConstructioÌ„ The principall line LN fouÌ„de * Because the lines AF and â—G are proued commensurable in length * By the first oâ— the sixth and tenth of the tenth The first part of the construction The first part of the demonstration Note AK rational Note DK mediall AI and FK incommensurable The second part of the construction The second part of the demonstration LN the chiefe line of this theoreme founde Demonstration The line LN Demonstration The fiueth Senary These sixe propositions following are the conuâ—rses of the sixe former propositions Construction Demonstration * By the 20. of the tenth ** By the 21. of the tenth * By the 22. of the tenth â—F coÌ„cluded a residual line Construction Demonstration CF conclâ—ded a residuall line Construction Demonstration CF concluded a residual line Construction Demonstration CF proued a residuall line CF proued a residuall line Construction Demonstration CF â—roued â— residuall The sixt Senary Construction Demonstration CD coÌ„cluded a residuall line Note Construction Demonstration CD proued a mediall Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Seuenth Senary Constraction Demonstration Construction Demonstration Demoâ—stratioâ— Construction Demonstration on leading to an impossibilitie A Corollary The determination hath sundry partes orderly to be proued Construction Demonstration This is an Assumpt problematicall artificially vsed and demonstrated * Therfore those three lines are in continuall proportion FE concluded a residuall liâ—â— which is sâ—â—what prepâ—â—icroâ—sly in respect oâ— the â—â—der propounded both in the propositioÌ„ and also in the determinatioÌ„ Construction Demonstration Construction Demonsâ—ration Here are the â—ower partes of the propositiâ— more orderly hâ—dled theâ— in the former demöstration Construction Demonsâ—ration An Assumpt An other demonstratioÌ„ after Flussas Construction Demonsâ—ration This is in a maner the conuerse of both the former propositions ioyntly Construction Demonstration Construction Demonstration Demonstration An other demonstration DemonstratioÌ„ leading to an impossibiliâ—ie An other demonstration leading to an impossibiliâ—ie The argument of the eleuenth booke A point the beginning of all quantitie continuall The methode vsed by Euclide in the ten â—â—â—mer booâ—es â—irst boâ—â—e Second â—â—oâ—e Third booâ—e â—ourth bâ—oâ—e â—iueth boâ—â—e Sixth booâ—e Seuenth bookâ— â—ight booâ—â— Ninth booke Tenth booâ—e What is entreaâ—ea of in the fiâ—e booâ—es follâ—wiâ—◠〈◊〉 â—â—â—ular bodiesâ— the â—â—all ende 〈…〉 oâ— I uâ—â—â—â—es â—eomeâ—â—â—all â—â—ementes Coâ—â—aâ—isâ—n â—â— the 〈◊〉 â—â—oâ—e and 〈◊〉 booke 〈◊〉 First difâ—inition A solide the most perfectest quantitie No science of thinges infinite Second diffinition Third diffinition Two difâ—initions included in this diâ—â—inition DeclaratioÌ„ of the first part Declaration of the second part Fourth diffinition Fifth diffinition Sixth diffinition Seuenth defâ—inition Eighth diâ—finition Ninth diâ—â—iâ—ition Tenth diffinition Eleuenth diffinition An other diffinition of a prisme which is a speciall diffinition of a prisme as it is commonly called and vsed This bodie called Figura Serratilis Psellus Twelueth diffinition What is to be taâ—â—n heede of in the diffinition of a sphere geuen by Iohannes de Sacro Busco Theodosiuâ— diâ—â—inition of a sphere The circumference of a sphere Galens diffinition 〈◊〉 a sphâ—râ— The digâ—itie of a sâ—here A sphere called a Globe Thirtenth diffinition Theodosius diffinition of the axe of a sphere Fourtenth diffinition Theodosius diffinition of the center of a sphere Flussas diffinition of the center of a sphere Fiuetenth diffinition Difference betwene the diameter axe of a sphere Seuententh diffinition First kinde of Cones A Cone called of Campane a roâ—â—de Piramis Seuententh diffinition A conicall superficies Eightenth diffinition Ninetenth diffinition A cillindricall superficies Corollary A roundâ— Columne or sphere A Corollary added by Campane Twenty diffinition Twenty one diffinitioâ— Twenty two diffinition A Tetrahedron one of the fiue regular bodyes Diâ—â—erence betwene a Tetrahedron and a Piramis Psellus calleth a Tetrahedron a Piramis Twenty three definition TweÌ„ty â—oâ—er definition Twenty fiue diffinition Fiue regular bodies The dignity of these bodies A Tetrahedron ascribed vnto the fire An octohedron ascribed vnto the ayre An Ikosahedron assigned vnto the water A cube assigned vnto the earth A dodecahedron assigned to heauen Diffinition of a parallelipipedon A Dâ—dâ—â—â—â—edron An Icosaâ—edron Demonstration leading to an impossibilitie An other demonstration after Flussas Construction Demonstration leading to an impossibilitie Demonstration leading to an impossibilitie Construction Demonstration Demonstration leading to an impossibilitie Construction * An Assumpt as M. Dee prâ—ueth it Demonstration Demonstration leading to an impossibilitie This proposition is as it were the conuerse of the sixth Construction Demonstration Construction Demonstration Construction Demonstration Construction Two cases in this proposition The first case Iohn Dee * This requireth the imagination of a plaine superficies passing by the pointe A and the straight line BC. And so helpe your selfe in the lyke cases either Mathematically imagining or Mechanically practising Second
casâ— Demonstâ—atâ—on Construction Demonstration Demonstration leading to an impossibilitie Note this maner of imagination Mathematicall Demonstration leading to an impossibilitie Construction Demonstration Demonstration leading to an absurditieâ— In tâ—is â—ronoâ—â—â—oâ—â—â— must vndâ—rsâ—and the propâ—rtioâ—â—ll â—artes or sâ—â—â—ions to be thâ—se which are câ—ntaiâ—ed 〈◊〉 the parallel superâ—â—cies Construction Demâ—nstration Construction Demonstration Demonstration leading to an impossibilitie Demonstration Construction Demonstraâ—ion Two cases in this proposition The first case Second case Constructiâ—â—â— Demonstration An other demonstration Construction Demonstraâ—â—â—n Construction Three cases in this proposition The first case A necessary thing to be proued before he pâ—oceede any â—arther in the construction of the Problemâ— * Which how to finde out is taught at the end of this demonstration and also was taught in the asâ—umpt put before the 14. proposition of the teÌ„th boke Demonstration of the first case Second case Third case An other demoÌ„stratioÌ„ to proue that thâ— line AB is not lesse theÌ„ the line LX. This was before taâ—ght in the tenth booke in the assumpt put before the 14. proposition * M. Dee to auoide cauillation addeth to Euclides proposition this worde sixe whome I haue followed accordingly and not Zamberts in this This kinde of body mencioned in the proposition is called a ParallelipipedoÌ„ according to the diâ—finition before geuen thereof Demonstration that the opposite sides are parallelogrammes DemonstratioÌ„ that the opposite superficies are equall * AB is equall to DC because the superficies AC is proued a parallelograÌ„me and by the same reason is BH equall to CF because the superficies FB is proued a parallelogramme therefore the 34. of the first is our proofe Fâ—â—â—â— Corollary Second Corolry These solides which he speaketh of in this Corollary are of some called sidâ—d columnâ—s Third Corollary Constrâ—ction Demonstrâ—tion * Looke at the end of the demonstratioâ— what is vnderstanded by staÌ„ding lines Iohn d ee his figure By this figure it appâ—arâ—th why â—uch Prismes were called â—â—edges of 〈◊〉 vâ—ry shape of a wedge as is the solide DEFGAC c. StaÌ„ding lines Construction Demonstration I. Dees figure Two cases in this proposition Thâ— first case Construction We are beholding to M. d ee for inuenting this figure with other which till his reforming were as much mishappen as this was and so both in the Greeke and Latine copies remaine Demonstration * Note now how the base respectiuely is takenâ— for so may alteratioÌ„ of respects alter the name of the bowndes eyther of solides or playnes Second case * There you perceaue how the base is diuersly considered chosen as before we aduertised you Construction Demonstration Construction Demonstration * * Note this famous Lâ—mma The doubling oâ—â—he Cube * Note * 〈…〉 Lemma Note what iâ— yet lacking requisite to the doubling of the Cube The conuerse of both the partes of the first case The conuerse of the second case The generall conclusion Construction DemonstratioÌ„ of the first part DemonstratioÌ„ of the second part To finde two middâ—e proportionals betwene two numbers geueÌ„ Note the practise of appâ—oching to precieâ—es in Cubik rootes * This is the way to apply any square geuen to a line also geueÌ„ sufficienty extended A probleme worth the searching for Construction Demonstration Construction Demonstration * It is euideÌ„t that those perpendiculars are all one with the staâ—ding lines of the solides if their solide angles be made of superâ—icâ—â—il right angles onely Dâ—uâ—ling of the 〈◊〉 c. * Dâ—mâ—â—â—â—ratioâ—â—f pâ—sâ—ibilitie in the â—â—oblem An other argumeâ—t to comâ—oâ—t the studious Demonstration of the first part Demonstration of the second part which is the conuerse of the first part Demonstration leading to an impossibilitie Construction Demonstration Construction Demonstration * Which of some are called sided Columnes Sided Columnes Construction Demonstration The squaring of the circle Demonstratiâ—n â—eading to an impossibiâ—itie Two cases in this proposition The first case That a square within any circle described is bigger than halfe the circle That the Isosceles triangles without the square are greater then halfe the segments wherin they are Second case * This As 〈…〉 afteâ—ward at the end of the demâ—straâ—ion proued Construction Demonstration Construction Demonstration Diâ—â—reence betwene the first probleme and the second Construction Demonstration Consâ—ruction Demonstration Consâ—ruction Demonstration Consâ—â—uction Note this well for it iâ— of great vse An other way of demonstâ—atâ—oÌ„ of the fâ—â—st â—robleme of thâ—â— addition Note this properâ—ie of a triangle rectangle Construction Demonstration * Though I say without the square yet you must thinke that it may be also within the square that diuersly Wherfore this Probleme may haue diuerse cases so but briefly to aâ—â—yde all may thus be said cut any side of that square into 3. partsâ— in the proportion of X to Y. Note the maner of the drift in this demonstration and construction mixtly and with no determination to the constructioÌ„â— as commoÌ„ly iâ— in problemeâ—â— which is here of me so vsedâ— for an example to young studeÌ„tes of variety in art Construction Demonstration * Note and remember one â—eâ—th in these solids The conclusion of the first part DemonstratioÌ„ of the second part namely that it is deuided moreouer into two equal Prismes Conclusion of the second part DemonstratioÌ„ of the last part that the two Prismes are greater then the halfe of the whole Pyramis Conclusion of the last part Conclusion of the whole proposition * An AssuÌ„pt An Assumpt Conclusion of the whole Demonstration leading to an impossibilitie * In the Assuâ—pâ—â—â—llowinâ— the second â—ropâ—â—ition of this bâ—â—ke Construction Demonstration Demonstration Note Sided Columnes sometime called prismes are triple to pyramids hauing one base and equall heâ—th with them Note â—arallelipipedons treble to pyramids of one base and heith with them Construction Demonstration An addition by Campane and Flussas Demonstration of the first part Demonstration â—f the second part which iâ— the conueâ—se of the first Constr 〈◊〉 〈◊〉 Parallelipipedons called Prismes * By this it is manifest that Euclide comprehended sided Columnes also vnder the name of a Prisme * A prisme hauing for his base a poligonon figure as we haue often before noted vnto you Note M. d ee his chiefe purpose in his additions Demonstration touching cylinders Second case Second parâ— which concerneth Cillinders Construction Demonstraâ—ion Construction Demonstration touching Cylinders Demonstration touching Cones First part of the propositioÌ„ demonstrated touching Cones Two cases in this proposition The first case Second case Construction Demonstration touching cylinders Second part demonstraâ—ed Construction * Note this LM because of KZ in the next proposition and here the point M for the point Z in the next demonstration I. Dee * For that the sections were made by the number two that is by taking halues and of the residue the halâ—eâ— and so to LD being an halfe and a residue which shall be a coÌ„mon measure backe againe to make sides of the Poligonon figure Construction
Demonstration * The circles so made or so considered in the sphere are called the greatest circles All other not hauing the center of the sphere to be their center alsoâ— are called lesse circles Note these descriptions * An other Corollary * An other Corollary Construction * This is also proued in the Asâ—umpt before added out oâ— Flussas Note what a greater or greatest circle in a Spere is First part of the Construction Noteâ— * You know full well that in the superficies of the sphere â—â—â—ly the circumferences of the circles are but by thâ—se circumferences the limitatioÌ„ and assigning of circles is vsed and so the circumference of a circle vsually called a circle which in this place can not offend This figure is restored by M. Dee his diligence For in the greeke and Latine Euclides the line GL the line AG and the line KZ in which three lynes the chiefe pinch of both the demonstrations doth stand are vntruely drawen as by comparing the studious may perceaue Note You must imagine 〈◊〉 right line AX to be perpeÌ„dicular vpon the diameters BD and CE though here AC the semidiater seme to be part of AX. And so in other pointes in this figure and many other strengthen your imagination according to the tenor of constructions though in the delineatioÌ„ in plaine sense be not satisfied Note BO equall to BK in respect of M. Dee his demonstration following †Note â—his point Z that you may the better vnderstand M. Dee his demoÌ„stration Second part of the construction Second part of the demonstration ✚ Which of necessity shall fall vpon Z as M. Dee proueth it and his profe is set after at this marke ✚ following I. Dee * But AZ is greater theÌ„ AG as in the former propositioÌ„ KM was euident to be greater then KG so may it also be made manifest that KZ doth neyther touch nor cut the circle FGâ—H An other proue that the line AY is greater theÌ„ the line AG. * This as an assumpt is presently proued Two cases in this proposition The first case Demonstration leading to an impossibilitie Second case * As it is â—asiâ— to gather by the â—â—â—umpt put after the secoâ—â— of this booâ—â— Note a generall rule The second part of the Probleme two wayes executed An vpright Cone The second part of the Probleme The second â—aâ—â— oâ— the â—robleme ☜ * This may easely be demonstrated as in thâ— 17. proposition the section of a sphere was proued to be a circle * For taking away all doubt this aâ— a Lemma afterward is demâ—strated A Lemma as it were presently demonstrated Construction Demonstration The second part of the Probleme * Construction Demonstration An other way of executing this probleme The conuerse of the assuÌ„pt A great error commonly maintained Betwene straight and croked all maner of proportioÌ„ may be geuen Construction Demonstration The diffiniâ—ioÌ„ of a circâ—e â—â—apâ—â—d in a spâ—erâ— Construction Demonstration This is manifest if you consider the two triangles rectangles HOM and HON and then with all vse the 47. of the first of Euclide Construction DemonstratioÌ„ Construction DemonstratioÌ„ This in maner of a Lemmâ— is presently proued Note here of Axe base soliditie more then I nede to bring any farther proofe for Note * I say halfe a circâ—lar reuolution for that suâ—â—iseth in the whole diameter ST to describe a circle by iâ— it be moued â—â—out his center Q c. Lib 2 prop 2. de Spheâ—a Cylindrâ— Note * A rectangle parallelipipedon geuâ—n equall to a Sphere geuen To a Sphere or to any part of a Sphere assigned as a third fourth fifth c to geue a parallelipipedon equall Sided Columes Pyramids and prismes to be geuen equall to a Sphere or to any certayne part thereof To a Sphere or any segment or sector of the same to geue a cone or cylinder equall or in any proportion assigned Farther vse of Sphericall Geometrie The argument of the thirtenth booke Construction Demonstration * The AssuÌ„pt proued * Because AC is supposed greater then AD therefore his residue is lesse then the residue of AD by the common sentence Wherefore by the supposition DB is greater then â—C The chieâ—e line in all Euclides Geometrie What is ment here by A section in one onely poiâ—t Construction Demonstration * Note how CE and the gnonom XOP are proued equall for it serueth in the conuerse demonstrated by M. Dee here next after This proposition â—the conuerse of the former * As we haâ—e noted the place of the peculier proâ—e there â—in the demoÌ„stration of the 3. * Therefore by my second Theoreme added vpon the second proposition DC is deuided by extreame and meane proportion in the point A. And because AC is bigger then CB therfore DA is greater then AC wherefore if a right line c. as in the proposition Which was to be demonstrated * Therefore by my second Theoreme added vpon the second proposition DC is deuided by extreame and meane proportion in the point A. And because AC is bigger then CB therfore DA is greater then AC wherefore if a right line c. as in the proposition Which was to be demonstrated Construction * Though I say perpeÌ„dicular yes you may perceue how infinite other pâ—sâ—â—ioÌ„s will serue so that DI and AD make an angle for a triangle to haue his sides proportionally cut c. Demonstration Demonstration I. Dee This is most euident of my second Theoreme added to the third propositioÌ„ For to adde to a whole line a line equall to the greater segmeÌ„t to adde to the greater segment a line equall to the whole line is all one thing in the line produced By the whole line I meane the line diuided by extreme and meane proportion This is before demonstrated most euidently and briefly by M. Dee after the 3. proposition Note Note 4. Proportional lines Note two middle proportionals Note 4. wayes of progresâ—ion in the proportion of a line deuided by extreme and middle proportion What resolution and composition is hath before bene taught in the beginning of the first booke * Proclus in the Greeke in the 58. page Construction Demonstration Two cases in this proposition Construction Thâ— first case Demonstration The second case Construction Demonstration Construction Demonstration This Corollary is the 3. proposition of the â—4 booke after Campane Demonstration of the first part Demonstration of the second part Construction Demâ—nstration Construstion Demonstration Constrâ—yction Demonstration This Corollary is the 11. propâ—sition of the 14. booke after Campane This Corollary is the 3. Corollary after the 17. proposition of the 14 booke after Campane * By the name oâ— a Pyramis both here iâ— this booke following vnderstand a Tetrahedron An other construction and demonstration of the second part after Fâ—ussas Third part of the demonstration This Corollary is the 15. proposition of the 14. booke after Campane This Corollary Campane putteth as a Corollary after
the 17. proposition of the 14. booke Construction â—rist part of the demonstration * For the 4. angles at the point K are equall to fower right angles by the Corollary of tâ—e 15. of the first and those 4. angle are equall the one to the other by the â— of the â—irst and theâ—efore ech is a right angle Second part of the demoÌ„stratioÌ„ * For the square of the line AB which is proued equall to the square of the line LM is double to the square of the line BD which is also equall to the square of the line LE. This Corollary is the 16. proposition of the 14. booke after Campane First part of the demonstration Second part of the ConstructioÌ„ Second part of the demonstraâ—ion * By the 2. Assumpt of the 13. of this booke Third part of the demonstration Second part of the demoÌ„stratioÌ„ For the line QW is equall to the line IZ the line ZW is commoÌ„ to them both This part is againe afterward demonstrated by Flussas The pentagon VBWCR proued to be in one and the selfe same playne superficies The pentagon VBWCZ it proued equiangle * Looke for a farther construction after Flussas at the ende of the demonstration That the side of the dodecahedron is a residuall line Draw in the former figure these lines ctA ctL câ—D The side of a pyramis The side of a cube The sides of a dodecahedron Comparison of the fiue sides of the foresayd bodies An other demoÌ„monstration to proue that the side of the IcosahedroÌ„ is greater then the side of the dodecahedron That 3. squares of the line FB are greater theÌ„ 6. squares of the line NB. That there can be no other solide besids these fiue contayned vnder equilater and equiangle bases That the angle of an equilater and equiangle Pentagon is one right angle and a 〈◊〉 part 〈◊〉 which thing was also before proued in the 〈◊〉 of the 32. of the â—irst The sides of the angle of the inclâ—â—â—tion of the 〈◊〉 of the 〈◊〉 are 〈◊〉 rationall The sides of the angle of the inclination of the 〈◊〉 â—f tâ—e 〈…〉 That the plaines of an octohedron are in liâ—e sort inclined That the plaines of an Icosahedron are in like sort inclined That the plaines of a Dâ—â—â—â—â—hedron are 〈◊〉 like sort inclined The sides of the angle of the inclination of the supeâ—ficieces of the TetrahedroÌ„ are proued rationall The sides of the angle of the inclination of the superficieces of the cube proued rationall The sides of thâ— angle c. of the octohedron proued rationall The sides of the angle c. of the Icosahedron proued irratioâ—all How to know whether the angle of the inclinatioÌ„ be a right angle an acute angle or an oblique angle The argument of the fourtenth booke First proposition after Flussas Construction Demonstration * This is manifest by the 12. propositioÌ„ of the thirtenh booke as Campane well gathereth in a Corollary of the same The 4. pâ—posâ—tioÌ„ after Flussas * This is afterward proued in the 4. proposition This Assumpt is the 3. proposition after Flussas Construction of the Assumpt Demonstration of the Assumpt Construction of the proposition Demonstration of the â—â—opâ—sition * Thâ— 5. proposition aâ—tâ—r 〈◊〉 Construction Demoâ—stration The 5. proposition aâ—ter Fâ—ussas Demonstration * This is the reason of the Corollary following A Corollary which also Flussas putteth as a Corollary after the 5. proposition in his order The 6. pâ—â—positioÌ„â—â—ter Flussas Coâ—struction Demonstration * This is not hard to proue by the 15. 16. and 19. of the â—â—â—eth †In the Corollary of the 17. of the tâ—irteÌ„th * 〈◊〉 againe is required the Assumpt which is afterward proued in this 4 proposition †But first the Assumpt following the construction whâ—reâ—f here beginneâ—h is to be proued The Assumpt which also Flussas putteth as an Assumpt aâ—ter the 6. propositioÌ„ Demonstration of the Assumpt Construction pertaining to the second dâ—monstration of the 4. propositioÌ„ Second demonstration oâ— the 4. proposition The 7â— proposition after Flussas Construction Demonstration * Here againe is required the Assumpt afterward proued in this 4. proposition †As may by the Assumpt afterward in this propositioÌ„ be plainely proued The 8. proâ—ition aâ—ter Flussas †By the Corollary added by Flussas after has Assumpt put after the 17. proposition of the 12. booke Corollary of the 8. after Flussas This Assumpt is the 3. propositioÌ„ aâ—ter â—lussas and is it which 〈◊〉 times hath bene taken aâ— gâ—aunted in this booke and oâ—ce also in the last proposition of the 13. booke as we haue beâ—ore noted Demonstration * In the 4. section â—f this proposition †In the 1. and 3 section of the same propositioÌ„ †In the 5. sectioÌ„ of the same proposition A Corollary The first proposition after Campane Construction Demonstration The 2. proposition after Campane Demonstration leading to an impossibilitie The 4. proposition after Campane Construâ—tâ—on Demonstration This Corollary Campane also â—utteth after the 4. proposition in his order The 5. proposition after Campane Construction Demonstration This is the 6. and 7. propositions after Campane Construction Demonsâ—ration This Corollary Campane also addeth after the 7. proposition iâ— his order The 5. proposition aâ—ter Campane Construction Demonstration This Assumpt Campane also hath after the 8. proposition in his order Construction Demonstration The 9. proposition after Campane Construction Demonstration This Campane putteth aâ— a Corollary in the 9. proposition after his order This Corollary is the 9. proposition after Campane The 12. proposition after Campane Construction Demonstration The 13. proposition after Campane The 14. proposition after Campane Demonstration of the first part Demonstration of the second part The 17. proposition after Campane Firâ—t part of the construction First part of the Demonstration Second parâ— of the câ—nstruction Second part of the Dâ—monstration The 18. proposition after Campane Demonstration of the first part Demonstration of the second part The Corollary of the 8. proposition after Campane Demâ—nstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration The argument of the 15. booke †In this proposition as also in all the other following by the name of a pyramis vnderstand a tetrahedron as I haue before admonished Construction Demonstration Construction Demonstration Construction Demonstrâ—tion Construction Demonstration Construction Demonstration That which here followeth concerning the inclination of the plaines of the fiue solides was before tought â—hough not altogether after the same maner out of Flussas in the latter â—nde of the 13 booke Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration First part of the construction First part of the demonsâ—ration Second part of the construction Second part of the Demonstration Third part of the construction Third part of the demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Produce in the figure the line TF to the point B. Construction Demonstration This proposition Campane hath is the last also in order of the 15. booke with him The argument of the 16. booke Construction Demonstration * By a Pyramis vnderstand a Tetrahedron throughout all this booke Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration †That is aâ— 18. to 1. Demonstration †That iâ— as 9. to 2. * That is as 18. to 2. or 9. to 1. Draw in the figure a line from B to H. * What the duple of an extreme and meane proportion is Construction Demonstration Demonstration Constrution Demonstration Construction Demonstration Demonstration * That is at 13. 1 â— is to â— Demonstration Demonstration Construction Demonstration Construction Demonstration Extend in the figure a line â—rom the point E to the point B. Extend in the figure a line from the point E to the point B. Demonstration Construction Demonstration Second part of the Demonstration Icosidodecahedron ExoctohedroÌ„ That the â—xoctohedron is contayned in a sphere That the exoctohedâ—on is contayned in the sphere geuen That the diaâ—â—â—ter of the sâ—here is doâ—ble to the side â—f the exoctohedron That the Icosidodecahedron is contayned in the sphere geuen * That is as 8. 103 â— Faultes escaped â—cl â—ag Line Faultesâ— Co 〈◊〉 〈◊〉    Errata Lib. 1.  1 2 41 point B. at Campane point C aâ— Campane 3 1 22 aâ—l lines drawne all righ◠〈…〉 3 1 28 lines drawen to the superficies right lines draweÌ„ to the circumference 9 1 42 liâ—es AB and AC lines AB and BC 15 1 35 are equall are proued equall 20 2 28 by the first by the fourth 21 1 39 tâ—e centre C. the centre E 2â— 2 â— Iâ—â—ower right If two right 25 2 3 fâ—â—â— petition fiueth petition 49 2 7 14. â—â— 32.64 c. 4.8.16.32.64 53 1 39 the triangle NG the triangle Kâ— 54 2 25 by the 44 by the 42 57 2 23 and Câ—G in the and CGB is thâ— Â Â Â In stede of â—lussates through out 〈◊〉 whole booke read â—lusâ—as    Errata Lib. 2.  60 2 29 Gnomon FGEH Gnomon AHKD   30 Gnomon EHFG Gnomon â—CKD 69 1 18 the whole line the whole â—igure 76 2 9 the diameter CD the diameter AHF    Errata Lib. 3.  82 2 36 angle equall to the angle 92 1 last the line AC is the line AF 〈◊〉    Errata Lib. 4.  110 2 10 CD toucheth the ED toucheth the   12 side of the other angle of the other 115 1 21 and HB and HE 117 2 44 the angle ACD the angle ACB 118 1 2 into ten equall into two equall 121 1 3â— CD and EA CD DE EA â—â—â— 1 29 the first the third    Errata Lib. 5.  126 1 43 it maketh 12. more then 17. by 5. it maketh 24. more then 17. by 7. 129 1  In stede of the figure of the 6. definition draw in the magâ—â— a figure like vnto thâ—s 134 2 4 As AB is to A so is CD to C As AB is to B so is CD to D 141 2 last But if K excede M But if H excede M LIEFE IS DEATHE AND DEATH IS LIEFE AETATIS SVAE XXXX AT LONDON Printed by Iohn Daye dwelling ouer Aldersgate beneath Saint Martins ¶ These Bookes are to be solde at his shop vnder the gate 1570.
coÌ„pareth them all with Triangles also together the one with the other In it also is taught how a figure of any forme may be chaunged into a Figure of an other forme And for that it entreateth of these most common and generall thynges thys booke is more vniuersall then is the seconde third or any other and therefore iustly occupieth the first place in order as that without which the other bookes of Eâ—clide which follow and also the workes of others which haue written in Geometry cannot be perceaued nor vnderstanded And forasmuch â—s all the demonstrations and proofes of all the propositions in this whole booke depende of these groundes and principles following which by reason of their playnnes neede no greate declaration yet to remoue all be it neuer so litle obscuritie there are here set certayne shorte and manifesâ— expositions of them Definitions 1. A signe or point is that which hath no part The better to vnderstand what manâ—r of thing a signe or point is ye must note that the nature and propertie of quantitie wherof Geometry entreateth is to be deuided so that whatsoeuer may be deuided into sundâ—y partes is called quantitie But a point although it pertayne to quantitie and hath his beyng in quantitie yet is it no quantitie for that it cannot be deuided Because as the definition saith it hath no partes into which it should be deuided So that a pointe is the least thing that by minde and vnderstanding can be imagined and conceyued then which there can be nothing lesse as the point A in the margent A signe or point is of Pithagoras Scholers after this manner defined A poynt is an vnitie which hath position NuÌ„bers are conceaued in mynde without any forme figure and therfore without matter wheron to 〈◊〉 figure consequently without place and position Wherfore vnitie beyng a part of number hath no position or determinate place Wherby it is manifest that â—umbâ—â— iâ— more simple and pure then is magnitude and also immateriall and so vnity which iâ— the bâ—ginning of number is lesse materiall then a â—igne or poyâ—â— which is the beginnyng of magnitude For a poynt is maâ—eriall and requireth position and place and â—â—â—rby differeth from vnitie â— A line is length â—ithout breadth There pertaine to quantiâ—â—e three dimensions length bredth thicknes or depth and by these thre are all quaÌ„titieâ— measured made known There are also according to these three dimensions three kyndes of continuall quantities a lyne a superficies or plaine and a body The first kynde namely a line is here defined in these wordes A lyne is length without breadth A point for that it is no quantitie nor hath any partes into which it may be deuided but remaineth indiuisible hath not nor can haue any of these three dimensions It neither hath length breadth nor thickenes But to a line which is the first kynde of quantitie is attributed the first dimension namely length and onely that for it hath neither breadth nor thicknes but is conceaued to be drawne in length onely and by it it may be deuided into partes as many as ye list equall or vnequall But as touching breadth it remaineth indiuisible As the lyne AB which is onely drawen in length may be deuided in the pointe C equally or in the point D vnequally and so into as many partes as ye list There are also of diuers other geuen other definitions of a lyne as these which follow A lyne is the mouyng of a poynte as the motion or draught of a pinne or a penne to your sence maketh a lyne Agayne A lyne is a magnitude hauing one onely space or dimension namely length wantyng breadth and thicâ—â—es 3 The endes or limites of a lyne are pointes For a line hath his beginning from a point and likewise endeth in a point so that by this also it is manifest that pointes for their simplicitie and lacke of composition are neither quantitie nor partes of quantitie but only the termes and endes of quantitie As the pointes A B are onely the endes of the line AB and no partes thereof And herein differeth a poynte in quantitie from vnitie in numberâ— for that although vnitie be the beginning of nombers and no number as a point is the beginning of quantitie and no quantitie yet is vnitie a part of number For number is nothyng els but a collection of vnities and therfore may be deuided into them as into his partes But a point is no part of quantitie or of a lyneâ— neither is a lyne composed of pointes as number is of vnities For things indiuisible being neuer so many added together can neuer make a thing diuisible as an instant in time is neither tyme nor part of tyme but only the beginning and end of time and coupleth ioyneth partes of tyme together 4 A right lyne is that which lieth equally betwene his pointes As the whole line AB lyeth straight and equally betwene the poyntes AB without any going vp or comming downe on eyther side A right line is the shortest extension or draught that is or may be from one poynt to an other Archimedes defineth it thus Plato defineth a right line after this maner A right line is that whose middle part shadoweth the exâ—remeâ— As if you put any thyng in the middle of a right lyne you shall not see from the one ende to the other which thyng happeneth not in a crooked lyne The Ecclipse of the Sunne say Astronomers then happeneth when the Sunne the Moone our eye are in one right line For the Moone then being in the midst betwene vs and the Sunne causeth it to be darkened Diuers other define a right line diuersly as followeth A right lyne is that which standeth firme betwene his extremes Agayne A right line is that which with an other line of lyke forme cannot make a figure Agayne A right lyne is that which hath not one part in a plaine superficies and an other erected on high Agayne A right lyne is that all whose partes agree together with all his other partes Agayne A right lyne is that whose extremes abiding cannot be altered Euclide doth not here define a crooked lyne for it neded not It may easely be vnderstand by the definition of a right lyne for euery contrary is well manifested set forth by hys contrary One crooked lyne may be more crooked then an other and from one poynt to an other may be drawen infinite crooked lynes but one right lyne cannot be righter then an other and therfore from one point to an other there may be drawen but one tight lyne As by figure aboue set you may see 5 A superficies is that which hath onely length and breadth A superficies is the second kinde of quantitie and to it are attributed two â—imensions namely length and breadth As in the
about the diameter together with the two supplementes make a gnomon As the parallelograme EBKH with the two supplementes AEGK and KHFD make the gnomon FGEH Likewise the parallelogramme GKCF with the same two supplementes make the gnomon EHFG And this diffinition of a gnomon extendeth it selfe and is generall to all kyndes of parallelogrammes whether they be squares or figures of one side longer or Rhombus or Romboides To be shorte if you take away from the whole parallelogramme one of the partiall parallelogrammes which are about the diameter whether ye will the rest of the figure is a gnomon Campaâ—e after the last proposition of the first booke addeth this propositioÌ„ Two squares being geuen to adioyne to one of them a Gnomon equall to the other square which for that as then it was not taught what a Gnomon is I there omitted thinking that it might more aptly be placed here The doing and demonstration whereof is thus Suppose that there be two squares AB and CD vnto one of which namely vnto AB it is required to adde a Gnomon equall to the other square namely to CD Produce the side BF of the square AB directly to the point E. and put the line FE equall to the side of the square CD And draw a line from E to A. Now then forasmuch as EFA is a rectangle triangle therefore by the 47. of the first the square of the line EA is equall to the squares of the lines EF FA. But the square of the line EF is equall to the square CD the square of the side FA is the square AB Wherefore the square of the line AE is equall to the two squares CD and AB But the sides EF and FA are by the 21. of the first longer then the side AE and the side FA is equall to the side FB Wherfore the sides EF and FB are longer theÌ„ the side AE Wherefore the whole line BE is longer then the line AE From the line BE cut of a line equall to the line AE which let be BC. And by the 46. proposition vpon the line BC describe a square which let be BCGH which shal be equal to the square of the line AE but the square of the line AE is equal to the two squares AB and DC Wherefore the square BCGH is equal to the same squares Wherfore forasmuch as the square BCGH is composed of the square AB and of the gnomon FGAH the sayde gnomon shal be equall vnto the square CD which was required to be done An other more redy way after Pelitarius Suppose that there be two squares whose sides let be AB and BC. It is required vnto the square of the line AB to adde a gnomon equall to the square of the line BC. Set the lines AB and BC in such sort that they make a right angle ABC And draw a line froÌ„ A to C. And vpoÌ„ the line AB describe a square which let be ABDE And produce the line BA to the point F and put the line BF equall to the line AC And vpon the line BF describe a square which let be BFGH which shal be equal to the square of the line AC wheÌ„ as the lines BF and AC are equal and therefore it is equal to the squares of the two lines AB and BC. Now forasmuch as the square BFGH is made complete by the square ABDE and by the gnomon FEGD the gnomon FEGD shal be equal to the square of the line BC which was required to be done The 1. Theoreme The 1. Proposition If there be two right lines and if the one of them be deuided into partes howe many soeuer the rectangle figure comprehended vnder the two right lines is equall to the rectangle figures whiche are comprehended vnder the line vndeuided and vnder euery one of the partes of the other line SVppose that there be two right lynes A and BC and let one of them namely BC be deuided at all aduentures in the pointes D and E. Then I say that the rectangle figure comprehended vnder the lines A and BC is equall vnto the rectangle figure comprehended vnder the lines A and BD vnto the rectangle figure which is coÌ„prehended vnder the lines A and DE and also vnto the rectangle figure which is comprehended vnder the lines A and EC For from the pointe Brayse vp by the 11. of the first vnto the right line BC a perpendiculer line BF vnto the line A by the third of the first put the line BG equall and by the point G by the 31. of the first draw a parallel line vnto the right line BC and let the same be GM and by the selfe same by the points D E and C draw vnto the line BG these parallel lines DK EL and CH. Now then the parallelograme BH is equall to these parallelogrammes BK DL and EH But the parallelograme BH is equall vnto that which is contayned vnder the lines A and BC. For it is compreheÌ„ded vnder the lines GB BC and the line GB is equall vnto the line A And the parallelograme BK is equall to that which is contayned vnder the lines A and BD for it is comprehended vnder the line GB and BD and BG is equall vnto A And the parallelograme DL is equall to that which is contayned vnder the lines A and DE for the line DK that is BG is equal vnto A And moreouer likewise the parallelograme EH is equall to that which is contained vnder the lines A EC VVherfore that which is compreheÌ„ded vnder the lines A BC is equall to that which is comprehended vnder the lines A BD vnto that which is compreheÌ„ded vnder the lines A and DE and moreouer vnto that which is comprehended vnder the lines A and EC If therfore there be two right lines and if the one of them be deuided into partes how many soeuer the rectangle figure comprehended vnder the two right lines is equall to the rectangle figures which are comprehended vnder the line vndeuided and vnder euery one of the partes of the other line which was required to be demonstrated Because that all the Propositions of this second booke for the most part are true both in lines and in numbers and may be declared by both therefore haue I haue added to euery Proposition conuenient numbers for the manifestation of the same And to the end the studious and diligent reader may the more fully perceaue and vnderstand the agrement of this art of Geometry with the science of Arithmetique and how nere deare sisters they are together so that the one cannot without great blemish be without the other I haue here also ioyned a little booke of Arithmetique written by one Barlaam a Greeke authour a man of greate knowledge In whiche booke are by the authour demonstrated many of the selfe same proprieties and passions in number which Euclide in this his second boke hath demonstrated in magnitude
namely the first ten propositions as they follow in order VVhich is vndoubtedly great pleasure to coÌ„sider also great increase furniture of knowledge VVhose PropositioÌ„s are set orderly after the propositioÌ„s of Euclide euery one of Barlaam correspoÌ„dent to the same of Euclide And doubtles it is wonderful to see how these two coÌ„trary kynds of quantity quantity discrete or number and quantity continual or magnitude which are the subiectes or matterâ— of Arithmitique and Geometry shoulde haue in them one and the same proprieties common to them both in very many points and affections although not in all For a line may in such sort be deuided that what proportion the whole hath to the greater parte the same shall the greater part haue to the lesse But that can not be in number For a number can not so be deuided that the whole number to the greater part thereof shall haue that proportion which the greater part hath to the lesse as Iordane very playnely proueth in his booke of Arithmetike which thynge Campane also as we shall afterward in the 9. booke after the 15. proposition see proueth And as touching these tenne firste propositions of the seconde booke of Euclide demonstrated by Barlaam in numbers they are also demoÌ„strated of Campane after the 15. proposition of the 9. booke whose demonstrations I mynde by Gods helpe to set forth when I shal come to the place They are also demoÌ„strated of Iordane that excellet learned authour in the first booke of his Arithmetike In the meane tyme I thought it not amisse here to set forth the demonstrations of Barlaam for that they geue great light to the seconde booke of Euclide besides the inestimable pleasure which they bring to the studious considerer And now to declare the first Proposition by numbers I haue put this example following Take two numbers the one vndeuided as 74. the other deuided into what partes and how many you list as 37. deuided into 20. 10. 5. and 2â— which altogether make the whole 37. Then if you multiply the number vndeuided namely 74 into all the partes of the number deuided as into 20. 10. 5. and 2. you shall produce 1480. 740. 370. 148. which added together make 2738 which self number is also produced if you multiplye the two numbers first geuen the one into the other As you see in the example on the other side set So by the aide of this Proposition is gotten a compendious way of multiplication by breaking of one of the numbers into his partes which oftentimes serueth to great vse in workingâ— chiâ—â—ly in the rule of proportions The demonstration of which proposition followeth in Barlaam But â—irst are put of the author these principles following ¶ Principles 1. A number is sâ—yd to multiply an other number when the number multiplied is so oftentymes added to it selfe as there be vnities in the number which multiplieth wherby is produced a certaine number which the number multiplied measureth by the vnities which are in the number which multipliâ—th 2. And the number produced of that a multiplication is called a plaine or superficiall number 3. A square number is that which is produced of the multiplicatian of any number into it selfe 4. Euery lesse number compared to a greater is sayd to be a part of the greater whether the lesse measure the greater or measure it not 5. Numbers whome one and the selfe same number measureth equally that is by one and the selfe same number are equall the one to the otheâ— 6. Numbers that are equemultiplâ—ces to one and the selfe same number that is which contayne one and the same number equally and alike are equall the one to the other The first Proposition Two numbers beyng geuen if the one of them be deuided into any numbers how many soeuer the playne or superficiall number which is produced of the multiplication of the two numbers first geuen the one into the other shall be equall to the superficiall nuÌ„bers which are produced of the multiplication of the number not deuided into euery part of the number deuided Suppose that there be two numbers AB and C. And deuide the number AB into certayne other numbers how many soeuer as into AD DE and EB Then I say that the superficiall number which is produced of the multiplication of the number C into the number AB is equall to the superficiall numbers which are produced of the multiplication of the number C into the nuÌ„ber AD and of C into DE and of C into EB For let F be the superficiall number produced of the multiplication of the number C into the number AB and let GH be the superficiall number produced of the multiplication of C into AD And let HI be produced of the multiplication of C into DE aâ—d finally of the multiplication of C into EB let there be produced the number IK Now forasmuch as AB multiplying the number C produced the number F therefore the number C measureth the number F by the vnities which are in the number AB And by the same reason may be proued that the number C doth also measure the number GH by the vnities which are in the number AD and that it doth measure the number HI by the vnities which are in the nuÌ„ber DF and finally that it measureth the number IK by the vnities which are in the number EB Wherefore the nuÌ„ber C measureth the whole number GK by the vnities which are in the number AB But it before measured the number F by the vnities which are in the number AB wherfore either of these numbers F and GK is equemultiplex to the number C. But numbers which are equemultiplices to one the selfe same numbers are equall the one to the other by the 6. definition Wherfore the number F is equall to the number GK But the number F is the superficiall number produced of the multiplication of the nuÌ„ber C into the number AB and the number GK is composed of the superficiall numbers produced of the multiplication of the nuÌ„ber C not deuided into euery one of the numbers AD DE and EB If therefore there be two numbers geuen and the one of them be deuided c. Which was required to be proued The 2. Theoreme The 2. Proposition If a right line be deuided by chaunce the rectangles figures which are comprehended vnder the whole and euery one of the partes are equall to the square whiche is made of the whole SVppose that the right line AB be by chaunse denided in the point C. Then I say that the rectangle figure comprehended vnder AB and BC together with the rectangle comprehended vnder AB and AC is equall vnto the square made of AB Describe by the 46. of the first vpon AB a square ADEB and by the 31 of the first by the point C draw a line parallel vnto either of these lines AD aâ—d BE and let the same
the double of B is prime vnto the double of B then the two numbers whereof the number is composed namely the number composed of A and C and the double of B shall be prime the one to the other by the 30 of the seuenth And therefore the number composed of A and C shall be prime to B taken once For if any number should measure the two numbers namely the number composed of A and C and the number B it should also measure the number composed of A and C and the double of B by the 5. common sentence of the seuenth which is not possible for that they are proued to be prime numbers Here haue I added an other demonstration of the former Proposition after Campane which proueth that in nuÌ„bers how many soeuer which is there proued onely touching three numbers and the demonstration seement somwhat more perspicous then Theons demonstration And thus he putteth the proposition If numbers how many soeuer being in continuall proportion be the least that haue one the same proportion with them euery one of them shal be to the number composed of the rest prime Secondly I say that this is so in euery one of them namely that C is a prime number to the number composed of A B D. For if not then as before let E measure C and the number composed of A B D which E shal be a number not prime either to F or to G by the former proposition added by Campane wherefore let H measure them And forasmuch as H measureth E it shall also measure the whole A B C D whom E measureth And forasmuch as H measureth one of these numbers F or G it shall measure one of the extreames A or D which are produced of F or G by the second of the eight if they be multiplâ—ed into the meanes L or K. And moreouer the same H shall measure the meames BC by the 5. common sentence of the seuenth when as by supposition it measureth either F or G. which measure B C by the second of the eight But the same H measureth the whole A B C D as we haue proued for that it measureth E. Wherefore it shall also measure the residue namely the number composed of the extreames A and D by the 4. common sentence of the seuenth And it measureth one of these A or D for it measureth one of these F or G which produce A and D wherefore the same H shall measure one of these A or D and also the other of them by the former common sentence which numbers A and D are by the 3. of the eight prime the one to the other Which were absurd This may also be proued in euery one of these numbers A B C D. Wherefore no number shall measure one of these numbers A B C D and the numbers composed of the rest Wherefore they are prime the one to the other If therefore numbers how many soeuer c which was required to be proued Here as I promised I haue added Campanes demonstrations of those Propositions in numbers which Euclâ—de in the second booke demonstrated in lines And that in thys place so much the rather for that Theon as we see in the demonstration of the 15. Proposition seemeth to alledge the 3. 4. Proposition of the second boke which although they concerne lines onely yet as we there declared and proued are they true also in numbers ¶ The first Proposition added by Campane That number which is produced of the multiplication of one number into numbers how many soeuer is equall to that number which is produced of the multiplication of the same number into the number composed of them This proueth that in numbers which the first of the second proued touching lines Suppoâ—â— that the number A being multiplyed into the number B and into the number C and into the number D doo produce the numbers E F and G. Then I say that the number produced of A multiplyed into the number composed of B C and D is equall to the number composed of E F and G. For by the conuerse of the definition of a number multiplyed what part vnitie is of A the selfe same part is B of E and C of F and also D of G. Wherefore by the 5. of the seuenth what part vnitie is of A the selfe same part is the number composed of B C and D of the number composed of E F and G. Wherfore by the definition that which is produced of A into the number composed of B C D is equall to the number composed of E F G which was required to be proued The second Proposition That number which is produced of the multiplication of numbers how many soeuer into one nuÌ„ber is equall to that number which is produced of the multiplication of the number composed of them into the same number This is the conuerse of the former As if the â—â—â—bers â— and G and D multiplyed into the number A doo produce the numbers E and F and G. Then the number composed of B C D. multiplyed into the number A shall produce the number composed of the numbers E F G. Which thing is easly proued by the 16. of the seuenth and by the former proposition ¶ The third Proposition That number which is produced of the multiplication of numbers how many soeuer into other numbers how many soeuer is equall to that number which is produced of the multiplication of the number composed of those first numbers into the number composed of these latter numbers As if the numbers A B C doo multiply the numbers D E F ech one eche other and if the numbers produced be added together Then I say that the number composed of the numbers produced is equall to the number produced of the number composed of the numbers A B C into the number composed of the numbers D E F. For by the former propositioÌ„ that which is produced of the number composed of A B C into D is equall to that which is produced of euery one of the sayd numbers into D and by the same reason that which is produced of the number composed of A B C into E is equal to that which is produced of euery one of the sayd numbers into E and so likewise that which is produced of the number composed of A B C into F is equall to that which is produced of euery one of the sayd numbers into F. But by the first of these propositions thâ—â— which is produced of the number composed of these numbers A B C into euery one of these numbers D E F is equall to that which is produced of the number composed into the number composed wherefore that is manifest which was required to be proued ¶ The fourth Proposition If a number be deuâ—â—â—d into partes how many soeuer that nuÌ„ber which is produced of the whole into him selfe is equall to that number which is produced of
therof 〈◊〉 This iâ— also to be noted that of lines some are commensurable in length the one to the other and some are commensurable the one to the other in power Of lines commensurable in length the one to the other was geuen an example in the declaration of the first diffinitioÌ„ namely the lines A and B which were commensurable in length one and the selfe measure namely the line C measured the length of either of them Of the other kinde is geuen this diffinition here set for the opening of which take this example Let there be a certaine line namely the line BC and let the square of that line be the square BCDE Suppose also an other line namely the line FH let the square thereof be the square FHIK and let a certayne superficies namely the superficies A measure the square BCDE taken 16. times which is the number of the litle areas squares plats or superficieces coÌ„tained and described within the sayd squares ech of which is equall to the superficie A. Agayne let the same superficies A measure the square FHIK 9. times taken according to the number of the fieldâ—s or superficieces contayned and described in the same Ye see theÌ„ that one and the selfe same superficies namely the superficies A is a common measure to both these squares and by certayne repeticions thereof measureth them both Wherefore the two lines BC and FH which are the sides or lines producing these squares and whose powers these squares are are by this diffinition lines commensurable in power 4 Lines incommensurable are such whose squares no one plat or superficies doth measure This diffinition is easy to be vnderstanded by that which was sayd in the diffinition last set before this and neadeth no farther declaration And thereof take this example If neither the superficies A nor any other superficies doo measure the two squares B CDE and FHIK or if it measure the one â—â—rely BCDE and not the other FHIK or if it measure the square FHIK and not the square BCDE the two lines BC and FH are in power incommensurable and therfore also incommeÌ„surable in length For whatsoeuer lines are incommeÌ„surable in power the same are also incommensurable in length as shall afterward in the 9. proposition of this booke be proued And therfore such lines are here defined to be absolutely incommensurable These thinges thus standing it may easely appeare that if a line be assigned and layd before vs there may be innumerable other lines commensurable vnto it and other incommensurable vnto it of commensurable lines some are commensurable in length and power and some in power onely 5 And that right line so set forth is called a rationall line Thus may ye see how to the supposed line first set may be compared infinite lines some commensurable both in length power and some commensurable in power onely and incommensurable in length and some incommensurable both in power in length And this first line so set whereunto and to whose squares the other lines and their squares are compared is called a rationall line commonly of the most part of writers But some there are which mislike that it should be called a rationall line that not without iust cause In the Greeke copy it is called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 rete which signifieth a thing that may be spokeÌ„ expressed by word a thing certayne graunted and appoynted Wherefore Flussates a man which bestowed great trauell and diligence in restoring of these elementes of Euclide leauing this word rationall calleth this line supposed and first set a line certaine because the partes thereof into which it is deuided are certaine and known and may be expressed by voyce and also be coumpted by number other lines being to this line incommensurable whose parts are not distinctly known but are vncertayne nor can be expressed by name nor assignd by number which are of other men called irrationall he calleth vncertaine and surd lines Petrus Montaureus although he doth not very wel like of the name yet he altereth it not but vseth it in al his booke Likewise wil we doo here for that the word hath bene and is so vniuersally receiued And therefore will we vse the same name and call it a rationall line For it is not so great a matter what names we geue to thinges so that we fully vnderstand the thinges which the names signifie This rationall line thus here defined is the ground and foundation of all the propositions almost of this whole tenth booke And chiefly from the tenth proposition forwardes So that vnlesse ye first place this rationall line and haue a speciall and continuall regard vnto it before ye begin any demonstration ye shall not easely vnderstand it For it is as it were the touch and triall of all other lines by which it is known whether any of them be rationall or not And this may be called the first rationall line the line rationall of purpose or a rationall line set in the first place and so made distinct and seuered from other rationall lines of which shall be spoken afterwarde And this must ye well commit to memory 6 Lines which are commensurable to this line whether in length and power or in power onely are also called rationall This definition needeth no declaration at all but is easily perceiued if the first definition be remembred which â—heweth what magnitudes are commensurable and the third which â—heweth what lines are commensurable in power Here notâ— how aptly naturally Euclide in this place vseth these wordes commensurable either in length and power or in power onely Because that all lines which are commensurable in length are also commensurable in powerâ— when he speaketh of lines commensurable in leÌ„gth he euer addeth and in power but when he speaketh of lines commensurable in power he addeth this worde Onely and addeth not this worde in length as he in the other added this worde in power For not all lines which are commensurable in power are straight way commensurable also in length Of this definition take this example Let the first line rationall of purpose which is supposed and laide forth whose partes are certaine known and may be expressed named and nuÌ„bred be AB the quadrate wherof let be ABCD then suppose againe an other lyne namely the line EF which let be commensurable both in length and in power to the first rationall line that is as before was taught let one line measure the length of eche line and also lâ—t one superâ—icies measure the two squares of the said two lines as here in the example is supposed and also appeareth to the eie then is the line E F also a rationall line Moreouer if the lyne EF be commensurable in power onely to the rationall line AB first set and supposed so that no one line do measure the two lines AB and EF As in example yâ— see to be for
that the line EF is made equall to the line AD which is the diameter of the square ABCD of which square the line AB is a side it is certayne that the â—ide of a square is incoÌ„meÌ„surable in leÌ„gth to the diameter of the same square if there be yet founde any one superficies which measureth the two squares ABCD and EFGH as here doth the triangle ABD or the triangle ACD noted in the square ABCD or any of the foure triangles noted in the square EFGH as appeareth somwhat more manifestly in the second example in the declaration of the last definition going before the line EF is also a rational line Note that these lines which here are called rationall lines are not rational lines of purpose or by supposition as was the first rationall line but are rationall onely by reason of relation and comparison which they haue vnto it because they are commensurable vnto it either in length and power or in power onely Farther here is to be noted that these wordes length and power and power onely are ioyned onely with these wordeâ— commensurable or incommensurable and are neuer ioyned with these woordes rationall or irrationall So that no lines can be called rational in length or in power nor like wise can they be called irrationall in length or in power Wherin vndoubtedly Campanus was deceiued who vsing those wordes speaches indifferently caused brought in great obscuritie to the propositions and demonstrations of this boke which he shall easily see which marketh with diligence the demonstrations of Campanus in this booke 7 Lines which are incommensurable to the rationall line are called irrationall By lines incommensurable to the rationall line supposed in this place he vnderstandeth such as be incommensurable vnto it both in length and in power For there are no lines incommensurable in power onely for it cannot be that any lines should so be incommenâ—urable in power onely that they be not also incommensurable in length What so euer lines be incommeâ—surable in power the same be also incommensurable in length Neither can Euclide here in this place meane lines incommensurable in length onely for in the diffinition before he called them rationall lines nâ—ither may they be placed amongst irrationall lines Wherfore it remayneth that in this diffintion he speaketh onely of those lines which are incommensurable to the rationall line first geuen and supposed both in length and in power Which by all meanes are incommensurable to the rationall line therfore most aptly are they called irrationall lines This diffinition is easy to be vnderstanded by that which hath bene sayd before Yet for the more plainenes see this example Let the â—â—rst rationall line supposed be the line AB whose square or quadrate let be ABCD. And let there be geuen an other line EF which lâ—t be to the rationall line incommensurable in length and power so that let no one line measure the length of the two lines AB and EF and let the square of the line EF be EFGH Now if also there be no one superficies which measureth the two squares ABCD and EFGH as is supposed to be in this example theÌ„ is the line EF an irrationall line which word irrational As before did this word rational misliketh many learned in this knowledge of Geometry Flussates as he left the word rationall and in steade thereof vsed this word certaine so here he leaueth the word irrationall and vseth in place thereof this word vncertaine and euer nameth these lines vncertaine lines Petrus Montaureus also misliking the word irrationall would rather haue them to be called surd lines yet because this word irrationall hath euer by custome and long vse so generally bene receiuedâ— he vseth continually the same In Greeke such lines are called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 alogoi which signifieth nameles vnspeakeable vncertayne in determinate and with out proportion not that these irrationall lines haue no proportion at all either to the first rationall line or betwene them selues but are so named for that theyr proportions to the rationall line cannot be expressed in number That is vndoubtedly very vntrue which many write that their proportions are vnknowne both to vs and to nature Is it not thinke you a thing very absurd to say that there is any thing in nature and produced by nature to be hidde from nature and not to be knowne of nature it can not be sayd that their proportions are vtterly hidde and vnknowne to vs much lesse vnto nature although we cannot geue them their names and distinctly expresse them by numbers otherwise should Euclide haue taken all this trauell and wonderfull diligence bestowed in this bookeâ— in vaine and to no vseâ— in which he doth nothing ellâ— but teach the proprieties and passions of these irrationall linesâ— and sheweth the proportions which they haue the one to the other Here is also to be noted which thing also Tartalea hath before diligently notedâ— that Campanus and many other writers of Geometryâ— ouer much â—â—â—ed and were deceiued in that they wrote and taught that all these lines whose squares were not sâ—gnified and mought be expressed by a square number although they mighâ— by any other number as by 11. 12. 14. and such others not square numbers are irrationall lines Which is manifestly repugnant to the groundes and principles of Euclide who wil that all lines which are commensurable to the rationall line whether it be in length and power or in power onely should be rationall Vndoubtedly this hath bene one of the chiefest and greatest causes of the wonderfull confusion and darkenes of this booke which so hath tossed and tormoyled the wittes of all both writers and readers masters and scholers and so ouerwhelmed them that they could not with out infinite trauell and sweate attayne to the truth and perfect vnderstanding thereof 8 The square which is described of the rationall right line supposed is rationall Vntill this diffinition hath Euclide set forth the nature and proprietie of the first kinde of magnitude namely of lines how they are rationall or irrationall now he bâ—ginneth to â—hew how the second kinde of magnitudes namely superficies are one to the other rationall or irrationall This diffinition is very playne Suppose the line AB to be the rationall line hauing his parts and diuisions certaynely knowne the square of which line let be the square ABCD. Now because it is the square of the rationall line AB it is also called rationall and as the line AB is the first rationall line vnto which other lines compared are coumpted rationall or irrationall so is the quadrat or square thereof the â—irst rationall superficies vnto which all other squares or figures compared are coumpted and named rationall or irrationall 9 Such which are commensurable vnto it are rationall In this diâ—â—inition where it is sayd such as are commensurable to the square of the rationall line are not vnderstand onely other squares or
parallelogramme shall afterward be taught in the 27. and 28. Propositions of thys booke ¶ A Corollary Hereby it is manifest that a rectangle parallelogramme contayned vnder two right lines is the meane proportionall betwene the squares of the sayd lines As it was manifest by the first of the sixt that that which is contayned vnder the lines AB and BC is the meane proportionall betwene the squares AD and CX This Corollary is put after the 53. Proposition of this booke as an Assumpt and there demonstrated which there in his place you shall finde But because it followeth of this Proposition so euidently and briefly without farther demonstration I thought it not amisse here by the way to note it ¶ The 23. Theoreme The 26. Proposition A mediall superficies excedeth not a mediall superficies by a rationall superficies FOr if it be possible let AB being a mediall superficies exceede AC being also a mediall superficies by DB being a rationall superficies And let there be put a rationall right line EF. And vpon the line EF apply a rectangle parallelogramme FH equall vnto the mediall superficies AB whose other side let be EH and from the parallelogramme FH take away the parallelogramme FG equall vnto the mediall superficies AC Wherefore by the third common sentence the residue BD is equall to the residue KH But by supposition the superficies DB is rationall Wherfore the superficies KH is also rationall And forasmuch as either of these superficieces AB and AC is mediall and AB is equall vnto FH AC vnto FG therefore either of these superficieces FH and FG is mediall and they are applyed vpon the rationall line EF. Wherefore by the 22. of the tenth either of these lines HE and EG is rationall incommensurable in length vnto the line EF. And forasmuch as the superficies DB is rationall and the superficies KH is equall vnto it therefore KH is also rationall and it is applied vpoÌ„ the rationall line EF for it is applied vpon the line GK which is equall to the line EF Wherefore by the 20. of the tenth the line GH is rationall and commensurable in length vnto the line GK But the line GK is equall to the line EF. Wherfore the line GH is rationall and commensurable in length vnto the line EF. But the line EG is rationall and incommensurable in length to the line EF. Wherefore by the 13. of the tenth the line EG is incommensurable in length vnto the line GH And as the line EG is to the line GH so is the square of the line EG to the parallelogramme contayned vnder the lines EG and GH by the Assumpt put before the 21. of the tenth Wherefore by the 10. of the tenth the square of the line EG is incommensurable vnto the parallelogramme contayned vnder the lines EG and GH But vnto the square of the line EG are commensurable the squares of the lines EG and GH for either of them is rationall as hath before bene proued Wherefore the squares of the lines EG and GH are incommensurable vnto the parallelogramme contayned vnder the lines EG and GH But vnto the parallelogramme contayned vnder the lines EG and GH is commensurable that which is contayned vnder the lines FG and GH twise for they are in proportion the one to the other as number is to number namely as vnitie is to the number 2 or as 2. is to 4 and therefore by the 6. of this booke they are commensurable Wherefore by the 13. of the tenth the squares of the lines EG and GH are incommensurable vnto that which is contayned vnder the lines EG and GH twise This is more brieâ—ly concluded by the corollary of the 13. of the tenth But the squares of the lines EG and GH together with that which is contayned vnder the lines EG and GH twise are equall to the square of the line EH by the 4. of the second Wherefore the square of the line EH is incoÌ„mensurable to the squares of the lines EG and GH by the 16. of the tenth But the squares of the lines FG GH are rationall Wherfore the square of the line EH is irrationall Wherefore the line also EH is irrationall But it hath before bene proued to be rationall which is impossible Wherefore a mediall superficies exceedeth not a mediall superficies by a rationall superficies which was required to be proued ¶ The 4. Probleme The 27. Proposition To finde out mediall lines commensurable in power onely contayning a rationall parallelogramme LEt there be put two rationall lines commensurable in power onely namely A and B. And by the 13. of the six take the meane proportionall betwene the lines A and B and let the same line be C. And as the line A is to the line B so by the 12. of the sixt let the line C be to the line D. And forasmuch as A and B are rationall lines commensurable in power onely therfore by the 21. of the tenth that which is contayned vnder the lines A and B that is the square of the line C. For the square of the line C is equall to the parallelogramme contayned vnder the lines A anâ— B by the 17. of the sixth is mediall â—herfore C also is a mediall line And for that as the line A is to the line B so is the line C to the line D therfore as the square of the line A is to the square of the lyne B so is the square of the line C to the square of the line D by the 22. of the sixth But the squares of the lines A and B are commensurable for the liâ—â—s A and B aâ—e supposed to be rationall commeÌ„surable in power onely Wherefore also the squares of the lines C and D are commensurable by the 10. of the tenth wherfore the lines C and D are commensurable in power onely And C is a mediall line Wherfore by the 23. of the tenth D also is a mediall line Wherfore C and D are mediall lynes commensurable in power onely Now also I say that they contayne a rationall parallelogramme For for that as the line A is to the line B so is the line C to the line D therfore alternately also by the 16. of the fift as the line A is to the line C so is the lyne B to the lyne D. But as the lyne A is to the lyne C so is the line C to the lyne B wherfore as the line C is to the line B so is the line B to the lyne D. Wherfore the parallelograÌ„me coÌ„tayned vnder the lines C and D is equal to the square of the line B. But the square of the lyne B is rationall Wherfore the parallelograme which is contayned vnder the lynes C and D is also rationall Wherfore there are found out mediall lines commensurablâ— in powâ—r onâ—ly contayning a rationall parallelogrammeâ— which 〈◊〉 required to be done The 5. Probleme The
power a rationall and a mediall which was required to be demonstrated An other demonstration of the same after Campane Supose that AB be a line contayning in power a rationall and a mediall whereunto let the line GD be commensurable either in length and power or in power onely Then I say that the line GD is a line contayning in power a rationall and a mediall Take a rational line EZ vpoÌ„ which by the 45. of the first apply a rectangle parallelograÌ„me EZFC equall to the square of the line AB and vpon the line CF which is equall to the line EZ applye the parallelogramme FCHI equall to the square of the line GDâ— and let the breadths of the sayd parallelogrammes be the lines EG and CH. And forasmuch as the line AB is commensurable to the line GD at the least in power onely therefore the parallelogrammes EF and FH which are equall to their squares shal be commensurable Wherefore by the 1. of the sixt the right lines EC and CH are coÌ„meÌ„surable in leÌ„gth And forasmuch as the parallelogramme EF which is equall to the square of the line Aâ— which contayneth in power â— rationall and a mediall is applyed vpon the rationall EZ making in breadth the line EC therefore the line EC is a fifth binomiall line by the 64. of this booke vnto which line EC the line CH is coÌ„meÌ„surable in length wherefore by the 66. of this booke the line CH is also a fifth binomiall line And forasmuch as the superficies CI is contayned vnder the rationall line EZ that is CF and a fifth binomall line CH therefore the line which contayneth in power the superficies CI which by supposition is the line GD is a line contayning in power a rationall and a mediall by the 58. of this booke A line therefore commensurable to a line contayning in power a rationall and a mediall c. ¶ The 52. Theoreme The 70. Proposition A line commensurable to a line contayning in power two medialls is also a line contayning in power two medialls SVppose that AB be a line contayning in power two medialls And vnto the line AB let the line CD be commensurable whether in length power or in power onely Then I say that the line CD is a line contayning in power two medialls Forasmuch as the line AB is a line contayning in power two medialls let it be deuided into his partes in the point E. Wherefore by the 41. of the tenth the lines AE and EB are incommensurable in power hauing that which is made of the squares of them added together mediall and that also which is contained vnder them mediall and that which is made of the squares of the lines AE EB is incommensurable to that which is contained vnder the lines AE and EB Let the selfe same construction be in this that was in the former And in like sort may we proue that the lines CF FD are incommensurable in power and that that which is made of the squares of the lines AE and EB added together is commensurable to that which is made of the squares of the lines CF and FD added together and that that also which is contained vnder the lines AE and EB is commensurable to that which is contained vnder the lines CF and FD. Wherefore that which is made of the squares of the lines CF and FD is mediall by the Corollary of the 23. of the tenth and that which is contayned vnder the lines CF and FD is mediall by the same Corollary â— and moreouer that which is made of the squares of the lines CF FD is incommensurable to that which is contained vnder the lines CF and FD. Wherefore the line CD is a line containing in power two medialls which was required to be proued ¶ An Assumpt added by Montaureus That that which is made of the squares of the lines CF and FD added together is incommensurable to that which is contained vnder the lines CF and FD is thus proued For because as that which is made of the squares of the lines AE and EB added together is to the square of the line AE so is that which is made of the squares of the lines CF and FD added together to the square of the line CF as it was proued in the Propositions going before therefore alternately as that which is made of the squares of AE and EB added together is to that which is made of the squares of CF and FD added together so is the square of the line AE to the square of the line CF. But before namely in the 68. Proposition it was proued that as the square of the line AE is to the square of the line CF so is the parallelograÌ„me contained vnder the lines AE and EB to the parallelogramme contained vnder the lines CF and FD. Wherefore as that which is made of the squares of the lines AE and EB is to that which is made of the squares of the lines CF and FD so is the parallelogramme contained vnder the lines AE and EB to the parallelogramme contained vnder the lines CF and FD. Wherefore alternately as that which is made of the squares of the lines AE and EB is to the parallelogramme contained vnder the lines AE and EB so is that which is made of the squares of the lines CF and FD to the parallelogramme contained vnder the lines CF and FD. But by supposition that which is made of the squares of the lines AE and EB is incommâ—nsurable to the parallelogramme contained vnder the lines AE EB Wherefore that which is made of the squares of the lines CF and FD added together is incommensâ—rable to the parallelogramme contained vnder the lines CF and FD which was required to be proued An other demonstration after Campane Suppose that AB be a line contayning in power two medialls wherunto let the line GD be commensurable either in length and in power or in power onely Then I say that the line GD is a line coÌ„tayning in power two medialls Let the same construction be in this that was in the former And forasmuch as the parallelogramme EF is equall to the square of the line AB and is applyed vpon a rationall line EZ it maketh the breadth EC a sixt binomiall line by the 65. of this booke And forasmuch as the parallelogrammes EF CI which are equall vnto the squares of the lines AB and GD which are supposed to be commensurable are commensurable therefore the lines EC and CH are commensurable in length by the first of the sixt But EC is a sixt binomiall line Wherefore CH also is a sixt binomiall line by the 66. of this booke And forasmuch as the superficies CI is contayned vnder the rationall line CF and a sixt binomiall line CH therefore the line which coÌ„tayneth in power the superficies CI namely the line GD is a line contayning in power two medialls by the 59. of
