namely the part greater then the whole which is impossible Wherefore the circumference shall not cut the line CD Now I say that it shall not passe aboue the line CD and not touch it in the point D. For if it be possible let it passe aboue it and extend the line CD to the circuÌference and let it cut it in the point â And draw the lines FB and FA and it shall followe as before that the line CD is greater then the line CF which is impossible Wherefore that is manifest which was required to be proued ¶ Second Assumpt If there be a right angle vnto which a base is subtended and if vpon the same be described a semicircle the circumference thereof shall passe by the point of the right angle The conuerse of this was added after the demonstration of the 31. of the third out of Pelitarius And these two Assumptes of Campane are necessary for the better vnderstanding of the demonstration of the secoâd part of this 13. Proposition wherein is proued that the pyramis is contained in the Sphere geuen ¶ Certaine Corollaryes added by Flussas First Corollary The diameter of the Sphere is in power quadruple sesquialtera to the line which is drawen from the centre to the circumference of the circle which containeth the base of the pyramis For forasmuch as it hath bene proued that the diameter KL is in power sesquialter to the side EF and it is proued also by the 12. of this booke that the side EF is in power triple to the line EH which is drawen from the centre of the circle contayning the triangle EFG But the proportion of the extremes namely of the diameter to the line EH consisteth of the proportions of the meanes namely of the proportion of the diameter to the line EF and of the proportion of the line EF to the line EH by the 5. definition of the sixt which proportions namely triple and sesquialter added together make quadruple sesquialter as it is easie to proue by that which was taught in the declaration of the 5. definition of the sixt booke Wherefore the Corollary is manifest ¶ Second Corollary Onely the line which is drawen from the angle of the pyramis to the base opposite vnto it passing by the center of the Sphere is perpendicular to the base and falleth vpon the centre of the circle which containeth the base For if any other line then the line KMH which is drawen by the centre of the Sphere to the centre of the circle should fall perpendicularly vpon the plaine of the base then from one and the selfe same point should be drawen to one and the selfe same plaine two perpendicular lines contrary to the 13. of the eleuenth which is impossible Farther if from the toppe K should be drawen to the center of the base namely to the point H any other right line not passing by the centre M two right liues shoulde include a superficies contrary to the last common sentence which were absurde Wherefore onely the line which is drawen by the center of the Sphere to the centre of the base is perpendicular to the sayd base And the line which is drawen from the angle perpendicularly to the base shall passe by the centre of the Sphere Third Corollary The perpendicular line which is drawen from the centre of the Sphere to the base of the pyramis is equall to the sixt part of the diameter of the Sphere For it is before proued that the line MH which is drawen from the centre of the Sphere to the centre of the base is equall to the line NC which line NC is the sixt part of the diameter AB and therfore the line MH is the sixt part of the diameter of the Sphere For the diameter AB is equall to the diameter of the Sphere as hath also before bene proued ¶ The 2. Probleme The 14. Proposition To make an octohedron and to coÌprehend it in the sphere geuen namely that wherein the pyramis was comprehended and to proue that the diameter of the sphere is in power double to the side of the octohedron TAke the diameter of the former sphere geuen which let be the line AB and diuide it by the 10. of the first into two equall partes in the point C. And describe vpon the line AB a semicircle ADB And by the 11. of the first froÌ the point C rayse vp vnto the line AB a perpendicular line CD And draw a right line from the point D to the point B. And describe a square EFGH hauing euery one of his sides equall to the line BD. and draw the diagonal lines FH EG cutting the one the other in the point K. And by the 12. of the eleuenth from the point K namely the point where the lines FH and EG cut the one the other rayse vp to the playne superficies wherein the square EFGH is a perpendicular line KL and extend the line KL on the other side of the playne superficies to the point M. And let eche of the lines KL and KM be put equall to one of these lines KE KF KH or KG And draw these right lines LE LF LG LH ME MF MG and MH Now forasmuch as the line KE is by the corollary of the 34. of the first equall to the line KH and the angle EKH is a right angle therefore the square of HE is double to the square of EK by the 47. of the first Agayne forasmuch as the line LK is equall to the line KE by position and the angle LKE is by the second diffinition of the eleuenth a right angle therefore the square of the line EL is double to the square of the line EK and it is proued that the square of the line HE is double to the square of the line EK Wherefore the square of the line LE is equall to the square of the line EH Wherefore also the line LE is equall to the line EH And by the same reason the line LH is also equall to the line HE. Wherefore the triangle LHE is equilater In like sort may we proue that euery one of the rest of the triangles whose bases are the sides of the square EFGH and toppes the pointes L and M are equilater The sayd eight triangles also are equall the one to the other for euery side of eche is equall to the side of the square EFGH Wherfore there is made an octohedron coÌtained vnder eight triangles whose sides are equall Now it is required to comprehend it in the sphere geuen and to proue that the diameter of the sphere is in power double to the side of the octohedron Forasmuch as these three lines LK KM and KE are equall the one to the other therfore a semicircle described vpon the line LM shall passe also by the point E. And by the same reason if the semicircle be turned round about vntill it returne vnto the selfe same
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 beâng quadrilater and equilater and in one and the selfe same playne is a square by the diââinition 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 ceâtres 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 ãâã ABâDâ euery one of whose sâdes ãâã be diuidâd into two equall partes in the poyâââââ M K L P N. And iâ euery one of the bâses of that pyramis descride the trianglââ Lââ PMN NKL and ãâ¦ã which triangles shall be equilater by the 4. of the firât âor the sides subâend 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 âiuided by an extreame and meane proporââon 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 fourteÌth and therfore they make the liâe sectioÌs equall by the â part of the ninth of the fiueth Now I say that the foresayd poynâes doâ recâaue the angles of the Icosahedron inscribed in the pyramis ABâD 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 subâââd 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 Icosahâdronâ 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 âPNK is a square cutting the pyramis ABâD into two equall paââââ 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 nuÌber shal be set 12. angles of the IcosahedroÌ coÌtained 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 ABâD by the first diââânition 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 âche 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 coÌcentricall 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
proportion which is in power duple to EF the side of the OctohedroÌ inscribed in the Dodecahedron Draw the diameters EL and FK of the Octohedron Now they couple the midle sections of the opposite sides of the dodecahedron AB and GD by the 9. of the fiuetenth 3. corollary of the 17. of the thirteÌth euery one of those diameters being diuided by an extreame and meane proportion doo make the lesse segment the side of the dodecahedron by the 4. corollary of the same Wherefore the side AB is the lesse segment of the line FK But the line FK contayneth in power the two equall lines EF EK by the 47. of the first for the angle FEK is a right angle of the square FEKL of the Octohedron Wherfore the line FK is in power duple to the line EF. Wherefore the line AB the side of the dodecahedron is the lesse segment of the line FK which is in power duple to EF the sidâ of the Octohedron The side therefore of a Dodecahedron iâ the lesse segment of that right line which is in power duple to the side of the Octohedron inscribed in the same Dodâcahedron ¶ The 22. Proposition The diameter of an Icosahedron is in power sesquitertia to the side of the same Icosahedron and also is in power sesquialter to the side of the Pyramis inscribed in the Icosahedron FOr forasmuch as it hath bene proued by the 10. of this booke that if froÌ the power of the diameter of the IcosahedroÌ be taken away the triple of the power of the side of the cube inscribed in it there shal be left a square sesquitertia to the square of the side of the Icosahedron But the power of the side of the cube tripled is the diameter of the same cube by the 15. of the thirteÌth And the cube the pyramis inscribed in it are contained in one the self same sphere by the first of this booke and in one the self same Icosahedron by the corollary of the same Wherfore one and the selfe same diameter of the cube or of the sphere which coÌtaineth the cube and the pyramis is in power sesquialter to the side of the pyramis by the 13. of the thirtenth Wherfore it followeth that if from the diameter of the Icosahedron be taken away the triple power of the side of the cube or the sesquialter power of the side of the pyramis which are the powers of one and the selfe same diameter there shall be left the sesquitertia power of the side of the Icosahedron The diameter therefore of an Icosahedron is in power sesquitertia to the side of the same Icosahedron and also is in power sesquialter to the side of the Pyramis inscribed in the Icosahedron The 23. Proposition The side of a Dodecahedron is to the side of an Icosahedron inscribed in it as the lesse segment of the perpendicular of the PentagoÌ is to that line which is drawne from the centre to the side of the same pentagon LEt there be taken a Dodecahedron ABGDFSO Whose side let be AS or SO and let the Icosahedron inscribed in it be KLNMNE whose side let be KL From the two angles of the pentagonâ BAS and FAS of the Dodecahedron namely from the angleââ and F let there be drawne to the common base AS perpendicular lines BC FC which shal passe by the centres K L of the sayd pentagons by the corollary of the 10. of the thirteÌth Draw the lines BF and RO. Now forasmuche as the line RO subtendeth the angle OFR of thâ pentagon of the dodecahedron it shall cut the line FC by an extreme and meane proportion by the 3. of this booke let it cut it in the poynt I. And forasmuche as the line KL is the side of the Icosahedron inscribed in the Dodecahedron it coupleth the ceÌtres of the bases of the dodecahedron for the angles of the Icosahedron are set in the centres of the bases of the dodecahedron by the 7. of the fiuetenth Now I say that SO the side of the dodecahedron is to KL the side of the Icosahedron as the lesse segment IF of the perpendicular line CF is to the line LC which is drawne from the centre L to AS the side of the pentagon For forasmuch as in the triangle BCF the two sides CB and CF are in the centres L and K cut like proportionally the lines BF and KL shal be parellels by the 2. of the sixth Wherefore the triangles BCF and KCL shall be equiangle by the corollary of the same Wherfore as the line CL is to the line KLâ so is the line CF to the line BF by the 4. of the sixth But CF maketh the lesse segment the line IF by the 3. of this booke and the linâ BF maketh the lesse segment the line SO namely the side of the Dodecahedron by the 2. corollary of the 13. of the fiuetenth For the line BF which coupleth the angles B and F of the bases of the dodecahedron is equall to the side of the cube which contayneth the dodecahedron by the .13 of the fiuetenth Wherefore as the whole line Câ is to the whole line BF so is the lesse segment IF to the lesse segment SO by the 2. of the 14 But as the line CF is to the line BF so is the line CL proued to be to the line KL Wherefore as the line IF is to the line SO so is the line CL to the line KL Wherefore alternately by the 16. of the fiueth as the line IF the lesse segment of the perpendicular of the pentagon FAS is to the line LC which is drawne from the centre of the pentagon to the base so is the line SO the side of the Dodecahedron to thâ line KL the side of the Icosahedron inscribed in it The side therfore of a Dodecahedron is to the side of an Icosahedron inscribed in it as the lesse segment of the perpendicular of the pentagon is to that line which is drawne from the cenâre to the side of the same pentagon ¶ The 24. Proposition If halfe of the side of an Icosahedron be deuided by an extreme meane proportion and if the lesse segment thereof be taken away from the whole side and againe from the residue be taken away the third part that which remaineth shall be equal to the side of the Dodecahedron inscribed in the same Icosahedron SVppose that ABGDF be a pentagon containing fiue sides of the Icosahedron by the 16. of the thirtenth and let it be inscribed in a circle whose centre let be the point E. And vpon the sides of the pentagon let there be reared vp triangles making a solide angle of the Icosahedron at the poynt I by the 16. of the thirtenth And in the circle ABD inscribe an equilater triangle AHK From the centre E drawe to HK the side of the triangle and GD the side of the pentagon a perpendicular line which
same is treble it is manifest by the 12. of the fiueth that all the Prismes are to all the Pyramids treble Wherefore Parallelipipedons are treble to Pyramids set vpon the selfe same base with them and vnder the same altitude for that they contayne two Prismes Third Corollary If two Prismes being vnder one and the selfe same altitude haue to their bases either both triangles or both parallelogrammes the Prismes are the one to the other as their bases are For forasmuch as those Prismes are equemultiqlices vnto the Pyramâds vpon the selfe same bases and vnder the same altitude which Pyramids are in proportion as their bases it is manifest by the 15. of the fift that the Prismes are in the proportion of the bases For by the former Corollary the Prismes are treble to the Pyramids sât vpon the triangular bases Fourth Corollary Prismes are in sesquealtera proportion to Pyramids which haue the selfe same quadrangled base that the Prismes haue and are vnder the selfe same altitude For that Pyramis contayneth two Pyramids set vpon a triangular base of the same Prisme for it is proued that that Prisme is treble to the Pyramis which is set vpon the halfe of his quadrangled base vnto which the other which is set vpon the whole base is double by the sixth of this booke Fiueth Corollary Wherefore we may in like sort conclude that solides mencioned in the second Corollary which solids Campane calleth sided Columnes being vnder one and the selfe same altitude are in proportion the one to the other as their bases which are poligonon figures For they are in the proportion of the Pyramids or Prismes set vpon the selfe same bases and vnder the selfe same altitude that is they are in the proportioÌ of the bases of the sayde Pyramids or Prismes For those solids may be deuided into Prismes hauing the selfe same altitude when as their opposite bases may be deuided into triangles by the 20 of the sixth Vpon which triangles Prismes beyng set are in proportion as their bases By this 7. Proposition it playnely appeareth that âuâlide as it was before noted in the diffinitionââ vnder the diffinition of a Prisme comprehended also those kinds of solids which Campane calleth sided Columnes For in that he sayth Euery Prisme hauing a triangle to his base may be deuidedâ c. he neded not taking a Prisme in that sense which Campane and most men take it to haue added that particle hauing to his base a triangle For by their sense there is no Prisme but it may haue to his base a triangleâ and so it may seeme that Euclide ought without exceptioâ haue sayd that euery prisme whatsoeuer may be deuided into three pyramids equall the one to the other hauing also triangles to âheir bases For so do Campane and Flussas put the proposition leauing out the former particle hauing to his base a triangle which yet is red in the Greeke copye not leât out by any other interpreters knowne abroade except by Campane and Flussas Yea and the Corollary following of this proposition added by Theon or Euclide and ameÌded by M. Dee semeth to confirme this sence Of this âs ãâã made manifest that euery pyramis is the third part of the prisme hauing the same base with it and equall altitude For and if the base of the prisme haue any other right lined figure then a triangle and also the superficies opposite to the base the same figure that prisme may be deuided into prismes hauing triangled bases and the superficieces to those bases opposite also triangled a ââike and equally For there as we see are put these wordes âor and if the base of the prisme be any other right lined figureâ c. whereof a man may well inferre that the base may be any other rectiline figure whatsoeuer not only a triangle or a parallelogramme and it is true also in that sence as it is plaine to see by the second corollary added out of Flussas which corollary as also the first of his corollaries is in a maner all one with the Corollary added by Theon or Euclide Farther Theon in the demonstration of the 10. proposition of this booke as we shall aâterward see most playnely calleth not onely sided columnes prismes but also parallelipipedons And although the 40. proposition of the eleuenth booke may seme hereunto to be a lât For that it can be vnderstanded of those prismes onely which haue triangles to their like equall opposite and parallel sides or but of some sided columnes and not of all yet may that let be thus remoued away to say that Euclide in that propositioÌ vsed genus pro specie that is the generall word for some special kinde therof which thing also is not rare not only with him but also with other learned philosophers Thus much I thought good by the way to note in farther defence of Euclide definition of a Prisme The 8. Theoreme The 8. Proposition Pyramids being like hauing triangles to their bases are in treble proportion the one to the other of that in which their sides of like proportion are SVppose that these pyramids whose bases are the triangles GBC and HEF and toppes the poyntes A and D be like and in like sort described and let AB and DE be sides of like proportion Then I say that the pyramis ABCG is to the pyramis DEFH in treble proportioÌ of that in which the side AB is to the side DE. Make perfect the parallelipipedons namely the solides BCKL EFXO And forasmuch as the pyramis ABCG is like to the pyramis DEFH therfore the angle ABC is equall to the angle DEF the angle GBC to the angle HEF and moreouer the angle ABG to the angle DEH and as the line AB is to the line DE so is the line BC to the line EF and the line BG to the line EH And for that as the line AB is to the line DE so is the line BC to the line EF and the sides about the equall angles are proportionall therefore the parallelogramme BM is like to the parallelograÌme EP and by the same reason the parallelogramme BN is like to the parallelogramme ER and the parellelogramme BK is like vnto the parallelogramme EX Wherefore the three parallelogrammes BM KB and BN are like to the three parallelogrammes EP EX and ER. But the three parallelogrammes BM KB and BN are equall and like to the three opposite parallelogrammes and the three parallelogrammes EP EX and ER are equall and like to the three opposite parallelogrammes Wherefore the parallelipipedons BCKL and EFXO are comprehended vnder playne superficieces like and equall in multitude Wherefore the solide BCKL is like to the solide EFXO But like parallelipipedons are by the 33. of the eleuenth in treble proportion the one to the other of that in which side of like proportion is to side of like proportion Wherefore the solide BCKL is to the solide EFXO in treble
FD. Wherfore cones cylinders consisting vpon equal bases are in proportion the one to the other as their altitudes which was required to be demonstrated ¶ The 15. Theoreme The 15. Proposition In equall Cones and Cylinders the bases are reciprokall to their altitudes And cones and Cylinders whose bases are reciprokall to their altitudes are equall the one to the other SVppose that these cones ACL EGN or these cylinders AX EO whose bases are the circles ABCD EFGH and axes KL and MN which axes are also the altitudes of the cones cylinders be equall the one to the other TheÌ I say that the bases of the cylinders XA EO are reciprokal to their altitudes that is that as the base ABCD is to the base EFGH so the altitude MN to the altitude KL For the altitude KL is either equall to the altitude MN or not First let it be equall But the cylinder AX is equal to the cylinder EQ But cones and cylinders consisting vnder one and the selfe same altitude are in proportion the one to the other as their bases are by the 11. of the twelueth Wherfore the base ABCD is equall to the base EFGH Wherefore also they are reciprokal as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL But now suppose that the altitude LK be not equall to the altitude M. N but let the altitude MN be greater And by the 3. of the first from the altitude MN take away PM equall to the altitude KL so that let the line PM be put equal to the line KL And by the point P let there be extended a playne superâicies TVS which let cut the cylinder EO and be a parallell to the two opposite playne superâicieces that is to the circles EFGH and RO. And making the base the circle EFGH the altitude MP imagine a cylinder ES. And for that the cylinder AX is equall to the cylinder EO and there is an other cylinder ES therfore by the 7. of the fift as the cylinder AX is to the cylinder ES so is the cylinder EO to the cylinder ES. But as the cylinder AX is to the cylinder ES so is the base ABCD to the base EFGH For the cylinders AX and ES are vnder one and the selfe same altitude And as the cylinder EO is to the cylinder ES so is the altitude MN to the altitude MP For cylinders coÌsisting vpoÌ equall bases are in proportion the one to the other as their altitudes Wherfore as the base ABCD is âo the base EFGH so is the altitude MN to the altitude MP But the altitude PM is equall to the altitude KL Wherefore as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL Wherefore in the equall cylinders AX and EO the bases are reciprokall to their altitudes But now suppose that the bases of the cylinders AX and EO be reciprokal to their altitudes that is as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL Then I say that the cylinder AX is equall to the cylinder EO For the selfe same order of constructioÌ remayning for that as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL but the altitude KL is equall to the altitude PM Wherefore as the base ABCD is to the base EFGH so is the altitude MN to the altitude PM But as the base ABCD is to the base EFGH so is the cylinder AX to the cylinder ES for they are vnder equall altitudes and as the altitude MN is to the altitude PM so is the cylinder EO to the cylinder ES by the 14. of the twelueth Wherefore also as the cylinder AX is to the cylinder ES so is the cylinder EO to the cylinder ES. Wherefore the cylinder AX is equall to the cylinder EO by the 9. of the fift And so also is it in the cones which haââ the selfe same bases and altitudes with the cylinders Wherefore in equall cones and cylinders the bases are reciprokall to their altitudes c. which was required to be demonstrated A Corrollary added by Campane and Flussas Hitherto hath bene shewed the passions and proprieties of cones and cylinders whose altitudes fall perpendicularly vpon the bases Now will we declare that cones and cilinders whose altitudes fall obliquely vpon their bases haue also the selfe same passions and proprieties which the foresayd cones and cilinders haue Forasmuch as in the tenth of this booke it was sayd that euery Cone is the third part of a cilinder hauing one and the selfe same base one the selfe same altitude with it which thing was demoÌstrated by a cilinder geuen whose base is cut by a square inscribed in it and vpon the sides of the square are described Isosceles triangles making a poligonon figure and againe vpon the sides of this poligonon figure are infinitely after the same maner described other Isosceles triangles taking away more theÌ the halfe as hath ofteÌtimes bene declared therfore it is manifest that the solides set vpon these bases being vnder the same altitude that the cilinder inclined is and being also included in the same cilinder do take away more then the halfe of the cilinder and also more theÌ the halfe of the residue as it hath bene proued in erected cylinders For these inclined solides being vnder equall altitudes and vpon equall bases with the erected solides are equall to the erected solides by the corollary of the â0 of the eleuenth Wherfore they also in like sort as the erected take away more then the halfe If therfore we coÌpare the inclined cilinder to a cone set vpon the selfe same base and hauing his altitude erected and reason by an argument leading to an impossibilitie by the demonstration of the tenth of this booke we may proue that the sided solide included in the inclined cylinder is greater then the triple of his pyramis and it is also equall to the same which is impossible And this is the first case wherein it was proued that the cilinder not being equall to the triple of the cone is not greater then the triple of the same And as touching the second case we may after the same maner conclude that that âided solide contayned in the cylinâer is greater then the cylinder which is very absurdâ Wherefore if the cylinder be neither greater then the triple of the cone nor lesse it must nedes be equall to the same The demonstration of these inclined cylinders most playnely followeth the demonstration of the erected cylinders for it hath already bene proued that pyramids and sided solides which are also called generally Prismes being set vpon equall bases and vnder one and the selfe same altitude whether the altitude be erected or inclined are equall the one to the other namely are in proportion as their bases are by
