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.
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
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
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
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
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 ¶
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
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
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
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
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
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
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
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