measureth the square number produced of EF. Wherefore also by the 14. of the eight the number G measureth the number EF and the number G also measureth it selfe Wherefore the number G measureth these numbers EF and G when yet they are prime the one to the other which is impossible Wherefore the diameter A is not commensurable in length to the side B. Wherefore it is incommensurable which was required to be demonstrated An other demonstration after Flussas Suppose that vppon the line AB be described a square whose diameter let be the line AC Then I say that the side AB is incommensurable in length vnto the diameter AC Forasmuch as the lines AB and BC are equall therefore the square of the line AC is double to the square of the line AB by the 47. of the first Take by the 2. of the eight nuÌ„bers how many soeuer in continuall proportion froÌ„ vnitie and in the proportion of the squares of the lines AB and AC Which let be the numbers D E F G. And forasmuch as the first from vnitie namely E is no square number for that it is a prime number neither is also any other of the sayd numbers a square number except the third from vnitie and so all the rest leuing one betwene by the 10. of the ninth Wherefore D is to E or E to F or F to G in that proportion that a square number is to a number not square Wherefore by the corrollary of the 25. of the eight they are not in that proportion the one to the other that a square number is to a square number Wherefore neither also haue the squares of the lines AB and AC which are in the same proportion that porportion that a square number hath to a square number Wherefore by the 9. of this booke their sides namely the side AB and the diameter AC are incommensurable in length the one to the other which was required to be proued This demonstration I thought good to adde for that the former demonstrations seme not so full and they are thought of some to be none of Theons as also the proposition to be none of Euclides Here followeth an instruction by some studious and skilfull Grecian perchance Theon which teacheth vs of farther vse and fruite of these irrationall lines Seing that there are founde out right lines incommensurable in length the one to the â—ther as the lines A and B there may also be founde out many other magnitudes hauing leÌ„gth and breadth such as are playne superficieces which shal be incommeÌ„surable the one to the other For if by the 13. of the sixth betwene the lines A and B there be taken the meane proportionall line namely C then by the second corrollary of the 20. of the sixth as the line A is to the line B so is the figure described vpon the line A to the figure described vpon the line C being both like and in like sort described that is whether they be squares which are alwayes like the one to the other or whether they be any other like rectiline figures or whether they be circles aboute the diameters A and C. For circles haue that proportion the one to the other that the squares of their diameters haue by the 2. of the twelfth Wherfore by the second part of the 10. of the tenth the â—igures described vpon the lines A and C being like and in like sort described are incommensurable the one to the other Wherfore by this meanes there are founde out superficieces incommensurable the one to the other In like sort there may be founde out figures coÌ„meÌ„surable the one to the other if ye put the lines A and B to be coÌ„mensurable in leÌ„gth the one to the other And seing that it is so now let vs also proue that euen in soliâ—es also or bodyes there are some commensurable the one to the other and other some incommensurable the one to the other For if from eche of the squares of the lines A and B or from any other rectiline figures equal to these squares be â—rected solides of equall altiâ—ude whether those solides be compâ—sed of equidistant supersicieces or whether they be pâ—ramids or prismes thosâ— solides sâ— erâ—câ—ed shal be in that proportioÌ„ the one to the other that theyr bases are by the 32. oâ— the eleuenth and 5. and 6. of the twelfth Howbeit there is no such proposition concerning prismes And so if the bases of the solides bâ— commensurable the one to the other the solides also shall be commensurable the one to the other and if the bases be incommensurable the one to the other the solides also shall be incommensurable the one to the other by the 10. of the tenth And if there be two circles A and B and vpon ech of the circles be erected Cones or Cilinders of equal altitude those Cones Cilinders sâ—all be in that proportion the one to the other that the circles are which are their bases by the 11. of the twelfth and so if the circles be commensurable the one to the other the Cones and Cilinders also shall be commensurable the one to the other But if the circles be incoÌ„mensurable the one to the other the Cones also and Cilinders shal be incoÌ„mensurable the one to the other by the 10. of the tenth Wherefore it is manifest that not onely in lines and superâ—icieces but also in solides or bodyes is found commensurabilitie or incommensurability An aduertisement by Iohn Dee Although this proposition were by Euclide to this booke alotted as by the auncient grecian published vnder the name of Aristoteles 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 it would seme to be and also the property of the same agreable to the matter of this booke and the proposition it selfe so famous in Philosophy and Logicke as it was would in maner craue his elemeÌ„tal place in this teÌ„th boke yet the dignitie perfectionâ— of Mathematicall Method can not allow it here as in due order following But most aptly after the 9. propositioÌ„ of this booke as a Corrollary of the last part thereof And vndoubtedly the propoâ—itioÌ„ hath for this 2000. yeares bene notably regarded among the greke Philosophers and before Aristotles time was concluded with the very same inconuenience to the gaynesayer that the first demonstration here induceth namely Odde number to be equall to euen as may appearâ— in Aristotles worke named Analitica prima the first booke and 40. chapter But els in very many places of his workes he maketh mention of the proposition Euident also it is that Euclide was about Aristotles time and in that age the most excellent Geometrician among the Grekes Wherefore seing it was so publike in his time so famous and so appertayning to the property of this booke it is most likely both to be knowne to Euclide and also to haue bene by him in apt order placed But of the disordring of it can remayne no doubt if ye consider in Zamberts translation
at all aduentures namely D V G S and a right line is drawen from the point D to the point G and an other from the point V to the point S. Wherefore by the 7. of the eleuenth the lines DG and VS are in one and the selfe same plaine superficies And forasmuch as the line DE is a parallel to the line BG therefore by the 24. of the first the angle EDT is equall to the angle BGT for they are alternate angles and likewise the angle DTV is equall to the angle GTS Now then there are two triangles that is DTV and GTS hauing two angles of the one equall to two angles of the other and one side of the one equall to one side of the other namely the side which subtendeth the equall angles that is the side DV to the side GS for they are the halfes of the lines DE and BG Wherefore the sides remayning are equall to the sides remayning Wherfore the line DT is equall to the line TG and the line VT to the line T S If therefore the opposite sides of a Parallelipipedon be deuided into two equall partes and by their sections be extended plaine superficieces the common section of those plaine superficieces and the diameter of the Parallelipipedon do deuide the one the other into two equall partes which was required to be demonstrated A Corollary added by Flussas Euery playne superficies extended by the center of a parallelipipedon diuideth that solide into two equall partes and so doth not any other playne superficies not extended by the center For euery playne extended by the center cutteth the diameter of the parallelipipedon in the center into two equall partes For it is proued that playne superficieces which cutte the solide into two equall partes do cut the dimetient into two equall partes in the center Wherefore all the lines drawen by the center in that playne superficies shall make angles with the dimetient And forasmuch as the diameter falleth vpon the parallel right lines of the solide which describe the opposite sides of the sayde solide or vpon the parallel playne superficieces of the solide which make angels at the endes of the diameter the triangles contayned vnder the diameter and the right line extended in that playne by the center and the right line which being drawen in the opposite superficieces of the solide ioyneth together the endes of the foresayde right lines namely the ende of the diameter and the ende of the line drawen by the center in the superficies extended by the center shall alwayes be equall and equiangle by the 26. of the first For the opposite right lines drawen by the opposite playne superficieces of the solide do make equall angles with the diameter forasmuch as they are parallel lines by the 16. of this booke But the angles at the ceÌ„ter are equall by the 15. of the first for they are head angles one side is equall to one side namely halfe the dimetient Wherefore the triangles contayned vnder euery right line drawen by the center of the parallelipipedon in the superficies which is extended also by the sayd center and the diameter thereof whose endes are the angles of the solide are equall equilater equiangle by the 26. of the first Wherfore it followeth that the playne superficies which cutteth the parallelipipedon doth make the partes of the bases on the opposite side equall and equiangle and therefore like and equall both in multitude and in magnitude wherefore the two solide sections of that solide shal be equall and like by the 8. diffinition of this booke And now that no other playne superficies besides that which is extended by the center deuideth the parallelipipedon into two equall partes it is manifest if vnto the playne superficies which is not extended by the center we extend by the center a parallel playne superficies by the Corollary of the 15. of this booke For forasmuch as that superficies which is extended by the center doth deuide the parallelipipedoÌ„ into two equall parâ—â— it is manifest that the other playne superficies which is parallel to the superficies which deuideth the solide into two equall partes is in one of the equall partes of the solide wherefore seing that the whole is euer greater then his partes it must nedes be that one of these sections is lesse then the halfe of the solide and therefore the other is greater For the better vnderstanding of this former proposition also of this Corollary added by Flussas it shal be very nedefull for you to describe of pasted paper or such like matter a parallelipipedoÌ„ or a Cube and to deuide all the parallelograÌ„mes therof into two equall parts by drawing by the câ—Ì„ters of the sayd parallelogrammes which centers are the poynts made by the cutting of diagonall lines drawen froÌ„ thâ— opposite angles of the sayd parallelograÌ„mes lines parallels to the sides of the parallelograÌ„mes as in the former figure described in a plaine ye may see are the sixe parallelograÌ„mes DE EH HA AD DH and CG whom these parallel lines drawen by the ceÌ„ters of the sayd parallelograÌ„mes namely XO OR PR and PX do deuide into two equall parts by which fower lines ye must imagine a playne superficies to be extended also these parallel lynes KL LN NM and Mâ— by which fower lines likewise yâ— must imagine a playne superficies to be extended ye may if ye will put within your body made thus of pasted paper two superficieces made also of the sayd paper hauing to their limites lines equall to the foresayde parallel lines which superficieces must also be deuided into two equall partes by parallel lines drawen by their centers and must cut the one the other by these parallel lines And for the diameter of this body exteÌ„d a thred from one angle in the base of the solide to his opposite angle which shall passe by the center of the parallelipipedon as doth the line DG in the figure before described in the playne And draw in the base and the opposite superficies vnto it Diagonall lines from the angles from which is extended the diameter of the solide as in the former description are the lines BG and DE. And when you haue thus described this body compare it with the former demonstration and it will make it very playne vnto you so your letters agree with the letters of the figure described in the booke And this description will playnely set forth vnto you the corollary following that proposition For where as to the vnderstanding of the demonstration of the proposition the superficieces put within the body were extended by parallel lynes drawen by the ceÌ„ters of the bases of the parallelipipedon to the vnderstanding of the sayd Corollary ye may extende a superficies by any other lines drawen in the sayd bases so that yet it passe through the middest of the thred which is supposed to be the center of the parallelipipedon ¶ The 35. Theoreme The 40. Proposition If there be
a triangle and if the parallelogramme be double to the triangle those Prismes are by the 40. of the eleuenth equall the one to the other therefore the Prisme contained vnder the two triangles BKF and EHG and vnder the three parallelogrammes EBFG EBKH and KHFG is equall to the Prisme contained vnder the two triangles GFC and HKL and vnder the three parallelogrammes KFCL LCGH and HKFG And it is manifest that both these Prismes of which the base of one is the parallelogramme EBFG and the opposiâ—e vnto it the line KH and the base of the other is the triangle GFC and the opposite side vnto it the triangle KLH are greater then both these Pyramids whose bases are the triangles AGE and HKL and toppes the pointes H D. For if we drawe these right lines EF and EK the Prisme whose base is the parallelogramme EBFG and the opposite vnto it the right line HK is greater then the Pyramis whose base is the triangle EBF toppe the point K. But the Pyramis whose base is the triangle EBF and toppe the point K is equall to the Pyramis whose base is the triangle AEG and toppe the point H for they are contained vnder equall and like plaine superficieces Wherefore also the Prisme whose base is the parallelogramme EBFG and the opposite vnto it the right line HK is greater then the Pyramis whose base is the triangle AEG and toppe the point H. But the prisme whose base is the parallelogramme EBFG and the opposite vnto it the right line HK is equall to the prisme whose base is the triangle GFC and the opposite side vnto it the triangle HKL And the Pyramis whose base is the triangle AEG and toppe the point H is equall to the Pyramis whose base is the triangle HKL and toppe the point D. Wherefore the foresaid two prismes are greater then the foresaid two Pyramids whose bases are the triangles AEG HKL and toppes the pointes H and D. Wherefore the whole Pyramis whose base is the triangle ABC and toppe the point D is deuided into two Pyramids equall and like the one to the other and like also vnto the whole Pyramis hauing also triangles to their bases and into two equall prismes and the two prismes are greater then halfe of the whole Pyramis which was required to be demonstrated If ye will with diligence reade these fower bookes following of Euclide which concerne bodyes and clearely see the demonstrations in them conteyned it shall be requisite for you when you come to any proposition which concerneth a body or bodies whether they be regular or not first to describe of pâ—sâ—ed paper according as I taught you in the end of the definitions of the eleuenth booke such a body or bodyes as are there required and hauing your body or bodyes thus described when you haue noted it with letters according to the figure set forth vpoÌ„ a plaine in the propositioÌ„ follow the construction required in the proposition As for example in this third propositioÌ„ it is sayd that Euery pyramis hauing a triangle to â—is base may be deuided into two pyramids c. Here first describe a pyramis of pasted paper haâ—ing his base triangled it skilleth not whether it be equilater or equiangled or not only in this proposition is required that the base be a triangle Then for that the proposition supposeth the base of the pyramis to be the triangle ABC note the base of your pyramis which you haue described with the letters ABC and the toppe of your pyramis with the letter D For so is required in the proposition And thus haue you your body ordered ready to the construction Now in the construction it is required that ye deuide the lines AB BC CA. c namely the sixe lines which are the sids of the fower triangles contayning the piramis into two equall partes in the poyntet â— F G c. That is ye must deuide the line AB of your pyramis into two equall partes and note the poynt of the deuision with the letter E and so the line BC being deuided into two equall partes note the poynt of the deuision with the letter F. And so the rest and this order follow ye as touching the rest of the construction there put and when ye haue finished the construction compare your body thus described with the demonstration and it will make it very playne and easy to be vnderstaÌ„ded Whereas without such a body described of matter it is hard for young beginners vnlesse they haue a very deepe imagination fully to conceaue the demonstration by the sigâ—e as it is described in a plaine Here for the better declaration of that which I haue sayd haue I set a figure whose forme if ye describe vpon pasted paper noted with the like letters and cut the lines â—A DA DC and folde it accordingly it will make a Pyramis described according to the construction required in the proposition And this order follow ye as touching all other propositions which concerne bodyes ¶ An other demonstration after Campane of the 3. proposition Suppose that there be a Pyramis ABCD hauing to his base the triangle BCD and let his toppe be the solide angle A from which let there be drawne three subtended lines AB AC and AD to the three angles of the base and deuide all the sides of the base into two equall partes in the three poyntes E F G deuide also the three subteÌ„ded lines AB AC and AD in two equall partes in the three points H K L. And draw in the base these two lines EF and EG So shall the base of the pyramis be deuided into three superficieces whereof two are the two triangles BEF and EGD which are like both the one to the other and also to the whole base by the 2 part of the secoÌ„d of the sixth by the definitioÌ„ of like superâ—iciecâ—s they are equal the one to the other by the 8. of the first the third superficies is a quadrangled parallelogramme namely EFGC which is double to the triangle EGD by the 40. and 41. of the first Now then agayne from the poynt H draw vnto the points E and F these two subtendent lines HE and HF draw also a subtended line from the poynt K to the poynt G. And draw these lines HK KL and LH Wherefore the whole pyramis ABCD is deuided into two pyramids which are HBEF and AHKL and into two prismes of which the one is EHFGKC and is set vpon the quadrangled base CFGE the other is EGDHKL and hath to his base the triangle EGD Now as touching the two pyramids HBEF and AHKL that they are equall the one to the other and also that they are like both the one to the other and also to the whole it is manifest by the definition of equall and like bodyes and by the 10. of the eleuenth and by 2. part of the second of the sixth And that the two Prismes are equall it
it comprehendeth Wherfore the pyramis whose base is the square ABCD and altitude the self same that the cone hath is greater then the halfe of the cone Deuide by the 30. of the third euery one of the circumferences AB BC CD and DA into two equall partes in the pointes E F G and H and drawe these right lines AE EB BF FC CG GD DH and HA. Wherefore euery one of these triangles AEB BFC CGD and DHA is greater then the halfe part of the segment of the circle described about it Vppon euery one of these triangles AEB BFC CGD and DHA describe a pyramis of equall altitude with the cone and after the same maner euery one of those pyramids so described is greater then the halfe part of the segment of the cone set vpon the segment of the circle Now therefore diuiding by the 30 of the third the circumferences remaining into two equall parts drawing right lines raysing vp vpon euery one of those triangles a pyramis of equall altitude with the cone and doing this continually we shal at the length by the first of the tenth leaue certayne segmentes of the cone which shal be lesse then the excesse whereby the cone excedeth the third part of the cylinder Let those segmentes be AE EB BF FC CG GD DH and HA. Wherefore the pyramis remayning whose base is the poligonoÌ„ figure AEBFCGDH and altitude the self same with the cone is greater then the third part of the cylinder But the pyramis whose base is the poligonon figure AEBFCGDH and altitude the self same with the cone is the third part of the prisme whose base is the poligonoÌ„ figure AEBFCGDH and altitude the self same with the cylinder Whefore the prisme whose base is the poligonon figure AEBFCGDH and altitude the self same with the cylinder is greater then the cylinder whose base is the circle ABCD. But it is also lesse for it is contayned of it which is impossible Wherfore the cylinder is not in lesse proportion to the cone then in treble proportion And it is proued that it is not in greater proportion to the cone then in treble proportion wherefore the cone is the third part of the cylinder Wherfore euery cone is the third part of a cylinder hauing one the self same base and one and the selfe same altitude with it which was required to be demonstrated ¶ Added by M. Iohn Dee ¶ A Theoreme 1. The superficies of euery vpright Cylinder except his bases is equall to that circle whose semidiameter is middell proportionall betwene the side of the Cylinder and the diameter of his base ¶ A Theoreme 2. The superficies of euery vpright or Isosceles Cone except the base is equall to that circle whose semidiameter is middell proportionall betwene the side of that Cone and the semidiameter of the circle which is the base of the Cone My entent in additions is not to amend Euclideâ— Method which nedeth little adding or none at all But my desire is somwhat to furnish you toward a more general art Mathematical theÌ„ Euclides ElemeÌ„ts remayning in the termes in which they are written can sufficiently helpe you vnto And though Euclides Elementes with my Additions run not in one Methodicall race toward my marke yet in the meane space my Additions either geue light where they are annexed to Euclides matter or geue some ready ayde and shew the way to dilate your discourses Mathematicall or to inuent and practise thinges Mechanically And in deede if more leysor had happened many more straunge matters Mathematicall had according to my purpose generall bene presently published to your knowledge but want of due leasour cauâ—eth you to want that which my good will toward you most hartely doth wish you As concerning the two Theoremes here annexed their veritie is by Archimedes in his booke of the Sphere and Cylinder manifestly demonstrated and at large you may therefore boldly trust to them and vse them as suppositions in any your purposes till you haue also their demoÌ„strations But if you well remember my instructions vpon the first proposition of this booke and my other addition vpon the second with the suppositions how a Cylinder and a Cone are Mathematically produced you will not neede Archimedes demonstration nor yet be vtterly ignoraunt of the solide quantities of this Cylinder and Cone here compared the diameter of their base and heith being knowne in any measure neither can their croked superficies remayne vnmeasured Whereof vndoubtedly great pleasure and commoditie may grow to the sincere student and precise practiser ¶ The 11. Theoreme The 11. Proposition Cones and Cylinders being vnder one and the selfe same altitude are in that proportion the one other that their bases are In like sorte also may we proueâ— that as the circle EFGH is to the circle ABCD so is not the cone EN to any solide lesse then the cone AL. Now I say that as the circle ABCD is to the circle EFGH so is not the cone AL to any solide greater then the cone EN For if it be possible let it be vnto a greater namely to the solide X. Wherefore by conuersion as the circle EFGH is to the circle ABCD so is the solide X to the cone AL but as the solide X is to the cone AL so is the cone EN to some solide lesse then the cone AL as we may see by the assumpt put after thâ— second of this booke Wherefore by the 11. of the fift as the circle EFGH is to the circle ABCGâ— so is the cone EN to some solide lesse then the cone AL which we haue proued to be impossible Wherefore as the circle ABCD is to the circle EFGH so is not the cone AL to any solide greater then the cone EN And it is also proued that it is not to any lesse Wherefore as the circle ABCD is to the circle EFGH so is the cone AL to the cone EN But as the cone is to the cone so is the cylinder to the cylinder by the 15. of the fift for the one is in treble proportion to the other Wâ—erefore by the 11. of the fift as the circle ABCD is to the circle EFGH so are the cylinders which are set vpon them the one to the other the said cylinders being vnder equall altitudes with the cones Cones therefore and cylinders being vnder one the self same altitude are in that proportion the one to the other that their bases are which was required to be demonstrated ¶ The 12. Theoreme The 12. Proposition Like Cones and Cylinders are in treble proportion of that in which the diameters of their bases are Now also I say that the cone ABCDL is not to any solide greater then the cone EFGHN in treble proportion of that in which the diameter BD is to the diameter FH For if it be possible let it be to a greater namely to the solide X. Wherefore by conuersion by the
line which subtendeth the angle ZOB to the third line which subtendeth the angle ZKB But by construction BO is equall to BK therefore OZ is equall to KZ And the third alâ—o is equall to the third Wherefore the point Z in respecte of the two triangles rectangles OZB and KZB determineth one and the same magnitude iâ— the line BZ Which can not be if any other point in the line BZ were assigned nearer or farther of from the point B. One onely poynt therefore is that at which the two perpendiculars KZ and OZ fall But by construction OZ falleth at Z the point and therefore at the same Z doth the perpendicular drawen from K fall likewyse Which was required to be demonstrated Although a briefe monition mought herein haue serued for the pregnant or the humble learner yet for them that are well pleased to haue thinges made plaine with many wordes and for the stiffenecked busie body it was necessary with my controlment of other to annexe the cause reason therof both inuincible and also euident A Corollary 1. Hereby it is manifest that two equall circles cutting one the other by the whole diameter if from one and the same end of their common diameter equall portions of their circumferences be taken and from the pointes ending those equall portions two perpendiculars be let downe to their common diameter those perpendiculars shall fall vpon one and the same point of their common diameter 2. Secondly it followeth that those perpendiculars are equall ¶ Note From circles in our first supposition eche to other perpendicularly erected we procede and inferre now these Corollaries whether they be perpendicularly erected or no by reasou the demonstration hath a like force vpon our suppositions here vsed ¶ The 16. Theoreme The 18. Proposition Spheres are in treble proportion the one to the other of that in which their diameters are SVppose that there be two spheres ABC and DEF and let their diameters be BC and EF. Then I say that the sphere ABC is to the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF. For if not then the sphere ABC is in treble proportion of that in which BC is to EF either to some sphere lesse then the sphere DEF or to some sphere greater First let it be vnto a lesse namely to GHK And imagine that the spheres DEF and GHK be both about one and the selfe same centre And by the proposition next going before describe in the greater sphere DEF a polihedron or a solide of many sides not touching the superficies of the lesse sphere GHK And suppose also that in the sphere ABC be inscribed a polihedron like to the polihedron which is in the sphere DEF Wherefore by the corollary of the same the polihedron which is in the sphere ABC is to the polihedron which is in the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF. But by supposition the sphere ABC is to the sphere GHK in treble proportion of that in which the diameter BC is to the diameter EF. Wherefore as the sphere ABC is to the sphere GHK so is the polihedroÌ„ which is described in the sphere ABC to the polihedroÌ„ which is described in the sphere DEF by the 11. of the fift Wherfore alternately by the 16. of the fift as the sphere ABC is to the polihedron which is described in it so is the sphere GHK to the polihedron which is in the sphere DEF But the sphere ABC is greater then the polihedroÌ„ which is described in it Wherfore also the sphere GHK is greater then the polihedroÌ„ which is in the sphere DEF by the 14. of the fift But it is also lesse for it is contayned in it which impossible Wherefore the sphere ABC is not in treble proportioÌ„ of that in which the diameter BC is to the diameter EF to any sphere lesse then the sphere DEF In like sort also may we proue that the sphere DEF is not in treble proportion of that in which the diameter EF is to the diameter BC to any sphere lesse then the sphere ABC Now I say that the sphere ABC is not in treble proportioÌ„ of that in which the diameter BC is to the diameter EF to any sphere greater theÌ„ the sphere DEF For if it be possible let it be to a greater namely to LMN Wherfore by conuersion the sphere LMN is to the sphere ABC in treble proportion of that in which the diameter EF is to the diameter BC. But as the sphere LMN is to the sphere ABC so is the sphere DEF to some sphere lesse theÌ„ the sphere ABC as it hath before bene proued for the sphere LMN is greater then the sphere DEF Wherfore the sphere DEF is in treble proportioÌ„ of that in which the diameter EF is to the diameter BC to some sphere lesse theÌ„ the sphere ABC which is proued to be impossible Wherefore the sphere ABC is not in treble proportion of that in which BE is to EF to any sphere greater theÌ„ the sphere DEF And it is also proued that it is not to any lesse Wherefore the sphere ABC is to the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF which was required to be demonstrated A Corrollary added by Flussas Hereby it is manifest that spheres are the one to the other as like Polihedrons and in like sort described in them are namely eche are in triple proportion of that in which the diameters A Corollary added by Mâ— Dee It is then euident how to geue two right lines hauing that proportion betwene them which any two spheres geuen haue the one to the other For if to their diameters as to the first and second lines of fower in continuall proportion you adioyne a third and a fourth line in continuâ—ll proportion as I haue taught before The first and fourth lines shall aunswere the Pâ—obleme How generall this rule is in any two like solides with their correspondent or Omologall lines I neede not with more wordes declare ¶ Certaine Theoremes and Problemes whose vse is manifolde in Spheres Cones Cylinders and other solides added by Ioh. Dee A Theoreme 1. The whole superficies of any Sphere is quadrupla to the greatest circle in the same sphere contayned It is needeles to bring Archimedes demonstration hereof into this place seing his boke of the Sphere and Cylinder with other his woâ—kes are euery where to be had and the demoÌ„stration therof easie A Theoreme 2. Euery sphere is quadruplâ— to that Cone whose base is the greatest circle height the semidiameter of the same sphere This is the 32. Proposition of Archimedes fiâ—st booke of the Sphere and Cylinder A Probleme 1. A Sphere being geuen to make an vpright Cone equall to the same or in any other proportioâ— geuen betwene two right lines And as concerning the other part of
third proposition we will vse the same suppositions and constructions there specified so farre as they shall serue our purpose Beginning therefore at the conclusion we must infer the part of the proposition before graunted It was concluded that the square of the line DB is quintuple to the square of the line DC his owne segment Therefore DN the square of DB is quintuple to GF the square of DC But the squaâ—e of AC the double of DC which is RS is quadruple to GF by the second Corollary of the 20. of the sixth and therefore RS with GF are quintuple to GF and so it is euident that the square DN is equall to the square RS together with the square GF Wherefore from those two equalles taking the square GF common to them both remayneth the square RS equall to the Gnomon XOP But to the Gnomon XOP the parallelogramme CE is equall Wherefore the square of the line AC which is RS is equâ—ll to the parallelograÌ„me Câ— Which parallelogamme is coÌ„tained vnder BE equall to AB and CB the part remayning of the first line gâ—uen which was DB. And the line AB is made of the double of the segment DC and of CBâ— the other part of the line DB first gouen Wherefore the double of the segment DC with CB the part remayning which altogether is the whole line AB is to AC the double of the segment DC as that same AC is to CB by the second part of the 16. of the sixth Therfore by the 3. definitioÌ„ of the sixth booke the whole line AB is deuided by an extreme and meane proportion AC the double of the segmeÌ„t DC being middell proportionall is the greater part therof Wheâ—efore if a right line be quintuple in power c. as in the proposition which was to be demonstrated Or thus it may be demonstrated Forasmuch as the square DN is quinâ—uple to the square GF I meane the square of DB the line geueÌ„ to the square oâ— DC the segmeÌ„t And the same square DN is equall to the parallelograÌ„me vnder AB CB with the square made of the line DC by the sixth of the second for vnto the line AC equally deuided the line CB is as it were adioyned Wherefore the parallelogramme vnder AB CB together with the square of DC which is GF is quintuple to the square GF made oâ— thâ— line DC Taking then that square GF â—rom the parallelogramme vnder AB CB that parallelogramme vnder AB CB remayning alone is but quadruple to the sayd square of the line DC But by the 4. of the second or the second Corollary of the 20. of the sixth RS â—he square of the line AC is quadrupla to the same square GFâ— Wherfore by the 7. of the fifth the square of the line AC is equall to the parallelogramme vnder AB CB and so by the second part of the 16. of the sixth AB AC and CB are three lines in continuall proportion And seing AB is greater theÌ„ AC the same AC the double of the line DC shall be greater then the part BC remayning Wherfore by the 3. definition of the sixth AB composed or made of the double of DC and the other part of DB remaining is deuided by an extreme and middel proportion and also his greater segment is AC the double of the segment DC Wherfore If a right line be quintuple in power c. as in the propositionâ— which was to be demonstrated A Theoreme 2. If a right line deuided by an extreme and meane proportion be geuen and to the great segment â—herof he directly adioyned a line equal to the whole line geuen that adioyned line and the said greater segment do make a line diuided by extreme and meane proportion whose greater segment is the line â—dioyned Suppose the line geuen deuided by extreame and meane proportion to be AB deuided in the point C and his greater segment let be AC vnto AC directly adioyne a line equall to AB let that be AD I say that AD together with AC that is DC is a deuided by extreme and middel proportion whose greater segment is AD the line adioyned Deuide AD equally in the point E. Now forasmuch as AE is the halfe of AD by construction it is also the halfe of AB equall to AD by construction Wherfore by the 1. of the thirtenth the square of the line composed of AC and AE which â—ne is EC is quintuple to the square of the line AE Wherefore the double of AE and the line AC composed as in one right line is a line deuided by extreme and meane proportion by the conuerse of this third by me demonstrated and the double of AE is the greater segment But DC is the line composed of the double of AE the line AC and with all AD is the double of AE Wherfore DC is a line deuided by extreme and meane proportion and AD iâ— hiâ— greater segment If a right line therefore deuided by extreme and meane proportion be geuen and to the greater segment thereof be directly adioyned a line equall to the whole line geuen that adioyned line and the sayd greater segment do make a line diuided by extreame and meane proportion whose greater segment is the line adioyned Which was required to be demonstrated Two other briefe demonstrations of the same Forasmuch as AD is to AC as AB is to AC because AD is equall to AB by construction but as AB is to AC so is AC to CB by supposition Therefore by the 11. of the fifth as AC is to CB so is AD to AC Wherefore as AC and CB which is AB is to CB so is AD and AC which is DC to AC Therefore euersedly as AB is to AC so is DC to AD. And it is proued AD to be to AC as AC is to CB. Wherefore as AB is to AC and AC to CB so is DC to AD and AD to AC But AB AC and CB are in continuall proportion by supposition Wherfore DC AD and AC are in continuall proportion Wherefore by the 3. definition of the sixth booke DC is deuided by extreme and middell proportion and his greatest segment is AD. Which was to be demonstrated Note from the marke how this hath two demonstrations One I haue set in the margent by ¶ A Corollary 1. Vpon Euclides third proposition demonstrated it is made euident that of a line deuided by extreame and meane proportion if you produce the lesse segment equally to the length of the greater the line therby adioyned together with the sayd lesse segment make a new line deuided by extreame and middle proportion Whose lesse segment is the line adioyned For if AB be deuided by extreme and middell proportion in the point C AC being the greater segment and CB be produced from the poynt B making a line with CB equall to AC which let be CQ and the
two lines HIF and TIO cutting the one the other are in one and the selfe same ' plaine by the 2. of the eleuenth And therefore the poyntes H T F O are in one the selfe same plaine Wherforeâ— the rectangle figure HOFT beâ—ng quadrilater and equilater and in one and the selfe same playne is a square by the diâ—â—inition of a square And by the same reason may the rest of the bases of the solide be proued to be squares equall and plaine or superficial Now then the solide is comprehended of 6. equal squares which are contained of 12. equal sides which squares make 8. solide angles of which foure are in the ceâ—tres of the bases oâ— the pyramis and the other 4. are in the midle sections of the foure perdendiculars Wherfore the solide HOFTPGRN is a cube by the 21. diffinition of the eleuenth and is inscribed in the pyramis by the first definition of this boke Wherfore in a trilater equilater pyramis geuen is inscribed a cube ¶ A Corrollary The line which cutteth into two equall partes the opposite sides of the Pyramis is triple to the side of the cube inscribed in the pyramis and passeth by the centre of the cube For the line SEV whose third part the line SI is cutteth the opposite sides CD and AB into two equll partes but the line EI which is drawne from the centre of the cube to the base is proued to be a third part of the line ES wherefore the side of the cube which is double to the line EI shall be a third part of the whole line VS which is as hath bene proued double to the line ES. The 19. Probleme The 19. Proposition In a trilater equilater Pyramis geuen to inscribe an Icosahedron SVppose that the pyramis is geuen 〈◊〉 ABâ—Dâ— euery one of whose sâ—des 〈◊〉 be diuidâ—d into two equall partes in the poyâ—â—â—â—â— M K L P N. And iâ— euery one of the bâ—ses of that pyramis descride the trianglâ—â— Lâ—â— PMN NKL and 〈…〉 which triangles shall be equilater by the 4. of the firâ—t â—or the sides subâ—end equall angles of the pyramis contayned vnder the halues of the sides of the same pyramisâ— wherfore the sides of the said triangles are equall Let those sides be â—iuided by an extreame and meane proporâ—â—on by the 30. of the sixth in the poyntes C E Q R S T H I O V Y X. Now then those sides are cutte into the selfe same proportions by the 2. of the fourteÌ„th and therfore they make the liâ—e sectioÌ„s equall by the â— part of the ninth of the fiueth Now I say that the foresayd poynâ—es doâ— recâ—aue the angles of the Icosahedron inscribed in the pyramis ABâ—D In the foresayd triangles let there agayne be made other triangles by coupling the sections and let those triangles be TRS IOH CEQ and VXY which shall be equilater for euery one of their sides doo subâ—â—â—d equall angles of equilater triangles and those sayd equall angles are contayned vnder equall sideâ— namely vnder the greater segmenâ— and the lesse â— and therefore the sides which subtend those angles are equall by the 4. of the first Now let vs proue that at eche of the foresayd poynts as for example at T is set the solide angle of an Icosahâ—dronâ— Forasmuch as the triangles TRS and TQO are equilater and equall the 4. right lines TR T S TQ and TO shall be equall And forasmuch as â—PNK is a square cutting the pyramis ABâ—D into two equall paâ—â—â—â— by the corollay of the second of this bookeâ— the line TH shall be in power duple to the line TN or NH by the 47. of the first For the lines TN or NH are equall for that by construction they are eche lesse segmentes and the line RT or T S is in power duple to the same line TN or NH by the corollary of the 16. of this booke for it subtendeth the angle of the triangle contayned vnder the two segmentes Wherfore the lines TH T S TR TQ and TO are equall and so also are the lines HS SR RQ QO and OH which subtend the angles at the poynt T equall For the line QR contayneth in power the two lines PQ and PR the lesse segmentes which two lines the line TH also contayned in power And the rest of the lines doo subtend angles of equilater triangles contayned vnder the greater segment and the lesse Wherefore the fiue triangles TRS TSH THO TOQ TQR are equilater and equall making the solide angle of an Icosahedron at the poynt T by the 16. of the thirtenth in the side PN of the triangle P NM And by the same reason in the other sides of the 4. triangles PNM NKL FMK LFP which are inscribed in the bases of the pyramis which sides are 12â— in nuÌ„ber shal be set 12. angles of the IcosahedroÌ„ coÌ„tained vnder 20. equal equilater triangles of which fowere are set in the 4. bases of the pyramis namely these fower triangles TRS HOI CEQ VXY 4. triangles are vnder 4. angles of the pyramis that is the fower triangles CIX YSH ERV TQO and vnder euery one of the sixe sides of the pyramis are set two triangles namely vnder the side of the triangles THS and THOâ— vnder the side DB the triangles RQE and RQT vnder the side DA the triangles COQ and COI vnder the side AB the triangles EXC and EXVâ— vnder the side BG the triangles SVR and SVY and vnder the side AG the triangles IYH and IYX. Wherefore the solide being contayned vnder 20. equilater and equall triangles shall be an Icosahedron by the 23. diffinition of the eleuenth and shall be inscribed in the pyramis ABâ—D by the first diâ—â—â—nition of this booke for all his angles doo at one time touch the bases of the pyramis Wherefore in a trilater equilater pyramis geuen we haue inscribed an Icosahedron ¶ The 20. Proposition The 20. Probleme In a trilater equilater Pyramis geuen to inscribe a dodecahedron SVppose that the pyramis geuen be ABGD â—che of whose sides let be cutte into two equall partes and draw the lines which couple the sections which being diuided by an extreame and meane proportion and right lines being drawne by the sections shall receaue 20. triangles making an Icosahedron as in the former proposition it was manifest Now then if we take the centres of those triangles we shall there finde the 20. angles of the dodecahedron inscribed in it by the 5. of this booke And forasmuch as 4. bases of the foresayd IcosahedroÌ„ are coÌ„centricall with the bases of the pyramis as it was proued in the 2. corollary of the 6. of this boke there shal be placed 4â— angles of the dodecahedroÌ„ namely the 4. angles E F H D in the 4. centres of the bases and of the other 16. angles vnder euery one of the 6. sides of the pyramis are subtended two namely vnder the side AD the angles CK vnder the side BD the angles LI vnder the
be wood Copper Tinne Lead Siluer c. being as I sayd of like nature condition and like waight throughout And you may by Say Balance haue prepared a great number of the smallest waightes which by those Balance can be discerned or tryed and so haue proceded to make you a perfect Pyle company Number of waightes to the waight of six eight or twelue pound waight most diligently tryed all And of euery one the Content knowen in your least waight that is wayable They that can not haue these waightes of precisenes may by Sand Vniforme and well dusted make them a number of waightes somewhat nere precisenes by halfing euer the Sand they shall at length come to a least common waight Therein I leaue the farder matter to their discretion whom nede shall pinche The Venetians consideration of waight may seme precise enough by eight descentes progressionall halfing from a grayne Your Cube Sphaere apt Balance and conuenient waightes being ready fall to worke â— First way your Cube Note the Number of the waight Way after that your Sphaere Note likewise the NuÌ„ber of the waight If you now find the waight of your Cube to be to the waight of the Sphaere as 21. is to 11 Then you see how the Mechanicien and Experimenter without Geometrie and Demonstration are as nerely in effect tought the proportion of the Cube to the Sphere as I haue demonstrated it in the end of the twelfth boke of Euclide Often try with the same Cube and Sphaere Then chaunge your Sphaere and Cube to an other matter or to an other bignes till you haue made a perfect vniuersall Experience of it Possible it is that you shall wynne to nerer termes in the proportion When you haue found this one certaine Drop of Naturall veritie procede on to Inferre and duely to make assay of matter depending As bycause it is well demonstrated that a Cylinder whose heith and Diameter of his base is aequall to the Diameter of the Sphaere is Sesquialter to the same Sphaere that is as 3. to 2 To the number of the waight of the Sphaere adde halfe so much as it is and so haue you the number of the waight of that Cylinder Which is also Comprehended of our former Cube So that the base of that Cylinder is a Circle described in the Square which is the base of our Cube But the Cube and the Cylinder being both of one heith haue their Bases in the same proportion in the which they are one to an other in their Massines or Soliditie But before we haue two numbers expressing their Massines Solidities and Quantities by waight wherfore we haue the proportion of the Square to the Circle inscribed in the same Square And so are we fallen into the knowledge sensible and Experimentall of Archimedes great Secret of him by great trauaile of minde sought and found Wherfore to any Circle giuen you can giue a Square aequall as I haue taught in my Annotation vpon the first proposition of the twelfth boke And likewise to any Square giuen you may giue a Circle aequall If you describe a Circle which shall be in that proportion to your Circle inscribed as the Square is to the same Circle This you may do by my Annotations vpon the second proposition of the twelfth boke of Euclide in my third Probleme there Your diligence may come to a proportion of the Square to the Circle inscribed nerer the truth then is the proportion of 14. to 11. And consider that you may begyn at the Circle and Square and so come to conclude of the Sphaere the Cube what their proportion is as now you came from the Sphâ—ere to the Circle For of Siluer or Gold or Latton Lamyns or plates thorough one hole draweÌ„ as the maner is if you make a Square figureâ— way it and then describing theron the Circle inscribed cut of file away precisely to the Circle the ouerplus of the Square you shall then waying your Circle see whether the waight of the Square be to your Circle as 14. to 11. As I haue Noted in the beginning of Euclides twelfth boke c. after this resort to my last proposition vpon the last of the twelfth And there helpe your selfe to the end And here Note this by the way That we may Square the Circle without hauing knowledge of the proportion of the Circumference to the Diameter as you haue here perceiued And otherwayes also I can demonstrate it So that many haue cumberd them selues superfluously by trauailing in that point first which was not of necessitie first and also very intricate And easily you may and that diuersly come to the knowledge of the Circumference the Circles Quantitie being first knowen Which thing I leaue to your consideration making hast to despatch an other Magistrall Probleme and to bring it nerer to your knowledge and readier dealing with then the world before this day had it for you that I can tell of And that is A Mechanicall Dubblyng of the Cube c. Which may thus be done Make of Copper plates or Tyn plates a foursquare vpright Pyramis or a Cone perfectly fashioned in the holow within Wherin let great diligence be vsed to approche as nere as may be to the Mathematicall perfection of those figures At their bases let them be all open euery where els most close and iust to From the vertex to the Circumference of the base of the Cone to the sides of the base of the Pyramis Let 4. straight lines be drawen in the inside of the Cone and Pyramis makyng at their fall on the perimeters of the bases equall angles on both sides them selues with the sayd perimeters These 4. lines in the Pyramis and as many in the Cone diuide one in 12. aequall partes and an other in 24. an other in 60 and an other in 100. reckenyng vp from the vertex Or vse other numbers of diuision as experience shall reach youâ— Then set your Cone or Pyramis with the vertex downward perpendicularly in respect of the Base Though it be otherwayes it hindreth nothyng So let theÌ„ most stedily be stayed Now if there be a Cube which you wold haue Dubbled Make you a prety Cube of Copper Siluer Lead Tynne Wood Stone or Bone. Or els make a hollow Cube or Cubiâ— coffen of Copper Siluer Tynne or Wood c. These you may so proportioÌ„ in respect of your Pyramis or Cone that the Pyramis or Cone will be hable to conteine the waight of them in waâ—â— 3. or 4. times at the least what stuff so euer they be made ofâ— Let not your Solid angle at the vertex be to sharpe but that the water may come with ease to the very vertex of your hollow Cone or Pyramis Put one of your Solid Cubes in a Balance apt take the waight therof exactly in water Powre that water without losse into the hollow Pyramis or Cone quietly
ACD and ADC shall also be equall by the fifth propositionâ— likewise for asmuch as the lines BC and BD are equal the angles vnder the base namely the angles FDC and ECD are equall by the seconde part of the same proposition And for as much as the angle ECD is lesse then the angle ACD It followeth that the angle FDC is lesse theÌ„ the angle ADC which is impossible for that the angle AD C is a part of the angle FD C. And the same inconuenience will follow if the poynt D fall within the triangle ABC The fift Theoreme The 8. Proposition If two triangles haue two sides of th' one equall to two sides of the other eche to his correspondent side haue also the base of the one equall to the base of the other they shall haue also the angle contained vnder the equall right lines of the one equall to the angle contayned vnder the equall right lynes of the other SVppose that there be two triangles ABC and DEF let these two sides of the one AB and AC be equall to these two sides of the other DE and DF ech to his correspondent side that is AB to D E and AC to DF let the base of the one namely BC be equal to the base of the other namely to EF. Then I say that the angle BAC is equall to the angle EDF For the triangle ABC exactly agreing with the triangle DE F and the point B being put vpon the point E and the right line BC vpon the right line EF the point C shall exactly agree with the point F for the line BC is equall to the line EF And BC exactly agreeing with EF the lines also BA and AC shall exactly agree with the lines ED DF. For if the base BC do exactly agree with the base FE but the sides BA AC doo not exactly agree with the sides ED DF but differ as FG GF do theÌ„ from the endes of one lyne shal be drawn two right lines to a poynt from the self same endes on the same side shal be drawn two other lines equal to the two first lines the one to the other and vnto an other poynt but that is impossible by the seuenth propositioÌ„ VVherfore the base BC exactly agreeing with the base EF the sides also BA and AC do exactly agre with the sides ED and DF. VVherfore also the angle BAC shall exactly agre with the angle EDF and therfore shall also be equal to it If theâ—fore two triangles haue two sides of the one equall to two sides of the other ech to his correspondent side and haue also the base of the one equall to the base of the other they shall haue also the angle contayned vnder the equall right lines of the one equall to the angle contayned vnder the equall right lines of the other which was required to be proued This Theoreme is the conuerse of the fourth but it is not the chiefest and principall kind oâ— conuersion For it turneth not the whole supposition into the conclusion and the whole conclusion into the supposition For the fourth propositioÌ„ whose conuerse this is is a coÌ„pound â—heoreme hauing two things geueÌ„ or sâ—pposed which are these the one that two sides of the one triaÌ„gle be equal to two sides of the other triaÌ„gle th' other that the angle coÌ„tained of the two sides of th' one is equal to the angle contained of the two sides of th' one but hath amongest other one thing required whiche is that the base of the one is equal to the base of the other Now in this 8. propositioÌ„ being the conuerse therofâ— that the base of the one is equal to the base of th' other is the supposition or the thing geueÌ„ which in the former propositioÌ„ was the conclusioÌ„ And this that two sides of the one are equall to two sides of the other is in this proposition also a supposition like as it was in the former proposition so that it is a thing geuen in either proposition The conclusion of this proposition is that the angle enclosed of the two equall sides of the one triangle is equall to the angle enclosed of the two equall sides of the other triangle which in the former proposition was one of the things geuen Philo and his scholas demonstrate this proposition without the helpe of the former proposition in this maner First let it fall directlye And forasmuche as the line DE is equall to the line E G and DFG is one righte line therfore DEG is an Isosceles triaÌ„gle and so by the fifth proposition the angle at the point D is equal to the angle at the poynt G which was required to be proued The 4. Probleme The 9. Proposition To deuide a rectiline angle geuen into two equall partes SVppose that the rectiline angle geuen be BAC It is required to deuide the angle BAC into two equal partes In the line AB take a point at all aduentures let the same be D. And by the third proposition from the lyne AC cutte of the line AE equall to AD. And by the first peticion draw a right line from the point D to the point E. And by the first proposition vpon the line DE describe an equilater triangle and let the same be DFE and by the first peticion drawe a right line from the poynte A to the point F. Then I say that the angle BAC is by the line AF deuided into two equal partes For forasmuch as AD is equall to AE and AF is coÌ„mon to them both therfore these two DA and AF are equall to these two EA and AF the one to the other But by the first proposition the base DF is equall to the base EF wherfore by the 8. proposition the angle DAF is equal to the angle FAE VVherfore the rectiline angle geuen namely B AC is deuided into two equal partes by the right line AFâ— VVhich was required to be done In this proposition is not taught to deuide a right lined angle into mo partes then two albeit to deuide an angle so it be a right angle into three partes it is not hard And it is taught of Viâ—ellio in his first boke of Perspectiue the 28. Proposition â—or to deuide an acute angle into three equal partes is as saith Proclus impossible vnles it be by the helpe of other lines which are of a mixt nature Which thing Nicomedes did by such lines which are called Concoideâ— linea who first serched out the inuention nature properties of such lines And others did it by other meanes as by the helpe of quadrant lines inuented by Hippias Nicomedes Others by Helices or Spiral lines inuented of Archimedes But these are things of much difficulty and hardnes and not here to be intreated of Here against this proposition may of the aduersary be brought an instance For he may cauill that the
sydes VVherfore the angle BHA is equall to the angle EFD But the angle EFD is equall to the angle BCA VVherefore the angle BHA is equal to the angle BCA VVherefore the outward angle of the triangle AHC namely the angle BHA is equall to the inward and opposite angle namely to the angle HCA which by the 16 proposition is impossible VVherfore the syde EF is not vnequall to the syde BC wherefore it is equall And the syde AB is equall to the syde DE wherefore these two sydes AB and BC are equall to these two sydes DE and EF the one to the other and they contayne equall angles VVherfore by the 4. proposition the base AC is equall to the base DF and the triangle ABC is equall to the triangle DEF and the angle remayning namely the angle BAC is equall to the angle remayning that is to the angle EDF If therefore two triangles haue two angles of the one equall to two angles of the other ech to his correspondent angle and haue also one syde of the one equall to oâ—e syde of the other either that syde which lieth betwene the equall angles or that which is subtended vnder one of the equall angles the other sydes also of the one shal be equall to the other sydes of the other eche to his correspondent side and the other angle of the one shal be equall to the other angle of the other which was required to be proued VVhereas in this proposition it is sayde that triangles are equall which hauing two angles of the one equall to two angles of the other the one to the other haue also one side of the one equall to one side of the other either that side which lieth betwene the equall angles or that side which subtendeth one of the equall angles this is to be noted that without that caution touching the equall side the proposition shall not alwayes be true As for example The reason wherof is for that the equal side in one triangle subteÌ„deth one of the equall angles and in the other lieth betwene the equal angles So that you see that it is of necessitie that the equall side do in both triangles either subtend one of the equall angles or lie betwene the equall angles Of this proposition was Thales Milesius the inuentor as witnesseth Eudemus in his booke of Geometricall enarrations The 18. Theoreme The 27. Proposition If a right line falling vpon two right lines do make the alternate angles equall the one to the other those two right lines are parallels the one to the other SVppose that the right line EF falling vppon these two right lines AB and CD do make the alternate angles namely the angles AEF EFD equall the one to the other Then I say that AB is a parallel line to CD For if not then these lines produced shall mete together either on the side of B and D or on the syde of A C. Let them be produced therfore and let them mete if it be possible on the syde of B and D in the point G. VVherfore in the triangle GEF the outward angle AEF is equal to the inward and opposite angle EFG which by the 16. proposition is impossible VVherfore the lines AB and CD beyng produced on the side of B and D shall not meete In like sorte also may it be proued that they shall not mete on the syde of A and C. But lines whiche being produced on no syde meete together are parrallell lines by the last definition wherfore AB is a parrallel line to CD If therfore a right line falling vpon two right lines do make the alternate angles equall the one to the other those two right lines are parrallels the one to the other which was required to be demonstrated This worde alternate is of Euclide in diuers places diuersly taken somtimes for a kind of situation in place and somtime for an order in proportion in which signification he vseth it in the v. booke and in his bokes of numbers And in the first signification he vseth it here in this place and generally in all hys other bokes hâ—uing to do with lines figures And those two angles he calleth alternate which beyng both contayned within two parallel or equidistant lynes are neither angles in order nor are on the one and selfe same side but are seperated the one from the other by the line which falleth on the two lines the one angle beyng aboue and the other beneath The 19. Theoreme The 28. Proposition If a right line falling vpon two right lines make the outward angle equall to the inward and opposite angle on one and the same syde or the inwarde angles on one and the same syde equall to two right angles those two right lines shall be parallels the one to the other SVppose that the right line EF fallyng vppon these two right lines AB and CD do make the outward angle EGB equall to the inward and opposite angle GHD or do make the inward angles on one and the same side that is the angles BGH and GHD equall to two right angles Then I say that the lyne AB is a parallel line to the lyne CD For forasmuch as the angle EGB is by supposition equall to the angle GHD and the angle EGB is by the 15. proposition equall to the angle AGH therfore the angle AGH is equall to the angle GHD and they are alternate angles VVherfore by the 27. proposition AB is a parallel line to CD Agayne forasmuch as the angles BGH and GHD are by supposition equall to two right angles by the 13. proposition the angles AGH and BGH are also equall to two right angles wherefore the angles AGH and BGH are equall to the angles BGH and GHD take away the angle BGH which is common to them both VVherfore the angle remainyng namely AGH is equall to the angle remayning namely to GHD And they are alternate angles VVherfore by the former proposition AB is a parallell line to CD If therfore a right line fallyng vpon two right lines do make the outward angle equall to the inward and opposite angle on one and the same side or the inwarde angles on one and the same side equall to two right angles those two right lines shall be parallels the one to the other which was required to be proued Ptolomeus demonstrateth the second part of this proposition namely that the two inward angles on one and the same side being equall the right lines are parellels after this manner The 20. Theoreme The 29. Proposition A right line line falling vppon two parallel right lines maketh the alternate angles equall the one to the other and also the outwarde angle equall to the inwarde and opposite angle on one and the same side and moreouer the inwarde angles on one and the same side equall to two right angles And the angle AGH is by the 15. proposition equall to
diameter is double to that square whose diameter it is The 34. Theoreme The 48. Proposition If the square which is made of one of the sides of a triangle be equall to the squares which are made of the two other sides of the same triangle the angle comprehended vnder those two other sides is a right angle SVppose that ABC be a triangle and let the square which is made of one of the sides there namely of the side BC be equall to the squares which are made of the sides BA and AC Then I say that the angle BAC is a right angle Rayse vp by the 11. propositioÌ„ from the point A vnto the right line AC a perpendicular line AD. And by the thirde proposition vnto the line AB put an equall line AD. And by the first peticion draw a right line from the point D to the poinâ— C. And forasmuch as the line DA is equall to the line AB the square which is made of the line DA is equall to the square whiche is made of the line AB Put the square of the line AC common to them both VVherefore the squares of the lines DA and AC are equal to the squares of the lines BA and AC But by the proposition going before the square of the line DC is equal to the squares of the lines AD and AC For the angle DAC is a right angle and the square of BC is by supposition equall to the squares of AB and AC VVherefore the square of DC is equall to the square of BC wherefore the side DC is equall to the side BC. And forasmuch as AB is equall to AD â—nd AC is common to them both therefore these two sides DA and AC are equall to these two sides BA and AC the one to the other and the base DC is equall to the base BCâ— wherfore by the 8. proposition the angle DAC is equall to the angle BAC But the angle DAC is a right angle wherefore also the angle BAC is a right angle If therefore the square which is made of one of the sides of a triangle be equall to the squares which are made of the two other sides of the same triangle the angle comprehended vnder those two other sides is a right angle which was required to be proued This proposition is the conuerse of the former and is of Pelitarius demonstrated by an argument leading to an impossibilitie after this maner The ende of the first booke of Euclides Elementes ¶ The second booke of Euclides Elementes IN this second booke Euclide sheweth what is a GnomoÌ„ and a right angled parallelogramme Also in this booke are set forth the powers of lines deuided euenly and vneuenly and of lines added one to an other The power of a line is the square of the same line that is a square euery side of which is equall to the line So that here are set forth the qualities and proprieties of the squares and right lined figures which are made of lines of their parts The Arithmetician also our of this booke gathereth many compendious rules of reckoning and many rules also of Algebra with the equatioÌ„s therein vsed The groundes also of those rules are for the most part by this second booke demonstrated This booke moreouer contayneth two wonderfull propositions one of an obtuse angled triangle and the other of an acute which with the ayde of the 47. proposition of the first booke of Euclide which is of a rectangle triangle of how great force and profite they are in matters of astronomy they knowe which haue trauayled in that arte VVherefore if this booke had none other profite be side onely for these 2. propositions sake it were diligently to be embraced and studied The definitions 1. Euery rectangled parallelogramme is sayde to be contayned vnder two right lines comprehending a right angle A parallelogramme is a figure of fower sides whose two opposite or contrary sides are equall the one to the other There are of parallelogrammes fower kyndes a square a figure of one side longer a Rombus or diamond and a Romboides or diamond like figure as before was sayde in the 33. definition of the first booke Of these fower sortes the square and the figure of one side longer are onely right angled Parallelogrammes for that all their angles are right angles And either of them is contayned according to this definition vnder two right lynes whiâ—h concurre together and cause the right angle and containe the same Of which two lines the one is the length of the figure the other the breadth The parallelogramme is imagined to be made by the draught or motion of one of the lines into the length of the other As if two numbers shoulde be multiplied the one into the other As the figure ABCD is a parallelograme and is sayde to be contayned vnder the two right lines AB and AC which contayne the right angle BAC or vnder the two right lines AC and CD for they likewise contayne the right angle ACD of which 2. lines the one namely AB is the length and the other namely AC is the breadth And if we imagine the line AC to be drawen or moued directly according to the leÌ„gth of the line AB or contrary wise the line AB to be moued directly according to the length of the line AC you shall produce the whole rectangle parallelogramme ABCD which is sayde to be contayned of them euen as one number multiplied by an other produceth a plaine and righte angled superficiall number as ye see in the figure here set where the number of sixe or sixe vnities is multiplied by the number of fiue or by fiue vnities of which multiplication are produced 30. which number being set downe and described by his vnities representeth a playne and a right angled number VVherefore euen as equall numbers multipled by equal numbers produce numbers equall the one to the other so rectangle parallelogrames which are comprehended vnder equal lines are equal the one to the other 2. In euery parallelogramme one of those parallelogrammes which soeuer it be which are about the diameter together with the two supplementes is called a Gnomon Those perticuler parallelogrames are sayde to be about the diameter of the parallelograme which haue the same diameter which the whole parallelograme hath And supplementes are such which are without the diameter of the whole parallelograme As of the parallelograme ABCD the partial or perticuler parallelogrames GKCF and EBKH are parallelogrames about the diameter for that ech of them hath for his diameter a part of the diameter of the whole parallelogramme As CK and KB the perticuler diameters are partes of the line CB which is the diameter of the whole parallelogramme And the two parallelogrammes AEGK and KHFD are supplementes because they are wythout the diameter of the whole parallelogramme namely CB. Now any one of those partiall parallelogrammes
nombers of the nombers AD and DB are double to the square nombers of AC and CD For forasmuch as the nomber AD is deuided into the nombers AB and BD therefore the square nombers of the nombers AD and DB are equall to the superficiall nomber produced of the multiplication of the nombers AD and DB the on into the other twise together with the square of the nomber AB by the 7 propositiō But the square of the nomber AB is equal to fower squares of either of the nombers AC or CB for AC is equall to the nomber CB wherfore also the squares of the nombers AD and DB are equall to the superficiall nomber produced of the multiplication of the nombers AD and DB the one into the other twise and to fower squares of the nomber BC or CA. And forasmuch as the superficiall nomber produced of the multiplication of the nombers AD and DB the one into the other together with the square of the nomber CB is equal to square of the nomber CD by the 6 propositiō therfore the nomber produced of the multiplication of the nomber◠AD and DB the one into the other twise together with two squares of the nomber CB is equall to two squares of the nomber CD Wherefore the squares of the nombers AD and DB are equall to two squares of the nomber CD and to two squares of the nomber AC Wherefore they are double to the squares of the numbers AC and CD And the square of the nomber AD is the square of the whole and of the nomber added And the square of DB is the square of the nomber added the square also of the nomber CD is the square of the nomber composed of the halfe and of the nomber added If therefore an euen nomber be deuided c. Which was required to be proued The 1. Probleme The 11. Proposition To deuide a right line geuen in such sort that the rectangle figure comprehended vnder the whole and one of the partes shall be equall vnto the square made of the other part SVppose that the right line geuen be AB Now it is required to deuide the line AB in such sort that the rectangle figure contayned vnder the whole and one of the partes shall be equall vnto the square which is made of the other part Describe by the 46. of the first vpon AB a square ABCD. And by the 10. of the first deuide the line AC into two equall partes in the point E and draw a line from B to E. And by the second petition extend CA vnto the point F. And by the 3. of the first put the line EF equall vnto the line BE. And by the 46. of the first vpon the line AF describe a square FGAH And by the 2. petition extend GH vnto the point K. Then I say that the line AB is deuided in the point H in such sort that the rectangle figure which is comprehēded vnder AB and BH is equall to the square which is made of AH For forasmuch as the right line AC is deuided into two equall partes in the poynt E and vnto it is added an other right line AF. Therefore by the 6. of the second the rectangle figure contayned vnder CF and FA together with the square which is made of AE is equall to the square which is made of EF. But EF is equall vnto EB VVherefore the rectangle figure contayned vnder CF and FA together with the square which is made of EA is equall to the square which is made of EB But by 47. of the first vnto the square which is made of EB are equall the squares which are made of BA and AE For the angle at the poynt A is a right angle VVherefore that which is contayned vnder CF and FA together with the square which is made of AE is equall to the squares which are made of BA and AE Take away the square which is made of AE which is common to them both VVherfore the rectangle figure remayning contayned vnder CF and FA is equall vnto the square which is made of AB And that which is contained vnder the lines CF and FA is the figure FK For the line FA is equall vnto the line FG. And the square which is made of AB is the figure AD. VVherefore the figure FK is equall vnto the figure AD. Take away the figure AK which is common to them both VVherefore the residue namely the figure FH is equall vnto the residue namely vnto the figure HD But the figure HD is that which is contayned vnder the lines AB and BH for AB is equall vnto BD. And the figure FH is the square which is made of AH VVherfore the rectangle figure comprehended vnder the lines AB and BH is equall to the square which is made of the line HA. VVherefore the right line geuen AB is deuided in the point H in such sort that the rectangle figure contayned vnder AB and BH is equall to the square which is made of AH which was required to be done Thys proposition hath many singular vses Vpon it dependeth the demonstration of that worthy Probleme the 10. Proposition of the 4. booke which teacheth to describe an Isosceles triangle in which eyther of the angles at the base shall be double to the angle at the toppe Many and diuers vses of a line so deuided shall you finde in the 13. booke of Euclide Thys is to be noted that thys Proposition can not as the former Propositions of thys second booke be reduced vnto numbers For the line EB hath vnto the line AE no proportion that can be named and therefore it can not be expressed by numbers For forasmuch as the square of EB is equall to the two squares of AB and AE by the 47. of the first and AE is the halfe of AB therefore the line BE is irrationall For euen as two equall square numbers ioyned together can not make a square number so also two square numbers of which the one is the square of the halfe roote of the other can not make a square number As by an example Take the square of 8. which is 64. which doubled that is 128. maketh not a square number So take the halfe of 8. which is 4. And the squares of 8. and 4. which are 64. and 16. added together likewyse make not a square number For they make 80. who hath no roote square Which thyng must of necessitie be if thys Probleme should haue place in numbers But in Irrational numbers it is true and may by thys example be declared Let 8. be so deuided that that which is produced of the whole into one of his partes shall be equall to the square number produced of the other part Multiply 8. into him selfe and there shall be produced 64. that is the square ABCD. Deuide 8. into two equall partes that is into 4 and 4. as the line
AE or EC And multiply 4. into hym selfe and there is produced 16 which adde vnto 64 and there shall be produced 80 whose roote is √ â— 80 which is the line EB or the line EF by the 47. of the first And forasmuch as the line EF is √ â— 80. the lyne EA is 4. therfore the lyne AF is √ â— 80 4. And so much shall the line AH be And the line BH shall be 8 √ â— 80 4 that is 12 √ â— 80. Now theÌ„ 12 √ â— 80 multiplied into 8 shal be as much as √ â— 80 4. multiplied into it selfe For of either of them is produced 96 √ 5120. The 11. Theoreme The 12. Proposition In obtuseangle triangles the square which is made of the side subtending the obtuse angle is greater then the squares which are made of the sides which comprehend the obtuse angle by the rectangle figure which is comprehended twise vnder one of those sides which are about the obtuse angle vpon which being produced falleth a perpendicular line and that which is outwardly taken betwene the perpendicular line and the obtuse angle SVppose that ABC be an obtuseangle triangle hauing the angle BAC obtuse and from the point B by the 12. of the first draw a perpendicular line vnto CA produced and let the same be BD. Then I say that the square which is made of the side BC is greater then the squares which are made of the sides BA and AC by the rectangle figure comprehended vnder the lines CA and AD twise For forasmuch as the right line CD is by chaunce deuided in the poynt A therefore by the 4. of the second the square which is made of CD is equall to the squares which are made of CA and AD and vnto the rectangle figure contayned vnder CA and AD twise Put the square which is made of DB common vnto them both VVherefore the squares which are made of CD and DB are equall to the squares which are made of the lines CA AD and DB and vnto the rectangle figure contayned vnder the lines CA and AD twise But by the 47. of the first the square which is made of CB is equall to the squares which are made of the lines CD and DB. For the angle at the point D is a right angle And vnto the squares which are made of AD and DB by the selfe same is equall the square which is mâ—de of AB VVherfore the square which is made of CB is equall to the squares which are made of CA and AB and vnto the rectangle figure contayned vnder the lines CA and AD twise VVherfore the square which is made of CB is greater then the squares which are made of CA and AB by the rectangle figure contayned vnder the lines CA and AD twise In obtuseangle triangles therefore the square which is made of the side subtending the obtuse angle is greater then the squares which are made of the sides vvhich comprehend the obtuse angle by the rectangle figure vvhich is comprehended twise vnder one of those sides which are about the obtuse angle vpon which being produced falleth a perpendiculer lyne and that which is outwardly taken betwene the perpendiculer lyne and the obtuse angle which was required to be demonstrated Of what force thys Propositionâ— and the Proposition following touching the measuring of the obtuseangle triangle and the acuteangle triangle with the ayde of the 47. Proposition of the first booke touching the rightangle triangle he shall well perceaue which shall at any time neede the arte of triangles in which by thre thinges knowen is euer searched out three other thinges vnknowen by helpe of the table of arkes and cordes The 12. Theoreme The 13. Proposition In acuteangle triangles the square which is made of the side that subtendeth the acute angle is lesse then the squares which are made of the sides which comprehend the acute angle by the rectangle figure which is coÌ„prehended twise vnder one of those sides which are about the acuteangle vpoÌ„ which falleth a perpendiculer lyne and that which is inwardly taken betwene the perpendiculer lyne and the acute angle SVppose that ABC be an acuteangle triangle hauyng the angle at the point B acute by the 12. of the first from the point A draw vnto the lyne BC a perpendiculer lyne AD. Then I say that the square which is made of the lyne AC is lesse then the squares which are made of the lyne CB and BA by the rectangle figure conteyned vnder the lines CB and BD twise For forasmuch as the right lyne BC is by chaunce deuided in the point D therfore by the 7. of the second the squares which are made of the lines CB and BD are equall to the rectangle figure contained vnder the lines CB and DB twise and vnto the square whiche is made of line CD Put the square which is made of the line DA common vnto them both VVherfore the squares which are made of the lines CB BD and DA are equall vnto the rectangle figure contayned vnder the lines CB and BD twise and vnto the squares which are made of AD and DC But to the squares whiche are made of the lines BD and DA is equal the square which is made of the line AB for th'angle at the point D is a right angle And vnto the squares whiche are made of the lines AD and DC is equall the square whiche is made of the line AC by the 47. of the first wherfore the squares which are made of the lines CB and BA are equal to the square which is made of the line AC and to that which is contained vnder the lines CB and BD twise VVherfore the square which is made of the line AC beyng taken alone is lesse then the squares which are made of the lines CB and BA by the rectangle figure which is contained vnder the lines CB and BD twise In rectangle triangles therfore the square which is made of the side that subtendeth the acute angle is lesse then the squares which are made of the sides which comprehend the acute angle by the rectangle figure which is comprehended twise vnder one of those sides which are about the acute angle vpon which falleth a perpendicular line and that which is inwardly taken betwene the perpendicular line and the acute angle which was required to be proued ¶ A Corollary added by Orontius Hereby is easily gathered that such a perpendicular line in rectangle triangles falleth of necessitie vpon the side of the triangle that is neyther within the triangle nor without But in obtuseangle triangles it falleth without and in acuteangle triangles within For the perpendicular line in obtuseangle triangles and acuteangle triangles can not exactly agree with the side of the triangle for then an obtuse an acuteangle should be equal to a right angle contrary to the eleuenth and twelfth definitions of the first booke Likewise in obtuseangle
triangles it can not fall within nor in acuteangle triangles without for then the outward angle of a triangle should be lesse then the inward and opposite angle which is contrary to the 16. of the first And this is to be noted that although properly an acuteangle triangle by the definition therof geueÌ„ in the first booke be that triangle whose angles be all acute yet forasmuch as there is no triangle but that it hath an acute angle this proposition is to be vnderstanded is true generally in all kindes of triangles whatsoeuer and may be declared by them as you may easily proue The 2. Probleme The 14. Proposition Vnto a rectiline figure geuen to make a square equall SVppose that the rectiline figure geuen be A. It is required to make a square equall vnto the rectiline figure A. Make by the 45. of the first vnto the rectiline figure A an equall rectangle parallelogramme BCDE Now if the line BE be equall vnto the line ED then is the thyng done whiche was required for vnto the rectiline figure A is made an equal square BD. But if not one of these lines BE is ED the greater Let BE be the greater and let it be produced vnto the point F. And by the 3. of the first put vnto ED an equall line EF. And by the 10. of the first deuide the line BF into two equall partes in the point G. And making the centre the point G and the space GB or GF describe a semicircle BHF And by the 2. peticion extend the line DE vnto the point H. And by the 1. peticion draw a line from G to H. And forasmuch as the right line FB is deuided into two equall partes in the point G and into two vnequal partes in the point E therfore by the 5. of the second the rectangle figure comprehended vnder the lines BE and EF together with the square which is made of the line EG is equall to the square which is made of the line GF But the line GF is equall vnto the line GH VVherfore the rectangle figure comprehended vnder the lines BE and EF together with the square which is made of the line GE is equall to square which is made of the line GH But vnto the square which is made of the line GH are equall the squares whiche are made of the lines HE and GE by the 47. of the first VVherfore that which is contained vnder the lines BE and EF together with the square which is made of GE is equall to the squares which are made of HE and GE. Take away the square of the line EG common to them both VVherfore the rectangle figure contained vnder the lines BE EF is equall to the square which is made of the line EH But that whiche is contained vnder the lines BE and EF is the parallelogramme BD for the line EF is equall vnto the line ED. VVherfore the parallelogramme BD is equall to the square whiche is made of the line HE. But the parallelograme BD is equall vnto the rectiline figure A. VVherfore the rectiline figure A is equall to the square which is made of the line HE. VVherfore vnto the rectiline figure geuen A is made an equall square described of the line EH which was required to be done ¶ The ende of the second Booke of Euclides Elementes ¶ The third booke of Euclides Elementes THis third booke of Euclide entreateth of the most perfect figure which is a circle Wherefore it is much more to be estemed then the two bookes goyng before in which he did set forth the most simple proprieties of rightlined figures For sciences take their dignities of the worthynes of the matter that they entreat of But of al figures the circle is of most absolute perfection whose proprieties and passions are here set forth and most certainely demoÌ„strated Here also is entreated of right lines subtended to arkes in circles also of angles set both at the circumference and at the centre of a circle and of the varietie and differences of them Wherfore the readyng of this booke is very profitable to the attayning to the knowledge of chordes and arkes It teacheth moreouer which are circles contingeÌ„t and which are cutting the one the other and also that the angle of contingence is the least of all acute rightlined angles and that the diameter in a circle is the longest line that can be drawen in a circle Farther in it may we learne how three pointes beyng geuen how soeuer so that they be not set in a right line may be drawen a circle passing by them all three Agayne how in a solide body as in a Sphere Cube or such lyke may be found the two opposite pointes Whiche is a thyng very necessary and commodious chiefly for those that shall make instrumentes seruyng to Astronomy and other artes Definitions Equall circles are such whose diameters are equall or whose lynes drawen from the centres are equall The circles A and B are equal if theyr diameters namely EF and CD be equall or if their semidiameters whiche are lynes drawen from the center to the circumferenceâ— namely AF and BD be equall By thys also is knowen the definition of vnequall circles Circles whose diameters or semidiameters are vnequall are also vnequall And that circlâ— which hath the greater diameter or semidiameter is the greater circle and that circle which hath the lesse diameter or semidiameter is the lesse circle A right line is sayd to touch a circle which touching the circle and being produced cutteth it not As the right lyne EF drawen from the point E and passyng by a point of the circle namely by the point G to the point F only toucheth the circle GH and cutteth it not nor entreth within it For a right line entryng within a circle cutteth and deuideth the circle As the right lyne KL deuideth and cutteth the circle KLM and entreth within it and therfore toucheth it in two places But a right lyne touchyng a circle which is commonly called a coÌ„tingent lyne toucheth the circle onely in one point Circles are sayd to touch the one the other which touching the one the other cut not the one the other As the two circles AB and BC touch the one the other For theyr circumferences touch together in the poynt B. But neither of them cutteth or deuideth the other Neither doth any part of the one enter within the other And such a touch of circles is euer in one poynt onely which poynt onely is common to them both As the poynt B is in the conference of the circle AB and also 〈…〉 the â—â—â—â—â—ference of the circle BC. Circles may touch together two maner of wayes either outwardly the one wholy without the other or els the one being contayned within the other As the circles DE and DF of which the one DE contayneth the other namely DF and touch the one