place from whence first it began to be moued it shal passe by the pointes F G H and the octohedron shall be comprehended in a sphere I say also that it is comprehended in the sphere geuen For forasmuch as the line LK is equal to the line KM by position and the line KE is common to them both and they contayne right angles by the 3. diffinition of the eleuenth therefore by the 4. of the first the base LE is equall to the base EM And forasmuch as the angle LEM is a right angle by the 31. of the third for it is in a semicircle as hath bene proued therefore the square of the line LM is double to the square of the line LE by the 47. of the first Againe forasmuch as the line AC is equall to the line BC therefore the line AB is double to the line BC by the diâfinition of a circle But as the line AB is to the line BC so is the square of the line AB to the square of the line BD by the corollaries of the 8. and â0 of the sixt Wherefore the square of the line AB is double to the square of the line BD. And it is proued that the square of the line LM is double to the square of the line LE. Wherefore the square of the line BD is equall to the square of the line LE. For the line EH which is equall to the line LF is put to be equall to the line DB. Wherefore the square of the line AB is equall to the square of the line LM Wherefore the line AB is equall to the line LM And the line AB is the diameter of the sphere geueÌ wherefore the line LM is equall to the diameter of the sphere geuen Wherefore the octoedron is contayned in the sphere geuen and it is also proued that the diameter of the sphere is in power double to the side of the octohedron Wherefore there is made an octohedron and it is comprehended in the sphere geuen wherein was comprehended the Pyramis and it is proued that the diameter of the sphere is in power double to the side of the octohedrn which was required to be doone and to be proued Certayne Corollaries added by Flussas First Corollary The side of a Pyramis is in power sesquitertia to the side of an octâhedron inscribed in the same Sphere For forasmch as the diameter is in power double to the side of the octohedron therefore of what partes the diameter contayneth in power 6. of the same the side of the octohedron coÌtayneth in power 3. but of what partes the diameter contayneth 6. of the same the side of the pyramis contayneth 4. by the 13. of this booke Wherefore of what partes the side of the pyramis contayneth 4. of the same the side of the octohedron contayneth 3. Second Corollary An octohedron is deuided into two equall and like Pyramids The common bases of these Pyramids are set vpon euery square contayned of the sides of the octohedron vpon which square are set the ââ triangles of the octohedron which pyramids are by the â diffinition of the eleuenth equall and like And the foresayd square common to those Pyramids is the halfe of the square of the diameter of the sphere for it is the square of the side of the octohedron Third Corollary The three diameters of the octohedron do cutte the one the other perpendicularly into two equall parts in the center of the sphere which contayneth the sayd octohedron As it is manifest by the three diameters EG FH and LM which cutte the one the other in the center K equally and perpendicularly ¶ The 3. Probleme The 15. Proposition To make a solide called a cube and to comprehend it in the sphere geuen namely that Sphere wherein the former two solides were comprehendâdâ and to proue that the diameter of the sphere is in power treble to the side of the cube TAke the diameter of the sphere geuen namely AB and diuide it in the point Câ So that let the line AC be double to the line BC by the 9. of the sixt And vpon the line AB describe a semicircle ADB And by the 11. of the first from the pâynt C râyse vp vnto the line AB a perpeÌdicular line CD And draw a right linâ DB. And describe a squarâ EFGH hauing euery one of his sides equall to the line DB And from the pointes E F G H rayse vp by the 12. of the eleuenth vnto the playne superficies of the square EFGH perpendicular lines EK FL GM and HN and let euery one of the lines EK FL GM and HN be put equall to one of the lines EF FG GH or HE which are the sides of the square and draw these right lines KL LM MN and NK Wherfore there is made a cube namely FN which is contayned vnder six equall squares Now it is required to comprehend the same cube in the sphere geuen and to proue that the diameter of the sphere is in power ble to the side of the cube Draw these right lines KG and EG And forasmuch as the angle KEG is a right angle for that the line KE is erected perpendicularly to the playne superficies Eâ and therefore also to the right line EG by the 2. diffinitioÌ of the eleuenth wherefore a semicircle described vpon the line KG shall passe by the poynt E. Agayne forasmuch as the line FG is erected perpendicularly to either of these lines FL and FE by the diffinition of a square by the 2. diffinition of the eleuenth therefore the line FG is erected perpendicularly to the playne superficies FK by the 4. of the eleuenth Wherefore if we draw a right line from the point F to the point K the line GF shall be erected perpendicularly to the line KF by the 2. diffinition of the eleuenth And by the same reason agayne a semicircle described vpon the line GK shall passe also by the point F. And likewise shall it passe by the rest of the pointes of the angles of that cube If now the diameter KG abiding fixed the semicircle be turned round about vntill it returne into the selfe same place from whence it began first to be moued the cube shal be compreheÌded in a sphere I say also that it is comprehended in the sphere geuen For forasmuch as the line GF is equall to the linââE and the angle F is a right angle therefore the square of the line EG is by the 47. of the first double to the square of the line âF But the line EF is equall to the line EK Wherefore the square of the line EG is double to the square of the line EK Wherfore the squares of EG and EK that is the square of the line GK by the 47. of the first are treble to the square of the line EK And forasmuch as the line AB is treble to the line BC but
as the line AB is to the line BC so is the square of the line AB to the square of the line BD by the corollaries of the 8. and 20. of the sixt Wherefore the square of the line AB is treble to the square of the line BD. And it is proued that the square of the line GK is treble to the square of the line KE and the line KE is put equall to the line BD. Wherefore the line KG is also equall to the line AB And the line AB is the diameter of the sphere geuen Wherefore the line KG is equall to the diameter of the sphere geuen Wherfore the cube is coÌpreheÌded in the sphere geuen and it is also proued that the diameter of the Sphere is in power treble to the side of the cube which was required tââe doone and to be proued An other demonstration after Flussas Suppose that the diameter of the Sphere geuen in the former Propositions be the line Aâ And let the center be the point C vpon which describe a semicircle ADB And from the diameter AB cut of a third part BG by the 9. of the sixt And from the point G raise vp vnto the line AB a perpendicular line DG by the 11. of the first And draw these right lines DA DC and DB. And vnto the right line DB put an equall right line ZI and vpon the line ZI describe a square EZIT And froÌ the pointes E Z I T erecte vnto the superficies EZIT perpendicular lines EK ZH IM TN by the 12. of the eleuenth and put euery one of those perpendicular lines equall to the line ZI And drawe these right lines KH HM MN and NK ech of which shall be equall and parallels to the line ZI and to the rest of the lines of the square by the 33. of the first And moreouer they shall containe equall angles by the 10. of the eleuenth and therefore the angles are right angles for that EZIT is a square wherfore the rest of the bases shall be squares Wherfore the solide EZITKHMN being coÌtained vnder 6. equall squares is a cube by the 21. definition of the eleuenth Extend by the opposite sides KE and MI of the cube a plaine KEIM and againe by the other opposite sides NT and HZ extend an other plaine HZTN Now forasmuch as ech of these plaines deuide the solide into two equall partes namely into two Prismes equall and like by the 8. definition of the eleuenth therfore those plaines shall cut the cube by the centre by the Corollary of the 39. of the eleuenth Wherefore the coÌmon section of those plaines shall passe by the centre Let that common section be the line LF And forasmuch as the sides HN and KM of the superficieces KEIM and HZTN do diuide the one the other into two equall partes by the Corollary of the 34. of the first and so likewise do the sides ZT and EI therefore the common section LF is drawen by these sections and diuideth the plaines KEIM and HZTN into two equall partes by the first of the sixt for their bâses are equall and the altitude is one and the âame namely the altitude of the cube Wherefore the line LF shall diuide into two equall partes the diameters of his plaines namely the right lines KI EM ZN and NT which are the diameters of the cube Wherfore those diameters shall concurre and cut one the other in one and the selfe same poynt let the same be O. Wherfore the right lines OK OE OI OM OH OZ OT and ON shall be equâll the onâ to the other for that they are the halfes of the diameters of equall and like rectangle parallelograÌmes Wherefore making the centre the point O and the space any of these lines OE or OK c. a Sphere described shall passe by euery one of the angles of the cube namely which are at the pointes E Z I T K H M N by the 12. definition of the eleuenth for that all the lines drawen from the point O to the angles of the cube are equall But the right line EI containeth in power the two equall right lines EZ and ZI by the 47. of the first Wherefore the square of the line EI is double to the square of the line ZI And forasmuch as the right line KI subtendeth the right angle KEI for that the right line KEâ is erected perpendicularly to the plaiâe âuperficies of the right lines EZ and ZT by the 4. of the eleueÌth â therefore the square of the line KI is equall to the squares of the lines EI and EK but the square of the line EI is double to the square of the line EK for it is double to the square of the line ZI as hath bene proued and the bases of the cube are equall squares Wherefore the square of the line KI is triple to the square of the line KE that is to the square of the line ZI But the right line ZI is equall to thâ right line DB by position vnto whose square the square of the diâmeter AB is triple by that which was demonstrated in the 13. Proposition of this booke Wherefore the diameters KI DB are equall Wherefore there is described a cube KI and it is comprehended in the Sphere geuen wherin the other solides were contained the diameter of which Sphere is the line AB And the diameter KI or AB of the same Sphere is proued to be in power triple to the side of the cube namely to the line DB or ZI ¶ Corollaryes added by Flussas First Corollary Hereby it is manifest that the diameter of a Sphere containeth in power the sides both of a pyramis and of a cube inscribed in it For the power of the side of the pyramis is two thirdes of the power of the diameter by the 13. of this booke And the power of the side of the cube is by this Proposition one third of the power of the sayd diameter Wherefore the diameter of the Sphere contayneth in power the sides of the pyramis and of the cube .. ¶ Second Corollary All the diameters of a cube cut the one the other into two equall partes in the centre of the sphere which containeth the cube And moreouer those diameters do in the selfe same point cut into two equall partes the right lines which ioyne together the centres of the opposite bases As it is manifest to see by the right line LOF For the angles LKO and FIO are equall by the 29. of the first and it is proued that they are contained vnder equall sides Wherefore by the 4. of the first the bases LO and FO are equall In like sort may be proued that the rest of the right lines which ioyne together the centres of the opposite bases do cut the one the other into two equall partes in the centre O. ¶ The 4. Probleme The 16. Proposition To make an Icosahedron and to comprehend it in the Sphere
F. And draw these right lines FA FB FC FD FE Wherefore those lines do diuide the angles of the pentagon into two equall partes in the poyntes A B C D E by the 4. of the first And âorasmuch as the fiue angles that are at the poynt F aâe equall to fower right angles by the corollary of the 15. of the first and they are equall the one to the other by the 8. of the first therfore one of those angles as âor example sake the angle AFB is a fiâth part lesse then a right angle Wherfore the angles remayning namely FAB ABF are one right angle and a fifth part ouer But the angle FAB is equall to the angle FBC Wherefore the whole angle ABC being one of the angles of the pentagon is a right angle and a fifth part more then a right angle which was required to be proued ¶ A Corollary added by Flussas Now let vs teach how those fiue solides haue eche like inclinations of theyr bases âiâst let vs take a Pyramis and diuide one of the sides thereof into two equall parts and from the two angles opposite vnto that side dâaw perpeÌdiculars which shall fall vpon the section by the corollary of the 12. of the thirtenth and at the sayd poynt of diuision as may easily be proued Wherfore they shal containe the angâe of the inclination of the plaines by the 4. diffinition of the eleuenth which angle is subtended of the opposite side of the pyramis Now forasmuch as the rest of the angles of the inclination of the playnes of the Pyramis are contayned vnder two perpeÌdicular lines of the triangles and are subtended of the side of the Pyramis it foloweth by the 8. of the firât that those angles are equall Wherâfoâe by the 5. diffinition of the eleueÌth the superficieces are in like sort inclined the one to the other One of the sides of a Cube being diuided into two equall parts if from the sayd section be drawen in two of the bases thereof two perpendicular lines they shal be parallels and equall to the sides of the square which coÌtayne a right angle And forasmuch as all the angles of the bases of the Cube are right angles therefore those perpendiculars falling vpon the section of the side common to the two bases shall contâyne a right angle by the 10. of the eleuenth which selfe angle is the angle of inclination by the 4. diffinition of the eleuenth and is subtended of the diameter of the base of the Cube And by the same reason may we proue that the rest of the angles of the inclination of the bases of the cube are right angles Wherefore the inclinations of the superficieces of the cube the one to the other are equal by the 5. diffinition of the eleuenth In an Octohedron take the diameter which coupleth the two opposite angles And from those opposite angles draw to one and the selâe same side of the Octohedron in two bases thereof two perpendicular lines which shall diuide that side into two equall parts and perpendicularly by the Corollary of the 12. of the thirtenth Wherefore those perpendiculars shall contayne the angle of the inclination of the bases by the 4. diffinition of the eleueÌth and the same angle is subtended of the diameter of the OctohedroÌ Wherfore the rest of the angles after the same maner described in the rest of the bases being comprehended and subtended of equall sides shall by the 8. of the first be equall the one to the other And therefore the inclinations of the playnes in the Octohedron shal by the 5. diffinition of the eleuenth be equall In an Icosahedron let there be drawen from the angles of two of the bases to one side common to both the sayd bases perpendiculars which shall contayne the angle of the inclination of the bases by the 4. diffinition of the eleuenth which angle is subtended of the right line which subtendeth the angle of the pentagon which contayneth fiue sides of the Icosahedron by the 16. of this booke for it coupleth the twoo opposite angles of the triangles which are ioyned together Wherefore the rest of the angles of the inclination of the bases being after the same maner found out they shal be contayned vnder equall sides and subtended of equall bases and therefore by the 8. of the fiâst those angles shal be equall Wherfore also al the inclinations of the bases of the Icosahedron the one to the other shalbâ equall by the 5. diffinition of the eleuenth In a Dodecahedron from the two opposite angles of two next pentagons draw to theyr common side perpendicular lines passing by the centres of the sayd pentagons which shal where they fal diuide the side into two equall parts by the 3. of the third For the bases of a Dodecahedron are contayned in a circle And the angle contaynâd vnder those perpendicular lines is the inclination of those bases by the 4. diffinition of the eleuenth And the foresayd opposite angles are coupled by a right line equal to the right line which coupleth the opposite sections into two equall parts of the sides of the dodecahedroÌ by the 33. of the first For they couple together the halfe sids of the dodecahedroÌ which halfes are parallels and equall by the 3. corollary of the 17. of this booke which coupling lines also are equall by the same corollary Wherefore the angles being contayned of equal perpendicular lines and subtended of equall coupling lines shall by the 8. of the first be equal And they are the angles of the inclinations Wherefore the bases of the dodecahedron are in like sort inclined the one to the other by the 5. diffinition of the eleuenth Flussas after this teacheth how to know the rationality or irrationality of the sides of the triangles which contayne the angles of the inclinations of the superficieces of the foresayd bodies In a Pyramis the angle of the inclinatioÌ is contayned vnder two perpâdicular lines of the triangles and is subtended of the side of the Pyramis Now the side of the pyramis is in power sesquitertia to the perpendicular line by the corollary of the 12. of this booke and therfore the triangle coÌtained of those perpeÌdicular lines and the side of pyramis hath his sides rational commensurable in power the one to the other Forasmuch as the twoo sides of a Cube or right lines equall to them subtended vnder the diameter of one of the bases doo make the angle of the inclination and the diameter of the cube is in power sesquialter to the diameter of the base which diameter of the base is in power double to the side by the 47. of the first therefore those lines are rationall and commensurable in power In an Octohedron whose two perpendiculars of the bases contayne the angle of the inclination of the Octohedron which angle also is subtended of the diameter of the Octohedron the diameter is in power
one and the selfe same sphere LEt the diameter of the sphere geuen be AB and let the bases of the Icosahedron and Dodecahedron described in it be the triangle MNR and the pentagon FKH and about them let there be described circles by the 5. and 14. of the fourth And let the lines drawne from the centres of those circles to the circumferences be LN and OK Then I say that the lines LN and OK are equal and therfore one and the selfe same circle containeth both those figures Let the right line AB be in power quintuple to some one right line as to the line CG by the Corollary of the 6. of the tenth And making the ceÌtre the poynt C the space CG describe a circle DZG And let the side of a pentagon inscribed in that circle by the 11. of the fourth be the line ZG And let EG subtending halfe of the arke ZG be the side of a Decagon inscribed in that circle And by the 30. of the sixt diuide the line CG by an extreme meane proportion in the poynt I. Now forasmuche as in the 16. of the thirtenth it was proued that this line CG vnto whome the diameter AB of the sphere is in power quintuple is the line which is drawne from the centre of the circle which containeth fiue angles of the Icosahedron and the side of the pentagon described in that circle DZG namely the line ZG is side of the Icosahedron described in the Sphere whose diameter is the line AB therefore the right line ZG is equal to the line MN which was put to be the side of the IcosahedroÌ or of his triaÌgular base Moreouer by the 17. of the thirtenth it was manifest that the right line âH which subtendeth the angle of the pentagon of the Dodecahedron inscribed in the foresayde sphere is the side of the Cube inscribed in the self same sphere For vpon the angles of the cube were made the angles of the Dodecahedron Wherefore the diameter AB is in power triple to FH the side of the Cube by the 15. of the thirtenth But the same line AB is by supposition in power quintuple to the line CG Wherefore fiue squares of the line CG are equal to thre squares of the line FH for eche is equal to one and the self same square of the line AB And forasmuche as EG the side of the Decagon cutteth the right line CG by an extreme and meane proportion by the corollary of the 9. of the thirtenth Likewise the line HK cutteth the line FH the side of the Cube by an extreeme and meane proportion by the Corollary of the 17. of the thirtenth therfore the lines CG and FH are deuided into the self same proportions by the second of this booke and the right lines CI and EG which are the greater segmentes of one and the selfe same line CG are equal And forasmuche as fiue squares of the line CG are equal to thre squares of the lines FH therefore fiue squares of the line GE are equal to thre squares of the line HK for the lines GE and HK are the greater segmeÌts of the lines CG and FH Wherefore fiue squâreâ of the lineâ CG GE are equal to the squares of the ãâã âH HK by the 1â of the âift But vnto the squares of the lines CG and GEâ is âqual the squâre of thââine ZG by the 10. of the thirteÌth and vnto the line ZG the line MN was equal wherfore fiue squares of the line MN are equall to three squares of the lines FH HK But the squares of the lines ââ and HK ãâã quintuple to the square of the line OK which is drawne from the centre by the third of this booke Wherfore thre squares of the lines FH and HK make 15. squares of the line OK And forasmuch as the square of the line MN is triple to the square of the line LN which is drawne from the centre by the 12. of the thirtenth therfore fiue squares of the line MN are equal to 15. squares of the line LN But fiue squares of the line MN are equal vnto thre squares of the lines FH and HK Wherefore one square of the line LN is equall to one square of the line OK being eche the fiuetenth part of equal magnitudes by the 15. of the fifââ Wherfore the lines LN and OK which are drawne from the centers are equal Wherefore also the circles NRM and FKH which are described of those lines are equal And those circles contayne by supposition the bâses of the Dodecahedron and of the Icosahedron described in one and the selfe same sphere Wherfore one and the selfe same circle c. aâ in thâ proââsition which was required to be proued The 5. Proposition If in a circle be inscribed the pentagon of a Dodecahedron and the triangle of an Icosahedron and from the centre to one of theyr sides be drawne a perpendicular line That which is contained 30. times vnder the side the perpendicular line falling vpon it is equal to the superficies of that solide vpon whose side the perpendicular line falleth SVppose that in the circle AGE be described the pentagon of a Dodecahedron which let be ABGDE and the triangle of an Icosahedron described in the same sphere which let be AFH And let the centre be the poynt C. ââon which draw perpendicularly the line CI to the side of the Pentagon and the line CL to the side of the triangle Then I say that the rectangle figure contained vnder the lines CI and GD 30. times is equal to the superficies of the Dodecahedron and that that which is coÌtained vnder the lines CL AF 30. times is equal to the superâicies of the IcosahedroÌ described in the same sphere Draw these right lines CA CF CG and CD Now forasmuch as that which is coÌtained vnder the base GD the altitude IC is double to the triangle GCD by the 41. of the first And fiue triangles like and equal to the triangle GCD do make the pentagon ABGDE of the Dodecahedron wherfore that which is contained vnder the lines GD and IC fiue times is equal to two pentagoÌs Wherfore that which is contained vnder the lines GD and IC â0 times is equal to the 12. pentagons which containe the superficies of the Dodecahedron Againe that which is contained vnder the lynes CL and AF is double to the triangle ACF wherefore that which is contained vnder the lines CL and AF three times is equal to two suche triangles as AFH is which is one of the bases of the Icosahedron for the triangle ACF is the third part of the triangle AFH as it is easie to proue by the 8. 4. of the first Wherfore that which is coÌtained vnder the lines CL and AF. 30 times times is equall to 10. such triangles as AFH iâ which containe the superficies of the Icosahedron And forasmuch as one and the selfe same