the circle ABC Agayne for asmuch as in the selfe same circle ABC the right line NX that is the line ME deuideth the right line BG into two equall partes and maketh right angles therefore by the third of the third in the line NX is the centre of the circle ABC And it is proued that it is also in the line AC And these two right lines AC and NX meete together in no other poynt besides O. Wherefore the poynt O is the centre of the circle ABC And in like sort may we proue that the poynt O is the centre of the circle DEF Wherefore the two circles ABC and DEF deuiding the one the other haue one and the same centre which by the 5. of the third is impossible A circle therfore cutteth not a circle in moe poyntes then two which was required to be proued An other demonstration to proue the same Suppose that the circle ABC do cut the circle DGF in mo poyntes then two that is in B G F and H. And by the first of the third take the centre of the circle ABC and let the same be the poynt K. And draw these right lines KB KG and KF Now for asmuch as within the circle DEF is taken a certaine poynt K and from that poynt are drawen vnto the circumference moe then two equall right lines namely KB KG and KF therefore by the 9. of the third K is the centre of the circle DEF And the poynt K is the centre of the circle ABC Wherefore two circles cutting the one the other haue one and the same centre which by the 5. of the third is impossible A circle therfore cutteth not a circle in moe pointes then two which was required to be demonstrated The 10. Theoreme The 11. Proposition If two circles touch the one the other inwardly their centres being geuen a right line ioyning together their centres and produced will fall vpon the touch of the circles SVppose that these two circles ABC and ADE do touch the one the other in the poynt A. And by the first of the third take the centre of the circle ABC and let the same be F and likewise the centre of the circle ADE and let the same be G. Then I say that a right line drawen from F to G and being produced will fall vpon the poynt A. For if not then if it be possible let it fall as the line FGDH doth And draw these right lines AF AG. Now for asmuch as the lines AG and GF are by the 20. of the first greater then the line FA that is then the line FH take away the line GF which is common to them both Wherefore the residue AG is greater then the residue GH But the line DG is equall vnto the line GA by the 15. definition of the first Wherefore the line GD is greater then the line GH the lesse then the greater which is impossible Wherfore a right line drawen from the poynt F to the poynt G and produced falleth not besides the poynt A which is the point of the touch Wherefore it falletâ— vpon the touch If therefore two circles touch the one the other inwardly their centres being geuen a right line ioyning together their centres and produced will fall vpon the touch of the circles which was required to be proued An other demonstration to proue the same But now let it fall as GFC falleth and extend the line GFC to the poynt H and drawe these right lines AG and AF. And for asmuch as the lines AG and GF are by the 20. of the first greater then the line AF. But the line AF is equall vnto the line CF that is vnto the line FH Take away the line FG common to them both Wherfore the residue AG is greater then the residue GH that is the line GD is greater then the line GH the lesse greater then the greater which is impossible Which thing may also be proued by the 7. Proposition of this booke For for asmuch as the line HC is the diameter of the circle ABC in it is taken a poynt which is not the centre namely the poynt G therefore the line GA is greater then the line GH by the sayd 7. Proposition But the line GD is equall to the line GA by the definition of a circle Wherefore the line GD is greater then the line GH namely the part grâ—ater then the whole which is impossible The 11. Theoreme The 12. Proposition If two circles touch the one the other outwardly a right line drawen by their centres shall passe by the touch SVppose that these two circles ABC and ADE do touch the one the other outwardly in the poynt A. And by the third of the third take the centre of the circle ABC and let the same be the poynt F and likewise the centre of the circle ADE and let the same be the poynt G. Then I say that a right line drawen from the poynt F to the poynt G shall passe by the poynt of the touch namely by the poynt A. For if not then if it be possible let it passe as the right line FCDG doth And draw these right lines AF AG. And for asmuch as the poynt F is the centre of the circle ABC therfore the line FA is equall vnto the line FC Againe for asmuch as the poynt G is the centre of the circle ADE therefore the line GA is equall to the line GD And And it is proued that the line FA is equall to the line FC Wherefore the lines FA and AG are equall vnto the lines FC and GD Wherefore the whole line FG is greater then the lines FA and AG. But it is also lesse by the 20. of the first which is impossible Wherfore a right line drawen from the poynt F to the poynt G shall passe by the poynt of the touch namely by the poynt A. If therefore two circles touch the one the other outwardly a right line drawen by their centres shall passe by the touch which was required to be demonstrated ¶ An other demonstration after Pelitarius Suppose that the two circles ABC and DEF do touch the one the other outwardly in the poynt A And let G be the centre of the circle ABC From which poynt produce by the touch of the circles the line GA to the poynt F of the circumference DEF Which for asmuch as it passeth not by the centre of the circle DEF as the aduersary affirmeth draw from the same centre G an other right line GK which if it be possible let passe by the centre of the circle DEF namely by the poynt H cutting the circumference ABC in the poynt B the circuÌ„ference DEF in the poynt D let the opposite poynt therof be in the point K. And for asmuch as froÌ„ the poynt G taken without the circle DEF is drawen the line GK passing by the centre H and
when perpendicular lines drawen frō the centre to those lines are equall by the 4. definition of the third Wherfore the lines AB and CD are equally distant from the centre But now suppose that the right lines AB and CD be equally distant from the centre that is let the perpendicular line EF be equall to the perpendicular line EG Then I say that the line AB is equall to the line CD For the same order of construction remayning we may in like sort proue that the line AB is double to the line AF and that the line CD is double to the line CG And for asmuch as the line AE is equall to the line CE for they are drawen from the centre to the circumference therfore the square of the line AE is equall to the square of the line CE. But by the 47. of the first to the square of the line AE are equall the squares of the lines EF and FA. And by the selfe same to the square of the line CE are equall the squares of the lines EG and GC Wherfore the squares of the lines EF and FA are equall to the squares of the lines EG and GC Of which the square of the line EG is equall to the square of the line EF for the line EF is equall to the line EG Wherefore by the third common sentence the square remayning namely the square of the line AF is equall to the square of the line CG Wherefore the line AC is equall vnto the line CG But the line AB is double to the line AF and the line CD is double to the line CG Wherefore the line AB is equall to the line CD Wherefore in a circle equall right lines are equally distant from the centre And lines equally distant from the centre are equall the one to the other which was required to be proued ¶ An other demonstration for the first part after Campane Suppose that there be a circle ABDC whose centre let be the poynt E. And draw in it two equall lines AB and CD Then I say that they are equally distant from the centre Draw from the centre vnto the lines AB and CD these perpendicular lines EF and EG And by the 2. part of the 3. of this booke the line AB shall be equally deuided in the poynt F. and the line CD shall be equally deuided in the poynt G. And draw these right lines EA EB EC and ED. And for asmuch as in the triangle AEB the two sides AB and AE are equall to the two sides CD and CE of the triangle CED the base EB is equall to the base ED. therefore by the 8. of the first the angle at the point A shall be equall to the angle at the point C. And for asmuch as in the triangle AEF the two sides AE and AF are equall to the two sides CE and CG of the triangle CEG and the angle EAF is equall to the angle CEG therefore by the 4. of the first the base EF i◠equall to the base EG which for asmuch as they are perpendicular lines therefore the lines AB CD are equally distant frō the centre by the 4. definition of this booke The 14. Theoreme The 15. Proposition In a circle the greatest line is the diameter and of all other lines that line which is nigher to the centre is alwayes greater then that line which is more distant SVppose that there be a circle ABCD and let the diameter thereof be the line AD and let the centre thereof be the poynt E. And vnto the diameter AD let the line BC be nigher then the line FG. Then I say that the line AD is the greatest and the line BC is greater then the line FG. Draw by the 12. of the first from the centre E to the lines BC and FG perpendicular lines EH and EK And for asmuch as the line BC is nigher vnto the centre then the line FG therfore by the 4. definition of the third the line EK is greater then the line EH And by the third of the first put vnto the line EH an equall line EL. And by the 11. of the first from the point L raise vp vnto the line EK a perpendicular line LM and extend the line LM to the poynt N. And by the first petition draw these right lines EM EN EF and EG And for asmuch as the line EH is equall to the line EL therefore by the 14. of the third and by the 4. definition of the same the line BC is equall to the line MN Againe for asmuch as the line AE is equall to the line EM and the line ED to the line EN therefore the line AD is equall to the lines ME and EN But the lines ME and EN are by the 20. of the first greater then the line MN Wherefore the line AD is greater then the line MN And for asmuch as these two lines ME and EN are equall to these two lines FE and EG by the 15. definition of the first for they are drawen from the centre to the circumference and the angle MEN is greater then the angle FEG therefore by the 24. of the first the base MN is greater then the base FG. But it is proued that the line MN is equall to the line BC Wherefore the line BC also is greater then the line FG. Wherefore the diameter AD is the greatest and the line BC is greater then the line FG. Wherefore in a circle the greatest line is the diameter and of all the other lines that line which is nigher to the centre is alwaies greater then that line which is more distant which was required to be proued ¶ An other demonstration after Campane In the circle ABCD whose centre let be the poynt E draw these lines AB AC AD FG and HK of which let the line AD be the diameter of the circle Then I say that the line AD is the greatest of all the lines And the other lines eche of the one is so much greater then ech of the other how much nigher it is vnto the centre Ioyne together the endes of all these lines with the centre by drawing these right lines EB EC EG EK EH and EF. And by the 20. of the first the two sides EF and EG of the triangle EFG shall be greater then the third side FG. And for asmuch as the sayd sides EF EG are equall to the line AD by the definition of a circle therefore the line AD is greater then the line FG. And by the same reason it is greater then euery one of the rest of the lines if they be put to be bases of triangles for that euery two sides drawen frō the centre are equall to the line AD. Which is the first part of the Proposition Agayne for asmuch as the two sides EF and EG of the triangle EFG are equall to the
and it is in the segment ABC which is greater then the semicircle And forasmuch as in the circle there is a figure of foure sides namely ABCD. But if within a circle be described a figure of foure sides the angles therof which are opposite the one to the other are equall to two right angles by the 22. of the third Wherfore by the same the angles ABC and ADC are equall to two right angles But the angle ABC is lesse then a right angle Wherfore the angle remayning ADC is greater then a right angle and it is in a segment which is lesse then the semicircle Againe forasmuch as the angle comprehended vnder the right lines AC and AF is a right angle therfore the angle comprehended vnder the right line CA and the circumference ADC is lesse then a right angle Wherfore in a circle an angle made in the semicircle is a right angle but an angle made in the segment greater then the semicircle is lesse then a right angle and an angle made in the segment lesse then the semicircle is greater then a right angle And moreouer the angle of the greater segment is greater then a right angle the angle of the lesse segment is lesse then a right angle which was required to be demonstrated An other demonstration to proue that the angle BAC is a right angle Forasmuch as the angle AEC is double to the angle BAE by the 32. of the first for it is equall to the two inward angles which are opposite But the inwarde angles are by the 5. of the first equall the one to the other and the angle AEB is double to the angle EAC Wherfore the angles AEB and AEC are double to the angle BAC But the angles AEB and AEC are equall to two right angles Whâ—rfore the angle BAC is a right angle Which was required to be demonstrated Correlary Hereby it is manifest that if in a triangle one angle be equall to the two other angles remayning the same angle is a right angle for that the side angle to that one angle namely the angle which is made of the side produced without the triangle is equall to the same angles but when the side angles are equall the one to the other they are also right angles ¶ An addition of Pelitarius If in a circle be inscribed a rectangle triangle the side opposite vnto the right angle shall be the diameter of the circle Suppose that in the circle ABC be inscribed a rectangle triangle ABC whose angle at the point B let be a right angle Then I say that the side AC is the diameter of the circle For if not then shall the centre be without the line AC as in the point E. And draw a line from the poynt A to the point E produce it to the circumference to the point D and let AED be the diameter and draw a line from the point B to the point D. Now by this 31. PropositioÌ„ the angle ABD shall be a right angle and therefore shall be equall to the right angle ABC namely the part to the whole which is absurde Euen so may we proue that the centre is in no other where but in the line AC Wherfore AC is the diameter of the circle which was required to be proued ¶ An addition of Campane By thys 31. Proposition and by the 16. Proposition of thys booke it is manifest that although in mixt angles which are contayned vnder a right line and the circumference of a circle there may be geuen an angle lesse greater then a right angle yet can there neuer be geueÌ„ an angle equall to a right angle For euery section of a circle is eyther a semicircle or greater then a semicircle or lesse but the angle of a semicircle is by the 16. of thys booke lesse then a right angle and so also is the angle of a lesse section by thys 31. Proposition Likewise the angle of a greater section is greater then a right angle as it hath in thys Proposition bene proued The 28. Theoreme The 32. Proposition If a right line touch a circle and from the touch be drawen a right line cutting the circle the angles which that line and the touch line make are equall to the angles which consist in the alternate segmentes of the circle SVppose that the right line EF do touch the circle ABCD in the point B and from the point B let there be drawen into the circle ABCD a right line cutting the circle and let the same be BD. Then I say that the angles which the line BD together with the touch line EF do make are equall to the angles which are in the alternate segmentes of the circle that is the angle FBD is equall to the angle which consisteth in the segment BAD and the angle EBD is equall to the angle which consisteth in the segment BCD Raise vp by the 11. of the first from the point B vnto the right line EF a perpendicular line BA And in the circumference BD take a point at all aduentures and let the same be C. And draw these right lines AD DC and CB. And for asmuch as a certaine right line EF toucheth the circle ABC in the point B and from the point B where the touch is is raysed vp vnto the touch line a perpendicular BA Therfore by the 19. of the third in the line BA is the centre of the circle ABCD. Wherfore the angle ADB being in the semicircle is by the 31. of the third a right angle Wherefore the angles remayning BAD and ABD are equall to one right angle But the angle ABF is a right angle Wherefore the angle ABF is equall to the angles BAD and ABD Take away the angle ABD which is common to them both Wherefore the angle remayning DBF is equall to the angle remayning BAD which is in the alternate segment of the circle And for asmuch as in the circle is a figure of fower sides namely ABCD therfore by the 22. of the third the angles which are opposite the one to the other are equall to two right angles Wherfore the angles BAD and BCD are equall to two right angles But the angles DBF and DBE are also equall to two right angles Wherefore the angles DBF and DBE are equall to the angles BAD and BCD Of which we haue proued that the angle BAD is equall to the angle DBF Wherefore the angle remayning DBE is equall to the angle remayning DCB which is in the alternate segment of the circle namely in the segment DCB If therfore a right line touch a circle and from the touch be drawen a right line cutting the circle the angles which that line and the touch line make are equall to the angles which consist in the alternate segmentes of the circle which was required to be proued In thys Proposition may be two cases For the line drawen from
the touch and cutting the circle may eyther passe by the centre or not If it passe by the centre then is it manifest by the 18. of thys booke that it falleth perpendicularly vpon the touch line and deuideth the circle into two equall partes so that all the angles in eche semicircle are by the former Proposition right angles and therfore equall to the alternate angles made by the sayd perpendicular line and the touch line If it passe not by the centre then followe the construction and demonstration before put The 5. Probleme The 33. Proposition Vppon a right lyne geuen to describe a segment of a circle which shall contayne an angle equall to a rectiline angle geueÌ„ SVppose that the right line geuen be AB and let the rectiline angle geâ—â—n be C. It is required vpon the right line geueÌ„ AB to describe a segment of a circle which shall contayne an angle equall to the angle C. Now the angle C is either an acute angle or a right angle or an obtuse angle First let it be an acute angle as in the first description And by the 23 of the first vpon the right line AB and to the point in it A describe an angle equal to the angle C and let the same be DAB Wherfore the angle DAB is an acute angle From the point A raise vp by the 11. of the first vnto the line AD a perpendiculer line AF. And by the 10. of the first deuide the line AB into two equall partes in the point F. And by the 11. of the same from the point F raise vp vnto the line AB a perpendicular lyne FG and draw a line from G to B. And forasmuch as the line AF is equall to the line FB and the line FG is common to them both therfore these two lines AF and FG are equall to these two lines FB and FG and the angle AFG is by the 4. peticion equall to the angle GFB Wherfore by the 4. of the same the base AG is equall to the base GB Wherfore making the centre G and the space GA describe by the 3. peticion a circle and it shall passe by the point B describe such a circle let the same be ABE And draw a line from E to B. Now forasmuch as from the ende of the diameter AE namely from the point A is 〈◊〉 a right line AD making together with the right line AE a right angle therfore by the correllary of the 16. of the third the line AD toucheth the circle ABE And forasmuch as a certaine right line AD toucheth the circle ABE from the point A where the touch is is drawen into the circle a certaine right line AB therfore by the 32. of the third the angle DAB is equall to the angle AEB which is in the alternate segment of the circle But the angle DAB is equall to the angle C wherfore the angle C is equall to the angle AEB Wherfore vpon the right line geuen AB is described a segment of a circle which contayneth the angle AEB which is equall to the angle geuen namely to C. But now suppose that the angle C be a right angle It is againe required vpon the right line AB to describe a segment of a circle which shall contayne an angle equal to the right angle C. Describe againe vpon the right line AB and to the point in it A an angle BAD equal to the rectiline angle geuen C by the 23. of the first as it is set forth in the second description And by the 10. of the first deuide the line AB into two equall partes in the point F. And making the centre the point F and the space FA or FB describe by the 3. peticion the circle AEB Wherfore the right line AD toucheth the circle AEB for that the angle BAD is a right angle Wherfore the angle BAD is equall to the angle which is in the segment AEB for the angle which is in a semicircle is a right angle by the 31. of the third But the angle BAD is equal to the angle C. Wherfore there is againe described vpon the line AB a segment of a circle namely AEB which containeth an angle equall to the angle geuen namely to C. But now suppose that the angle C be an obtuse angle Vpon the right line AB and to the point in it A describe by the 23. of the first an angle BAD equall to the angle C as it is in the third description And from the point A rayse vp vnto the line AD a perpendiculer line AE by the 11. of the first And agayne by the 10. of the first deuide the line AB into two equall partes in the point F. And from the point F. raâ—se vp vnto the line AB a perpeÌ„dicular line FG by the 11. of the same drawe a line from G to B. And now forasmuch as the line AF is equal to the line FB and the line FG is common to them both therfore these two lines AF and FG are equall to these two lines BF and FG and the angle AFG is by the 4. peticion equall to the angle BFG wherfore by the 4. of the same the base AG is equall to the base GB Wherfore making the centre G and the space GA describe by the 3. peticion a circle and it shall passe by the point B let it be described as the circle AEB is And forasmuch as from the ende of the diameter AE is drawen a perpendiculer line AD therefore by the correllary of the 16. of the third the line AD toucheth the circle AEB from the point of the touche namely A is extended the line AB Wherfore by the 32. of the third the angle BAD is equall to the angle AHB which is in the alternate segment of the circle But the angle BAD is equall to the angle C. Wherefore the angle which is in the segment AHB is equall to the angle C. Wherfore vpon the right line geuen AB is described a segment of a circle AHB which contayneth an angle equall to the angle geuen namely C which was required to be done The 6. Probleme The 34. Proposition From a circle geuen to cut away a section which shal containe an angle equall to a rectiline angle geuen SVppose that the circle geuen be AC and let the rectiline angle geuen be D. It is required froÌ„ the circle ABC to cut away a segment which shall contayne an angle equall to the angle D. Draw by the 17. of the third a line touching the circle and let the same be EF and let it touche in the point B. And by the 23. of the first vpon the right line EF and to the point in it B describe the angle FBC equall to the angle D. Now forasmuch as a certayne right line EF toucheth the circle ABC in the point B and frora
the point of the touche namely B is drawn into the circle a certaine right line BC thereâ—ore by the 32. of the third the angle FBC is equall to the angle BAC which is in the alternate segment But the angle FBC is equall to the angle D. Wherfore the angle BAC which consisteth in the segment BAC is equall to the angle D. Wherefore from the circle geuen ABC is cut away a segment BAC which containeth an angle equall to the rectiline angle geuen which was required to be done The 29. Theoreme The 35. Proposition If in a circle two right lines do cut the one the other the rectangle parallelograme comprehended vnder the segmentes or parts of the one line is equall to the rectangle parallelograme comprehended vnder the segment or partes of the other line LEt the circle be ABCD and in it let these two right lines AC and BD câ—t the one the other in the point E. Then I say that the rectangle parallelogramme contayned vnder the partes AE and EC is equall to the rectangle parallelogramme contained vnder the partes DE and EB For if the line AC and BD be drawen by the centre then is it manifest that for as much as the lines AE and EC are equall to the lines DE and EB by the definition of a circle the rectangle parallelograme also contayned vnder the lines AE and EC is equall to the rectangle parallelograme contained vnder the lines DE and EB But now suppose that the lines AC and DB be not extended by the centre and take by the 1. of the third the centre of the circle ABCD and let the same be the point F and from the point F draw to the right lines AC and DB perpendicular lines FG and FH by the 12. of the first and draw these right lines FB FC and FE And forasmuch as a certaine right line FG drawen by the centre cutteth a certaine right line AC not drawen by the centre in such sorte that it maketh right angles it therfore deuideth the line AC into two equall partes by the 3. of the third Wherfore the line AG is equall to the line GC and forasmuch as the right line AC is deuided into two equall partes in the point G and into two vnequall partes in the point E therfore by the 5. of the second the rectangle parallelogramme contained vnder the lines AE and EC together with the square of the line EG is equall to the square of the line GC Put the square of the line GF common to them both wherfore that which is contained vnder the lines AE EC together with the squares of the lines EG and GF is equall to the squares of the lines GF GC But vnto the squares of the lines EG GF is equall the square of the line FE by the 47. of the first and to the squares of the lines GC and GF is equall the square of the line FC by the same Wherfore that which is contained vnder the lines AE and EC together with the square of the line FE is equall to the square of the line FC But the line FC is equall to the line FB For they are drawen from the centre to the circumference Wherfore that which is contained vnder the lines AE and Eâ— together with the square of the lyne FE is equal to the square of the line FB And by the same demonstration that which is contained vnder the lines DE and EB together with the square of the line FE is equall to the square of the line FB Wherfore that which is contained vnder the lines AE and EC together with the square of the line EF is equall to that which is contayned vnder the lines DE and EB together with the square of the line EF. Take away the square of the line FE which is common to them both Wherfore the rectangle parallelogramme remayning which is contayned vnder the lines AE and EC is equall to the rectangle parallelogramme remayning which is contayned vnder the lines DE and EB If therefore in a circle two right lines do cut the one the other the rectangle parallelogramme comprehended vnder the segmentes or parts of the one line is equall to the rectangle parallelograme comprehended vnder the segmentes or parts of the other line which was required to be demonstrated In thys Proposition are three cases For eyther both the lines passe by the centre or â—eyther of them passeth by the centre or the one passeth by the centre and the other not The two first cases are before demonstrated Amongest all the Propositions in this third booke doubtles thys is one of the chiefest For it setteth forth vnto vs the wonderfull nature of a circle So that by it may be done many goodly conclusions in Geometry as shall afterward be declared when occasion shall serue The 30. Theoreme The 36. Proposition If without a circle be taken a certaine point and from that point be drawen to the circle two right lines so that the one of them do cut the circle and the other do touch the circle the rectangle parallelogramme which is comprehended vnder the whole right line which cutteth the circle and that portion of the same line that lieth betwene the point and the vtter circuÌ„ference of the circle is equall to the square made of the line that toucheth the circle SVppose that the circle be ABC and without the same circle take any point at all aduentures and let the same be D. And from the point D let there be drawen to the circle two right lines DCA and DB and let the right line DCA cut the circle ACB in the point C and let the right line BD touch the same Then I say that the rectangle parallelogramme contayned vnder the lines AD and DC is equall to the square of the line BD. Now the line DCA is either drawen by the centre or not First let it be drawen by the centre And by the first of the third let the poynt F be the centre of the circle ABC and drawe a line from F to B. Wherefore the angle FBD is a right angle And for asmuch as the right line AC is deuided into two equall partes in the poynt F and vnto it is added directly a right line CD therfore by the 6. of the second that which is contayned vnder the lines AD and DC together with the square of the line CF is equall to the square of the line FD. But the line FC is equall to the line FB for they are drawen from the centre to the circumference Wherfore that which is contayned vnder the lines AD and DC together with the square of the line FB is equall to the square of the line FD. But the square of the line FD is by the 47. of the first equall to the squares of the lines FB and BD for the angle FBD is a right angle Wherefore
that which is contayned vnder the lines AD and DC together with the square of the line FB is equall to the squares of the lines FB and BD. Take away the square of the line FB which is common to them both Wherefore that which remayneth namely that which is contayned vnder the lines AD and DC is equall to the square made of the line DB which toucheth the circle But now suppose that the right line DCA be not drawen by the centre of the circle ABC And by the first of the third let the point E be the centre of the circle ABC And from the poynt E draw by the 12. of the first vnto the line AC a perpendicular line EF and draw these right lines EB EC and ED. Now the angle EFD is a right angle And â—or asmuch as a certaine right line EF drawen by the centre cutteth a certayne other right line AC not drawen by the centre in such sort that it maketh right angles it deuideth it by the third of the third into two equall partes Wherefore the line AF is equall to the line FC And for asmuch as the right line AC is deuided into two equall partes in the poynt F vnto it is added directly an other right line making both one right line therefore by the 6. of the second that which is contayned vnder the lines DA and DC together with the square of the line FC is equall to the square of the line FD put the square of the line FE common to them both Wherefore that which is contayned vnder the lines DA and DC together with the squares of the lines CF and FE is equall to the squares of the lines FD and FE But to the squares of the lines FD and FE is equall the square of the line DE by the 47. of the first for the angle EFD is a right angle And to the squares of the lines CF and FE is equall the square of the line CE by the same Wherfore that which is contayned vnder the lines AD and DC together with the square of the line EC is equall to the square of the line ED. But the line EC is equall to the line EB for they are drawen from the centre to the circumference Wherefore that which is contayned vnder the lines AD and DC together with the square of the line EB is equall to the square of the line ED. But to the square of the line ED are equall the squares of the lines EB and BD by the 47. of the first for the angle EBD is a right angle Wherefore that which is contayned vnder the lines AD and DC together with the square of the line EB is equall to the squares of the lines EB and BD. Take away the square of the line EB which is common to them both Wherefore the residue namely that which is contayned vnder the lines AD and DC is equall to the square of the line DB. If therfore without a circle be taken a certaine point and from that poynt be drawen to the circle two right lines so that the one of them do cut the circle and the other do â—ouch the circle the rectangle parallelogramme which is comprehended vnder the whole right line which cutteth the circle and that portion of the same line that lieth betwene the poynt and the vtter circumference of the circle is equall to the square made of the line that toucheth the circle which was required to be demonstrated ¶ Two Corollaries out of Campane If from 〈◊〉 and the selfe same poynt taken without a circle be drawen into the circle lines how many soeuer the rectangle Parallelogrammes contayned vnder euery one of them and hys outward parâ— are equall the one to the other And thys is hereby manifest for that euery one of those rectangle Parallelogrammes are equall to the square of the line which is drawen from that poynt and toucheth the circle by thys 36. Proposition Hereunto he addeth If two lines drawen from one and the selfe same point do touch a circle they are equall the one to the other Which although it neede no demonstration for that the square of eyther of them is equall to that which is contayned vnder the line drawen from the same poynt and hys outward part yet he thus proueth it The same may be proued an other way Draw a line from B to D. And by the 5. of the first the angle EBD is equall to the angle EDB And for asmuch as the two angles ABE and ADE are equall namely for that they are right angles if you take from them the equall angles EBD EDB the two other angles remayning namely the angles ABD and ADB shall be equall Wherefore by the 6. of the first the line AB is equall to the line AD. ¶ Hereunto also Pelitarius addeth this Corollary From a poynt geuen without â— circle can be drawen vnto a circle onely two touch lines The former description remayning I say that from the poynt A can be drawen vnto the circle BCD no more touch lines but the two lines AB and AD. For if it be possible let AF also be in the former figure a touch line touching the circle in the poynt F. And prawe a line from E to F. And the angle at the point F shall be a right angle by the 18. of this booke Wherefore it is equall to the angle EBA which is contrary to the 20. of the first This may also be thus proued For asmuch as all the lines drawen from one and the selfe same poynt touching a circle are equall as we haue before proued but the lines AB and AF can not be equall by the 8. Proposition of this booke therefore the line AF can not touch the circle BCD The 31. Theoreme The 37. Proposition If without a circle be taken a certaine point and from that point be drawen to the circle two right lines of which the one doth cut the circle and the other falleth vpon the circle and that in such sort that the rectangle parallelogramme which is coÌ„tayned vnder the whole right line which cutteth the circle and that portion of the same line that lieth betwene the point and the vtter circumfereÌ„ce of the circle is equall to the square made of the line that falleth vpon the circle then that line that so falleth vpon the circle shall touch the circle LEt the circle be ABC and without the same circle take a point and let the same be D from the point D let there be drawen to the circle ABC two right lines DCA and DB and let DCA cut the circle and DB fall vpon the circle And that in such sort that that which is contayned vnder the lines AD and DC be equall to the square of the line DB. Then I say that the line DB toucheth the circle ABC Drawe by the 17. of the third from the poynt D a right line
such a multiplex as is next greater then FG and againe let the same be N which let be quadruple to D. And in like sort as before may we proue that the whole FH is vnto AB equemultiplex as GH is to EB and also that FH K are equemultiplices vnto AB â— C and finally that GH is equall vnto K. And forasmuch as the multiplex N is next greater then FG therefore FG is not lesse then M. But GH is greater then D by construction Wherefore the whole FH is greater then D and M and so consequently is greater then N. But K excedeth not N for K is equall to GH for how multiplex K is to EB the lesse so multiplex is FG to Aâ— the greater Bâ—t those magnitudes which are equemultipliceâ— vnto vnequall magnitudâ—s are according to the same proportion vnequall Wherefore K is lesse then FG and therefore iâ— much lesse then N. Wherefore againe the multiplex of the first exceedeth the multiplex of the second but the multiplex of the third excedeth not thâ— multiplex of the fourth Wherefore by the 8. definition of the fift Aâ— the first hath to D the second a greater proportion then hath C the third to D the fourth But now if AE be equall vnto EB eyther of them shall be equall vnto C. Wherfore vnto either of thosâ— three magnitudes take equemultiplices greater then D. So that let FG be multiplex to AE and GH vnto EB and K agayne to C which by the 6. coÌ„mon sentence shall be equall the one to the other Let N also be multiplex to D and be next greater then euery one of them namely let it be qâ—adruplâ— to D. This coÌ„struction finished we may again proue that FH and K are equemultiplices to AB and C and that FH the multiplex of thâ— first magnitude exceedeth N the multiplex of the second magnitudeâ—â—nd thaâ— K tâ—â—â—ultiplex of the third excedeth not the multiplex of the fourth Wherfore we may conclude that AB hath vnto D a greater proportion then hath C to D. Now also I say that the self same magnitude D hath vnto the lesse magnitude C a greater proportion theÌ„ it hath to the greater AB And this may plainly be gathered by the foresayd discourse without chaunging the order of the magnitudes of the equemultiplices For seing that euery way it is before proued that FH excedeth N and K is exceeded of the selfe same N therefore conuersedly N excedeth K but doth not excede FH But N is multiplex to D that is to the first and third magnitude and K is multiplex to the second namely to C â— and FH is multiplex to the fourth namely to AB â— Wherefore the multiplex of the first excedeth the multiplex of the second but the multiplex of the third excedeth not the multiplex of the fourth Wherefore by the 8. definition of this fift booke D the first hath vnto C the second a greater proportion then hath D the third to AB the fourth which was required to be proued The 9. Theoreme The 9. Proposition Magnitudes which haue to one and the same magnitude one and the same proportion are equall the one to the other And those magnitudes vnto whome one and the same magnitude hath one and the same proportion are also equall SVppose that either of these two magnitudes A and B haue to C one and the same proportion Then I say that A is equall vnto B. For if it be not then either of these A and B should not haue to C one the same proportioÌ„ by the 8. of the fifth but by supposition they haue wherefore A is equall vnto B. Againe suppose that the magnitude C haue to either of these magnitudes A and B one and the same proportion Then I say that A is equall vnto B. For if it be not C should not haue to either of these A and B one and the same proportion by the former proposition but by supposition it hath wherfore A is equall vnto B. Wherfore magnitudes which haue to one and the same magnitude one and the same proportion are equall the one to the other And thosâ— magnitudes vnto whome one and the same magnitude hath one and the same proportion are also equall which was required to be proued The 10. Theoreme The 10. Proposition Of magnitudes compared to one and the same magnitude that which hath the greater proportion is the greater And that magnitude wherunto one and the same magnitude hath the greater proportion is the lesse SVppose that A haue to C a greater proportion then B hath to C. Then I say that A is greater then B. For if it be not then either A is equall vnto B or lesse then it But A cannot be equal vnto B for then either of these A and B should haue vnto C one and the same proportion by the 7 of the fifth but by supposition they haue not wherfore A is not equall vnto B. Neither also is A lesse then B for theÌ„ should A haue to C a lesse proportion then hath B to C by the 8. of the fifth but by supposition it hath not Wherfore A is not lesse then B. And it is also proued that it is not equall wherfore A is greater then B. Agayne suppose that C haue to B a greater proportion then C hath to A. Then I say that B is lesse then A. For if it be not then is it either equall vnto it or els greater but B cannot be equall vnto A for then should C haue to either of these A and B one and the same proportion by the 7. of the fifth but by supposition it hath not wherfore B is not equall vnto A. Neither also is B greater then A for then should C haue to B a lesse proportion then it hath to A by the 8. of the fifth but by supposition it hath not wherefore B is not greater then A. And it was proued that it is not equall vnto A wherfore B is lesse then A. Wherfore of magnitudes compared to one and the same magnitude that which hath y greater proportion is the greater And that magnitude wherunto one and the same magnitude hath the greater proportion is the lesse Which was required to be proued The 11. Theoreme The 11. Proposition Proportions which are one and the selfe same to any one proportion are also the selfe same the one to the other SVpppose that as A is to B so is C to D and as C is to D so is E to F. Then I say that as A is to B so is E to F. Take equemultiplices to A C and E which let be G H K. And likewise to B D and F take any other equemultiplices which let be L M and N. And because as A is to B so is C to D and to A and G are taken equemultiplices G H to B and D are take certaine other equemultiplices L
therefore alternately as AB is to E so is CD to F by the 16. of the fift But AB is greater then E Wherfore also CD is greater then F Which thing may also be proued by the 14. of the same Now for that as AB is to CD so is E to F but E is equall vnto AG and F is equall vnto CH therefore as AB is to CD so is AG to CH and forasmuch as the whole AB is to the whole CD so is the part taken away AG to the part taken away CH therefore the residue GB by the 1â— of the fift is vnto the residue HD as the whole AB is to the whole CD But AB the first is greater then CD the third Wherfore GB the second is greater then HD the fourth by the 14. of the fift And forasmuch as AG is equall vnto E CH is equall vnto F therefore AG and F are equall vnto CH and E. And forasmuch as if vnto thinges vnequall be added thinges equall all shall be vnequall by the fourth common sentence therefore seing that GB and DH are vnequall and GB is the greater if vnto GB be added AG and F and likewise if vnto HD be added CH E there shall be produced AB and F greater then CD E. If therefore there be foure magnitudes proportionall the greatest and the least of them shall be greater theÌ„ the other remayning which was required to be demonstrated Here follow certayne propositions added by Campane which are not to be contemned and are cited euen of the best learned namely of Iohannes Regio montanus in the Epitome which he writeth vpon Ptolome ¶ The first Proposition If there be foure quantities and if the proportion of the first to the second be greater then the proportion of the third to the fourth then contrariwise by conuersion the proportion of the second to the first shall be lesse then the proportion of the fourth to the third It may also be demonstrated directly For let E be vnto B as C is to D. Then coÌ„uersedly B is to E as D is to C. And forasmuch as A is greater then E by the first part of the tenth of this booke therfore by the second part of the 8 of the same B hath vnto A a lesse proportion then hath B to E. Wherfore by the 13. of the same B hath vnto A â— lesse proportion then hath D to C which was required to be proued ¶ The second Proposition If there be foure quantities and if the proportion of the first to the second be greater then the proportion of the third to the fourth then alternately the proportion of the first to the third shall be greater then the proportion of the second to the fourth This may also be demonstrated affirmatiuely let E be vnto B as C is to D. Now theÌ„ by the first part of the tenth of this booke E is lesse then A wherfore by the first parte of the 8. of the same the proportion of A to C is greater then the proportion of E to C. But alternately E is to C as B is to D. Wherfore by the 13. of the same A hath to C a 〈…〉 ¶ The third Proposition If there be foure quantities and if the proportion of the first tâ— the second be greater then the proportion of the third to the fourth then by composition also the proportioÌ„ of the fâ—â—th and second to the second shall be greater then the proportioâ— of the third and fourthâ— to the fourth This may also be demonstrated aâ—firmatâ—â—ely Forasmuch as the proportion of A to B is greater then the proportion of C to D let E be vnto B as C is to D. And so by the first part of the 10. of this booke E shall be lesse then A. And therfore by the common sentence EB shall be lesse then AB Wherfore by the first part of the 8. of the same AB hath vnto B a greater proportion then hath EB to B. But by composition EB is to B as CD is to D. For by supposition E is vnto B as is to D. Wherfore by the 12. of this booke AB hath to B a greater proportion then hath CD to D which was required to be proued ¶ The fourth Proposition If there be foure quantities and if the proportion of the first and the second to the second be greater then the proportion of the third and fourth to the fourth then by diuision also the proportion of the first to the second shall be greater thâ—n the proportion of the thirde to the fourth Suppose that the proportion of AB to B be greater then the proportion of CD to D. Then I say that by diuision also the proportion of A to B is greater then the proportion of C to D. For it cannot be the same For then by composition AB should be to B as CD is to D. Neither also can it be lesse for if the proportion of C to D be greater then the proportion of A to B then by the former proposition the proportion of CD to D should be greater then the proportion of AB to B which is contrary also to the suppositioÌ„ Wherfore the proportion of A to B is neither one and the same with the proportion of C to D 〈…〉 it Wherefore it is greater then it which was required to be proued The same may also be proued affirmatiuely Suppose that EB be vnto B as CD is to D. Now then by the first part of the 10. of the fifth EB shall be lesse then AB and therefore by the common sentence E is lesse then A wherfore by the first part of the 8. of this booke the proportion of E to B is lesse then the proportion of A to B but as E is to B so is C to D wherfore the proportion of C to D is lesse then the proportion of A to B. Wherfore the proportion of A to B is greater then the proportion of C to D which was required to be proued ¶ The fifth Proposition If there be foure quantities and if the proportion of the first and the second to the second be greater then the proportion of the third and the fourth to the fourth then by euersion the proportion of the first and second to the first shall be lesse then the proportion of the third and fourth to the third Suppose that the proportion of AB to B be greater then the proportion of CD to D. Then I say that by euersion the proportion of AB to A is lesse then the proportion of CD to C. For by diuision by the former proposition the proportion of A to B is greater then the proportion of C to D. Wherefore by the first of these propositions conuersedly B hath vnto A a lesse proportioÌ„ theÌ„ hath D to C. Wherfore by the 3. of the same by composition the proportion of AB to A is lesse theÌ„ the
proportion then hath BC to EF. Wherefore A hath to D a greater proportion then hath BC to EF. Wherfore alternately A hath to BC a greater proportion then hath D to EF wherfore by composition ABC hath to BC a greater proportion then hath DEF to EF. Wherfore agayne alternately ABC hath to DEF a greater proportion then hath BC to EF. Wherefore by the former proposition the proportion of A to D is greater then the proportion of ABC to DEF Which was required to be proued The end of the fifth booke of Euclides Elementes ¶ The sixth booke of Euclides Elementes THIS SIXTH BOOKE is for vse and practise a most speciall booke In it are taught the proportions of one figure to an other figure of their sides the one to the other and of the sides of one to the sides of an other likewise of the angles of the one to the angles of the other Moreouer it teacheth the description of figures like to â—igures geuen and marueilous applications of figures to lines euenly or with decrease or excesse with many other Theoremes not onely of the propoâ—tions of right lined figures but also of sectors of circles with their angles On the Theoremes and Problemes of this Booke depend for the most part the compositions of all instrumentes of measuring length breadth or deâ—peâ—es and also the reason of the vse of the same instrumentes as of the Geometricall â—quarâ— the Scale of the Astrolabe the quadrant the staffe and such other The vse of which instrumentes besides all other mechanicall instrumentes of raysing vp of mouing and drawing huge thinges incredible to the ignorant and infinite other ginnes which likewise haue their groundes out of this Booke are of wonderfull and vnspeakeable profite besides the inestimable pleasure which is in them Definitions 1. Like rectiline figures are such whose angles are equall the one to the other and whose sides about the equall angles are proportionall As if ye take any two rectiline figures As for example two triangles ABC and DEF 〈…〉 of the one triangle be equall to the angles of the other namely if the angle A be equall to the angle D and the angle B equall to the angle E also the angle C equall to the angle F. And moreouer iâ— the sides which containe the equall angles be proportionall As if the side AB haue that proportion to the side BC whâ—ch the side DE hath to the side EF and also if the side BC be vnto the side CA as â—he side EF is to the side FD and morâ—ouer if the side CA be to the side AB as the side FD is to the side DE then are these two triangles sayd to be like and so iudge ye of any other kinde of figures As if in the parallelogrammes ABCD and EFGH the angle A be equall to the angle E and the angle B equall to the angle F and the angle C equall to the angle G and the angle D equall to the angle H. And farthermore if the side AC haue that proportion to the side CD which the side EG hath to the side GH and if also the side CD be to the side DB as the side GH is to the side HF and moreouer if the side DB be to the side BA as the side HF is to the side FE and finally if the side BA be to the side AC as the side FE is to the side EG then are these parallelogrammes like 2. Reciprocall figures are those when the termeâ— of proportion are both antecedentes and consequentes in either figure As if ye haue two parallelogrammes ABCD and EFGH If the side AB to the side EF an antecedent of the first figure to a consequent of the second figure haue mutually the same proportion which the side EG hath to the side AC an antecedent of the second figure to a consequent of the first figure then are these two figures Reciprocal They are called of some figures of mutuall sides and that vndoubtedly not amisse nor vnaptly And to make thys definition more plaine Campane and Pestitarius and othersâ— thus put it Reciprocall figures are when the sides of other 〈◊〉 mutually proportionall as in the example and declaration before geuen Among the barbarous they are called Mutekesia reseruing still the Arabike worde 3. A right line is sayd to be deuided by an extreme and meane proportion when the whole is to the greater part as the greater part is to the lesse As if the line AB be so deuided in the point C that the whole line AB haue the same proportion to the greater part thereof namely to AC which the same greater part AC hath to the lesse part therof namely to CB then is the line AB deuided by an extreme and meane proportion Commonly it is called a line deuided by proportion haâ—ing a meane and two extremes How to deuide a line in such sort was taught in the 11. Proposition of the second Booke but not vnder this forme of proportion 4. The alitude of a figure is a perpendicular line drawen from the toppe to the base As the altitude or hight of the triangle ABC is the line AD being drawen perpendicularly from the poynt A being the toppe or highest part of the triangle to the base therof BC. So likewise in other figures as ye see in the examples here set That which here â—ee calleth the altitude or height of a figure in the first booke in the 35. Proposition and certaine other following he taught to be contayned within two equidistant lines so that figures to haue one altitude and to be contayned within two equidistant lines is all one So in all these examples if from the highest point of the figure ye draw an equidistant line to the base therof and then froÌ„ that poynt draw a perpendicular to the same base that perpendicular is the altitude of the figure 5. A Proportion is said to be made of two proportions or more when the quantities of the proportions multiplied the one into the other produce an other quantitie An other example where the greater inequalitie and the lesse inequalitie are mixed together 6. 4. 2. 3. the denomination of the proportion of 6. to 4 is 1 â— â— of 4. to 2 is â— â— and of 2. to 3 is â— â— now if ye multiply as you ought all these denominations together ye shall produce 12. to 6 namely dupla proportion Forasmuch as so much hath hetherto bene spoken of addition of proportions it shall not be vnnecessary somewhat also to say of substraction of them Where it is to be noted that as addition of them is made by multiplicatioÌ„ of their denominations the one into the other so is the substraction of the one from the other done by diuision of the denomination of the one by the denomination of the other As if ye will from sextupla proportion subtrahe dupla proportion take
the denominations of them both The denomination of sextupla proportion is 6 the denomination of dupla proportion is 2. Now deuide 6. the denomination of the one by 2. the denomination of the other the quotient shall be 3 which is the denomination of a new proportion namely tripla so that when dupla proportion is subtrahed from sextupla there shall remayne tripla proportion And thus may ye do in all others 6. A Parallelogramme applied to a right line is sayd to want in forme by a parallelogramme like to one geuen wheÌ„ the parallelograÌ„me applied wanteth to the filling of the whole line by a parallelogramme like to one geuen and then is it sayd to exceede when it exceedeth the line by a parallelogramme like to that which was geuen As let E be a ParallelograÌ„me geuen and let AB be a right line to whom is applied the parallelogramme ACDF Now if it want of the filling of the line AB by the parallelograÌ„me DFGB being like to the parallelogramme geuen E then is the parallelogramme sayd to want in forme by a parallelogramme like vnto a parallelogramme geuen Likewise if it exceede as the parallelogramme ACGD applyed to the linâ— ABâ— if it exceede it by the parallelogramme FGBD being like to the parallelogramme F which was geuen then is the parallelogramme ABGD sayd to exceede in forme by a parallelogramme like to a parallelogramme geuen This definition is added by Flussates as it seemeth it is not in any coÌ„mon Greke booke abroad nor in any Commentary It is for many Theoremes following very necessary The 1. Theoreme The 1. Proposition Triangles parallelogrammes which are vnder one the self same altitude are in proportion as the base of the one is to the base of the other And forasmuch as the lines CB BG and GH are equall the one to the other therfore the triangles also AHG AGB and ABC are by the 38. of the first equall the one to the other Wherfore how multiplex the base HC is to the base BC so multiplex also is the triangle AHC to the triangle ABC And by the same reason also how multiplex the base LC is to the base DC so multiplex also is the triangle ALC to the triangle ADC Wherfore if the base HC be equall vnto the base CL then by the 38. of the first the triangle AHC is equall vnto the triangle ACL And if the base HC exceede the base CL then also the triangle AHC excedeth the triangle ACL and if the base be lesse the triaÌ„gle also shall be lesse Now then there are foure magnitudes namely the two basâ—s BC and CD and the two triangles ABC and ACD and to the base BC and to the triangle ABC namely to the first and the third are taken equemulâ—iplices namely the base HC and the triangle AHC and likewise to the base CD and to the triangle ADC namely to the second and the fourth are taken certaine other equemultiplices that is the base CL and the triangle ALC And it hath bene proued that if the multiplex of the first magnitude that is the base HC do exceede the multiplex of the second that is the base CL the multiplex also of the third that is the triangle AHC excedeth the multiplex of the fourthâ— that is the triangle ALC and if the said base HC be equall to the said baâ—â— CL the triangle also AHC is equall to the triangle ALC and if it be lesse it iâ— lesse Wherfore by the sixt definiâ—ion of the fifth as the first of the foresaid magnitudes is to the second so is the third to the fourth Wherfore as the base BC is to the base CD so is the triangle ABC to the triangle ACD And because by the 41. of the first the parallelogramme EC is double to the triangle ABC and by the same the parallelogramme FC is double to the triangle ACD therfore the parallelogrammes EC and FC are equemultiplices vnto the triangles ABC and ACD But the partes of equemultiplices by the 15. of the fifth haue one and the same proportion with theiâ— equemultiplices Wherfore as the triangle ABC is to the triangle ACD so is the parallelograme EC to the parallelogramme FC And forasmuch as it hath bene demonstrated that as the base BC is to the base CD so is the triangle ABC to the triangle ACD and as the triangle ABC is to the triangle ACD so is the parallelogramme EC to the parallelogramme FC Wherefore by the 11. of the fifth as the base BC is to the base CD so is the parallelogramme EC to the parallelogramme FC The parallelogrammes may also be demonstrated a part by themselues as the triangles are if we describe vpon the bases BG GH and DK KL parallelogrammes vnder the self same altitude that the parallelogrammeâ— geuen are Wherfore triangles and parallelogrammes which are vnder one and the selfe same altitude are in proportioâ— as the base of the one is to the base of the other which was required to be demonstrated Here Flussates addeth this Corollary If two right lines being geuen the one of them be deuided how so euer the rectangle figures contayned vnder the whole line vndeuided and eche of the segmentes of the line deuided are in proportion the one to the other as the segmentes are the one to the other For imaginyng the figures BA and AD in the former description to be rectangled the rectangle figures contayned vnder the whole right lyne AC and the segments of the right line BD which is cuâ— in the poynt C namely the parallelogrammes BA and AD are in proportion the one to the other as the segmeÌ„tes BC and CD are The 2. Theoreme The 2. Proposition If to any one of the sides of a triangle be drawen a parallel right line it shall cut the sides of the same triangle proportionally And if the sides of a triangle be cut proportionally a right lyne drawn from section to section is a parallel to the other side of the triangle SVppose that there be a triangle ABC vnto one of the sides whereof namely vnto BC let there be drawen a parallel line DE cuttyng the sides AC and AB in the pointes E and D. Then I say first that as BD is to DA so is CE to EA Draw a line from B to E also from C to D. Wherfore by the 37. of the first the triangle BDE is equall vnto the triangle CDE for they are set vpon one and the same base DE and are contained within the selfe same parallels DE and BC. Consider also a certaine other triangle ADE Now thinges equall by the 7. of the fifth haue to one selfe thing one and the same proportion Wherfore as the triangle BDE is to the triangle ADE so is the triangle CDE to the triangle ADE But as the triangle BDE is to the triangle ADE so is the base BD to the base DA by the first of this booke For they are vnder one
and the selfe same toppe namely E and therfore are vnder one and the same altitude And by the same reason as the triangle CDE is to the triangle ADE so is the lyne CE to the lyne EA Wherfore by the 11. of the fifth as the line BD is to the line DA so is the line CE to the line EA But now suppose that in the triangle ABC the sides AB AC be cut proportionally so that as BD is to DA so let CE be to EA draw a line from D to E. Then secondly I say that the line DE is a parallel to the lyne BC. For the same order of construction being kept for that as BD is to DA so is CE to EA but as BD is to DA so is the triangle BDE to the triangle ADE by the 1. of the sixt as CE is to EA so by the same is the triangle CDE to the triangle ADE therfore by the 11. of the fifth as the triangle BDE is to the triangle ADE so is the triangle CDE to the triangle ADE Wherfore either of these triangles BDE and CDE haue to the triangle ADE one and the same proportion Wherefore by the 9. of the fifth the triangle BDE is equall vnto the triangle CDE and they are vpon one and the selfe base namely DE. But triangles equall and set vpon one base are also contained within the same parallel lines by the 39. of the first Wherfore the line DE is vnto the line BC a parallel If therfore to any one of the sides of a triangle be drawn a parallel line it cutteth the other sides of the same triangle proportionally And if the sides of a triangle be cut proportionally a right lyne drawen from section to section is parallel to the other side of the triangle which thing was required to be demonstrated ¶ Here also Flussates addeth a Corollary If a line parallel to one of the sides of a triangle do cut the triangle it shall cut of from the whole triangle a triangle like to the whole triangle For as it hath bene proued it deuideth the sides proportionally So that as EC is to EA so is BD to DA wherfore by the 18. of the fifth as AC is to AE so is AB to AD. Wherfore alternately by the 16. of the fifth as AC is to AB so is AE to AD wherefore in the two triangles EAD and CAB the sides about the common angle A are proportional The sayd triangles also are equiangle For forasmuch as the right lynes AEC and ADB do fall vpon the parallel lynes ED and CB therefore by the 29. of the firs◠they make the angles AED and ADE in the triangle ADE equall to the angles ACB and ABC in the triangle ACB Wherefore by the first definition of this booke the whole triangle ABC is like vnto the triangle cut of ADE The 3. Theoreme The 3. Proposition If an angle of a triangle be deuided into two equall partes and if the right line which deuideth the angle deuide also the base the segmentes of the base shall be in the same proportion the one to the other that the other sides of the triangle are And if the segmētes of the base be in the same proportion that the other sides of the sayd triangle are a right drawen from the toppe of the triangle vnto the section shall deuide the angle of the triangle into two equall partes SVppose that there be a triangle ABC and by the 9. of the first let the angle BAC be deuided into two equall partes by the right lyne AD which let cut also the base BC in the point D. Then I say that as the segment BD is to the segment DC so is the side BA to the side AC For by the point C by the 31. of the first draw vnto the line DA a parallel line CE and extende the line BA till it concurre with the line CE in the point E and do make the triangle BEC But the lyne BA shall concurre with the line CE by the 5. peticion for that the angles EBC and BCE are lesse then two right angles For the angle ECB is equall to the outwarde and opposite angle ADB by the 29. of the first And the two angles ADB and DBA of the triangle BAD are lesse then two right angles by the 17. of the first Now forasmuch as vpon the parallels AD and EC falleth the right line AC therefore by the 29. of the first the angle ACE is equall vnto the angle CAD But vnto the angle CAD is the angle BAD supposed to be equall Wherfore the angle BAD is also equall vnto the angle ACE Againe because vpon the parallels AD and EC falleth the right line BAE the outward angle BAD by the 28. of the first is equall vnto the inward angle AEC But before it was prouell that the angle ACE is equall vnto the angle BAD wherfore the angle ACE is equall vnto the angle AEC Wherefore by the 6. of the first the side AE is equall vnto the side AC And because to one of the sides of the triangle BCE namely to EC is drawen a parallel line AD therfore by the 2. of the sixt as BD is to DC so is BA to AE But AE is equall vnto AC therfore as BD is to DC so is BA to AC But now suppose that as the segment BD is to the segment DC so is the side BA to the side AC draw a line from A to D. Then I say that the angle BAC is by the right line AD deuided into two equall partes For the same order of construction remayning for that as BD is to DC so is BA to AC but as BD is to DC so is BA to AE by the 2. of the sixt for vnto one of the sides of the triangle BCE namely vnto the side EC is drawn a parallel line AD. Wherefore also as BA is to AC so is BA to AE by the 11. of the fifth Wherfore by the 9. of the fifth AC is equall vnto AE Wherfore also by the 5. of the first the angle AEC is equall vnto the angle ACE but the angle AEC by the 29. of the first is equall vnto the outward angle BAD and the angle ACE is equall vnto the angle CAD which is alternate vnto him wherefore the angle BAD is equall vnto the angle CAD Wherfore the angle BAC is by the right line AD deuided into two equall partes Wherefore if an angle of a triangle be deuided into two equall partes and if the right line which deuideth the angle cut also the base the segmentes of the base shall be in the same proportion the one to the other that the other sides of the said triangle are And if the segmentes of the base be in the same proportion that the other sides of the sayd triangle are a right line drawen from the toppe of the
produce the lyne CD till it concurre with the line AB produced vnto the pointe D. Then I say that the line BD is a third lyne proportionall with the lines AB and BC which thing is manifest by the corollary of the 8. of this booke The 4. Probleme The 12. Proposition Vnto three right lines geuen to finde a fourth in proportion with them SVppose that the three right lines geuen be A B C. It is required to finde vnto A B C a fourth line in proportioÌ„ with them Let there be taken two right lines DE DF comprehending an angle as it shall happen namely EDF And by the 2. of the first vnto the line A put an equall line DG And vnto the line B by the same put an equall line GE. And moreouer vnto the line C put an equall line DH Then draw a line froÌ„ G to H. And by the poynt E by the 31. of the first draw vnto the line GH a parallell line EF. Now forasmuch as vnto one of the sides of the triangle DEF namely vnto y side EF is drawen a parallell line GH therefore by the 2. of the sixt as the line DG is to the line GE so is the line DH to the line HF. But the line DG is equall vnto the line A and the line GE is equall vnto the line B and the line DH vnto the line C. Wherfore as the line A is vnto the line B so is the line C vnto the line HF. Wherfore vnto the three right lines geuen A B C is found a fourth line HF in proportion with them which was required to be done ¶ An other way after Campane Suppose that there be three right lines AB BC and BD. It is required to adde vnto them a fourth line in proportion with them Ioyne AB the first with BD the third in such sort that they both make one right line namely AD. And vpon the said lyne AB erect from the point B the second line BC making an angle at all aduentures And draw a line from A to C. Then by the point D draw the lyne DE parallel to the line AC which produce vntill it concurre in the point E with the line CB being likewise produced to the point E. Then I say that the line BE is the fourth line in proportion with the lines AB BC and BD so that as AB is to BC so is BD to BE. For forasmuch as by the 15 and 29. of the first the two triangles ABC and DBE are equiangle therfore by the 4. of this booke AB is to BC as BD is to BE which was required to be done The 5. Probleme The 13. Proposition Vnto two right lines geuen to finde out a meane proportionall SVppose the two right lines geuen to be AB and BC. It is required betwene thâ—se two lines AB and BC to finde out a meane line proportionall Let the lines AB and BC be so ioyned together that they both make one right line namely AC And vpon the line AC describe a semicircle ADC and from the poynt B raise vp vnto the line AC by the 11. of the first a perpendicular line BD cutting the circumference in the point D and draw a line from A to D and an other from D to C. Now forasmuch as by the 31. of the third the angle in the semicircle ADC is a right angle and for that in the rectangle triangle ADC is drawen from the right angle vnto the base a perpendicular line DB therefore by the Corollary of the 8. of the sixt the line DB is a meane proportionall betwene the segmeÌ„tes of the base AB BC. Wherefore betwene the two right lines geuen AB BC is found a meane proportionall DB which was required to be done ¶ A Proposition added by Pelitarius A meane proportionall beyng geuen to finde out in a line geuen the two extremes Now it behoueth that the meane geuen be not greater then the halfe of the lyne geuen Flussates putteth this Proposition added by Pelitarius as a corollary following of this 13. proposition The 9. Theoreme The 14. Proposition In equall parallelogrammes which haue one angle of the one equall vnto one angle of the other the sides shall be reciprokall namely those sides which containe the equall angles And if parallelogrammes which hauing one angle of the one equal vnto one angle of the other haue also their sides reciprokal namely those which contayne the equall angles they shall also be equall SVppose that there be two equall Parallelogrammes AB and BC hauing the angle B of the one equall vnto the angle B of the other And let the lines DB and DE be set directly in such sort that they both make one right line namely DE. And then by the 14. of the first shall the lines FB and BG be so set that they shall make also one right line namely GF Then I say that the sides of the parallelogrammes AB and BC which containe the equall angles are reciprocally proportionall that is as BD is to BE so is GB to BF Make complete the parallelogramme FE by producing the sides AF and CE till they concurre in the poynt H. Now forasmuch as the parallelogramme AB is by supposition equall vnto the parallelogramme BC and there is a certaine other parallelogramme FE therfore by the 7. of the fift as the parallelograÌ„me AB is to the parallelogramme FE so is the parallelogramme BC to the parallelogramme FE But as the parallelogramme AB is to the parallelogramme FE so is the side DB to the side BE by the first of this booke And by the same as the parallelogramme BC is to the parallelogramme FE so is the side GB to the side BF Wherefore also by the 11. of the fift as the side DB is to the side BE so is the side GB to the side BF Wherefore in the parallelogrammes AB and BC the sides which containe the equall angles are reciprokally proportionall which was first required to be proued But now suppose that the sides about the equall angles be reciprokally proportionall so that as the side DB is to the side BE so let the side GB be to the side BF Then I say that the parallelogramme AB is equall vnto the parallelogramme BC. For for that as the side DB is to the side BE so is the side GB to the side BF but as the side DB is to the side BE so by the 1. of the sixt is the parallelogramme AB to the parallelogramme FE and as the side GB is to the side BF so is the parallelogramme BC to the parallelogramme FE Wherefore also by the 11. of the fift as the parallelogramme AB is to the parallelograÌ„me FE so is the parallelogramme BC to the same parallelogramme FE Wherefore the parallelogramme AB is equall vnto the parallelogramme BC by the 9. of the fift Wherefore in equall and equiangle parallelogrammes the sides
and the same proportion wherfore by the 9. of the fifth the figure NH is equal vnto the figure SR And it is vnto it like and in like sort situate But in like and equall rectiline figures beyng in like sort situate the sides of like proportion on which they are described are equall Wherfore the line GH is equall vnto the line QR And because as the lyne AB is to the line CD so is the line EF to the line QR but the line QR is equall vnto the line GH therfore as the line AB is to he line CD so is the line EF to the line GH If therefore there be foure right lines proportionall the rectiline figures also described vpon them beyng like and in lyke sort situate shall be proportionall And if the rectiline figures vpon them described beyng like and in like sort situate be proportionall those right lines also shall be proportional which was required to be proued An Assumpt And now that in like and equall figures being in like sort situate the sides of like proportion are also equall which thing was before in this proposition taken as graunted may thus be proued Suppose that the rectiline figures NH and SR be equall and like and as HG is to GN so let RQ be to QS and let GH and QR be sides of like proportion Then I say that the side RQ is equall vnto the side GH For if they be vnequall the one of them is greater then the other let the side RQ be greater then the side HG And for that as the line RQ is to the line QS so is the line HG to the line GN and alternately also by the 16. of the fifth as the line RQ is to the line HG so is the line QS to the lyne GN but the line RQ is greater then the line HG Wherfore also the line QS is greater then the line GN Wherefore also the figure RS is greater then the figure HN but by supposition it is equall vnto it which is impossible Wherfore the line QR is not greater then the line GH In like sorte also may we proue that it is not lesse then it wherfore it is equall vnto it which was required to be proued Flussates demonstrateth this second part more briefly by the first corollary of the â—0 of this boke thus Forasmuch as the rectiline figures are by supposition in one and the same proportion and the same proportion is double to the proportion of the sides AB to CD and EF to GH by the foresaid corollary the proportion also of the sides shall be one and the selfe same by the 7. common sentence namely the line AB shall be vnto the line CD as the line EF is to the line GH The 17. Theoreme The 23. Proposition Equiangle Parallelogrammes haue the one to the other that proportion which is composed of the sides Flussates demonstrateth this Theoreme without taking of these three lines K L M after this maner Forasmuch as sayth he it hath bene declared vpon the 10. definition of the fift booke and â—ift definition of this booke that the proportions of the extremes consist of the proportions of the meanes let vs suppose two equiangle parallelograÌ„mes ABGD and GEZI and let the angles at the poynt G in eyther be equall And let the lines BG and GI be set directly that they both make one right line namely BGI. Wherefore EGD also shall be one right line by the conuerse of the 15. of the first Make complete the parallelogramme GT Then I say that the proportion of the parallelogrammes AG GZ is composed of the proportions of the sides BG to GI and DG to GE. For forasmuch as that there are three magnitudes AG GT and GZ and GT is the meane of the sayd magnitudes and the proportion of the extremes AG to GZ consisteth of the meane proportions by the 5. definition of this booke namely of the proportion of AG to GT and of the proportion GT to GZ But the proportion of AG to GT is one and the selfe same with the proportion of the sides BG to GI by the first of this booke And the proportion also of GT to GZ is one and the selfe same with the proportion of the other sides namely DG to GE by the same Proposition Wherefore the proportion of the parallelogrammes AG to GZ consisteth of the proportions of the sides BG to GI and DC to GE. Wherefore equiangle parallelogrammes are the one to the other in that proportion which is composed of theyr sides which was required to be proued The 18. Theoreme The 24. Proposition In euery parallelogramme the parallelogrammes about the dimecient are lyke vnto the whole and also lyke the one to the other SVppose that there be a parallelogramme ABCD and let the dimecient therof be AC and let the parallelogrammes about the dimecient AC be EG and HK Then I say that either of these parallelogrames EG and HK is like vnto the whole parallelogramme ABCD and also are lyke the one to the other For forasmuch as to one of the sides of the triangle ABC namely to BC is drawen a parallel lyne EF therfore as BE is to EA so by the 2. of the sixt is CF to FA. Agayne forasmuch as to one of the sides of the triangle ADC namely to CD is drawen a parallel lyne Fâ— therefore by the same as CF is to FA so is DG to GA. But as CF is to FA so is it proâ—ued that BE is to EA Wheâ—fore as BE is to EA so by the 11. of the fifth â— is DG to GA. Wherfore by composition by the 18. of the fifth as BA is to AEâ— so is DA to AG. And alternately by the 16. of the fifth as BA is to AD so is EA to AG. Wherfore in the parallelogrammesâ— ABCD and EG the sides which are about the common angle BAD are proportionall And because the line GF is a parallel vnto the lyne DCâ— therfore the angle AGF by the 29â— of theâ— first is equall vnto the angle ADC â— the angle GFA equall vnto the angle DCA and the angle DAC is common to the two triangles ADC and AFG Wherfore the triangle DAC is equiangle vnto the triangle AGF And by the same reason the triangle ABC is equiangle vnto the triangle AEF Wherfore the whole parallelogramme ABCD is equiangle vnto the parallelograÌ„me EG Wherfore as AD is in proportion to DC so by the 4. of the sixth is AG to GF and as DC is to CA so is GF to FA. And as AC is to CB so is AF to FE And moreouer as CB is to BA so is FE to EA And forasmuch as it is proued that as Dâ— is to CA so is GF to FA but as AC is to Câ— so is AF to FE Wherfore of equalitie by the 22. of the fifth as DC is to CB so is GF to FE Wherefore in the parallelogrammes ABCD and