and hauing the same altitude with it namelye the altitude of the parallel bases as it is manifest by the former is equal to thre of those pyramids of the Octohedron by the first corollary of the seueÌth of the twelft Wherefore that prisme shall haue to the other prisme vnder the same altitude composed of the 4. pyramids of the whole octohedron the proportion of the triangular bases by the 3. corollary of the same And forasmuch as 4. pyramids are vnto 3. pyramids in sesquitercia proportion therefore the trianguler base of the prisme which containeth 4. pyramids is in sesquitertia proportion to the base of the prisme which containeth thre pyramids of the same octohedron and are set vpon the base of the Octohedron and vnder the altitude thereof that is in sesquitercia proportion to the base of the Octohedron But the base of the same octohedron is in sesquitertia proportion to the base of the pyramis by the âenth of this booke Wherefore the triangular bases namely of the prisme which coÌtaineth four pyramids of the octohedron and is vnder the altitude thereof are equal to the triangular bases of the prisme which containeth three pyramids vnder the altitude of the pyramis EFGH But the prisme of the octohedron is equal to the octohedron and the prisme of the pyramis EFGH is proued triple to the same pyramis EFGH Now then the prismes set vpoÌ equal bases are the one to the other as their altitudes are by the corollary of the 25. of the eleuenth namely as are the parallelipidedons their doubles by the corollary of the 31. of the eleuenth But the altitude of the Octohedron is equal to the side of the cube contained in the same sphere by the corollary of the 13. of this booke And the side of the cube is in power to the altitude of the Tetrahedon in that proportion that 12. is to 16 by the 18. of the thirtenth And the side of the octohedron is to the side of the pyramis in that proportion that 18. is to 24. by the same 18. of the thirteÌth which proportion is one the self same with the proportioÌ of 12. to 16. Wherfore that prisme which is equal to the Octohedron is to the prisme which is triple to the Tetrahedron in that proportioÌ that the altitudes or that the sides are Wherfore an octohedroÌ is to the triple of a Tetrahedron coÌtained in one and the selfe same sphere in that proportion that their sides are which was required to be demonstrated A Corollary The sides of a Tetrahedron of an OctohedroÌ are proportionall with their altitudes For the sides altitudes were in power sesquitercia Moreouer the diameter of the sphere is to the side of the Tetrahedron as the side of the Octohedron is to the ââde of the cubeâ namely the powers of eche is in sesquialter proportion by the 18. of the thirtenth The 15. Proposition If a rational line containing in power two lines make the whole and the greater segment and again containing in power two lines make the whole and the lesse segment the greater segment shal be the side of the Icosahedron and the lesse segment shal be the side of the Dodecahedron contayned in one and the selfe same sphere SVppose that AG be the diameter of the sphere which containeth the Icosahedron ABGC And let BG subtend the sides of the pentagon described of the sides of the Icosahedron by the 16. of the thirteÌth Moreouer vpon the same diameter AG or DF equal vnto it let ther be described a dodecahedron DEFH by the 1â of the thirtenth whose opposites sides ED and FH let be cut into two equal partes in the poynts I and K and draw a line from I to K. And let the line EF couple two of the opposite angles of the bases which are ioyned together TheÌ I say that AB the side of the Icosahedron is the greater segment which the diameter AG containeth in power together with the whole line and line ED is the lesse segment which the same diameter AG or DF containeth in power together with the whole For forasmuche as the opposite sides AB and GC of the Icosahedron being coupled by the diameters AG and BC are equal parallels by the 2. corollary of the 16. of the thirteÌth the right lines BG AC which couple theÌ together are equal parallels by the 33. of the first Moreouer the angles BAC ABG being subtended of equal diamâters shall by the 8. of the first be equal by the 29 of the ãâã they shal be right angles Wherfore the right line AG ãâã in power the âwo lines AB and BG by the 47. of ãâ¦ã And forasmuch as the line BG subtendeth the angle of the pentagon composed of the sides of the Icosahedron the greater segment of the right line BG shal be the right line AB by the â of the thirtenth which line AB togeâher with the whole line BG the line AG containeth in power And forasmuch aâ the line IK coupling the opposite and parallel sides ED and FH of the Dodecahedron maketh at those poyntes right angles by the 3. corollary of the 17. of tâe thirtenth the right line EF which coupleth together equal and parallel lines EI FK shal be equal to the same line IK by the 33. of the first Wherfore the angle DEF shal be â right angle by the 29. of the first Wherefore the diameter DF coÌtaineth in power the two lines ED and EF. But the lesse segment of the line IK is ED the side of the Dodecahedron by the 4. corollary of the 17â of the thirtenth Wherfore the same line ED is also the lesse segment of the line EF which is equal vnto the line IK wherfore the diamâter DF containing in power the two lines ED and EF by the 47. of the first containeth in powârâ ED the side of the dodecahedron the lesse segment together with the whole If therfore a rational line AG or DF containing in power two lines AB and BG doo make the whole line and the greater sâgment and agaiâe containing in power two lines EF and ED do make the whole line and the lesse segment the greater segment AB shall be the side of the Icosahedron and the lesse segment ED shall be the side of the Dodecahedron contained in one and the selfe same sphere The 16. Proposition If the power of the side of an Octohedron be expressed by two right lineâ ioyned together by an extreme and meâne proportion the side of the Icosahedron contained in the same sphere shal be duple to the lesse segment LEt AB the side of the Octohedron ABG containe in power the two lines C and H which let haue that proportion that the whole hath to the greater segment by the corollarye of the first proposition added by Flussas after the last propositioÌ of the sixth booke And let the Icosahedron contained in the same sphere be
bene demonstrated FD is the diameter of the Sphere which containeth the Icosahâdron which diameter is in power sesquialter to AB the side of the Tetrahedron inscribed in theâ same Sphere by the 13. of the thirtenth Wherfore the line ED the side of the Icosahedron is in power sesquialter to Gâ the greater segment or lesse line If therefore the side of a Tetrahedron containe in power two right lines ioyned together an extreme and meane proportion the side of an Icosahedron described in the selfe same Sphere is in power sesquialter to the lesse right line ¶ The 19. Proposition The superficies of a Cube is to the superficies of an Octohedron inscribed in one and the selfe same Sphere in that proportion that the solides are SVppose that ABCDE be a Cube whose fower diameters let be the lines AC BC DC and EC produced on ech side Let also the Octohedron inscribed in the selfe same Sphere be FGHK whose three diameters let be FH GK and ON Then I say that the cube ABD is to the Octohedron FGH as the superficies of the cube is to the superficies of the Octohedron Drawe from the centre of the cube to the base ABED a perpendicular line CR. And from the centre of the Octohedron draw to the base GNH a perpendicular line âL And forasmuch as the three diameters of the cube do passe by the ãâã C therefore by the 2. Corollary of the 15. of the thirtenth âhere shall be made of the cube sixe pyramids as thys pyramis ABDEC equall to the whole cube For there are in the cube âixe bases vpon which fall equall perpendiculars from the cenâââ by the Corollary of the Assumpâ of the 16. of the twelfth for the bases are contained in equall circlâ of the Sphere But in the Octohedron the three diameters do make vpon the 8. bases 8. pyramids hauing their toppes in the centre by the 3. Corollary of the 14â of the thirtenth Now the bases of the cube and of the Octohedron are contained in equall circles of the Sphere by the 13. of this booke Wherefore they shall be equally distant from the centre and the perpendicular lines CR and â shall be equall by the Corollary of the Assumpt of the 16. of the twelfth Wherefore the pyramids of the cube shall be vnder one and the selfe same altitude with the pyramids of the Octohedron namely vnder the perpendicular line drawen from the centre to the bases Wherfore sixe pyramids of the cube are to 8. pyramids of the Octohedron being vnder one and the same altitude in that propoâtion that their bases are by the 6. of the twelfth that is one pyramis set vpon sixe bases of the cube and hauing to his altitude the perpendicular line which pyramis is equall to the sixe pyramids by the same 6. of the twelfth is to one pyramis set vpon the 8. bases of the Octohedron being equall to the Octohedron and also vnder onâ and the selfe same altitude in that proportion that sixe bases of the cube which containe the whole superficies of the cube are to 8. bases of the Octohedronâ which containe the whole superficies of the Octohedron For the solides of those pyramids are in proportion the one to the other as their bases are by the selfe same 6. of the twelfth Wherefore âhe superficies of the cube is to the superficies of the Octohedron inscribed in one and the selfe same Sphere in that proportion that the solides are which was required to be proued ¶ The 20. Proposition If a Cube and an Octohedron be contained in one 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 SVppose that the Octohedron AECDB be inscribed in the Sphere ABCD and let the cube inscribed in the same Sphere be FGHIM whose diameter let be HI which is equall to the diameter AC by the 15. of the thirtenth let the halfe of the diameter be AE Then I say that the cube FGHIM is to the Octohedron AECDB as the side MG is to the semidiameter AE Forasmuch as the diameter AC is in power double to BK the side of the Octohedron by the 14. of the thirtenth and is in power triple to MG the side of the cube by the 15. of the same therefore the square BKDL shall be sesquialâer to FM the square of the cube From the line AE cut of a third part AN and froÌ the line MG cut of likewise a third part GO by the 9. of the sixth Now then the line EN shall be two third partes of the line AE and so also shall the line MO be of the line MG Wherefore the parallelipipedon set vpon the base BKDL and hauing his altitude the line EA is triple to the parallelipipedon set vpon the same base and hauing his altitude the line AN by the Corollary of the 31. of the eleuenth but it is also triple to the pyramis ABKDL which is set vpon the same base and is vnder the same altitude by the second Corollary of the 7. of the twelfth Wherefore the pyramis ABKDL is equall to the parallelipipedon which is set vpon the base BKDL and hath to his altitude the line AN. But vnto that parallelipipedoÌ is double the parallelipipedon which is set vppon the same base BKDL and hath to his altitude a line double to the line EN by the Corollary of the 31. of the first and vnto the pyramis is double the Octohedron ABKLDC by the 2. Corollary of the 14. of the thirtenth Wherefore the Octohedron ABKDLC is equall to the parallelipipedon set vpon the base BKLD hauing his altitude the line EN by the 15. of the fifth But the parallelipipedon set vpon the base BKDL which is sesquialter to the base FM and hauing to his altitude the line MO which is two third partes of the side of the cube MG is equall to the cube FG by the 2. part of the 34. of the eleuenth For it was before proued that the base BKDL is sesquialter to the base FM Now then these two parallelipipedons namely the parallelipipedon which is set vpoÌ the base BKDL which is sesquialter to the base of the cube and hath to his altitude the line MO which is two third partes of MG the side of the cube which parallelipipedon is proued equall to the cube and the parallelipipedon set vpon the same base BKDL and hauing his altitude the line EN which parallelipipedon is proued equall to the Octohedron these two parallelipipedons I say are the one to the other as the altitude MO is to the altitude EN by the Corollary of the 31. of the eleuenth Wherefore also as the altitude MO is to the altitude EN so is the cube FGHIM to the Octohedron ABKDLC by the 7. of the fifth But as the line MO is to the line EN so is
ouerthrowne and ouerwhelmed the whole world he was vtterly rude and ignorant in the Greke tongue so that certenly he neuer redde Euclide in the Greke nor of like translated out of the Greke but had it translated out of the Arabike tonge The Arabians were men of great study and industry and commonly great Philosophers notable Phisitions and in mathematicall Artes most expert so that all kinds of good learning flourished and raigned amongst them in a manner only These men turned whatsoeuer good author was in the Greke tonge of what Art and knowledge so euer it were into the Arabike tonge And froÌ thence were many of theÌ turned into the Latine and by that meanes many Greeke authors came to the handes of the Latines and not from the first fountaine the Greke tonge wherin they were first written As appeareth by many words of the Arabike tonge yet remaining in such bokes as are Zenith nadir helmuayn helmuariphe and infinite suche other Which Arabians also in translating such Greke workes were accustomed to adde as they thought good for the fuller vnderstanding of the author many things as is to be sene in diuers authors as namely in Theodosius de Sphera where you see in the olde translation which was vndoubteldy out of the Arabike many propositions almost euery third or fourth leafe Some such copye of Euclide most likely did Campanus follow wherein he founde those propositioÌs which he hath more aboue those which are found in the Greke set out by Hypsicles and that not only in this 15. boke but also in the 14. boke wherin also ye finde many propositions more theÌ are founde in the Greeke set out also by Hypsicles Likewise in the bookes before ye shall finde many propositions added and manye inuerted and set out of order farre otherwise then they are placed in the Greeke examplars Flussas also a diligent restorer of Euclide a man also which hath well deserued of the whole Art of Geometrie hath added moreouer in this booke as also in the former 14. boke he added 8. proâositioÌs 9. propositioÌs of his owne touching the inscription and circumscriptâon ãâ¦ã bodies very siâgular ândoubtedly and wittye All which for that nothing should want to the desirous louer of knowledge I haue faithfully with no small paines turned And whereas Flâssââ in the beginning of the eleuenth booke namely in the end of the diffinitions there âeâ putteth two diffinitions of the inscription and circumscription of solides or corporall figures within or about the one the other which certainely are not to be reiected yet for that vntill this present 15. boke there is no mention made of the inscription or circumscription of these bodyes I thought it not so conuenient thârâ to place them but to referre theÌ to the beginning of this 15. booke where they are in maner of necessitie required to the elucidation of the Proposiâions and dâmonstrationâ of the same The diffinitions are these Diffinition 1. A solide figure is then âaid to be inscribed in a solide figure when the angles of the figure inscribed touche together at one time either the angles of the figure circumscribed or the superficieces or the sides Diffinition 2. A solide figure is then said to be circumscribed about a solide figure when together at one time either the angles or the superficieces or the sides of the figure circumscribed âouch the angles of the figure inscribed IN the fourââ booke in the diffinitions of the inscription or circumscription of playne rectiline figures one with in or about an other was requâred that all the angles of the figuââ inscribed should at one time touch all the sides of the figure circumscribed but in the fiue regular solides âo whome chefely these two diffinitions pertaine for that the nomber of their angles superficieces sides are not equal one compared to an other it is not of necessitie that all the angles of the solide inscribed should together at one time touch either all the angles or all the superficieces or all the sides of the solide circumscribed but it is sufficient that those angles of the inscribed solide which touch doe at one time together eche touch some one angle of the figure circumscribed or some one base or some one side so that if the angles of the inscribed figure do at one time touche the angles of the figure circumscribed none of them may at the same time touche either the bases or the sides of the same circumscribed figure and so if they touch the bases they may touche neither angles nor sides and likewise if they touche the sides they may touch neither angles nor bases And although sometimes all the angles of the figure inscribed can not touch either the angles or the bases or the sides of the figure circumscribed by reason the nomber of the angles bases or sides of the said figure circumscribed wanteth of the nomber of the angles of the âigure inscribed yet shall those angles of the inscribed figure which touch so touch that the void places left betwene the inscribed and circumscribed figures shal on euery side be equal and like As ye may afterwarde in this fiftenth booke most plainely perceiue ¶ The 1. Proposition The 1. Probleme In a Cube geuen to describe a trilater equilater Pyramis SVppose that the cube geuen be ABCDEFGH In the same cube it is required to inscribe a Tetrahedron Drawe these right lines AC CE AE AH EH HC Now it is manifest that the triangles AEC AHE AHC and CHE are equilater for their sides are the diameters of equall squares Wherfore AECH is a trilater equilater pyramis or Tetrahedron it is inscribed in the cube geueÌ by the first definition of this booke which was required to be done ¶ The 2. Proposition The 2. Probleme In a trilater equilater Pyramis geuen to describe an Octohedron SVppose that the trilater equilater pyramis geueÌ be ABCD whose sides let be diuided into two equall partes in the pointes E Z I K L T. And draw these 12. right lines EZ ZI IE KL LT TK EK KZ ZL LI IT and TE Which 12. right lines are by the 4. of the first equall For they subtend equall plaine angles of the bases of the pyramis and those equall angles are contained vnder equall sides namely vnder the halfes of the sides of the pyramis Wherefore the triangles TKL TLI TIE TEK ZKL ZLI ZIB ZEK are equilater and they limitate and containe the solide TKLEZI Wherefore the solide TKLEZI is an Octohedron by the 23. definition of the eleuenth And the angles of the same Octohedron do touch the sides of the pyramis ABCD in the pointes E Z I T K L. Wherefore the Octohedron is inscribed in the pyramis by the 1. definition of this booke Wherefore in the trilater equilater pyramis geuen is inscribed an Octohedron which was required to be done A Corollary added by Flussas Hereby it is manifest that a pyramis is cut into two
to the bases of the Octohedron For forasmuch as the sides of the bases of the Pyramis touching the one the other are parallels to the sides of the Octohedron which also touch the one the other as for example HL was proued to be a parallel to GI and LC to DI therefore by the 15. of the eleuenth the plaine superficies which is drawen by the lines HL and LC is a parallel to the plaine superficies drawen by the lines GI and DI. And so likewise of the rest Second Corollary A right line ioyning together the centres of the opposite bases of the Octohedron is sesquialter to the perpendicular line drawen from the angle of the inscribed pyramis to the base thereof For forasmuch as the pyramis and the cube which containeth it do in the selfe same pointes end their angles by the 1. of this booke therefore they shall both be inclosed in one and the selfe same Octohedron by the 4. of this booke But the diameter of the cube ioyneth together the centres of the opposite bases of the Octohedron and therefore is the diameter of the Sphere which containeth the cube and the pyramis inscribed in the cube by the 13. and 14. of the thirtenth which diameter is sesquialter to the perpendicular which is drawen from the angle of the pyramis to the base thereof for the line which is drawen from the centre of the sphere to the base of the pyramis is the sixth part of the diameter by the 3. Corollary of the 13. of the thirtenth Wherefore of what partes the diâmeter containeth sixe of the same partes the perpendicular containeth fower ¶ The 7. Proposition The 7. Probleme In a dodecahedron geuen to inscribe an Icosahedron SVppose that the dodecahedron geuen be ABCDE And let the centres of the circles which coÌtayne sixe bases of the same dodecahedron be the polnes L M N P Q O. And draw these right lines OL OM ON OP OQ and moreouer these right lines LM MN NP PQ QL And now forasmuch as equall and equilater pentagons are contayned in equall circles therefore perpendicular lines drawne from their centres to the sides shall be equall by the 14. of the third and shall diuide the sides of the dodecahedron into two equall partes by the 3. of the same Wherefore the foresayde perpendicular lines shall coâoutre in the point of the section wherein the sides are diuided into two equall partes as LF and MF doo And they also containe equall angles namely the inclination of the bases of the dodecahedron by the 2. corollary of the 18. of the thirtenth Wherfore the right lines LM MN NP PQ QL and the rest of the right lines which ioyne together two centres of the bases and which subtende the equall angles âontayned vnder the sayd equall perpendicular lines are equall the one to the other by the 4. of the first Wherefore the triangles OLM OMN ONP OPQ OQL and the rest of the triangles which are set at the centres of the pentagons are equilater and equall Now forasmuch as the 12. pentagons of a dodecahedron containe 60. plaine superficiall angles of which 60. eueây âhre make one solide angle of the dodecahedron it followeth that a dodecahedron hath 20. solide angles but eche of those solide angles is subteÌded of ech of the triangles of the Icosahedron namely of ech of those triangles which ioyne together the centres of the pentagoÌs which make the solide angle as we haue before proued Wherefore the 20 equall and equilater triangles which subtende the 20. solide angles of the dodecahedron and haue their sides which are drawne from the centres of the pentagons common doo make an Icosahedron by the 25. diffinition of the eleuenth and it is inscribed in the dodecahedron geuen by the first diffinition of this booke for that the angles thereof doo all at one time touch the bases of the dodecahedron Wherefore in a dodecahedron geuenâ iâ inscribed an Icosahedron which was required to be done ¶ The 8. Proposition The 8. Probleme In a dodecahedron geuen to include a cube DEscribe by the 17. of the thirtenth a dodecahedron And by the same take the 12. sides of the cube eche of which subtend one angle of eche of the 12. bases of the dodecahedron for the side of the cube subtendeth the angle of the pentagon of the dodecahedron by the 2. corollary of the 17. of the thirtenth If therefore in the dodecahedron described by the selfe same 17. proposition we draw the 12. right lines subâended vnder the foresayd 12. angles and ending in 8. angles of the dodecahedron and concurring together in such sort that they be in like sort situate as it was plainely proued in that proposition then shall it be manifest that the right lines drawne in this dodecahedron from the foresayd 8. angles thereof doo make the foresayd cube which therefore is included in the dodecahedron for that the sides of the cube are drawne in the sides of the dodecahedron and the angles of the same cube are set in the angles of the said dodecahedron As for example take 4. pentagons of a dodecahedron namely AGIBO BHCNO CKEDN and DFAON And draw these right lines AB BC CD DA. Which fower right lines make a square for that eche of those right lines doo subtend equall angles of equall pentagons the angles which those 4. right lines coÌtaine are right angles as we proued in the construction of the dodecahedron in the 17. propositioÌ before alledged Wherfore the sixe bases being squares do make a cube by the 21. diffinition of the eleuenth and for that the 8. angles of the sayd cube are set in 8. angles of the dodecaheeron therefore is the sayd cube inscribed in the dodecahedron by the first diffinition of this booke Wherefore in a dodecahedron is inscribed a cube which was required to be doone ¶ The 9. Proposition The 9. Probleme In a Dodecahedron geuen to include an Octohedron SVppose that the dodecahedron geuen be ABGD Now by the 3. correllary of the 17. of the thirteÌth take the 6. sides which are opposite the one to the other those 6. sides I saye whose sections wherin they are deuided into two equal partes are coupled by three right lines which in the centre of the sphere wherin the Dodecahedron is contained doe cut the one the other perpendicularly And let the poyntes wherin the forsayde sides are cut into two equal partes be A B G D C I. And let the foresaid thre right lines ioyning together the saide sections be AB GD and CI. And let the centre of the sphere be E. Now forasmuch as by the foresaid correllary those thre right lines are equal it foloweth by the 4. of the first that the right lines subteÌding the right angles which they make at the centre of the sphere whiche right angles are contained vnder the halues of the said three right lines are equal the one to the other