triangle ABC is by the 6. of the sixt equiangle vnto the trianglâ— DCE Wherefore the angle ABC is equall vnto the angle DCE And it is proued that the angle ACD is equall vnto the angle BAC Wherefore the whole angle ACE is equall vnto the two angles ABC and BAC Put the angle ACB common to them both Wherefore the angles ACE and ACB are equall vnto the angles CAB ACB CBA But the angles CAB ACB and CBA are by the 32. of the first equall vnto two right angles Wherefore also the angles ACE and ACB are equall to two right angles Now then vnto the right line AC and vnto the point in it C are drawen two right lines BC and CE not on one and the same side making the side angles ACE ACB equall to two right angles Wherefore the lines BC and CE by the 14. of the first are set directly and do make one right line If therefore two triangles be set together at one angle hauing two sides of the one proportionall to two sides of the other so that their sides of like proportion be also parallels then the sides remayning of those triangles shall be in one right line which was required to be proued Although Euclide doth not distinctly set forth the maner of proportion of like rectiline figures as he did of lines in the 10. PropositioÌ„ of this Booke and in the 3. following it yet as Flussates noteth is that not hard to be done by the 22. of thys Booke â—or two like rectiline figures being geuen to finde out a third proportionallâ— also betwene two rectiline superficieces geuen to finde out a meane proportionall which we before taught to do by Pelitarius after the 24. Proposition of this booke and moreouer three like rectiline figures being geuen to finde out a fourth proportionall like and in like sort described and such kinde of proportions are easie to be found out by the proportions of lines As thus If vnto two sides of like proportion we should find out a third proportionall by the 11. of this bokeâ— the rectiline figure described vpon that line shall be the third rectiline figure proportionall with the two first figures geuen by the 22. of thys booke And if betwene two sides of like proportion be taken a meane proportionall by the 13. of thys Booke the rectiline â—igure described vpon the sayd meane shall likewise be a meane proportionall betwene the two rectiline figures geueÌ„ by the same 22. of the sixt And so if vnto three sides geâ—en be found out the fourth side proportionall by the 12. of this booke the rectiline â—igure described vpon the sayd fourth line shall be the fourth rectiline figure proportionall For if the right lines be proportionall the rectiline figures described vpon them shall also be proportionall so that the said rectiline â—igures be like in like sort described by the said 22. of the sixt The 23. Theoreme The 33. Proposition In equal circles the angles haue one and the selfe same proportion that the circumfereÌ„ces haue wherin they coÌ„sist whether the angles be set at the centres or at the circumferences And in like sort are the sectors which are described vppon the centres SVppose the equall circles to be ABC and DEF whose centres let be G and H and let the angles set at their centres G and H be BGC and EHF and let the angles set at their circumferences be BAC and EDF Then I say that as the circumference BC is to the circumference EF so is the angle BGC to the angle EHF and the angle BAC to the angle EDF and moreouer the sector GBC to the sector HEF Vnto the circumference BC by the 38. of the third put as many equall circumferences in order as you will namely CK and KL and vnto the circumference EF put also as many equall circumferences in number as you will namely FM and MN And drawe these right lines GK GL HM and HN. Now forasmuch as the circumferences BC CK and KL are equall the one to the other the angles also BGC and CGK and KGL are by the 27. of the third equall the one to the other Therfore how multiplex the circumference BL is to the circumference BC so multiplex is the angle BGL to the angle BGC And by the same reason also how multiplex the circumference NE is to the circumference EF so multiplex is the angle NHE to the angle EHF Wherfore if the circumference BL be equal vnto the circumference EN the angle BGL is equall vnto the angle EHN and if the circumference BL be greater then the circumference EN the angle BGL is greater then the angle NHE and if the circumference be lesse the angle is lesse Now then there are foure magnitudes namely the two circumferences BC and EF and the two angles that is BGC and EHF and to the circumference BC and to the angle BGC that is to the first and third are takeÌ„ equemultiplices namely the circumference BL and the angle BGL and likewise to the circumference EF and to the angle EHF that is to the second and fourth are taken certayne other equemultiplices namely the circumference EN and the angle EHN. And it is proued that if the circumference BL exceede the circumference EN the angle also BGL exceedeth the angle EHN. And if the circumference be equall the angle is equall and if the circumfereÌ„ce be lesse the angle also is lesse Wherfore by the 6. definition of the fifth as the circumference BC is to the circumference EF so is the angle BGC to the angle EHFâ— But as the angle BGC is to the angle EHF so is the angle BAC to the angle EDF for the angle BGC is double to the angle BAC and the angle EHF is also double to the angle EDF by the 20. of the third Wherfore as the circumference BC is to the circumference EF so is the angle BGC to the angle EHF and the angle BAC to the angle EDF Wherfore in equall circles the angles are in one and the selfe same proportion that their circumferences are whether the angles be set at the centres or at the circumferences which was required to be proued I say moreouer that as the circumference BC is to the circumference EF so is the sector GBC to the sector HEF Draw these lines BC and CK And in the circumferences BC and CK take pointes at all aduentures namely P and O. And draw lines from B to P and from P to C from C to O and from O to K. And forasmuch as by the 15. definition of the first the two lines BG and GC are equall vnto the two lines CG and GK and they also comprehend equall angles therfore by the 4. of the first the base BC is equall vnto the base CK the triangle GBC is equall vnto the triangle GCK. And seing that the circumference BC is equall vnto the circumference CK therefore the circumference remayning of the whole circle ABC
proportion that the first of the three lines put is to the 〈◊〉 â—or tâ—e 〈◊〉 line to the third namely the line AE to the line EB is in double propoâ—tion that it is to the second by the 10. deâ—inition of the fiâ—t ¶ The second Corollary Hereby may we learne how from a rectiline â—igure geuen to take away a part appointed leaâ—ing the rest of the rectiline â—igure like vnto the whole For if froÌ„ the right line AB be cut of a part appoynted namely EB by the 9. of this booke as the line AE is to the line EB so is the rectiline â—igure described of the line AF to the rectiline figure described of the line FB the sayd â—igures being supposed to be like both the one to the other and also to the rectiline â—igure described of the line AB and being also in like sort situate Wherfore taking away â—rom the rectiline â—igure described of the line AB the rectiline figure described of the line FB the residue namely the rectiline figure described of the line AF shall be both like vnto the whole rectiline â—igure geuen described of the line AB and in like sort situate ¶ The third Corollary To compose two like rectiline â—igures into one rectiline figure like and equall to the same figures Let their sides of like proportioÌ„ be set so that they make a right angle as the lines AF and FB are And vpoÌ„ the line subtending the said angle namely the line AB describe a rectiline â—igure like vnto the rectiline figures geuen and in like sort situate by the 18. of this booke and the same shall be equall to the two rectiline figures geuen by the 31. of this booke ¶ The second Proposition If two right lines cut the one the other obtuseangled wise and from the endes of the lines which â—ut the one the other be drawen perpendicular lines to either line the lines which are betwene the endes and the perpendicular lines are cut reciprokally Suppose that there be two right lines AB and GD cutting the one the other in the point E and making an obtuse angle in the section E. And from the endes of the lines namely A and G let there be drawen to either line perpendicular lines namely from the point A to the line GD which let be AD and from the point G to the right line AB which let be GB Then I say that the right lines AB and GD do betwene the end A and the perpendicular B and the end G and the perpendicular D cut the one the other reciprokally in the point E so that as the line AE is to the line ED so is the line GE to the line EB For forasmuch as the angles ADE and GBE are right angles therfore they are equall But the angles AED and GEB are also equall by the 15. of the first Wherefore the angles remayning namely EAD EGB are equall by the Corollary of the 32. of the first Wherefore the triangles AED and GEH are equiangle Wherfore the sides about the equall angles shall be proportionall by the 4. of the sixt Wherfore as the line AE is to the line ED so is the line GE to the line EB If therefore two right lines cut the one the other obtuseangled wife c which was required to be proued ¶ The third Proposition If two right lines make an acute angle and from their endes be drawen to ech line perpendicular lines cutting them the two right lines geuen shall be reciprokally proportionall as the segmentes which are about the angle Suppose that there be two right lines AB and GB making an acute angle ABG And from the poyntes A and G let there be drawen vnto the lines AB and GB perpendicular lines AC and GE cutting the lines AB and GB in the poyntes E and â— Then I say that the lines namely AB to GB are reciprokally proportionall as the segmentes namely CB to EB which are about the acute angle B. For forasmuch as thâ— right angles ACB and GER are equall and the angleâ— ABG is common to the triangles ABC and GBE â— therefore the angles remayning BAC and EGB are equall by the Corollary of the 32. of the first Wherfore the triangles ABC and GBE are equiangle Wherefore the sideâ— about the equall angles are proportionall by the 4. of the sixe â— so that as the line AB is to the line FC so is the line GB to the line BE. Wherefore alternately as the line AB is to the line GB so is the line CB to the line BE. If therefore two right lines makâ— aâ—â—câ—te angleâ— câ— which was required to be proued â— The fourth Propositionâ— If in a circle be drawen two right lines cutting the one the other the sections of the one to the sections of the other shall be reciprokally proportionall In the circle AGB let these two right lines 〈…〉 one the other in the poynt E. Thâ—â— I say that reciprokally 〈◊〉 â—hâ— line AE is to the line ED so is the line GE to the line EB For forasmuch as by the 35. of the third the rectangle figure contayned vnder the lines AE and EB is equall to the rectangle figure contayned vnder the lines GE and ED but in equall rectangle parallelogrammes the sides about the equall angles are reciprokall by the 14. of the sixt Therefore the line AE is to the line ED reciprokally as the line GE is to the line EB by the second definition of the sixt If therefore in a circle be drawen two right lines c which was required to be proued ¶ The fift Proposition If from a poynt geuen be drawen in a plaine superâ—icies two right lines to the concaue circumference of a circle they shall be reciprokally proportionall with their partes takeÌ„ without the circle And moreouer a right line drawen from the sayd poynt touching the circle shall be a meane proportionall betwene the whole line and the vtter segment Suppose that there be a circle ABD and without it take a certayne poynt namely G. And from the point G drawe vnto the concaue circumference two right lines GB and GD cutting the circle in the poyntes C and E. And let the line GA touch the circle in the point A. TheÌ„ I say that the lines namely GB to GD are reciprokally as their parts taken without the circle namely as GC to GE. For forasmuch as by the Corollary of the 36. of the third the rectangle figure contayned vnder the lines GB and GE is equall to the rectangle figure contayned vnder the lines GD and GC therefore by the 14. of the sixt reciprokally as the line GB is to the line GD so is the line GC to the line GE for they are sides contayning equall angles I say moreouer that betwene the lines GB and GE or betwene the lines GD and GC the touch line GA is a meane proportionall For forasmuch as the rectangle
the parts added together are not equal to the whole nor make the whole but make either more or lesse Wherefore of imperfect numbers there are two kindes the one is called abundanâ— or a building the other 〈◊〉 or wanting A number abunding is that whose partes being all added together make more then the wâ—â—ds number whose partes they are as 12. is an abundant number For all the parteâ— of 12. namely 6.4.3.2 and 1. added together make 16 which are more then 12. Likewise 18. is a number abunding all his partâ— nâ—mely 9.6.3.2 and 1. added together make 20. which are more then 18 and so of others A number diminute or wanting is that whose partes being all added together make lesse then the whole or number whose partes they are As 9. is a diminute or wanting number for all his partes namelyâ— 3. and 1. moe partes he hath not added together make onely 4 which are lesse then 9. Also 26. is a diminute nuÌ„ber all his partes namely 13.2.1 added togethâ—r make onely 16 which is a number much lesse then 26. And so of such like CAmpane and Flussates here adde certayne common sentences some of whichâ— for that they are in these three bookes following somtimes alledged I thought good here to annexe 1 The lesse part is that which hath the greater denomination and the greater part is that which hath the lesse denomination As the numbers 6. and 8. are either of them a part of the number 24 6. is a fourth part 4. times 6. is 24 and 8. is a third part 3. times 8. is 24. Now forasmuch as 4 which denominateth what part 6. is of 24 is greater then 3. which denominateth what part 8. is of 24. therefore is 6. a lesse part of 24â— then is 8. and so is 8. a greater part of 24. theÌ„ 6. is And so in others 2 Whatsoeuer numbers are equemultiplices to one the selfe same nuÌ„ber or to equall numbers are also equall the one to the other As if vnto the number 3 be taken two numbers containing the same number foure times that is being equemultiplices to the same number three the sayd two numbers shal be equall For 4. times 3. will euer be 12. So also will it be if vnto the two equal numbers 3. 3. be taken two numbers the one coÌ„taining the one number 3. foure times the other containing the other number 3. also foure times that is being equemultiplices to the equall numbers 3. and 3. 3 Those numbers to whome one and the selfe same number is equmultiplex or whose equemultiplices are equall are also equall the on to the other As if the number 18. be equemultiplex to any two numbers that is contayne any two numbers twise thrise fower times c As for example 3. times then are the sayd two numbers equall For 18. deuided by 3. will euer bring forth 6. So that that diuision made twise will bring forth 6. and 6. two equall numbers So also would it follow if the two numbers had equall equemultiplices namely if 18. and 18. which are equall numbers contayned any two numbers 3. times 4 If a number measure the whole and a part taken away it shall also measure the residue As if from 24. be taken away 9. there remaineth 15. And for as much as the number 3 measureth the whole number 24 also the number takeÌ„ away namely 9. it shall also measure the residue which is 15â— For 3. measureth 15 by fiue fiue times 3. is 15. And so of others 5 If a number measure any number it also measureth euery number that the sayd number measureth As the number 6. measuring the number 12. shall also measure all the numbers that 1â— measurethâ— as the numbers 24.36.48.60 and so forth which the number 12. doth measure by the numberâ— 2.3.4 and 5. And for as much as the number 12. doth measure the numbers 24.36.48 and 60. And the nuÌ„ber 6 doth measure the number 12. namely by 2. It followeth by this commoÌ„ sentence that the number 6. measureth eche of thâ—se numbers 24. 36.48 and 60. And so of others 6 If a number measure two numbers it shall also measure the number composed of them As the number 3 measureth these two numbers 6. and 9â— it measureth 6. by 2â— and 9. by 3. And therefore by this common sentence it measureth the number 15. which is composed of the numbers 6. and 9 namely it measureth it by 5. 7 If in numbers there be proportions how manysoeuer equall or the selfe same to one proportion they shall alâ—o be equall or the selfe same the one to the other As yf the proportion of the number 6. to the number 3. be as the proportion of the number 8. to the number 4 if also the proportion of the number 10. to the number 5. be as the proportion of the number 8. to the number 4 then shall the proportion of the number 6. to the number 3. be as the proportion of the number 10. is to the number 5 namely eche proportion is duple And so of others Euclide in his â— booke the 11. proposition demonstrated this also in continuall quantitie which although as touching that kinde of quantitie it might haue bene put also as a principle as in numbers he taketh it yet for that in all magnitudes theyr proportion can not be expressed as hath before bene noted shal be afterward in the tenth booke more at large made manifest therefore he demonstrateth it there in that place and proueth that it is true as touching all proportions generally whither they be rational or irrationall ¶ The first Proposition The first Theoreme If there be geuen two vnequall numbers and if in taking the lesse continually from the greater the number remayning do not measure the number going before vntill it shall come to vnitie then are those numbers which were at the beginning geuen prime the one to the other SVppose that there be two vnequal numbers AB the greater and CD the lesse and from AB the greater take away CD the lesse as oâ—ten as you can leauing FA and from CD take away FA as often as you can leauing the number GC And from FA take away GC as often as you can and so do continually till there remayne onely vnitie which let be HA. Then I say that no number measureth the numbers AB and CD For if it be possible let some number measure them and let the same be E. Now CD measuring AB leaueth a lesse number then it selfe which let be FA â— And FA measuring DC leaueth also a lesse then it selfe namely GC And GC measâ—ring FA leaueth vnitie HA. And forasmuch as the number E measureth DC and the number CD measureth the number BF therfore the number E also measureth BF and it measureth the whole number BA wherfore it also measureth that which remayneth namely the number FA by the
other number DE is of an other nuÌ„ber F and let AB be lesse then DE. Then I say that alternately also what part or partes AB is of DE the selfe same partes or part is C of F. Forasmuch as what partes AB is of C the selfe same partes is DE of F therefore how many partes of C there are in AB so many partes of F also are there in DE. Deuide AB into the partes of C that is into AG and GB And likewise DE into the partes of F that is DH and HE. Now then the multitude of these AG and GB is equall vnto the multitude of these DH and HE. And forasmuch as what part AG is of C the selfe same part is DH of F therefore alternately also by the former what part or partes AG is of DH the selfe same part or partes is C of F. And by the same reason also what part or partes GB is of HE the same part or partes is C of F. Wherefore what part or partes AG is of DH the selfe same part or partes is AB of DE by the 6. of the seuenth But what part or partes AG is of DH the selfe same part or partes is it proued that C is of F. Wherefore what partes or part AB is of D E the selfe same partes or part is C of F which was required to be proued ¶ The 9. Theoreme The 11. Proposition If the whole be to the whole as a part taken away is to a part taken away then shall the residue be vnto the residue as the whole is to the whole SVppose that the whole number AB be vnto the whole number CD as the part takeÌ„ away AE is to the part takeÌ„ away CF. TheÌ„ I say that the residue EB is to the residue FD as the whole AB is to the whole CD For forasmuch as AB is to CD as AE is to CF therfore what part or partes AB is of CD the selfe same part or partes is AE of CF. Wherfore also the residue EB is of the residue FD by the 8. of the seuenth the selfe same parte oâ— partes that AB is of CD Wherefore also by the 21. definition of this booke as EB is to FD so is AB to CD which was required to be proued ¶ The 10. Theoreme The 12. Proposition If there be a multitude of numbers how many soeuer proportionall as one of the antecedentes is to one of the consequentes so are all the antecedentes to all the consequentes SVppose that there be a multitude of nuÌ„bers how many soeuer proportional namely A B C D so that as A is to B so let C be to D. Then I say that as one of the antecedentes namely A is to one of the consequentes namely to B or as C is to D so are all the antecedentes namely A and C to all the consequentes namely to B and D. For forasmuch as by supposition as A is to B so is C to D therfore what parte or partes A is of B the selfe same part or partes is C of D by the 21. definition of this booke wherefore alternately what part or partes A is of C the selfe same parte or partes is B of D by the ninth and tenth of the seuenth wherefore both these numbers added together A and C are of both these numbers B and D added together the selfe same part or partes that A is of B by the 5. and 6. of the seuenth wherfore by the 21. definition of the seuenth as one of the antecedents namely A is to one of the consequentes namely to B so are all the antecedentes A and C to all the consequentes B D. Which was required to be proued ¶ The 11. Theoreme The 13. Proposition If there be foure numbers proportionall then alternately also they shall be proportionall SVppose that there be foure numbers proportional A B C D so that as A is to B so let C be to D. Then I say that alternately also they shal be proportional that is as A is to C so is B to D. For forasmuch as by supposition as A is to B so is C to D therfore by the 21. definition of this booke what part or partes A is of B the selfe same part or partes is C of D. Therfore alternately what part or partes A is of C the selfe same part or partes is B of D by the 9. of the seuenth also by the 10. of the same wherfore as A is to C so is B to D by the 21. definition of this booke which was required to be proued Here is to be noted that although in the foresayd example and demonstration the number A be supposed to be lesse then the number B and so the number C is lesse then the number D yet will the same serue also though A be supposed to be greater then B wherby also C shall be greater then D as in thâ—s example here put For for that by supposition as A is to B so is C to D and A is supposed to be greater then B and C greater then D therefore by the 21. definition of this Booke how multiplex A is to B so multiplex is C to D and therefore what part or partes B is of A the selfe same part or partes is D of C. Wherefore alternately what part or partes B is of D the selfe same part or partes is A of C and therefore by the same definition B is to D as A is to C. And so must you vnderstand of the former Proposition next going before ¶ The 12. Theoreme The 14. Proposition If there be a multitude of numbers how many soeuer and also other numbers equall vnto them in multitude which being compared two and two are in one and the same proportion they shall also of equalitie be in one and the same proportion SVppose that there be a multitude of numbers how many soeuer namely A B C and let the other numbers equall vnto them in multitude be D E F which being compared two and two let be in one and the same proportion that is as A to B so let D be to E and as B is to C so let E be to F. Then I say that of equalitie as A is to C so is D to F. For forasmuch as by supposition as A is to B so is D to E therefore alternately also by the 13 of the seuenth as A is to D so is B to E. Againe for that as B is to C so is E to F therfore alternately also by the self same as B is to E so is C to F. But as B is to E so is A to D. VVherfore by the seuenth common sentence of the seuenth as A is to D so is C to F. Wherâ—ore alternately by the 13. of the seuenth as A is to C so is D to F which
was required to be demonstrated After this Proposition Campane demonstrateth in numbers these foure kindes of proportionalitie namely proportion conuerse composed deuided and euerse which were in continual quantitie demonstrated in the 4. 17. 18. and 19. propositions of the fift booke And first he demonstrateth conuerse proportion in this maner But if A be greater then B C also is greater then D and what part or partes B is of A the selfe same part or partes is D of C. Wherefore by the same definition as B is to A so iâ— D to C which was required to be proued Proportionalitie deuided is thus demonstrated Suppose that the number AB be to the number B as the number CD is to the number D. Then I say that deuided also as A is to B so is C to D. For for that as AB is to B so is CD to D therâ—fore alternately by the 14. of this booke as AB is to CD so is B to D. Wherefore by the 11. of this booke as AB is to CD so is A to C. Wherefore as B is to D so is A to C and for that as A is to C so is B to D theâ—efore alternately as A is to B so is C to D. Proportionalitie composed is thus demonstrated If A be vnto B as C is to D then shall AB be to B as CD is to D. For alternately A is to C as B is to D. Wherefore by the 13. of this booke as AB namely all the antecedentes are to CD namely to all the consequentes so is B to D namely one of the antecedentes to one of the consequentes Wherfore alteâ—nately as AB is to B so is CD to D. Euerse proportionalitie is thus proued Supposâ— that AB be to B as CD is to D then shall AB be to A as CD is to C. For alternately AB is to CD aâ— B is to D. Wherefâ—râ— by the 13. of this booâ—â— Aâ— is â— CD as A is to C. Wherefore alternately AB iâ— to A aâ— CD iâ— to C whiâ—h was required to be proued ¶ A proportion here added by Campane If the proportion of the first to the second be as the proportion of the third to the foâ—rth and if the proportiâ—n of â—he fift to the second be as the propâ—rtion of the sixt to the fourth then the proportion of the first and the fifth taken together shall be to the second as the proportion â—f the third and the sixt taken together to the fourth And after the same maner may you proue the conuerse of this Proposition If B be to A as D is to Câ— and if also B be vnto E as D is to F Then shall B be to AE as D is to CF. For by conuerse proportionalitie A is to B as C is to D. Wherefore of equalitie A is to E as C is to F. Wherefore by composition A and E are to E as C and F are to F. Wherefore conuersedly E is to A and E as F is to C and F. But by supposition B is to E as D is to F. Wherefore agayne by Proportion of equalitie B is to A and E as D is to C and F which was required to be proued A Corollary By this also it is manifest that if the proportion of numbers how many soeuer vnto the first be as the proportion of as many other numbers vnto the second then shall the proportion of the numbers composed of all the numbers that were antecedentes to the first be to the first as the number composed of all the numbers that were antecedentes to the second is to the second And also conuersedly if the proportion of the first to nuÌ„bers how many soeuer be as the proportion of the second to as many other numbers then shall the proportion of the first to the number composed of all the numbers that were consequentes to it selfe be as the proportion of the second to the number composed of all the numbers that were consequenâ—es to it selfe ¶ The 13. Theoreme The 15. Proposition If vnitie measure any number and an other number do so many times measure an other number vnitie also shall alternately so many times measure the third number as the second doth the fourth SVppose that vnitie A do measure the number BC and let an other nuÌ„ber D so many times measure some other nuÌ„ber namely EF. Then I say that alternately vnitie A shall so many times measure the number D as the number BC doth measure the number EF. For forasmuch as vnitie A doth so many times measure BC as D doth EF therefore how many vnities there are in BC so many numbers are there in EF equall vnto D. Deuide I say BC into the vnities which are in it that is into BG GH and HC And deuide likewise EF into the numbers equall vnto D that is into EK KL and LF Now then the multitude of these BG GH and HC is equall vnto the multitude of these EK KL LF And forasmuch as these vnities BG GH and HC are equall the one to the other and these numbers EK KL LF are also equall the one to the other and the multitude of the vnities BG GH and HC are equall vnto the multitude of the numbers EK KL LF therefore as vnitie BG is to the number EK so is vnitie GH to the number KL and also vnitie HC to the number LF Wherfore by the 12â— of the seueâ—th as one of the antecedeâ—tâ—â—â—s to one of the consequentes so are all the antecedenâ—es to all the consequentes Wherfore as vnitie BG is to the number EK so is the number BC to the number EF. But vnitie BG is equall vnto vnitie A and the number EK to the number D. VVherefore by the 7. common sentence as vnitie A is to the number D so is the number BC to the number EF. VVherefore vnitie A measureth the nuÌ„ber D so many times as BC measureth EF by the 21 definition of this booke which was required to be proued ¶ The 14. Theoreme The 16. Proposition If two numbers multiplying them selues the one into the other produce any numbers the numbers produced shall be equall the one into the other SVppose that there be two numbers A and B and let A multiplying B produce C and let B multiplying A produce D. Thâ—n I say that the number Câ— equall vnto the nâ—mber D. Take any vnitie namelyâ— E. And forasmuch as A multiplying B produced C therefore B measureth C by the vnities which are in A. And vnitie E measureth the number A by those vnities which are in the number A. VVhereâ—ore vnitie E so many times measureth A as B measureth C. VVherefore alternately by the 15. of the seuenth vnitie E measureth the number B so many times as A measureth C. Againe for that B multiplying A produced D therefore A measureth D by thâ— vnities which are in B. And vnitie E
measureth B by the vnities which are in B. VVherefore vnitie E so many times measureth the number B as A measureth D. But vnitie E so many times measureth the number B as A measureth C. VVherefore A measureth either of these numbers C and D a like VVherefore by the 3. common sâ—ntence of this booke C is equall vnto D which was required to be demonstrated The 15. Theoreme The 17. Proposition If one number multiply two numbers and produce other numbers the numbers produced of them shall be in the selfe same proportion that the numbers multiplied are SVppose that the number A multiplieng two numbâ—rs B and C do produce the numbers D and E. Then I say that as B is to C so is D to E. Take vnitie namely F. And â—orasmuch as A multiplieng B produced D therfore B measureth D by those vnities that are in A. And vnitie F measureth A by thâ—sâ— vâ—itiâ—â— whih are in A. Wherfore vnitie F so many times measureth the number A as B measureth D. VVherfore as vnitie I is to the number A. so is the number B to the number D by the 21 definition of this booke And by the same reason as vnitie F is to the number A so is the number C to the number E wherefore also by the 7. common sentence of this booke as B is to D so is C to E. VVherfore alternately by the 15. of the seuenth as B is to C so is D to E. If therfore one number multiply two numbersâ— and produce other numbers the numbers produced of them shall be in the selfe same proportion that the numbers multiplied are which was required to be proued Here Fluâ—â—tes addeâ—h thiâ— Coâ—ollary If two numberâ— hauing one and the samâ— proportiâ—â— with two other numbers do multiply thâ— oâ—e the other alternately and produce any numbers the numbers produced of them shall be equall the one to the other Suppose that there be two numberâ— â— and B and also two other numbers C and D hauing thâ— same proportion that the numbers A and B haue and let the numbers A and B multiply the numberâ— C D alternately that is let A multiplieng D produce F and let B multiplieng C produce E. Then I say that the numbers produced namely E F are equall Let A and B multiply the one the other in such sort that let A multiplieng B produce G and let B multiplieng A produce H Now then the numbers G and H are equal by the 16. of this bookeâ— And forasmuch as A multipliâ—ng the two numbers B and D produced the numbers G and F therfore G is to â— as B is to D by this proposition So likewise B multiplieng the two numbers A and C produced the two numbers H and E. Wherfore by the same H is to E as A is to C. But alternately by the 13. of this booke A is to C as B is to D but as A is to C so is H to E and as â— is to D so is G to â— Wherfore by the seuenth common sentence as H is to E â— so is G to F. Wherfore aâ—ternately by the 13. of this booke H is to G as E is to F. But it is proued that G H are equall Wherfore E and F which haue the same proportion that A and B haue are equall If therefore there be two numbers c. Which was required to be proued ¶ The 16. Theoreme The 18. Proposition If two numbers multiply any number produce other numbers the numbers of them produced shall be in the same proportion that the numbers multiplying are SVppose that two numbers A and B multiplieng the number C doo produce the numbers D and E. Then I say that as A is to B so is D to E. For forasmuch as A multiplieng C produced D therfore C multiplieng A produceth also D by the 16. of this booke And by the same reason C multiplieng B produceth E. Now then one number C multiplieng two numbers A and B produceth the numbers D and E. VVherfore by the 17. of the seuenth as A is to B so is D to E which was required to be demonstrated This Proposition and the former touching two numbers may be extended to numbers how many soeuer So that if one number multiply numbers how many soeuer and produce any numbers the proportion of the numbers produced and of the numbers multiplied shall be one and the selfe same Likewise if numbers how many soeuer multiply one number and produce any numbers the proportion of the numbers producedâ— and of the numbers multiplieng shall be one and the selfe same which thing by this and the former proposition repeted as often as is needefull is not hard to proue ¶ The 17. Theoreme The 19. Proposition If there be foure numbers in proportion the number produced of the first and the fourth is equall to that number which is produced of the second and the third And if the number which is produced of the first and the fourth be equall to that which is produced of the second the third those foure numbers shall be in proportion But now againe suppose that E be equall vnto F. Then I say that as A is to B so is C to D. For the same order of construction remayning still forasmuch as A multiplieng C D produced G and E therfore by the 17. of the seuenth as C is to D so is G to E but E is equall vnto F But if two numbers be equall one number shall haue vnto them onâ— and the same proportion wherfore as G is to E so is G to F. But as G is to E so is C to D. Wherefore as C is to D so is G to F but as G is to F so is A to B by the 18. of the seuenth wherfore as A is to B so is C to D which was required to be proued Here Campane addeth that it is needeles to demonstrate that if one number haue to two numbers one and the same proportion the said two numbers shall be equall or that if they be equal one number hath to them one and the same proportion For saith he if G haue vnto E and F one and the same proportion theÌ„ either what part or partes G is to E the same part or parts is G also of F or how multiplex G is to E so multiplex is G to F by the 21. definition And therfore by the 2 and 3 common sentence the said numbers shall be equall And so conuersedly if the two numbers E and F be equal then shall the numbers E and F be either the selfe same parte or partes of the number G or they shall be equemultiplices vnto it And therfore by the same definition the number G shall haue to the numbers E and F one and the same proportion ¶ The 18. Theoreme The 20. Proposition If there be three numbers in proportion the number
wherefore A is a plaine number and the sides therof are D and F by the 17. definition of the seuenth Againe forasmuch as D and E are the lest numbers that haue one the same proportion with C B therefore by the 21. of the seuenth how many times D measureth C so many times doth E measure B. How often E measureth B so many vnities let there be in G. Wherefore E measureth B by those vnities which are in G wherefore G multiplying E produceth B wherefore B is a plaine number by the 17. definition of the seuenth And the sides thereof are E and G. Wherefore those two numbers A and B are two plaine numbers I say moreouer that they are like For forasmuch as F multiplying E produced C and G multiplying E produced B therefore by the 17. of the seuenth as F is to G so is C to B but as C is to B so is D to E wherefore as D is to E so is F to G. Wherefore A and B are like plaine numbers for their sides are proportionall which was required to be proued ¶ The 19. Theoreme The 21. Proposition If betwene two numbers there be two meane proportionall numbers those numbers are like solide numbers SVppose that betwene two numbers A and B there be two meane proportionall numbers C D. Then I say that A and B are like solide numbers Take by the 3â— of the seuenth or 2. of the eight three of the least numbers that haue one and the same proportion with A C D B and let the same be E F G. Wherefore by the 3. of the eight their extremes E G are prime the one to the other And forasmuch as betwene the numbers E and G there is one meane proportionall number therfore by the 20 of the eight they are like plaine numbers Suppose that the sides of E be H and K. And let the sides of G be L and M. Now it is manifest that these numbers E F G are in continuall proportion and in the same proportion that H is to L and that K is to M. And forasmuch aâ— E F G are the least numbers that haue one and the same proportion with A C D therefore of equalitie by the 14. of the seuenth as E is to G so is A to D. But E G are by the 3. of the eight prime numbers yea they are prime and the least but the least numbers by the 21. of the seuenth measure those numbers that haue one the same proportion with them equally the greater the greater and the lesse the lesse that is the antecedent the antecedent the consequent the consequent therfore how many timeâ— E measureth A so many times G measureth D. How many times E measureth A so many vnities let there be in N. Wherefore N multiplieng E produceth A. But E is produced of the numbers H K. Wherfore N multiplieng that which is produced of H K produceth A. Wherefore A is a solide number and the sides therof are H K N. Agayne forasmuch as E F G are the least numbers that haue one and the same proportion with C D B therefore how many times E measureth C so many times G measureth B. How ofteÌ„times G measureth B so many vnities let there be in X. Wherfore G measureth B by those vnities which are in X. Wherfore X multiplieng G produceth B. But G is produced of the numbers L M. Wherefore X multiplieng that number which is produced of L and M produceth B. Wherfore B is a solide number and the sides therof are L M X. Wherfore A B are solide numbers I say moreouer that they are like solide numbers For forasmuch as N and X multiplieng E produced A and C therfore by the 18. of the seuenth as N is to X so iâ— A to C that is E â—o F. But as E is to F so is H to L and K to M therefore as H is to L so is K to M and N to X. And H K N are the sides of A and likewise L M X aâ—â— thâ— sides of Bâ— wherfore A Bâ— are like solide numbers which was required to be proued ¶ The 20. Theoreme The 22. Proposition If three numbers be in continuall proportion and if the first be a square number the third also shall be a square number SVppose that there be three numbers in continuall proportion A B C and let the first be a square number Then I say that the third is also a square number For forasmuch as betwene A and C there is one meane proportionall numberâ— namely B therefore by the 20. of the eight A and C are like playne numbers But A is a square number Wherefore C also is a square number which was required to be proued ¶ The 21. Theoreme The 23. Proposition If foure numbers be in continuall proportion and if the first be a cube nuÌ„ber the fourth also shall be a cube number SVppose that there be foure numbers in continuall proportion A B C D. And let A be a cube number TheÌ„ I say that D also is a cube number For forasmuch as betwene A and D there are two meane proportionall numbers Bâ— C. Therfore A D are like solide numbers by the 21. of this booke But A is a cube number wherfore D also is a cube numberâ— which was required to be demonstrated ¶ The 22. Theoreme The 24. Proposition If two numbers be in the same proportioÌ„ that a square number is to a square number and if the first be a square number the second also shall be a square number SVppose that two numbers A and B be in the same proportion that the square number C is vnto the squâ—â—â— nuÌ„ber D. And let A be a square nuÌ„ber Then I say that B also is a square number For forasmuch as C and D are square numbers Therfore G and D are like plaine numbers Wherfore by the 18. of the eight betwene C and D there is one meane proportionall number But as C is to D so is A to B. Wherfore betwene A and B there is one meane proportionall number by the 8. of the eight But A is a square number Wherfore by the 22. of the eight B also is a square number which was â—equired to be proued ¶ The 23. Theoreme The 25. Proposition If two numbers be in the same proportion the one to the other that a cube number is to a cube number and if the first be a cube number the second also shall be a cube number SVppose that two numbers A and B be in the same proportioÌ„ the one to the other that the cube nuÌ„ber C is vnto the cube number D. And let A be a cube number Then I say that B also is a cube nuÌ„ber For forasmuch as C D are cube nuÌ„bers therfore C D are like solide numbers wherfore by the 19. of the eight betwene C and D
there are two proportionall numbers But how many numbers fall in continual proportion betwene C and D so many by the 8. of the eight fal there betwene the numbers that haue the same proportion with them Wherefore betwene A and B there are two meane proportionall numbers which let be E and F. And forasmuch as there are foure numbers in continuall proportion namely A E F B and A is a cube number therefore by the 2â— of the eight B also is a cube number which was required to be demonstrated A Corollary added by Flussates Betwene a square number and a number that is not a square number falleth not the proportion of one square number to an other For if the first be a square number the second also should be a square number which is contrary to the supposition Likewisâ— betwene a cube number and a number that is no cube number falleth not the proportion of one cube number to an other For if the first be a cube number the second also should be a cube number which is contrary to the supposition therfore impossible ¶ The 24. Theoreme The 26. Proposition Like playne numbers are in the same proportion the one to the other that a square number is to a square number SVppose that A and B be like plaine numbers Then I say that A is vnto B in the same proportioÌ„ that a square number is to a square number For forasmuch as A B are like plaine numbers therefore betwene A and B there falleth one meane proportional nuÌ„ber by the 18. of the eight Let there fal such a number and let the same be C. And by the 35. of the seuenth take the three least numbers that haue one and the same proportion with A C B and let the same be D E F wherefore by the corollary of the 2 â— of the eight their 〈◊〉 that is D F are square numbers And for that as D is to F so is A to B by the 14. of the seueÌ„th and D F are square numbers Therfore A is vnto B in that proportion that a square nuÌ„ber is vnto a square numâ—er which was required to be proued The 25. Theoreme The 27. Proposition Like solide numbers are in the same proportion the one to the other that a cube number is to a cube number SVppose that a A and B be like solide numbers Then I say that A is vnto B in the same proportion that a cube numbe is to to a cube number For forasmuch as A B are like solide numbers Therefore by the 19. of the eight betwene A and B there fall two meane proportionall numbers Let there fall two such numbers and let the same be C and D. And take by the 35. of the seuenth the least numbers that haue one and the same proportion with A C D B and equall also with theÌ„ in multitude and let the same be E F G H. Wherfore by the corollary of the 2. of the eight their extreames that is EH are cube numbers But as E is to H so is A to B. Wherefore A is vnto B in the same proportion that a cube number is to a cube number which was required to be demonstrated ¶ A Corollary added by Flussates If two nnmbers be in the same proportion the one to the other that a square number is to a square number those two numbers shall be like superficiall numbers And if they be in the same proportion the one to the other that a cube number is to a cube number they shall be like solide nuÌ„bers First let the number A haue vnto the number B the same proportion that the square number C hath to the square number D. Then I say that A and B are like superficiall nuÌ„bers For forasmuch as betwene the square numbers C and D there falleth a meane proportionall by the 11. of this booke there shall also betwene A and B which haue one and the same proportion with C and D fall a meane proportionall by the 8. of this booke Wherefore A and B are like superficiall nuÌ„bers by the 20. of this booke But if A be vnto B as the cube number C is to the cube number D. Then are A B like solide numbers For forasmuch as C and D are cube numbers there falleth betwene them â—wo meane proportionall numbers by the 12. of this booke And therefore by the 8. of the same betwene A and B which are in the same proportion that C is to D there falleth also two meane proportionall numbers Wherfore by the 21. of this booke A and B are like solide numbers An other Corollary added also by Flussates If a number multiplying a square number produce not a square number the sayd number multiplying shall bâ— no square number For if it should be a square number then should it and the number multiplied being like superficiall numbers by reason they are square numbers haue a meane proportionall by the 18. of thys booke And the number produced of the sayd meane should be equall to the number contayned vnder the extremes which are square numbers by the 20. of the seuenth Wherefore the number produced of the extremes being equall to the square number produced of the meane should be a square number But the sayd number by supposition is no square number Wherefore neither is the number multiplying the square number a square number The first part of the first Corollary is the conuerse of the 26. Proposition of this booke and hath some vse in the tenth booke The second part of the same also is the conuerse of the 27. Proposition of the same The end of the eighth booke of Euclides Elementes ¶ The ninth booke of Euclides Elementes IN THIS NINTH BOOKE Euclide continueth his purpose touching numbers partly prosecuting thynges more fully which were before somewhat spoken of as of square and cube numbers and partly setting out the natures and proprieties of such kindes of number as haue not yet bene entreated of which yet are most necessary to be knowen As are numbers euen and odde whose passions and conditions are in this booke largely taught with their compositions and subductions of the one from the other with many other generall and speciall thinges to be noted worthy the knowledge ¶ The 1. Theoreme The 1. Proposition If two like plaine numbers multiplying the one the other produce any number the number of them produced shall be a square number SVppose that A and B be two like plaine numbers And let A multiplying B produce the number C. Then I say that C is a square number For let A multiplying him selfe produce D. Wherefore D is a square number And forasmuch as A multiplying him selfe produced D and multiplying B produced C therefore by the 17. of the seuenth as A is to B so is D to C. And forasmuch as A B are like plaine numbers therefore by the 18. of
the eight betwene A and B there falleth a meane proportionall number But if betwene two numbers fall numbers in continuall proportion how many numbers fall betwene them so many also by the 8. of the eight shall there fall betwene the numbers that haue the same proportion with them Wherfore betwene C and D also there falleth a meane proportionall number But D is a square number Wherfore by the 22. of the eight C also is a square number which was required to be proued ¶ The 2. Theoreme The 2. Proposition If two numbers multiplying the one the other produce a square number those numbers are like plaine numbers SVppose that two numberâ— A and B multiplying the one the other do produce C a square number Then I say that A and B are like plaine numbers For let A multiplying him selfe produce D. Wherefore D is a square number And forasmuch as A multiplying him selfe produced D and multiplying B produced C therefore by the 17. of the seâ—enth as A is to B so is D to C. And forasmuch as D is a square number and so likewise is C therefore D and C are like plaine numbers Wherefore betwene D and C there is by the 18. of the eight one meane proportionall number But as D is to C so is A to B. Wherefore by the 8. of the eight betwene A and B there is one meane proportionall number But if betwene two numbers there be one meane proportionall number those numbers are by the 20. of the eight like plaine numbers Wherefore A and B are like plaine numbers which was required to be proued A Corollary added by Campane Hâ—â—â—â— it is manifest thâ—t two squarâ— numbers multiplyed the one into the other do alwayes produce a squaâ—â— numâ—â—r For they are like superficiall numbers and therefore the number produced of them is by the first of this booke a square number But a square number mulâ—â—plyeâ— into a number not square produceth a number not square For if they should produce a square number they should be like superficiall numbers by this Proposition But they are not Wherefore they produce a number not square But if a square number multiplyed into an other number produce a square number that other number shall be a square number For by this Proposition that other number is like vnto the square number which multiplyeth it and therefore is a square number But if a square number multiplyed into an other number produce a number not square neither shall that other number also be a square number For if it should be a square number then being multiplyed into the square number it should produce a square number by the first part of this Corollary The 3. Theoreme The 3. Proposition If a cube number multiplying himselfe produce a number the number produced shall be a cube number SVppose that A being a cube number multiplieng himselfe do produce the number B. Then I say that B is a cube number Take the side of A and let the same be the number C and let C multiplieng himselfe produce the number D. Now it is manifest that C multiplieng D produceth A by the 20. definition of the seueÌ„th And forasmuch as C multiplieng himselfe produced D therfore C measureth D by those vnities which are in C. But vnitie also measureth C by those vnities which are in C. Wherfore as vnitie is to C so is C to D. Againe forasmuch as C multiplieng D produceth A therefore D measureth A by those vnities which are in C. But vnitie measureth C by those vnities which are in C wherefore as vnitie is to C so is D to A. But as vnitie is to C so is C to D wherfore as vnitie is to C so is C to D D to A. Wherefore betwene vnitie A there are two meane proportionall numbers namely C D. Againe forasmuch as A multiplieng himselfe produced B therefore A measureth B by those vnities which are in A. But vnitie also measureth A by those vnities which are in A. Wherfore as vnitie is to A so is A to B. But betwene A and vnitie there are two meane proportionall numbers Wherfore betwene A and B also there are two meane proportionall numbers by the 8. of the eight But if betwene two numbers there be two meane proportionall numbers and if the first be a cube number the fourth also shall be a cube number by the 21. of the eight But A is a cube number wherefore B also is a cube number which was required to be proued ¶ The 4. Theoreme The 4. Proposition If a cube number multiplieng a cube number produce any number the number produced shall be a cube number SVppose that the cube number A multiplieng the cube number B do produce the nuÌ„ber C. Then I say that C is a cube number For let A multiplieng himselfe produce D. Wherefore D is a cube number by the proposition going before And forasmuch as A multiplieng himselfe produced D and multiplieng B it produced C therfore by the 17. of the seuenth as A is to B so is D to C. And forasmuch as A and B are cube numbers therfore A and B are like solide numbers Wherfore betwene A and B by the 19. of the eight there are two meane proportionall numbers Wherefore also by the 8. of the same betwene D and C there are two meane proportionall numbers But D is a cube number Wherfore C also is a cube number by the 23. of the eight which was required to be demonstrated ¶ The 5. Theoreme The 5. Proposition If a cube number multiplying any number produce a cube nuÌ„ber the number multiplyed is a cube number SVppose that the cube number A multiplying the number B do produce a cube number namely C. Then I say that B is a cube number For let A multiplying him selfe produce D. Wherefore by the 3. of the ninth D is a cube nuÌ„ber And forasmuch as A multiplying him selfe produced D and multiplying B it produced C therefore by the 17. of the seuenth as A is to B so is D to C. And forasmuch as D and C are cube numbers they are also like solide nuÌ„bers Wherefore by the 19. of the eight betwene D and C there are two meane proportionall numbers But as D is to C so is A to B. Wherefore by the 8. of the eight betwene A and B there are two meane proportionall numbers But A is a cube number Wherefore B also is a cube number by the 23. of the eight which was required to be proued ¶ A Corollary added by Campane Hereby it is manifest that if a cube number multiply a number not cube it shall produce a number not cube For if it should produce a cube number then the number multiplyed should also be a cube number by this Proposition which is contrary to the supposition For it is supposed to be no cube number
And if a cube number multiplying a number produce a number not cube the number multiplyed shall be no cube number For if the number multiplied should be a cube number the number produced should also be a cube number by the 4. of this booke which is contrary to the supposition and impossible ¶ The 6. Theoreme The 6. Proposition If a number multiplieng himselfe produce a cube number then is that number also a cube number SVppose that the number A multiplieng himself do pâ—oduce B a cube nuÌ„ber Then I say that A also is a cube number For let A multiplieng B produce C. And forasmuch as A multiplieng himselfe produced B multiplieng B it produced C therfore C is a cube number And for that A multiplieng himselfe produced B and multiplieng B it produced C therfore by the 17. of the seuenth as A is to B so is B to C. And for that B and C are cube numbers they are also like solide numbers Wherfore by the 19. of the eight betwene C and B there are two meane proportional numbers But as B is to C so is A to B wherfore by the 8. of the eight betwene A and B there are two meane proportional nuÌ„bers But B is a cube number Wherefore A also is a cube number by the 23. of the eight which was required to be demonstrated ¶ The 7. Theoreme The 7. Proposition If a composed number multiplieng any number produce a number the nuÌ„ber produced shall be a solide number SVppose that the composed number A multiplieng the nuÌ„ber B do produce the number C. Then I say that C is a solide number For forasmuch as A is a composed nuÌ„ber therfore some number measureth it by the 14. definition Let D measure it And how oâ—ten D measureth A so many vnities let there be in E. Wherefore E multiplieng D produceth A. And forasmuch as two numbers D and E multiplieng themselâ—es produce A which A againe multiplieng B produceth C therfore C produced of three numbers multiplieng the one the other namely D E and B is by the 18. definition of the seuenth a solide number And the sides therof are the numbers D E B. If therefore a composed number c which was required to be proued ¶ The 8. Theoreme The 8. Proposition If from vnitie there be numbers in continuall proportion how many soeuer the third number from vnitie is a square number and so are all forwarde leauing one betwene And the fourth number is a cube number and so are all forward leauing two betwene And the seuenth is both a cube number and also a square number and so are all forward leauing fiue betwene SVppose that from vnitie there be these numbers in continuall proportion A B C D E F. Then I say that the third number from vnitie namely B is a square number and so are all forward leauing one betwene namely D and F. And that C the fourth number is a cube number and so are all forwarde leauyng two betwene And that F the seuenth number is both a cube number and also a square number and so are all forward leauing fiue betwene For for that as vnitie is to A so is A to B. Therefore how many times vnitie measureth A so many times A measureth B. But vnitie measureth A by those vnities which are in A wherefore A measureth B by those vnities which are in A. And forasmuch as A measureth B by those vnities which are in A. Therfore A multiplieng himselfe produceth B. Wherfore B is a square number And forasmuch as these numbers B C D are in continuall proportion and B is a square number therfore by the 22. of the eight D also is a square number And by the same reason also F is a square number And in like sort may we proue that leauing alwayes one betwene all the rest forward are square numbers Now also I say that the fourth number from vnitie that is C is a cube number and so are all forward leauing two betwene For for that as vnitie is to the number A so is B to C therefore how many times vnitie measureth the number A so many times B measureth C. But vnitie measureth A by those vnities which are in A wherfore B measureth C by those vnitâ—â—s which are in A. Wherfore A multiplieng B produceth C. And forasmuch as A multipliâ—ng himselfe produced B and multiplieng B it produced C therefore C is a cube number And forasmuch as C D E F are in continuall proportion But G is a cube number therefore by the 23. of the eight F also is a cube number And it is proued that F being the seuenth number from vnitie is also a square number Wherfore F is both a cube number and also a square number In like sort may we proue that leaâ—ing alwaies fiue betwene all the rest forwarde are numbers both cube and also square which was required to be proued ¶ The 9. Theoreme The 9. Proposition If fromâ— vnitie be numbers in continuall proportion how many soeuer and if thâ—â— number which followeth next after vnitie be a square number then all the rest following also be square numbers And if that number which followeth next after vnitie be a cube number then all the rest following shall be cube numbers SVppose that from vnitie there be these numbers in continuall proportion A B C D E F. And let A which followeth next vnto vnitie be a square number Then I say that all the rest following also are square numbers That the third number namely B is a square number so all forward leauing one betwene it is plaine by the Proposition next going before I say also that all the rest are square numbers For forasmuch as A B C are in continuall proportion and A is a square number therfore by the 22. of the eight C also is a square number Againe forasmuch as B C D are in continuall proportion and B is a square number therfore D also by the 22. of the eight is a square number In like sort may we proue that all the rest are square numbers But now suppose that A be a cube number Then I say that all the rest following are cube numbers That the fourth from vnitie that is C is a cube number and so all forward leauing two betwene it is plaine by the Proposition going before Now I say that all the rest also are cube numbers For for that as vnitie is to A so is A to B therefore how many times vnitie measureth A so many times A measureth B. But vnitie measureth A by those vnities which are in A. Wherefore A also measureth B by those vnities which are in A. Wherefore A multiplying him selfe produceth B. But A is a cube number But if a cube number mutiplying him selfe produce any number the number produced is by the 3. of the ninth a cube number
E produced D wherfore A measureth D but it also measureth it not which is impossible Wherfore it is impossible to finde out a fourth number proportionall with these numbers A B C whensoeuer A measureth not D. ¶ The 20. Theoreme The 20. Proposition Prime numbers being geuen how many soeuer there may be geuen more prime numbers SVppose that the prime numbers geuen be A B C. Then I say that there are yet more prime numbers besides A B C. Take by the 38. of the seuenth the lest number whom these numbers A B C do measure and let the same be DE. And vnto DE adde vnitie DF. Now EF is either a prime number or not First let it be a prime number then are there found these prime numbers A B C and EF more in multitude then the prime numbers â—irst geuen A B C. But now suppose that EF be not prime Wherefore some prime number measureth it by the 24. of the seuenth Let a prime number measure it namely G. Then I say that G is none of these numbers A B C. For if G be one and the same with any of these A B C. But A B C measure the nuÌ„ber DE wherfore G also measureth DE and it also measureth the whole EF. Wherefore G being a number shall measure the residue DF being vnitieâ— which is impossible Wherefore G is not one and the same with any of these prime numbers A B C and it is also supposed to be a prime number Wherefore there are â—ound these prime numbers A B C G being more in multitude then the prime numbers geuen A B C which was required to be demonstrated A Corollary By thys Proposition it is manifest that the multitude of prime numbers is infinite ¶ The 21. Theoreme The 21. Proposition If euen nuÌ„bers how many soeuer be added together the whole shall be eueÌ„ SVppose that these euen numbers AB BC CD and DE be added together Then I say that the whole number namely AE is an euen number For forasmuch as euery one of these numbers AB BC CD and DE is an euen number therefore euery one of them hath an halfe Wherefore the whole AE also hath an halfe But an euen number by the definition is that which may be deuided into two equall partes Wherefore AE is an euen number which was required to be proued ¶ The 22. Theoreme The 22. Proposition If odde numbers how many soeuer be added together if their multitude be euen the whole also shall be euen SVppose that these odde numbers AB BC CD and DE being euen in multitude be added together Then I say that the whole AE is an euen number For forasmuch as euery one of these numbers AB BC CD and DE is an odde number is ye take away vnitie from euery one of them that which remayneth oâ— euery one of theÌ„ is an euen number Wherefore they all added together are by the 21. of the ninth an euen number and the multitude of the vnities taken away is euen Wherefore the whole AE is an euen number which was required to be proued ¶ The 23. Theoreme The 23. Proposition If odde numbers how many soeuer be added together and if the multitude of them be odde the whole also shall be odde SVppose that these odde numbers AB BC and CD being odde in multitude be added together Then I say that the whole AD is an odde number Take away from CD vnitie DE wherefore that which remaineth CE is an euen number But AC also by the 22. of the ninth is an euen number Wherfore the whole AE is an euen number But DE which is vnitie being added to the euen number AE maketh the whole AD aâ— odde number which was required to be proued◠¶ The 24. Theoreme The 24. Proposition If from an euen number be takeÌ„ away an euen number that which remaineth shall be an euen number SVppose that AB be an euen number and from it take away an euen number CB. Then I say that that which remayneth namely AC is an euen number For forasmuch as AB is an euen euen number it hath an halfe and by the same reason also BC hath an halfe Wherfore the residue CA hath an halfe Wherfore AC is an euen number which was required to be demonstrated ¶ The 25. Theoreme The 25. Proposition If from an euen number be taken away an odde number that which remaineth shall be an odde number SVppose that AB be an euen number and take away from it BC an odde number Then I say that the residue CA is an odde number Take away from BC vnitie CD Wherfore DB is an euen number And AB also is an euen number wherefore the residue AD is an euen number by the â—ormer proposition But CD which is vnitie being taken away from the euen nuÌ„ber AD maketh the residue AC an odde number which was required to be proued ¶ The 26. Theoreme The 26. Proposition If from an odde number be taken away an odde number that which remayneth shall be an euen number SVppose that AB be an odde number and from it take away an odde number BC. TheÌ„ I say that the residue CA is an euen number For forasmuch as AB is an odde number take away from it vnitie BD. Wherfore the residue AD is euen And by the same reason CD is an euen number wherfore the residue CA is an euen number by the 24. of this booke â— which was required to be proued ¶ The 27. Theoreme The 27. Proposition If from an odde number be taken a way an euen number the residue shall be an odde number SVppose that AB be an odde number and from it take away an euen number BC. Then I say that the residue CA is an odde number Take away froÌ„ AB vnitie AD. Wherfore the residue DB is an eueÌ„ number BC is by supposition euen Wherfore the residue CD is an euen number Wherefore DA which is vnitie beyng added vnto CD which is an euen number maketh the whole AC an â—dde number which was required to be proued ¶ The 28. Theoreme The 28. Proposition If an odde number multiplieng an euen number produce any number the number produced shall be an euen number SVppose that A being an odde number multiplieng B being an euen number do produce the number C. Then I say that C is an euen number For forasmuch as A multiplieng B produced C therfore C is composed of so many numbers equall vnto B as there be in vnities in A. But B is an euen nuÌ„ber wherfore C is composed of so many euen numbers as there are vnities in A. But if eueÌ„ numbers how many soeuâ—r be added together the whole by the 21. of the ninth is an euen number wherfore C is an euen number which was required to be demonstrated ¶ The 29. Theoreme The 29. Proposition Iâ— an odde number multiplying an
odde number produce any number the number produced shal be an odde number SVppose that A being an odde number multiplying B being also an odde number doo produce the number C. Then I say that C is an odde number For forasmuch as A multiplying B produced C therefore C is composed of so many numbers equall vnto B as there be vnities in A. But either of these numbers A and B is an odde number Wherefore C is composed of odde numbers whose multitude also is odde Wherfore by the 23. of the ninth C is an odde nuÌ„ber which was required to be demonstrated A proposition added by Campane If an odde number measure an euen number it shall measure it by an euen number For if it should measure it by an odde number then of an odde number multiplyed into an odde number should be produced an odde number which by the former proposition is impossible An other proposition added by him If an odde number measure an odde number it shall measure it by an odde number For if it should measure it by an euen number then of an odde number multiplyed into an euen number should be produced an odde number which by the 28. of this booke is impossible ¶ The 30. Theoreme The 30. Proposition If an odde number measure an euen number it shall also measure the halfe thereof SVppose that A being an odde number doo measure B being an euen number Thâ—â— I say that it shall measure the halfe thereof For forasmuch as A measureth B let iâ— measure it by C. TheÌ„ I say that C is an euen number For if not then if it be possible leâ— iâ— be odde And forasmuch as A measureth B by C therfore A multiplying C produceth B. Wherfore B is composed of odde numbers whose multitude also is odde Wherfore B is an odde number by the 29. of this booke which is absurdâ— for it is supposed to be euen wherefore C is an euen numâ—er Wherefore A measureth B by an euen number and C measureth B by A. But either of these numbers C and B hath an halfe part wherfore as C is to B so is the halfe to the halfe But C measureth B by A. Wherefore the halfe of C measureth the halfe of B by A wherfore A multiplying the halfe of C produceth the halfe of B. Wherfore A measureth the halfe of B and it measureth it by the halfe of C. Wherefore A measureth the halfe of the number B which was required to be demonstrated ¶ The 31. Theoreme The 31. Proposition If an odde number be prime to any number it shal also be prime to the double thereof SVppose that A being an odde number be prime vnto the number B and let the double of B be C. Then I say that A is prime vnto C. For if A and C be not prime the one to the other some one number measureth them both Let there be such a number which measureth them both and let the same be D. But A is an odde number Wherefore D also is an odde number For if D which measureth A should be an euen number then should A also be an euen number by the 21. of this booke which is coÌ„trary to the supposition For A is supposed to be an odde nuÌ„ber therefore D also is an odde number And forasmuch as D being an odde number measureth C but C is an eueÌ„ number for that it hath an halfe namely B Wherfore by the Proposition next going before D measureth the halfe of C. But the halfe of C is B. Wherefore D measureth B and it also measureth A. Wherefore D measureth A and B being prime the one to the other which is absurde Wherefore no number measureth the numbers A C. VVherfore A is a prime number vnto C. VVherefore these numbers A and C are prime the one to the other which was required to be proued ¶ The 32. Theoreme The 32. Proposition Euery nuÌ„ber produced by the doubling of two vpward is euenly euen onely SVppose that A be the number two and from A vpward double numbers how many soeuer as B C D. Then I say that B C D are numbers euenly euen onely That euery one of them is euenly euen it is manifest for euery one of them is produced by the doubling of two I say also that euery one of them is euenly euen onely Take vnitie E. And forasmuch as from vnitie are certaine numbers in continuall proportion A which followeth next after vnitie is a prime number therefore by the 13. of the third no number measureth D being the greatest number of these numbers A B C D besides the selfe same numbers in proportion But euery one of these numbers A B C is euenly euen VVherefore D is euenly euen onely In like sort may we proue that euery one of these numbers A B C is euenly euen onely which was required to be proued ¶ The 33. Theoreme The 33. Proposition A number whose halfe part is odde is euenly odde onely SVppose that A be a number whose halfe part is odde Then I say that A is euenly od onely That it is euenly odde it is manifest for his halfe being odde measureth him by an eueÌ„ number namely by 2. by the definition I say also that it is euenly odde onely For if A be euenly euen his halfe also is euen For by the definition an euen number measureth him by an euen number Wherefore that euen number which measureth him by an euen number shall also measure the halfe thereof being an odde number by the 4. common sentence of the seuenth which is absurd Wherfore A is a number euenly odde onely which was required to be proued An other demonstration to proue the same Suppose that the number A haue to his halfe an od nuÌ„ber namely B. TheÌ„ I say that A is eueÌ„ly od onely That it is euenly odde needeth no profe forasmuch as the number 2. an euen number measureth it by the halfe thereof which is an odde number Let C be the number 2. by which B measureth A for that A is supposed to be double vnto B And let an euen number namely D measure A which is possible for that A is an euen number by the definition by F. And forasmuch as that which is produced of C into B is equall to that which is produced of D into F therefore by the 19. of the seuenth as C is to D so is B to F. But C the number two measureth D being an euen number wherfore F also measureth B which is the halfe of A. Wherfore F is an odde number For if F were an euen number then should it in the B whome it measureth an odde number also by the 21. of this booke which is contrary to the supposition And in like maner may we proue that all the eueÌ„ nuÌ„bers which measure the number Aâ— do measure it by odde numbers Wherefore A is a number euenly odde onely
which was required to be proued ¶ The 34. Theoreme The 34. Proposition If a number be neither doubled from two nor hath to his half part an odde number it shall be a number both euenly euen and euenly odde SVppose that the nuÌ„ber A be a nuÌ„ber neither doubled froÌ„ the nuÌ„ber two neither also let it haue to his halfe part an odde nuÌ„ber Then I say that A is a nuÌ„ber both euenly euen and euenly odde That A is euenly euen it is manifest for the halfe therof is not odde and is measured by the number 2. which is an euen number Now I say that it is euenly odde also For if we deuide A into two equall partes and so continuing still we shall at the length light vpon a certaine odde number which shall measure A by an euen number For if we should not light vpon such an odde nuÌ„ber which measureth A by an euen number we should at the length come vnto the number two and so should A be one of those numbers which are doubled from two vpward which is contrary to the supposition Wherfore A is euenly odde And it is proued that it is euenly eueÌ„ wherfore A is a number both euenly euen and euenly odde whiche was required to be demonstrated This proposition and the two former manifestly declare that which we noted vppon the tenth definition of the seuenth booke namely that Campane and Flussates and diuers other interpreters of Euclide onely Theon except did not rightly vnderstand the 8. and 9. definitions of the same booke concerning a number euenly euen and a number euenly odde For in the one definition they adde vnto Euclides wordes extant in the Greeke this word onely as we there noted and in the other this word all So that after their definitions a number can not be euenly euen vnlesse it be measured onely by euen numbers likewise a number can not be euenly odde vnlesse all the euen numbers which doo measure it doo measure it by an odde number The contrary whereof in this proposition we manifestly see For here Euclide proueth that one number may be both euenly euen and euenly odde And in the two former propositions he proued that some numbers are euenly euen onely and some euenly odde onely which word onely had bene in vaine of him added if no number euenly euen could be measured by an odde number or if all the numbers that measure a number euenly odde must needes measure it by an odde number Although Campane and Flussates to auoyde this absurdity haue wreasted the 32. proposition of this booke froÌ„ the true sence of the Greeke and as it is interpreted of Theon So also hath Flussates wreasted the 33. proposition For wheras Euclide sayth Euery nuÌ„ber produced by the doubling of two vpward is eueÌ„ly eueÌ„ only they say onely the numbers produced by the doubling of two are euenly euen Likewise whereas Euclide saith A number whose hafle part is odde is euenly odde onely Flussates sayth onely a number whose halfe part is od Is eueÌ„ly od Which their interpretatioÌ„ is not true neither can be applyed to the propositions as they are extaÌ„t in the Greeke In dede the sayd 32. and 33. propositions as they put theÌ„ are true touching those numbers which are euenly euen onely or eueÌ„ly od onely For no number is euenly euen onely but those onely which are doubled from two vpward Likewise no numbers are euenly odde onely but those onely whose halfe is an odde number But this letteth not but that a number may be euenly euen although it be not doubled from two vpward also that a number may be eueÌ„ly odde although it haue not to his halfe an odde number As in this 34. propositioÌ„ Euclide hath plainly proued Which thing could by no meanes be true if the foresayd 32. 33. propositons of this booke should haue that sence and meaning wherein they take it ¶ The 35. Theoreme The 35. Proposition If there be numbers in continuall proportion how many soeuer and if from the second and last be taken away numbers equall vnto the first as the excesse of the second is to the first so is the excesse of the last to all the nuÌ„bers going before the last SVppose that these numbers A BC D and EF be in continuall proportion beginning at A the least And from BC which is the second take away CG equall vnto the first namely to A and likewise from EF the last take away FH equall also vnto the first namely to A. Then I say that as the excesse BG is to A the first so is HE the excesse to all the numbers D BC and A which go before the last number namely EF. Forasmuch as EF is the greater for the second is supposed greater then the first put the number FL equall to the number D and likewise the number FK equall to the number BC. And forasmuch as FK is equall vnto CB of which FH is equall vnto GC therefore the residue HK is equall vnto the residue GB And for that as the whole Fâ— is to the whole FL so is the part taken away FL to the part taken away FK therefore the residue LE is to the residue KL as the whole â—E is to the whole FL by the 11. of the seuenth So likewise for that FL is to FK as FK is to FH KL shall be to HK as the whole FL is to the whole FK by the same Proposition But as FE is to FL and as FL is to FK and FK to FH so were FE to D and D to BC and BC 〈◊〉 A. Wherefore as LE is to KL and as KL is to HK so is D to BC. Wherefore alternately by the 23. of the seuenth as LE is to D so is KL to be BC and as KL is to BC so is HK to A. Wherefore also as one of the antecedentes is to one of the consequentes so are all the antecedentes to all the consequentes Wherefore as KH is to A so are HK KL and LE to D BC and A by the 12. of the seuenth But it is proued that KH is equall vnto BG Wherefore as BG which is the excesse of the second is to A so is EH the excesse of the last vnto the numbers going before D BC and A. Wherefore as the excesse of the second is vnto the first so is the excesse of the last to all the numbers going before the last which was required to be proued ¶ The 36. Theoreme The 36. Proposition If from vnitie be taken numbers how many soeuer in double proportion continually vntill the whole added together be a prime number and if the whole multiplying the last produce any number that which is produced is a perfecte number SVppose that from vnitie be taken these numbers A B C D in double proportion continually so that all those numbers A B C D vnitie added together make a prime number and let E be the number composed of
the square of the line A haue not vnto the square of the line B the same proportion that a square number hath to a square number Then I say that the lines A and B are incommensurable in length For if the lines A and B be commensurable in length then the square of the line A should haue vnto the square of the line B the same proportion that a square number hath to a square number by the first part of this proposition but by supposition it hath not wherfore the lines A and B are not commensurable in length Wherfore they are incomensurable in length Wherfore squares made of right lines commensura in length haue that proportion the one to the other that a square number hath to a square number And squares which haue that proportion the one to the other that a square number hath to a squaâ—e number shall also haue the sides commensurable in length But squares described of right lines incommensurable in length haue not that proportion the one to the other that a square number hath to a square number And squares which haue not that propoâ—tion the one to the other that a sâ—uare number hath to a square number haue not also their sides commâ—nsurable in length which was all that was required to be proued ¶ Corrollary Hereby it is manifest that right lines coÌ„mensurable in length are also euer commensurable in power But right lines commensurable in power are not alwayes commensurable in length And right lines incoÌ„mensurable in leÌ„gth are not alwayes incommensurable in power But right lines incommensurable in power are euer also incommensurable in length For forasmuâ—h as squares made of right lines commensurable in length haue that proportion the one to the other that a square number hath to a square number by the first part of this proposition but magnitudes which haue that proportion the one to the other that number simply hath to number are by the sixt of the tenth commensurable Wherfore right lines commensurable in length are commensurable not onely in length but also in power Againe forasmuch as there are certaine squares which haue not that proportion the one to the other that a square number hath to a square number but yet haue that proportion the one to the other which number simply hath to number their sides in dede are in power commensurable for that they describe squares which haue that proportion which number simply hath to number which squares are therfore commensurable by the 6. of this booke but the said sides are incommensurable in length by the latter part of this proposition Wherâ—fore it is tâ—ue that lines commensurable in power are not straight way commensurable in length also And by the selâ—e same reason is proued also that third part of the corollary that lines incommensurable in length are not alwayes incommensurable in power For they may be incommâ—nsurable in length but yet commensurable in power As in those squares which are in proportion the one to the other as number is to number but not as a square number is to a square number But right lines incommensurable in power are alwayes also incommensurable in length For iâ— they be commensurable in length they shal also be commensurable in power by the first part of this Corollary But they are supposed to be incommensurable in length which is absurde Wherâ—ore right lines incommensurable in power are euer incommensurable in lengthâ— For the better vnderstanding of this proposition and the other following I haue here added certayne annotacions taken out of Montaureus And first as touching the signiâ—ication oâ— wordes and termes herein vsed whâ—ch arâ— such that vnlesse they be well marked and peysed the matter will be obscure and hard and in a maner inexplicable First this ye must note that lines to be commensurable in length and lines to be in proportion the one to the other as number is to number is all one So that whatsoeuer lines are commensurable in length are also in proportion the one to the other as number is to number And conuersedly what so euer lynes are in proportion the one to the other as number is to number are also commeÌ„surable in length as it is manifest by the 5 and 6 of this booke Likewise lines to be incommensurable in length and not to be in proportion the one to the other as number is to number is all one as it is manifest by the 7. and 8. of this booke Wherfore that which is sayd in this Theoreme ought to be vnderstand of lines commensurable in length and incommensurable in length This moreouer is to be noted that it is not all one numbers to be square numbers and to be in proportioÌ„ the one to the other as a square number is to a square number For although square numbers be in proportion the one to the other as a square number is to a square number yet are not all those numbers which are in proportion the one to the other as a square number is to a square number square numbers For they may be like superficiall numbers and yet not square numbers which yet are in proportioÌ„ the one to the other as a square number is to a square number Although all square numbers are like superficiall numbers For betwene two square numbers there â—alleth one meane proportionall number by the 11. of the eight But if betwene two numbers there fall one meane proportionall number those two numbers are like superficiall numbers by the 20. of the eight So also if two numbers be in proportion the one to the other as a square number is to a square number they shall be like superficiall nuÌ„bers by the first corollary added after the last proposition of the eight booke And now to know whether two superficiall numbers geuen be like superficiall numbers or no it is thus found out First if betwene the two numbâ—rs geuen there fall no meane proportionall then are not these two numbers like superficiall numbers by the 18. of the eight But if there do fall betwene them a meane proportionall then are they like superâ—iciall numbers by the 20. of the eight Moreouer two like superficiall numbers multiplied the one into the other do produce a square number by the firsâ— of the ninth Wherfore if they do not produce a square number then are they not like superficiall numbers And if the one being multiplied into the other they produce a square number then are they like superficiall by the 2. of the ninth Moreouer if the said two superficial numbers be in superperticular or superbipartient proportion then are they not like superficiall numbers For if they should be like then should there be a meane proportionall betwene them by the 20. of the eight But that is contrary to the Corollary of the 20. of the eight And the easilier to conceiue the demonstrations following take this example of that which we haue sayd ¶