side GD the angles M N vnder the side AB the angles T S vnder the side BG the angles P O and vnder the side AG the angles R Q so there rest 4. angles whose true place we will now appoynt Forasmuch as a cube contayned in one and the selfe same sphere with a dodecahedron is inscribed in the same dodecahedron as it was manifest by the 17. of the thirtenth and 8. of this booke it followeth that a cube and a dodecahedron circumscribed about it are contayned in one and the selfe same bodies for that their angles concurre in one and the selfe same poyntes And it was proued in the 18. of this booke that 4. angles of the cube inscribed in the pyramis are set in the middle sections of the perpendicularâ which are drawne from the solide angles of the pyramis to the opposite bases wherefore the other 4. angles of the dodecahedron are also as the angles of the cube set in those middle sections of the perpendiculars Namely the angle V is set in the middest of the perpendicular AHâ the angle Y in the middest of the perpendicular BF the angle X in the middest of the perpendicular GE and lastly the angle D in the middest of the perpendicular D which is drawne from the toppe of the pyramis to the opposite base Wherefore those 4. angles of the dodecahedron may be sayd to be directly vnder the solide angles of the pyramis or they may be sayd to be set at the perpendiculars Wherefore the dodecahedron after this maner set is inscribed in the pyramis geuen by the first diffinition of this booke for that vpoÌ euery one of the bases of the pyramis are set an angle of the dodecahedroÌ inscribed Wherefore in a trilater equilater pyramis is inscribed a dodecahedron The 21. Probleme The 21. Proposition In euery one of the regular solides to inscribe a Sphere IN the 13. of thâ thirtenth and thâ other 4. propositioââ following iâ was declared that âhe ââ regular solidesââre so contaââed in a sphere that âight linââ drawne from the cenâââ oâ the ãâ¦ã of ãâã solide inscribed are equall Which right lines therefore make pyramids whose âoppes are the centre of the sphere or of the solide and the basââââe cuâââ one of the bases of those solides And ãâ¦ã solide âquall and like the one to the other and described in equall circles those cirâles shall cutte the sphere for the angles which touch the circumference of the circle touch also the superficies of the sphere Wherefore perpeÌdiculars drawne from the centre of the sphere to the bases or to the playne superficieces of the equall circles are equall by the corollary of the assumpt of the 1â of the twelfth Wherefore making the centre the ãâã of the sphere which ãâã the solide and thâ space some one of the equall perpendicularâ dâscribâ a sphere and it shall touch euery one of the bases of ãâã solide ãâ¦ã perficies of the sphere passe beyond those bases when as those pââpeâdiculars ãâ¦ã are drawne from the centre to the bases by the 3. corollary of the saâââââumpt Wherâfore âe haue iâ euery one of the regular bodies inscribed a sphere which regular boâââ are in number one iâ ãâã by the corollary of the 1â of the ãâã A Corollary The regular figures inscribed in spheres and also the spheres circumscribed about them or contayning them haue one and the selfe same centre Namely their pyramids the ângles of whose bâses touch the superââââââ of thâââhere doo from those angles cause equall right lines to be drawââ to one and âhe selfe ãâã poynâ making the topâââ of the pyramidâ in the same poynt and therefore they ãâ¦ã thâ cââtres of the spheres in the selfe same toppes when ãâã the right lines drawne from those angles to the croââed superficies whârein are ãâã the angles of the bases of the pyramidâ are equallâ An adueâââsment of Flussas â Of these solides onely the Octohedron receaueth the other solides inscribed one with ãâ¦ã other For the Octohedron contayneth the Icosahedron inscribed in it and the same Icosahedron contayneth the Dodecahedron inscribed in the same Icosahedron and the same dodecahedron contayneth the cube inscribed in the same Octohedron and ãâ¦ã âârââmscribeth the Pyramis inscribed in the sayd Octohedron But this happâneth not in the other solides The ende of the fiuetenth Booke of Euclides Elemenâââ after Caâpaââ and ãâã ¶ The sixtenth booke of the Elementes of Geometrie added by Flussas IN the former fiuetenth booke hath bene taught how to inscribe the fiue regular solides one with in an other Now semeth to rest to coÌpare those solidâ so inscribed one to an other and to set forth their passionâ and proprieties which thing Flussas considering in this sixteÌth booke added by him hath excellently well and most conningly performed For which vndoubtedly he hath of all them which haue a loue to the Mathematicals deserued much prayse and commendacion both for the great traâailes and paynâs which it is most likely he hath taâân in iâuenting such straunge and wonderfull propositions with their demonstrations in this booke contayned as also for participating and communicating abrode the same to others Which booke also that the reader should want nothing conducing to the perfection of Euclides Elements I haue with some trauaile translated for the worthines âhereof haue added it aâ a sixtenth booke to the 15. bookes of Euclide Vouchsafe therefore gentle reader diligently to read and peyse it for in it shall you finde noâ onely matter strange and delectable but also occasion of inuention of greater things pertayning to the natures of the fiue regular solidâsâ ¶ The 1. Proposition A Dodecahedron and a cube inscribed in it and a Pyramis inscribed in the same cube are contained in one and the selfe same sphere FOr the angles of the pyramiâ are seâ in the angâes of the cube wherein it is inscribed by the first of the fiuetenthâ and all the angles of the cube are set in the angles of the dodecahedâââ circumscribed ãâ¦ã ãâã the 8. of the fiuetenth And all the angles of the Dodecahedron are set in the superficies of the sphere by the 17. of the thirtenth Wherefore those three solides inscribed one within an other are contained in one and the selfe same sphere by the first diffinition of the fiuetenth A dodecahedron therfore and a cube inscribed in it and a pyramis inscribed in the same cube are contained ãâ¦ã ââlfe same sphere ãâ¦ã These three solides li ãâ¦ã elfe same Icosahedron or Octohedron or Pyramis ãâ¦ã me Icosahedron by the 5.11 12. of the fiuetenth and they ar ãâ¦ã ctohedron by the 4. 6. and 16. of the same lastly they are inscribed in ãâ¦ã the first 18. and 19. of the same For the angles of all these solide ãâ¦ã the circumscribed Icosahedron or octohedron or pyramis ¶ The ãâ¦ã The proportion of a Dodecahedron circumscribed about a cube to a DodecahedroÌ inscribed in the same cube is
same the cube containeth 12 namely is sesquialter to the pyramis Wherefore of what partes the cube containeth 12 of the same the whole Octohedron which is double to the pyramis ABDFC containeth 54. Which 54. hath to 12. quadruple sesquialter proportion Wherefore the whole Octohedron is to the cube inscribed in it in quadruple sesquialter proportion Wherefore we haue proued that an Octohedron geuen is quadruple sesquialter to a cube inscribed in it ¶ A Corollary An Octohedron is to a cube inscribed in it in that proportion that the squares of their sides are For by the 14. of this booke the side of the Octohedron is in power quadruple sesquialter to the side of the cube inscribed in it ¶ The 29. Proposition To proue that an octohedroÌ geueÌ is tredecuple sesquialter to a trilater equilater pyramis inscribed in it LEt the octohedron geâen be AB in which let there be inscribed a cube FCED by the 4. of the fiuetenth and in the cube let there be inscribed a pyramis FEGD by the â of the fiuetenth And forasmuche as the angles of the pyramis are by the same first of the fiuetenth set in the angles of the cube and the angles of the cube are set in the centres of the bases of the Octohedron namely in the poyntes F E C D G by the 4. of the fiuetenth Wherfore the angles of the pyramis are set in the centres F C E D of the octohedron Wherefore the pyramis FEDG is inscribed in the octohedron by the 6. of the fiuetenth And forasmuche as the octohedron AB is to the cube FCED inscribed in it quadruple sesquialter by the former propositioÌ and the cube CDEF is to the pyramis FEDG inscribed in it triple by the 25. of booke wherefore three magnitudes being geuen namely the octohedron the cube and the pyramis the proportion of the extremes namely of the octohedron to the piramis is made of the proportions of the meanes namely of the octohedron to the cube and of the cube to the pyramis as it is easie to see by the declaration vpon the 10. diffinition of the fiueth Now then multiplying the quantities or denominations of the proportions namely of the octohedron to the cube which is 4 1 â and of the cube to the pyramis which is 3 as was taught in the diffinition of the sixth there shal be produced 13 1 â namely the proportion of the octohedron to the pyramis inscribed in it For 4 ½ multiplyed by 3. produce 13 ½ Wherefore the Octohedron is to the pyramis inscribed in it in tredecuple sesquialter proportion Wherefore we haue proued that an Octohedron is to a trilater equilater pyramis inscribed in it in tredecuple sesquialter proportion ¶ The 30. Proposition To proue that a trilater equilater Pyramis is noncuple to a cube inscribed in it SVppose that the pyramis geuen be ABCD whose two bases let be ABC and DBC and let their centres be the poynts G and I. And from the angle A draw vnto the base BC a perpendicular AE likewise from the angle D draw vnto the same base BC a perpendicular DE and they shal concurre in the section E by the 3. of the third and in them shal be the ceÌtres G and I by the corollary of the first of the third And forasmuch as the line AD is the side of the pyramis the same AD shall be the diameter of the base of the cube which coÌtaineth the pyramis by the 1 of the fiueteÌth Draw the line GI And forasmuch as the line GI coupleth the centreâ of the bases of the pyramis the saide line GI shal be the diameter of the base of the cube inscribed in the pyramis by the 18. of the fiuetenth And forasmuche as the line AG is double to the line GE by the corollarye of the twelueth of the thirtenth the whole line AE shal be triple to the line GE and so is also the line DE to the line IE Wherefore the lines AD and GI are parallels by the 2. of the sixth And therefore the triangles AED and GEI are likeâ by the corollary of the same And forasmuch as the triangles AED and GEI are like the line ADâ shal be triple to the line GI by the 4. of the sixth But the line AD is the diameter of the base of the cube circumscribed about the pyramis ABCD and the line GI is the diameter of the base of the cube inscribed in the pyramis ABCD but the diameters of the bases are equemultiplices to the sides namely are in power duple Wherfore the side of the cube circumscribed about the pyramis ABCD is triple to the side of the cube inscribed in the same piramis by the 15. of the fiueth but like cubes are in triple proportion the one to the other of that in which their sides are by the 33. of the eleuenth and the sides are in triple proportion the one to the other Wherfore triple taken thre times bringeth forth twenty seuencuple which is 27. to 1 for the 4. termes 27.9.3.1 being set in triple proportion the proportion of the first to the fourth namely of 27. to 1. shal be triple to the proportion of the first to the second namely of 27. to 9 by the 10. diffinition of the fiueth which proportion of 27. to 1. is the proportioÌ of the sides tripled which proportioÌ also is found in like solides Wherefore of what partes the cube circumscribed containeth 27. of the same the cube inscribed containeth one but of what partes the cube circumscribed containeth 27. of the same the pyramis inscribed in it containeth 9. by the 25. of this booke wherfore of what partes the pyramis AB CD containeth 9. of the same the cube inscribed in the pyramis containeth one Wherefore we haue proued that a trilater and equilater pyramis is nonâcuple to a cube inscribed in it ¶ The 31. Proposition An Octohedron hath to an Icosohedron inscribed in it that proportion which two bases of the Octohedron haue to fiue bases of the Icosahedron SVppose that the octohedron geuen be ABCD and let the Icosahedron inscribed in it be FGHMKLIO Then I say that the octohedron is to the Icosahedron as two bases of the octohedron are to fiue bases of the Icosahedron For forasmuche as the solide of the octohedron consisteth of eight pyramids set vpon the bases of the octohedron and hauing to theyr altitude a perpendicular line drawne from the centre to the base let that perpendicular be ER or ES being drawne from the centre E which centre is common to either of the solides by the corollary of the 21. of the fiuetenth to the centres of the bases namely to the poyntes R and S. Wherefore for that thre pyramids are equal and like they shal be equal to a prisme set vpon the selfe same base and vnder the selfe same altitude by the corollary of the seuenth of the twelueth But
in the poynt E. And vnto the line CG put the line CL equall Now forasmuch as the lines AG and GC are the greater segâââtes of halfe the line AB for âche of them is the halfe of the greater segment of the whole line AB the lines EB and EC shall be the lesse segmentes of halfe the line AB Wherefore the whole line Câ is the greater segment and the line CE is the lesse segment But as the line CL is to the line CE so is the line CE to the residue EL. Wherfore the line EL is the greater segment of the line CE or of the line EB which is equall vnto it Wherfore the residue LB is the lesse segment of the same EB which is the lesâe segment of halfâ the side of the cube But the lines AG GC and CL are three greater segmentes of the halfe of the whole line AB which thre greater segmentes make the altitude of the foresayd solide wherefore the altitude of the sayd solide wanteth of AB the side of the cube by the line LB which is the lesse sâgment of the line BE. Which line BE agayne is the lesse segment of halfe the side AB of the cube Wherefore the foresayd solide consisting of the sixe solides whereby the dodecahedron exceedeth the cube inscribed in it is set vpon a base which wanteth of the base of the cube by a third part of the lesse segment and is vnder an altitude wanting of the side of the cube by the lesse segment of the lesse segment of halfe the side of the cube The solide therefore of a dodecahedron exceedeth the solide of a cube inscribed in it by a parallelipipedon whose base wanteth of the base of the cube by a third part of the lesse segment and whose altitude wanteth of the altitude of the cube by the lesse segment of the lesse segment of halfe the side of the cube ¶ A Corollary A Dodecahedron is double to a Cube inscribed in it taking away the third part of the lesse segment of the cube and moreouer the lesse segment of the lesse segment of halfe of that excesse For if there be geuen a cube from which is cut of a solide set vpon a third part of the lesse segment of the base and vnder one and the same altitude with the cube that solide taken away hath to the whole solide the proportion of the section of the base to the base by the 32. of the eleuenth Wherefoâe from the cube is taken away a third âart of the lesse segment Farther forasmuch as the residue wanteth of the altitude of the cube by the lesse segment of the lesse segment of halfe the altitude or side and that residue is a parallelipipedon if it be cut by a plaine superficies parallel to the opposite plaine superficieces cutting the altitude of the cube by a point it shall take away from that parallelipipedon a solide hauing to the whole the proportion of the section to the altitude by the 3. Corollary of the 25. of the eleuenth Wherefore the excesse wanteth of the same cube by the thiâd part of the lesse segment and moreouer by the lesse segment of the lesse segment of halfe of that excesse ¶ The 34. Proposition The proportion of the solide of a Dodecahedron to the solide of an Icosahedron inscribed in it consisteth of the proportion tripled of the diameter to that line which coupleth the opposite bases of the Dodecahedron and of the proportion of the side of the Cube to the side of the Icosahedron inscribed in one and the selfe same Sphere SVppose that AHBCK be a Dodecahedronâ whose diametâr let be AB and let the line which coupleth the ceÌtres of the opposite bases be KHâ and let the Icosahedron inscribed in the Dodecahedron ABC be d ee whose diameter let be DE. Now forasmuch aâ oâe and the selfe same circle coÌtaineth the pentagon of a Dodecahedron the triangle of an Icosahedroâ described in one and the selfe same Sphere by the 14. of the fourtenth Let that circle be IGO. Wherfore IO is the side of the cube and IG the side of the Icosahedron by the same TheÌ I say that the proportion of the Dodecahedron AHBCK to the Icosahedron DEF inscribed in it coÌsisteth of the proportioÌ tripled of the line AB to the line KH and of the proportion of the line IO to the line IG For âoâasmuch as the Icosahedron DEF is inscribed in the DodecahedroÌ ABC by suppositioÌ the diameter DE shal be equal to the line KH by the 7. of the fiuetenth Wherefore the Dodecahedron set vpoÌ the diameter KH shall be inscribed in the same Sphere wherein the Icosahedron DEF is inscribed but the Dodecahedron AHBCK is to the Dodecahedron vpon the diameter KH in triple proportion of that in which the diameter AB is to the diameter KH by the Corollary of the 17. of the twelfth and the same Dodecahedron which is set vpon the diameter KH hath to the Icosahedron DEF which is set vpon the same diameter or vpon a diameter equall vnto it namely DE that proportion which IO the side of the cube hath toâ IG the side of the Icosahedron inscribed in one the selfe same Sphere by the 8 of the fouretenth Wherefore the proportion of the Dodecahedron AHBCK to the Icosahedron DEF inscribed in it consisteth of the proportion tripled of the diameter AB to the line KH which coupleth the centres of the opposite bases of the Dodecahedron which proportion is that which the Dodecahedron AHBCK hath to the Dodecahedron set vpon the diameter KH and of the proportion of IO the side of the cube to IG the side of the Icosahedron which is the proportion of the Dodecahedron set vpon the diameter KH to the Icosahedron DEF described in one and the selfe same Sphere by the 5. definition of the sixth The proportion therefore of the solide of a Dodecahedron to the solide of an Icosahedron inscribed in it conâisteth of the proportion tripled of the diameter to that line which coupleth the opposite bases of the Dodecahedron and of the propââtion of the side of the cube to the side of the Icosahedron inscribed in one and the selfe same Sphere The 35. Proposition The solide of a Dodecahedron containeth of a Pyramis circumscribed about it two ninth partes taking away a third part of one ninth part of the lesse segment of a line diuided by an extreme and meane proportion and moreouer the lesse segment of the lesse segment of halfe the residue IT hath bene proued that the Dodecahedron together with the cube inscribed in it is contained in one and the selfe same pyramis by the Corollary of the first of this booke And by the Corollary of the 33. of this booke it is manifest that the Dodecahedron is double to the same cube taking away the third part of the lesse segment and moreouer the lesse segment of the lesse segment of halfe
word or two hereafter shall be sayd How Immateriall and free from all matter Number is who doth not perceaue yeâ who doth not wonderfully woÌder at it For neither pure Element nor Aristoteles Quinta Essentia is hable to serue for Number as his propre matter Nor yet the puritie and simplenes of Substance Spirituall or Angelicall will be found propre enough thereto And therefore the great godly Philosopher Anitius Boetius sayd Omnia quaecunque a primaua rerum naââra constructa sunt Numerorum videntur ratione formata Hoc enim fuit principale in animo Conditoris Exemplar That is All thinges which from the very first originall beâng of thinges haue bene framed and made do appeare to be Formed by the reason of Numbers For this was the principall example or patterne in the minde of the Creator O comfortable alluremenâ O rauishing perswasion to deale with a Science whose Subiect iâ so Auncient so pure so excellent so surmounting all creatures so vsed of the Almighty and incomprehensible wisdome of the Creator in the distinct creation of all creaturesâ in all their distinct partes properties natures and vertues by order and most absolute number brought from Nothing to the Formalitie of their being and state By Numbers propertie therefore of vs by all possible meanes to the perfection of the Science learned we may both winde and draw our selues into the inward and deepe search and vâw of all creatures distinct vertues natures properties and Former And also farder arise clime ascend and mount vp with Speculatiue winges in spirit to behold in the Glas of Creation the Forme of Formes the Exemplar Number of all thinges Numerable both visible and inuisibleâ mortall and immortall Corporall and Spirituall Part of this profound and diuine Science had Ioachim the Prophesier atteyned vnto by Numbers Formall Naturall and Rationall forseyng concludyng and forshewyng great particular euents long before their comming His bookes yet remainyng hereof are good profe And the noble Earle of Mirandula besides that a sufficient witnesse that Ioachim in his prophesies proceded by no other way then by Numbers Formall And this Earle hym selfe in Rome set vp 900. Conclusions in all kinde of Sciences openly to be disputed of and among the rest in his Conclusions Mathematicall in the eleuenth Conclusion hath in Latin this English sentence By Numbers a way is had to the searchyng out aud vnderstandyng of euery thyng hable to be knowen For the verifying of which Conclusion I promise to aunswere to the 74. Quaestions vnder written by the way of Numbers Which CoÌclusions I omit here to rehearse aswell auoidyng superfluous prolixitie as bycause Ioannes Picus workes are commonly had But in any case I would wish that those Conclusions were red diligently and perceiued of such as are earnest Obseruers and Considerers of the constant law of nuÌbers which is planted in thyngs Naturall and Supernaturall and is prescribed to all Creatures inuiolably to be kept For so besides many other thinges in those Conclusions to be marked it would apeare how sincerely within my boundes I disclose the wonderfull mysteries by numbers to be atteyned vnto Of my former wordes easy it is to be gathered that Number hath a treble state One in the Creator an other in euery Creature in respect of his complete constitution and the third in Spirituall and Angelicall Myndes and in the Soule of maÌ In the first and third state Number is termed Number Numbryng But in all Creatures otherwise Number is termed NuÌber Numbred And in our Soule NuÌber beareth such a swaye and hath such an affinitie therwith that some of the old Philosophers taught Mans Soule to be a Number mouyng it selfe And in dede in vs though it be a very Accident yet such an Accident it is that before all Creatures it had perfect beyng in the Creator Sempiternally Number Numbryng therfore is the discretion discerning and distincting of thinges But in God the Creator This discretion in the beginnyng produced orderly and distinctly all thinges For his Numbryng then was his Creatyng of all thinges And his Continuall Numbryng of all thinges is the Conseruation of them in being And where and when he will lacke an Vnit there and then that particular thyng shal be Discreated Here I stay But our Seuerallyng distinctyng and Numbryng createth nothyngâ but of Multitude considered maketh certaine and distinct determination And albeit these thynges be waighty and truthes of great importance yet by the infinite goodnes of the Almighty Ternarie Artificiall Methods and easy wayes are made by which the zelous Philosopher may wyn nere this Riuerish Ida â this Mountayne of Contemplation and more then Contemplation And also though Number be a thyng so Immateriall so diuine and aeternall yet by degrees by litle and litle stretchyng forth and applying some likenes of it as first to thinges Spirituall and then bryngyng it lower to thynges sensibly perceiued as of a momentanye sounde iterated then to the least thynges that may be seen numerable And at length most grossely to a multitude of any corporall thynges seen or felt and so of these grosse and sensible thynges we are trayned to learne a certaine Image or likenes of numbers and to vse Arte in them to our pleasure and proffit So grosse is our conuersation and dull is our apprehension while mortall Sense in vs ruleth the common wealth of our litle world Hereby we say Three Lyons are three or a Ternarie Three Egles are three or a Ternarie Which Ternaries are eche the Vnion knot and Vniformitie of three discrete and distinct Vnits That is we may in eche Ternarie thrise seuerally pointe and shew a part One One and One. Where in Numbryng we say One two Three But how farre these visible Ones do differre from our Indiuisible Vnits in pure Arithmetike principally considered no man is ignorant Yet from these grosse and materiall thynges may we be led vpward by degrees so informyng our rude Imagination toward the coceiuyng of Numbers absolutely Not supposing nor admixtyng any thyng created Corporall or Spirituall to support conteyne or represent those Numbers imagined that at length we may be hable to finde the number of our owne name gloriously exemplified and registred in the booke of the Trinitie most blessed and aeternall But farder vnderstand that vulgar Practisers haue Numbers otherwise in sundry Considerations and extend their name farder then to Numbers whose least part is an Vnit. For the common Logist Reckenmaster or Arithmeticien in hys vsing of Numbers of an Vnit imagineth lesse partesâ and calleth them Fractions As of an Vnit he maketh an halfe and thus noteth it ½ and so of other infinitely diuerse partes of an Vnit Yea and farder hath Fractions of Fractions c. And forasmuch as Addition Substraction Multiplication Diuision and Extraction of Rotes are the chief and sufficient partes of Arithmetike which is the Science that demonstrateth the properties of Numbers and all operatioÌs in numbers to be performed
diameter is double to that square whose diameter it is The 34. Theoreme The 48. Proposition If the square which is made of one of the sides of a triangle be equall to the squares which are made of the two other sides of the same triangle the angle comprehended vnder those two other sides is a right angle SVppose that ABC be a triangle and let the square which is made of one of the sides there namely of the side BC be equall to the squares which are made of the sides BA and AC Then I say that the angle BAC is a right angle Rayse vp by the 11. propositioÌ from the point A vnto the right line AC a perpendicular line AD. And by the thirde proposition vnto the line AB put an equall line AD. And by the first peticion draw a right line from the point D to the poinâ C. And forasmuch as the line DA is equall to the line AB the square which is made of the line DA is equall to the square whiche is made of the line AB Put the square of the line AC common to them both VVherefore the squares of the lines DA and AC are equal to the squares of the lines BA and AC But by the proposition going before the square of the line DC is equal to the squares of the lines AD and AC For the angle DAC is a right angle and the square of BC is by supposition equall to the squares of AB and AC VVherefore the square of DC is equall to the square of BC wherefore the side DC is equall to the side BC. And forasmuch as AB is equall to AD ând AC is common to them both therefore these two sides DA and AC are equall to these two sides BA and AC the one to the other and the base DC is equall to the base BCâ wherfore by the 8. proposition the angle DAC is equall to the angle BAC But the angle DAC is a right angle wherefore also the angle BAC is a right angle If therefore the square which is made of one of the sides of a triangle be equall to the squares which are made of the two other sides of the same triangle the angle comprehended vnder those two other sides is a right angle which was required to be proued This proposition is the conuerse of the former and is of Pelitarius demonstrated by an argument leading to an impossibilitie after this maner The ende of the first booke of Euclides Elementes ¶ The second booke of Euclides Elementes IN this second booke Euclide sheweth what is a GnomoÌ and a right angled parallelogramme Also in this booke are set forth the powers of lines deuided euenly and vneuenly and of lines added one to an other The power of a line is the square of the same line that is a square euery side of which is equall to the line So that here are set forth the qualities and proprieties of the squares and right lined figures which are made of lines of their parts The Arithmetician also our of this booke gathereth many compendious rules of reckoning and many rules also of Algebra with the equatioÌs therein vsed The groundes also of those rules are for the most part by this second booke demonstrated This booke moreouer contayneth two wonderfull propositions one of an obtuse angled triangle and the other of an acute which with the ayde of the 47. proposition of the first booke of Euclide which is of a rectangle triangle of how great force and profite they are in matters of astronomy they knowe which haue trauayled in that arte VVherefore if this booke had none other profite be side onely for these 2. propositions sake it were diligently to be embraced and studied The definitions 1. Euery rectangled parallelogramme is sayde to be contayned vnder two right lines comprehending a right angle A parallelogramme is a figure of fower sides whose two opposite or contrary sides are equall the one to the other There are of parallelogrammes fower kyndes a square a figure of one side longer a Rombus or diamond and a Romboides or diamond like figure as before was sayde in the 33. definition of the first booke Of these fower sortes the square and the figure of one side longer are onely right angled Parallelogrammes for that all their angles are right angles And either of them is contayned according to this definition vnder two right lynes whiâh concurre together and cause the right angle and containe the same Of which two lines the one is the length of the figure the other the breadth The parallelogramme is imagined to be made by the draught or motion of one of the lines into the length of the other As if two numbers shoulde be multiplied the one into the other As the figure ABCD is a parallelograme and is sayde to be contayned vnder the two right lines AB and AC which contayne the right angle BAC or vnder the two right lines AC and CD for they likewise contayne the right angle ACD of which 2. lines the one namely AB is the length and the other namely AC is the breadth And if we imagine the line AC to be drawen or moued directly according to the leÌgth of the line AB or contrary wise the line AB to be moued directly according to the length of the line AC you shall produce the whole rectangle parallelogramme ABCD which is sayde to be contayned of them euen as one number multiplied by an other produceth a plaine and righte angled superficiall number as ye see in the figure here set where the number of sixe or sixe vnities is multiplied by the number of fiue or by fiue vnities of which multiplication are produced 30. which number being set downe and described by his vnities representeth a playne and a right angled number VVherefore euen as equall numbers multipled by equal numbers produce numbers equall the one to the other so rectangle parallelogrames which are comprehended vnder equal lines are equal the one to the other 2. In euery parallelogramme one of those parallelogrammes which soeuer it be which are about the diameter together with the two supplementes is called a Gnomon Those perticuler parallelogrames are sayde to be about the diameter of the parallelograme which haue the same diameter which the whole parallelograme hath And supplementes are such which are without the diameter of the whole parallelograme As of the parallelograme ABCD the partial or perticuler parallelogrames GKCF and EBKH are parallelogrames about the diameter for that ech of them hath for his diameter a part of the diameter of the whole parallelogramme As CK and KB the perticuler diameters are partes of the line CB which is the diameter of the whole parallelogramme And the two parallelogrammes AEGK and KHFD are supplementes because they are wythout the diameter of the whole parallelogramme namely CB. Now any one of those partiall parallelogrammes
nombers of the nombers AD and DB are double to the square nombers of AC and CD For forasmuch as the nomber AD is deuided into the nombers AB and BD therefore the square nombers of the nombers AD and DB are equall to the superficiall nomber produced of the multiplication of the nombers AD and DB the on into the other twise together with the square of the nomber AB by the 7 propositioÌ But the square of the nomber AB is equal to fower squares of either of the nombers AC or CB for AC is equall to the nomber CB wherfore also the squares of the nombers AD and DB are equall to the superficiall nomber produced of the multiplication of the nombers AD and DB the one into the other twise and to fower squares of the nomber BC or CA. And forasmuch as the superficiall nomber produced of the multiplication of the nombers AD and DB the one into the other together with the square of the nomber CB is equal to square of the nomber CD by the 6 propositioÌ therfore the nomber produced of the multiplication of the nomberâ AD and DB the one into the other twise together with two squares of the nomber CB is equall to two squares of the nomber CD Wherefore the squares of the nombers AD and DB are equall to two squares of the nomber CD and to two squares of the nomber AC Wherefore they are double to the squares of the numbers AC and CD And the square of the nomber AD is the square of the whole and of the nomber added And the square of DB is the square of the nomber added the square also of the nomber CD is the square of the nomber composed of the halfe and of the nomber added If therefore an euen nomber be deuided c. Which was required to be proued The 1. Probleme The 11. Proposition To deuide a right line geuen in such sort that the rectangle figure comprehended vnder the whole and one of the partes shall be equall vnto the square made of the other part SVppose that the right line geuen be AB Now it is required to deuide the line AB in such sort that the rectangle figure contayned vnder the whole and one of the partes shall be equall vnto the square which is made of the other part Describe by the 46. of the first vpon AB a square ABCD. And by the 10. of the first deuide the line AC into two equall partes in the point E and draw a line from B to E. And by the second petition extend CA vnto the point F. And by the 3. of the first put the line EF equall vnto the line BE. And by the 46. of the first vpon the line AF describe a square FGAH And by the 2. petition extend GH vnto the point K. Then I say that the line AB is deuided in the point H in such sort that the rectangle figure which is compreheÌded vnder AB and BH is equall to the square which is made of AH For forasmuch as the right line AC is deuided into two equall partes in the poynt E and vnto it is added an other right line AF. Therefore by the 6. of the second the rectangle figure contayned vnder CF and FA together with the square which is made of AE is equall to the square which is made of EF. But EF is equall vnto EB VVherefore the rectangle figure contayned vnder CF and FA together with the square which is made of EA is equall to the square which is made of EB But by 47. of the first vnto the square which is made of EB are equall the squares which are made of BA and AE For the angle at the poynt A is a right angle VVherefore that which is contayned vnder CF and FA together with the square which is made of AE is equall to the squares which are made of BA and AE Take away the square which is made of AE which is common to them both VVherfore the rectangle figure remayning contayned vnder CF and FA is equall vnto the square which is made of AB And that which is contained vnder the lines CF and FA is the figure FK For the line FA is equall vnto the line FG. And the square which is made of AB is the figure AD. VVherefore the figure FK is equall vnto the figure AD. Take away the figure AK which is common to them both VVherefore the residue namely the figure FH is equall vnto the residue namely vnto the figure HD But the figure HD is that which is contayned vnder the lines AB and BH for AB is equall vnto BD. And the figure FH is the square which is made of AH VVherfore the rectangle figure comprehended vnder the lines AB and BH is equall to the square which is made of the line HA. VVherefore the right line geuen AB is deuided in the point H in such sort that the rectangle figure contayned vnder AB and BH is equall to the square which is made of AH which was required to be done Thys proposition hath many singular vses Vpon it dependeth the demonstration of that worthy Probleme the 10. Proposition of the 4. booke which teacheth to describe an Isosceles triangle in which eyther of the angles at the base shall be double to the angle at the toppe Many and diuers vses of a line so deuided shall you finde in the 13. booke of Euclide Thys is to be noted that thys Proposition can not as the former Propositions of thys second booke be reduced vnto numbers For the line EB hath vnto the line AE no proportion that can be named and therefore it can not be expressed by numbers For forasmuch as the square of EB is equall to the two squares of AB and AE by the 47. of the first and AE is the halfe of AB therefore the line BE is irrationall For euen as two equall square numbers ioyned together can not make a square number so also two square numbers of which the one is the square of the halfe roote of the other can not make a square number As by an example Take the square of 8. which is 64. which doubled that is 128. maketh not a square number So take the halfe of 8. which is 4. And the squares of 8. and 4. which are 64. and 16. added together likewyse make not a square number For they make 80. who hath no roote square Which thyng must of necessitie be if thys Probleme should haue place in numbers But in Irrational numbers it is true and may by thys example be declared Let 8. be so deuided that that which is produced of the whole into one of his partes shall be equall to the square number produced of the other part Multiply 8. into him selfe and there shall be produced 64. that is the square ABCD. Deuide 8. into two equall partes that is into 4 and 4. as the line
the other in the poynt D and that onely poynt is common to them both neither doth the one enter into the other If any part of the one enter into any part of the other then the one cutteth and deuideth the other and toucheth the one the other not in one poynt onely as in the other before but in two pointâs and haue also a superficies common to them both As the circles GHK and HLK cut the one the other in two poyntes H and K and the one entreth into the other Also the superficies HK is common to them both For it is a part of the circle GHK and also it is a part of the circle HLK. Right lines in a circle are sayd to be equally distant from the centre when perpendicular lines drawen from the centre vnto those lines are equall And that line is sayd to be more distant vpon whom falleth the greater perpendicular line As in the circle ABCD whose centre is E the two lynes AB and CD haue equall distance from the centre E bycause that the lyne EF drawen from the centre E perpendicularly vpon the lyne AB and the lyne EG drawen likewise perpendilarly from the centre E vpon the lyne CD are equall the one to the other But in the circle HKLM whose centre is N the lyne HK hath greater distance from the centre N then hath the lyne LM for that the lyne ON drawen from the centre N perpendicularly vppon the lyne HK is greater then the lyne NP which is drawen froÌ the centre N perpendicularly vpon the lyne LM So likewise in the other figure the lynes AB and DC in the circle ABCD are equidistânt from the centre G â bycause the lynes OG and GP perpendicularly drawen from the centre G vppon the sayd lynes AB and DC are equall And the lyne AB hath greater distance from the centre G then hath the the lyne EF bycause the lyne OG perpendiculârly drâwen from the centre G to the lyne AB is greâter then the lyne HG whiche is perpendicularly drawen from the cââtre G to the lyne EF. A section or segment of a circle is a figure coÌprehended vnder a right line and a portion of the circumference of a circle As the figure ABC is a section of a circle bycause it is comprehended vnder the right lyne AC and the circumference of a circle ABC Likewise the figure DEF is a section of a circle for that it is comprehended vnder the right lyne DF and the circuÌference DEF And the figure ABC for that it coÌtaineth within it the centre of the circle is called the greater section of a circle and the figure DEF is the lesse section of a circle bycause it is wholy without the centre of the circle as it was noted in the 16. Definition of the first booke An angle of a section or segment is that angle which is contayned vnder a right line and the circuÌference of the circle As the angle ABC in the section ABC is an angle of a section bycause it is contained of the circumference BAC and the right lyne BC. Likewise the angle CBD is an angle of the section BDC bycause it is contayned vnder the circumference BDC and the right lyne BC. And these angles are commonly called mixte angles bycause they are contayned vnder a right lyne and a crooked And these portions of circumferences are commonly called arkes and the right lynes are called chordes or right lynes subtended And the greater section hath euer the greater angle and the lesse section the lesse angle An angle is sayd to be in a section wheÌ in the circumference is taken any poynt and from that poynt are drawen right lines to the endes of the right line which is the base of the segment the angle which is contayned vnder the right lines drawen from the poynt is I say sayd to be an angle in a section As the angle ABC is an angle is the section ABC bycause from the poynt B beyng a poynt in the circumference ABC are drawen two right lynes BC and BA to the endes of the lyne AC which is the base of the section ABC Likewise the angle ADC is an angle in the section ADC bycause from the poynt D beyng in the circuÌference ADC are drawen two right lynes namely DC DA to the endes of the right line AC which is also the base to the sayd section ADC So you see it is not all one to say an angle of a section and an angle in a section An angle of a section coÌsisteth of the touch of a right lyne and a crooked And an angle in a section is placed on the circumference and is contayned of two right lynes Also the greater section hath in it the lesse angle and the lesse section hath in it the greater angle But when the right lines which comprehend the angle do receaue any circumference of a circle then that angle is sayd to be correspondent and to pertaine to that circumference As the right lynes BA and BC which containe the angle AB C and receaue the circumference ADC therfore the angle ABC is sayd to subtend and to pertaine to the circuÌference ADC And if the right lynes whiche cause the angle concurre in the centre of a circle then the angle is sayd to be in the centre of a circle As the angle EFD is sayd to be in the centre of a circle for that it is comprehended of two right lynes FE and FD whiche concurre and touch in the centre F. And this angle likewise subtendeth the circumference EGD whiche circumference also is the measure of the greatnes of the angle EFD A Sector of a circle is an angle being set at the centre of a circle a figure contayned vnder the right lines which make that angle and the part of the circumference receaued of them As the figure ABC is a sector of a circle for that it hath an angle at the centre namely the angle BAC is coÌtained of the two right lynes AB and AC whiche contayne that angle and the circumference receaued by them Like segmentes or sections of a circle are those which haue equall angles or in whom are equall angles Here are set two definitions of like sections of a circle The one pertaineth to the angles whiche are set in the centre of the circle and receaue the circumfereÌce of the sayd sections the other pertaineth to the angle in the section whiche as before was sayd is euer in the circumference As if the angle BAC beyng in the centre A and receaued of the circumference BLC be equall to the angle FEG beyng also in the centre E and receaued of the circumference FKG then are the two sections BCL and FGK lyke by the first definition By the same definition also are the other two sections like namely BCD and FGH for that the angle BAC is equall to the
say that the lyne GFH which by the correllary of the 16. of this booke toucheth the circle is a parallel vnto the line AB For forasmuch as the right line CF fallyng vpon either of these lines AB GH maketh all the angles at the point â right angles by the 3. of this boke and the two angles at the point Fare supposed to be right angles therfore by the 29. of the first the lines AB and GH are parallels which was required to be done And this Probleme is very commodious for the inscribing or circumscribing of figures in or about circles The 16. Theoreme The 18. Proposition If a right lyne touch a circle and from the centre to the touch be drawen a right line that right line so drawen shal be a perpendicular lyne to the touche lyne SVppose that the right line DE do touch the circle ABC in the point C. And take the centre of the circle ABC and let the same be F. And by the first petition from the poynt F to the poynt C drawe a right line FC Then I say that CF is a perpendicular line to DE. For if not draw by the 12. of the first from the poynt F to the line DE a perpendicular line FG. And for asmuch as the angle FGC is a right angle therefore the angle GCF is an acute angle Wherefore the angle FGC is greater then the angle FCG but vnto the greater angle is subtended the greater side by the 19. of the first Wherefore the line FC is greater then the line FG. But the line FC is equall to the line FB for they are drawen from the centre to the circumference Wherfore the line FB also is greater then the line FG namely the lesse then the greater which is impossible Wherefore the line FG is not a perpendicular line vnto the line DE. And in like sort may we proue that no other line is a perpendicular line vnto the line DE besides the line FC Wherfore the line FC is a perpendicular line to DE. If therefore a right line touch a circle from y centre to the touch be drawen a right line that right line so drawen shall be a perpendicular line to the touch line which was required to be proued ¶ An other demonstration after Orontius Suppose that the circle geuen be ABC which let the right lyne DE touch in the point C. And let the centre of the circle be the point F. And draw a right line from F to C. Then I say that the line FC is perpendicular vnto the line DE. For if the line FC be not a perpeÌdiculer vnto the line DE then by the conuerse of the x. definition of the first boke the angles DCF FCE shal be vnequall therfore the one is greater then a right angle and the other is lesse then a right angle For the angles DCF and FCE are by the 13. of the first equall to two right angles Let the angle FCE if it be possible be greater then a right angle that is let it be an obtuse angle Wherfore the angle DCF âhal be an acute angle And forasmuch as by suppositioÌ the right line DE toucheâh the circle ABC therefore it cutteth not the circle Wherefore the circumference BC falleth betwene the right lines DC CF therfore the acute and rectiline angle DCF shall be greater then the angle of the semicircle BCF which is contayned vnder the circumfereÌce BC the right line CF. And so shall there be geueÌ a rectiline acute angle greater then the angle of a semicircle which is contrary to the 16. proposition of this booke Wherfore the angle DCF is not lesse then a right angle In like sort also may we proue that it is not greater then a right angle Wherfore it is a right angle and therfore also the angle FCE is a right angle Wherefore the right line FC is a perpendicular vnto the right line DE by the 10. definition of the firstâ which was required to be proued The 17. Theoreme The 19. Proposition If a right lyne doo touche a circle and from the point of the touch be raysed vp vnto the touch lyne a perpendicular lyne in that lyne so raysed vp is the centre of the circle SVppose that the right line DE do touch the circle ABC in the point C. And from C raise vp by the 11. of the first vnto the line DE a perpendicular line CA. Then I say that in the line CA is the centre of the circle For if not then if it be possible let the centre be without the line CA as in the poynt F. And by the first petition draw a right line from C to F. And for asmuch as a certaine right line DE toucheth the circle ABC and from the centre to the touch is drawen a right line CF therefore by the 18. of the third FC is a perpendicular line to DE. Wherefore the angle FCE is a right angle But the angle ACE is also a right angle Wherefore the angle FCE is equall to the angle ACE namely the lesse vnto the greater which is impossibleâ Wherefore the poynt F is not the centre of the circle ABC And in like sort may we proue that it is no other where but in the line AC If therefore a right line do touch a circle and from the point of the touch be raised vp vnto the touch line a perpendicular line in that line so raised vp is the centre of the circle which was required to be proued The 18. Theoreme The 20. Proposition In a circle an angle set at the centre is double to an angle set at the circumference so that both the angles haue to their base one and the same circumference SVppose that there be a circle ABC and at the centre thereof namely the poynt E let the angle BEC be set at the circumference let there be set the angle BAC and let them both haue one and the same base namely the circumference BC. Then I say that the angle BEC is double to the angle BAC Draw the right line AE and by the second petition extend it to the poynt F. Now for asmuch as the line AE is equall to the line EB for they are drawen from the centre vnto the circumference the angle EAB is equall to the angle EBA by the 5. of the first Wherefore the angles EAB and EBA are double to the angle EAB But by the 32. of the same the angle BEF is equall to the angles EAB and EBA Wherefore the angle BEF is double to the angle EAB And by the same reason the angle FEC is double to the angle EAC Wherefore the whole angle BEC is double to the whole angle BAC Againe suppose that there be set an other angle at the circumference and let the same be BDC And by the âirst petition draw a line from D to E. And by the second petition extend
Poligonon figure of 24. sides Likewyse of the Hexagon AB and of the Pentagon AC shall be made a Poligonon figure of 30. sides one of whose sides shall subtend the arke BC. For the denomination of AB which is 6. excedeth the denomination of AC which is 5. onely by vnitie So also forasmuch as the denomination of AB which is 6. excedeth the denomination of AE which is 3. by 3. therefore the arke BE shall contayne 3. sides of a Poligonon figure of .18 sides And obseruing thys selfe same methode and order a man may finde out infinite sides of a Poligonon figure The end of the fourth booke of Euclides Elementes ¶ The fifth booke of Euclides Elementes THIS FIFTH BOOKE of Euclide is of very great commoditie and vse in all Geometry and much diligence ought to be bestowed therin It ought of all other to be throughly and most perfectly and readily knowne For nothyng in the bookes followyng can be vnderstand without it the knowledge of them all depende of it And not onely they and other writinges of Geometry but all other Sciences also and artes as Musike Astronomy Perspectiue Arithmetique the arte of accomptes and reckoning with other such like This booke therefore is as it were a chiefe treasure and a peculiar iuell much to be accompted of It entreateth of proportion and Analogie or proportionalitie which pertayneth not onely vnto lines figures and bodies in Geometry but also vnto soundes voyces of which Musike entreateth as witnesseth Boetius and others which write of Musike Also the whole arte of Astronomy teacheth to measure proportions of tymes and mouinges Archimides and Iordan with other writing of waightes affirme that there is proportion betwene waight and waight and also betwene place place Ye see therefore how large is the vse of this fift booke Wherfore the definitions also thereof are common although hereof Euclide they be accommodate and applied onely to Geometry The first author of this booke was as it is affirmed of many one Eudoxus who was Platos scholer but it was afterward framed and put in order by Euclide Definitions A parte is a lesse magnitude in respect of a greater magnitude when the lesse measureth the greater As in the other bookes before so in this the author first setteth orderly the definitions and declarations of such termes and wordes which are necessarily required to the entreatie of the subiect and matter therof which is proportion and comparison of proportions or proportionalitie And first he sheweth what a parte is Here is to be considered that all the definitions of this fifth booke be general to Geometry and Arithmetique and are true in both artes euen as proportion and proportionalitie are common to them both and chiefly appertayne to number neither can they aptly be applied to matter of Geometry but in respect of number and by number Yet in this booke and in these definitions here set Euclide semeth to speake of them onely Geometrically as they are applied to quantitie continuall as to lines superficieces and bodies for that he yet continueth in Geometry I wil notwithstanding for facilitie and farther helpe of the reader declare theÌ both by example in number and also in lynes For the clearer vnderstandyng of a parte it is to be noted that a part is taken in the Mathematicall Sciences two maner of wayes One way a part is a lesse quantitie in respect of a greater whether it measure the greater oâ no. The second way a part is onely that lesse quantitie in respect of the greater which measureth the greater A lesse quantitie is sayd to measure or number a greater quantitie when it beyng oftentymes taken maketh precisely the greater quantitie without more or lesse or beyng as oftentymes taken from the greater as it may there remayneth nothyng As suppose the line AB to contayne 3. and the lyne CD to contayne 9. theÌ doth the line AB measure the line CD for that if it be take certayne times namely 3. tymes it maketh precisely the lyne CD that is 9. without more or lesse Agayne if the sayd lesse lyne AB be taken from the greater CD as often as it may be namely 3. tymes there shall remayne nothing of the greater So the nuÌber 3. is sayde to measure 12. for that beyng taken certayne tymes namely foure tymes it maketh iust 12. the greater quantitie and also beyng taken from 12. as often as it may namely 4. tymes there shall remayne nothyng And in this meaning and signification doth Euclide vndoubtedly here in this define a part saying that it is a lesse magnitude in comparison of a greater when the lesse measureth the greater As the lyne AB before set contayning 3. is a lesse quantitie in comparison of the lyne CD which containeth 9. and also measureth it For it beyng certayne tymes taken namely 3. tymes precisely maketh it or taken from it as often as it may there remayneth nothyng Wherfore by this definition the lyne AB is a part of the lyne CD Likewise in numbers the number 5. is a part of the number 15. for it is a lesse number or quantitie compared to the greater and also it measureth the greater for beyng taken certayne tymes namely 3. tymes it maketh 15. And this kynde of part is called commonly pars metiens or mensurans that is a measuryng part some call it pars multiplicatina and of the barbarous it is called pars aliquota that is an aliquote part And this kynde of parte is commonly vsed in Arithmetique The other kinde of a part is any lesse quantitie in comparison of a greater whether it be in number or magnitude and whether it measure or no. As suppose the line AB to be 17. and let it be deuided into two partes in the poynt C namely into the line AC the line CB and let the lyne AC the greater part containe 12. and let the line BC the lesse part contayne 5. Now eyther of these lines by this definition is a part of the whole lyne AB For eyther of them is a lesse magnitude or quaÌtity in coÌparisoÌ of the whole lyne AB but neither of theÌ measureth the whole line AB for the lesse lyne CB contayning 5. taken as ofteÌ as ye list will neuer make precisely AB which contayneth 17. If ye take it 3. tymes it maketh only 15. so lacketh it 2. of 17. which is to litle If ye take it 4. times so maketh it 20. theÌ are there thre to much so it neuer maketh precisely 17. but either to much or to litle Likewise the other part AC measureth not the whole lyne AB for takeÌ once it maketh but 12. which is lesse then 17. and taken twise it maketh 24. which are more then 17. by â So it neuer precisely maketh by takyng therof the whole AB but either more or lesse And this kynde of part they commonly call pars