An Assumpt Forasmuch as in the eight booke in the 26. proposition it was proued that like playne numbers haue that proportion the one to the other that a square number hath to a square number and likewise in the 24. of the same booke it was proued that if two numbers haue that proportion the one to the other that a square number hath to a square number those numbers are like plaine numbers Hereby it is manifest that vnlike plaine numbers that is whose sides are not proportionall haue not that proportion the one to the other that a square number hath to a square number For if they haue then should they be like plaine numbers which is contrary to the supposition Wherfore vnlike plaine numbers haue not that proportion the one to the other that a square number hath to a square nuÌ„ber And therfore squares which haue that proportion the one to the other that vnlike plaine numbers haue shall haue their sides incommensurable in length by the last part of the former proposition for that those squares haue not that proportion the one to the other that a square number hath to a square number ¶ The 8. Theoreme The 10. Proposition If foure magnitudes be proportionall and if the first be commensurable vnto the second the third also shal be commensurable vnto the fourth And if the first be incommensurable vnto the second the third shall also be incommensurable vnto the fourth SVppose that these foure magnitudes A B C D be proportionall As A is to B so let C be to D and let A be commensurable vnto B. Then I say that C is also commensurable vnto D. For forasmuch as A is commensurable vnto B it hath by the fift of the tenth that proportion that number hath to number But as A is to B so is C to D. Wherfore C also hath vnto D that proportion that number hath to number Wherfore C is commensurable vnto D by the 6. of the tenth But now suppose that the magnitude A be incommensurable vnto the magnitude B. Then I say that the magnitude C also is incommensurable vnto the magnitude D. For forasmuch as A is incommensurable vnto B therfore by the 7. of this booke A hath not vnto B such proportion as number hath to number But as A is to B so is C to D. Wherefore C hath not vnto D such proportion as number hath to number Wherfore by the 8. of the tenth C is incommensurable vnto D. If therefore there be foure magnitudes proportionall and if the first be commensurable vnto the second the third also shall be commensurable vnto the fourth And if the first be incommensurable vnto the second the third shall also be incommensurable vnto the fourth which was required to be proued ¶ A Corollary added by Montaureus If there be foure lines proportionall and if the two first or the two last be commensurable in power onely the other two also shall be commensurable in power onely This is proued by the 22. of the sixt and by this tenth proposition And this Corollary Euclide vseth in the 27. and 28. propositions of this booke and in other propositions also ¶ The 3. Probleme The 11. Proposition Vnto a right line first set and geuen which is called a rationall line to finde out two right lines incommensurable the one in length onely and the other in length and also in power SVppose that the right line first set and geuen which is called a rationall line of purpose be A. It is required vnto the said line A to finde out two right lines incommensurable the one in length onely the other both in length and in power Take by that which was added after the 9. proposition of this booke two numbers B and C not hauing that proportion the one to the other that a square number hath to a square number that is let them not be like plaine numbers for like plaine numbers by the 26. of the eight haue that proportion the one to the other that a square number hath to a square number And as the number B is to the number C so let the square of the line A be vnto the square of an other line namely of D how to do this was taught in the assumpt put before the 6. proposition of this booke Wherfore the square of the line A is vnto the square of the line D commensurable by the sixt of the tenth And forasmuch as the number B hath not vnto the number C that proportion that a square number hath to a square nuÌ„ber therfore the square of the line A hath not vnto the square of the line D that proportioÌ„ that a square number hath to a nuÌ„ber Wherfore by the 9. of the tenth the line A is vnto the line D incommensurable in length onely And so is found out the first line namely D incommensurable in length onely to the line geuen A. Agayne take by the 13. of the sixt the meane proportionall betwene the lines A and D and let the same be E. Wherfore as the line A is to the line D so is the square of the line A to the square of the line E by the Corollary of the 20. of the sixt But the line A is vnto the line D incommensurable in length Wherfore also the square of the line A is vnto the square of the line E incommensurable by the second part of the former proposition Now forasmuch as the square of the line A is incoÌ„meÌ„surable to the square of the line E it followeth by the definition of incommensurable lynes that the line A is incommensurable in power to the line E. Wherfore vnto the right line geuen and first set A which is a rationall line and which is supposed to haue such diuisions and so many partes as ye list to conceyue in minde as in this example 11 whereunto as was declared in the 5. definition of this booke may be compared infinite other lines either commensurable or incommensurable is found out the line D incommensurable in length onely Wherfore the line D is rationall by the sixt definitioÌ„ of this booke for that it is incommensurable in length onely to the line A which is the first line set and is by suppositioÌ„ rational There is also found out the line E which is vnto the same line A incommensurable not onely in length but also in power which line E compared to the rationall line A is by the definition irrationall For Euclide alwayes calleth those lines irrationall which are incommensurable both in length and in power to the line first set and by supposition rationall ¶ The 9. Theoreme The 12. Proposition Magnitudes commensurable to one and the selfe same magnitude are also commensurable the one to the other SVppose that either of these magnitudes A and B be commensurable vnto the magnitude C Then I say that the magnitude A is commensurable vnto the magnitude B. For â—orasmuch as the
magnitude A is commensurable vnto the magnitude C therefore by the 5. of the tenth A hath vnto C such proportion as number hath to number Let A haue vnto C that proportion that the number D hath to the number E. Againe forasmuch as B is commensuâ—able vnto C therefore by the selfe same C hath vnto B that proportion that number hath to number Let C haue vnto B that proportion that the number F hath vnto the number G. Now then take the least numbers in continuall proportion and in these proportions geuen namely that the number D hath to the number E and that the number F hath to the number G by the 4. of the eight which let be the numbers â— K L. So that as the number D is to the number E so let the number H be to the number K and as the nuÌ„ber F is to the nuÌ„ber G so let the nuÌ„ber K be to the nuÌ„ber L. Now for that as A is to C so is D to E but as D is to E so is H to K therfore as A is to C so is H to K. Againe for that as C is to B so is F to G but as F is to G so is K to L therefore as C is to B so is K to L. But it is now proued that as A is to C so is H to K. Wherefore of equalitie by the 22. of the fift as A is to B so is the number H to the number L. Wherefore A hath vnto B such proportion as number hath to number Wherefore by the sixt of the tenth the magnitude A is commensurable vnto the magnitude B. Magnitudes therefore commensurable to one and the selfe same magnitude are also commensurable the one to the other which was required to be proued ¶ An Assumpt If there be two magnitudes compared to one and the selfe same magnitude and if the one of them be commensurable vnto it and the other incommensurable those magnitudes are incommensurable the one to the other SVppose that there be two magnitudes namely A and B and let C be a certayne other magnitude And let A â—e commensurable vnto C and let B be commeÌ„surable vnto the selfe same C. Then I say that the magnitude A is incommensurable vnto B. For if A be commensurable vnto B forasmuch as A is also commeÌ„surable vnto C therefore by the 12. of the tenth B is also commeÌ„surable vnto C which is contrary to the supposition ¶ The 10. Theoreme The 13. Proposition If there be two magnitudes commensurable and if the one of them be incommensurable to any other magnitude the other also shall be incommensurable vnto the same SVppose that these two magnitudes A B be commensurable the one to the other and let the one of them namely A be incommensurable vnto an other magnitude namely vnto C. Then I say that the other magnitude also namely B is incommensurable vnto C. For if B be commensurable vnto C then forasmuch as A is commensurable vnto B therefore by the 12. of the tenth the magnitude A also is commensurable vnto the magnitude C. But it is supposed to be incommensurable vnto it which is impossible Wherefore the magnitudes B and C are not commensurable Wherefore they are incommensurable If therefore there be two magnitudes commensurable and if the one of them be incommensurable to any other magnitude the other also shal be incommensurable vnto the same which was required to be proued ¶ A Corollary added by Montaureus Magnitudes commensurable to magnitudes incommeÌ„surable are also incommensurable the one to the other Suppose that the magnitudes A and B be incommensurable the one to the other and let the magnitudâ— C be coÌ„mensurable to A and let the magnitude D be coÌ„mensurable vnto B. Then I say that the magnituâ—â—s C and D are incommensurable the one to the other For A and C are commensurable of which the magnitude A is incommensurable vnto B wherefore by this 13. proposition the magnitudes C and B are also incommensurable but the magnitudeâ— B and D are coÌ„mensurable wherefore by the same or by the former assumpt the magnitudes C and D are incommensurable the one to the other This corollary Theon vseth often times as in the 22. 26. and 36 propositions of this booke and in other propositions also ¶ An Assumpt Two vnequall right lines being geuen to fiâ—de out how much the greater is in power more then the lesse And like in sorte two right lines being geuen by this meanes may be founde out a right lyne which contayneth them both in power Suppose that the two right lines geuen be AD and DB. It is required to â—inde out a right lyne that contayneth them both in power Let the lines AB and DB be so put that they comprehend a right angle ADB and draw a right line from A to B. Now agayne it is manifest by the 47. of the â—irst that the line AB contayneth in power the lines AD and DB. ¶ The 11. Theoreme The 14. Proposition If there be sower right lines proportionall and if the first be in power more then the second by the square of a right line commensurable in length vnto the first the third also shal be in power more then the fourth by the square of a right line commensurable vnto the third And if the first be in power more then the second by the square of a right line incommensurable in length vnto the first the third also shall be in power more then the fourth by the square of a right line incommensurable in length to the third SVppose that these foure right lines A B C D be proportionall As A is to B so let C be to D. And let A be in power more then B by the square of the line E. And likewise let C be in power more then D by the square of the line F. Then I say that if A be commensurable in length vnto the line E C also shall be commensurable in length vnto the line F. And if A be incommensurable in length to the line E C also shall be incommensurable in length to the line F. For for that as A is to B so is C to D therefore as the square of the line A is to the square of the line B so is the square of the line C to the square of the line D by the 22. of the sixt But by supposition vnto the square of the line A are equall the squares oâ— the lines E and B and vnto the square of the line C are equall the squares of the of the lines D and F Wherefore as the squares of the lines E and B which are equall to the square of the line A are to the square of the line B so are the squares of the lines D and F which are equall to the square of the line C to the square of the line D by the seuenth of the fift Wherfore
Now forasmuch as D measureth AB and BC it also measureth the whole magnitude AC And it measureth AB Wherefore D measureth these magnitudes CA and AB Wherefore CA AB are commensurable And they are supposed to be incoÌ„mensurableâ— which is impossible Wherfore no magnitude measureth AB and BC. Wherefore the magnitudes AB and BC are incommensurable And in like sort may they be proued to be incommensurable if the magnitude AC be supposed to be incommensurable vnto BC. If therefore there be two magnitudes incommensurable composed the whole also shall be incommensurable vnto either of the two partes component and if the whole be incommensurable to one of the partes component those first magnitudes shall be incommensurable which was required to be proued ¶ A Corollary added by Montaureus If an whole magnitude bee incommensurable to one of the two magnitudes which make the whole magnitude it shall also be incommensurable to the other of the two magnitudes For if the whole magnitude AC be incoÌ„mensurable vnto the magnitude BC then by the 2 part of this 16. Theorâ—me the magnitudes AB and BC shall be incommensurable Wherefore by the first part of the same Theoreme the magnitude AC shall be incommensurable to either of these magnitudes AB and BC. This Corollary 〈◊〉 vseth in the demonstration of the â—3 Theoreme also of other Propositions ¶ An Assumpt If vpon a right line be applied a parallelogramme wanting in figure by a square the parallelogramme so applied is equall to that parallelogramme which is contayned vnder the segmentes of the right line which segmentes are made by reason of that application This Assumpt I before added as a Corollary out of Flussates after the 28. Proposition of the sixt booke ¶ The 14. Theoreme The 17. Proposition If there be two right lines vnequall and if vpon the greater be applied a parallelogramme equall vnto the fourth part of the square of the lesse line and wanting in figure by a square if also the parallelogramme thus applied deuide the line where vpon it is applied into partes commensurable in length then shall the greater line be in power more then the lesse by the square of a line commensurable in length vnto the greater And if the greater be in power more then the lesse by the square of a right line commensurable in length vnto the greater and if also vpon the greater be applied a parallelograÌ„me equall vnto the fourth part of the square of the lesse line and wanting in figure by a square then shall it deuide the greater line into partes commensurable But now suppose that the line BC be in power more then the line A by the square of a line commensurable in length vnto the line BC. And vpon the line BC let there be applied a rectangle parallelograme equall vnto the fourth part of the square of the line A and wanting in figure by a square and let the sayd parallelograme be that which is contained vnder the lines BD and DC Then must we proue that the line BD is vnto the line DC commensurable in length The same constructions and suppositions that were before remayning we may in like sort proue that the line BC is in power more then the line A by the square of the line FD. But by suppositioÌ„ the line BC is in power more theÌ„ the line A by the square of a line coÌ„mensurable vnto it in length Wherfore the line BC is vnto the line FD coÌ„mensurable in length Wherefore the line composed of the two lines BF and DC is coÌ„mensurable in length vnto the line FD by the second part of the 15. of the tenth Wherefore by the 12. of the tenth or by the first part of the 15. of the tenth the line BC is commensurable in length to the line composed of BF and DC But the whole line conposed BF and DC is commensurable in length vnto DC For BF as before hath bene proued is equall to DC Wherefore the line BC is commensurable in length vnto the line DC by the 12. of the tenth Whâ—â—â—fore also the line BD is commensurable in length vnto the line DC by the second part of thâ— 15. of the teâ—th If therfore there be two right lines vnequall and if vpon the greater be appliâ—d a parallelograme equall vnto the fourth part of the square of the lesse and wanting in figure by a square if also the parallelograme thus applied deuide the line whereupon it is applied into partes commensurable in length then shall the greater line be in power more then the lesse by the square of a line commensurable in length vnto the greater And if the greater be in power more then the lesse by the square of a line commeÌ„surable in length vnto the greater and if also vpon the greater be applied a parallelograme equall vnto the fourth part of the square made of the lesse and wanting in figure by a square then shall it deuide the greater line into partes commensurable in length which was required to be proued Campanâ— after this proposition reacheth how we may redily apply vpon the line BC a parallelograme equall to the fourth part of the square of halfe of the line A and wanting in figure by a square after this maner Deuide the line BC into two lines in such sort that halfe of the line A shal be the meane proportionall betwene those two lines which is possible when as the line BC is supposed to be greater then the line A and may thus be done Deuide the line BC into two equal partes in the point E and describe vpon the line BC a semicircle BHC And vnto the line BC and from the point C erect a perpâ—dicular line CK and put the line CK equall to halfe of the line Aâ— And by the point K draw vnto the line EC a parallel line KH cutting the semicircle in the point H which it must needes cut foâ—asmuch as the line BC is greater then the line A And froÌ„ the point H draw vnto the line BC a perpendicular liâ—e HD which line HDâ— forasmuch as by the 34 of the first it is equall vnto the line KC shall also be equall to halfe of the line A draw the lines BH and HC Now then by the â—â— of the third the angle BHC is a right aâ—gle Wherefore by the corollary of the eight of the sixt booke the line HD is the meane proportionall betwene the lines BD and DC Wherefore the halfe of the line A which is equall vnto the line HD is the meane proportionall betwene the lines BD and DC Wherefore that which is contained vnder the lines BD and DC is equall to the fourth part of the square of the line A. And so if vpon the line BD be described a rectangle parallelograme hauing his other side equall to the line DC there shal be applied vpon the line BC a rectangle parallelograme equall vnto the square of halfe of the line A and wanting in figure by
first set they also must needes be at the least commensurable in power the one to the other For forasmuch as their squares are rationall they shall bee commensurable to the square of the rationall line first set Wherfore by the 12. of this booke they are also commensurable the one to the other Wherefore their lines are at the least commensurable in power the one to the other And it is possible also that they may be commeÌ„surable in leÌ„gth the one to the other For suppose that A be a rationall liâ—e first set and let the line B be vnto the same rationall line A commensurable in power onely that is incommensurable in length vnto it Let there be also an other line C commensurable in length to the lyne B which is possible by the principles of this booke Now by the 13. of the tenth it is manifest that the line C is incommensurable in length vnto the line A. But the square of the line A is coÌ„meÌ„surable to the square of the line B by supposition and the square of the line C is also commensurable to the square of the line B by supposition Wherefore by the 12. of this booke the square of the line C is commensurable to the square of the line A. Wherfore by the definition the line C shall be rationall commensurable in power onely to the line A as also is the line B. Wherefore there are geuen two rationall lines commensurable in power onely to the rationall line first set and commeÌ„surable in length the one to the other Here is to be noted which thing also we before noted in the definitions that Campane and others which followed him brought in these phrases of speaches to call some lynes rationall in power onely and other some rationall in length and in power which we cannot finde that Euclide euer vsed For these wordes in length and in power are neuer referred to rationalitie or irrationalitie but alwayes to the commensurabilitie or incommensurablitie of lines Which peruerting of wordes as was there declared hath much increased the difficulty and obscurenes of this booke And now I thinke it good agayne to put you in minde that in these propositions which follow we must euer haue before our eyes the rationall line first set vnto which other lines compared are either rationall or irrationall according to their commensurability or incommensurabilitie ¶ The 16. Theoreme The 19. Proposition A rectangle figure comprehended vnder right lines commensurable in lengthe being rationall according to one of the foresaide wayes is rationall SVppose that this rectangle figure AC be comprehended vnder these right lines AB and BC being commensurable in length and rationall according to one of the foresaid wayes Then I say that the superficies AC is rationall describe by the 46. oâ— the first vpon the line AB a square AD. Wherfore that square AD is rationall by the definition And forasmuch as the line AB is commensurable in length vnto the line BC and the line AB is equall vnto the lyne BD therefore the lyne BD is commensurable in length vnto the line BC. And as the line BD is to the line BC so is the square DA to the superficies AC by the first of the sixt but it is proued that the line BD is commensurable vnto the line BC wherfore by the 10. of the tenth the square DA is commensurable vnto the rectangle superficies AC But the square DA is rationall wherfore the rectangle superficies AC also is rationall by the definition A rectangle figure therfore comprehended vnder right lines commensurable in length beyng rationall accordyng to one of the foresayd wayes is rationall which was required to be proued Where as in the former demonstration the square was described vpon the lesse line we may also demonstrate the Proposition if we describe the square vpon the greater line and that after thys maner Suppose that the rectangle superficies BC be contayned of these vnequall lines AB and AC which let be rationall commensurable the one to the other in length And let the line AC be the greater And vpon the line AC describe the square DC Then I say that the parallelogramme BC is rationall For the line AC is commensurable in length vnto the line AB by supposition and the line DA is equall to the line AC Wherefore the line DA is commensurable in length to the line AB But what proportion the line DA hath to the line AB the same hath the square DC to the paraâ—lelogramme Câ— by the first of the sixt Wherefore by the 10. of this booke the square DC is commensurable to the parallelogramme CB. But it is manifest that the square DC is rationall for that it is the square of a rationall line namely AC Wherefore by the definition the parallelograÌ„me also CB is rationall Moreouer forasmuch as those two former demonstrations seeme to speake of that parallelograÌ„me which is made of two lines of which any one may be the liâ—e first set which is called the first rationall line from which we sayd ought to be taken the measures of the other lines compared vnto it and the other is commeÌ„surable in length to the same first rationall line which is the first kinde of rationall lines coÌ„mensurable in length I thinke it good here to set an other case of the other kinde of rationall lines of lines I say rationall coÌ„mensurable in length compared to an other rationall line first set to declare the generall truth of this Theoreme and that we might see that this particle according to any of the foresayd wayes was not here in vaine put Now then suppose first a rationall line AB Let there be also a parallelograÌ„me CD contayned vnder the lines CE and ED which lines let be rationall that is commensurable in length to the â—irst rationall line propounded AB Howbeit let those two lines CE and ED be diuers and vnequall lines vnto the first rationall line AB Then I say that the parallelogramme CD is rationall Describe the square of the line DE which let be DF. First it is manifest by the 12. of this booke that the lines CE ED are commensurable in leÌ„gth the one to the other For either of them is supposed to be commensurable in length vnto the line AB But the line ED is equall to the line EF. Wherefore the line CE is commensurable in length to the line BF But 〈◊〉 the line CE is â—o the line â— F â—o is the parallelogramme CD to the square DF by the first of the sixt Whâ—refore by the 10. of this booke the parallelogramme CD shall be commensurable to the square DF. But the square DF is commensurable to the square of the line AB which is the first rationall line propounded Wherfore by the 12. of this booke the parallelogramme CD is commensurable to the square of the line AB But the square of
the line AB is rationall by the definition Wherfore by the definition also of rationall figures the parallelogramme CD shall be rationall Now resteth an other caâ—e of the thirde kinde of rationall lines commensurable in length the one to the other which are to the rationall line AB first set commensurable in power onely and yet are therfore rationall lines And let the lines CE and ED be coÌ„mensurable in length the one to the other Now then let the selfe same construction remaine that was in the former so that let the lines CE and ED be rationall commensurable in power onely vnto the line AB But let them be commensurable in length the one to the other Then I say that in this case also the parallelogramme CD is rationall First it may be proued as before that the parallelogramme CD is commensurable to the square DF. Wherfore by the 12. of this booke the parallelogramme CD shall be commensurable to the square of the line ABâ— But the square of the line AB is rationall Wherefore by the definition the parallelograÌ„me CD shall be also rationall This case is well to be noted For it serueth to the demonstration and vnderstanding of the 25. Proposition of this booke ¶ The 17. Theoreme The 20. Proposition If vpon a rationall line be applied a rationall rectangle parallelogramme the other side that maketh the breadth thereof shall be a rationall line and commensurable in length vnto that line wherupon the rationall parallelogramme is applied SVppose that this rationall rectangle parallelogramme AC be applied vpon the line AB which let be rationall according to any one of the foresaid wayes whether it be the first rationall line set or any other line commensurable to the rationall line first set and that in length and in power or in power onely for one of these three wayes as was declared in the Assumpt put before the 19. Proposition of this booke is a line called rationall and making in breadth the line BC. Then I say that the line BC is rationall and commensurable in length vnto the line BA Desrcribe by the 46. of the first vpon the line BA a square AD. Wherfore by the 9. definitioÌ„ of the tenth the square AD is rationall But the parallelogramme AC also is rationall by supposition Wherefore by the conuersion of the definition of rationall figures or by the 12. of this booke the square DA is commensurable vnto the parallelogramme AC But as the square DA is to the parallelogramme AC so is the line DB to the line BC by the first of the sixt Wherfore by the 10. of the tenth the line DB is commensurable vnto the line BC. But the line DB is equall vnto the line BA Wherefore the line AB is coÌ„mensurable vnto the line BC. But the line AB is rationall Wherefore the line BC also is rationall and commensurable in length vnto the line BA If therefore vpon a rationall line be applied a rationall rectangle parallelogramme the other side that maketh the breadth therof shall be a rationall line commensurable in length vnto that line whereupon the rationall parallelogramme is applied which was required to be demonstrated ¶ An Assumpt A line contayning in power an irrationall superficies is irrationall Suppose that the line AB coÌ„taine in power an irrationall superficies that is let the square described vpon the line AB be equall vnto an irrationall superficies Then I say that the line AB is irrationall For if the line AB be rationall theÌ„ shall the square of the line AB be also rationall For so was it put in the definitions But by supposition it is not Wherefore the line AB is irrationall A line therefore contayning in power an irrationall superficies is irrationall ¶ The 18. Theoreme The 21. Proposition A rectangle figure comprehended vnder two rationall right lines commensurable in power onely is irrationall And the line which in power contayneth that rectangle figure is irrationall is called a mediall line SVppose that this rectangle figure AC be comprehended vnder these rationall right lines AB and BC commensurable in power onely Then I say that the superficies AC is irrationall and the line which contayneth it in power is irrationall and is called a mediall line Describe by the 46. of the first vpon the line AB a square AD. Wherefore the square AD is rationall And forasmuch as the line AB is vnto the line BC incommensurable in length for they are supposed to be commensurable in power onely and the line AB is equall vnto the line BD therefore also the lineâ— BD is vnto the line BC incommensurable in length And 〈◊〉 â—hâ— lin◠〈…〉 is to the line â— C so 〈◊〉 the square AD to the parallelogramme AC by the first of the fiuâ— Wherefore by the 10. of the tenth the square DA is vnto the parallelogramme AC incommensurable But the square DA is rationall Wherefore the parallelogramme AC is irrationall Wherefore also the line that contayneth the superficies AC in power that is whose square is equall vnto the parallelogramme AC is by the Assumpt going before irrationall And it is called a mediall line for that the square which is made of it is equall to that which is contayned vnder the lines AB and BC and therefore it is by the second part of the 17. of the sixt a meane proportionall line betwene the lines AB and BC. A rectangle figure therefore comprehended vnder rationall right lines which are commensurable in power onely is irrationall And the line which in power contayneth that rectangle figure is irrationall and is called a mediall line At this Proposition doth Euclide first entreate of the generation and production of irrationall lines And here he searcheth out the first kinde of them which he calleth a mediall line And the definition therof is fully gathered and taken out of this 21. Proposition which is this A mediall line is an irrationall line whose square is equall to a rectangled figure contayned of two rationall lines commensurable in power onely It is called a mediall line as Theon rightly sayth for two causes first for that the power or square which it producethâ— is equall to a mediall superficies or parallelogramme For as that line which produceth a rationall square is called a rationall line and that line which produceth an irrationall square or a square equall to an irrationall figure generally is called an irrationall line so iâ— thaâ— line which produceth a mediall square or a square equall to a mediall superficies called by speciall name a mediall line Secondly it is called a mediall line because it is a meane proportionall betwene the two lines coÌ„mensurable in power onely which comprehend the mediall superficies ¶ A Corollary added by Flussates A rectangle parallelogramme contayned vnder a rationall line and an â—rrationall line is irrationall For if the line AB be rationall and
to the number CE so is the square of the line AB to the square of the line AF therefore by conuersion by the corollary of the 19. of the fifte as the number CD is to the number DE so is the square of the line AB to the square to the line FB But the number C D hath not to the number DE that proportion that a squarâ— nâ—mbeâ— hâ—th to a square number Wherefore neither also the square of the line AB hath to the square of the line BF that proportion that a square number hath to a square number Wherefore the line AB is by the 9 of the tenth incommensurable in length to the line BF And the line AB is in power more then the line AF by the square of the right line BF which is incommensurable in length vnto the line AB Wherfore the lines AB and AF are rationall commensurable in power onely And the line AB is in power more then the line AF by the square of the line FB which is commensurable in length vnto the line AB â— which was required to be done ¶ An Assumpt If there be two right lines hauing betwene them selues any proportion as the one right line is to the other so is the parallelograme contained vnder both the right lines to the square of the lesse of those two lines Suppose that these two right AB and BC be in some certaine proportion Then I say that as the line AB is to the line BC so is the parallelograme contained vnder AB and BC to the square of BC. Describe the square of the line BC and let the same be CD and make perfect the parallelograme AD now it is manifest that as the line AB is to the line BC so is the parallelograme AD to the parallelograme or square BE by the first of the sixt But the parallelograme AD is that which is bontained vnder the lines AB and BC for the line BC is equall to the line BD and the parallelograme BE is the square of the line BC. Wherefore as the line AB is to the line BC so is the parallelograme coutained vnder the lines AB and BC to the square of the line BC which was required to be proued ¶ The 8. Probleme The 31. Proposition To finde out two mediall lines commensurable in power onely comprehending a rationall superficies so that the greater shall be in power more then the lesse by the square of a line commensurable in length vnto the greater LEt there be taken by the 29. of the tenth two rationall lines commensurable in power onely A and B so that let the line A being the greater be in power more then the line B being the lesse by the square of a line commensurable in length vnto the line A â— And let the square of the line C be equall to the parallelograme contained vnder the lines A and B which is done by finding out the meane proportionall line namely the line C betwene the lines A and B by the 13. of the sixt Now the parallelograme contained vnder the lines A and B is mediall by the 21. of this booke Wherefore by the corollary of the 23. of the tenth the square also of the line C is mediall Wheâ—fore the line C also is mediall Vnto the square of the line B let the parallelograme contained vnder the lines C and D be equall by finding out a third line proportionall namely the line D to the two lines C and B by the 11. of the sixt But the square of the line B is rationall Wherfore the parallelograme contained vnder the line C and D is rationall And for that as the line A is to the line B so is the parallelograme contained vnder the lines A and B to the square of the line B by the assumpt going before But vnto the parallelograme contained vnder the lines A and B is equall the square of the line C and vnto the square of the line B is equal the parallelograme contained vnder the lines C and D as it hath now bene proued therefore as the line A is to the line B so is the square of the line C to the parallelograme contained vnder the lines C D. But as the square of the line C is to that which is contayned vnder the lines C and D so is the line C to the line D. Wherefore as the line A is to the line B so is the line C to the line D. But by supposition the line A is commensurable vnto the line B in power onely Wherefore by the 11. of the tenth the line C also is vnto the line D commensurable in power onely But the line C is mediall Wherefore by the 23â— of the tenth the line D also is mediall And for that as the line A is to the line B so is the line C to the line D but the line A is in power more then the line B by the square of a line commensurable in length vnto the line A by supposition Wherefore the line C also is in power more then the line D by the square of a line commensâ—rable in length vnto the line C. Wherefore there are found out two mediall lines C and D commensurable in power onely comprehending a rationall superfiâ—ies and the line C is in power more then the line D by the square of a line commensurable in length vnto the line C. And in like sort may be found out two mediall lines commensurable in power onely contayning a rationall superficies so that the greater shal be in power more theÌ„ the lesse by the square of a line incoÌ„mensurable in leÌ„gth to the greater namely when the line A is in power more theÌ„ the line B by the square of a line incoÌ„mensuraâ—le in length vnto the line A which to do is taught by the 30. of this booke The selfe same construction remaining that part of this proposition froÌ„ these wordes And for that as the line A is to the line B to these wordes But by supposition the line A is commensurable vnto the line B may more easely be demonstrated after this maner The lines C B D are in continuall proportion by the second part of the 17. of the sixt But the lines A C D are also in continuall proportion by the same Wherefore by the 11. of the fifth as the line A is to the line C so is the line B to the line D. Wherfore alternately as the line A is to the line B so is the line C to the line D. c. which was required to be doone ¶ An assumpt If there be three right lines hauing betwene them selues any proportion as the first is to the third so is the parallelograme contained vnder the first and the second to the parallelograme contained vnder the second and the third Suppose that these three lines AB B C and CD be in some certayne proportion Then I say that as the line AB is to
line is when the square of the greater part excedeth the square of the lesse part by the square of a line coÌ„mensurable in length vnto it And neither part is commensurable in length to the rationall line geueÌ„ As suppose the line CE to be a binomiall line whose partes are ioyned together in the poynt D and let the square of the line CD the greater part exceede the square of the lesse part DE by the square of the line FG and let the line FG be commensurable in length to the line CD the greater part of the binomiall Moreouer let neither the greater part CD nor the lesse part DE be commensurable in length to the rationall line AB then is the line CE by this definition a third binomiall line A fourth binomiall line is when the square of the greater part exceedeth the square of the lesse by the square of a line incommensurable in length vnto the greater part And the greater is also commensurable in length to the rationall line As let the line CE be a binomiall line whose partes let be CD DE let the square of the line CD the greater part exceede the square of the line DE the lesse by the square of the line FG. And let the line FG be incommensurable in length to the line CD the greater Let also the line CD the greater part be commensurable in length vnto the ratâ—onall line AB Then by this definition the line CE is a fourth binomiall line A fift binomiall line is when the square of the greater part exceedeth the square of the lesse part by the square of a line incommensurable vnto it in length And the lesse part also is commensurable in length to the rationall line geuen As suppose that CE be a binomiall line whose greater part let be CD and let the lesse part be DE. And let the square of the line CD excede the square of the line DE by the square of the line FG which let be incoÌ„mensurable in length vnto the line CD the greater part of the binomiall line And let the line DE the second part of the binomiall line be commensurable in length vnto the rationall line AB So is the line CE by this definition a fift binomiall line A sixt binomiall line is when the square of the greater part exceedeth the square of the lesse by the square of a line incommensurable in length vnto it And neither part is commensurable in length to the rationall line geuen As let the line CE be a binomiall line deuided into his names in the point D. The square of whose greater part CD let exceede the square of the lesse part DE by the square of the line FG and let the line FG be incommensurable in length to the line CD the greater part of the binomiall line Let also nâ—ither CD the greater part nor DE the lesse part be commensurable in length to the rationall line AB And so by this definition the line CE is a sixt binomiall line So ye see that by these definitions their examples and declarations all the kindes of binomiall lines are made very playne This is to be noted that here is nothing spoken of those lines both whose portions aâ—e comâ—ensurable in length vnto the rationall line first set for that such lines cannot be binomiall lines â—or binomiall lines are composed of two rationall lines commensurable in Power onely by the 36. of this booke But lines both whose portions are commâ—nsurable in length to the rationall line first set are not binomiall lines For that the partes of such lines should by the 12. of this booke be commensurable in length the one to the other And so should they not be such lines as are required to the composition of a binomiall line Moreouer such lines should not be irrationall but rationall for that they are commensurable tâ—â—ch of the parts whereof they are coÌ„posed by the 15. oâ— this booke And therefore they should be rationall for that the lines which composâ— them are rationall ¶ The 13. Probleme The 48. proposition To finde out a first binomiall line TAke two numbers AC and CB let them be such that the number which is made of them both added together namely AB haue vnto one of them 〈◊〉 vnto BC that proportion that a square number hath to a square numbâ—r â—ut vnto the other namely vnto CA let it not haue that proportion that a square number hath to a square number such as is euery square number which may be deuided into a square number and into a number not square Take also a certayne rationall line and let the same be D. And vnto the line D let the line EF be commensurable in length Wherefore the line EF is rationall And as the number AB is to the nuÌ„ber AC so let the square of the line EF be to the square of an other â—iâ—e namely of FG by the corollary of the sixt of the teÌ„th Wherfore the square of the line EF hath to the square of the line FG that proportion that number hath to number Wherefore the square of the line EF is commensurable to the square of the line FG by the 6. of this booke And the line EF is rationall Wherefore the line FG also is rationall And forasmuch as the number AB hath not to the number AC that proportion that a square number hath to a square number neither shal the square of the line EF haue to the square of the line FG that proportion that a square number hath to a square number Wherefore the line EF is incommensurable in length to the line FG by the 9. of this booke Wherefore the lines EF and FG are rationall commensurable in power onely Wherefore the whole line EG is a binomiall line by the 36. of the tenth I say also that it is a â—irst binomiall line For for that as the 〈◊〉 BA is to the number AC so is the square of the line EF to the square of the line â—G but the number BA is greater then the number AC wherefore the square of the line â—F is also greater then the square oâ— the line FG. Vnto the square of the line EF let the squares of the lines FG and H be equall which how to finde out is taught in the assumpt put â—ftâ—r the 13. of the tâ—nth And fâ—r that as thâ— number BA is to the number AC so is the square of the line EF to the square of the line FG therefore by coâ—uersion or eâ—ersion of proportion by the corollary of the 19. of the fift as the number AB is to the number BC so is the square of the line EF to the square of the line H. But the number AB hath to the number BC that proportion that a square number hath to a square number Wherefore also the square of the line EF hath to the square of the line H
the definition of a first binomiall line seâ— before the 48. proposition of this booke the line DG is a first binomiall line which was required to be proued This proposition and the fiue following are the conuerses of the sixe former propositions ¶ The 43. Theoreme The 61. Proposition The square of a first bimediall line applied to a rationall line maketh the breadth or other side a second binomiall line SVppose that the line AB be a first bimediall line and let it be supposed to be deuided into his partes in the point C of which let AC be the greater part Take also a rationall line DE and by the 44. of the first apply to the line DE the parallelograÌ„me DF equall to the square of the line AB making in breadth the line DG Then I say that the line DG is a second binomiall line Let the same constructions be in this that were in the Proposition going before And forasmuch as the line AB is a first bimediall line and is deuided into his partes in the point C therefore by the 37. of the tenth the lines AC and CB are mediall commensurable in power onely coÌ„prehending a rationall superficies Wherfore also the squares of the lines AC and CB are mediall Wherefore the parallelogramme DL is mediall by the Corollary of the 23. of the tenth and it is applied vppon the rationall line DE. Wherefore by the 22. of the tenth the line MD is rationall and incommensurable in length to the line DE. Againe forasmuch as that which is coÌ„tayned vnder the lines AC and CB twise is rationall therefore also the parallelogramme MF is rationall and it is applied vnto the rationall line ML Wherefore the line MG is rationall and commensurable in length to the line ML that is to the line DE by the 20. of the tenth Wherefore the line DM is incommensurable in length to the line MG and they are both rationall Wherefore the lines DM and MG are rationall commensurable in power onely Wherefore the whole line DG is a binomiall line Now resteth to proue that it is a second binomiall line Forasmuch as the squares of the lines AC and CB are greater then that which is contayned vnder the lines AC and CB twise by the Assumpt before the 60. of this booke therefore the parallelogramme DL is greater then the parallelogrrmme MF Wherefore also by the first of the sixt the line DM is greater then the line MG And forasmuch as the square of the line AC is commensurable to the square of the line CB therefore the parallelogramme DH is commensurable to the parallelogramme KL Wherefore also the line DK is commensurable in length to the line KM And that which is contayned vnder the lines DK and KM is equall to the square of the line MN that is to the fourth part of the square of the line MG Wherefore by the 17. of the tenth the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM and the line MG is commensurable in length to the rationall line put namely to DE. Wherefore the line DG is a second binomiall line which was required to be proued ¶ The 44. Theoreme The 62. Proposition The square of a second bimediall line applied vnto a rationall line maketh the breadth or other side therof a third binomiall lyne SVppose that AB be a second bimediall line and let AB be supposed to be deuided into his partes in the point C so that let AC be the greater part And take a rationall line DE. And by the 44. of the first vnto the line DE apply the parallelogramme DF equall to the square of the line AB and making in breadth the line DG Then I say that the line DG is a third binomiall line Let the selfe same constructions be in this that were in the propositions next going before And forasmuch as the line AB is a second bimediall line and is deuided into his partes in the point C therfore by the 38. of the tenth the lines AC and CB are medials commensurable in power only compreheÌ„ding a mediall superficies Wherfore that which is made of the squares of the lines AC and CB added together is mediall and it is equall to the parallelogramme DL by construction Wherefore the parallelogramme DL is mediall and is applied vnto the rationall line DE wherfore by the 22. of the tenth the line MD is rationall and incommensurable in length to the line DE. And by the lyke reason also the line MG is rationall and incommensurable in length to the line ML that is to the line DE. Wherfore either of these lines DM and MG is rational and incommensurable in length to the line DE. And forasmuch as the line AC is incommensurable in length to the line CB but as the line AC is to the line CB so by the assumpt going before the 22. of the tenth is the square of the line AC to that which is contained vnder the lines AC and CB. Wherfore the square of the line AC is incâ—mmmensurable to that which is contayned vnder the lines AC and CB. Wherfore that that which is made of the squares of the lines AC and CB added together is incommensurable to that which is contained vnder the lines AC and CB twise that is the parallelogramme DL to the parallelogramme MF Wherfore by the first of the sixt and 10. of the tenth the line DM is incommensurable in length to the line MG And they are proued both rationall wherfore the whole line DG is a binomiall line by the definition in the 36. of the tenth Now resteth to proue that it is a third binomiall line As in the former propositions so also in this may we conclude that the line DM is greater then the line MG and that the line DK is commensurable in length to the line KM And that that which is contained vnder the lines DK and KM is equall to the square of the line MN Wherfore the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM and neither of the lines DM nor MG is commensurable in length to the rational line DE. Wherfore by the definition of a third binomiâ—ll line the line DG is a third binomiall line which was required to be proued ¶ Here follow certaine annotations by M. Dee made vpon three places in the demonstration which were not very euident to yong beginners †The squares of the lines AC and Câ— are medials 〈◊〉 iâ— taught after the 21â— of this tenth and therâ—ore forasmuch as they are by supposition commeÌ„surable th' one to the other by the 15. of the teÌ„th the compound of them both is commensurable to ech part But the partes are medials therfore by the coâ—ollary of the 23. of the tenth the compound shall be
line Deuide the line AB into his partes in the point E. and let the rest of the construction be in this as it was in the former And for that as the line AB is to the line CD so is the line AE to the lyne CF and the line EB to the line FD therfore as the line AE is to the lyne CF so is the line EB to the line FD but the line AB is commensurable to the line CD Wherfore also the lyne AE is commensurable to the lyne CF and likewise the line EB to the line FD. And for thâ—â— as the line AE is to the line CF so is the line EB to the line FD therfore alternately as the lyne AE is to the line EB so is the line CF to the lyne FD. Wherfore by the 22. of the sixt as the square of the lyne AE is to the square of the line EB so is the square of the line CF to the square of the line FD. Wherfore by composition by the 18. of the fift as that which is made of the squares of the lynes Aâ— and Eâ— added together is to the square of the lyne EB so is that which is made of the squareâ— of the lyneâ— Câ— and FD added together to the square of the lyne FD. Wherefore by contrary proportion as the square of the line EB is to that which is made of the squares of the lines AE and Eâ— added together so is the square of the lyne FD to that which is made of the squares of the lynes CF and FD added together Wherfore alternately as the square of the line EB is to the square of the lyne FD so is that which is made of the squares 〈◊〉 the lâ—nes AE and EB added together to that whiche is made of the squares of the lynes CF and FD added together But the square of the lyne EB is coÌ„mensurable to the square of the lyne FD for it hath already bene proued that the lines EB and FD are coÌ„meÌ„surable Wherfore that which is made of the squares of the lines AE EB added together is commeÌ„surable to that which is made of the squares of Câ— FD added together But that which is made of the squares of the lines AE and EB added together is rationall by suppositioÌ„ Wherfore that which is made of the squares of the lynes CF and FD added together is also rationall And as the lyne AE is to the lyne EB so is the line CF to the lyne FD But as the lyne AE is to the lyne EB so is the square of the line A 〈…〉 contayned vnder the lynes AE and EB therfore at the lyne CF is to the lyne FD so is the square of the lyne AE to the parallelogramme contayned vnder the lines AE and EB as the lyne CF is to the lyne FD so is the square of the lyne CF to the parallelograÌ„me contayned vnder the lynes â—F FD. Wherfore as the square of the lyne AE is to the parallelograÌ„me conâ—â—â—â—ed vnder the lines AE and EB so is the square of the lyne CF to the parallelogramme coÌ„tayned vnder the lynes CF and FD. Wherâ—or◠〈◊〉 â—s the square of the line AE is to the square of the lyne CF so is the parallelogramme contained vnder the lynes AE and EB to the parallelogramme 〈◊〉 vndeâ— the lines â—â— and â—â— But the square of the lyne AE is commensurable to the square of the lyne CF for it is already prâ—â—â—d that the lynes AE and CF are commeÌ„surable Wherefore the parallelogramme contayned vnder the lynes AE and EB is commensurable to the parallelogramme contayned vnder the lynes CF and FD. But the parallelogramme contayned vnder the lines AE and EB is mediall by suppoâ—ition Wherfore the parallelogramme contayned vnder the lynes CF and â—D also is mediall And as it hath already bene proued as the line AE is to the lyne EB so is the lyne CF to the lyne FD. But the lyne AE was by supposition incommensurable in power to the line EB Wherfore by the 10. of the tenth the lyne CF is incommensurable in power to the lyne FD. Wherfore the lynes CF and FD are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contayned vnder them mediall Wherfore the whole lyne CD is by the 39. of the tenth a greater lyne Wherfore a lyne commensurable to a greater lyne is also a greater lyne which was required to be demonstrated An other more briefe demonstration of the same after Campane Suppose that A be a greater line vnto which let the line B be commeÌ„surable either in length and power or in power onely And take a rational line CD And vpon it apply the superficies Câ— equall to the square of the line A and also vpoÌ„ the line FE which is equall to the rationall line CD apply the parallelogramme FG equall to the square of the line B. And forasmuch as the squares of the two lines A and â— are commensurable by supposition the superficies Câ— shal be commensurable vnto the superficies FG and therefore by the first of the sixt and tenth of this booke the line DE is commensurable in length to the line GB And forasmuch as by the â—3 of this booke the line DE is a fourth binomiall line therefore by the â—6 of this booke the line GE is also a fourth binomiall line wherefore by the 57. of this booke the line B which contayneth in power the superficies FG is a greater line ¶ The 51. Theoreme The 69. Proposition A line commensurable to a line contayning in power a rationall and a mediall is also a line contayning in power a rationall and a mediall SVppose that AB be a line contayning in power a rationall and a mediall And vnto the line AB let the line CD be commensurable whether in length and power or in power onely TheÌ„ I say that the line CD is a line coÌ„tayning in power a rationall a mediall Duide the line AB into his parts in the poynt E. Wherfore by the 40. of the tenth the lines AE and EB are incommensurable in power hauing that which is made of the squares of them added together medial and that which is contayned vnder theÌ„ nationall Let the same construction be in this that was in the former And in like sort we may proue that the lines CF and FD are incommensurable in power and that that which is made of the square of the lines AE and EB is commensurable to that which is made of the squares of the lines CF and FD and that that also which is contayned vnder the lines AE and EB is commeÌ„surable to that which is contayned vnder the lines CF and FD. Wherefore that which is made of the squares of the lines CF and FD is mediall and that which is contayned vnder the lines CF and FD is rationall Wherefore the whole line CD is a line contayning in
BC are rationall for the lines AB and BC are put to be rationall wherfore the line AC is irrationall and is called a residuall line which was required to be proued An other demonstration after Campane Campane demonstrateth this Proposition by a figure more briefly after this mâ—ner Let the superficies EG be equall to the squares of the lines AB and BC added together which shall be rationall for that the lines AB and BC are supposed to be rationall coÌ„mensurable in power onely FroÌ„ which superficies take away the superficies DF equall to that which is conâ—â—yaâ—d vnder the lines AB DC twise which shall be mediall by the 21. of this booke Now by the 7. of the second the superficies FG is equall to the square of the line AC And forasmuch as the superficies EG is incommensurable to the superficies DF for that the one is rationall and the other mediall therefore by the 16. of this booke the 〈◊〉 superficies EG is incommensurable to the superficies FG. Wherfore the superficies FG is irrationall And therefore the line AC which contayneth it in power is irrationall which was required to be proued An annotation of P. Montaâ—reâ—s This Theoreme teacheth nothing els but that that portion of the greater name of a binomiall line which remayneth after the taking away of the lesse name from the greater name is irrationall which is called a residuall line that is to say if from the greater name of a binomiall line which greater name is a rationall line coÌ„mensurable in power onely to the lesse name be taken away the lesse name which selfe lesse name is also commensurable in power onely to the greater name which greater name this Theoreme calleth the whole line the rest of the line which remaineth is irrational which he calleth a residuall line Wherfore all the lines which are intreated in this Theoreme and in the fiue other which follow are the portions remayning of the greater partes of the whole lines which were intreated of in the 36.37.38.39.40.41 propositioÌ„s after the taking away the lesse part from the greater In this proposition is set forth the nature of the eight kinde of irrationall lines which is called a residuall line the definition whereof by this proposition is thus A residuall line is an irrationall line which remayneth when from a rationall line geuen is taken away a rationall line commensurable to the whole line in power onely ¶ The 56. Theoreme The 74. Proposition If from a mediall line be taken away a mediall line commensurable in power onely to the whole line and comprehending together with the whole line a rationall superficies the residue is an irrationall line and is called a first mediall residuall line Out of this proposition is taken the definition of the ninth kinde of irrationall lines which is called a first residuall mediall line the difinition whereof is thus A first residuall mediall line is an irrationall line which remayneth when from a mediall line is taken away a mediall line commensurable to the whole in power onely and the part taken away and the whole line contayne a mediall superficies An other demonstration after Campane Let the line DE be rationall vpon which apply the superficies DF equall to that which is contâ—ined vnder the lynes AB and BC twise and let the superficies GE be equal to that which is composed of the squares of the lynes AB and BC wherfore by the 7. of the second the superficies FG is equal to the square of the lyne AC And forasmuch as by supposition the superficies EG is mediall therfore by the 22. of the tenth the lyne DG is rationall coÌ„mensurable in power onely to the rational lyne DE. And forasmuch as by supposition the superficies EH is rational therfore by the 20. of the tenth the line DH is rational commensurable in length vnto the rationall line DE. Wherfore the lynes DG and DH are rationall commensurable in power only by the assumpt put before the 13. of this boke Wherfore by the 73 of this boke the lyne GH is a residuall lyne and is therefore irrationall Wherfore by the corollary of the 21. of this boke the superficies FG is irrational And therfore the line AC which coÌ„tayneth it in power is irrationall and is called a first medial residuall lyne ¶ The 57. Theoreme The 75. Proposition If from a mediall lyne be taken away a mediall lyne commensurable in power only to the whole lyne and comprehending together with the whole lyne a mediall superficies the residue is an irrationall lyne and is called a second mediall residuall lyne SVppose that AB be a mediall line and from AB take away a mediall line CB commensurable in power onely to the whole line AB and comprehending together with the whole line AB a mediall superficies namely the parallelogramme contained vnder the lines AB and BC. Then I say that the residue namely the line AC is irrationall and is called a second mediall residuall line Take a rationall line DI and by the 44. of the first vnto the line DI apply the parallelogramme DE equall to the squares of the lines AB BC and making in bredth the line DG And vnto the same line DI apply the parallelogramme DH equall to that which is coÌ„tained vnder the lines AB BC twise and makyng in breadth the line DF. Now the parallelogramme DH is lesse then the parallelogramme DE for that also the square of the lines AB and BC are greater then that which is contained vnder the lines AB and BC twise by the square of the line AC by the 7. of the second Wherfore the parallelogramme remayning namely FE is equal to the square of the line AC And forasmuch as the squares of the lines AB and BC are mediall therfore also the parallelogramme DE is mediall and is applied to the rationall line DI making in bredth the line DG Wherfore by the 22. of the tenth the line DG is rational and incommensurable in length to the line DI. Againe forasmuch as that which is contained vnder the lines AB and BC is mediall therfore also that which is contained vnder the lines AB and BC twise is mediall but that which is contained vnder the lines AB and BC twise is equall to the parallelogramme DH Wherfore the parallelogramme DH is mediall and is applied to the rationall line DI making in breadth the line DF. Wherfore the line DF is rationall and incommensurable in length to the line DI. And forasmuch as the lines AB and BC are coÌ„mensurable in power onely therfore the line AB is incommeÌ„surable in leÌ„gth to the line BC. Wherfore by the assumpt going before the 22. of the tenth and by the 10. of the tenth the square of the line AB is incommensurable to that which is contained vnder the lines AB and BC. But vnto the square of the line AB are commensurable
to the same and so the line BD is a sixt residuall line and the line KH is a sixt binomiall line Wherfore KH is a binomiall line whose names KF and FH are commensurable to the names of the residuall line BD namely to BC and CD and in the selfe same proportion and the binomiall line KH is in the selfe same order of binomiall lines that the residuall BD is of residuall lines Wherefore the square of a rationall line applied vnto a residuall line maketh the breadth or other side a binomiall line whose names are commensurable to the names of the residuall line and in the selfe same proportion and moreouer the binomiall line is in the selfe same order of binomiall lines that the residuall line is of residuall lines which was required to be demonstrated The Assumpt confirmed Now let vs declare how as the line KH is to the line EH so to make the line HF to the line FE Adde vnto the line KH directly a line equall to HE and let the whole line be KL and by the tenth of the sixt let the line HE be deuided as the whole line KL is deuided in the point H let the line HE be so deuided in the point F. Wherfore as the line KH is to the line HL that is to the line HE so is the line HF to the line FE An other demonstration after Flussas Suppose that A be a rationall line and let BD be a residuall line And vpon the line BD apply the parallelogramme DT equall to the square of the line A by the 45. of the first making in breadth the line BT Then I say that BT is a binominall line such a one as is required in the proposition Forasmuch as BD is a residuall line let the line coÌ„ueniently ioyned vnto it be GD Wherfore the lines BG and GD are rationall commensurable in power onely Vpon the rationall line BG apply the parallelogramme BI equall to the square of the line A and making in breadth the line BE. Wherefore the line BE is rationall and commensurable in length to the line BG by the 20. of the tenth Now forasmuch as the parallelogrammes BI and TD are equall for that they are eche equall to the square of the line A therfore reciprokally by the 14. of the sixth as the line BT is to the line BE so is the line BG to the line BD. Wherefore by conuersion of proportion by the corrollary of the 19. of the fifth as the line BT is to the line TE so is the line BG to the line GD As the line BG is to the line GD so let the line TZ be to the line ZE by the corrollary of the 10. of the sixth Wherefore by the 11. of the fifth the line BT is to the line TE as the line TZ is to the line ZE. For either of them are as the line BG is to the line GD Wherefore the residue BZ is to the residue ZT as the whole BT is to the whole TE by the 19. of the fifth Wherefore by the 11. of the fifth the line BZ is to the line ZT as the line ZT is to the line ZE. Wherfore the line TZ is the meane proportionall betwene the lines BZ and ZE. Wherefore the square of the first namely of the line BZ is to the square of the second namely of the line ZT as the first namely the line BZ is to the third namely to the line ZE by the corollary of the 20. of the sixth And for that as the line BG is to the line GD so is the line TZ to the line ZE but as the line TZ is to the line ZE so is the line BZ to the line ZT Wherefore as the line BG is to the line GD so is the line BZ to the line ZT by the 11. of the fifth Wherfore the lines BZ and ZT are commensurable in power onely as also are the lines BG and GD which are the names of the residuall line BD by the 10. of this booke Wherfore the right lines BZ and ZE are coÌ„mensurable in length for we haue proued that they are in the same proportion that the squares of the lines BZ and ZT are And therefore by the corollary of the 15. of this booke the residue BE which is a rationall line is commensurable in length vnto the same line BZ Wherefore also the line BG which is commensurable in length vnto the line BE shall also be commensurable in length vnto the same line EZ by the 12. of the tenth And it is proued that the line RZ is to the line ZT commensurable in power onely Wherefore the right lines BZ and ZT are rationall commensurable in power onely Wherefore the whole line BT is a binomiall line by the 36. of this booke And for that as the line BG is to the line GD so is the line BZ to the line ZT therefore alternately by the 16. of the fifth the line BG is to the line BZ as the line GD is to the line ZT But the line BG is commensurable in length vnto the line BZ Wherefore by the 10. of this booke the line GD is commensurable in length vnto the line ZT Wherefore the names BG and GD of the residuall line BD are commensurable in length vnto the names BZ and ZT of the binomial line BT and the line BZ is to the line ZT in the same proportion that the line BG is to the line GD as before it was more manifest And that they are of one and the selfe same order is thus proued If the greater or lesse name of the residuall line namely the right lines BG or GD be coÌ„mensurable in length to any rationall line put the greater name also or lesse namely BZ or ZT shal be commensurable in length to the same rationall line put by the 12. of this booke And if neither of the names of the residuall line be commensurable in length vnto the rationall line put neither of the names of the binomiall line shal be commensurable in length vnto the same rationall line put by the 13. of the tenth And if the greater name BG be in power more then the lesse name by the square of a line commensurable in length vnto the line BG the greater name also BZ shal be in power more then the lesse by the square of a line commensurable in length vnto the line BZ And if the one be in power more by the square of a line incommensurable in length the other also shal be in power more by the square of a line incommensurable in length by the 14. of this booke The square therefore of a rationall line c. which was required to be proued ¶ The 90. Theoreme The 114. Proposition If a parallelograÌ„me be coÌ„tained vnder a residuall line a binomiall lyne whose names are commensurable to the names of the residuall line and in the selâ—e same proportion the lyne which contayneth in power
that superficies is rationall SVppose that a parallelogramme be contained vnder a residuall line AB and a binomiall line CD and let the greater name of the binomiall line be CE and the lesse name be ED and let the names of the binomiall line namely CE and ED be commensurable to the names of the residuall line namely to AF and Fâ— and in the selfe same proportion And let the line which containeth in power that parallelograÌ„me be G. TheÌ„ I say that the line G is rational Take a rational line namely H. And vnto the line CD apply a parallelograÌ„me equal to the square of the line H and making in breadth the line KL Wherefore by the 112. of the tenth KL is a residuall line whose names let be KM and ML which are by the same coÌ„mensurable to the names of the binomiall line that is to CE and ED and are in the selfe same proportioÌ„ But by position the lines CE and ED are coÌ„mensurable to the lines AF and FB and are in the selfe same proportion Wherfore by the 12. of the tenth as the line AF is to the line FBâ— so is the line KM to the line ML Wherfore alternately by the 16. of the fift as the line AF is to the line KM so is the line BF to the line LM Wherfore the residue AB is to the residue KL as the whole AF is to the whole KM But the line AF is commensurable to the line KM for either of the lines AF and KM is commensurable to the line CE. Wherfore also the line AB is commensurable to the line KL And as the line AB is to the line KL so by the first of the sixt is the parallelogramme contained vnder the lines CD and AB to the parallelogramme contained vnder the lines CD and KL Wherfore the parallelogramme contained vnder the lines CD and AB is commensurable to the parallelogramme contained vnder the lines CD and KL But the parallelogramme contained vnder the lines CD and KL is equall to the square of the line H. Wherfore the parallelograÌ„me coÌ„tained vnder the lines CD AB is coÌ„mensurable to the square of the line H. But the parallelograÌ„me contained vnder the lines CD and AB is equall to the square of the line G. Wherfore the square of the line H is commensurable to the square of the line G. But the square of the line H is rationall Wherfore the square of the line G is also rationall Wherfore also the line G is rational and it containeth in power the parallelogramme contained vnder the lines AB and CD If therfore a parallelogramme be contained vnder a residuall line and a binomiall line whose names are commensurable to the names of the residuall line and in the selfe same proportion the line which containeth in power that superficies is rationall which was required to be proued ¶ Corollary Hereby it is manifest that a rationall parallelogramme may be contained vnder irrationall lines ¶ An otâ—â—r 〈…〉 Flussas 〈…〉 line â—D whosâ— names Aâ— and â—D let be commensurable in length vnto the names of the residuall line Aâ— which let be AF and FB And let the liâ—e AEâ— be to the line EDâ— in the same proportion that the line AF is to the line Fâ— And let the right line â— contayne in power the superficies Dâ— Then I say thaâ— the liâ—e â— is a rationall lin◠〈…〉 lâ—ne which lâ—â— bâ—â— And vpon the line â—â— describe by the 4â— of the first a parallelogramme eqâ—all to the squarâ— of the line â—â— and making in breadth the line DC Wherefore by the â—12 of this booke CD is a residuâ—ll lineâ— whose names Which let be â—â— and OD shall be coâ—mensurablâ— in leâ—gth vnto the names Aâ— and â—D and the line C o shall be vnto the line OD in the same proporâ—ion that the line AE is to the line EDâ— But as the line Aâ— is to the line â—D so by supposition is the line AF to the line FE Wherfore as the line CO is to the line OD so is the line AF to the line Fâ—â— Wherefore the lines CO and OD are commensurable with the lines Aâ— and â—â— by the â—â— of this boke Wherfore the residue namely the line CD is to the residue namely to the line Aâ— as the line CO is to the line AF by the 19. of the fifth But it is proued that the line CO is coÌ„mensurable vnto the line AF. Wherefore the line CD is commensurable vnto the line AB Wherefore by the first of the sixth the parallelogramme CA is commensurable to the parallelogramme Dâ— But the parallelogramme â—â— iâ— by construction rationall for it is equall to the square of the rationall line â— Whâ—refore the parallelogramme â—D â—s also ratâ—â—nâ—llâ— Wherâ—fore the line â— which by supposition coÌ„tayneth in power the superficies â—Dâ— is also rationall If therfore a parallelograÌ„me be contayned c which was required to be proued ¶ The 91. Theoreme The 115. Proposition Of a mediall line are produced infinite irrationall lines of which none is of the selfe same kinde with any of those that were before SVppose that A be a mediall line Then I say that of the line A may be produced infinite irrationall lines of which none shall be of the selfe same kinde with any of those that were before Take a rationall line B. And vnto that which is contained vnder the lines A and B let the square of the line C be equall by the 14. of the second â— Wherefore the line C is irrationall For a superficies contained vnder a rationall line and an irrationall line is by the Assumpt following the 38. of the tenth irrationall and the line which containeth in power an irrationall superficies is by the Assumpt going before the 21. of the tenth irrationall And it is not one and the selfe same with any of those thirtene that were before For none of the lines that were before applied to a rationall line maketh the breadth mediall Againe vnto that which is contained vnder the lines B and C let the square of D be equall Wherefore the square of D is irrationall Wherefore also the line D is irrationall and not of the self same kinde with any of those that were before For the square of none of the lines which were before applied to a rationall line maketh the breadth the line C. In like sort also shall it so followe if a man proceede infinitely Wherefore it is manifest that of a mediall line are produced infinite irrationall lines of which none is of the selfe same kinde with any of those that were before which was required to be proued An other demonstratioâ— Suppose that AC be a mediall line Then I say that of the line AC may be produced infinite irrationall lines of which none shall be of the selfe same kinde with any of those irrationall lines before named Vnto the line AC and from the point A
two other propositions going next before it so farre misplaced that where they are word for word before duâ—ly placed being the 105. and 106. yet here after the booke ended they are repeated with the numbers of 116. and 117. proposition Zambert therein was more faythfull to follow as he found in his greke example than he was skilfull or carefull to doe what was necessary Yea and some greke written auncient copyes haue them not so Though in deede they be well demonstrated yet truth disorded is halfe disgracedâ— especially where the patterne of good order by profession is auouched to be But through ignoraunce arrogancy and â—emerltie of vnskilfull Methode Masters many thinges remayne yet in these Geometricall Elementes vnduely tumbled in though true yet with disgrace which by helpe of so many wittes and habilitie of such as now may haue good cause to be skilfull herein will I hope ere long be taken away and thinges of importance wanting supplied The end of the tenth booke of Euclides Elementes ¶ The eleuenth booke of Euclides Elementes HITHERTO HATH â—VCLIDâ— IN THâ—Sâ— former bookes with a wonderfull Methode and order entreated of such kindes of figures superficial which are or may be described in a superficies or plaine And hath taught and set forth their properties natures generations and productions euen from the first roote ground and beginning of them namely from a point which although it be indiuisible yet is it the beginning of all quantitie and of it and of the motion and slowing therof is produced a line and consequently all quantitie coÌ„tinuall as all figures playne and solide what so euer Euclide therefore in his first booke began with it and from thence went he to a line as to a thing most simple next vnto a point then to a superficies and to angles and so through the whole first booke he intreated of these most simple and plaine groundes In the second booke he entreated further and went vnto more harder matter and taught of diuisions of lines and of the multiplication of lines and of their partes and of their passions and properties And for that rightlined â—igures are far distant in nature and propertie from round and circular figures in the third booke he instructeth the reader of the nature and conditioÌ„ of circles In the fourth booke he compareth figures of right lines and circles together and teacheth how to describe a figure of right lines with in or about a circle and contraâ—iwiâ—e a circle with in or about a rectiline figure In the fifth booke he searcheth out the nature of proportion a matter of wonderfull vse and deepe consideration for that otherwise he could not compare â—igure with figure or the sides of figures together For whatsoeuer is compared to any other thing is compared vnto it vndoubtedly vnder some kinde of proportion Wherefore in the sixth booke he compareth figures together one to an other likewise their sides And for that the nature of proportion can not be fully and clearely sene without the knowledge of number wherein it is first and chiefely found in the seuenth eight and ninth bookes he entreatâ—th of number of the kindes and properties thereof And because that the sides of solide bodyes for the most part are of such sort that compared together they haue such proportion the one to the other which can not be expresâ—ed by any number certayne and therefore are called irrational lines he in the teÌ„th boke hath writteÌ„ taught which lineâ— are coÌ„meÌ„surable or incoÌ„meÌ„surable the one to the other and of the diuersitie of kindes of irrationall lines with all the conditions proprieties of them And thus hath Euclide in these ten foresayd bokes fully most pleÌ„teously in a meruelous order taught whatsoeuer semed necessary and requisite to the knowledge of all superficiall figures of what sort forme so euer they be Now in these bookes following he entreateth of figures of an other kinde namely of bodely figures as of Cubes Piramids Cones Columnes Cilinders Parallelipipedons Spheres and such othersâ— and sheweth the diuersitie of theÌ„ the generation and production of them and demonstrateth with great and wonderfull art their proprieties and passions with all their natures and conditions He also compareth one oâ— them to an other whereby to know the reason and proportion of the one to the other chiefely of the fiue bodyes which are called regular bodyes And these are the thinges of all other entreated of in Geometrie most worthy and of greatest dignitie and as it were the end and finall entent of the whole are of Geometrie and for whose cause hath bene written and spoken whatsoeuer hath hitherto in the former bookes bene sayd or written As the first booke was a ground and a necessary entrye to all the râ—st â—ollowing so is this eleuenth booke a necessary entrie and ground to the rest which follow And as that contayned the declaration of wordes and definitions of thingeâ— requisite to the knowledge of superficiall figures and entreated of lines and of their diuisions and sections which are the termes and limites of superficiall figures so in this booke is set forth the declaration of wordes and definitions of thinges pertayning to solide and corporall figures and also of superficieces which are the termes limites of solides moreouer of the diuision and intersection of them and diuers other thinges without which the knowledge of bodely and solide formes can not be attayned vnto And first is set the definitions as followâ—th Definitions A solide or body is that which hath length breadth and thicknes and the terme or limite of a solide is a superficies There are three kindes of continuall quantitie a line a superficies and a solide or body the beginning of all which as before hath bene sayd is a poynt which is indiuisible Two of these quantities namely a line and a superficies were defined of Euclide before in his first booke But the third kinde namely a solide or body he there defined not as a thing which pertayned not then to his purpose but here in this place he setteth the definitioÌ„ therof as that which chiefely now pertayneth to his purpose and without which nothing in these thinges can profitably be taught A solide sayth he is that which hath leÌ„gth breadth and thicknes or depth There are as before hath bene taught three reasons or meanes of measuring which are called coÌ„monly dimensions namely lâ—ngth breadth and thicknes These dimensions are ascribed vnto quantities onely By these are all kindes of quantitie deâ—ined â—â— are counted perfect or imperfect according as they are pertaker of fewer or more of them As Euclide defined a line ascribing vnto it onely one of these dimensions namely length Wherefore a line is the imperfectest kinde of quantitie In defining of a superficies he ascribed vnto it two dimensions namely length and breadth whereby a superficies is a quantitie of
but not euery Pyramis a Tetrahedron And in dede Psellus in numbring of these fiue solides or bodies calleth a Tetrahedron a Pyramis in manifest wordes This I say might make Flussas others as I thinke it did to omitte the definition of a Tetrahedron in this place as sufficiently comprehended within the definition of a Pyramis geuen before But why then did he not count that deâ—inition of a Pyramis faultie for that it extendeth it selfe to large and comprehendeth vnder it a Tetrahedron which differeth from a Pyramis by that it is contayned of equall triangles as he not so aduisedly did before the definition of a Prisme 23 An Octohedron is a solide or bodily figure coÌ„tained vnder eight equall and equilater triangles As a Cube is a solide figure contayned vnder sixe superficiall figures of foure sides or squares which are equilater equiangle and equall the one to the other so is an Octohedron a solide figure contained vnder eight triangles which are equilater and equall the one to the other As ye may in these two figures here set beholde Whereof the first is drawen according as this solide is commonly described vpon a plaine superficies The second is drawen as it is described by arte vpon a plaine to shewe bodilike And in deede although the second appeare to the eye more bodilike yet as I before noted in a Cube for the vnderstanding of diuers Propositions in these fiue bookes following is the first description of more vse yea of necessitie For without it ye can not coÌ„ceaue the draught of lines and sections in any one of the eight sides which are sometimes in the descriptions of some of those Propositions required Wherefore to the consideration of this first description imagine first that vppon the vpper face of the superficies of the parallelogramme ABCD be described a Pyramis hauing his fower triangles AFB AFC CFD and DFB equilater and equiangle and concurring in the point F. TheÌ„ coÌ„ceaue that on the lower face of the superâ—icies of the former parallelogramme be described an other Pyramis hauing his fower triangles AEB AEC CED DEB equilater and equiangle and concurring in the point E. For so although somewhat grosly by reason the triangles can not be described equilater you may in a plaine perceaue the forme of this solide and by that meanes conceaue any lines or sections required to be drawen in any of the sayd eight triangles which are the sides of that body 24 A Dodecahedron is a solide or bodily figure coÌ„tained vnder twelue equall equilater and equiangle Pentagons As a Cube a Tetrahedron and an Octohedron are contayned vnder equall plaine figures a Cube vnder squares the other two vnder triangles so is this solide figure contained vnder twelue equilater equiangle and equall Pentagons or figures of fiue sides As in these two figures here set you may perceaue Of which the first which thinge also was before noted of a Cube a Tetrahedron and an Octohedron is the common description of it in a plaine the other is the description of it by arte vppon a plaine to make it to appeare somwhat bodilike The first description in deede is very obscure to conceaue but yet of necessitie it must so neyther can it otherwise be in a plaine described to vnderstaÌ„d those Propositions of Euclide in these fiue bokes a following which concerne the same For in it although rudely may you see all the twelue Pentagons which should in deede be all equall equilater and equiangle And now how you may somewhat conceaue the first figure described in the plaine to be a body Imagine first the Pentagon ABCDE â—o be vpon a ground plaine superficies then imagine the Pentagon FGHKL to be on high opposite vnto the Pentagon ABCDE And betwene those two Pentagons there will be ten Pentagons pulled vp fiue froÌ„ the fiue sides of the ground Pentagon namely from the side AB the Pentagon ABONM from the side BC the Pentagon BCQPO from the side CD the Pentagon CDSRQ from the side DE the Pentagon DEVTS from the side EA the Pentagon EAMXV the other fiue Pentagons haue eche one of their sides common with one of the sides of the Pentagon FGHKL which is opposite vnto the Pentagon in the ground superficies namely these are the other fiue Pentagons FGNMX GHPON HKRQP KLRST LFXVT So here you may behold twelue Pentagons which if you imagine to be equall equilater equiangle and to be lifted vp ye shall although somewhat rudely conceaue the bodily forme of a Pentagon And some light it will geue to the vnderstanding of certaine Propositions of the fiue bookes following concerning the same 25 An Icosahedron is a solide or bodily figure contained vnder twentie equall and equilater triangles These â—iue solides now last defined namely a Cube a TetrahedroÌ„ an Octohedron a Dodecahedron and an IcosahedroÌ„ are called regular bodies As in plaine superficieces those are called regular figures whose sides and angles are equal as are equilater triangles equilater pentagons hexagons such lyke so in solides such only are counted and called regular which are coÌ„preheÌ„ded vnder equal playne superficieces which haue equal sides and equal angles as all these fiue foresayd haue as manifestly appeareth by their definitions which were all geuen by this proprietie of equalitie of their superficieces which haue also their sides and angles equall And in all the course of nature there are no other bodies of this condition and perfection but onely these fiue Wherfore they haue euer of the auncient Philosophers bene had in great estimation and admiration and haue bene thought worthy of much contemplacion about which they haue bestowed most diligent study and endeuour to searche out the natures properties of them They are as it were the ende and perfection of all Geometry for whose sake is written whatsoeuer is written in Geometry They were as men say first inuented by the most witty Pithagoras then afterward set forth by the diuine Plato and last of all meruelously taught and declared by the most excellent Philosopher Euclide in these bookes following and euer since wonderfully embraced of all learned Philosophers The knowledge of them containeth infinite secretes of nature Pithagâ—ras Timeus and Plato by them searched out the coÌ„position of the world with the harmony and preseruation therof and applied these â—iue solides to the simple partes therof the Pyramis or TetrahedroÌ„ they ascribed to the â—ire for that it ascendeth vpward according to the figure of the Pyramis To the ayre they ascribed the Octohedron for that through the subtle moisture which it hath it extendeth it selfe euery way to the one side and to the other accordyng as that figure doth Vnto the water they assigned the Ikosahedron for that it is continually flowing and mouing and as it were makyng angle◠〈…〉 â—ide according to that figure And to the earth they attributed a Cube as to a thing stable◠〈◊〉 and sure as the figure
angles BAC CAD DAB be equall the one to the other then is it manifest that two of them which two so euer be taken are greater then the third But if not let the angle BAC be the greater of the three angles And vnto the right line AB and from the poynt A make in the playne superficies BAC vnto the angle DAB an equall angle BAE And by the 2. of the first make the line AE equall to the line AD. Now a right line BEC drawne by the poynt E shall cut the right lines AB and AC in the poyntes B and C draw a right line from D to B and an other from D to C. And forasmuch as the line DA is equall to the line AE and the line AB is common to theÌ„ both therefore these two lines DA and AB are equall to these two lines AB and AE and the angle DAB is equall to the angle BAE Wherefore by the 4. of the first the base DB is equall to the base BE. And forasmuch as these two lines DB and DC are greater then the line BC of which the line DB is proued to be equall to the line BE. Wherefore the residue namely the line DC is greater then the residue namely then the line EC And forasmuch as the line DA is equall to the line AE and the line AC is common to them both and the base DC is greater then the base EC therefore the angle DAC is greater then the angle EAC And it is proued that the angle DAB is equall to the angle BAE wherfore the angles DAB and DAC are greater then the angle BAC If therefore a solide angle be contayned vnder three playne superficiall angles euery two of those three angles which two so euer be taken are greater then the third which was required to be proued In this figure ye may playnely behold the former demonstration if ye eleuate the three triangles ABD Aâ—C and ACD in such â—orâ—that they may all meete together in the poynt A. The 19. Theoreme The 21. Proposition Euery solide angle is comprehended vnder playne angles lesse then fower right angles SVppose that A be a solide angle contayned vnder these superficiall angles BAC DAC and DAB Then I say that the angles BAC DAC and DAB are lesse then fower right angles Take in euery one of these right lines ACAB and AD a poynt at all aduentures and let the same be B C D. And draw these right lines BC CD and DB. And forasmuch as the angle B is a solide angle for it is contayned vnder three superficiall angles that is vnder CBA ABD and CBD therefore by the 20. of the eleuenth two of them which two so euer be taken are greater then the third Wherefore the angles CBA and ABD are greater then the angle CBD and by the same reason the angles BCA and ACD are greater then the angle BCDâ— and moreouer the angles CDA and ADB are greater then the angle CDB Wherefore these sixe angles CBA ABD BCA ACD CDA and ADB are greater theÌ„ these thre angles namely CBD BCD CDB But the three angles CBD BDC and BCD are equall to two right angles Wherefore the sixe angles CBA ABD BCA ACD CDA and ADB are greater theÌ„ two right angles And forasmuch as in euery one of these triangles ABC and ABD and ACD three angles are equall two right angles by the 32. of the first Wherefore the nine angles of the thre triangles that is the angles CBA ACB BAC ACD DAC CDA ADB DBA and BAD are equall to sixe right angles Of which angles the sixe angles ABC BCA ACD CDA ADB and DBA are greater then two right angles Wherefore the angles remayning namely the angles BAC CAD and DAB which contayne the solide angle are lesse then sower right angles Wherefore euery solide angle is comprehended vnder playne angles lesse then fower right angles which was required to be proued If ye will more fully see this demonstration compare it with the figure which I put for the better sight of the demonstration of the proposition next going before Onely here is not required the draught of the line AE Although this demonstration of Euclide be here put for solide angles contayned vnder three superficiall angles yet after the like maner may you proceede if the solide angle be contayned vnder superficiall angles how many so euer As for example if it be contayned vnder fower superficiall angles if ye follow the former construction the base will be a quadrangled figure whose fower angles are equall to fower right angles but the 8. angles at the bases of the 4. triangles set vpon this quadrangled figure may by the 20. proposition of this booke be proued to be greater then those 4. angles of the quadrangled figure As we sawe by the discourse of the former demonstration Wherefore those 8. angles are greater then fower right angles but the 12. angles of those fower triangles are equall to 8. right angles Wherefore the fower angles remayning at the toppe which make the solide angle are lesse then fower right angles And obseruing this course ye may proceede infinitely ¶ The 20. Theoreme The 22. Proposition If there be three superficiall plaine angles of which two how soeuer they be taken be greater then the third and if the right lines also which contayne those angles be equall then of the lines coupling those equall right lines together it is possible to make a triangle SVppose that there be thre superficial angles ABC DEF and GHK of which let two which two soeuer be taken be greater then the third that is let the angles ABC and DEF be greater then the angle GHK and let the angles DEF and GHK be greater then the angle ABC and moreouer let the angles GHK and ABC be greater then the angle DEF And let the right lines AB BC DE EF GH and HK be equall the one to the other and draw a right line from the point A to the point C and an other from the point D to the point F and moreouer an other from the point G to the point K. Then I say that it is possible of three right lines equall to the lines AC DF and GK to make a triangle that is that two of the right lynes AC DF and GK which two soeuer be taken are greater then the third Now if the angles ABC DEF and GHK be equall the one to the other it is manifest that these right lines AC DF and GK being also by the 4. of the first equall the one to the other it is possible of three right lines equall to the lines AC DF and GK to make a triangle But if they be not equall let them be vnequall And by the 23. of the first vnto the right line HK and at the point in it H make vnto the angle ABC an equall angle KHL. And by the â— of the first to one of the lines
reformed by M. Dee described for it in the playne especially if ye remember the forme of the figure of the 29. proposition of this booke Only that which there ye conceaue to be the base imagine here in both the figures of this second case to be the vpper superficies opposite to the base and that which was there supposed to be the vpper superficies conceaue here to be the base Ye may describe them vpon pasted paper for your better sight taking hede ye note the letters rightly according as the construction requireth Flussas demonstrateth this proposition an otherway taking onely the bases of the solides and that after this maner Take equall bases which yet for the surer vnderstanding let be vtterly vnlike namely AEBF and ADCH and let one of the sides of eche concurre in one the same right line AED the bases being vpon one and the selfe same playne let there be supposed to be set vpon theÌ„ parallelipipedons vnder one the selfe same altitude Then I say that the solide set vpoÌ„ the base AB is equal to the solide set vpon the base AH By the poynt E draw vnto the line AC a parallel line EG which if it fall without the base AB produce the right line HC to the poynt I. Now forasmuch as AB and AH are parallelogrmaes therefore by the 24. of this booke the triangles ACI and EGL shall be equaliter the one to the other and by the 4. of the first they shal be equiangle and equall and by the first definition of the sixth and fourth Proposition of the same they shall be like Wherfore Prismes erected vppon those triangles and vnder the same altitude that the solides AB and AH aâ—e shall be equall and like by the 8. definition of this booke For they are contayned vnder like playne superficieces equall both in multitude and magnitude Adde the solide set vpon the base ACLE common to them both Wherefore the solide set vppon the base AEGC is equall to the solide set vpon the base AELI And forasmuch as the superficieces AEBF and ADHC are equall by supposition and the part taken away AG is equall to the part taken away AL therefore the residue BI shall be equall to the residue GD Wherefore as AG is to GD as AL is to BI namely equalls to equalls But as AG is to GD so iâ— the solide set vpon AG to the solide set vpon GD by the 25. of this booke for it is cut by a playne superficies set vpon the line GE which superficies is parallel to the opposite superficieces Wherefore as AL is to BI so is the solide set vpon AL to the solide set vpon BI Wherefore by the 11. of the fifth as the solide set vpon AG or vpon AL which is equall vnto it is to the solide set vpon GD so is the same solide set vpon AG or AL to the solide set vpon BI Wherefore by the 2. part of the 9. of the fifth the solides set vpon GD and BI shall be equall Vnto which solides if ye adde equall solides namely the solide set vpon AG to the solide set vpon GD and the solide set vpon AL to the solide set vpon BI the whole solides set vpon the base AH and vpon the base AB â—hall be equall Wherefore Parallelipedons consisting vpon equall bases and being vnder one and the selfe same altitude are equall the one to the other which was required to be proued ¶ The 27. Theoreme The 32. Proposition Parallelipipedons being vnder one and the selfe same altitude are in that proportion the one to the other that their bases are SVppose that these parallelipipedons AB and CD be vnder one the selfe same altitude Then I say that those parallelipipedons AB and CD are in that proportion the one to the other that their bases are that is that as the base AE is to the base CF so is the parallelipipedon AB to the parallelipipedon CD Vpon the line FG describe by the 45. of the first the parallelogramme FH equall to the parallelogramme AE and equiangle with the parallelogramme CF. And vpon the base FH describe a parallelipipedoÌ„ of the selfe same altitude that the parallelipipedoÌ„ CD is let the same be GK Now by the 31. of the eleuenth the parallelipipedon AB is equall to the parallelipipedon GK for they consist vpon equall bases namely AE and FH and are vnder one and the selfe same altitude And forasmuch as the parallelipipedon CK is cut by a plaine superficies DG being parallel to either of the opposite plaine superâ—icieces therfore by the 25. of the eleuenth as the base HF is to the base FC so is the parallelipipedon GK to parallelipipedon CD but the base HF is equal to the base AE and the parallelipipedon GK is proued equall to the parallelipipedon AB Wherfore as the base Aâ—E is to the base CF so is the parallelipedon AB to the parallelipipedon CD Wherfore parallelipipedons being vnder one and the selfe same altitude are in that proportion the one to the other that their bases are which was required to be demonstrated I neede not to put any other figure for the declaration of this demonstration for it is easie to see by the figure there described Howbeit ye may for the more full sight therof describe solides of pasted paper according to the construction there set forth which will not be hard for you to do if ye remember the descriptions of such bodies before taught A Corollary added by Flussas Equall parallelipipedons coÌ„tained vnder one and the selfe same altitude haue also their bases equal For if the bases should be vnequall the parallelipipedons also should be vnequal by this 32 propositioÌ„ And equall parallelipipedons hauing equall bases haue also one and the selfe same altitude For if they should haue a greater altitude they should exceede the equall parallelipipedons which haue the selfe same altitude But if they should haue a lesse they should want so much of those selfe same equal parallelipipedons The 28. Theoreme The 33. Proposition Like parallelipipedons are in treble proportion the one to the other of that in which their sides of like proportion are SVppose that these parallelipipedons AB and CD be like let the sides AE and CF be sides of like proportion Then I say the parallelipipedon AB is vnto the parallelipipedon CD in treble proportion of that in which the side AE is to the side CF. Extend the right lines AE GE and HE to the pointes K L M. And by the 2. of the first vnto the line CF put the line EK equal and vnto the line FN put the line EL equall and moreouer vnto the line FR put the line EM equall and make perfect the parallelogramme KL and the parallelipipedon KO Now forasmuch as these two lines EK and EL are equall to these two lines CF and FN but the angle KEL is equall to the angle CFN for the angle
fourth The second and third are to be found which may betwene A B be two meanes in continuall propoâ—tion as now suppose such two lines found and let them be C and D. Wherefore by Euclides Corollary as A is to B if A were taken as first so shall the Parallelipipedon described of A be to the like Parallelipipedon and in like sort described of C being the second of the fower lines in continuall proportion it is to weâ—e A C D and B. Or if B shall be taken as first and that thus they are orderly in continuall proportion B D C A then by the sayd Corollary as B is to A so shall the Parallelipipedon described of B be vnto the like Parallelipipedon and in like sort described oâ— D. And vnto a Parallelipipedon of A or B at pleasure described may an other of C or D be made like and in like sort situated or described by the 27. of this eleuenth booke Wherefore any two right lines being geueÌ„ c which was required to be done Thus haue I most briefly brought to your vnderstanding if first B were double to A then what Parallelipipedon soeuer were described of A the like Parallelipipedon and in like sort described of C shall be double to the Parallelipipedon described of A. And so likewise secondly if A were double to B the Parallelipipedon of D shoulde be double to the like of B described both being like situated Wherefore if of A or B were Cubes made the Cubes of C and D are proued double to them as that of C to the Cube of A and the Cube of D to the Cube of B in the second case And so of any proportion els betwene A and B. Now also do you most clerely perceaue the Mathematicall occasion whereby first of all men Hippocrates to double any Cube geuen was led to the former Lemma Betwene any two right lines geuen to finde two other right lines which shall be with the two first lines in continuall proportion After whose time many yeares diuine Plato Heron Philo Appollonius Diâ—â—lâ—â— Pappus Sporus Menechâ—us Archytas Tarentinus who made the wodden doue to slye Eratoâ—â—hene Nicomedes with many other to their immortall fame and renowme published diuers their witty deuises methods and engines which yet are extant whereby to execute thys Problematicall Lemma But not withstanding all the trauailes of the â—oresayd Philosophers and Mathematiciens yea and all others doinges and contriuinges vnto this day about the sayd Lemma yet there remaineth sufficient matter Mathematically so to demonstrate the same that most exactly readily it may also be Mechanically practisâ—d that who soeuer shall achieue that feate shall not be counted a second Archimedes but rather a perâ—les Mathematicien and Mathematicorum Princeps I will sundry wayes in my briefe additions and annotations vpon Euclide excite you thereto yea and bring before your eyes sundry new wayes by meinuented and in this booke so placed as matter thereof to my inuentions appertayning may geue occasion Leauing the farther full absolute my concluding of the Lemma to an other place and time which will now more coÌ„pendiously be done so great a part therof being before hand in thys booke published ¶ A Corollary added by Flussas Parallelipipedons consisting vpon equall bases are in proportion the one to the other as their altitudes are For if those altitudes be cut by a plaine superficies parallel to the bases the sections shall be in proportion the one to the other as the sections of the bases cut by the 25. of this booke Which sections of the bases are the one to the other in that proportion that their sides or the altitudes of the solides are by the â—irst of the sixt Wherefore the solides are the one to the other as their altitudes are But if the bases be vnlike the selfe same thing may be proued by the Corollary of the 25. of this booke which by the 25. Proposition was proued in like bases ¶ The 29. Theoreme The 34. Proposition In equall Parallelipipedons the bases are reciprokall to their altitudes And Parallelipipedons whose bases are reciprokall to their altitudes are equall the one to the other But now againe suppose that the bases of the Parallelipipedons AB and CD be reciprokall to their altitudes that is as the base EH is to the base NP so let the altitude of the solide CD be to the altitude of the solide AB Then I say that the solide AB is equall to the solide CD For againe let the standing lines be erected perpendicularly to their bases And now if the base EH be equall to the base NP but as the base EH is to the base NP so is the altitude of the solide CD to the altitude of the solide AB Wherefore the altitude oâ— the solide CD is equall to the altitude of the solide AB But Parallelipipedons consisting vpon equall bases and vnder one and the selfe same altitude are by the 31. of the eleuenth equall the one to the other Wherefore the solide AB is equall to the solide CD But now suppose that the base EH be not equall to the base NP but let the base EH be the greater Wherefore also the altitude of the solide CD that is the line CM is greater then the altitude of the solide AB that is then the line AG. Put againe by the 3. of the first the line CT equall to the line AG and make perfecte the solide CZ Now for that as the base EH is to the base NP so is the line MC to the line AG. But the line AG is equall to the line CT Wherefore as the base EH is to the base NP so is the line CM to the line CT But as the base EH is to the base NP so by the 32. of the eleuenth is the solide AB to the solide CZ â—or the solides AB and CZ are vnder equall altitudes And as the line CM is to the line CT so by the 1. of the sixt is the base MP to the base Pâ—T and by the 32. of the eleuenth the solide CD to the solide CZ Wherefore also by the 11. and 9. of the fift as the solide AB is to the solide CZ so is the solide CD to the solide CZ Wherfore either of these solides AB and CD haue to the solide CZ one and the same proportion Wherefore by the 7. of the fift the solide AB is equall to the solide CD which was required to be demonstrated Againe suppose that the bases of the Parallelipipedons AB and CD be reciprokall to their altitudes that is as the base EH is to the base NP so let the altitude of the solide CD be to the altitude of the solide AB Then I say that the solide AB is equall to the solide CD For the same order of construction remayning for that as the base EH is to the base NP so is the altitude of the solide CD to the altitude
two Prismes vnder equall altitudes the one haue to his base a parallelogramme and the other a triangle and if the parallelogramme be double to the triangle those Prismes are equall the one to the other SVppose that these two Prismes ABCDEF GHKMON be vnder equall altitudes and let the one haue to his base the parallelogramme AC and the other the triangle GHK and let the parallelogramme AC be double to the triangle GHK Then I say that the Prisme ABCDEF is equall to the Prisme GHKMON Make perfecte the Parallelipipedons AX GO And forasmuch as the parallelogramme AC is double to the triangle GHK but the parallelogramme GH is also by the 41. of the first double to the triangle GHK wherefore the parallelogramme AC is equall to the parallelogramme GH But Parallelipipedons consisting vpon equall bases and vnder one and the selfe same altitude are equall the one to the other by the 31. of the eleuenth Wherefore the solide AX is equall to the solide GO But the halfe of the solide AX is the Prisme ABCDEF and the halfe of the solide GO is the Prisme GHKMON Wherfore the Prisme ABCDEF is equall to the Prisme GHKMON If therefore there be two Prismes vnder equall altitudes and the one haue to his base a parallelogramme the other a triangle and if the parallelogramme be double to the triangle those Prismes are equall the one to the other which was required to be proued This Proposition and the demonstration thereof are not hard to conceaue by the former figures but ye may for your fuller vnderstanding of theÌ„ take two equall Parallelipipedons equilateâ— and equiangle the one to the other described of pasted paper or such like matter and in the base of the one Parallelipipedon draw a diagonall line and draw an other diagonall line in the vpper superficies opposite vnto the said diagonall line drawen in the base And in one of the parallelogrammes which are set vpon the base of the other Parallelipipedon draw a diagonall line and drawe an other diagonall line in the parallelogramme opposite to the same For so if ye extend plaine superficieces by those diagonall lines there will be made two Prismes in ech body Ye must take heede that ye put for the bases of eche of these Parallelipipedons equall parallelograÌ„mes And then note theÌ„ with letters according to the letters of the figures before described in the plaine And coÌ„pare theÌ„ with the demonstration and they will make both it and the Proposition very clere vnto you They will also geue great light to the Corollary following added by Flussas A Corollary added by Flussas By this and the former propoâ—itions it is manifest that Prismes and solides contayned vnder two poligoâ—on figures equall like and parallels and the rest parallelogrammes may be compared the one to the other after the selfe same maner that parallelipipedons are For forasmuch as by this proposition and by the second Corollary of the 2â— of this booke it is manifest that euery parallelipipedon may be resolued into two like and equal Prismes of one and the same altitude whose base shal be one and the selfe same with the base of the parallelipipedon or the halfe thereof which Prisâ—es also shal be contâ—yned vnder the selfe same sideâ— with the parallelipipedoÌ„ the sayde sideâ— beyng also sides of like proportion I say that Prismeâ— may be compared together after the like maner that their Parallelipipedonâ— areâ— For if we would deuide a Prisme like vnto his foliâ—e by the 25. of this booke ye shall finde in the Corollaryes of the 25. propoâ—â—tioÌ„ that that which is set forth touching a parallelipipedon followeth not onely in a Prisme but also in any sided columne whose opposite bases are equall and like and his sides parallelogrammes If it be required by the 27. proposition vpon a right line geuen to describe a Prisme like and in like sorte situate to a Prisme geuen describe â—â—â—st the whole parallelipipedon whereof the prisme geuen is the halfe which thing ye see by this 40. proposition may be done And vnto that parallelipipedoÌ„ describe vpon the right line geuen by the sayd 27. proposition an other parallelipipedon like and the halfe thereof shal be the prisme which ye seeke for namely shal be a prisme described vpon the right line geuen and like vnto the prisme geuen In deede Prismes can not be cut according to the 28. proposition For that in their opposite sides can be drawen no diagonall lines howbeit by that 28. proposition those Prismes are manifestly confirmed to be equall and like which are the halues of one and the selfe same parallelipipedon And as touching the 29. proposition and the three following it which proueth that parallelipipedons vnder one and the selfe same altitude and vpon equall bases or the selfe same bases are equal or if they be vnder one and the selfe same altiâ—â—dâ— they are in proportion the one to the other as their bases areâ— to apply these comparisons vnto 〈◊〉 it is to 〈◊〉 required that the bases of the Prismes compared together be either all parallelogrammes or all triaâ—gles For so one and the selfe altitude remayning the comparison of thinges equall 〈◊〉 one and thâ—â— selfe same and the halfes of the bases are euer the one to the other in the same proportion that their wholes are Wherfore Prismes which are the halues of the parallelipipedons and which haue the same proportion the one to the other that the whole parallelipipedons haue which are vnder one and thâ— selfâ—â—ame altitude must needes cause that their bases being the halues of the baseâ— of the parallelipâ—pâ—â—â—â—â—â—e in the same proportioÌ„ the one to the other that their whole parallelipipâ—donâ— are If thereâ—oâ—â— the wâ—ole parallelipipedons be in the proportion of the whole bases their hâ—lâ—â—â— also which are Prismes shal be in the proportion either of the wholes if their bases be parallelâ—grâ—mmâ—â—â— or of the halâ—â—â—â—f they be triangles which is euer all one by the 15. of the fiueth And forasmuch as by the 33. proposition like parallelipipedons which are the doubles of their Prismes are in treble proportion the one to the other that their sides of like proportion are it is manifest that Prismes being their halues which haue the one to the other the same proportion that their wholes haue by the 15 of the fiueth and hauing the selfe same sides that theiâ— parallelipipedons haue are the one to the other in treble proportion of that which the sides of like proportion are And for that Prismes are the one to the other in the same proportion that their parallelipipedons are and the bases of the Prismes being all either triangles or parallelograÌ„mes are the one to the other in the same proportion that the bases of the parallelipipedons are whose altitudes also are alwayes equall we may by the 34. proposition conclude that the bases of the prismes and the bases of the parallelipipedons their doubles being ech the one to the
other in one and the selfe same proportion are to the altitudes in the same proportion that the bases of the double solides namely of the parallelipipedons are For if the bases of the equall parallelipipedoÌ„s be reciprokall with their altitudes then their halues which are Prismes shall haue their bases reciprokall with their altitudes By the 36. proposition we may conclude that if there be three right lines proportionall the angle of a Prisme made of these three lines being common with the angle of his parallelipipedon which is double doth make a prisme which is equall to the Prisme described of the middle line and contayning the like angle consisting also of equall sides For aâ— in the parallelipipedon so also in the Prisme this one thing is required namely that the three dimensions of the proportionall lines do make an angle like vnto the angle contayned of the middle line taken three tymes Now then if the solide angle of the Prisme be made of those three right lines there shall of them be made an angle like to the angle of the parallelipipedon which is double vnto it Wherefore it followeth of necessitie that the Prismes which are alwayes the halues of the Parallelipipedons are equiangle the one to the other as also are their doubles although they be not equilater and therefore those halues of equall solides are equall the one to the other namely that which is described of the middle proportionall line is equall to that which is described of the three proportionall lines By the 37. proposition also we may conclude the same touching Prismes which was concluded touching Parallelipipedons For forasmuch as Prismes described like in like sort of the lines geuen are the halues of the Parallelipipedons which are like and in like sort described it followeth that these Prismes haue the one to the other the same proportion that the solides which are their doubles haue And therfore if the lines which describe them be porportionall they shal be proportionall and so conuersedly according to the rule of the sayd 37. proposition But forasmuch as the 39. proposition supposeth the opposite superficiall sides of the solide to be parallelogrammes and the same solide to haue one diameter which thinges a Prisme can not haue therefore this proposition can by â—o meanes by applyed to Prismes But as touching solides whose bases are two like equall and parallel poligonon figures and their sides are parallelogrammes forasmuch as by the second Corollary of the 25. of this booke it hath bene declared that such solides are composed of Prismes it may easely be proued that their nature is such as is the nature of the Prismes whereof they are composed Wherefore a parallelipipedon being by the 27. proposition of this booke described there may also be described the halfe thereof which is a Prisme and by the description of Prismes there may be composed a solide like vnto a solide geuen composed of Prismes So that it is manifest that that which the 29. 30. 31. 32. 33. 34. and 37. propositions set forth touching parallelipipedons may well be applyed also to these kyndes of solides The end of the eleuenth booke of Euclides Elementes ¶ The twelueth booke of Euclides Elementes IN THIS TVVELVETH BOOKE EVCLIDE setteth forth the passions and proprieties of Pyramids Prismes Cones Cylinders and Spheres And compareth Pyramids first to Pyramids then to Prismes so likewise doth he Cones and Cylinders And lastly he compareth Spheres the one to the other But before he goeth to the treatie of those bodies he proueth that like Poligonon figures inscribed in circles and also the circles theÌ„ selues are in proportion the one to the other as the squares of the diameters of those circles are Because that was necessary to be proued for the confirmation of certayne passions and proprieties of those bodies ¶ The 1. Theoreme The 1. Proposition Like Poligonon figures described in circles are in that proportion the one to the other that the squares of their diameters are SVppose that there be two circles ABCDE and FGHKL and in them let there be described like Poligonon figures namely ABCDE and FGHKL and let the diameters of the circles be BM and GN Then I say that as the square of the line BM is to the square of the line GN so is the Poligonon figure ABCDE to the Poligonon figure FGHKL Drawe these right lines BE AM GL and FN And forasmuch as the Poligonon figure ABCDE is like to the Poligonon figure FGHKL therefore the angle BAE is equall to the angle GFL and as the line BA is to the line AE so is the line GF to the line FL by the definition of like Poligonon figures Now therefore there are two triangles BAE and GFL hauing one angle of the one equall to one angle of the other namely the angle BAE equall to the angle GFL and the sides about the equall angles are proportionall Wherefore by the first definition of the sixt the triangle ABE is equiangle to the triangle FGL Wherefore the angle AEB is equall to the angle FLG But by the 21. of the third the angle AEB is equall to the angle AMB for they consiste vpon one and the selfe same circumference and by the same reason the angle FLG is equall to the angle FNG Wherfore the angle AMB is equall to the angle FNG And the right angle BAM is by the 4. petition equall to the right angle GFN Wherefore the angle remayning is equall to the angle remayning Wherefore the triangle AMB is equiangle to the triangle FNG Wherefore proportionally as the line BM is to the line GN so is the line BA to the line GF But the square of the line BM is to the square of the line GN in double proportion of that which the line BM is to the line GN by the Corollary of the 20. of the sixt And the Poligonon figure ABCDE is to the Poligonon figure FGHKL in double proportion of that which the line BA is to the line GF by the â—0 of the sixt Wherefore by the 11. of the fift as the square of the line BM is to the square of the line GN so is the Poligonon figure ABCDE to the Poligonon figure FGHKL Wherefore like Poligonon figures described in circles are in that proportion the one to the other that the squares of the diameters are which was required to be demonstrated ¶ Iohn Dee his fruitfull instructions with certaine Corollaries and their great vse WHo can not easily perceaue what occasion and ayde Archimedes had by these first second Propositions to â—inde the nere Area or Content of a circle betwene a Poligonon figure within the circle and the like about the same circle described Whose precise quantities are most easily knowen being comprehended of right lines Where also to auoyde all occasions of errour it is good in numbers not hauing precise square rootes to vse the Logisticall processe according to the rules with √ â— 12 √ â— 19
is manifest by the last of the eleuenth And now that both the prismes taken together are greater then the halfe of the whole pyramis hereby it is manifest for that either of them may be deuided into two pyramids of which the one is a triangular pyramis equall to one of the two pyramids into which together with the two prismes is deuided the whole pyramis and the other is a quadrangled pyramis double to the other pyramis Wherefore it is playne that the two prismes taken together are three quarters of the whole pyramis deuided But if ye are desirous to know the proportioÌ„ betwen them reade the â— of this booke But now here to this purpose it shall be sufficient to know that the two prismes taken together do exceede in quantity the two partial pyramids taken together into which together with the two prismes the whole pyramis was deuided ¶ The 4. Theoreme The 4. Proposition If there be two Pyramids vnder equall altitudes hauing triangles to their bases and either of those Pyramids be deuided into two Pyramids equall the one to the other and like vnto the whole and into two â—quall Prismes and againe if in either of the Pyramids made of the two first Pyramids be still obserued the same order and maner then as the base of the one Pyramis is to the base of the other Pyramis so are all the Prismes which are in the one Pyramis to all the Prismes which are in the other being equall in multitude with them SVppose that there be two Pyramids vnder equall altitudes hauing triangles to their bases namely ABC and DEF and hauing to their toppes the pointes G and H. And let either of these pyramids be diuided into two pyramids equall the one to the other and like vnto the whole and into two equall prismes according to the methode of the former Proposition And againe let either of those pyramids soâ— made of the two first pyramids be imagined to be after the same order deuided and so do continually Then I say that as the base ABC is to the base DEF so are all the prismes which are in the pyramis ABCG to all the prismes which are in the pyramis DEFH being equall in multitude with them For forasmuch as the line BX is equall to the line XC and the line AL to the line LC For as we saw in the construction pertayning to the former Proposition al the sixe sides of the whole pyramids are ech deuided into two equall parts the like of which construction is in this proposition also supposed therefore the line XL is a parallel to the line AB the triangle ABC is like to the triangle LXC by the Corollary of the second of the sixth and by the same reason the triangle DEF is like to the triangle RWF And forasmuch as the line BC is double to the line CX and the line FE to the line FW therefore as the line BC is to the line CX so is the line EF to the line FW And vpon the lines BC and CX are described rectiline figures like and in like sort set namely the triangles ABC and LXC and vpon the lines EF and FW are also described rectiline figures like and in like sort set namely the triangles DEF RWFâ— But if there be fower right lines proportionall the rectiline figures described of them being like und in like sort set shall also be proportionall by the 22. of the sixt Wherefore as the triangle ABC is to the triangle LXC so is the triangle DEF to the triangle RWF Wherefore alternately by the 16. of the fift as the triangle ABC is to the triangle DEF so is the triangle LXC to the triangle RWF But as the triangle LXC is to the triangle RWF so is the prisme whose base is the triangle LXC and the opposite side vnto it the triangle OMN to the prisme whose base is the triangle RWF and the opposite side vnto it the triangle STV by the Corollary of the 40. of the eleuenth For these prismes are vnder one the selfe same altitude namely vnder the halfe of the altitude of the whole Pyramids which Pyramids are supposed to be vnder one and the selfe same altitude this is also proued in the Assumpt following Wherefore by the 11. of the fift as the triangle ABC is to the triangle DEF so is the prisme whose base is the triangle LXC and the opposite side vnto it the triangle OMN to the prisme whose base is the triangle RWF and the opposite side vnto it the triangle STV And forasmuch as there are two prismes in the pyramis ABCG equal the one to the other two prismes also in the pyramis DEFH equall the one to the other therefore as the prisme whose base is the parallelograÌ„me BKLX and the opposite side vnto it the line MO is to the prisme whose base is the triangle LXC and the opposite side vnto it the triangle OMN so is the prisme whose base is the parallelogramme PERW and the opposite vnto it the line ST to the prisme whose base is the triangle RWF and the opposite side vnto it the triangle STV Wherfore by coÌ„position by the 18. of the fift as the prismes KBXLMO LXCMNO are to the prisme LXCMNO so are the prismes PEWRST and RWFSTV to the prisme RWFSTV Wherefore alternately by the 16. of the fift as the two prismes KBXLMO and LXCMNO are to the two prismes PEWRST and RWFSTV so is the prisme LXCMNO to the prisme RWFSTV But as the prisme LXCMNO is to the prisme RWFSTV so haue we proued that the base LXC is to the base RWF and the base ABC to the base DEF Wherefore by the 16. of the fift as the triangle ABC is to the triangle DEF so are both the prismes which are in the pyramis ABCG to both the prismes which are in the pyramis DEFH And in like sort if we diuide the other pyramids after the selfe same maner namely the pyramis OMNG and the pyramis STVH as the base OMN is to the base STV so shall both the prismes that are in the pyramis OMNG be to both the prismes which are in the pyramis STVH But as the base OMN is to the base STV so is the base ABC to the base DEF Wherefore by the 11. of the fift as the base ABC is to the base DEF so are the two prismes that are in the pyramis ABCG to the two prismes that are in the pyramis DEFH and the two prismes that are in the pyramis OMNG to the two prismes that are in the pyramis STVH and the fower prismes to the fower prismes And so also shall it followe in the prismes made by diuiding the two pyramids AKLO and DPRS and of all the other pyramids in generall being equall in multitude And that as the triangle LXC is to the triangle RWF so is the prisme whose base is the triangle LXC and the opposite side OMN to the prisme whose base is the triangle RWF and the
same is treble it is manifest by the 12. of the fiueth that all the Prismes are to all the Pyramids treble Wherefore Parallelipipedons are treble to Pyramids set vpon the selfe same base with them and vnder the same altitude for that they contayne two Prismes Third Corollary If two Prismes being vnder one and the selfe same altitude haue to their bases either both triangles or both parallelogrammes the Prismes are the one to the other as their bases are For forasmuch as those Prismes are equemultiqlices vnto the Pyramâ—ds vpon the selfe same bases and vnder the same altitude which Pyramids are in proportion as their bases it is manifest by the 15. of the fift that the Prismes are in the proportion of the bases For by the former Corollary the Prismes are treble to the Pyramids sâ—t vpon the triangular bases Fourth Corollary Prismes are in sesquealtera proportion to Pyramids which haue the selfe same quadrangled base that the Prismes haue and are vnder the selfe same altitude For that Pyramis contayneth two Pyramids set vpon a triangular base of the same Prisme for it is proued that that Prisme is treble to the Pyramis which is set vpon the halfe of his quadrangled base vnto which the other which is set vpon the whole base is double by the sixth of this booke Fiueth Corollary Wherefore we may in like sort conclude that solides mencioned in the second Corollary which solids Campane calleth sided Columnes being vnder one and the selfe same altitude are in proportion the one to the other as their bases which are poligonon figures For they are in the proportion of the Pyramids or Prismes set vpon the selfe same bases and vnder the selfe same altitude that is they are in the proportioÌ„ of the bases of the sayde Pyramids or Prismes For those solids may be deuided into Prismes hauing the selfe same altitude when as their opposite bases may be deuided into triangles by the 20 of the sixth Vpon which triangles Prismes beyng set are in proportion as their bases By this 7. Proposition it playnely appeareth that â—uâ—lide as it was before noted in the diffinitionâ—â— vnder the diffinition of a Prisme comprehended also those kinds of solids which Campane calleth sided Columnes For in that he sayth Euery Prisme hauing a triangle to his base may be deuidedâ— c. he neded not taking a Prisme in that sense which Campane and most men take it to haue added that particle hauing to his base a triangle For by their sense there is no Prisme but it may haue to his base a triangleâ— and so it may seeme that Euclide ought without exceptioâ— haue sayd that euery prisme whatsoeuer may be deuided into three pyramids equall the one to the other hauing also triangles to â—heir bases For so do Campane and Flussas put the proposition leauing out the former particle hauing to his base a triangle which yet is red in the Greeke copye not leâ—t out by any other interpreters knowne abroade except by Campane and Flussas Yea and the Corollary following of this proposition added by Theon or Euclide and ameÌ„ded by M. Dee semeth to confirme this sence Of this â—s 〈◊〉 made manifest that euery pyramis is the third part of the prisme hauing the same base with it and equall altitude For and if the base of the prisme haue any other right lined figure then a triangle and also the superficies opposite to the base the same figure that prisme may be deuided into prismes hauing triangled bases and the superficieces to those bases opposite also triangled a â—â—ike and equally For there as we see are put these wordes â—or and if the base of the prisme be any other right lined figureâ— c. whereof a man may well inferre that the base may be any other rectiline figure whatsoeuer not only a triangle or a parallelogramme and it is true also in that sence as it is plaine to see by the second corollary added out of Flussas which corollary as also the first of his corollaries is in a maner all one with the Corollary added by Theon or Euclide Farther Theon in the demonstration of the 10. proposition of this booke as we shall aâ—terward see most playnely calleth not onely sided columnes prismes but also parallelipipedons And although the 40. proposition of the eleuenth booke may seme hereunto to be a lâ—t For that it can be vnderstanded of those prismes onely which haue triangles to their like equall opposite and parallel sides or but of some sided columnes and not of all yet may that let be thus remoued away to say that Euclide in that propositioÌ„ vsed genus pro specie that is the generall word for some special kinde therof which thing also is not rare not only with him but also with other learned philosophers Thus much I thought good by the way to note in farther defence of Euclide definition of a Prisme The 8. Theoreme The 8. Proposition Pyramids being like hauing triangles to their bases are in treble proportion the one to the other of that in which their sides of like proportion are SVppose that these pyramids whose bases are the triangles GBC and HEF and toppes the poyntes A and D be like and in like sort described and let AB and DE be sides of like proportion Then I say that the pyramis ABCG is to the pyramis DEFH in treble proportioÌ„ of that in which the side AB is to the side DE. Make perfect the parallelipipedons namely the solides BCKL EFXO And forasmuch as the pyramis ABCG is like to the pyramis DEFH therfore the angle ABC is equall to the angle DEF the angle GBC to the angle HEF and moreouer the angle ABG to the angle DEH and as the line AB is to the line DE so is the line BC to the line EF and the line BG to the line EH And for that as the line AB is to the line DE so is the line BC to the line EF and the sides about the equall angles are proportionall therefore the parallelogramme BM is like to the parallelograÌ„me EP and by the same reason the parallelogramme BN is like to the parallelogramme ER and the parellelogramme BK is like vnto the parallelogramme EX Wherefore the three parallelogrammes BM KB and BN are like to the three parallelogrammes EP EX and ER. But the three parallelogrammes BM KB and BN are equall and like to the three opposite parallelogrammes and the three parallelogrammes EP EX and ER are equall and like to the three opposite parallelogrammes Wherefore the parallelipipedons BCKL and EFXO are comprehended vnder playne superficieces like and equall in multitude Wherefore the solide BCKL is like to the solide EFXO But like parallelipipedons are by the 33. of the eleuenth in treble proportion the one to the other of that in which side of like proportion is to side of like proportion Wherefore the solide BCKL is to the solide EFXO in treble
proportion of that in which the side of like proportion AB is to the side of like proportion DE. But as the solide BCKL is to the solide EFXO so is the pyramis ABCG to the pyramis DEFH by the 15. of the fifth for that the pyramis is the sixth part of this solide for the prisme being the halfe of the parallelipipedon is treble to the pyramis Wherefore the pyramis ABCG is to the pyramis DEFH in treble proportion of that in which the side AB is to the side DE. Which was required to be proued Corollary Hereby it is manifest that like pyramids hauing to their bases poligonon figures are in treble proportion the one to the other of that in which side of like proportion is to side of like proportion For if they be deuided into pyramids hauing triangles to their bases for like poligonoÌ„ figures are deuided into like triangles and equal in multitude and the sides are of like proportion as one of the pyramids of the one hauing a triangle to his base is to one of the pyramids of the other hauing also a triangle to his base so also are all the pyramids of the one pyramis hauing triangles to their bases to all the pyramids of the other pyramis hauing also triangles to their bases That is the pyramis hauing to his base a poligonoÌ„â—igure to the pyramis hauing also to his base a poligonoÌ„â—igure But a pyramis hauing a triangle to his base is to a pyramis hauing also a triangle to his base being like vnto it in treble proportioÌ„ of that in which side of like proportioÌ„ is to side of like proportioÌ„ Wherfore a pyramis hauing to his base a poligonoÌ„ figure is to a pyramis hauing also a poligonon figure to his base the sayd pyramids being like the one to the other in treble proportion of that in which side of like proportion is to side of like proportion Likewise Prismes and sided columnes being set vpon the bases of those pyramids and vnder the same altitude forasmuch as they are equemultiplices vnto the pyramids namely triples by the corollary of the 7. of this booke shal haue the â—ormer porportion that the pyramids haue by the 15 of the fifth and therefore they shall be in treble proportion of that in which the sides of like proportion are ¶ The 9. Theoreme The 9. Proposition In equall pyramids hauing triangles to their bases the bases are reciprokall to their altitudes And pyramids hauing triangles to their bases whose bases are reciprokall to their altitudes are equall the one to the other SVppose that BCGA and EFHD be equall pyramids hauing to their bases the triangles BCG and EFH and the tops the pointes A and D. Then I say that the bases of the two pyramids BCGA and EFHD are reciprokall to their altitudes that is as the base BCG is to the base EFH so is the altitude of the pyramis EFHD to the altitude of the pyramis BCGA Make perfect the parallelipipedons namely BGML and EHPO And forasmuch as the pyramis BCGA is equall to the pyramis EFHD the solide BGML is sextuple to the pyramis BCGA For the parallelipipedon is duple to the Prisme set vpon the base of the Pyramis the Prisme is triple to the pyramis and likewise the solide EHPO is sextuple to the pyramis EFHD Wherefore the solide BGML is equal to the solide EHPO But in equall parallelipipedons the bases are by the 34. of the eleueÌ„th reciprokall to their altitudes Wherfore as the base BN is to the base EQ so is the altiâ—â—de of the solide EHPâ— to the altitude of the solide BGML But as the base BN is to the base EQ so is the triangle GBC to the triangle HEF by the 15. of the â—ifth for the triangles GBC HEF are the halues of the parallelogrammes BN and EQ â— Wherfore by the 11. of the fifth as the triangle GBC is to the triangle HEF so is the altitude of the solide EHPO to the altitude of the solide BGML But the altitude of the solide EHPO is one and the same with the altitude of the pyramis EFHD and the altitude of the solide BGML is one and the same with the altitude of the pyramis BCGA Wherefore as the base GBC is to the base HEF so is the altitude of the pyramis EFHD to the altitude of the pyramis BCGA Wherefore the bases of the two pyramids BCGA and EFHD are reciprokall to their altitudes But now suppose that the bases of the pyramids BCGA and EFHD be reciprokall to their altitudes that is as the base GBC is to the base HEF so let the altitude of the pyramis EFHD be to the altitude of the pyramis BCGA Then I say that the pyramis BCGA is equall to the pyramis EFHD For the selfe same order of construction remaining for that as the base GBC is to the base â—EF so is the altitude of the pyramis EFHD to the altitude of the pyramis BCGA But as the base GBC is to the base HEF so is the parallelogramme GC to the parallelogramme HF. Wherefore by the 11. of the fifth as the parallelogramme GC is to the parallegoramme HF so is the altitude of the pyramis EFHD to the altitude of the pyramis BCGA But the altitude of the pyramis EFND and of the solide EHPO is one and the selfe same and the altitude of the pyramis BCGA and of the solide BGML is also one and the same Wherefore as the base GC is to the base HF so is the altitude of the solide EHPO to the altitude of the solide BGML But parallelipipedons whose bases are reciprokall to their altitudes are by the 34. of the eleuenth equall the one to the other Wherefore the parallelipipedon BGML is equall to the parallelipipedon EHPO But the pyramis BCGA is the sixth part of the solide BGML and likewise the pyramis EFHD is the sixth part of the solide EHPO Wherefore the pyramis BCGA is equall to the pyramis EFHD Wherefore in equall pyramids hauing triangles to their bases the bases are reciprokall to their altitudes And pyramids hauing triangles to their bases whose bases are reciprocall to their altitudes are equall the one to the other which was required to be demonstrated A Corrollary added by Campane and Flussas Hereby it is manifest that equall pyramids hauing to their bases Poligonon figures haue their bases reciprokall with their altitudes And Pyramids whose bases being poligonon figures are reciprokall with their altitudes are equall the one to the other Suppose that vpon the poligonon figures A and B be set equall pyramids Then I say that their bases A and B are reciprokall with their altitudes Describe by the 25. of the sixth triangles equall to the bases A and B. Which let be C and D. Vpon which let there be set pyramids equall in altitude with the pyramids A and B. Wherfore the pyramids C and D being set vpoÌ„ bases equall with the bases of the pyramids A and B and hauing also their altitudes equall
with the altitudes of the sayd pyramids A and B shall be equall by the 6. of this booke Wherefore by the first part of this proposition the bases of the pyramids C to D are reciprokall with the altitudes of D to C. But in what proportion are the bases C to D in the same are the bases A to B forasmuch as they are equall And in what proportion are the altitudes of D to C in the same are the altitudes of B to A which altitudes are likewise equall Wherefore by the 11. of the fifth in what proportion the bases A to B are in the same reciprokally are the altitudes of the pyramids B to A. In like sort by the second part of this proposition may be proued the conuerse of this corollary The same thing followeth also in a Prisme and in a sided columne as hath before at large bene declared in the corollary of the 40. proposition of the 11. booke For those solides are in proportioÌ„ the one to the other as the pyramids or parallelipipedons for they are either partes of equemultiplices or equemultiplices to partes The 10. Theoreme The 10. Proposition Euery cone is the third part of a cilinder hauing one and the selfe same base and one and the selfe same altitude with it SVppose that there be a cone hauing to his base the circle ABCD and let there be a cilinder hauing the selfe same base and also the same altitude that the cone hath Then I say that the cone is the third part of the cilinder that is that the cilinder is in treble proportion to the cone For if the cilinder be not in treble proportion to the cone then the cilinder is either in greater proportions then triple to the cone or els in lesse First let it be in greater then triple And describe by the 6. of the fourth in the circle ABCD a square ABCD. Now the square ABCD is greater then the halfe of the circle ABCD For if about the circle ABCD we describe a square the square described in the circle ABCD is the halfe of the square described about the circle And let there be Parallelipipedon prismes described vpon those squares equall in altitude with the cilinder But prismes are in that proportion the one to the other that their bases are by the 32. of the eleuenth and 5. Corollary of the 7. of this booke Wherefore the prisme described vpon the square ABCD is the halfe of the prisme described vpon the square that is described about the circle Now the clinder is lesse then the prisme which is made of the square described abouâ— the circle ABCD being equal in altitude with it for it contayneth it Wherfore the prisme described vpon the square ABCD and being equall in altitude with the cylinder is greater then half the cylinder Deuide by the 30. of the third the circumferences AB BC CD and DA into two equall parts in the points E F G H And draw these right lines AE EB BF FC CG GD DH HA. Wherfore euery one of these triangles AEB BFC CGD and DHA is greater then halfe of that segment of the circle ABCD which is described about it as we haue before in the 2. proposition declared Describe vpon euery one of these triangles AEB BFC CGD and DHA a prisme of equall altitude with the cylinder Wherefore euery one of these prismes so described is greater then the halfe part of the segment of the cylinder that is set vpon the sayd segments of the circle For if by the pointes E F G H be drawen parallell lines to the lines AB BC CD and DA and then be made perfect the parallelogrammes made by those parallell lines and moreouer vpon those parallelograÌ„mes be erected parallelipipedons equall in altitude with the cylinder the prismes which are described vpon eche of the triangles AEB BFC CGD and DHA are the halfes of euery one of those parallelipipedons And the segments of the cylinder are lesse then those parallelipipedons so described Wherefore also euery one of the prismes which are described vpon the triangles AEB BFC CGD and DHA is greater then the halfe of the segment of the cylinder set vpon the sayd segment Now therefore deuiding euery one of the circumferences remaining into two equall partes and drawing right lines and raysing vp vpon euery one of these triangles prismes equall in altitude with the cylinder and doing this continually we shall at the length by the first of the tenth leaue certaine segments of the cylinder which shal be lesse then the excesse whereby the cylinder excedeth the cone more then thrise Let those segments be AE EB BF FC CG GD DH and HA. Wherfore the prisme remayning whose base is the poligonon â—igure AEBFCGDH and altitude the selfe same that the cylinder hath is greater then the cone taken three tymes But the prisme whose base is the poligonon figure AEBFCGDH and altitude the selfe same that the cylinder hath is treble to the pyramis whose base is the poligonon figure AEBFCGDA and altitude the selfe same that the cone hath by the corollary of the 3. of this booke Wherfore also the pyramis whose base is the poligonon figure AEBFCGDH and toppe the self same that the cone hath is greater then the cone which hath to his base the circle ABCD. But it is also lesse for it is contayned of it which is impossible Wherefore the cylinder is not in greater proportion then triple to the cone I say moreouer that the cylinder is not in lesse proportion then triple to the coneâ— For if it be possible let the cylinder be in lesse proportion then triple to the cone Wherefore by conuersion the cone is greater then the third part of the cylinder Describe now by the sixth of the fourth in the circle ABCD a square ABCD. Wherefore the square ABCD is greater then the halfe of the circle ABCD vpon the square ABCD describe a pyramis hauing one the selfe same altitude with the cone Wherfore the pyramis so described is greater theÌ„ halfe of the cone For if as we haue before declared we describe a square about the circle the square ABCD is the halfe of the square described about the circle and if vppon the squares be described parallelipipedons equall in altitude with the cone which solides are also called prismes the prisme or parallelipipedon described vpoÌ„ the square ABCD is the halfe of the prisme which is described vpoÌ„ the square described about the circle for they are the one to the other in that proportioÌ„ that their bases are by the 32. of the eleueÌ„th 5. corollary of the 7. of this booke Wherfore also their third parts are in the self same proportion by the 15. of the fift Wherfore the pyramis whose base is the square ABCD is the halfe of the pyramis set vpon the square described about the circle But the pyramis set vpon the square described about the circle is greater then the cone whome
to GR by the 14. of the twelfth and to ED is the cylinder AB equall by construction and to EF we haue proued the cone QEK equall wherefore by the 7. of the fifth AB is to QEK as GT iâ— to GR. Wherfore an vpright cone a cylinder being geuen we haue found two right lines hauing the same proportion betwene them which the cone and the cylinder haue one to the other which was requisite to be done An assumpt If a coâ—e and a cylinder being both on one base are equall one to the other the heith of the cone â—s tripla to the heith of the cylinder And if a cone and a cylinder being both on one base the heith of the cone be tripla to the heith of the cylinder the cone and the cylinder are equall We will vse the cylinder AB the cone PBH in the â—oâ—mer probleme with their base heithes so noted as before I say if PBH be equall to AB that CP the heith of PBH is tripla to CS the heith of the cylinder AB Suppose vpon the base BH a cone to be rered of the heith of CS which let be SBH it is manifest that AB is tripla to that cone SBH by the 10. of this twelfth Wherfore a cone equal to AB the cylinder is tripla to SBH the cone by the 7. of the fifth but PBH is supposed equall to AB Therefore PBH is tripla to SBH therefore the heith of PBH shall be tripla to the heith of SBH by the 14. of the twelfth But the heith of PBH is CP and of SBH the heith is CS wherefore CP is tripla to CS And CS is the heith of the cylinder AB by supposition Therefore a cone and a cylinder being both on one base and equall the heith of the cone is tripla to the heith of the cylinder And the second part as easely may be confirmed For if AB a cylinder and PBH a cone haue one base as the circle about BH and the heith of PBH be tripla to the heith of AB I say that PBH and AB are equal The heith of AB let be as afore CS and of PBH the heith let be CP of the heith CS imagine a Cone vpon the same base BH by the 10. of this twelfth AB shall be triple to that cone And the cone PBH hauing heith CP by supposition tripla to CS shall also be tripla to that cone SBH by the 14. of this twelâ—th Wherefore by the 7. of the fifth AB and PBH are equall Therefore if a cone and a cylinder being both on one base the heith of the cone be tripla to the heith of the cylinder the cone and the cylinder are equall So haue we demonstrated both partes as was required ¶ A Theoreme 4. The superficies of the segment or protion of any sphere is equall to the circle whose semidiameter is equall to that right line which is drawne from the toppe of that segment to the circumference of the circle which is the base of that portion or segment This hath Archimedes demonstrated in this first booke of the Sphere and Cylinder in his 40. and 41 propositions and I remitte them thether that will herein demonstratiuely be certified I would wish all Mathematiciens as well of verities easy as of verities rare and obscure to seeke the causes demonstratiue the finall fruite thereof is perfection in this art Note Besides all other vses and commodities that are of the Croked superficieces of the Cone Cylinder and Sphere so easely and certaynely of vs to be dealt with all this is not the least that a notable Error which among Sophisticall brablers and vngeometricall Masters and Doctors hath a long time bene vpholden may most euidently hereby be confuted and vtterly rooted out of all mens fantasies for euer The Error is this Curui ad rectum nulla est proportio that is Betwene croked and straight is no proportionâ— This error in lineâ— supâ—â—ficieces and solides may with more true demonstrations be ouerthrowne then the fauourers of that fond faâ—tasie are able with argument either probable or Sophisticall to make shew or pretence to the contrary In lines I omitte as now Archimedes two wayes for the finding of the proportion of the circles circumference to a straight line I meane by the inscription and ciâ—cumscription of like poligonon figures and that other by spirall lines And I omitte likewise as now in solides of a parallelipipedon equall to a Sphere Cone or Cylinderâ— or any segment or sector of the sayd solides And onely here require you to consider in this twelfth booke the wayes brought to your knowledge how to the croked superficies of a cone and cylinder and of a sphere the whole any segment or sector thereof a playne and straight superficies may be geuen equall Namely a Circle to be geuen equall to any of the sayd croked superficieces assigned and geuen And then farther by my Additions vpon the second proposition you haue meanes to proceede in all proportions that any man can in right lines geueâ— or assigne Theâ—fore Curui ad rectum proportio omnimoda potest dari One thing it is to demonstrate that betwene a croked line and a straight or a croked superficies and a playne or straight superficies c. there is proportion And an other thing it is to demonstrate a particular and speciall kinde of proportion being betwene a croked superficies and a straight or playne superficies For this also confiâ—meth the first This short warning will cause you to auoyde the sayd error and make you also hable to cure them which are infected therewith A Theoreme 5. Any two Spheres being geuen as the Sphericall Superficies of the one is to the Sphiricall Superficies of the other So is the greatest Circle conteyned in that one to the greatest Circle conteyned in that other And as greatest Circle is to greatest Circle so is Sphericall superficies to Sphericall superficies For the superficies of euery sphere is quadrupla to his greatest circle by my first Theoreme wherfore of two geuen spheres as the sphericall superâ—icies of the one is to his greatest circle so is the sphericall superficies of the other to his greatest circle therfore by alternate proportion as spherical superficies is to spherical superficies so is greatest circle to greatest circle And therfore also as greatest circle is to greatest circle so is spherical superficies to sphericall superficies which was to be demoÌ„strated A Probleme 9. A Sphere being geuen to geue an other Sphere to whose Sphericall superficies the superficies Sphericall of the Sphere geuen shall haue any proportion betwene two right lines geuen Suppose A to be a sphere geuen and the proportion geuen to be that which is betwene the right lines X and Y. I say that a sphere is to be gâ—uen to whose sphericall superficies the superficies sphericall of A shall haue that proportion which X hath to Y. Let the greatest circle
conteyned in A the sphere be the circle BCD And by the probleme of my additions vpon the second proposition of this booke as X is to Y so let the circle BCD be to an other circle found let that other circle be EFG and his diameter EG I say that the sphericall superficies of the sphere A hath to the sphericall superficies of the sphere whose greatest circle is EFG or his equall that proportion which X hath to Y. For by construction BCD is to EFG as X is to Y and by the theoreme next beforeâ— as BCD is to â—FG so is the spherical superficies of A whose greatest circle is BCD by supposition to the sphericall superficies of the sphere whose greatest circle is EFG wherefore by the 11. of the fifth as X is to Y So is the sphericall superficies of A to the sphericall superficies of the sphere whose greatest circle is EFG wherefore the sphere whose diameter is EG the diameter also of EFG is the sphere to whose sphericall superficies the sphericall superficies of the sphere A hath that proportion which X hath to Y. A sphere being geuen therefore we haue geuen an other sphere to whose sphericall superficies the superficies sphericall of the sphere geuâ— hath any proportion geuen betwene two right lines which ought to be done A Probleme 10. A sphere being geuen and a Circle lesse then the greatest Circle in the same Sphere conteyned to coapt in the Sphere geuen a Circle equall to the Circle geuen Suppose A to be the sphere geuen and the circle geuen lesse then the greatest circle in A conteyned to be FKG I say that in the Sphere A a circle equall to the circle FKG is to be coapted First vnderstand what we meane here by coapting of a circle in a Sphere We say that circle to be coapted in a Sphere whose whole circumference is in the superficies of the same Sphere Let the greatest circle in the Sphere A conteyned be the circle BCD Whose diameter suppose to be BD and of the circle FKG let FG be the diameter By the 1. of the fourth let a line equall to FG be coapted in the circle BCD Which line coapted let be BE. And by the line BE suppose a playne to passe cutting the Sphere A and to be perpendicularly erected to the superficies of BCD Seing that the portion of the playne remayning in the sphere is called their common section the sayd section shall be a circle as before is proued And the common section of the sayd playne and the greatest circle BCD which is BE by supposition shall be the diameter of the same circle as we will proue For let that circle be BLEM Let the center of the sphere A be the point H which H is also the ceÌ„ter of the circle BCD because BCD is the greatest circle in A conteyned From H the center of the sphere A let a line perpendicularly be let fall to the circle BLEM Let that line be HO and it is euident that HO shall fall vpon the common section BE by the 38. of the eleuenth And it deuideth BE into two equall parts by the second part of the third proposition of the third booke by which poynt O all other lines drawne in the circle BLEM are at the same pointe O deuided into two equall parts As if from the poynt M by the point O a right line be drawne one the other side comming to the circumference at the poynt N it is manifest that NOM is deuided into two equall partes at the poynt O by reason if from the center H to the poyntes N and M right lines be drawne HN and HM the squares of HM and HN are equall for that all the semidiameters of the sphere are equal and therefore their squares are equall one to the other and the square of the perpendicular HO is common wherefore the square of the third line MO is equall to the square of the third line NO and therefore the line MO to the line NO So therefore is NM equally deuided at the poynt O. And so may be proued of all other right lines drawne in the circle BLEM passing by the poynt O to the circumference one both sides Wherefore O is the center of the circle BLEM and therefore BE passing by the poynt O is the diameter of the circle BLEM Which circle I say is equal to FKG for by construction BE is equall to FG and BE is proued the diameter of BLEM and FG is by supposition the diameter of the circle FKG wherefore BLEM is equall to FKG the circle geuen and BLEM is in A the sphere geueÌ„ Wherfore we haue in a sphere geuen coapted a circle equall to a circle geuen which was to be done A Corollary Besides our principall purpose in this Probleme euidently demonstrated this is also made manifest that if the greatest circle in a Sphere be cut by an other circle erected vpon him at right angles that the other circle is cut by the center and that their common section is the diameter of that other circle and therefore that other circle deuided is into two equall partes A Probleme 11. A Sphere being geuen and a circle lesse then double the greatest circle in the same Sphere contained to cut of a segment of the same Sphere whose Sphericall superficies shall be equall to the circle geuen Suppose K to be a Sphere geuen whose greatest circle let be ABC and the circle geuen suppose to be DEF I say that a segment of the Sphere K is to be cut of so great that his Sphericall superficies shall be equall to the circle DEF Let the diameter of the circle ABC be the line AB At the point A in the circle ABC coapt a right line equall to the semidiameter of the circle DEF by the first of the fourth Which line suppose to be AH From the point H to the diameter AB let a perpendicular line be drawen which suppose to be HI Produce HI to the other side of the circumference and let it come to the circumference at the point L. By the right line HIL perpendicular to AB suppose a plaine superficies to passe perpendicularly erected vpon the circle ABC and by this plaine superficies the Sphere to be cut into two segmentes one lesse then the halfe Sphere namely HALI and the other greater then the halfe Sphere namely HBLI I say that the Sphericall superficies of the segment of the Sphere K in which the segment of the greatest circle HALI is contayned whose base is the circle passing by HIL and toppe the point A is equall to the circle DEF For the circle whose semidiameter is equall to the line AH is equall to the Sphericall superficies of the segment HAL by the 4. Theoreme here added And by construction AH is equall to the semidiameter of the circle DEF therefore the Sphericall superficies of the segment of the Sphere K cut of by the
the 7. of the fift our conclusion is inferred the superficies Sphericall of the segment CAE to be to the superficies Sphericall of the segment FGH as AD is to GI A Theoreme 6. To any solide sector of a Sphere that vpright Câ—â—e is equall whose base is equall to the câ—nnex Sphericall superficies of that sector and heith equall to the semidiameter of the same Sphere Hereof the demonstration in respect of the premises and the common argument of inscriptioâ— and circumscription of figures is easy and neuerthelesse if your owne write will not helpe you sufficiently you may take helpe at Archimedes hand in his first booke last proposition of the sphere and cylinder Whether if ye haue recourse you shalâ— perceaue how your Theoreme here amendeth the common translation there and also our delinâ—ation geueth more sâ—uâ—y shew of the chiefe circumstances necessary to the construction then there you shall finde Of the sphere here imagined to be A we note a solide sector by the letterâ— PQRO. So that PQR doth signifie the sphericall superficies to that solide sector belonging which is also common to the segment of the same sphere PRQ and therefore a line drawne from the toppe of that segmentâ— which toppe suppose to be Q is the semidiameter of the circle which is equall to the sphericall superficies of the sayd solide sector or segmentâ— as before is taught Let that line be QP By Q draw a line contingent which let be SQT. At the poynt Q from the line QS cut a line equall to PQ which let be SQ And vnto SQ make QT equall then draw the right lines OSOâ— and OQ About which OQ as an axe faâ—â—ened if you imagine the triangle OST to make an halfe circular reuolution you shall haue the vpright cone OST whose heith is OQ the semidiameter of the sphere and base the circle whose diameter is ST equall to the solide sector PQRO. A Theoreme 7. To any segment or portion of a Sphere that cone iâ— equall which hath that circle to his base which is the bâ—se of the segmeÌ„t and heith a right line which vnto the heith of the segmeÌ„t hath that proportioÌ„ which the semidiameter of the Sphere together with the heith of the other segment remayâ—â—â—g hath to the heith of the same other segment remaynâ—ng This is well demonstrated by Archiâ—â—des therefore nedeth no inuention of myne to confirme the same and for that the sayd demonstration is ouer long here to be added I will refere you thether for the demonstration and here supply that which to Archimedes demonstration shall geue light and to your farther speculation and practise shal be a great ayde and direction Suppose K to be a sphere the greatest circle K in coÌ„teyned let be ABCE and his diameter BE ceÌ„ter D. Let the sphere K be cutte by a playne superficies perpâ—ndicularly erected vpon the sayd greatest circle ABCE let the section be the circle about AC And let the segmentes of the sphere be the one that wherein is ABC whose â—oppe is Bâ— and the other let be that wherein is AEC and his toppe let be E I say that a cone which hath his base the circle about AC held a line which to BF the heith of the segment whose toppe is B hath that proportion that a line compoâ—ed of DE the semidiameter of the sphere and EF the heith of the other remayning segment whose toppe is E hath to EF the heith of that other segment remayning is equall to the segment of the sphere K whose toppe is B. To make this cone take my easy order thus Frame your worke for the findâ—ng of the fourth proportionall lineâ— by making EF the first and a line composed of DE and EF the secondâ— and the third let be BF then by the 12. of the sixâ—h let the fourth proportionall line be found which let be FGâ— vpon F the center of the base of the segment whose toppe is B erect a line perpendicular equall to FG found and drawe the lines GA and GC and so make perfect the cone GAC I say that the cone GAC is equall to the segment of the sphere K whose toppe is B. In like maner for the other segmeÌ„t whose toppe is E to finde the heith due for a cone equal to it by the order of the Theoreme you must thus frame your lines let the first be BF the second DB and BF composed in one right line and the third must be EF where by the 12. of the sixth finding the fourth it shall be the heith to rere vpon the base the circle about AC to make an vpright cone equall to the segment whose toppe is E. ¶ Logistically ¶ The Logisticall finding hereof is most easy the diameter of the sphere being geuen and the portions of the diameter in the segmentes conteyned or axes of the segmentes being knowne Then order your numbers in the rule of proportion as I here haue made most playne in ordring of the lines for the â—ought heith will be the producte A Corollary 1. Hereby and other the premises it is euident that to any segment of a Sphere whose whole diameter is knowne and the Axe of the segment geuen An vpright cone may be made equall or in any proportion betwene two right lines assignedâ— and therefore also a cylinder may to the sayd segment of the Sphere be made equall â—r in any proportion geuen betwene two right lines A Corollary 2. Manifestly also of the former theoreme it may be inferred that a Sphere and his diameter being deuided by one and the same playne superficies to which the sayd diameter is perpendicularâ— the two segmentes of the Sphere are one to the other in that proportion in which a rectangle parallelipipedon hauing for his base the square of the greater part of the diameter and his heith a line composed of the lesse portion of the diameter and the semidiameter to the rectangle parallelipâ—pedon hauing for his base the square of the lesse portion of the diameter his heith a line composed of the semidiameter the greater part of the diameter A Theoreme 8. Euery Sphere to the cube made of his diameter is in maner as 11. to 21. As vpon the first and second propositioÌ„s of this booke I began my additions with the circle being the chiefe among playne figures and therein brought manifold considerations about circles as of the proportion betwene their circumferences and their diameters of the content or Area of circles of the proportion of circles to the squares described of their diameters of circles to be geuen in al proâ—portions to other circles with diuerse other most necessary problemes whose vse is partly there specified So haue I in the end of this booke added some such Problemes Theoremesâ— about the sphere being among solides the chiefe as of the same either in it selfe considered or to cone and cylinder compared by reason of