EG the sides which include the equall angles are proportionall Wherefore the parallelogramme ABCD is by the first definition of the sixth like vnto the parallelogramme EG And by the same reason also the parallelogramme ABCD is like to the parallelogramme KH wherefore either of these parallelogrammes EG and KH is like vnto the parallelogramme ABCD. But rectiline figures which are like to one and the same rectiline figure are also by the 21. of the sixth like the one to the other Wherefore the parallelogramme EG is like to the parallelogramme HK Wherfore in euery parallelogramme the parallelogrammes about the dimecient are like vnto the whole and also like the one to the other Which was required to be proued ¶ An other more briefe demonstration after Flussates Suppose that there be a parallelograÌme ABCD whose dimeâient let bâ Aâ about which let consist these parallelogrammes EK and TI hauing the angles at the pointes â and ãâ¦ã with the whole parallelogramme ABCD. Then I say that those parallelogrammes EK and TI are like to the whole parallelogramme DB and also alâ like the one to the other For forasmuch as BD EK and TI are parallelogrammes therefore the right line AZG falling vpon these parallell lines AEB KZT and DI G or vpon these parallell lines AKD EZI and BTG maketh these angles equall the one to the other namely the angle EAZ to the angle KZA the angle EZA to the angle KAZ and the angle TZG to the angle ZGI and the angle TGZ to the angle IZG and the angle BAG to the angle AGD and finally the angle BGA to the angle DAG Wherefore by the first Corollary of the 32. of the first and by the 34. of the first the angles remayning are equall the one to the other namely the angle B to the angle D and the angle E to the angle K and the angle T to the angle I. Wherefore these triangles are equiangle and therefore like the one to the other namely the triangle ABG to the triangle GDA and the triangle AEZ to the triangle ZKA the triangle ZTG to the triangle GIZ. Wherefore as the side AB is to the side BG so is the side AE to the side EZ and the side ZT to the side TG Wherefore the parallelogrammes contayned vnder those right lines namely the parallelogrammes ABGD EK TI are like the one to the other by the first definition of this booke Wherefore in euery parallelogramme the parallelogrammes c. as before which was required to be demonstrated ¶ A Probleme added by Pelitarius Two equiangle Parallelogrammes being geuen so that they be not like to cut of from one of them a parallelogramme like vnto the other Suppose that the two equiangle parallelogrammes be ABCD and CEFG which let not be like the one to the other It is required from the Parallelogramme ABCD to cut of a parallelogramme like vnto the parallelogramme CEFG Let the angle C of the one be equall to the angle C of the other And let the two parallelogrammes be so ãâã that the lines BC CG may make both one right line namely BG Wherefore also the right lines DC and CE shall both make one right line namely DE. And drawe a line from the poynt F to the poynt C and produce the line FC till it coÌcurre with the line AD in the poynt H. And draw the line HK parallell to the line CD by the 31. of the first Then I say that from the parallelogramme AC is cut of the parallelograÌme CDHK like vnto the parallelograÌme EG Which thing is manifest by thys 24. Proposition For that both the sayd parallelogrammes are described about one the selfe same dimetient And to the end it might the more plainly be seene I haue made complete the Parallelogramme ABGL ¶ An other Probleme added by Pelitarius Betwene two rectiline Superficieces to finde out a meane superficies proportionall Suppose that the two superficieces be A and B betwene which it is required to place a meane superficies proportionall Reduce the sayd two rectiline figures A and B vnto two like parallelograÌmes by the 18. of this booke or if you thinke good reduce eyther of them to a square by the last of the second And let the said two parallelogrammes like the one to the other and equall to the superficieces A and B be CDEF and FGHK And let the angles F in either of them be equall which two angles let be placed in such sort that the two parallelogrammes ED and HG may be about one and the selfe same dimetient CK which is done by putting the right lines EF and FG in such sort that they both make one right line namely EG And make coÌplete the parallelograÌme CLK M. Then I say that either of the supplements FL FM is a meane proportionall betwene the superficieces CF FK that is betwene the superficieces A and B namely as the superficies HG is to the superficies FL so is the same superficies FL to the superficies ED. For by this 24. Proposition the line HF is to the line FD as the line GF is to the line FE But by the first of this booke as the line HF is to the line FD so is the superficies HG to the superficies FL and as the line GF is to the line FE so also by the same is the superficies FL to the superficies ED. Wherfore by the 11. of the fift as the superficies HG is to the superficies FL so is the same superficies FL to the superficies ED which was required to be done The 7. Probleme The 25. Proposition Vnto a rectiline figure geuen to describe an other figure lyke which shal also be equall vnto an other rectiline figure geuen SVppose that the rectiline figure geueÌ wherunto is required an other to be made like be ABC and let the other rectiline figure whereunto the same is required to be made equal be D. Now it is required to describe a rectiline figure like vnto the figure ABC and equall vnto the figure D. Vppon the line BC describe by the 44. of the first a parallelogramme BE equall vnto the triangle ABC and by the same vpon the line CE describe the parallelogramme C M equall vnto the rectiline figure D and in the said parallelogramme let the angle FCE be equall vnto the angle CBL And forasmuch as the angle FCE is by construction equall to the angle CBL adde the angle BCE common to them both Wherefore the angles LBC and BCE are equall vnto the angles BCE and ECF but the angles LBC and BCE are equall to two right angles by the 29. of the first wherfore also the angles BCE and ECF are equall to two right angles Wherfore the lines BC and CF by the 14. of the first make both one right line namely BF and in like sort do the lines LE and EM make both one right line namely LM Then by the
namely the circumference BLAKOC is equall vnto the circumâerence remayning of the selfe same circle ABC namely to the circumference CPBLAK Wherfore the angle BPC is equall vnto the angle COK by the 27. of the third Wherfore by the 10. definition of the third the segment BPC is like vnto the segment COK and they are set vpon equall right lines BC and KC But like segmentes of circles which consist vpon equall right lines are also equall the one to the other by the 24. of the third Wherfore the segment BPC is equall vnto the segment COK. And the triangle GBC is equall vnto the triangle GCK. Wherfore the sector GBC is equall vnto the sector GCK. And by the same reason also the sector GKL is equall vnto either of the sectors GBC and GCK. Wherfore the three sectors GBC and GCK and GKL are equall the one to the other And by the same reason also the sectors HEF and HFM and HMN are equall the one to the other Wherefore how multiplex the circumference BL is to the circumference BC so multiplex is the sector GLB to the sector GBC And by the same reason how multiplex the circumference NE is to the circumference EF so multiplex is the sector HEN to the sector HEF If therfore the circumference BL be equall vnto the circumference EN the sector also BGL is equall vnto the sector EHN. And if the circumfereÌce BL excede the circumference EN the sector also BGL excedeth the sector EHN. And if the circumference be lesse the sector also is lesse Now theÌ there are foure magnitudes namely the two circumferences BC and EF and the two sectors GBC HEF and to the circumference BC to the sector GBC namely to the first and the third are taken equemultiplices that is the circumference BL and the sector GBL and likewise to the circumference EF and to the sector HEF namely to the second and fourth are taken certayne other equimultiplices namely the circumference EN and the sector HEN. And it is proued that if the circumference BL excede the circumference EN the sector also BGL excedeth the sector EHN. And if the circumference be equall the segment also is equall and if the circumference be lesse the segment also is lesse Wherfore by the conuersion of the sixt definition of the fifth as the circumference BC is to the circumference EF so is the sector GBC vnto the sector HEF which is all that was required to be proued Corollary And hereby it is manifest that as the sector is to the sector so is angle to angle by the 11. of the fifth Flussates here addeth fiue Propositions wherof one is a Probleme hauing three Corollaryes following of it and the rest are Theoremes which for that they are both witty also serue to great vse as we shall afterward see I thought not good to omitte but haue here placed them but onely that I haue not put them to followe in order with the Propositions of Euclide as he hath done ¶ The first Proposition added by Flussates To describe two rectiline figures equall and like vnto a rectiline figure geuen and in like sort situate which shall haue also a proportion geuen Suppose that the rectiline figure geuen be ABH And let the proportion geuen be the proportion of the lines GC and CD And by the 10. of this booke deuide the line AB like vnto the line GD in the poynt E so that as the line GC is to the line CD so let the line AE be to the line EB And vpon the line AB describe a semicircle AFB And from the poynt E erect by the 11. of the first vnto the line AB a perpendicular line EF cutting the circumference in the poynt F. And draw these lines AF and FB And vpon either of these lines describe rectiline figures like vnto the rectiline figure AHB and in like sort situate by the 18. of the sixt which let be AKF FIB Then I say that the rectiline figures AKF and FIB haue the proportion geueÌ namely the proportion of the line GC to the line CD and are equall to the rectiline figure geuen ABH vnto which they are described like and in like sort situate For forasmuch as AFB is a semicircle therefore the angle AFB is a right angle by the 31. of the third and FE is a perpendicular line Wherefore by the 8. of this booke the triangles AFE and FBE are like both to the whole triangle AFB and also the one to the other Wherefore by the 4. of this booke as the line AF is to the line FB so is the line AE to the line EF and the line EF to the line EB which are sides coÌtayning equall angles Wherfore by the 22. of this booke as the rectiline figure described of the line AF is to the rectiline figure described of the line FB so is the rectiline figure described of the line AE to the rectiline figure described of the line EF the sayd rectiline figures being like and in like sort âituate But as the rectiline figure described of the line AE being the âirst is to the rectiline âigure described of the line EF being the second so is the line AE the first 10. the line âB the third by the 2. Corollary of the 20. of thys booke Wherfore the rectiline figure described of the line AF is to the rectiline figure described of the line FB as the line Aâ is to the line EB But the line AE is to the line EB by construction as the line GC is to the line CD Wherefore by the 11. of the fift as the line GC is to the line CD so is the rââtiline âigure described of the line AF to the rectiline âigure described ãâã the line âB the sayd rectiline figures being like and in like sort described But the ãâ¦ã described oâ the lines AF and FB are equall to the rectiline âigure dâââââbed oâ the line AB vnto which they are by construction described lyke and in like sort situate Wherefore there are described two rectiline figures AKF and FIB equâll and like vnto the rectiline figure geuen ABH and in like sort situate and they haâe also the one to the other the proportion geuen namely the proportion of the line GC to the line CD which was required to be done ¶ The first Corollary To resolue a rectiline figure geue into two like rectiline âigures which shall hauâ also a proportiâ geâeÌ For iâ there be put three right lines in the proportioÌ geueÌ and if the line AB be cuâ in the same proportion that the first line is to the third the rectiline âigures described of the lineâ Aâ and FB which figures haue the same proportion that the lines AE and EB haue shall be in double proportion to that which the lines AF and FB are by the âirsââorollary oâ the 20. oâ this booke Wherefore the right lines AF and FB are the oâe to the other in the same
number euenly euen is that which may be diuided into two euen partes and that part agayne into two euen partes and so continually deuiding without stay ãâã come to vnitie As by exampleâ 64. may be deuided into 31 and. 32. And either of these partes may be deuided into two euen partes for 32 may be deuided into 16 and 16. Againe 16 may be deuided into 8 and 8 which are euen partes and 8 into 4 and 4. Againe 4 into â and â and last of all may â be deuided into one and one 9 A number euenly odde called in latine pariter impar is that which an euen number measureth by an odde number As the number 6 which 2 an euen number measureth by 3 an odde number thre times 2 is 6. Likewise 10. which 2. an euen number measureth by 5 an odde number In this diffinition also is found by all the expositors of Euclide the same want that was found in the diffinition next before And for that it extendeth it selfe to large for there are infinite numbers which euen numbers do measure by odde numbers which yet after their mindes are not eueÌly odde nuÌbers as for example 12. For 4 an eueÌ nuÌber measureth 12 by â an odde numberâ three times 4 is 12. yet is not 12 as they thinke an euenly odde number Wherfore Campane amendeth it after his thinking by adding of this worde all as he did in the first and defineth it after this maner A number euenly odde is when all the euen numbers which measure it do measure it by vneuen tymes that is by an odde number As 10. is a number euenly odde for no euen number but onely 2 measureth 10. and that is by 5 an odde number But not all the euen numbers which measure 12. do measure it by odde numbers For 6 an euen number measureth 12 by 2 which is also euen Wherfore 12 is not by this definition a number euenly odde Flussates also offended with the ouer large generalitie of this definition to make the definition agree with the thing defined putteth it after this maner A number euenly odde is that which an odde number doth measure onely by an euen number As 14. which 7. an odde number doth measure onely by 2. which is an euen number There is also an other definition of this kinde of number commonly geuen of more plaines which is this A number euenly odde in that which may be deuided into two equall partes but that part cannot agayne be deuided into two equall partes as 6. may be deuided into two equall partes into 3. and 3. but neither of them can be deuided into two equall partes for that 3. is an odde number and suffereth no such diuision 10 A number oddly euen called im lattin in pariter par is that which an odde number measureth by an euen number As the number 12 for 3. an odde number measureth 12. by 4. which is an euen number three times 4. is 12. This definition is not founde in the greeke neither was it doubtles euer in this maner written by Euclide which thing the slendernes and the imperfection thereof and the absurdities following also of the same declare most manifestly The definition next before geuen is in substance all one with this For what number soeuer an euenâ number doth measure by an odde the selfe same number doth an odde number measure by an euen As 2. an eueÌ number measureth 6. by 3. an odde number Wherfore 3. an odde number doth also measure the same number 6. by 2. an eueÌ nuÌber Now if these two definitions be definitions of two distinct kindes of numbers then is this number 6. both euenly euen and also euenly odde and so is contayned vnder two diuers kindes of numbers Which is directly agaynst the authoritie of Euclide who playnely pâouoâh here after in the 9. booke that euery nomber whose halfe is an odde number is a number euenly odde onely Flussates hath here very well noted that these two euenly odde and oddely euen were taken of Euclide for on and the selfe same kinde of nomber But the number which here ought to haue bene placed is called of the best interpreters of Euclide numerus pariter par nupar that is a number eueÌly eueÌ and eueÌly odde Yeâ and it is so called of Euclide him selfe in the 34. proposition of his 9. booke which kinde of number Campanus and Flussates in steade of the insufficient and vâapt definition before geuen assigne this definition A number euenly euen and euenly âdde is that which an euen number doth measure sometime by an euen number and sometime by an odde As the number 12 for 2. an euen number measureth 12. by 6. an euen number two times 6. iâ 12. Also 4. an euen number measureth the same number 12. by 3. an odde number Add therefore is 12. a number euenly euen and euenly odde and so of such others There is also an other definition geuen of this kinde of number by Boetius and others commonly which is thus A number eueâly euen and euenly odde is that which may be deuided into two equall partes and eche of them may aâayne be deuided into two equall partes and so forth But this deuision is at lenghth stayd and continueth not till it come to vnitie As for example 48 which may be deuided into two equall partes namely into 24. and 24. Agayne 24. which is on of the partes may be deuided into two equall partes 12. and 12. Agayne 12. into 6. and 6. And agayne 6 may be deuided into two equall partes into 3. and 3 but 3. cannot be deuided into two equall partes Wherefore the deuision there stayeth and continueth not till it come to vnitie as it did in these numbers which are euenly euen only 11 A number odly odde is that which an odde number doth measure by an odde number As 25 which 5. an odde number measureth by an odde number namely by 5. Fiue times fiue is 25 Likewise 21. whom 7. an odde number doth measure by 3 which is likewise an odde number Three times 7. is 21. Flussatus geueth this definition following of this kinde of number which is all one in substance with the former definition A number odly odde it that which onely an odde number doth measure As 15. for no number measureth 15. but onely 5. and 3 also 25 none measureth it but onely 5. which is an odde number and so of others 12 A prime or first number is that which onely vnitie doth measure As 5.7.11.13 For no number measureth 5 but onely vnitie For v. vnities make the number 5. So no number measureth 7 but onely vnitie .2 taken 3. times maketh 6. which is lesse then 7 and 2. taken 4. times is 8 which is more then 7. And so of 11.13 and such others So that all prime numbers which also are called first numbers and numbers vncomposed haue
no part to measure theÌ but onely vnitie 13 Numbers prime the one to the other are they which onely vnitie doth measure being a common measure to them As 15. and 22. be numbers prime the one to the other .15 of it selfe is no prime number for not onely vnitie doth measure it but also the numbeââ 5. and 3 for â times 5. is lx Likewise 22. is of it selfe no prime number for it is measured by 2. and 11 besides vnitie For 11. twise or 2. eleuen times make 22. So that although neither of these two numbers 15. and 22. be a prime or incomposed number but eyther haue partes of his owne whereby it may be measured beside vnitie yet compared together they are prime the one to the other for no one number doth as a common measure measure both of them but onely vnitie which is a common measure to all numbers The numbers 5. and 3. which measure 15. will not measure 22 againe the numbers 2. and 11. which measure 22 do not measure 15. 14 A number composed is that which some one number measureth A number composed is not measured onely by vnitie as was a prime number but hath some number which measureth it As 15 the number 3. measureth 15 namely taken 5. times Also the number 5. measureth 15 namely taken 3. times 5. times â and 3. times 5 is 15. Likewise 18. is a composed number it is measured by these numbers 6.3.9.2 and so of others These numbers are also called commonly second numbers as contrary to prime or first numbers 15 Numbers composed the one to the other are they which some one number being a common measure to them both doth measure As 12. and 8. are two composed numbers the one to the other For the number 4 is a common measure to them both 4. taken three times maketh 12 and the same 4. taken two tymes maketh 8. So are 9. and 15 3. measureth them both Also 10. and 25 for 5. measureth both of them and so infinitely of others In thys do numbers composed the one to the other or second numbers differre from numbers prime the one to the other for that two numbers being composed the one to the other ech of them seuerally is of necessitie a composed number As in the examples before 8. and 12. are composed numbers likewise 9. and 15 also 10 and 25 but if they be two numbers prime the one to the other âit is not of necessitie that ech of them seuerally be a prime number As 9. and 22. are two numbers prime the one to the other no one number measureth both of them and yet neither of them in it selfe and in his owne nature is a prime number but ech of them is a composed number For 3. measureth 9 and 11. and 2. measure 22. 16 A number is sayd to multiply a number when the number multiplyed is so oftentimes added to it selfe as there are in the number multiplying vnities and an other number is produced In multiplication are euer required two numbers the one is whereby ye multiply commonly called the multiplier or multiplicant the other is that which is multiplied The number by which an other is multiplied namely the multiplyer is sayd to multiply As if ye will multiply 4. by 3 then is three sayd to multiply 4 therefore according to this definition because in 3. there are three vnities adde 4,3 times to it selfe saying 3. times 4 so shall ye bring forth an other number namely 12 which is the summe produced of that multiplication and so of all other multiplications 17 When two numbers multiplying them selues the one the other produce an other the number produced is called a plaine or superficiall number And the numbers which muliply them selues the one by the other are the sides of that number As let these two numbers 3. and 6 multiply the one the other saying 3. times 6 or sixe tymes 3 they shall produce 18. Thys number 18. thus produced is called a plaine number or a superficiall number And the two multiplying numbers which produced 12 namely 3. and 6 are the sides of the same superficiall or plaine number that is the length and breadth thereof Likewise if 9. multiply 11 or eleuen nine there shal be produced 99. a plaine number whose sides are the two numbers 9. and 11 â as the length and breadth of the same They are called plaine and superficiall numbers because being described by their vnities on a plaine superficies they represent some superficiall forme or figure Geometricall hauing length and breadth As ye see of this example and so of others And all such plaine or superficiall numbers do euer represent right angled figures as appeareth in the example 18 When three numbers multiplyed together the one into the other produce any number the number produced is called a solide number and the numbers multiplying them selues the one into the other are the sides therof As taking these three nuÌbers 3.4 5. multiply the one into the other First 4. into 5. saying foure times 5. is 20 then multiply that number produced nameây 20. into 3 which is the third number so shall ye produce 60. which is a solide number and the three numbers which produced the number namely 3.4 and 5. are the sides of the same And they are called solide numbers because being described by their vnities they represent solide and bodylicke figures of Geometry which haue length breadth and thicknes As ye see this number 60. expressed here by hys vnities Whose length is hys side 5 his breadth is 3 and thicknes 4. And thus may ye do of all other three nuÌbers multiplying the one the other 19 A square number is that which is equally equall or that which is contayned vnder two equall numbers As multiply two equall numbers the one into the other As 9. by 9. ye shall produce 81 which is a square number Euclide calleth it a number equally equall because it is produced of the multiplication of two equall numbers the one into the other Which numbers are also sayd in the second definition to contayne a square number As in the definitions of the second booke two lines are sayd to containe a square or a parallelogramme figure It is called a square number because being described by his vnities it representeth the figure of a square in Geometry As ye here see doth the number 81. whose sides that is to say whose length and breadth are 9. and 9 equall numbers which also are sayd to contayne the square number 81 and so of others 20 A cube number is that which is equally equall equally or that which is contayned vnder three equall numbers As multiply three equall numbers the one into the other as 9 9 and 9 first 9. by 9 so shall ye haue 81 which agayne multiply by 9 so shall ye produce 729. which is a cube number And Euclide calleth