superficies or soliditie in the hole or in partâ— such certaine knowledge demonstratiue may arise and such mechanical exercise thereby be deuised that sure I am to the sincere true student great light ayde and comfortable courage farther to wade will enter into his hart and to the Mechanicall witty and industrous deuiser new maner of inuentions executions in his workes will with small trauayle for fete application come to his perceiueraunce and vnderstanding Therefore euen a manifolde speculations practises may be had with the circle his quantitie being not knowne in any kinde of smallest certayne measure So likewise of the sphere many Problemes may be executed and his precise quantitie in certaine measure not determined or knowne yet because both one of the first humane occasioÌ„s of inuenting and stablishing this Arte was measuring of the earth and therfore called Geometria that is Earthmeasuring and also the chiefe and generall end in deede is measure and measure requireth a determination of quantitie in a certayne measure by nuÌ„ber expressed It was nedefull for Mechanicall earthmeasures not to be ignorant of the measure and contents of the circle neither of the sphere his measure and quantitie as neere as sense can imagine or wish And in very deede the quantitie and measure of the circle being knowne maketh not onely the cone and cylinder but also the sphere his quantitie to be as precisely knowne and certayne Therefore seing in respect of the circles quantitie by Archimedes specified this Theoreme is noted vnto you I wil by order vpon that as a supposition inferre the conclusion of this our Theoremes Note 1. Wherfore if you deuide the one side as TQ of the cube TX into 21. equall partes and where 11. partes do end reckening from T suppose the point P and by that point P imagine a plaine passing parallel to the opposite bases to cut the cube TX and therby the cube TX to be deuided into two rectangle parallelipipedons namely TN and PX It is manifest TN to be equall to the Sphere A by construction and the 7. of the fift Note 2. Secondly the whole quantitie of the Sphere A being coÌ„tayned in the rectangle parallelipipedon TN you may easilie transforme the same quantitie into other parallelipipedons rectangles of what height and of what parallelogramme base you list by my first and second Problemes vpon the 34. of this booke And the like may you do to any assigned part of the Sphere A by the like meanes deuiding the parallelipipedon TN as the part assigned doth require As if a third fourth fifth or sixth part of the Sphere A were to be had in a parallelipipedon of any parallelograâ—â—e base assigned or of any heith assigned then deuiding TP into so many partes as into 4. if a fourth part be to be transformed or into fiue if a fifth part be to be transformed c. and then proceede â—s you did with cutting of TN from TX And that I say of parallelipipedons may in like sort by my â—â—yd two problemes added to the 34. of this booke be done in any sided columnes pyramids and prismeâ— so thâ—â— in pyramids and some prismes you vse the cautions necessary in respect of their quan 〈…〉 odyes hauing parallel equall and opposite bases whose partes 〈…〉 re in their propositions is by Euclide demonstrated And finally 〈…〉 additions you haue the wayes and orders how to geue to a Sphere or any segmeâ—â— oâ— the same Cones or Cylinders equall or in any proportion betwene two right lines geuen with many other most necessary speculations and practises about the Sphere I trust that I haue sufficiently â—raughted your imagination for your honest and profitable studie herein and also geuen you reaâ—â— â—â—tter wheâ—â— with to sâ—â—p the mouthes of the malycious ignorant and arrogant despisers of the most excellent discourses trauayles and inuentions mathematicall Sting aswel the heauenly spheres sterres their sphericall soliditie with their conueâ—e spherical superficies to the earth at all times respecting and their distances from the earth as also the whole earthly Sphere and globe it selfe and infinite other cases concerning Spheres or globes may hereby with as much ease and certainety be determined of as of the quantitie of any bowle ball or bullet which we may gripe in our handes reason and experience being our witnesses and without these aydes such thinges of importance neuer hable of vs certainely to be knowne or attayned vnto Here ende M. Iohn d ee his additions vpon the last proposition of the twelfth booke A proposition added by Flussas If a Sphere touche a playne superficiesâ— a right line drawne from the center to the touche shall be erected perpendicularly to the playne superficies Suppose that there be a Sphere BCDL whose centre let be the poynt A. And let the playne superficies GCI touch the Spere in the poynt C and extend a right line from the centre A to the poynt C. Then I say that the line AC is erected perpendicularly to tâ—e playne GIC. Let the sphere be cutte by playne superficieces passing by the right line LAC which playnes let be ABCDL and ACEL which let cut the playne GCI by the right lines GCH and KCI Now it is manifest by the assumpt put before the 17. of this booke that the two sections of the sphere shall be circles hauing to their diameter the line LAC which is also the diameter of the sphere Wherefore the right lines GCH and KCI which are drawne in the playne GCI do at the poynt C fall without the circles BCDL and ECL. Wherefore they touch the circles in the poynt C by the second definition of the third Wherefore the right line LAC maketh right angles with the lines GCH and KCI by the 16. of the third Wherefore by the 4. of the eleuenth the right line AC is erected perpendicularly to to the playne superficies GCI wherein are drawne the lines GCH and KCI If therefore a Sphere touch a playne superficies a right line drawne from the centre to the touche shall be erected perpendicularly to the playne superficies which was required to be proued The ende of the twelfth booke of Euclides Elementes ¶ The thirtenth booke of Euclides Elementes IN THIS THIRTENTH BOOKE are set forth certayne most wonderfull and excellent passions of a lyne deuided by an extreme and meane proportion a matter vndoubtedly of great and infinite vse in Geometry as ye shall both in thys booke and in the other bookes following most euidently perceaue It teacheth moreouer the composition of the fiue regular solides and how to inscribe them in a Sphere geuen and also setteth forth certayne comparisons of the sayd bodyes both the one to the other and also to the Sphere wherein they are described The 1. Theoreme The 1. Proposition If a right line be deuided by an extreme and meane proportion and to the greater segment be added the halfe of the whole line the square made of those two
lines added together shal be quintuple to the square made of the halfe of the whole lyne SVppose that the right line AB be deuided by an extreme and meane proportioÌ„ in the point C. And let the greater segment therof be AC And vnto AC adde directly a ryght line AD and let AD be equall to the halfe of the line AB Then I say that the square of the line CD is quintuple to the square of the line DA. Describe by the 46. of the first vpon the lines AB and DC squares namely AE DF. And in the square DF describe and make complete the figure And extend the line FC to the point G. And forasmuch as the line AB is deuided by an extreme and meane proportion in the point C therefore that which is contayned vnder the lines AB and BC is equall to the square of the line AC But that which is contayned vnder the lines AB and BC is the parallelogramme CE and the square of the line AC is the square HF. Wherefore the parallelogramme CE is equall to the square HF. And forasmuch as the line BA is double to the line AD by constructâ—on 〈◊〉 the lyne BA is equall to the line KA and the line AD to the lyne AH therefore also the lyne KA is double to the line AH But as the lyne KA is to the line AH so is the parallelogramme CK to the parallelogramme CH Wherefore the parallelogramme CK is double to the parallelogramme CH. And the parallelogrammes LH and CH are double to the parallelogramme CH for supplementes of parallelogrammes are bâ— the 4â— of the first equall the one to the other Wherefore the parallelogramme CK is equall to the parallelogrammes LH CH. And it is proued that the parallelogramme CE is equall to the square FH Wherefore the whole square AE is equall to the gnâ—mon MXN And forasmuch as the line BA iâ— double to the line AD therefore the square of the line BA is by the 20. of the sixth quadruple to the square of the line DA that is the square AE to the square DH But the square AE is equall to the gnomoÌ„ MXN wherefore the gnomoÌ„ MXN is also quadruple to the square DH Wherefore the whole square DF is quintuple to the square DH But the square DF iâ— the square of the line CD and the square DH is the square of the line DA. Wherefore the square of the line CD is quintuple to the square of the line DA. If therefore a right line be deuided by an extreame and meane proportion and to the greater segment be added the halfe of the whole line the square made of those two lines added together shal be quintuple to the square made of the halfe of the whole line Which was required to be demonstrated Thys proposition is an other way demonstrated after the fiueth proposition of this booke The 2. Theoreme The â— Proposition If a right line be in power quintuple to a segment of the same line the double of the sayd segment is deuided by an extreame and meane proportion and the greater segment thereof is the other part of the line geuen at the beginning Now that the double of the line AD that is AB is greater then the line AC may thus be proued For if not then if if it be possible let the line AC be double to the line AD wherefore the square of the line AC is quadruple to the square of the line AD. Wherefore the squares of the lines AC and AD are quintuple to the squares of the line AD. And it is supposed that the square of the line DC is quintuple to the square of the line AD wherefore the square of the line DC is equall to the square of the lines AC and AD which is impossible by the 4. of the second Wherefore the line AC is not double to the line AD. In like sorte also may we proue that the double of the line AD is not lesse then the line AC for this is much more absurd wherefore the double of the line AD is greater theÌ„ the line ACâ— which was required to be proued This proposition also is an other way demonstrated after the fiueth proposition of this booke Two Theoremes in Euclides Method necessary added by M. Dee A Theoreme 1. A right line can be deuided by an extreame and meane proportion but in one onely poynt Suppose a line diuided by extreame and meane proportion to be AB And let the greater segment be AC I say that AB can not be deuided by the sayd proportion in any other point then in the point C. If an aduersary woulde contend that it may in like sort be deuided in an other point let his other point be supposed to be D making AD the greater segment of his imagined diuision Which AD also let be lesse then our AC for the first discourse Now forasmuch as by our aduersaries opinion AD is the greater segment of his diuided lineâ— the parallelogramme conteyned vnder AB and DB is equall to the square of AD by the third definition and 17. proposition of the sixth Booke And by the same definition and proposition the parallelogramme vnder AB and CB conteyned is equall to the square of our greater segment AC Wherefore as the parallelogramme vnder AB and Dâ— is to the square of AD so i◠〈◊〉 parallelogramme vnder AB and CB to the square of AC For proportion of equality is concluded in them both But forasmuch as Dâ—â— iâ— byâ— supposition greater theÌ„ CB the parallelograÌ„me vnder AB and DB is greater then the parallelogramme vnder AC and CB by the first of the sixth for AB is their equall heith Wherefore the square of AD shal be greater then the square of AC by the 14. of the fifth But the line AD is lesse then the line AC by supposition wherefore the square of AD is lesse then the square of AC And it is concluded also to be greater then the square of AC Wherefore the square of AD is both greater then the square of ACâ— and also lesse Which is a thing impossible The square therefore of AD is not equall to the parallelogramme vnder AB and DB. And therefore by the third definition of the sixth AB is not deuided by an extreame and meane proportion in the point D as our aduersary imagined And Secondly in like sort will the inconueniency fall out if we assigne AD our aduersaries greater segment to be greater then our AC Therefore seing neither on the one side of our point C neither on the other side of the same point C any point can be had at which the line AB can be deuided by an extreame and meane proportion it followeth of necâ—ssitie that AB can be deuided by an extreame and meane proportion in the point C onely Therefore a right line can be deuided by an extreame
and meane proportion but in one onely point which was requisite to be demonstrated A Theoreme 2. What right line so euer being deuided into two partes hath those his two partes proportionall to the two segmentes of a line deuided by extreame and meane proportion is also it selfe deuided by an extreame and meane proportion and those his two partes are his two segments of the sayd proportion Suppose AB to be a line deuided by an extreame and meane proportion in the point C and AC to be the greater segment Suppose also the right line DE to be deuided into two partes in the point F and that the part DF is to FE as the segment AC is to CB or DF to be to AC as FE is to CB. For so these paâ—tes are proportionall to the sayd segmentes I say now that DE is also deuided by an extreame and meane proportion in the point F. And that DF FE are his segmentes of the sayd proportion For seing as AC is to CB so is DF to FE by supposition Therfore as AC and CB which is AB are to CB so is DF and FE which is DE to FE by the 18. of the fifth Wherefore alternately as AB is to DE so is CB to FE And therefore the residue AC is to the residue DF as AB is to DE by the fifth of the fift And then alternately AC is to AB as DE is to DF. Now therefore backward AB is to AC as DE is to DF. But as AB is to AC so is AC to CB by the third definition of the sixth booke Wherefore DE is to DF as AC is to CB by the 11. of the fifth And by supposition as AC is to CB so is DF to FE wherefore by the 11. of the fifth as DE is to DF so is DF to FE Wherefore by the 3. definition of the sixth DE is deuided by an extreame and meane proportion in the point F. Wherefore DF and FE are the segmentes of the sayd proportion Therefore what right line so euer being deuided into two partes hath those his two partes proportionall to the two segmentes of a line deuided by extreame and meane proportionâ— is also it selfe deuided by an extreme and meane proportion and those his two partes are his two segmentes of the sayd proportionâ— which was requisite to be demonstrated Note Many wayes these two Theoremes may be demonstrated which I leaue to the exercise of young studentes But vtterly to want these two Theoremes and their demonstrations in so principall a line or rather the chiefe piller of Euclides Geometricall pallace was hetherto and so would remayne a great disgrace Also I thinke it good to note vnto you what we meant by one onely poynt We mâ—â—â—â— that the quantities of the two segmentes can not be altered the whole line being once geuen And though from either end of the whole line the greater segment may begin And so as it were the point of section may seeme to be altered yet with vs that is no alteration forasmuch as the quantities of the segmentes remayne all one I meane the quantitie of the greater segment is all one at which end so euer it be taken And therefore likewise the quantitie of the lesse segment is all one c. The like confideration may be had in Euclides tenth booke in the Binomiall lines c. Ioâ—n Dee 1569. Decemb. 18. The 3. Theoreme The 3. Proposition If a right line be deuided by an extreme and meane proportion and to the lesse segment be added the halfe of the gerater segment the square made of those two lines added together is quintuple to the square made of the half line of the greater segment SVppose that the right line AB be deuided by an extreme and meane proportion in the point C. And let the greater segment thereof be AC And deuide AC into two equall partes in the point D. Then I say that the square of the line BD it quintuple to the square of the line DC Describe by the 46. of the first vpon the line AB a square AE And describe and make perfect the figure that is diuide the lyne AT like vnto the diuision of the line AB by the 10. of the sixth in the pointes R H by which pointes drawe by the 31. of the first vnto the line AB parallel lines RM and HN. So likewise draw by the pointes D C vnto the line BE these parallel lines DL and CS draw the diameter BT And forasmuch as the line AC is double to the line DC therefore the square of AC is quadruple to the square of DC by the 20. of the sixth that is the square RS to the square FG And forasmuch as that which is contayned vnder the lines AB and BC is equall to the square of the line AC and that which is contayned vnder the lines AB and BC is equall to the parallelogramme CE the square of the line AC is the square RS wherefore the parallelogramme CE is equall to the square RS. But the square RS is quadruple to the square FG wherefore the parallelogramme CE also is quadruple to the square FG. Agayne forasmuch as the line AD is equall to the line DC therfore the line HK is equall the line KF wherefore also the square GF is equall to the square HL wherefore the line GK is equall to the line KL that is the line MN to the line NE wherefore the parallelogramme MF is equall to the parallelogramme FE But the parallelogramme MF is equall to the parallelogramme CG wherfore the parallelogramme CG is also equal to the parallelogramme FE Put the parallelogramme CN common to theÌ„ both Wherefore the gnomon XOP is equall to the parallelogramme CK But the parallelogramme CE is proued to be quadruple to Gâ— the square wherefore the gnomon XOP is quadruple to the square GF Wherefore the square DN is quintuple to the square FG And DN is the square of the line DB and GF the square of the line DC Wherefore the square of the line DB is quintuple to the square of the line DC If therefore a right line be deuided by an extreme and meane proportion and to the lesse segment be added the halfe of the greater segment the square made of those two lines added together is quintuple to the square made of the halfe line of the greater segmeÌ„t Which was required to be demonstrated Ye shall finde this proposition an other way demonstrated after the fiueth proposition of this booke Here foloweth M. Dee his additions ¶ A Theoreme 1. If a right line geuen be quintuple in power to the powre of a segment of him self the double of that segment and the other part remayning of the first geuen line make a line diuided by extreme and meane proportion and that double of the segment is the greater part thereof Forasmuch as this is the conuerse of Euclides
the square of ED to the square of DK but by construction ED is subquintuple to DF. Wherefore the square of ED is subquintuple to the square of DK And therefore the square of DK is quintuple to the square of ED. And ED is equall to ED by construction therefore the square of DK is quintuple to the square of E D. Wherefore the double of BD is deuided by an extreme and meane proportion whose greater segment is BK â— by the second of this thirteâ—th But by construction AB is the double of â—D â— Wherefore AB is diuided by extreme and meane proportion and his greater segment is BK and thereby K â— the point of the diuision We haue therefore deuided by extreme and meane proportion any right line geuen in length and position Which was to be done Noteâ— Ech of these wayes may well be executed But in the first you haue this auantage that the diameter is taken at pleasure Which â—n the second way is euer iust thrise so long as the line geuen to be deuided Iohn Dee ¶ The 4. Theoreme The 4. Proposition If a right line be deuided by an extreame and meane proportion the squares made of the whole line and of the lesse segmeÌ„t are treble to the square made of the greater segment SVppose that the right line AB be deuided by an extreame meane proportioÌ„ in the point C. And let the greater segment thereof be AC Then I say that the squares made of the lines AB and BC are treble to the square of the line AC Describe by the 46. of the first vpon the line AB a square ADEB And make perfect the figure Now forasmuch as the line AB is deuided by an extreame and meane proportion in the point C and the greater segmeÌ„t thereof is the line AC therefore that which is contayned vnder the lines AB and BC is equall to the square of the line AC But that which is contayned vnder the lines AB and CB is the parallelogramme AK and the square of the line AC is the square FD. Wherefore the parallelogramme AK is equall to the square FD. And the parallelogramme AF is equall to the parallelogramme FE put the square CK common to them both wherfore the whole parallelograÌ„me AK is equall to the whole parallelogramme CE Wherefore the parallelogrammes CE and AK are double to the parallelogramme AK Bâ—t the parallelogrammes AK and CE are the gnomon LMN and the square CK Wherefore the gnomon LMN and the square CK are double to the parallelogramme AK But it is proued that the parallelogramme AK is equal to the square DF. Wherefore the gnomon LMN and the square CK are double to the square DF. Wherefore the gnomon LMN and the squares CK and DF are treble to the square DF. But the gnomon LMN and the squares CK and DF are the whole square AE together with the square CK which are the squares of the lines AB and BC. And DF is the square of the line AC Wherefore the squares of the lines AB and BC are treble to the square of the line AC If therefore a right line be deuided by an extreame and meane proportion the squares made of the whole line and of the lesse segment are treble to the square made of the greater segment which was required to be proued Looke for an other demonstration of this proposition after the fifth proposition of this booke ¶ The 5. Theoreme The 5. Proposition If a right line be deuided by an extreame and meane proportion and vnto it be added a right â—ine equall to the greater segment the whole right line is deuided by an extreame and meane proportion and the greater segment thereof is the right line geuen at the beginning SVppose that the right line AB be deuided by an extreame and meane proportion in the point C and let the greater segment thereof be AC And vnto the line AB adde the line AD equall to the line AC Then I say that the line Dâ— is deuided by an extreame and meane proportion in the point A and the greater segment thereof is the right line pâ—t at the beginning namely AB Describe by the 46. of the first vpon on the line AB a square AE and make perfect the figure And forasmuch as the line AB is deuided by an extreame and meane proportion in the point C therefore that which is contayned vnder the lines AB and BC is equall to the square of the line AC But that which is contayned vnder the lines AB and BC is the parallelogramme CEâ— and the square of the â—â—ne AC is the square CH. Wherefore the parallelogramme CE is equall to the square CH. But vnto the square CH is equall the square DH by the first of the sixth and vnto the parallelogramme CE is equall the parallelogramme HE. Wherefore the parallelogramme DH is equall to the parallelogramme HE. Adde the parallelogramme HB common to them both Wherefore the whole parallelogramme DK is equall to the whole square AE And the parallelogramme DK is that which is contayned vnder the lines BD and DA for the line AD is equall to the line DL the square AE is the square of the line AB Wherfore that which is contayned vnder the lines AD and DB is equall to the square of the line AB Wherefore as the line DB is to the line BA so is the line BA to the line AD by the 17. of the sixth But the line DB is greater then the line BA Wherefore the line BA is greater then the line AD. Wherefore the line BD is deuided by an extreame and meane proportion in the point A and his greater segment is the line AB If therefore a right line be deuided by an extreame and meane proportion and vnto â—t be added a right line equall to the greater segment the whole right line is deuided by an extreame and meane proportion and the greater segment therof is the right line geuen at the beginning which was required to be demonstrated This proposition is agayne afterward demonstrated A Corollary added by Campane Hereby it is 〈…〉 from the greaâ—â—◠〈◊〉 of a line deuided by an extreame meane proportion be 〈◊〉 away 〈◊〉 segment the sayd great a segment shall be deuided by an extreame and meane proportion and the greater segment thereof shall be the line taken away As let the line â—â— be deuided by an extreame and meanâ— proportion in the point C. And leâ— the 〈…〉 line 〈…〉 A. D. I say that AC is also deuided by an extreame and meanâ— proportion in the point D and that his greater portion is DC For by the definitino of a line so deuided AB is to AC as AC is to CB. But as AC is to CB so is AC to DC by the 7. of the 〈…〉 is equall to CB wherefore by the 11. of the fifth as AB is to AC so is AC to CD
the line AB is diuided by an extreme and meane proportion in the poynt C and the greater segment thereof is the line ACâ— therfore that which is contayned vnder the lines AB and BC is equall to the square of the line AC Wherfore that which is coÌ„tayned vnder the lines AB and BC twise is double to the square of AC Wherfore that which is contayned vnder the lines AB and BC twise together with the square of the line AC is treble to the square of the line AC But that which is contayned vnder the lines AB and BC twise together with the square of the line AC is the squares of the lines AB and BC by the 7. of the second Wherefore the squares of the lines AB and BC are treble to the square of the line AC which was required to be demonstrated Resolution of the 5. Theoreme Suppose that a certaine right line AB be diuided by an extreme and meane proportion in the point C. And let the greater segment therof be the line AC And vnto the line AB adde a line equall to the line AC and let the same be AD. Theâ— I say that the line DB is diuided by an extreme and meane proportion in the point A. And the greater segment therof is the line AB For forasmuch as the line DB is diuided by an extreme meane proportion in the point A and the greater segment thereof is the line AB therfore as the line DB is to the line BA so is the line BA to the line AD but the line AD is equall to the line AC wherefore as the line DB is to the line BA so is the line BA to the line AC Wherfore by conuersioÌ„ as the line BD is to the line DA so is the line AB to the line BC by the corollary of the 19. of the fifth wherfore by diuision by the 17. of the fifth as the line BA is to the line AD â—o is the line AC to the line CB. But the line AD is equall to the line AC Wherfore as the line BA is to the line AC so is the line AC to the line CB. And so it is indeede for the line AB is by supposition diuided by an extreme and meane proportion in the point C. Composition of the 5. Theoreme Now forasmuch as the line AB is diuided by an extreme and meane proportion in the point C therefore as the line BA is to the line AC so is the line AC to the line CB but the line AC is equall to the line AD. Wherefore as the line BA is to the line AD so is the line AC to the line CB. Wherfore by composition by the 18. of the fifth as the line BD is to the line DA so is the line AB to the line BC. Wherefore by conuersion by the corollary of the 19. of the fiueth as the line DB is to the line BA so is the line BA to the line AC but the line AC is equall to the line AD. Wherefore as the line DB is to the line BA so is the line BA to the line AC Wherfore the line DB is deuided by an extreme and meane proportion in the point A and his greater segment is the line AB which was required to be demonstrated An Aduise by Iohn Dee added SEing it is doubteles that this parcel of Resolution and Composition is not of Euclides doyng it can not â—ustly be imputed to Euclide that he hath therby eyther superfluitie or any part disproportioned in his whole Composition Elementall And though for one thing one good demonstration well suffiseth for stablishing of the veritie yet oâ— one thing diuersly demonstrated to the diligent examiner of the diuerse meanes by which that varietie ariseth doth grow good occasions of inuenting demonstrations where matter is more straunge harde and barren Also though resolution were not in all Euclide before vsed yet thankes are to be geuen to the Greke Scholic writter who did leaue both the definition and also so short and easy examples of a Method so auncient and so profitable The antiquity of it is aboue 2000. yeares it is to weâ—e euer since Plato his time and the profite therof so great that thus I finde in the Greeke recorded 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Proclus hauing spoken of some by nature excellent in inuenting demonstrations pithy and breif sayeth Yet are there Methods geuen for that purpose And in dede that the best which by Resolution reduceth the thing inquired of to an vndoubted principle Which Method Plato taught Leodamas as iâ— reported And he is registred thereby to haue bene the inuenter of many things in Geometry And verely in Problemes it is the chief ayde for winning and ordring a demonstration first by Supposition of the thing inquired of to be done by due and orderly Resolution to bring it to a stay at an vndoubted veritie In which point of Art great abundance of examples are to be seen in that excellent and mighty Mathematiciâ—n Archimedes in his expositor Eutocius in Menaechmus likewise and in Diocles booke de Pytiâ—s and in many other And now for as much as our Euclide in the last six Propositions of this thirtenth booke propoundeth and concludeth those Problemes which were the ende Scope and principall purpose to which all the premisses of the 12. bookes and the rest of this thirtenth are directed and ordered It shall be artificially done and to a great commodity by Resolution backward from these 6. Problemes to returne to the first definition of the first booke I meane to the definition of a point Which is nothing hard to do And I do counsaile all such as desire to atteinâ— to the profound knowledge of Geometrie Arithmeticke or any braunche of the sciences Mathematicall so by Resolution discreatly and aduisedly to resolue vnlose vnioynt and disseauer euery part of any worke Mathematicall that therbyâ— aswell the due placing of euery verity and his proofe as also what is either superfluous or wanting may euidently appeare For so to inuent there with to order their writings was the custome of them who in the old time were most excellent And I for my part in writing any Mathematicall conclusion which requireth great discourse at length haue found by experience the commoditie of it such that to do other wayes were to me a confusion and an vnmethodicall heaping of matter together besides the difficulty of inuenting the matter to be disposed and ordred I haue occasion thus to geue you friendely aduise for your beâ—ofeâ— because some of late haue inueyed against Euclide or Theon in this place otherwise than I would wish they had The 6. Theoreme The 6. Propositionâ— If a rationall right line be diuided by an extreme and meane proportion eyther of the segments is an irrationall line of that kinde which is called a residuall line SVppose that AB beyng a rationall line be deuidedly 〈◊〉 extreme
as the line AB is to the line BC so is the square of the line AB to the square of the line BD by the corollaries of the 8. and 20. of the sixt Wherefore the square of the line AB is treble to the square of the line BD. And it is proued that the square of the line GK is treble to the square of the line KE and the line KE is put equall to the line BD. Wherefore the line KG is also equall to the line AB And the line AB is the diameter of the sphere geuen Wherefore the line KG is equall to the diameter of the sphere geuen Wherfore the cube is coÌ„preheÌ„ded in the sphere geuen and it is also proued that the diameter of the Sphere is in power treble to the side of the cube which was required tâ—â—e doone and to be proued An other demonstration after Flussas Suppose that the diameter of the Sphere geuen in the former Propositions be the line Aâ— And let the center be the point C vpon which describe a semicircle ADB And from the diameter AB cut of a third part BG by the 9. of the sixt And from the point G raise vp vnto the line AB a perpendicular line DG by the 11. of the first And draw these right lines DA DC and DB. And vnto the right line DB put an equall right line ZI and vpon the line ZI describe a square EZIT And froÌ„ the pointes E Z I T erecte vnto the superficies EZIT perpendicular lines EK ZH IM TN by the 12. of the eleuenth and put euery one of those perpendicular lines equall to the line ZI And drawe these right lines KH HM MN and NK ech of which shall be equall and parallels to the line ZI and to the rest of the lines of the square by the 33. of the first And moreouer they shall containe equall angles by the 10. of the eleuenth and therefore the angles are right angles for that EZIT is a square wherfore the rest of the bases shall be squares Wherfore the solide EZITKHMN being coÌ„tained vnder 6. equall squares is a cube by the 21. definition of the eleuenth Extend by the opposite sides KE and MI of the cube a plaine KEIM and againe by the other opposite sides NT and HZ extend an other plaine HZTN Now forasmuch as ech of these plaines deuide the solide into two equall partes namely into two Prismes equall and like by the 8. definition of the eleuenth therfore those plaines shall cut the cube by the centre by the Corollary of the 39. of the eleuenth Wherefore the coÌ„mon section of those plaines shall passe by the centre Let that common section be the line LF And forasmuch as the sides HN and KM of the superficieces KEIM and HZTN do diuide the one the other into two equall partes by the Corollary of the 34. of the first and so likewise do the sides ZT and EI therefore the common section LF is drawen by these sections and diuideth the plaines KEIM and HZTN into two equall partes by the first of the sixt for their bâ—ses are equall and the altitude is one and the â—ame namely the altitude of the cube Wherefore the line LF shall diuide into two equall partes the diameters of his plaines namely the right lines KI EM ZN and NT which are the diameters of the cube Wherfore those diameters shall concurre and cut one the other in one and the selfe same poynt let the same be O. Wherfore the right lines OK OE OI OM OH OZ OT and ON shall be equâ—ll the onâ— to the other for that they are the halfes of the diameters of equall and like rectangle parallelograÌ„mes Wherefore making the centre the point O and the space any of these lines OE or OK c. a Sphere described shall passe by euery one of the angles of the cube namely which are at the pointes E Z I T K H M N by the 12. definition of the eleuenth for that all the lines drawen from the point O to the angles of the cube are equall But the right line EI containeth in power the two equall right lines EZ and ZI by the 47. of the first Wherefore the square of the line EI is double to the square of the line ZI And forasmuch as the right line KI subtendeth the right angle KEI for that the right line KEâ— is erected perpendicularly to the plaiâ—e â—uperficies of the right lines EZ and ZT by the 4. of the eleueÌ„th â— therefore the square of the line KI is equall to the squares of the lines EI and EK but the square of the line EI is double to the square of the line EK for it is double to the square of the line ZI as hath bene proued and the bases of the cube are equall squares Wherefore the square of the line KI is triple to the square of the line KE that is to the square of the line ZI But the right line ZI is equall to thâ— right line DB by position vnto whose square the square of the diâ—meter AB is triple by that which was demonstrated in the 13. Proposition of this booke Wherefore the diameters KI DB are equall Wherefore there is described a cube KI and it is comprehended in the Sphere geuen wherin the other solides were contained the diameter of which Sphere is the line AB And the diameter KI or AB of the same Sphere is proued to be in power triple to the side of the cube namely to the line DB or ZI ¶ Corollaryes added by Flussas First Corollary Hereby it is manifest that the diameter of a Sphere containeth in power the sides both of a pyramis and of a cube inscribed in it For the power of the side of the pyramis is two thirdes of the power of the diameter by the 13. of this booke And the power of the side of the cube is by this Proposition one third of the power of the sayd diameter Wherefore the diameter of the Sphere contayneth in power the sides of the pyramis and of the cube .. ¶ Second Corollary All the diameters of a cube cut the one the other into two equall partes in the centre of the sphere which containeth the cube And moreouer those diameters do in the selfe same point cut into two equall partes the right lines which ioyne together the centres of the opposite bases As it is manifest to see by the right line LOF For the angles LKO and FIO are equall by the 29. of the first and it is proued that they are contained vnder equall sides Wherefore by the 4. of the first the bases LO and FO are equall In like sort may be proued that the rest of the right lines which ioyne together the centres of the opposite bases do cut the one the other into two equall partes in the centre O. ¶ The 4. Probleme The 16. Proposition To make an Icosahedron and to comprehend it in the Sphere
the same preface seme to import The Preface of Hypsicles before the fourtenth booke FRend Protarchus wheÌ„ that Basilides of Tire came into Alexandria hauing familiar frendship with my father by reason of his knowledge in the mathematicall sciences he remayned with him a long time yea euen all the time of the pestilence And sometime reasoning betwene themselues of that which Apollonius had written touching the comparison of a dodecahedron and of an Icosahedron inscribed in one and the selfe same sphere what proportion such bodies haue the one to the other they iudged that Apollonius had somewhat erred therein Wherefore they as my father declared vnto me diligently weighing it wrote it perfectly Howbeit afterward I happened to finde an other booke written of Apollonius which contayned in it the right demonstration of that which they sought for which when they saw they much reioysed As for that which Apollonius wrote may be sene of all men for that it is in â—uery mans hand And that which was of vs more diligently afterward written agayne I thought good to send and dedicate vnto you as to one whome I thought worthy commendation both for that deepe knowledge which I know you haue in all kindes of learning and chiefely in Geometrie so that you are able redily to iudge of those thinges which are spoken and also for the greate loue and good will which you beare towardes my father and me Wherfore vouchsafe gently to accept this which I send vnto you But now is it time to end our preface and to begin the matter ¶ The 1. Theoreme The 1. Proposition A perpendicular line drawen from the centre of a circle to the side of a Pentagon described in the same circle is the halfe of these two lines namely of the side of an hexagon figure and of the side of a decagon figure being both described in the selfe same circle SVppose that the circle be ABC And let the side of an equilater Pentagon described in the circle ABC be BC. And by the 1. of the third take the centre of the circle and let the same be D. And by the 12. of the first from the point D draw vnto the line BC a perpendicular line DE. And extend the right line DE directly to the point F. Then I say that the line DE which is drawen from the centre to BC the side of the pentagon is the halfe of the sides of an hexagon and of a decagon taken together and described in the same circle Draw these right lines DC and CF. And vnto the line EF put an equall line GE. And draw a right line from the point G to the point C. Now forasmuch as the circumference of the whole circle is quintuple to the circuÌ„ference BFC which is subtended of the side of the pentagon and the circumference ACF is the halfe of the circumference of the whole circle and the circumference CF which is subtended of the side of the decagon is the halfe of the circumference BCF therefore the circumference ACF is quintuple to the circumference CF by the 15. of the â—iâ—t Wherefore the circumference AC is qradruple to the circumference FC But as the circumference AC is to the circumference FC so is the angle ADC to the angle FDC by the last of the sixt Wherefore the angle ADC is quadruple to the angle FDC But the angle ADC is double to the angle EFC by the 20. of the third Wherefore the angle EFC is double to the angle GDC But the angle EFC is equall to the angle EGC by the 4. of the first Wherfore the angle EGC is double to the angle EDC Wherefore the line DG is equall to the line GC by the 32. and 6. of the first But the line GC is equall to the line CF by the 4. of the first Wherfore the line DG is equall to the line CF. And the line GE is equall to the line EF by construction Wherefore the line DE is equall to the lines EF and FC added together Vnto the lines EF and FC adde the line DE. Wherefore the lines DF and FC added together are double to the line DE. But the line DF is equall to the side of the hexagon and FC to the side of the decagon Wherefore the line DE is the halfe of the side of the hexagon and of the side of the decagon being both added together and described in one and the selfe same circle It is manifest by the Propositions of the thirtenth booke that a perpendicular line drawen from the centre of a circle to the side of an equilater triangle described in the same circle is halfe of the semidiameter of the circle Wherefore by this Proposition a perpendicular drawen from the câ—ntre of a circle to the side of a Pentagâ—n is equall to the perpendicular drawen from the centre to the side of the triangle â—nd to halfe of the side of the decagon described in the same circle ¶ The 2. Theoreme The 2. Proposition One and the selfe same circle comprehendeth both the Pentagon of a Dodecahedron and the triangle of an Icosahedron described in one and the selfe same Sphere THis Theoreme is described of Aristeus in that booke whose title is The comparison of the fiue figures and is described of Apollonius in his second edition of the coÌ„parison of a Dodecahedron to an Icosahedron which is that as the superâ—icies of a DodecahedroÌ„ is to the superficies of an Icosahedron so is the Dodecahedron to an Icosahedron for that a perpendicular line drawen from the centre of a sphere to the pentagon of a dodecahedron and to the triangle of an Icosahedron is one and the selfe same Now must we also proue that one and the selfe same circle comprehendeth both the pentagon of a Dodecahedron and also the triangle of an Icosahedron described in one and the selfe same sphere first this being proued If in a circle be described an equilater pentagon the squares which are made of the side of the pentagon and of that right line which is subtended vnder two sides of the pentagon are quintuple to the square of the semidiameter oâ— the circle Suppose that ABC be a circle And let the side of a pentagon in the circle ABC be AC And take by the 1. of the third the centre of the circle and let the same be D. And by the 12. of the first from the point D draw vnto the line AC a perpendicular line DF. And extend the line DF on either side to the pointes B and E. And draw a right line from the point A to the point B. Now I say that the squares of the lines BA and AC are quintâ—ple to the square of the line DE. Drawe a right line from the point A to the point E. Wherefore the line AE is the side of a decagon figure And forasmuch as the line BE is double to thâ— line DE therefore the square
triangle Wherefore sixe such triangles as DBC is are equall to that which is contayned vnder the lines DE and BC thrise But sixe sâ—ch triangles as DBC is are equall to two such triangles as ABC is Wherefore that which is contained vnder the lines DE and BC thrise is equall to two such triangles as ABC is But two of those triangles takeÌ„ ten times contayneth the whole Icosahedron Wherfore that which is contayned vnder the lines DE BC thirty times is equall to twenty such triangles as the triangle ABC is that is to the whole superficies of the IcosahedroÌ„ Wherefore as the superficies of the dodecahedron is to the supeâ—ficies of the Icosahedron so is that which is contayned vnder the lines CD and FG to that which is contayned vnder the lines BC and DE. ¶ Corollary By this it is manifest that as the superficies of the Dodecahedron is to the superficies of the Icosahedron so is that which is contained vnder the side of the Pentagon and the perpeÌ„dicular line which is drawen from the centre of the circle described about the Pentagon to the same side to that which is contained vnder the side of the Icosahedron and the perpendicular line which is drawen from the centre of the circle described about the triangle to the same side so that the Icosahedron and Dodecahedron be both described in one and the selfe same Sphere ¶ The 4. Theoreme The 4. Proposition This being done now is to be proued that as the superficies of the Dodecahedron is to the superficies of the Icosahedron so is the side of the cube to the side of the Icosahedron TAke by the 2. Theoreme of this booke a circle containing both the pentagon of a Dodecahedron and the triangle of an Icosahedron being both described in one and the selfe same sphere and let the same circle be DBC And in the circle DBC describe the side of an equilater triangle namely CD and the side of an equilater pentagon namely AC And take by the 1. of the third the centre of the circle and let the same be E. And from the point E drawe vnto the lines DC and AC perpendicular lines EF and EG And extend the line EG directly to the point B. And drawe a right line from the point B to the point C. And let the side of the cube be the line H. Now I say that as the superficies of the Dodecahedron is to the superficies of the Icosahedron so is the line H to the line CD Forasmuch as the line made of the lines EB and BC added together namely of the side of the hexagon and of the side of a decagon is by the 9. of the thirtenth diuided by an extreme and meane proportion and his greater segment is the line BE and the line EG is also by the 1. of the foâ—retenth the halfe of the same line and the line EF is the halfe of the line BE by the Corollary of the 12. of the thirtenth Wherefore the line EG being diuided by an extreme and meane proportion his greater segment shall be the line EF. And the line H also being diuided by an extreme meane proportion his greater segment is the line CA as it was proued in the Dodecahedron Wherefore as the line H is to the line CA so is the line EG to the line EF. Wherefore by the 16. of the sixt that which is contained vnder the lines H and EF is equall to that which is contained vnder the lines CA and EG And for that as the line H is to the line CD so is that which is contained vnder the lines H and EF to that which is contained vnder the lines CD and EF by the 1. of the sixt But vnto that which is contained vnder the lines H and EF is equall that which is contained vnder the lines CA and EG Wherefore by the 11. of the fift as the line H is to the line CD so is that which is contained vnder the lines CA and EG to that which is contained vnder the lines CD and EF that is by the Corollary next going before as the superficies of the Dodecahedron is to the superficies of the Icosahedron so is the line H to the line CD An other demonstration to proue that as the superficies of the Dodecahedron is to the superficies of the Icosahedron so is the side of the cube to the side of the Icosahedron LEt there be a circle ABC And in it describe two sides of an equilater pentagon by the 11. of the fift namely AB and AC and draw a right line from the point B to the point C. And by the 1. of the third take the centre of the circle and let the same be D. And draw a right line from the point A to the point D and extend it directly to the point E and let it cut the line BC in the point G. And let the line DF be halfe to the line DA and let the line GC be treble to the line HC by the 9. of the sixt Now I say that that which is contained vnder the lines AF and BH is equall to the pentagon inscribed in the circle ABC Draw a right line from the point B to the point D. Now forasmuch as the line AD is double to the line DF therefore the line AF is sesquialter to the line AD. Againe forasmuch as the line GC is treble to the line CH therefore the line GH is double to the line CH. Wherefore the line GC is sesquialter to the line HG Wherefore as the line FA is to the line AD so is the line GC to the line GH Wherefore by the 16. of the sixt that which is contained vnder the lines AF HG is equall to that which is contained vnder the lines DA and GC But the line GC is equall to the line BG by the 3. of the third Wherfore that which is contained vnder the lines AD and BG is equall to that which is contained vnder the lines AF and GH But that which is contained vnder the lines AD and BG is equall to two such triangles as the triangle ABD is by the 41. of the first Wherefore that which is contained vnder the lines AF and GH is equall to two such triangles as the triangle ABD is Wherefore that which is contained vnder the lines AF and GH â—iue times is equall to ten triangles But ten triangles are two pentagons Wherefore that which is contained vnder the lines AF and GH fiue times is equall to two pentagons And forasmuch as the line GH is double to the line HC therefore that which is contained vnder the lines AF and GH is double to that which is contained vnder the lines AF and HC by the 1. of the sixt Wherefore that which is contained vnder the lines AF and CH twise is equall to that which is contained vnder the lines
22. of the sixt Wherefore by coÌ„position by the 18. of the fifth as both the lines AB BC added the one to the other together with the line AC that is as two such lines as AB is are to the line AC so are both the lines DE and EF added the one to the other together with the line DF that is two such lines as DE is to the line DF. And in the same proportion are the halues of the antecedents by the 15. of the fifth Wherefore as the line AB is to the line AC so is the line DE to the line DF. And therefore by the 19. of the fifth as the line AB is to the line BC so is the line DF to the line FE Wherefore also by diuision by the 17. of the fifth as the line AC is to the line CB so is the line DF to the line DE Now that we haue proued that any right line whatsoeuer being diuided by an extreame and meane proportion what proportion the line contayning in power the squares made of the whole line and of the greater segment added together hath to the line contayning in power the squares made of the whole line and of the lesse segment added together the same proportion hath the side of the cube to the side of the Icosahedron Now also that we haue proued that as the side of the cube is to the side of the Icosahedron so is the superficies of the Dodecahedron to the superficies of the Icosahedron being both described in one and the selfe same sphere and moreouer seing that we haue proued that as the superficies of the Dodecahedron is to the superficies of the IcosahedroÌ„ so is the DodecahedroÌ„ to the Icosahedron for that both the pentagon of the Dodecahedron and the triangle of the Icosahedron are comprehended in one and the selfe same circle All these thinges I say being proued it is manifest that if in one and the selfe same sphere be described a Dodecahedron and an Icosahedron they shall be in proportion the one to the other as a right line whatsoeuer being diuided by an extreame and meane proportion the line contayning in power the squares of the whole line and of the greater segment added together is to the line containing in power the squares of the whole line and of the lesse segment added together For for that as the Dodecahedron is to the Icosahedron so is the superficies of the Dodecahedron to the superficies of the Icosahedron that is the side of the cube to the side of the Icosahedron but as the side of the cube is to the side of the Icosahedron so any right line what so euer being diuided by an extreame and meane proportion is the line contayning in power the squares of the whole line and of the greater segment added together to the line contayning in power the squares of the whole line and of the lesse segment added together Wherefore as a Dodecahedron is to an Icosahedron described in one and the selfe same sphere so any right line what so euer being diuided by an extreame and meane proportion is the line contayning in power the squares of the whole line of the greater segment added together to the line contayning in power the squares of the whole line and of the lesse segment added together The ende of the fourtenth Booke of Euclides Elementes after Hypsicles ¶ The fourtenth booke of Euclides Elementes after Flussas FOr that the fouretenth Booke as it is set forth by Flussas containeth in it moe Propositions then are found in Hypsicles also some of those Propositions which Hypsicles hath are by him somewhat otherwise demonstrated I thought my labour well bestowed for the readers sake to turne it also all whole notwithstanding my trauaile before taken in turning the same booke after Hypsicles Where note ye that here in this 14. booke after Flussas and in the other bookes following namely the 15. and 16. I haue in alleadging of the Propositions of the same 14. booke followed the order and number of the Propositions as Flussas hath placed them ¶ The first Proposition A perpendicular line drawen from the centre of a circle to the side of a Pentagon inscribed in the same circle is the halfe of these two lines taken together namely of the side of the hexagon and of the side of the decagon inscribed in the same circle TAke a circle ABC and inscribe in it the side of a pentagon which let be BC and take the centre of the circle which let be the point D and froÌ„ it draw vnto the side BC a perpendicular line DE which produce to the point â— And vnto the line E F put the line EG equall And draw these right lines CG CD and CF. Then I say that the right line DE which is drawen from the centre to BC the side of the pentagon is the halfe of â—he sideâ— of the decagon and hexagon taken together Forasmuch as the line DE is a perpendicular â—nto the line BC therefore the sections BE and EC shall be equall by the 3. of the third and the line EF is common vnto them both and the angles FEC and FEB are right angles by supposition Wherefore the bases BF and FC are equall by the 4. of the first But the line BC is the side of a pentagon by construction Wherefore FC which subtendeth the halfe of the side of the pentagon is the side of the decagon inscribed in the circle ABC But vnto the line FC is by the 4. of the first equall the line CG for they subtend right angles CEG and CEF which are contained vnder equall sides Wherefore also the angles CGE and CFE of the triangle CFG are equall by the 5. of the first And forasmuch as the arke FC is subtended of the side of a decagon the arke CA shall be quadruple to the arke CF Wherefore also the angle CDA shall be quadruple to the angle CDF by the last of the sixâ— And forasmuch as the same angle CDA which is set at the center is double to the angle CFA which is set at the circumference by the 20. of the third therefore the angle CFA or CFD is double to the angle CDâ— namely the halfe of quadruple But vnto the angle CFD or CFG is proued equall the angle CGF Wherefore the outward angle CGF is double to the angle CDF Wherefore the angles CDG and DCG shall be equall For vnto those two angles the angle CGF is equall by the 32. of the first Wherefore the sides GC and GD are equall by the 6. of the first Wherefore also the line GD is equall to the line FC which is the side of the decagon But vnto the right line FE is equall the line EG by construction Wherefore the whole line DE is equall to the two lines Câ— and FE Wherefore those lines taken together namely the lines DF and FC shall
spherâ— containeth the Dodecahedron of this pentagon and the Icosahedron of this triangle by the 4. of this booke â— and the line CL falleth perpendiculaâ—ly vpon the side of the Icosahedron and the line CI vpon the side of the Dodecahedron that which is 30. times contained vnder the side and the perpendicular line falling vpon it is equal to the â—uperficies of that solide vpon whose side the perpendiculâ—r falleth If therefore in a circle c. as in the proposition which was required to be demonstrated A Corollary The superficieces of a Dodecahedron and of an Icosahedron described in one and the selfe same sphere are the one to the other as that which is contained vnder the side of the one and the perpendicular line drawne vnto it from the centre of his base to that which is contained vnder the side of the other and the perpendicular line drawne to it from the centre of his base For aâ— thirtyâ— timâ—s is to thirty times so is once to once by the 15. of thâ— fifth The 6. Proposition The superficies of a Dodecahedron is to the superficies of an Icosahedron described in one and the selfe same sphere in that proportion that the side of the Cube is to the side of the Icosahedron contained in the self same sphere SVppose that there be a circle ABG in it by the 4. of this boke let there be inscribed the sideâ— of a Dodecahedron and of an Icosahedron contained in onâ— and the selfe same sphere And let the side oâ— the Dodecahedron be AG and the side of the Icosahedron be DG And let the centre be the poynt E from which draw vnto those sâ—des perpendicular lines EI and EZ And produce the line EI to the poynt B and draw the linâ— BG And let the side of the cube contained in the self same sphere be GC Then I say that the superficies of the Dodecahedron iâ— to the superficies of the Icosahedron as the line â—G iâ— to the liâ—â— GD For forasmuche as the line EI beinâ— diuided by an extreme and meane proportion the greater segment thâ—rof shall be the linâ— EZ by the corollary of the first of this booke and the line CG being diuided by an extreme and meane proportion his greater segment is the line AG by the corollary of the 17. of the thirtenth Wherefore the right lines EI and CG â—râ— cut proportionally by the second of this bâ—oke Whâ—râ—fore as the line CG is to the line AG so is the line EI to the line EZ Wherâ—fore that which it contained vnder the extreames CG and EZ is â—quall to that which iâ— contaynâ—d vnder the meanes AG and EI. by the 16. of the sixth But as that which iâ— contained vnder the linâ—â— CG and â—Z is to that which is contained vnder the lines DG and EZ so by the first of the sixth iâ— the linâ— CG to the line DG for both those parallelogrames haue oâ—â— and the selfe same altitude namely the line EZ Wherfore as that which is contained vnder the lines EI and AG which iâ— proued equal to that which is contained vnder the lineâ— CG and EZ is to that which is contained vnder the lines DG and EZ so is the line CG to the liâ—â— DG But as that which is contained vnder the lines EI and AG is to that which is contained vnder the lines DG and EZ so by the corollary of the former proposition is the superficies of the Dodecahedron to the superficies of the Icosahedron Wherfore as the superficies â—â— the Dodecahedron is to the superficies of the Icosahedron so is CG the side of the cube to GD the side of the Icosahedron The superficies therefore of a Dodecahedron is to the superficiesâ— c. as in the proposition which was required to be proued An Assumpt The Pentagon of a Dodecahedron is equall to that which is contained vnder the perpendicular line which falleth vpon the base of the triangle of the Icosahedron and fiue sixth partes of the side of the cube the sayd three solides being described in one and the selfe same sphere Suppose that in the circle ABEG the pentagon of a Dodecahedron be Aâ—CIG and let two sides thereof AB and AG be subtended of the right line BG And let the triangle of the Icosahedron inscribed in the selfe same sphere by the 4. of this booke be AFH And let the centre of the circle be the poynt D and let the diameter be ADE cutting FH the side of the triangle in the poynt Z and cutting the line BG in the poynt K. And draw the right line BD. And from the right line KG cut of a third part TG by the 9. of the sixth Now then the line BG subtending two sides of the Dodecahedron shal be the side of the cube inscribed in the same sphere by the 17. of the thirtenth and the triangle of the Icosahedron of the same sphere shal be Aâ—H by the 4. of this booke And the line AZ which passeth by the centre D shall fall perpendicularly vpon the side of the triangle For forasmuch as the angles GAE BAE are equall by the 27. of the thirdâ— for they are see vpon equall circumferences therefore the â—ases BK and KG are by the â— of the first equall Wherefore the line BT contayneth 5. sixth partes of the line BG Then I say that that which is contayned vnder the lines AZ and BT is equall to the pentagon Aâ—Câ—G For forasmuch as the line â—Z is sesqâ—ialter to the line AD for the line Dâ— is diuided into two equall partes in the poynt Z by the corollary of the â—2â— of the thirtenth Likewise by construction the line KG is sesquialter to the line KT therefore as the line AZ is to the line AD so is the line KG to the 〈◊〉 â—T Wherefore that which is contayned vndeâ— the 〈◊〉 AZ and KT is equall to that which is contayned vnder the meanes AD and KG by the 16. of the sixth But vnto the line KG is the line â—K â—roued equall Wherefore that which is contayned vnder the lines AZ and KT is equall to that which is contayned vnder the lines AD and BK But that which is contayned vnder the lines AD and BK is by the 41. of the first double to the triangle ABD Wherefore that which is contayned vnder the lines AZ and KT is double to the same triangle ABD And forasmuch as the pentagon ABCIG contayneth◠〈…〉 equall â—o the triangle ABD and that which is contayned vnder the lines AZ and KT contayneth two such triangles therefore the pentagon ABCIG is duple sesquialter to the rectangle parallelogramme contayned vnder the lines AZ and KT And 〈…〉 1. of the sixth that which is coÌ„teyned vnder the lines AZ and BT is to that which is contayned vnder the lines AZ and KT as the base BT is to the base â—â—Tâ— therefore that which is contayned vnder the lines AZ
together And a perpendicular line drawne from the centre of the sphere to any base of the cube is equall to halfe the side of the cube which was required to be prouâ—d ¶ A Corollary If two thirds of the power of the diameter of the sphere be multiplyed into the perpendicular line equall to halfe the side of the cube there shall be produced a solide equall to the solide of the cube For it is before manifest that two third partes of the power of the diameter of the sphere are equall to two bases of the cube If therefore vnto eche of those two thirds be applyed halfe the altitude of the cube they shall make eche of those solides equall to halfe of the cube by the 31. of the eleuenth for they haue equall bases Wherefore two of those solides are equall to the whole cube You shall vnderstand gentle reader that Campane in his 14. booke of Euclides Elementes hath 18. propositioÌ„s with diuers corollaries following of them Some of which propositions and corollaries I haue before in the twelfth and thirtenth bookes added out of Flussas as corollaries which thing also I haue noted on the side of those corollaries namely with what proposition or corollary of Campanes 14. booke they doo agree The rest of his 18. propositions and corollaries are contained in the twelue former propositions and corollaries of this 14. booke after Flussas where ye may see on the side of eche proposition and corollary with what proposition and corollary of Campanes they agree But the eight propositions following together with their corollaries Flussas hath added of him selfe as he him selfe affirmeth The 13. Proposition One and the self same circle containeth both the square of a cube and the triangle of an Octohedron described in one and the selfe same sphere SVppose that there be a cube ABG and an Octohedron DEF described in one and the selfe same sphere whose diameter let be AB or DH And let the lines drawne from the ceÌ„tres that is the semidiameters of the circles which ctoÌ„aine the bases of those solides â— be CA and ID Then I say that the lines CA and ID are equal Forasmuch as AB the diameter of the sphere which containeth the cube is in power triple to BG the side of the cube by the 15. of the thirtenth vnto which side AG the diameter of the base of the cube is in power double by the 47. of the first which line AG is also the diameter of the circle which coÌ„taineth the base by the 9. of the fourth therfore AB the diameter of the sphere is in power sesquialter to the line AG namely of what partes the line AB containeth in power 12. of the same the line AG shal containe in power 8. And therfore the right line AC whiche is drawn from the ceÌ„tre of the circle to the circumference conteineth in power of the same partes 2. Wherefore the diameter of the sphere is in power sextuple to the lyne which is drawne from the centre to the circumference of the circle whiche containeth the square of the cube But the Diameter of the selfe same Sphere whych containeth the Octohedron is one and the selfe same with the diameter of the cube namely DH is equall to AB and the same diameter is also the diameter of the square which is made of the sides of the Octohedron wherefore the saide diameter is in power double to the side of the same Octohedron by the 14. of the thirtenth But the side DF is in power triple to the line drawne from the centre to the circumference of the circle which containeth the triangle of the octohedron namely to the line ID by the 12. of the thirtenth Wherfore the selfe same diameter AB or DH which was in power sextuple to the line drawne from the centre to the circumference of the circle which containeth the square of the cube is also sextuple to the line ID drawne from the centre to the circumference of the circle which containeth the triangle of the Octohedron Wherefore the lines drawne from the centres of the circles to the circumferences which containe the bases of the cube and of the octohedron are equal And therfore the circles are equal by the first diffinition of the third Wherfore one and the selfe same circle containeth c. as in the proposition which was required to be proued A Corollary Hereby it is manifest that perpendiculars coupling together in a sphere the centres of the circles which containe the opposite bases of the cube and of the Octohedron are equal For the circles are equal by the second corollary of the assumpt of the 16. of the twelfth and the lines which passing by the centre of the sphere couple together the centres of the bases are also equal by the first corollary of the same Wherfore the perpendicular which coupleth together the opposite bases of the Octohedron is equal to the side of the cube For either of them is the altitude erected The 14. Proposition An Octohedron is to the triple of a Tetrahedron contained in one and the selfe same sphere in that proportion that their sides are SVppose that there be an octohedron ABCD and a Tetrahedron EFGH vpon whose base FGH erect a Prisme which is done by erecting from the angles of the base perpendicular lines equal to the altitude of the Tetrahedron which prisme shal be triple to the Tetrahedron EFGH by the first corollary of the 7. of the twelfth Then I say that the octohedron ABCD is to the prisme which is triple to the Tetrahedron EFGH as the side BC is to the side FG. For forasmuch as the sides of the opposite bases of the octohedron are right lines touching the one the other and are parellels to other right lines touching the one the other for the sides of the squares which are coÌ„posed of the sides of the octohedroÌ„ are opposite Wherfore the opposite plaine triangles namely ABC KID shal be parallels and so the rest by the 15. of the eleuenth Let the diameter of the Octohedron be the line AD. Now then the whole Octohedron is cut into foure equal and like pyramids set vpon the bases of the octohedron and hauing the same altitude with it being about the Diameter AD namely the pyramis set vpon the base BID and hauing his toppe the poynt A and also the pyramis set vppon the base BCD hauing his top the same poynt A. Likewise the pyramis set vpoÌ„ the base IKD hauing his toppe the same poynt A and moreouer the pyramis set vpon the base CKD and hauing his toppe the former poynt A which pyramids shal be equal by the 8. diffinition of the eleuenth for they eche consist of two bases of the octohedron and of two triangles contained vnder the diameter AD and two sides of the octohedroÌ„ Wherfore the prisme which is set vpon the base of the Octohedron
ouerthrowne and ouerwhelmed the whole world he was vtterly rude and ignorant in the Greke tongue so that certenly he neuer redde Euclide in the Greke nor of like translated out of the Greke but had it translated out of the Arabike tonge The Arabians were men of great study and industry and commonly great Philosophers notable Phisitions and in mathematicall Artes most expert so that all kinds of good learning flourished and raigned amongst them in a manner only These men turned whatsoeuer good author was in the Greke tonge of what Art and knowledge so euer it were into the Arabike tonge And froÌ„ thence were many of theÌ„ turned into the Latine and by that meanes many Greeke authors came to the handes of the Latines and not from the first fountaine the Greke tonge wherin they were first written As appeareth by many words of the Arabike tonge yet remaining in such bokes as are Zenith nadir helmuayn helmuariphe and infinite suche other Which Arabians also in translating such Greke workes were accustomed to adde as they thought good for the fuller vnderstanding of the author many things as is to be sene in diuers authors as namely in Theodosius de Sphera where you see in the olde translation which was vndoubteldy out of the Arabike many propositions almost euery third or fourth leafe Some such copye of Euclide most likely did Campanus follow wherein he founde those propositioÌ„s which he hath more aboue those which are found in the Greke set out by Hypsicles and that not only in this 15. boke but also in the 14. boke wherin also ye finde many propositions more theÌ„ are founde in the Greeke set out also by Hypsicles Likewise in the bookes before ye shall finde many propositions added and manye inuerted and set out of order farre otherwise then they are placed in the Greeke examplars Flussas also a diligent restorer of Euclide a man also which hath well deserued of the whole Art of Geometrie hath added moreouer in this booke as also in the former 14. boke he added 8. proâ—ositioÌ„s 9. propositioÌ„s of his owne touching the inscription and circumscriptâ—on 〈…〉 bodies very siâ—gular â—ndoubtedly and wittye All which for that nothing should want to the desirous louer of knowledge I haue faithfully with no small paines turned And whereas Flâ—ssâ—â— in the beginning of the eleuenth booke namely in the end of the diffinitions there â—eâ— putteth two diffinitions of the inscription and circumscription of solides or corporall figures within or about the one the other which certainely are not to be reiected yet for that vntill this present 15. boke there is no mention made of the inscription or circumscription of these bodyes I thought it not so conuenient thâ—râ— to place them but to referre theÌ„ to the beginning of this 15. booke where they are in maner of necessitie required to the elucidation of the Proposiâ—ions and dâ—monstrationâ— of the same The diffinitions are these Diffinition 1. A solide figure is then â—aid to be inscribed in a solide figure when the angles of the figure inscribed touche together at one time either the angles of the figure circumscribed or the superficieces or the sides Diffinition 2. A solide figure is then said to be circumscribed about a solide figure when together at one time either the angles or the superficieces or the sides of the figure circumscribed â—ouch the angles of the figure inscribed IN the fourâ—â— booke in the diffinitions of the inscription or circumscription of playne rectiline figures one with in or about an other was requâ—red that all the angles of the figuâ—â— inscribed should at one time touch all the sides of the figure circumscribed but in the fiue regular solides â—o whome chefely these two diffinitions pertaine for that the nomber of their angles superficieces sides are not equal one compared to an other it is not of necessitie that all the angles of the solide inscribed should together at one time touch either all the angles or all the superficieces or all the sides of the solide circumscribed but it is sufficient that those angles of the inscribed solide which touch doe at one time together eche touch some one angle of the figure circumscribed or some one base or some one side so that if the angles of the inscribed figure do at one time touche the angles of the figure circumscribed none of them may at the same time touche either the bases or the sides of the same circumscribed figure and so if they touch the bases they may touche neither angles nor sides and likewise if they touche the sides they may touch neither angles nor bases And although sometimes all the angles of the figure inscribed can not touch either the angles or the bases or the sides of the figure circumscribed by reason the nomber of the angles bases or sides of the said figure circumscribed wanteth of the nomber of the angles of the â—igure inscribed yet shall those angles of the inscribed figure which touch so touch that the void places left betwene the inscribed and circumscribed figures shal on euery side be equal and like As ye may afterwarde in this fiftenth booke most plainely perceiue ¶ The 1. Proposition The 1. Probleme In a Cube geuen to describe a trilater equilater Pyramis SVppose that the cube geuen be ABCDEFGH In the same cube it is required to inscribe a Tetrahedron Drawe these right lines AC CE AE AH EH HC Now it is manifest that the triangles AEC AHE AHC and CHE are equilater for their sides are the diameters of equall squares Wherfore AECH is a trilater equilater pyramis or Tetrahedron it is inscribed in the cube geueÌ„ by the first definition of this booke which was required to be done ¶ The 2. Proposition The 2. Probleme In a trilater equilater Pyramis geuen to describe an Octohedron SVppose that the trilater equilater pyramis geueÌ„ be ABCD whose sides let be diuided into two equall partes in the pointes E Z I K L T. And draw these 12. right lines EZ ZI IE KL LT TK EK KZ ZL LI IT and TE Which 12. right lines are by the 4. of the first equall For they subtend equall plaine angles of the bases of the pyramis and those equall angles are contained vnder equall sides namely vnder the halfes of the sides of the pyramis Wherefore the triangles TKL TLI TIE TEK ZKL ZLI ZIB ZEK are equilater and they limitate and containe the solide TKLEZI Wherefore the solide TKLEZI is an Octohedron by the 23. definition of the eleuenth And the angles of the same Octohedron do touch the sides of the pyramis ABCD in the pointes E Z I T K L. Wherefore the Octohedron is inscribed in the pyramis by the 1. definition of this booke Wherefore in the trilater equilater pyramis geuen is inscribed an Octohedron which was required to be done A Corollary added by Flussas Hereby it is manifest that a pyramis is cut into two
by the 13. of the first For the right lines TI and IK are set vpon the line MO. And by the same reason may the rest of the angles namely IKL KLT LTI be proued right angles and they are in one and the self same plaine superficies namely MNXO by the 7. of the eleueÌ„th Wherefore the right lines which ioyne together the centres of the plaine superficiall triangles which make the solide angle A do make the square ITKL And by the same reason may be proued that the plaine superficiall triangles of the rest of the fiue solide angles of the Octohedron set at the pointes B G Z D E do in the centres of their bases receaue squares So that there are in number sixe squares for euery Octohedron hath sixe solide angles and those squares are equall for their sides do containe equall angles of inclinations contained vnder equall sides namely vnder those sides which are drawen from the centre to the side of the equall triangles by the 2. Corollary of the 18. of the thirtenth Wherefore ITKLRPVS is a cube by the 21. definition of the eleuenth and hath his angles in the centres of the bases of the Octohedron and therefore is inscribed in it by the first definition of this booke Wherefore in an Octohedron geuen is described a cube which was required to be done The 5. Proposition The 5. Probleme In an Icosahedron geuen to describe a Dodecahedron TAke an Icosahedron one of whose solide angles let be Z. Now forasmuch as by those thinges which haue bene proued in the 16. of the thirtenth the bases of the triangles which contayne the angle of the Icosahedron doo make a pentagon inscribed in a circle let that pentagon be ABGDE which is made of the fiue bases of the triangles whose playne superficiall angles remayning make the solide angle geuen namely Z. And take the centres of the circles which contayne the foresaid triangles which centers let be the poyntes I T K M L and draw these right lines IT TK KM ML LI. Now then a perpendicular line drawne from the poynt Z to the playne superficies of the pentagon ABGDE shall fall vpon the centre of the circle which contayneth the pentagon ABGDE by those thinges which haue bene proued in the selfe same 16. of the thirtenth Moreouer perpendicular lines drawne from the centre to the sides of the pentagon ABGDE shall in the poyntes C N O where they fall cut the right lines AB BG GD into two equall partes by the 3. of the third Draw these right lines CN and NO And forasmuch as the angles CBN and NGO are equall and are contained vnder equall sides therefore the base CN is equall to the base NO by the 4. of the first Moreouer perpendicular lines drâ—â—â—e from the poynt Z to the bâ—sâ—â— of the pentagon ABGDE shall likewise cutte the bases into two equall partes by thâ—â— of the third For the perpendiculars passe by the centre by the corollary of the 12. of the thirteÌ„th Wherfore thâ—se perpendicular lines shall fal vpoÌ„ the points C N O. And now forasmuch as the right lines ZI IG are equall to the right lines ZT TN also to the right lines ZK KO by reason of the likenes of the equall triangles therefore the line IT is a parallell to the line CN and so also is the line TK to the line NO by the 2. of the sixt Wherfore the angles ITK and CNO are equal by the 11. of the eleuenth Agayne forasmuch as the triangles CBN and NGO are Isoscels triangles therefore the angles BCN and BNC are equall by the 5. of the first And by the same reason the angles GNO and GON are equall And moreouer the angles BCN and BNC are equall to the angles GNO and GON for that the triangles CBN and NGO are equall and like Bâ—â— the three angles BNC CNO ONG are equall to two right angles by the 13. of the first for that vpon the right line Bâ— are set the right lines CN ON And the three angles of the triangle CBN namely the angles BNC BCN or GNO for the angle GNO is equall to the angle BCN as it hath bene proued and NBC are also equall to two right angles by the 32. of the first Wherefore taking away the angles BNC GNO the angle remayning namely CNO is equall to the angle remayning namely to CBN Wherfore also the angle ITK which is proued to be equall to the angle CNO is equall to the angle CBN Wherefore ITK is the angle of a pentagon And by the same reason may be proâ—ed that the rest of the angles namelyâ— the angles TIL ILM LMK MKT are equall to the rest of the angles namely to BAE AED EDG DGB Wherefore ITKML is an equilater and equiangle pentagon by the 4. of the first For the equall bases of the pentagon ITKML doo subtend equall angles set at the point Z and comprehended vnder equall sides Moreouer it is manifest that the pentagon I TKML is in one and the selfe same playne superficies For forasâ—uch as the angles ONC and NCP are in one and the selfe sâ—me playne superficies namely in the superficies ABGDE But vnto the same playne superficies the playne superficieces of the angles KTI and TIL are parallels by the 15. of the eleuenth And the triangles KTI and TIL concurre wherefore they are in one and the selfe same playne superficies by the corollary of the 16. of the eleueÌ„th And by the same reasoÌ„ so may we proue that the triangles ILM LMK MKT are in the selfe same playne superficies wherein are the triangles KTI and TIL Wherefore the pentagon ITKML is in one and the selfe same playne superficies Wherefore the solide angle of the Icosahedron namely the solide angle at the poynt Z subtendeth an equilater and equiangle pentagon plaine superficies which pentagon hath his plaine superficiall angles in the centres of the triangles which make the solide angle Z. And in like sort may we proue that the other eleuen solide angles of the Icosahedron eche of which eleuen solide angles are equall and like to the solide angle Z by the 16. of the thirtenth are subtended vnto pentagons equall and like and in like sort set to the pentagon ITKML And forasmuch as in those peÌ„tagons the right lines which ioyne together the centers of the bases are common sides it followeth that those 12. pentagons include a solide which solide is therefore a dâ—decâ—hedron by the 24. diffinition of the eleuenth and is by the first diffinition of this booke described in the Icosahedron fiue sides whâ—reof 〈◊〉 set vpon the pentagon ABGDE Wherefore in an Icosahedron geuen iâ— inscribed a dodecahedron which was required to be done An annotation of Hypsiâ—les This is to be noted that if a man should demaund 〈◊〉 many sides an Icosahedron hath we may thus answere It is manifest that an Icosahâ—râ—n is contayned vnder 20. triangles and that euery triangle consisteth of three