denomination of the number B. For how often B measureth A so many vnities let there be in C. And let D be vnitie And forasmuch as B measureth A by those vnities which are in C and vnitie D measureth C by those vnities which are in C therefore vnitie D so many times measureth the number C as B doth measure A. Wherefore alternately by the 15. of the seuenth vnitie D so many times measureth B as C doth measure A. Wherfore what part vnitie D is of the number B the same part is C of A. But vnitie D is a part of B hauing his denomination of B. VVherfore C also is a part of A hauing his denomination of B. VVherfore A hath C as a part taking his denomination of B which was required to be proued The meaning of this Proposition is that if three measure any number that number hath a third part and if foure measure any number the sayd number hath a fourth part And so forth ¶ The 35. Theoreme The 40. Proposition If a number haue any part the number wherof the part taketh his denomination shall measure it SVppose that the number A haue a part namely B and let the part B haue his denomination of the number C. Then I say that C measureth A. Let D be vnitie And forasmuch as B is a part of A hauing his denomination of C and D being vnitie is also a part of the number C hauing his denominatioÌ of C therefore what part vnitie D is of the number C the same part is also B of A wherefore vnitie D so many âââes measureth the number C as B measurâââ A. Whârefâre âââernââely by ââe 15. âf the sââânth vnitie D so many ââmes mââââreth the nuââbeâ B ãâã C measââeth A. Wherefâââ C measureth A which was requiââd to bâ proued This Proposition is the conuerse of the former and the meaning therof is that euery number hauing a third part is measured of three and hauing a fourth part is measured of foure And so forth ¶ The 6. Probleme Th 41. Proposition To finde out the least number that containeth the partes geuen SVppose that the partes geuen be A B C namely let A be an halfe part B a third part C a fourth part Now it is required to finde out the least nuÌber which coÌtaineth the partes A B C. Let the said partes A B C haue their denominations of the numbers D E F. And take by the 38. of the seuenth âhe least number which the numbers D E F measure and let the same be G. And forasmuch as the numbers D E F measure the number G therfore the number G hath paâtes denominated of the nuÌbers D E F by the 39. of the seueÌth But the parts A B C haue their denominatioÌ of the numbers D E F. Wherfore G hath those partes A B C. I say also that it is the least number which hath these partes For if G be not the least number which containeth those partes A B C then let there be some number lesse then G which containeth the saide partes A B C. And suppose the same to be the number H. And forasmuch as H hath the said partes A B C therfore the numbers that the partes A B C take their denominations of shall measure H by the 40. of the seuenth But the numbers wheroâ the partes A B C take their denominations of are D E F. Wherfore the numbers D E F mâasuâe thâ number H which is lesse then G which is impossible For G is supposed to be tââ lââsâ number that the numbers D E F do measure Wherfore there is no number lesse then G which containeth these partes A B C which was required to be done Corrolary Hereby it is manifest that if there be taken the least number that numbers how many soeuer do measure the sayd number shall be the least which hath the partes denominated of the sayd numbers how many soeuer Campane after he hath taught to finde out the first least number that conâayneth the partes geuen teacheth also to finde out the second least number that is which except the least of all is lesse then all other and also the third least and the fourth c. The second is found out by doubling the number G. For the numbers which measure the nuÌber G âhall also measure the double therof by the 5. commoâ sentence of the seuenth But there cannot be geuen a number greater then the number G lesse then the double therof whom the partes geuen shall ââasureâ For forasmuch as the partes geuen do âeasurâ the whole namely which is lesse then the double and they also measure the part taken away namely the number G they should also measure the residue namely a number lesse then G which is proued to be the lest number that they do measure which is impossibleâ wherefore the second number which the said partes geuen do measureâ must exceeding G needes reache to the double of G and the third to the treble and the fourth to the quadruple and so inâinitely for those partes can neuer measure any number lesse then the number G. By this Proposition also it is easie to find out the least number containing the partes geuen of partes As if we would finde out the least number which contayneth one third ãâã ââ an halâe part and one fourth part of a third part reduce the said dââers fractiââ into simple fraction by the common ãâã of reducing of frâââions namely the ãâã of an halâe into a ãâã part of an wholeâ and âhe fourth of thiâd into a twelfth part of anâ whole And then by this Probleme search out the least number which contayneth a sixâ part and a twelfth part and so haue you done The end of the seuenth booke of Euclides Elementes ¶ The eighthe booke of Euclides Elementes AFter that Euclide hath in the seuenth booke entreated of the proprieties of numbers in generall and of certayne kindes thereof more specially and of prime and composed numbers with others now in this eight booke he prosecuteth farther and findeth out and demonstrateth the properties and passions of certayne other kindes of numbers as of the least numbers in proportion and how such may be found out infinitely in whatsoeuer proportion which thing is both delectable and to great vse Also here is entreated of playne numbers and solide and of theyr sides and proportion of them Likewise of the passions of numbers square and cube and of the natures and conditions of their sides and of the meane proportionall numbers of playne solide square and cube numbers with many other thinges very requisite and necessary to be knowne ¶ The first Theoreme The first Proposition If there be numbers in continuall proportion howmanysoeuer and if their extremes be prime the one to the other they are the least of all numbers that haue one and the same proportion with them SVppose that the numbers
Now forasmuch as D measureth AB and BC it also measureth the whole magnitude AC And it measureth AB Wherefore D measureth these magnitudes CA and AB Wherefore CA AB are commensurable And they are supposed to be incoÌmensurableâ which is impossible Wherfore no magnitude measureth AB and BC. Wherefore the magnitudes AB and BC are incommensurable And in like sort may they be proued to be incommensurable if the magnitude AC be supposed to be incommensurable vnto BC. If therefore there be two magnitudes incommensurable composed the whole also shall be incommensurable vnto either of the two partes component and if the whole be incommensurable to one of the partes component those first magnitudes shall be incommensurable which was required to be proued ¶ A Corollary added by Montaureus If an whole magnitude bee incommensurable to one of the two magnitudes which make the whole magnitude it shall also be incommensurable to the other of the two magnitudes For if the whole magnitude AC be incoÌmensurable vnto the magnitude BC then by the 2 part of this 16. Theorâme the magnitudes AB and BC shall be incommensurable Wherefore by the first part of the same Theoreme the magnitude AC shall be incommensurable to either of these magnitudes AB and BC. This Corollary ãâã vseth in the demonstration of the â3 Theoreme also of other Propositions ¶ An Assumpt If vpon a right line be applied a parallelogramme wanting in figure by a square the parallelogramme so applied is equall to that parallelogramme which is contayned vnder the segmentes of the right line which segmentes are made by reason of that application This Assumpt I before added as a Corollary out of Flussates after the 28. Proposition of the sixt booke ¶ The 14. Theoreme The 17. Proposition If there be two right lines vnequall and if vpon the greater be applied a parallelogramme equall vnto the fourth part of the square of the lesse line and wanting in figure by a square if also the parallelogramme thus applied deuide the line where vpon it is applied into partes commensurable in length then shall the greater line be in power more then the lesse by the square of a line commensurable in length vnto the greater And if the greater be in power more then the lesse by the square of a right line commensurable in length vnto the greater and if also vpon the greater be applied a parallelograÌme equall vnto the fourth part of the square of the lesse line and wanting in figure by a square then shall it deuide the greater line into partes commensurable But now suppose that the line BC be in power more then the line A by the square of a line commensurable in length vnto the line BC. And vpon the line BC let there be applied a rectangle parallelograme equall vnto the fourth part of the square of the line A and wanting in figure by a square and let the sayd parallelograme be that which is contained vnder the lines BD and DC Then must we proue that the line BD is vnto the line DC commensurable in length The same constructions and suppositions that were before remayning we may in like sort proue that the line BC is in power more then the line A by the square of the line FD. But by suppositioÌ the line BC is in power more theÌ the line A by the square of a line coÌmensurable vnto it in length Wherfore the line BC is vnto the line FD coÌmensurable in length Wherefore the line composed of the two lines BF and DC is coÌmensurable in length vnto the line FD by the second part of the 15. of the tenth Wherefore by the 12. of the tenth or by the first part of the 15. of the tenth the line BC is commensurable in length to the line composed of BF and DC But the whole line conposed BF and DC is commensurable in length vnto DC For BF as before hath bene proued is equall to DC Wherefore the line BC is commensurable in length vnto the line DC by the 12. of the tenth Whâââfore also the line BD is commensurable in length vnto the line DC by the second part of thâ 15. of the teâth If therfore there be two right lines vnequall and if vpon the greater be appliâd a parallelograme equall vnto the fourth part of the square of the lesse and wanting in figure by a square if also the parallelograme thus applied deuide the line whereupon it is applied into partes commensurable in length then shall the greater line be in power more then the lesse by the square of a line commensurable in length vnto the greater And if the greater be in power more then the lesse by the square of a line commeÌsurable in length vnto the greater and if also vpon the greater be applied a parallelograme equall vnto the fourth part of the square made of the lesse and wanting in figure by a square then shall it deuide the greater line into partes commensurable in length which was required to be proued Campanâ after this proposition reacheth how we may redily apply vpon the line BC a parallelograme equall to the fourth part of the square of halfe of the line A and wanting in figure by a square after this maner Deuide the line BC into two lines in such sort that halfe of the line A shal be the meane proportionall betwene those two lines which is possible when as the line BC is supposed to be greater then the line A and may thus be done Deuide the line BC into two equal partes in the point E and describe vpon the line BC a semicircle BHC And vnto the line BC and from the point C erect a perpâdicular line CK and put the line CK equall to halfe of the line Aâ And by the point K draw vnto the line EC a parallel line KH cutting the semicircle in the point H which it must needes cut foâasmuch as the line BC is greater then the line A And froÌ the point H draw vnto the line BC a perpendicular liâe HD which line HDâ forasmuch as by the 34 of the first it is equall vnto the line KC shall also be equall to halfe of the line A draw the lines BH and HC Now then by the ââ of the third the angle BHC is a right aâgle Wherefore by the corollary of the eight of the sixt booke the line HD is the meane proportionall betwene the lines BD and DC Wherefore the halfe of the line A which is equall vnto the line HD is the meane proportionall betwene the lines BD and DC Wherefore that which is contained vnder the lines BD and DC is equall to the fourth part of the square of the line A. And so if vpon the line BD be described a rectangle parallelograme hauing his other side equall to the line DC there shal be applied vpon the line BC a rectangle parallelograme equall vnto the square of halfe of the line A and wanting in figure by
if the line CB be irrationall they shall be incommensurable But as the line BD which is equall to the line BA is to the line BC so is the square AD to the parallelograÌme AC Wherefore the parallelogramme AC shall be incommensurable to the square AD which is rationall for that the line AB wherupon it is described is supposed to be rationall Wherefore the parallelogramme AC which is contayned vnder the rationall line AB and the irrationall line BC is irrationall ¶ An Assumpt If there be two right lines as the first is to the second so is the square which is described vpon the first to the parallelograme which is contained vnder the two right lines Suppose that there be two right lines AB and BC. Then I say that as the line AB is to the line BC so is the square of the line AB ââ that which is contained vnder the lines AB and BC. Describe by the 46. of the first vpon the line AB a square AD. And make perfect the parallelograme AC Now for that as the line AB is to the line BC for the line AB is equall to the line BD so is the square AD to the parallelograme CA by the first of the sixâ and AD is the square which is made of the line AB and AC is that which is contained vnder the lines BD and BC that is vnder the lines AB BC therfore as the line AB is to the line BC so is the square described vppon the the line AB to the rectangle figure contained vnder the lines AB BC. And conuersedly as the parallelograme which is contained vnder the lines AB and BC is to the square of the line AB so is the line CB to the line BA ¶ The 19. Theoreme The 22. Proposition If vpon a rationall line be applied the square of a mediall line the other side that maketh the breadth thereof shal be rationall and incommensurable in length to the line wherupon the parallelograme is applied SVppose that A be a mediall line and let BC be a line rationall and vpon the line BC describe a rectangle parallelograme equall vnto the square of the line A and let the same be BD making in breadth the line CD Then I say that the line CD is rationall and incoÌmensurable in length vnto the line CB. For forasmuch as A is a mediall line it containeth in power by the 21. of the tenth a rectangle parallelograme comprehended vnder rationall right lines commensurable in power onely Suppose that is containe in power the parallelograme GF and by supposition it also containeth in power the parallelograme BD. Wherefore the parallelograme BD is equall vnto the parallelograme GF and it is also equiangle vnto it for that they are ech rectaÌgle But in parallelogrames equall and equiangle the sides which containe the equall angles are reciprocall by the 14. of the sixt Wherfore what proportioÌ the line BC hath to the line EG the same hath the line EF to the line CD Therefore by the 22. of the sixt as the square of the line BC is to the square of the line EG so is the square of the line EF to the square of the line CD But the square of the line BC is commensurable vnto the square of the line EG by supposition For either of them is rationall Wherefore by the the 10. of the tenth the square of the line EF is commensurable vnto the square of the linâ CD But the square of the line EF is rationall Wherefore the square of the line CD is likewise rationall Wherefore the line CD is rational And forasmuch as the line EF is incoâmensurable in length vnto the line EG for they are supposed to be commensurable in power onely But as the line EF is to the line EG so by the assumpt going before is the square of the line EF to the parallelograme which is contained vnder the lines EF and EG Wherefore by the 10. of the tenth the square of the line EF is incommensurable vnto the parallelograme which is contained vnder the lines FE and EG But vnto the square of the line EF the square of the line CD is commensurable for it is proued that âither of them is a rationall linâ And that which is contained vnder the lines DC and CB is commensurable vnto that which is contained vnder the lines FE and EG For they are both equall to the square of the line A. Wherefore by the 13. of the tenth the square of the line CD is incommensurable to that which is contained vnder the lines DC and CB. But as the square of the line CD is to that which is contained vnder the lines DC and CB so by the assumpt going before is the line DC to the line CB. Wherefore the line DC is incommensurable in length vnto the line CB. Wherefore the line CD is rationall and incommensurable in length vnto the line CB. If therefore vpon a rationall line be applied the square of a mediall line the other side that maketh the breadth thereof shal be rationall and incommensurable in length to the line whereupon the parallelogramme is applied which was required to be proued A square is sayd to be applied vpon a line when it or a parallelograme equall vnto it is applied vpon the sayd line If vpon a rationall line geuen we will apply a rectangle parallelograme equall to the square of a mediall line geuen and so of any line geuen we must by the 11. of the sixt finde out the third line proportionall with the rationall line and the mediall line geuen so yet that the rationall line be the first and the mediall line geuen which containeth in power the square to be applied be the second For then the supeâficies contained vnder the first and the third shal be equall to the square of the midle line by the 17. of the sixt ¶ The 20. Theoreme The 23. Proposition A right line commensurable to a mediall line is also a mediall line SVppose that A be a mediall line And vnto the line A let the line B be commensurable either in length in power or in power only Then I say that B also is a mediall line Let there be put a rationall line CD And vpon the line CD apply a rectangle parallelograme CE equall vnto the square of the line A and making in breadth the line ED. Wherefore by the proposition going before the line ED is rationall and incommensurable in length vnto the line CD And againe vpon the line CD apply a recâangle parallelograme CF equall vnto the square of the line B and making in breadth the line DF. And forasmuch as the line A is commensurable vnto the line B therefore the square of the line A is commensurable to the square of the line B. But the parallelograme EC is equall to the square of the linâ A and the parallelograme CF is equall to
parallelogramme shall afterward be taught in the 27. and 28. Propositions of thys booke ¶ A Corollary Hereby it is manifest that a rectangle parallelogramme contayned vnder two right lines is the meane proportionall betwene the squares of the sayd lines As it was manifest by the first of the sixt that that which is contayned vnder the lines AB and BC is the meane proportionall betwene the squares AD and CX This Corollary is put after the 53. Proposition of this booke as an Assumpt and there demonstrated which there in his place you shall finde But because it followeth of this Proposition so euidently and briefly without farther demonstration I thought it not amisse here by the way to note it ¶ The 23. Theoreme The 26. Proposition A mediall superficies excedeth not a mediall superficies by a rationall superficies FOr if it be possible let AB being a mediall superficies exceede AC being also a mediall superficies by DB being a rationall superficies And let there be put a rationall right line EF. And vpon the line EF apply a rectangle parallelogramme FH equall vnto the mediall superficies AB whose other side let be EH and from the parallelogramme FH take away the parallelogramme FG equall vnto the mediall superficies AC Wherefore by the third common sentence the residue BD is equall to the residue KH But by supposition the superficies DB is rationall Wherfore the superficies KH is also rationall And forasmuch as either of these superficieces AB and AC is mediall and AB is equall vnto FH AC vnto FG therefore either of these superficieces FH and FG is mediall and they are applyed vpon the rationall line EF. Wherefore by the 22. of the tenth either of these lines HE and EG is rationall incommensurable in length vnto the line EF. And forasmuch as the superficies DB is rationall and the superficies KH is equall vnto it therefore KH is also rationall and it is applied vpoÌ the rationall line EF for it is applied vpon the line GK which is equall to the line EF Wherefore by the 20. of the tenth the line GH is rationall and commensurable in length vnto the line GK But the line GK is equall to the line EF. Wherfore the line GH is rationall and commensurable in length vnto the line EF. But the line EG is rationall and incommensurable in length to the line EF. Wherefore by the 13. of the tenth the line EG is incommensurable in length vnto the line GH And as the line EG is to the line GH so is the square of the line EG to the parallelogramme contayned vnder the lines EG and GH by the Assumpt put before the 21. of the tenth Wherefore by the 10. of the tenth the square of the line EG is incommensurable vnto the parallelogramme contayned vnder the lines EG and GH But vnto the square of the line EG are commensurable the squares of the lines EG and GH for either of them is rationall as hath before bene proued Wherefore the squares of the lines EG and GH are incommensurable vnto the parallelogramme contayned vnder the lines EG and GH But vnto the parallelogramme contayned vnder the lines EG and GH is commensurable that which is contayned vnder the lines FG and GH twise for they are in proportion the one to the other as number is to number namely as vnitie is to the number 2 or as 2. is to 4 and therefore by the 6. of this booke they are commensurable Wherefore by the 13. of the tenth the squares of the lines EG and GH are incommensurable vnto that which is contayned vnder the lines EG and GH twise This is more brieâly concluded by the corollary of the 13. of the tenth But the squares of the lines EG and GH together with that which is contayned vnder the lines EG and GH twise are equall to the square of the line EH by the 4. of the second Wherefore the square of the line EH is incoÌmensurable to the squares of the lines EG and GH by the 16. of the tenth But the squares of the lines FG GH are rationall Wherfore the square of the line EH is irrationall Wherefore the line also EH is irrationall But it hath before bene proued to be rationall which is impossible Wherefore a mediall superficies exceedeth not a mediall superficies by a rationall superficies which was required to be proued ¶ The 4. Probleme The 27. Proposition To finde out mediall lines commensurable in power onely contayning a rationall parallelogramme LEt there be put two rationall lines commensurable in power onely namely A and B. And by the 13. of the six take the meane proportionall betwene the lines A and B and let the same line be C. And as the line A is to the line B so by the 12. of the sixt let the line C be to the line D. And forasmuch as A and B are rationall lines commensurable in power onely therfore by the 21. of the tenth that which is contayned vnder the lines A and B that is the square of the line C. For the square of the line C is equall to the parallelogramme contayned vnder the lines A anâ B by the 17. of the sixth is mediall âherfore C also is a mediall line And for that as the line A is to the line B so is the line C to the line D therfore as the square of the line A is to the square of the lyne B so is the square of the line C to the square of the line D by the 22. of the sixth But the squares of the lines A and B are commensurable for the liââs A and B aâe supposed to be rationall commeÌsurable in power onely Wherefore also the squares of the lines C and D are commensurable by the 10. of the tenth wherfore the lines C and D are commensurable in power onely And C is a mediall line Wherfore by the 23. of the tenth D also is a mediall line Wherfore C and D are mediall lynes commensurable in power onely Now also I say that they contayne a rationall parallelogramme For for that as the line A is to the line B so is the line C to the line D therfore alternately also by the 16. of the fift as the line A is to the line C so is the lyne B to the lyne D. But as the lyne A is to the lyne C so is the line C to the lyne B wherfore as the line C is to the line B so is the line B to the lyne D. Wherfore the parallelograÌme coÌtayned vnder the lines C and D is equal to the square of the line B. But the square of the lyne B is rationall Wherfore the parallelograme which is contayned vnder the lynes C and D is also rationall Wherfore there are found out mediall lines commensurablâ in powâr onâly contayning a rationall parallelogrammeâ which ãâã required to be done The 5. Probleme The