right linâ—s Now then multiply the 20. triangles into the sides of one of the triangles and so shall there be produced 6â— â—he halfe of which is 30. And so many sides hath an Icosahedron And in like sort in a dodecahedron forasmuch as 12. pentagons make a dodecahedron and euery pentagon contayneth â— right linesâ— multiply â—â— into 12. and there shall be produced 60. the halfe of which is 30. And so many are the sides of a dodecahedron And the reason why we take the halfe iâ— for that euery side whether it be of a triangle or of a pentagon or of a square as in a cube â—s taken twise And by the same reason may you finde out how many sides are in a cube and in a pyramis and in an octohedron But now agayne if ye will finde out the number of the angles of euery one of the solide figures when ye haue done the same multiplication that ye did before diâ—idâ— the same sides by the number of the plaine superficieces which comprehend one of the angles of the solides As for example forasmuch as 5. triangles contayne the solide angle of an Icosahedron diuide 60. by 5. and there will come forth 12. and so many solide angles hath an Icosahedâ—on In a dodecahedron forasmuch as three pentagons comprehend an angle diuide 60. by 3. and there will come forth 20 and so many are the angles of a dodecahedron And by the same reason may you finde out how many angles are in eche of the rest of the solide figures If it be required to be knowne how one of the plaines of any of the fiue solides being geuen there may be found out the inclination of the sayd plaines the one to the other which contayne eche of the solides This as sayth Isidorus our greate master is foâ—â—d out after this maner It is manifest that in a cube the plaines which contayne iâ— do◠〈◊〉 the one the other by a right angle But in a Tetrahedron one of the triangles being geuen let the endes of one of the sides of the sayd triangle be the centers and let the space be the perpendicular line drawne from the toppe of the triangle to the base and describe circumferâ—nces of a circle which shall cutte the one the other and from the intersection to the centers draw right lines which shall containe the inclination of the plaines coÌ„tayning the Tetrahedron In an Octoâ—edron take one of the sides of the triangle therâ—of and vpon it describe a square and draw the diagonall line and making the centres the endes of the diagonall line and the space likewise the perpendicular line drawne from the toppe of the triangle to the base describe circumferences and agayne from the common section to the centres draw right lines and they shall contayne the inclination sought for In an Icosahedron vpon the side of one of the triangles thereof describe a pentagon and draw the line which subtendeth one of the angles of the sayd pentagon and making the centres the endes of that line and the space the perpendicular line of the triangle describe circumferences and draw from the common intersectioâ— of the circumferences vnto the centres right lines and they shall contayne likewise the inclination of the plaines of the icosahedron In a dodecahedron take one of the pentagons and draw likewise the line which subtendeth one of the angles of the pentagon and making the centres the endes of that line and the space the perpendicular line drawne from the section into two equall partes of that line to the side of the pentagon which is parallel vnto it describe circumferences and from the point of the intersection of the circumferences draw vnto the centres right lines and they shall also containe the inclination of the plaines of the dodecahedron Thus did this most singular learned man reason thinking the deâ—onstration in euery one of them to be plaine and cleare But to make the demonstration of them manifest I think it good to declare and make open his wordesâ— and first in a Tâ—trahedronâ— The ende of the fiuetenth Booke of Euclides Elementes after 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ¶ The 6. Proposition The 6. Probleme In an Octohedron geuen to inscribe a trilater equilater Pyramis SVppose thaâ— the Octohedron whereâ—â— the Tetrahedron is required to be insâ—riâ—â—â— be ABGDEI Take 〈…〉 bases of the Octoâ—â—dron that is 〈…〉 close in the loweâ—â— triangle BGD namely AEâ— HED IGD and let the fourth be AIB which is opposite to the lowest triangle before put namely to EGD And take the centres of those fower bases which let be the pointes H C N â— And vpon the triangle HCN erecte a pyramis HCNL Now â—orasmuch as these two bases of the Octohedron namely AGE and ABI are set vpon the right lines EG and BI which are opposite the one to the otherâ— in the square GEBI of the Octohedron from the poinâ— A draâ—e by the centres of the bases namely by the centres H L perpendicular lines AHF ALK cutting the lines EG and BI 〈◊〉 two equall partes in the pointes F K by the Corollary of the 1â— of the thirtenth Wherfore a right line drawen froÌ„ the point F to the point K shall be a parallel and equall to the sides of the Octohedron namely to â—â— and GI by the 33. of the first And the right line HL which cutteth the 〈…〉 AF AK proportionally for AH and AL are drawen from the centres of equall circles to the circumferences is a parallel to the right line FK by the 2. of the sixth and also to the sides of the Octohedron namely to Eâ— and IG by the 9. of the eleuenth Wherefore as the line AF is to the line AH so is the line FK to the line HL by the 4. of the sixth For the triangles AFK and AHL are like by thâ— Corollary of the 2. of the sixth But the line AF is in sesquialter proportion to the line AH for the side EG maketh HF the halfe of the right line AH by the Corollary of the 12. of the thirtenth Wherfore FK or GI the side of the Octohedron is sesquialter to the righâ—line HL. And by the same reason may we proue that the sides of the Octohedron are sesquialter to the rest of the right lines which make the pyramis HNCI namely to the right lineâ—â— N NC CI LN and CH wherefore those right lines are equall and therefore the triangleâ— which are described of them namely the triangles HCN HNL NCL and CHL. which make the pyramis HNCL are equall and equilater And forasmuch as the angles of the same pyramis namely the angles H C N L do end in the centres of the bases of the Octohedron therefore it is inscribed â—o the same Octohedron by the first definition of this booke Wherefore in an Octohedron â—euen is inscribed a trilâ—ter equilaâ—â—â—â—â—â—amisâ— which was required to â—e donâ— A Corollary The bases of a Pyramis inscribed in an Octohedron are parallels
to the bases of the Octohedron For forasmuch as the sides of the bases of the Pyramis touching the one the other are parallels to the sides of the Octohedron which also touch the one the other as for example HL was proued to be a parallel to GI and LC to DI therefore by the 15. of the eleuenth the plaine superficies which is drawen by the lines HL and LC is a parallel to the plaine superficies drawen by the lines GI and DI. And so likewise of the rest Second Corollary A right line ioyning together the centres of the opposite bases of the Octohedron is sesquialter to the perpendicular line drawen from the angle of the inscribed pyramis to the base thereof For forasmuch as the pyramis and the cube which containeth it do in the selfe same pointes end their angles by the 1. of this booke therefore they shall both be inclosed in one and the selfe same Octohedron by the 4. of this booke But the diameter of the cube ioyneth together the centres of the opposite bases of the Octohedron and therefore is the diameter of the Sphere which containeth the cube and the pyramis inscribed in the cube by the 13. and 14. of the thirtenth which diameter is sesquialter to the perpendicular which is drawen from the angle of the pyramis to the base thereof for the line which is drawen from the centre of the sphere to the base of the pyramis is the sixth part of the diameter by the 3. Corollary of the 13. of the thirtenth Wherefore of what partes the diâ—meter containeth sixe of the same partes the perpendicular containeth fower ¶ The 7. Proposition The 7. Probleme In a dodecahedron geuen to inscribe an Icosahedron SVppose that the dodecahedron geuen be ABCDE And let the centres of the circles which coÌ„tayne sixe bases of the same dodecahedron be the polnes L M N P Q O. And draw these right lines OL OM ON OP OQ and moreouer these right lines LM MN NP PQ QL And now forasmuch as equall and equilater pentagons are contayned in equall circles therefore perpendicular lines drawne from their centres to the sides shall be equall by the 14. of the third and shall diuide the sides of the dodecahedron into two equall partes by the 3. of the same Wherefore the foresayde perpendicular lines shall coâ—outre in the point of the section wherein the sides are diuided into two equall partes as LF and MF doo And they also containe equall angles namely the inclination of the bases of the dodecahedron by the 2. corollary of the 18. of the thirtenth Wherfore the right lines LM MN NP PQ QL and the rest of the right lines which ioyne together two centres of the bases and which subtende the equall angles â—ontayned vnder the sayd equall perpendicular lines are equall the one to the other by the 4. of the first Wherefore the triangles OLM OMN ONP OPQ OQL and the rest of the triangles which are set at the centres of the pentagons are equilater and equall Now forasmuch as the 12. pentagons of a dodecahedron containe 60. plaine superficiall angles of which 60. eueâ—y â—hre make one solide angle of the dodecahedron it followeth that a dodecahedron hath 20. solide angles but eche of those solide angles is subteÌ„ded of ech of the triangles of the Icosahedron namely of ech of those triangles which ioyne together the centres of the pentagoÌ„s which make the solide angle as we haue before proued Wherefore the 20 equall and equilater triangles which subtende the 20. solide angles of the dodecahedron and haue their sides which are drawne from the centres of the pentagons common doo make an Icosahedron by the 25. diffinition of the eleuenth and it is inscribed in the dodecahedron geuen by the first diffinition of this booke for that the angles thereof doo all at one time touch the bases of the dodecahedron Wherefore in a dodecahedron geuenâ— iâ— inscribed an Icosahedron which was required to be done ¶ The 8. Proposition The 8. Probleme In a dodecahedron geuen to include a cube DEscribe by the 17. of the thirtenth a dodecahedron And by the same take the 12. sides of the cube eche of which subtend one angle of eche of the 12. bases of the dodecahedron for the side of the cube subtendeth the angle of the pentagon of the dodecahedron by the 2. corollary of the 17. of the thirtenth If therefore in the dodecahedron described by the selfe same 17. proposition we draw the 12. right lines subâ—ended vnder the foresayd 12. angles and ending in 8. angles of the dodecahedron and concurring together in such sort that they be in like sort situate as it was plainely proued in that proposition then shall it be manifest that the right lines drawne in this dodecahedron from the foresayd 8. angles thereof doo make the foresayd cube which therefore is included in the dodecahedron for that the sides of the cube are drawne in the sides of the dodecahedron and the angles of the same cube are set in the angles of the said dodecahedron As for example take 4. pentagons of a dodecahedron namely AGIBO BHCNO CKEDN and DFAON And draw these right lines AB BC CD DA. Which fower right lines make a square for that eche of those right lines doo subtend equall angles of equall pentagons the angles which those 4. right lines coÌ„taine are right angles as we proued in the construction of the dodecahedron in the 17. propositioÌ„ before alledged Wherfore the sixe bases being squares do make a cube by the 21. diffinition of the eleuenth and for that the 8. angles of the sayd cube are set in 8. angles of the dodecaheeron therefore is the sayd cube inscribed in the dodecahedron by the first diffinition of this booke Wherefore in a dodecahedron is inscribed a cube which was required to be doone ¶ The 9. Proposition The 9. Probleme In a Dodecahedron geuen to include an Octohedron SVppose that the dodecahedron geuen be ABGD Now by the 3. correllary of the 17. of the thirteÌ„th take the 6. sides which are opposite the one to the other those 6. sides I saye whose sections wherin they are deuided into two equal partes are coupled by three right lines which in the centre of the sphere wherin the Dodecahedron is contained doe cut the one the other perpendicularly And let the poyntes wherin the forsayde sides are cut into two equal partes be A B G D C I. And let the foresaid thre right lines ioyning together the saide sections be AB GD and CI. And let the centre of the sphere be E. Now forasmuch as by the foresaid correllary those thre right lines are equal it foloweth by the 4. of the first that the right lines subteÌ„ding the right angles which they make at the centre of the sphere whiche right angles are contained vnder the halues of the said three right lines are equal the one to the other
that is the right lines AG GB BD DA CA CG CB CD and IA IG IB ID are equal the one to the other Wherfore also the 8. triangles CAG CGB CBD CDA IAG IGB IBD IDA are equal and equilater And therefore AGBDCI is an Octohedron by the 23. definition of the eleueÌ„th And the sayd Octahedron is included in the dodecahedron by the first definition of this booke for that all the angles thereof doe at one time touch the sides of the dodecahedron Wherefore in the dodecahedron geuen is included an Octohedron which was required to be done ¶ The 10. Proposition The 10. Probleme In a Dodecahedron geuen to inscribe an equilater trilater Pyramis SVppose that the Dodecahedron geuen be ABCD of which Dodecahedron take thre bases meting at the poynt S namely these thre bases ALSIK DNSLE and SIBRN and of those thre bases take the three angles at the poynts A B D and draw these right lines AB BD and DA and let the diameter of the sphere containing the dodecahedron be SO and then draw thse right lines AO BO and DO Now forasmuch as by the 17. of the thirtenth the angles of the dodecahedron are set in the superficies of the sphere described about the Dodecahedronâ— therefore if vpon the diameter SO and by the poynt A be described a semicircle it shall make the angle SAO a right angle by the 31. of the third And likewise if the same semicircle be drawne by the poyntes D and B it shall also make the angles SBO and SDO right angles Wherefore the diameter SO containeth in power bothe the lines SA AO or the lines SB BO or els SD DO but the lines SA SD SB are equal the one to the other for they eche subtend one of the angles of equal pentagoÌ„s Wherfore the other lines remaining namely AO BO DO are equal the one to the other And by the same reason may be proued that the diameter HD which subtendeth the two right lines HA AD containeth in power both the said two right lines and also containeth in power bothe the right lines HB and BD which two right lines it also suhtendeth And moreouer by the same reason the diameter AC which subtendeth the right lines CB and BA containeth in power both the said right lines Câ— and BA But the right lines HA HB and CB are equal the one to the other for that eche of them also subtendeth one of the angles of equal pentagonsâ— wherfore the right lines remaining namely AD BD and BA are equal the one to the other And by the same reason may be proued that eche of those right lines AD BD and BA is equal to eche of the right lines AO BO and DO Wherefore the sixe right lines AB BD DA AO BO DO are equal the one to the other And therefore the triangles which are made of theÌ„ namely the triangles ABD AOB AOD and BOD are equal and equilater which triangles therfore do make a pyramis ABDO whose base is ABD and toppe the poynt O. Eche of the angles of which pyramis namely the angles at the pointes A B D O doe in the selfe same pointes touche the angles of the Dodecahedron Wherfore the said pyramis is inscribed in the Dodecahedron by the first diffinition of this boke Wherefore in a Dodecahedron geuen is inscribed a trilater equilater pyramis which was required to be done ¶ The 11. Proposition The 11. Probleme In an Icosahedron geuen to inscribe a cube IT was manifest by the 7. of this booke that the angles of a Dodecahedron are set in the centres of the bases of the Icosahedron And by the 8. of this boke it was proued that the angles of a cube are set in the angles of a Dodecahedron Wherefore the selfe same angles of the cube shall of necessitie be set in the centres of the bases of Icosahedron Wherfore the cube shal be inscribed in the Icosahedron by the first diffinition of this boke Wherfore in an Icosahedron geuen is included a cube which was required to be done ¶ The 12. Proposition The 12. Probleme In an Icosahedron geuen to inscribe a trilater equilater pyramis BY the former proposition it was manifest that the angles of a cube are set in the centres of the bases of the Icosâ—hedron And by the first of this booke it was playne that the foure angles of a pyramis are set in foure angles of a cube Wherefore it is euident by the first diffinition of this booke that a pyramis described of right lines ioyning together these foure centres of the bases of the Icosahedron shal be inscribed in the same Icosahedron Wherefore in an Icosadron geuen is inscribed an equilater trilater pyramis which was required to be done ¶ The 13. Probleme The 13. Proposition In a Cube geuen to inscribe a Dodecahedron TAke a Cube ADFL And diuide euery one of the sides therof into two equall partes in the pointes T H K P G L M F and pkQs And drawe these right lines TK GF pQ Hk Ps and LM which lines againe diuide into two equall partes in the pointes N V Y I Z X. And draw these right lines NY VX and IZ Now the three lines NY VX and IZ together with the diameter of the cube shall cut the one the other into two equall partes in the centre of the cube by the 3â— of the eleuenth let that centre be the point O. And not to stand long about the demonstration vnderstand all these right lines to be equall and parallels to the sides of the cube and to cut the one the other right angled wise by the 29. of the first Let their halfes namely FV GV HI and kI and the rest such like be deuided by an extreme and meane proportion by the 30. of the sixth whose greater segmeÌ„ts let be the lines FS GB HC and kE c. and drawe these right lines GI GE BC and BE. Now forasmuch as the line GI is equall to the whole line GV which is the halfe of the side of the cube and the line IE is equall to the line BV that is to the lesse segmeÌ„t therfore the squares of the lines GI and IE are triple to the square of the line GB by the 4. of the thirtenth But vnto the squares of the lines GI and IE the square of the line GE is equall by the 47. of the firstâ— for the angle GIE is a right angle Wherefore the square of the line GE is triple to the square of the line GB And forasmuch as the line FG is erected perpendicularly to the plaine AGkL by the 4. of the eleuenth for it is erected perpendicularly to the two lines AG and GI therefore the angle BGE is a right angle for the line GE is drawen in the plaine AGkL Wherefore the line BE containing in power the two lines BG and GE by the 47. of the first is in power quadruple to the
wherfore the line Ce is to the line eg as the greater segment to the lesse and therefore their proportion is as the whole line IC is to the greater segment Ce and as the greater segment Ce is to the lesse segment eg wherefore the whole line Ceg which maketh the greater segment and the lesse is equall to the whole line IC or IE And forasmuch as two parallel plaine superficieces namely that which is extended by IOB and that which is extended by the line ag are cutte by the playne of the triangle BCE which passeth by the lines ag and IB their common sections ag and IB shall be parallels by the 16. of the eleuenth But the angle BIE or BIC is a right angle wherefore the angle agC is also a right angle by the 29. of the first and those right angles are contayned vnder equall sides namely the line gC is equall to the line CI and the line ag to the line BI by the 33. of the first wherfore the bases Ca and CB are equall by the 4. of the first But of the line CB the line CE was proued to be the greater segment wherefore the same line CE is also the greater segment of the line Ca but cn was also the greater segment of the same line Ca. Wherefore vnto the line CE the line cn which is the side of the dodecahedron and is set at the diameter is equall And by the same reason the rest of the sides which are set at the diameter may be proued eâ—uall to lines equall to the line CE. Wherfore the pentagon inscribed in the circle where in is contained the triangle BCE is by the 11. of the fourth equiangle and equilater And forasmch as two pentagons set vpon euery one of the bases of the cube doo make a dodecahedron and sixe bases of the cube doo receaue twelue angles of the dodecahedron and the 8. semidiameters doo in the pointes where they are cutte by an extreame and meane proportion receaue the rest therefore the 12. pentagon bases contayning 20. solide angles doo inscribe the dodecahedron in the cube by the 1. diffinition of this booke Wherefore in a cube geuen is inscribed a dodecahedron which was required to be done First Corollary The diameter of the sphere which containeth the dodecahedron containeth in power these two sides namely the side of the Dodecahedron and the side of the cube wherein the Dodecahedron is inscribed For in the first figure a line drawne from the centre O to the poynt B the angle of the Dodecahedron namely the line OB containeth in power these two lines OV the halfe side of the cube and VB the halfe side of the dodecahedron by the 47. of the first Wherefore by the 15. of the fiueth the double of the line OB which is the diameter of the sphere containing the Dodecahedron containeth in power the double of the other lines OV and VB which are the sides of the cube and of the dodecahedron ¶ Second Corollary The side of a cube diuided by an extreme and meane proportion maketh the lesse segment the side of the dodecahedron inscribed in it and the greater segment the side of the cube inscribed in the same Dodecahedron For it was before proued that the side of the dodecahedron is the greater segment of BE the side of the triangle BECâ— but the side BE which is equall to the lineâ— GB and SF is the greater segmeÌ„t of GF the side of the cube which line â—E subtending thâ— angle of the pentagon was by the â— of this booke the side of the cube inscribed in the dodecahedron Third Corollary The side of a cube is equal to the sides of a Dodecahedron inscribed in it and circumscribed about it For it was manifest by this proposition that the side of a cube maketh the lesse segment the side of a Dodecahedron inscribed in it namely as in the first figure the line BS the side of the Dodecahedron inscribed is the lesse segmeÌ„t of the line GF the side of the cube And it was proued in the 17. of the thirtenth that the same side of the cube subteÌ„deth the angle of the pentagon of the Dodecahedron circumscribed and therefore it maketh the greater segment the side of the Dodecahedron or of the pentagon by the first corollary of the same Wherefore it is equal to bothe those segments The 14. Probleme The 14. Proposition In a cube geuen to inscribe an Icosahedron SVppose that the cube geuen be ABC the Centres of whose bases let be the points D E G H I K by whiche poyntes draw in the bases vnto the other sids parallels not touching the one the other And deuide the lines drawn from the centres as the line DT c. by an extreme and meane proportion in the poyntes A F L M N B P Q R S C O by the 30. of the sixth and let the greater segmentes be about the ceÌ„tres And draw these right lines AL AG AM and TG And forasmuch as the lines cut are parallels to the sides of the cube they shall make right angles the one with the other by the 29. of the first and forasmuche as they are equal their sections shall be equal for that the sections are like by the 2. of the fourtenth Wherfore the line TG is equal to the line DT for they are eche halfe sides of the cube Wherfore the square of the whole line TG and of the lesse segment TA is triple to the square of the line AD the greater segment by the 4. of the thirteÌ„th But the line AG containeth in power the lines AT TG for the angle ATG is a right angle Wherefore the square of the line AG is triple to the square of the line AD. And forasmuch as the line MGL is erected perpendicularly to the plain passing by the lines AT which is parallel to the bases of the cube by the corollary of the 14. of the eleueÌ„th therfore the angle AGL is a right angle But the line LG is equal to the line AD for they are the greater segments of equal lines Wherfore the line AG which is in power triple to the line AD is in power triple to the line LG Wherefore adding vnto the same square of the line AG the square of the line LG the square of the line AL which by the 47. of the first containeth in power the two lines AG and GL shal be quadruple to the line AD or LG Wherefore the line AL is double to the line AD by the 20. of the sixth and therfore is equal to the line AF or to the line LM And by the same reason may we proue that euery one of the other lines which couple the next sections of the lines cut as the lines AM PF PM MQ and the rest are equal Wherfore the triangles ALM APF AMP PMQ and the rest such like are equal equiangle and equilater
subtend angles of triangles like vnto the triangles whose angles the lines AC BF and PL subtend are cut into two equall partes in the pointes Z and I by the 4. of the sixth so also are the other lines NV XM DS QT which are equall vnto the lines HK GE cut in like sort and they shall cut the lines AC BF and PL like Wherefore the line KO which is equall to RZ shall make the greater segment the line RO which is equall to the line ZK for the greater segment of the RZ was the line ZK and therefore the line OI shall be the lesse segment when as the whole line RI is equall to the whole line RZ Wherefore the squares of the whole line KO and of the lesse segment OI are triple to the square of the greater segment RO by the 4. of the thirtenth Wherfore the line KI which containeth in power the two lines KO and OI is in power triple to the line RO by the 47. of the first for the angle KOI is a right angle And forasmuch as the lines FE and FG which are the lesse segmentes of the sides of the Octohedron are equall and the line FK is coÌ„mon to them both and the angles KFG and KFE of the triangles of the Octohedron are equall the bases KG and KE shall by the 4 of the first be equall and therefore the angles KIE and KIG which they subtend are equall by the 8. of the first Wherefore they are right angles by the 13. of the first Wherefore the right line KE which containeth in power the two lines KI and â—E by the 47. of the first is in power quadruple to the line RO or IE for the line RI is proued to be in power triple to the same line RO But the line GE is double to the line IE Wherfore the line GE is also in power 〈…〉 PF And by the same reason may be proued that the â—est of the eleuen solide angles of the 〈◊〉 are 〈…〉 the sections of euery one of the sides of the Octohedron namely in the pointes E N V H â— M â— D S Q T. Wherefore there are 12. angles of the Icosahedron Moreouer forasmuch as euery one of the bases of the Octohedron do eche containe triangles of the Icosahedron 〈…〉 pyramiâ—â— ABCâ—FP which is the halfe of the Octohedron the triangle FCP receaueth in thâ— section of his sides the â— triangle GMS and the triangle CPB containeth the triangle NXS and thâ— triangle â—AP contayneth the triangle HND and moreouer the triangle APF containeth the triangle â—DG and the same may be proued in the opposite pyramis ABCFL Wherefore there shall be eight triangleâ— And forasmuch as besides these triangles to euery one of the solide angles of the Octohedron 〈◊〉 subtended two triangles as the triangles KEG amd MEG to the angle F and the triangles HNV and XNV to the angle B also the triangles NDS and â—DS to the angle P likewise the triangleâ— DHK and QHK to the angle A Moreouer the triangles EQT and VQT to the angle L and finally the triangles SXM and TXM to the angle C these 12. triangles being added to thâ—â— for 〈◊〉 triangles shall produce â—0 triangles equall and equilâ—ter coupled together which shall male an Icosahedron by the 25. definition of the eleuenth and it shall be inscribed in the Octohedron geuen ABCâ—â—L by the first definition of this booke for the 1â— angles thereof are set in 1â— like sections of the sides of the Octohedron Wherefore in an Octohedron geuen is inscribed an Icosahedron ¶ First Corollary The side of an equilater triangle being diuided by an extreme and meane proportion a right line subtending within the triangle the angle which is contained vnder the greater segment and the lesse is in power duple to the lesse segment of the same side For the line KE which subtendeth the angle KFE of the triangle AFL which angle KFE is contained vnder the two segmentes KF FE was proued equall 〈◊〉 the line HK which containeth in power the two lesse segmentes HA and AK by the 47. of the â—â—rst fo◠〈◊〉 angle HAK is 〈…〉 Second Corollary The bases of the Icosahedron are concentricall that is haue one and the selfe same centre with the bases of the Octohedron which contayneth it For suppose that 〈…〉 Octohedron 〈◊〉 ECD the base of an Icosahedron and let the centre of the base ABG be the point F. And drawe these right lines FA FB FC and FE Now then the 〈…〉 to the two lines FB and BC for they are lines drawen from the centre and are also lesse segmentes and they contayne the 〈…〉 ¶ The 17. Probleme The 17. Proposition In an Octohedron geuen to inscribe a Dodecahedron SVppose that the Octohedron geuen be ABGDEC whose 12. â—ides let be cut by an extreme and meane proportion as in the former Proposition It was manifest that of the right lines which couple thâ—se sections are made 20. triangles of which 8. are concentricall with the bases of the Octohedron by the second Corollary of the former Proposition If therefore in euery one of the centres of the 20. triangles be inscribed by the 1. of this booke euery one of the â—â— â—â—gles of the Dodecahedron we shall finde that â— angles of the Dodecahedron are set in the 8. centres of the bases of the Octohedron namely these angles I u ct O M a P and X and of the other 12. solide angles there are two in the centres of the two triangles which haue one side common vnder euery one of the solide angles of the Octohedron namely vnder the solide angle A the two solide angles K Z vnder the solide angle B the two solide angles H T vnder the solide angle G the two solide angles Y V vnder the solide angle D the two solide angles F L vnder the solide angle E the two solide angles S N vnder the solide angle C the two solide angles Q R and forasmuch as in the Octohedron are sixe solide angles vnder them shall be subtended 12. solide angles of the Dodâ—cahedron and so are mâ—de 20. solide angles composed of 12. equall and â—quilâ—ter superficiall pentagons as it was 〈◊〉 by the 5. of this booke which therefore containe a Dodecahedron by the 24. definition of the eleuenth And it is inscribed in the Octohedron by the 1. definition of this booke for that euery one of the bases of the Octohedron do receaue angles therof Wherefore in an Octohedron geuen is inscribed a Dodecahedron ¶ The 18. Probleme The 18. Proposition In a trilater and equilater Pyramis to inscribe a Cube SVppose that there be a trilater equilater Pyramis whose base let be ABC and â—oppe the point D. And let it be comprehended in a Sphereâ— by the 13. of the 〈◊〉 And lâ—â— the centre of that Sphere be the point E. And from the solide angles A B C D draw right lines passing by the centre E vnto the opposite bases of
side GD the angles M N vnder the side AB the angles T S vnder the side BG the angles P O and vnder the side AG the angles R Q so there rest 4. angles whose true place we will now appoynt Forasmuch as a cube contayned in one and the selfe same sphere with a dodecahedron is inscribed in the same dodecahedron as it was manifest by the 17. of the thirtenth and 8. of this booke it followeth that a cube and a dodecahedron circumscribed about it are contayned in one and the selfe same bodies for that their angles concurre in one and the selfe same poyntes And it was proued in the 18. of this booke that 4. angles of the cube inscribed in the pyramis are set in the middle sections of the perpendicularâ— which are drawne from the solide angles of the pyramis to the opposite bases wherefore the other 4. angles of the dodecahedron are also as the angles of the cube set in those middle sections of the perpendiculars Namely the angle V is set in the middest of the perpendicular AHâ— the angle Y in the middest of the perpendicular BF the angle X in the middest of the perpendicular GE and lastly the angle D in the middest of the perpendicular D which is drawne from the toppe of the pyramis to the opposite base Wherefore those 4. angles of the dodecahedron may be sayd to be directly vnder the solide angles of the pyramis or they may be sayd to be set at the perpendiculars Wherefore the dodecahedron after this maner set is inscribed in the pyramis geuen by the first diffinition of this booke for that vpoÌ„ euery one of the bases of the pyramis are set an angle of the dodecahedroÌ„ inscribed Wherefore in a trilater equilater pyramis is inscribed a dodecahedron The 21. Probleme The 21. Proposition In euery one of the regular solides to inscribe a Sphere IN the 13. of thâ— thirtenth and thâ— other 4. propositioâ—â— following iâ— was declared that â—he â—â— regular solidesâ—â—re so contaâ—â—ed in a sphere that â—ight linâ—â— drawne from the cenâ—â—â— oâ— the 〈…〉 of 〈◊〉 solide inscribed are equall Which right lines therefore make pyramids whose â—oppes are the centre of the sphere or of the solide and the basâ—â—â—â—e cuâ—â—â— one of the bases of those solides And 〈…〉 solide â—quall and like the one to the other and described in equall circles those cirâ—les shall cutte the sphere for the angles which touch the circumference of the circle touch also the superficies of the sphere Wherefore perpeÌ„diculars drawne from the centre of the sphere to the bases or to the playne superficieces of the equall circles are equall by the corollary of the assumpt of the 1â— of the twelfth Wherefore making the centre the 〈◊〉 of the sphere which 〈◊〉 the solide and thâ— space some one of the equall perpendicularâ— dâ—scribâ— a sphere and it shall touch euery one of the bases of 〈◊〉 solide 〈…〉 perficies of the sphere passe beyond those bases when as those pâ—â—peâ—diculars 〈…〉 are drawne from the centre to the bases by the 3. corollary of the saâ—â—â—â—â—umpt Wherâ—fore â—e haue iâ— euery one of the regular bodies inscribed a sphere which regular boâ—â—â— are in number one i◠〈◊〉 by the corollary of the 1â— of the 〈◊〉 A Corollary The regular figures inscribed in spheres and also the spheres circumscribed about them or contayning them haue one and the selfe same centre Namely their pyramids the â—ngles of whose bâ—ses touch the superâ—â—â—â—â—â— of thâ—â—â—here doo from those angles cause equall right lines to be drawâ—â— to one and â—he selfe 〈◊〉 poynâ— making the topâ—â—â— of the pyramidâ— in the same poynt and therefore they 〈…〉 thâ— câ—â—tres of the spheres in the selfe same toppes when 〈◊〉 the right lines drawne from those angles to the croâ—â—ed superficies whâ—rein are 〈◊〉 the angles of the bases of the pyramidâ— are equallâ— An adueâ—â—â—sment of Flussas â— Of these solides onely the Octohedron receaueth the other solides inscribed one with 〈…〉 other For the Octohedron contayneth the Icosahedron inscribed in it and the same Icosahedron contayneth the Dodecahedron inscribed in the same Icosahedron and the same dodecahedron contayneth the cube inscribed in the same Octohedron and 〈…〉 â—â—râ—â—mscribeth the Pyramis inscribed in the sayd Octohedron But this happâ—neth not in the other solides The ende of the fiuetenth Booke of Euclides Elemenâ—â—â— after Caâ—paâ—â— and 〈◊〉 ¶ The sixtenth booke of the Elementes of Geometrie added by Flussas IN the former fiuetenth booke hath bene taught how to inscribe the fiue regular solides one with in an other Now semeth to rest to coÌ„pare those solidâ— so inscribed one to an other and to set forth their passionâ— and proprieties which thing Flussas considering in this sixteÌ„th booke added by him hath excellently well and most conningly performed For which vndoubtedly he hath of all them which haue a loue to the Mathematicals deserued much prayse and commendacion both for the great traâ—ailes and paynâ—s which it is most likely he hath taâ—â—n in iâ—uenting such straunge and wonderfull propositions with their demonstrations in this booke contayned as also for participating and communicating abrode the same to others Which booke also that the reader should want nothing conducing to the perfection of Euclides Elements I haue with some trauaile translated for the worthines â—hereof haue added it aâ— a sixtenth booke to the 15. bookes of Euclide Vouchsafe therefore gentle reader diligently to read and peyse it for in it shall you finde noâ— onely matter strange and delectable but also occasion of inuention of greater things pertayning to the natures of the fiue regular solidâ—s◠¶ The 1. Proposition A Dodecahedron and a cube inscribed in it and a Pyramis inscribed in the same cube are contained in one and the selfe same sphere FOr the angles of the pyramiâ— are seâ— in the angâ—es of the cube wherein it is inscribed by the first of the fiuetenthâ— and all the angles of the cube are set in the angles of the dodecahedâ—â—â— circumscribed 〈…〉 〈◊〉 the 8. of the fiuetenth And all the angles of the Dodecahedron are set in the superficies of the sphere by the 17. of the thirtenth Wherefore those three solides inscribed one within an other are contained in one and the selfe same sphere by the first diffinition of the fiuetenth A dodecahedron therfore and a cube inscribed in it and a pyramis inscribed in the same cube are contained 〈…〉 â—â—lfe same sphere 〈…〉 These three solides li 〈…〉 elfe same Icosahedron or Octohedron or Pyramis 〈…〉 me Icosahedron by the 5.11 12. of the fiuetenth and they ar 〈…〉 ctohedron by the 4. 6. and 16. of the same lastly they are inscribed in 〈…〉 the first 18. and 19. of the same For the angles of all these solide 〈…〉 the circumscribed Icosahedron or octohedron or pyramis ¶ The 〈…〉 The proportion of a Dodecahedron circumscribed about a cube to a DodecahedroÌ„ inscribed in the same cube is
line AN is deuided by an extreme and meane proportion in the poynt H by the â— of the thirtenth But as the line AN is to the line AH so is the line AD to the line AE by the 2. of sixth for the lineâ— FH and ON are parallelâ— and againe as the line AD is to the line AE so by the same is the line AG to the line AQ and the line AI to the line AP for the lines PQ and GI are parallels Wherefore the lines AG and AI are deuided by an extreme and meane proportion in the points Q P the line AQ shal be the greater segmeÌ„t of the line AG or AB And forasmuch as the wholâ— line AG is to the greater segment AQ as the greater segment AI is to the residue AP the line Aâ— shal be the lesse segment of the whole line Aâ— or AG. Wherfore the liâ—â— PEQ which by the poynt E passeth parallelwise to the line GI cutteth the lines AG and BA by an extreme and meane proportion in the poynts Q and P. And by the same reason the line â—R which by the poynt C passeth parallelwise to the line AM shall fall vpon the sections P and R so also shal the line RQ which by the poynt D passeth parallelwise to the line BL fall vpoâ— the sections RQ Wherefore either of the lines PE and EQ shal be equal to the line CD in the parallelogrammes PD and QC by the 34. of the first And forasmuch as the lines PE and EQ are equal the lines PC CR RD and DQ shal be likewise equal Whâ—rfore the triangle PRQ iâ—â—quilater and cutteth the sides of the base of the pyramiâ— in the poyntes P Q R by an extreme and meane proportion And in it is inscribed the base ECD of the Icosahedron contained in the forâ—sayd pyramis If therefore from the angles of the base of a pyramis be drawne to the opposite sidâ—s right lines cutting the sayde sides by an extreme and meane proportion they shall containe the base of the Icosahedron inscribed in the pyramis which base shall be inscribed in an equilater triangle whose angles cut the sides of the base of the pyramis by an extreme meane propoâ—tion ¶ A Corollary The side of an Icosahedron inscribed in an Octohedron is the greater segment of the line which being drawen from the angle of the base of the Octohedron cutteth the opposite side by an extreame and meane proportion For by the 16. of the fiuetenth FKH is the base of the Octohedron which containeth the base of the Icosahedron CDE vnto which triangle FKH the triangle HKG is equall as hath bene proued By the point H draw vnto the line ME a parallel line HT cutting the line DN in the point S. Wherefore ES DT and ET are parallelogrammes and therefore the lines EH and MT are equall and the lines EM and HT are like cut in the pointes D and S by the 34. of the first Wherefore the greater segment of the line HT is the line HS which is equall to ED the side of the Icosahedron But by the 2. of the sixth the line TK is cut like to the line HK by the parallel DM And therefore by the 2. of the fourtenth it is diuided by an extreme and meane proportion But the line TM is equall to the line EH Wherefore also the line TK is equall to the line EF or DH Wherefore the residues EH and TG are equall For the whole lines FH and KG are equall Wherefore KG the side of the triangle HKG is in the point T diuided by an extreme and meane proportion in the point T by the right line HT and the greater segment thereof is the line ED the side of the Icosahedron inscribed in the Octohedron whose base is the triangle HKG or the triangle FKH which is equall to the triangle HKG by the 16. of the fiuetenth ¶ The 5. Proposition The side of a Pyramis diuided by an extreme and meane proportion maketh the lesse segment in power double to the side of the Icosahedron inscribed in it SVppose that ABG be the base of a pyramis and let the base of the Icosahedron inscribed in it be CDE described of three right lines which being drawen from the angles of the base ABG cut the opposite sides by an extreme and meane proportion by the former Proposition namely of these three lines AM BI and GI Then I say that AI the lesse segment of the side Aâ— is in power duple to CE the side of the Icosahedron For forasmuch as by the former Proposition it was proued that the triangle CDE is inscribed in an equilater triangle whose angles cut the sides of ABG the base of the pyramis by an extreme and meane proportion let that triangle be FHK cutting the line AB in the point F. Wherefore the lesse segment FA is equall to the segment AI by the 2. of the fouretenth for the lines AB and AG are cut like Moreouer the side FH of the triangle FHK is in the point D cut into two equall partes as in the former Proposition it was proued and FCED also by the same is a parallelogramme Wherefore the lines CE and FD are equall by the 33 of the first And forasmuch as the line FH subtendeth the angle BAG of an equilater triangle which angle is contained vnder the greater segment AH and the lesse segment AFâ— therefore the line FH is in power double to the line AF or to the line AI the lesse segment by the Corollary of the 16. of the fiuetenth But the same line FH is in power quadruple to the line CE by the 4. of the second for the line FH is double to the line CE Wherefore the line AI being the halfe of the square of the line FH is in power duple to the line CE to which the line FH was in power quadruple Wherefore the side AG of the pyramis being diuided by an extreme and meane proportion maketh thâ— lesse segment AI in power duple to the side CE of the Icosahedron inscribed in it ¶ A Corollary The side of an Icosahedron inscribed in a pyramis is a residuall line For the diameter of the Sphere which containeth the fiue regular bodies being rationall is in power sesquialtera to the side of the pyramis by the 13. of the thirtenth and therefore the side of the pyramis is rationall by the definition which side being diuided by an extreme and meane proportion maketh the lesse segment a residuall line by the 6. of the thirtenth Wherefore the side of the Icosahedron being commensurable to the same lesse segment for the square of the side of the Icosahedron is the halfe of the square of the said lesse segment is a residuall line by that which was added after the 103. of the tenth booke ¶ The 6. Proposition The side of a Cube containeth in power halfe the side of an equilater triangular Pyramis inscribed in the said
Cube FOr forasmuch as the side of the pyramis inscribed in the cube subteÌ„deth two sides of the cube which containe a right angle by the 1. of the fiuetenth it is manifest by the 47. of the first that the side of the pyramis subteÌ„ding the said sides is in power duple to the side of the cube Wherefore also the square of the side of the cube is the halfe of the square of the side of the pyramis The side therefore of a cube containeth in power halfe the side of an equilater triangular pyramis inscribed in the said cube ¶ The 7. Proposition The side of a Pyramis is duple to the side of an Octohedron inscribed in it FOrasmuch as by the 2. of the fiuetenth it was proued that the side of the Octohedron inscribed in a pyramis coupleth the midle sections of the sides of the pyramis Wherefore the sides of the pyramis and of the Octohedron are parallels by the Corollary of the 39. of the first and therefore by the Corollary of the 2. of the sixth they subtend like triangles Wherfore by the 4. of the sixth the side of the pyramis is double to the side of the Octohedron namely in the proportion of the sides The side therefore of a pyramis is duple to the side of an Octohedron inscribed in it ¶ The 8. Proposition The side of a Cube is in power duple to the side of an Octohedron inscribed in it IT was proued in the 3. of the fiuetenth that the diameter of the Octohedron inscribed in the cube coupleth the centres of the opposite bases of the cube Wherefore the said diameter is equall to the side of the cube But the same is also the diameter of the square made of the sides of the Octohedron namely is the diameter of the Sphere which containeth it by the 14. of the thirtenth Wherefore that diameter being equall to the side of the cube is in power double to the side of that square or to the side of the Octohedron inscribed in it by the 47. of the first The side therefore of a Cube is in power duple to the side of an Octohedron inscribed in it which was required to be proued ¶ The 9. Proposition The side of a Dodecahedron is the greater segment of the line which containeth in power halfe the side of the Pyramis inscribed in the sayd Dodecahedron SVppose that of the Dodecahedron ABGD the side be AB and let the base of the cube inscribed in the Dodecahedron be ECFH by the â—â— of the fiuetenth And let the side of the pyramis inscribed in the cube be CH by the 1. of the fiuetenth Wherefore the same pyramis is inscribed in the Dodecahedron by the 10. of the fiuetenth Then I say that AB the side of the Dodecahedron is the greater segment of the line which containeth in power halfe the line CH which is the side of the pyramis inscribed in the Dodecahedron For forasmuch as EC the side of the cube being diuided by an extreme and meane proportion maketh the greater segment the line AB the side of the Dodecahedron by the â—â—rst Corollary of the 17. of the thirtenth For they are contâ—ined in one and the selfe same Sphere by the first of this booke and the line EC the side of the cube contayneth in power the halfe of the side CH by the 6. of this booke Wherefore AB the side of the Dodecahedron is the greater segment of the line EC which containeth in power the halfe of the line CH which is the side of the Dodecahedron inscribed in the pyramis The side therefore of a Dodecahedron is the greater segment of the line which containeth in power halfe the side of the Pyramis inscribed in the said Dodecahedron ¶ The 10. Proposition The side of an Icosahedron is the meane proportionall betwene the side of the Cube circumscribed about the Icosahedron and the side of the Dodecahedron inscribed in the same Cube SVppose that there be a cube ABFD in which let there be inscribed an icosahedron CLIGOR by the 14. of the fiuetenth Let also the Dodecahedron inscribed in the same be EDMNPS by the 13. of the same Now forasmuch as CL the side of the Icosahedron is the greater segmeÌ„t of AB the side of the cube circumscribed about it by the 3. Corollary of the 14. of the fiuetenth and the side ED of the DodecahedroÌ„ inscribed in thesame cube is the lesse segmeÌ„t of the same side AB of the cube by the 2. Corollary of the 13. of the fiuetenth it followeth that AB the side of the cube being diuided by an extreme and meane proportion maketh the greater segment CL the side of the Icosahedron inscribed in it and the lesse segment ED the side of the Dodecahedron likewise inscribâ—d in it Wherefore as the whole line AB the side of the cube is to the greater segment CL the side of the Icosahedron so is the greater segment CL the side of the Icosahedron to the lesse segment EDâ— the side of the Dodecahedron by the third definition of the sixth Wherefore the side of an Icosahedron is the meane proportionall betwene the side of the cube circumscribed about the Icosahedron and the side of the Dodecahedron inscribed in the same cube ¶ The 11. Proposition The side of a Pyramis is in power Octodecuple to the side of the cube inscribed in it FOr by that which was demonstrated in the 18. of the fiuetenth the side of the pyramis is triple to the diameter of the base of the cube inscribed in it and therefore it is in power nonecuple to the same diameter by the 20. of the sixth But the diamer is in power double to the side of the cube by the 47. of the first And the double of nonecuple maketh Octodecuple Wherefore the side of the pyramis is in power Octodecuple to the side of the cube inscribed in it ¶ The 12. Proposition The side of a Pyramis is in power Octodecuple to that right line whose greater segment is the side of the Dodecahedron inscribed in the Pyramis FOrasmuch as the Dodecahedron and the cube inscribed in it are set in one and the sâ—lfâ— same pyramis by the Corollary of the first of this booke and the side of the pyramis circumscribed about the cube is in power octodecuple to the side of the cube inscribed by the former Proposition but the greater segment of the selfe same side of the cube is the side of the Dodecahedron which containeth the cube by the Corollary of the 17. of the thirtenth Wherfore the side of the pyramis is in power octodecuple to that right line namely to the side of the cube whose greater segment is the side of the Dodecahedron inscribed in the pyramis ¶ The 13. Proposition The side of an Icosahedron inscribed in an Octohedron is in power duple to the lesse segment of the side of the same
Octohedron FOrasmuch as in the 17. of the fiuetenth it was proued that the side of an Icosahedron inscribed in a pyramis coupleth together the two sections which are produced by an extreme and meane proportion of the side of the Octohedron which make a right angle and that right angle is contained vnder the lesse segmentes of the sides of the Octohedron and is subtended of the side of the Icosahedron inscribed it followeth therefore that the side of the Icosahedron which subtendeth the right angle being in power equall to the two lines which containe the said angle by the 47. of the first is in power duple to euery one of the lesse segmeÌ„tes of the side of the Octohedron which containe a right angle Wherefore the side of an Icosahedron inscribed in an Octohedron is in power duple to the lesse segment of the â—ide of the same Octohedron ¶ The 14. Proposition The sides of the Octohedron and of the Cube inscribed in it are in power the one to the other in quadrupla sesquialter proportion SVppose that ABGDE be an Octohedron and let the cube inscribed in it be FCHI Then I say that AB the side of the Octohedron is in power quadruple sesquialter to FI the â—ide of the cube Let there be drawen to BE the base of the triangle ABE a perpendicular AN and againe let there be drawen to the same base in the triangle Gâ—E the perpendicular GN which AN GN shall passe by the centres F and I and the line AF is duple to the line FN by the Corollary of the 12. of the thirtenth Wherfore the line AO is duple to the line OE by the 2. of the sixth For the lines FO and NE are parallels And therefore the diameter AG is triple to the line FI. Wherfore the power of AG is noncuple to the power of FI. But the line AG is in power duple to the side AB by the 14. of the thirtenth Wherefore the square of the line AB being ing the halfe of the square of the line AG which is noncuple to the square of the line FI iâ— quadruple sesquialter to the square of the line FI. The sides therefore of the Octohedâ—â—â—â—nd of the cube inscribed in itâ— are in power the one to the other in quadruple sesquialter proportion ¶ The 1â— Proposition The side of the Octohedron is in power quadruple sesquialter to that right line whose greater segment is the side of the Dodecahedron inscribed in the same Octohedron FOrasmuch as in the 14. of this booke it was proued that the side of the Octohedron is in power quadruple sesquialter to the side of the cube inscribed in it but the side of the cube being cut by an extreme and meane proportion maketh the greater segment the side of the Dodecahedron circumscribed about it by the 3. Corollary of the 13. of the fiuetenth therefore the side of the Octohedron is in power quadruple sesquialter to that right line namely to the side of the cube whose greater segment is the side of the Dodecahedron inscribed in the cube But the Dodecahedron and the cube inscribed one within an other arâ— inscribed in one and the selfe same Octohedron by the Corollary of the first of this booke The side therefore of the Octohedron is in power quadruple sesquialter to that right line whose greater segment is the side of the Dodecahedron inscribed in the same Octohedron ¶ The 16. Proposition The side of an Icosahedron is the greater segment of that right line which is in power duple to the side of the Octohedron inscribed in the same Icosahedron SVppose that there be an Icosahedron ABGDFHEC whose side let be BG or â—Câ— and let the Octohedron insâ—â—ibâ—d in it be AKDâ— and let the side therof be AL. Then I say that the side â—C is the greater segment of that right line which is in power duple to the side AL. For forasmuch as figures inscribed and circumscribed haue oâ—e the selfsame centre by the Corollary of the â—1 of the fiuetenth let the same be the point I. Now right lineâ— drawen by thâ—◠〈◊〉 to the midle sections of the opposite sides namely the lines AID and KIL do in the point I â—ut 〈…〉 the other inâ—â— two â—quall 〈◊〉 and perpendicularly by the Corollary of the 14. of the fiuetenth and forasmuch as they couple the midle sections of the opposite lines BG and HF therfore they cut them perpendiularly wherefore also the lines BG 〈…〉 are parallels by the 4. Corollary of the 14. of the 〈…〉 Now then draw a line from B to H and the sayd â—â—ne BH shall be equall and parallel to the line KL by the 33. of the first But the line BH subtendeth â—wâ— sides of the pentagon which is composed of the sides of the Icosahedron namely the sides BA and AH Wherfore the line BH being cut by an extreme and meane proportion maketh the greater segment the side of the pentagon by the 8. of the thirtenth which side is also the side of the Icosahedron namely EC And vnto the line BH the line KLâ— is equall and the line KL is in power duple to AL the side of the Octohedron by the 47. of the first for in the square AKDL the angle KAL is a right angle Wherefore EC the side of the Icosahedron is the greater segment of the line BH or KL which is in power duple to AL â—he side of the Octohedron inscribed in the Icosahedron Wherefore the side of an Icosahedron is the greater segment of that right line which is in power duple to the side of the Octohedron inscribed in the same Icosahedron ¶ The 17. Proposition The side of a Cube is to the side of a Dodecahedron inscribed in it in duple proportion of an extreame and meane proportion FOr it was manifesâ— by the â— corollary of the 13. of the fiuetenth that the side of a cube diuided by an extreame and meane prâ—portion maketh the lesse segment the side of the dodecahedron inscribed in it but the whole is to the lesse segment in duple proportion of that in which it is to the greater by the 10. diffinitioÌ„ of the fifth For the whole the greater segmeÌ„t and the lesse are lines in continuall proportion by the 3. diffinition of the sixth Wherefore the whole namely the side of the cube is to the side of the dodecahedron inscribed in it namely to his lesse segment in duple propoâ—tion of an extreame and meane proportion ' namely of that which the whole hath â—o the greater segmenâ— by the 2. of the fourtenth ¶ The 18. Proposition The side of a Dodecahedron is to the side of a Cube inscribed in it in conuerse proportion of an extreame and meane proportion IT was proued in the 3. corollary of the 13. of the fiuetenth that the side of a Dodecahedâ—on circumscribed about a Cube is the greater segment of the side of the same Cube Wherefore the whole
same the cube containeth 12 namely is sesquialter to the pyramis Wherefore of what partes the cube containeth 12 of the same the whole Octohedron which is double to the pyramis ABDFC containeth 54. Which 54. hath to 12. quadruple sesquialter proportion Wherefore the whole Octohedron is to the cube inscribed in it in quadruple sesquialter proportion Wherefore we haue proued that an Octohedron geuen is quadruple sesquialter to a cube inscribed in it ¶ A Corollary An Octohedron is to a cube inscribed in it in that proportion that the squares of their sides are For by the 14. of this booke the side of the Octohedron is in power quadruple sesquialter to the side of the cube inscribed in it ¶ The 29. Proposition To proue that an octohedroÌ„ geueÌ„ is tredecuple sesquialter to a trilater equilater pyramis inscribed in it LEt the octohedron geâ—en be AB in which let there be inscribed a cube FCED by the 4. of the fiuetenth and in the cube let there be inscribed a pyramis FEGD by the â— of the fiuetenth And forasmuche as the angles of the pyramis are by the same first of the fiuetenth set in the angles of the cube and the angles of the cube are set in the centres of the bases of the Octohedron namely in the poyntes F E C D G by the 4. of the fiuetenth Wherfore the angles of the pyramis are set in the centres F C E D of the octohedron Wherefore the pyramis FEDG is inscribed in the octohedron by the 6. of the fiuetenth And forasmuche as the octohedron AB is to the cube FCED inscribed in it quadruple sesquialter by the former propositioÌ„ and the cube CDEF is to the pyramis FEDG inscribed in it triple by the 25. of booke wherefore three magnitudes being geuen namely the octohedron the cube and the pyramis the proportion of the extremes namely of the octohedron to the piramis is made of the proportions of the meanes namely of the octohedron to the cube and of the cube to the pyramis as it is easie to see by the declaration vpon the 10. diffinition of the fiueth Now then multiplying the quantities or denominations of the proportions namely of the octohedron to the cube which is 4 1 â— and of the cube to the pyramis which is 3 as was taught in the diffinition of the sixth there shal be produced 13 1 â— namely the proportion of the octohedron to the pyramis inscribed in it For 4 ½ multiplyed by 3. produce 13 ½ Wherefore the Octohedron is to the pyramis inscribed in it in tredecuple sesquialter proportion Wherefore we haue proued that an Octohedron is to a trilater equilater pyramis inscribed in it in tredecuple sesquialter proportion ¶ The 30. Proposition To proue that a trilater equilater Pyramis is noncuple to a cube inscribed in it SVppose that the pyramis geuen be ABCD whose two bases let be ABC and DBC and let their centres be the poynts G and I. And from the angle A draw vnto the base BC a perpendicular AE likewise from the angle D draw vnto the same base BC a perpendicular DE and they shal concurre in the section E by the 3. of the third and in them shal be the ceÌ„tres G and I by the corollary of the first of the third And forasmuch as the line AD is the side of the pyramis the same AD shall be the diameter of the base of the cube which coÌ„taineth the pyramis by the 1 of the fiueteÌ„th Draw the line GI And forasmuch as the line GI coupleth the centreâ— of the bases of the pyramis the saide line GI shal be the diameter of the base of the cube inscribed in the pyramis by the 18. of the fiuetenth And forasmuche as the line AG is double to the line GE by the corollarye of the twelueth of the thirtenth the whole line AE shal be triple to the line GE and so is also the line DE to the line IE Wherefore the lines AD and GI are parallels by the 2. of the sixth And therefore the triangles AED and GEI are likeâ— by the corollary of the same And forasmuch as the triangles AED and GEI are like the line ADâ— shal be triple to the line GI by the 4. of the sixth But the line AD is the diameter of the base of the cube circumscribed about the pyramis ABCD and the line GI is the diameter of the base of the cube inscribed in the pyramis ABCD but the diameters of the bases are equemultiplices to the sides namely are in power duple Wherfore the side of the cube circumscribed about the pyramis ABCD is triple to the side of the cube inscribed in the same piramis by the 15. of the fiueth but like cubes are in triple proportion the one to the other of that in which their sides are by the 33. of the eleuenth and the sides are in triple proportion the one to the other Wherfore triple taken thre times bringeth forth twenty seuencuple which is 27. to 1 for the 4. termes 27.9.3.1 being set in triple proportion the proportion of the first to the fourth namely of 27. to 1. shal be triple to the proportion of the first to the second namely of 27. to 9 by the 10. diffinition of the fiueth which proportion of 27. to 1. is the proportioÌ„ of the sides tripled which proportioÌ„ also is found in like solides Wherefore of what partes the cube circumscribed containeth 27. of the same the cube inscribed containeth one but of what partes the cube circumscribed containeth 27. of the same the pyramis inscribed in it containeth 9. by the 25. of this booke wherfore of what partes the pyramis AB CD containeth 9. of the same the cube inscribed in the pyramis containeth one Wherefore we haue proued that a trilater and equilater pyramis is nonâ—cuple to a cube inscribed in it ¶ The 31. Proposition An Octohedron hath to an Icosohedron inscribed in it that proportion which two bases of the Octohedron haue to fiue bases of the Icosahedron SVppose that the octohedron geuen be ABCD and let the Icosahedron inscribed in it be FGHMKLIO Then I say that the octohedron is to the Icosahedron as two bases of the octohedron are to fiue bases of the Icosahedron For forasmuche as the solide of the octohedron consisteth of eight pyramids set vpon the bases of the octohedron and hauing to theyr altitude a perpendicular line drawne from the centre to the base let that perpendicular be ER or ES being drawne from the centre E which centre is common to either of the solides by the corollary of the 21. of the fiuetenth to the centres of the bases namely to the poyntes R and S. Wherefore for that thre pyramids are equal and like they shal be equal to a prisme set vpon the selfe same base and vnder the selfe same altitude by the corollary of the seuenth of the twelueth But
a right line coupling their centres being diuided by an extreame and meane proportion maketh the greater segment the right line which coupleth the centres of the next bases If by the centres of fiue bases set vppon one base be drawne a playne superficies and by the centres of the bases which are set vpon the opposite base be drawne also a playne superficies and then be drawne a right line coupling the centres of the opposite bases that right line is so cut that eche of his partes set without the playne superficies is the greater segment of that part which is contayned betwene the playnes The side of the dodecahedron is the greater segment of the line which subtendeth the angle of the pentagon A perpendicular line drawne from the centre of the dodecahedron to one of the bases is in power quintuple to half the line which is betwene the playnes And therfore the whole line which coupleth the centres of the opposite bases is in power quintuple to the whole line which is betwene the sayd playnes The line which subtâ—deth the angle of the base of the dodecahedroÌ„ together with the side of the base are in power quintuple to the line which is drawne from the ceÌ„tre of the circle which contayneth the base to the circumference A section of a sphere contayning three bases of the dodecahedron taketh a third part of the diameter of the sayd sphere The side of the dodecahedron and the line which subtendeth the angle of the pentagon are equall to the right line which coupleth the middle sections of the opposite sides of the dodecahedron ¶ The ende of the Elementes of Geometrie of the most auncient Philosopher 〈◊〉 of Megara The intent of this Preface Number Note the worde Vnit to expresse the Greke Monaâ— not Vnitie as we hauâ— all commonly till now vsed Magnitude A point A Line Magnitude Ano. 1488. ☞ Arithmetike Note * Anno. 1550. R. B. Note 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ☞ This noble Earle dyed Anno. 1554. skarse of 24. yeares of ageâ— hauing no issue by his wife Daughter to the Duke of Somerset Iustice. ☞ * Plato 7. de Rep. ☞ * Note The difference betwene Strataruhmetrie and Tacticie I.D. Fâ—endâ—â— you will finde it hard to performâ— my descriptiâ—n of â—his Fâ—ate But by Châ—râ—graphieâ— you may helpe your selfe some â—hat wherâ— thâ— Figures knowne in Sidâ—â—â—nd Angles are not Regular And whereâ— Resolution into Triangles can sâ—â—uâ— c. And yet you will finde it strange to deale thus generâ—lly with Arithmeticall figures and that for Battayle â—ay Their coâ—tentâ—â—â— differ so much from like Geometrâ—call Figurâ—s A marueilous Glasse ☞ S.W.P. ☞ Note 1. 2. 3. 5. 6. 7. 8. I.D. Read in Aristotle his 8. booke of Politikes the 5 6 and 7. chapters Where you shall haue some occasion farder to thinke of Musike than commonly is thought ☜ * Anno. 1548 and 1549. in Louayn Note I.D. The Cutting of a Sphare according to any proportion assigned may by this proposition be done Mechanically by tempering Lâ—quor to a certayne waight in respect of the waight of the Sphare 〈◊〉 Swyâ—â—â—ng A common errorâ— noted A paradox N. T. The wonderfull vse of these Propositions The practise Staticall to know the proportion betwene the Cube and the Sphare I. D. * For so haue you 256. partes of a Graine * The proportion of the Square to the Circle insâ—ribed * The Squaâ—ing of the Cirâ—le Mâ—â—haniâ—ally * To any Squirâ— gâ—uenâ— to 〈…〉 Note Squaring of the Circle without knowledge of the proportion betwene Circumference and Diameter To Dubble the Cube redily by Art Mechanicall depending vppon Demonstration Mathematicall I. D. The 4. sides of this Pyramiâ— must be 4. Isosceles Triangles â— likâ— and â—quall I. D. * In all workâ—ngeâ— with this Pyramis or Cone Let their Situationâ— be in all Pointeâ— and Conditions a like oâ— all one while you are about â—ne worke Els you will 〈◊〉 I. D. * Consider well whan you must put your wateâ—â— togyther and whan you must empty youâ— first waterâ— out of your Pyramiâ— or Cone Elâ— you will 〈◊〉 * Vitruuius Lib. 9. Cap. 3. ☞ God bâ— thanked â—or this Inuentionâ— the fruiâ—e â—nsuing * Note Note as concerning the Sphaericall Superâ—icies of the water ☞ * Note Note this Abridgeâ—ent of Dubbling 〈◊〉 Cube â—â— * Note * ☜ To giue Cubes one to the other in any proportion Rationall or Irrationall * Emptying the first The demonstrations of this Dubbling of the Cube and oâ— the rest I.D. * Here 〈…〉 of the water * By the 33. of the eleuenth books of Euclide I.D. * And your diligence in praâ—â—ise can â—o in waight of wateâ—â— peâ—forme it Therefore now you arâ— able to â—eue good reason of your whole doing * Note this Corollary * The great Commodities following of these new Inuentions * ☞ Such is the Fruite of the Mathematicall Sciences and Artes. MAN is the Lesse World. * ☜ Microcosmus * Lib. 3. Cap. 1. ☞ Saw Milles. * Atheneus Lib. 5. cap. 8. Proclus Pag. 18. To go to the bottom of the Sea without daunger Plutâ—â—â—bus in Marco Mâ—rcello Syâ—asius in Epistolis Polybius Plinius Quintâ—lianus T. Liuius * Athenaâ—s * Galeâ—us Anthemius Burning Glasses Gunnes 4. Reg. 20. A perpetuall Motion An obiection The Answer ☜ A Mathematicien Vitrunius VVho is an Architect * The Immaterialitie of perfect Architecture What Lineament is Note Anno. 1559. * Anno. 1551 De his quae Mundo mirabiliter eueniunt cap. 8. Tusc. â— * ☞ A Digression Apologeticall * A prouerb Fayre fisht and caught a Frog ☞ Psal. 140. Act. 7. C. Lib. 30. Cap. 1. ☜ R. B. ☞ Vniuersities ☜ ☞ The Ground platt of this Praeface in a Table The argumâ—â—â— of the first Booke Definition of a poynt Definition of a poynt after Pithagoras Definition of a liâ—â— An other definition of a line An other The endes of a line Difference of a point frâ—â—nity Vnitie is a part of number A poynt is no part of quantitie Definition of a right line DefinitioÌ„ therof after Plato An other definition An other An other An other An other VVhy Euclide here defineth not a crooked lyne Definition of a superficies A superficies may be deuided two wayes An other definition of a superficies The extremes of a superficies Another definition of a superficies Definition of a plaine superficies Another definition of a playne superficies NOTE Another definition of a playne superficies An other definition An other definition An other definition Definition of a playne angle Definition of a â—ecâ—ilined angle 〈◊〉 of angles VVhat a right angle VVhat also a perpendicular lyne iâ— VVhat an obtuse angle â—â— VVhat an acute angle is The limite of any thing No science of thinges infinite Definition of a figure Definition of a circle A circle the most perfect of all figures The centre of a circle Definition of a diameter Definition of a semicircle Definition of a section of a circle Definition of râ—â—â—â—lined figures
shall in like sort by the 14. of the tenth be in power more then the line DF by the square of a line incommensurable in length to the line CF and then if the line AE be commensurable in length to the rationall line the line CF shall also in like sort be commensurable in length to the same rationall line and so either of the lines AB and CD is a fourth residuall line And if the line BE be coÌ„meÌ„surable in leÌ„gth to the rationall line the line DF shall also be coÌ„mensurable in leÌ„gth to the same line and so either of the lines AB CD is a â—iâ—t residuall line And if neither of the lines AE nor BE be commensurable in length to the rationall line in like sort neither of the lines CF nor DF shall be coÌ„mensurable in leÌ„gth to the same rational line And so either of the lines AB CD is a sixt residual line Whereâ—ore the line CD is a residuall line of the selfe same order that the line AB is A line therfore commensurable in length to a residuall line is it selfe also a residuall line of the selfe same order which was required to be proued As before touching binomiall lines so also touching residuall lines this is to be noted that a line commensurable in length to a residuall line is alwayes a residuall line of the selfe same order that the residuall line is vnto whom it is coÌ„mensurable as hath before in this 103. propositioÌ„ bene proued But if a line be coÌ„mensurable in power only to a residuall lineâ— then followeth it not yea it is impossible that that line should be a residuall of the self same order that the residual line is vnto whom it is commensurable in power onely Howbeit those two lines shall of necessitie be both either of the three first orders of residâ—â—ll lines or of the three last orders which is not hard to proue if ye marke diligently the former demonstration and that which was spoken of binomiall lines as touching this matter ¶ The 80. Theoreme The 104. Proposition A line commensurable to a mediall residuall line is it selfe also a medial residuall line and of the selfe same order SVppose that AB be a mediall residuall line vnto whome let the line CD be commensurable in length and in power or in power onely Then I say that CD is also a mediall residuall line and of the selfe same order For forasmuch as the line AB is a mediall residuall line let the line conueniently ioyned vnto i◠〈◊〉 BE wherefore the lines AE and BE are mediall commensurable in power onely As AB is to CD so by the 22. of the sixth let BE be to DF. And in like sort as in the former so also in this may we proue that the line AE is commensurable in length and in power or in power onely vnto the line CF the line BE 〈◊〉 the line DF. Wherefore by the 23. of the tenth 〈◊〉 line CF is a mediall line and the line DF is also a mediall line for that it is commensurable to the mediall line BE. And in like sort the lines CF and DF are commensurable in power onely for that they haue the selfe same proportioÌ„ the one to the other that the lines AE and EB haue which are commensurable in power onely Wherefore the line CD is a mediall residuall line I say moreouer that it is of the selfe same order that the line AB is For for that as the line AE is to the line BE so is the line CF to the line DF. But as the line AE is to the line BE so is the square of the line AE to the parallelogramme contayned vnder the lines AE and BE by the first of the sixth and as the line CF is to the line DF so is the square of the line CF to the parallelogramme contayned vnder the lines CF and DF. Wherefore as the square of the line AE is to the parallelogramme contayned vnder the lines AE and BE so is the square of the line CF to the parallelogramme contayned vnder the lines CF and DF. Wherefore alternately as the square of the line AE is to the square of the line CF so is the parallelogramme contayned vnder the lines AE and BE to the parallelogramme contained vnder the â—ines CF and DF. But the square of the line AE is commensurable to the square of the line CF for the line AE is commensurable to the line CF Wherefore also the parallelogramme contayned vnder the lines AE and BE is commensurable to the parallelogramme contayned vnder the lines CF and DF. Wherefore if the parallelogramme contayned vnder the lines AE and EB be rationall the parallelogramme also contayned vnder the lines CF and FD shall be rationall And then either of the lines AB and CD is a first mediall residuall line But if the parallelogramme contayned vnder the lines AE and BE be mediall the parallelogramme also contayned vnder the lines CF and FD shall be also mediall by the corollary of the 23. of the teÌ„th and so either of the lines AB and CD is a second mediall residuall line Wherefore the line CD is a mediall residuall line of the selfe same order that the line AB is A line therefore commensurable to a mediall residuall line is it selfe also a mediall residuall line of the selfe same order which was required to be demonstrated This Theoreme is vnderstanded generally that whether a line be commensurable in length in power or in power onely to a mediall residuall line it is it selfe also a mediall residuall line and of the selfe same order which thing also is to be vnderstanded of the three Theoremes which follow An other demonstration after Campane Suppose that A be a mediall residuall line vnto whome let the line B be commensurable in length or in power onely And take a rationall line CD vnto which apply the parallelogramme CE equall to the square of the line A and vnto the line FE which is equall to the line CD apply the parallelogramme Fâ— equall to the square of the line B. Now then the parallelogrammes CE and FG shall be commensurable for that the lines A B are commensurable in power wherefore by the 1. of the sixth and 10. of this booke thâ— lines DE and FG are commensurable in length Now then if A be a first mediall residuall line then is the line DE a second residuall line by the 98. of this booke and if the line A be a sâ—cond mediall residuall line then is the line â— â— a third residuall line by the 99. of this booke But if DE be a second residuall line Gâ— also shall be a second residuall line by the â—03 of this boke And if DE be a third residuall line GE also shall by the same be also a third residuall line Wherefore it followeth by the 9â— and 93. of this booke that B is either a first
medial residuall line or a second mediall residuall line according as the line A is supposed to be which was required to be proued ¶ The 81. Theoreme The 105. Proposition A line commensurable to a lesse line is it selfe also a lesse line SVppose that AB be a lesse line vnto whom let the line CD be commensurable Then I say that the line CD is also a lesse line For let the same construction be in this that was in the former Propositions And forasmuch as the lines AE and EB are incommensurable in power therefore by the 22. of the sixt and 10. of the tenth the lines CF FD are incommensurable in power Againe by the 22. of the sixt as the square of the line AE is to the square of the line BE so is the square of the line CF to the square of the line DF. Wherfore by coÌ„position as the squares of the lines AE and BE are to the square of the line BE so are the squares of the lines CF and DF to the square of the line DF and alternately as the squares of the lines AE and BE are to the squares of the lines CF and DF so is the square of the line BE to the square of the line DF. But the square of the line BE is commensurable to the square of the line DF for the lines BE and DF are commensurable Wherefore that which is made of the squares of the lines AE and BE added together is commensurable to that which is made of the squares of the lines CF and DF added together But that which is made of the squares of the lines AE and BE added together is rationall Wherefore that which is made of the squares of the lines CF and DF added together is also rationall Againe for that as the square of the line AE is to the parallelogramme contained vnder the lines AE and BE so is the square of the line CF to the parallelogramme contained vnder the lines CF and DF as we declared in the Proposition next going before therefore alternately as the square of the line AE is to the square of the line CF so is the parallelogramme contained vnder the lines AE and BE to the parallelogramme contained vnder the lines CF and DF. But the square of the line AE is commensurable to the square of the line CF for the lines AE CF are commensurable Wherefore the parallelogramme contained vnder the lines AE and BE is commensurable to the parallelogramme contained vnder the lines CF and DF. But the parallelogramme contained vnder the lines AE and BE is mediall Wherefore the parallelogramme contained vnder the lines CF and DF is also mediall Wherefore the lines CF and DF are incommensurable in power hauing that which is made of their squares added together rationall and the parallelogramme contained vnder them mediall Wherefore the line CD is a lesse line A line therefore commensurable to a lesse line is it selfe also a lesse line which was required to be proued An other demonstration Suppose that A be a lesse line and vnto A let the line B be commensurable whether in length and power or in power onely Then I say that B is a lesse line Take a rationall line CD And vnto the line CD apply by the 44 of the first the parallelogramme CE equall to the square of the line A and making in bredth the line CF. Wherefore by the 100. proposition the line CF is a fourth residuall line Vnto the line FE apply by the same the parallelogramme EH equall to the square of the line B and making in breadth the line FH Now forasmuch as the line A is commensurable to the line B therefore also the square of the line A is coÌ„mensurable to the square of the line B. But vnto the square of the line A is equall the parallelogramme CE vnto the square of the line B is equal the parallelogramme EH Wherfore the parallelogramme CE is commensurable to the parallelogramme EH But as the parallelogramme CE is to the parallelogramme EH so is the line CF to the line FH Wherfore the line CF is commensurable in length to the line FH But the line CF is a fourth residuall line Wherfore the line FH is also a fourth residuall line by the 103. of the tenth and the line FE is rationall But if a superficies be contained vnder a rationall line and a fourth residuall lyne the line that containeth in power that superficies is by the 94. of the tenth a lesse lyne But the line B containeth in power the superficies EH Wherfore the line B is a lesse line which was required to be proued ¶ The 82. Theoreme The 106. Proposition A line commensurable to a lyne making with a rationall superficies the whole superficies mediall is it selfe also a lyne making with a rationall superficies the whole superficies mediall SVppose that AB be a line making with a rationall superficies the whole superficies mediall vnto whom let the line CD be commensurable Then I say that the line CD is a line making with a rationall superficies the whole superficies mediall Vnto the line AB let the line conueniently ioyned be BE. Wherefore the lines AE and EB are incommensurable in power hauing that which is made of their squares added together mediall and the parallelogramme contained vnder them rationall Let the construction be in this as it was in the former propositions And in like sort may we proue that as the line AE is to the line BE so is the line CF to the line DF and that that which is made of the squares of the lines AE and BE added together is commensurable to that which is made of the squares of the lines CF and DF added together and that that which is contained vnder the lynes AE and EB is in like sort commensurable to that which is contained vnder the lines CF and DF. Wherfore also the lines CF and DF are commensurable in power hauing that which is made of their squares added together mediall and that which is contained vnder them rationall Wherfore the line CD is a lyne making with a rationall superficies the whole superficies mediall Wherfore a line commensurable to a line making with a rationall superficies the whole superficies mediall is it selfe also a lyne making with a rational superficies the whole superficies mediall which was required to be demonstrated An other demonstration Suppose that A be a line making with a rationall superficies the whole superficies mediall and vnto it let the lyne B be commensurable either in length and in power or in power onâ—ly Then I say that B is a lyne making with a rationall superficies the whole superficies mediall Take a rational line CD and vnto the line CD apply the parallelogramme CE equal to the square of the line A and makyng in breadth the lyne GF Wherfore by the 101. proposition the lyne CF is a fift residuall lyne