28. Proposition To finde out mediall right lynes commensurable in power onely contayning a mediall parallelogramme LEt there be put three rationall right lines commensurable in power only namely A B and C and by the 13. of the sixt take the meane proportional betwene the lines A and B let thâ same be D. And as the line B is to the line C so by the 12. of the sixt let the line D be to the line E. And forasmuch as the lines A and B are rationall commensurable in power onely therefore by the 21. of the tenth that which is contained vnder the lines A and B that is the square of the line D is mediall Wherfore D is a mediall line And forasmuch as the lines B and C are commensurable in power onely and as the line B is to the line C so is the line D to the line E wherfore the lines D and E are commensurable in power onely by the corollary of the tenth of this booke but D is a mediall line Wherefore E also is a mediall line by the 23. of this booke Wherfore D E are mediall lines commensurable in power onely I say also that they containe a mediall parallelograme For for that as the line B is to the line C so is the line D to the line E therfore alternately by the 16 of the fift as the line B is to the line D so is the line C to the line E. But as the lyne B is to the line D so is the line D to the line Aâ by conuerse proportion which is proued by the corollary of the fourth of the fifth Wherfore as the line D is to the line A so is the line C to the line E. Wherfore that which is contained vnder the lines A C is by the 16. of the sixâ equall to that which is contayned vnder the lines D E. But that which is contained vnder the lines A and C is medial by the 21. of the tenth Wherfore that which is coÌtained vnder the lines D and E is mediall Wherfore there are found out mediall lines commensurable in power onely containing a mediall superficies which was required to be done An Assumpt To finde out two square numbers which added together make a square number Let there be put two like superficiall numbers AB and BC which how to finde out hath bene taught after the 9. proposition of this booke And let them both be either euen numbers or odde And let the greater number be AB And forasmuch as if from any euen number be taken away an euen number or froÌ an odde number be taken away an odde number the residue shall be euen by the 24. and 26 of the ninth If therfore from AB being an euen number be taken away BC an euen number or from AB being an odde number be taken away BC being also odde the residue AC shall be euen Deuide the number AC into two equall partes in D wherefore the number which is produced of AB into BC together with the square number of CD is by the sixt of the second as Barlaam demonstrateth it in numbers equall to the square number of BD. But that which is produced of AB into BC is a square nuÌber For it was proued by the first of the ninth that if two like plaine numbers multiplieng the one the other produce any nuÌber the number produced shal be a square number Wherfore there are found out two square numbers the one being the square number which is produced of AB into BC and the other the square number produced of CD which added together make a square number namely the square number produced of BD multiplied into himselfe forasmuch as they were demoÌstrated equall to it A Corollary And hereby it is manifest that there are found out two square numbers namely the ãâã the square number of BD and the other the square number of CD so that that numbâr wherin th one excedeth the other the number I say which is produced of AB into BC is also a square number namely when Aâ BC are like playne numbers But when they are not like playne numbers then are there found out two square numbers the square number of BD and the square number of DC whose excesse that is the number wherby the greater excedeth the lesse namely that which is produced of AB into BC is not a square number ¶ An Assumpt To finde out two square numbers which added together make not a square number Let AB and BC be like playne numbers so that by the first of the ninth that which is produced of AB into BC is a square number and let AC be an euen number And deuide Câ into two equall parâes in D. Now by that which hath before bene sayd in the former assumpt it is manifest that the square number produced of AB into BC together with the square number of CD is equall to the square number of BD. Take away from CD vnitie DE. Wherfore that which is produced of AB into BC together with the square of CE is lesse then the square number of BD. Now then I say that the square numâer produced of AB into BC added to the square number of CE make not a square number For if they do make a square number then that square number which they make is either greater theÌ the square number of BE or equall vnto it or lesse then it First greater it cannot be for it is already proued that the square number produced of AB into BC together with the square number of CE is lesse then the square number of BD. But betwene the square number of BD and the square number of BE there is no meane square number For the number BD excedeth the number BE onely by vnitie which vnitie can by no meanes be deuided into numbers Or if the number produced of AB into BC together with the square of the nuÌber CE should be greater then the square of the number BE then should the selfe same number produced of AB into BC together with the square of the number CE be equall to the square of the number BD the contrary wherof is already proued Wherfore if it be possible let that which is produced of AB into BC together with the square number of the number CE be equall to the square number of BE. And let GA be double to vnitie DE that is let it be the number two Now forasmuch as the whole number AC is by supposition double to the whole number CD of which the number AG is double to vnitie DE therfore by the 7. of the seuenth the residue namely the number GC is double to the residue namely to the number EC Wherfore the number GC is deuided into two equall partes in E. Wherefore that which is produced of GB into BC together with the square number of CE is equall to the square nuÌber
to the number CE so is the square of the line AB to the square of the line AF therefore by conuersion by the corollary of the 19. of the fifte as the number CD is to the number DE so is the square of the line AB to the square to the line FB But the number C D hath not to the number DE that proportion that a squarâ nâmbeâ hâth to a square number Wherefore neither also the square of the line AB hath to the square of the line BF that proportion that a square number hath to a square number Wherefore the line AB is by the 9 of the tenth incommensurable in length to the line BF And the line AB is in power more then the line AF by the square of the right line BF which is incommensurable in length vnto the line AB Wherfore the lines AB and AF are rationall commensurable in power onely And the line AB is in power more then the line AF by the square of the line FB which is commensurable in length vnto the line AB â which was required to be done ¶ An Assumpt If there be two right lines hauing betwene them selues any proportion as the one right line is to the other so is the parallelograme contained vnder both the right lines to the square of the lesse of those two lines Suppose that these two right AB and BC be in some certaine proportion Then I say that as the line AB is to the line BC so is the parallelograme contained vnder AB and BC to the square of BC. Describe the square of the line BC and let the same be CD and make perfect the parallelograme AD now it is manifest that as the line AB is to the line BC so is the parallelograme AD to the parallelograme or square BE by the first of the sixt But the parallelograme AD is that which is bontained vnder the lines AB and BC for the line BC is equall to the line BD and the parallelograme BE is the square of the line BC. Wherefore as the line AB is to the line BC so is the parallelograme coutained vnder the lines AB and BC to the square of the line BC which was required to be proued ¶ The 8. Probleme The 31. Proposition To finde out two mediall lines commensurable in power onely comprehending a rationall superficies so that the greater shall be in power more then the lesse by the square of a line commensurable in length vnto the greater LEt there be taken by the 29. of the tenth two rationall lines commensurable in power onely A and B so that let the line A being the greater be in power more then the line B being the lesse by the square of a line commensurable in length vnto the line A â And let the square of the line C be equall to the parallelograme contained vnder the lines A and B which is done by finding out the meane proportionall line namely the line C betwene the lines A and B by the 13. of the sixt Now the parallelograme contained vnder the lines A and B is mediall by the 21. of this booke Wherefore by the corollary of the 23. of the tenth the square also of the line C is mediall Wheâfore the line C also is mediall Vnto the square of the line B let the parallelograme contained vnder the lines C and D be equall by finding out a third line proportionall namely the line D to the two lines C and B by the 11. of the sixt But the square of the line B is rationall Wherfore the parallelograme contained vnder the line C and D is rationall And for that as the line A is to the line B so is the parallelograme contained vnder the lines A and B to the square of the line B by the assumpt going before But vnto the parallelograme contained vnder the lines A and B is equall the square of the line C and vnto the square of the line B is equal the parallelograme contained vnder the lines C and D as it hath now bene proued therefore as the line A is to the line B so is the square of the line C to the parallelograme contained vnder the lines C D. But as the square of the line C is to that which is contayned vnder the lines C and D so is the line C to the line D. Wherefore as the line A is to the line B so is the line C to the line D. But by supposition the line A is commensurable vnto the line B in power onely Wherefore by the 11. of the tenth the line C also is vnto the line D commensurable in power onely But the line C is mediall Wherefore by the 23â of the tenth the line D also is mediall And for that as the line A is to the line B so is the line C to the line D but the line A is in power more then the line B by the square of a line commensurable in length vnto the line A by supposition Wherefore the line C also is in power more then the line D by the square of a line commensârable in length vnto the line C. Wherefore there are found out two mediall lines C and D commensurable in power onely comprehending a rationall superfiâies and the line C is in power more then the line D by the square of a line commensurable in length vnto the line C. And in like sort may be found out two mediall lines commensurable in power onely contayning a rationall superficies so that the greater shal be in power more theÌ the lesse by the square of a line incoÌmensurable in leÌgth to the greater namely when the line A is in power more theÌ the line B by the square of a line incoÌmensuraâle in length vnto the line A which to do is taught by the 30. of this booke The selfe same construction remaining that part of this proposition froÌ these wordes And for that as the line A is to the line B to these wordes But by supposition the line A is commensurable vnto the line B may more easely be demonstrated after this maner The lines C B D are in continuall proportion by the second part of the 17. of the sixt But the lines A C D are also in continuall proportion by the same Wherefore by the 11. of the fifth as the line A is to the line C so is the line B to the line D. Wherfore alternately as the line A is to the line B so is the line C to the line D. c. which was required to be doone ¶ An assumpt If there be three right lines hauing betwene them selues any proportion as the first is to the third so is the parallelograme contained vnder the first and the second to the parallelograme contained vnder the second and the third Suppose that these three lines AB B C and CD be in some certayne proportion Then I say that as the line AB is to
contayned vnder the lines BD and DC is equall to the square of the line DA. As touching the fourth that the parallelogramme contained vnder the lines BC and AD is equall to the parallelogramme contained vnder the lines BA and AC is thus proued For forasmuch as as we haue already declared the triangle ABC is like and therefore equiangle to the triangle ABD therefore as the line BC is to the line AC so is the line BA to the line AD by the 4. of the sixt But if there be foure right lines proportionall that which is contained vnder the first and the last is equall to that which is contained vnder the two meanes by the 16. of the sixt Wherefore that which is contained vnder the lines BC and AD is equall to that which is contayned vnder the lines BA and AC I say moreouer that if there be made a parallelogramme complete contained vnder the lines BC and AD which let be EC and if likewise be made complete the parallelogramme contained vnder the lines BA and AC which let be AF it may by an other way be proued that the parallelogramme EC is equall to the parallelogramme AF. For forasmuch as either of them is double to the triangle ACB by the 41. of the first and thinges which are double to one and the selfe same thing are equall the one to the other Wherefore that which is contained vnder the lines BC and AD is equall to that which is contained vnder the lines BA and AC 2. ¶ An Assumpt If a right line be deuided into two vnequall partes as the greater part is to the lesse so is the parallelogramme contayned vnder the whole line and the greater part to the parallelogramme contayned vnder the whole line and the lesse part This Assumpt differeth litle from the first Proposition of the sixt booke 3. ¶ An Assumpt If there be two vnequall right lines and if the lesse be deuided into two equall partes the parallelogramme contained vnder the two vnequall lines is double to the parallelogramme contained vnder the greater line halfe of the lesse line Suppose that there be two vnequall right lines AB and BC of which leâ AB be the greater and deuide the line BC into two equall partes in the point D. Thân I say that the parallelogramme contained vnder the lines AB BC is double to the parallelogramme contained vnder the lines AB and BD. From the point B raise vp vpon the right line BC a perpendicular line BE and let BE be equall to the line BA And drawing from the point C and D the lines CG and DF parallels and equall to BE and then drawing the right line GFE the figure is complete Nââ for that aââhe line DB is to the line DC so is the parallelogramme BF to the parallelogramme DG by the 1. of the sixt therâore by composition of proportion as the whole line BC is to the line DC so is the parallelogramme BG to the parallelogramme DG by the 18. of the fift But the line BC is double to the line DC Wherefore the parallelogramme BG is double to the parallelogramme DG But the parallâlogramme BG is contained vnder the lines AB and BC for the line AB is equall to the line BE and the parallelogramme DG is contayned vnder the lines AB and BD for the line BD is equall to the line DC and the line AB to the line DF which was required to be demonstrated ¶ The 10. Probleme The 33. Proposition To âinde out two right lines incommensurable in power whose squares added together make a rationall superficies and the parallelogramme contained vnder them make a mediall superficies TAke by the 30. of the tenth two rationall right lines commensurable in power onely namely AB and BC so that let the line AB being the greater be in power more then the line BC being the lesse by the square of a line incommensurable in length vnto the line AB And by the 10. of the first deuide the line BC into two equall partes in the point D. And vpon the line AB apply a parallelogramme equall to the square either of the line BD or of the line DC and wanting in figure by a square by the 28. of the sixth and let that parallelogramme be that which is contained vnder the lines AE and EB And vpon the line AB describe a semicircle AFB And by the 11. of the first from the point E raise vp vnto the line AB a perpendiculer line EF cutting the circumference in the point F. And draw lines from A to F and from F to B. And forasmuch as there are two vnequall right lines AB and BC and the line AB is in power more then the line BC by the square of a line incommensurable in lângth vnto AB and vpon the line AB is applied a parallelograme equall to the fourth part oâ the square of the line BC that is to the square of the halfe of the line BC and wanting in âigure by a square and the said parallelogramme is that which is contained vnder the lines AE and EB wherfore by the 2. part of the 18. of the tenth the line AE is incommeÌsurable in length vnto the line EB But as the line AE is to the line EB so is the parallelogramme contained vnder the lines BA and AE to the parallelogramme contayned vnder the lines AB and BE by the second assumpt before put And that which is contained vnder the line BA and AE is equall to the square of the line AF by the second part of the first assumpt before put And that which is contained vnder the lines AB and BE is by the first part of the same assumpt equall to the square of the line BF Wherfore the square of the line AF is incommânsurable to the square of the line BF Wherfore the lines AF and BF are incommensurable in power And forasmuch as AB is a rationall line by supposition therfore by the 7 definition of the tenth the square of the line AB is rationall Wherefore also the squares of the lines AF and FB added together make a rationall superficies For by the 47. of the first they are equal to the square of the line AB Again forasmuch as by the third part of the first assumpt going before that which is contained vnder the lines AE and EB is equall to the square of the line EF. But by supposition that which is contained vnder the lines AE and EB is equall to the square of the line BD. Wherfore the line FE is equall to the line BD. Wherfore the linâ BG is double to the line â E. Wherfore by the third assumpt going before that which is contained vnder the lines AB and BC is double to that which is contained vnder the lines AB and EF. But that which is contained vnder the lines AB and BC is by supposition mediall
Wherfore by the corollary of the 23. of the tenth that which is contained vnder the lines AB and EF is also mediall but that whiche is contayned vnder the lines AB and EF is by the last parte of the first assumpt goyng before equall to that which is contained vnder the lines AF and FB Wherefore that which is contained vnder the lines AF FB is a mediall superficies And it is proued that that which is composed of the squares of the lines AF and FB added together is rationall Wherfore there are found out two right lines AF and FB incommensurable in power whose squares added together make a rationall superficies and the parallelogramme contained vnder them is a mediall superficiâsâ which was required to be done ¶ The 11. Probleme The 34. Proposition To finde out two right lines incââmensurable in power whose squares added together make a mediall superficies and the parallelogramme contayned vnder them make a rationall superficies TAke by the 31. of the tenth two mediall lines AB and BC commensurable in power onely comprehending a rationall superficies so that let the line AB be in power more then the line BC by the square of a line incommensurable in length vnto the line AB And describe vpon the line AB a semicircle ADB And by the 10. of the first deuide the line BC vnto two equall partes in the point E. And by the 28. of the sixt vpon the line AB apply a parallelogramme equall to the square of the line BE and wantyng in figure by a square and let that parallelogramme be that which is contayned vnder the lines AF and FB Wherfore the line AF is incommensurable in length vnto the line FB by the 2. part of the 18. of the tenth And from the point F vnto the right line AB raise vp by the 11. of the first a perpendiculer line FD and draw lines from A to D and from D to B. And forasmuch as the line AF is incommensurable vnto the line FB but by the second assumpt going before the 33. of the tenth as the line AF is to the line FB so is the parallelogramme contayned vnder the lines BA and AF to the parallelogramme contained vnder the lines BA and BF wherfore by the tenth of the tenth that which is contained vnder the lines BA and AF is incommensurable to that which is contayned vnder the lines AB and BF but that which is contained vnder the lines BA and AF is equall to the square of the line AD and that which is contained vnder the lines AB and BF is also equall to the square of the line DB by the second part of the first assumpt going before the 33. of the teÌth wherfore the square of the line AD is incoÌmensurable to the square of the line DB. Wherefore the lines AD and DB are incommensurable in power And forasmuch as the square of the line AB is mediall therefore also the superficies made of the squares of the lines AD and DB added together is mediall For the squares of the lines AD and DB are by the 47. of the first equall to the square of the line AB And forasmuch as the line BC is double to the line FD as it was proued in the proposition going before therefore the parallelogramme contained vnder the lines AB and and BC is double to the parallelogramme contained vnder the lines AB and FD by the third assumpt going before the 33. proposition wherefore it is also commensurable vnto it by the sixt of the tenth But that which is contained vnder the lines AB and BC is supposed to be rationall Wherfore that which is contained vnder the lines AB and FD is also rationall But that which is contained vnder the lines AB and FD is equall to that which is contained vnder the lines AD and DB by the last part of the first assumpt going before the 33. of the tenth Wherfore that which is contayned vnder the lines AD and DB is also rationall Wherefore there are âound out two right lines AD and DB incommensurable in power whose squares added together make a mediall superficies and the parallelogramme coÌtayned vnder them make a rationall superficies which was required to be done ¶ The 12. Probleme The 35. Proposition To finde out two right lines incommensurable in power whose squares added together make a mediall superficies and the parallelogramme contained vnder them make also a mediall superficies which parallelogramme moreouer shall be incommensurable to the superficies made of the squares of those lines added together TAke by the 32. of the tenth two mediall lines AB and BC commensurable in power onely comprehending a mediall superficies so that let the line AB be in power more then the line BC by the square of a line incommensurable in length vnto the line AB And vpon the line AB describe a semicircle ADB and let the rest of the construction be as it was in the two former propositions And forasmuch as by the 2 part of the 18. of the tenth the line AF is incommensurable in length vnto the line FB therfore the line AD is incommensurable in power vnto the line DB by that which was demonstrated in the propositioÌ going before And forasmuch as the square of the line AB is mediall therefore that also which is composed of the squares of the lines AD and DB which squares are equall to the square of the line AB by the 47. of the first is mediall And forasmuch as that which is contained vnder the lines AF and FB is equall to either of the squares of the lines EB and FD for by supposition the parallelogramme contained vnder the lines AF and FB is equall to the square of the line EB and the same parallelogramme is equall to the square of the line DF by the third part of the first assumpt going before the 33. of the teÌth Wherfore the line BE is equall to the line DF. Wherfore the line BC is double to the line FD. Wherefore that which is contained vnder the lines AB and BC is double to that which is contained vnder the lines AB and FD. Wherfore they are commensurable by the sixt of this boke but that which is contained vnder the lines AB and BC is mediall by supposition Wherfore also that which is contained vnder the lines AB and FD is mediall by the corollary of the 23 of the tenth but that which is contained vnder the lines AB and FD is by the fourth part of the first assumpt going before the 33. of the tenth equall to that which is contained vnder the lines AD and DB wherfore that which is contained vnder the lines AD and DB is also mediall And forasmuch as the line AB is incommensurable in length vnto the line BC. But the line BC is commensurable in length vnto the line BE. Wherfore by the 13â of
the tenth the line AB is incommensurable in length vnto the line BE. Wherefore the square of the line AB is incommensurable to that which is contained vnder the lines AB and BE by the first of the sixt and 10. of this booke But vnto the square of the line AB are equall the squares of the lines AD and DB added together by the 47. of the first and vnto that which is contayned vnder the lines AB and BE is equall that which is contained vnder the lines AB and FD that is which is contained vnder AD and DB. For the parallelogramme contained vnder the lines AB and FD is equall to the parallelogramme contained vnder the lines AD and DB by the last part of the first assumpt going before the 33. of this tenth booke Wherfore that which is composed of the squares of the lines AD and DB is incommensurable to that which is contained vnder the lines AB and DB. Wherefore there are found out two right lines AD and DB incommensurable in power whose squares added together make a mediall superficies and the parallelogramme contayned vnder them make also a mediall superficies which parallelogramme moreouer is incommensurable to the superficies composed of the squares of those lines added together which was required to be done The beginning of the Senaries by Composition ¶ The 2â Theoreme The 36. Proposition If two rationall lines commensurable in power onely be added together the whole line is irrationall and is called a binomium or a binomiall line ãâ¦ã B and BC is incommensurable to the square of the line BC. But vnto the parallelograme contained vnder the lines AB and BC is commensurable the parallelograme contained vnder AB and BC twise by the 6. of the tenth wherefore that which is contained vnder AB and BC twise is incommensurable to the square of the line BC by the 13 of the tenth But vnto the square of the line BC is commensurable that which is composed of the squares of the lines AB and BC by the 15. of the tenth for by supposition the lines AB and BC are commensurable in power onely Wherefore by the 13. of the tenth that which is composed of the squares of the lines AB and BC added together is incommensurable to that which is contained vnder the lines AB and BC twise Wherefore by the 16. of the tenth that which is contained vnder AB and BC twise together with the squares of the lines AB and BC which by the 4. of the second is equall to the square of the whole line AC is incommensurable to that which is composed of the squares of AB and BC added together But that which is composed of the squares of AB and BC added together is rationall for it is commensurable to either of the squares of the lines AB and BG of which either of them is rationall by supposition wherfore the square of the line AC is by the 10. definition of the tenth irrationall Wherefore the line AC also is irrationall and is called a binomiall line This proposition sheweth the generation and production of the second kinde of irrationall lines which is called a binomium or a binomial line The definition whereof is fully gathered out of this proposition and that thus A binomium or a binomiall line is an irrationall line composed of two rationall lines commensurable the one to the other in power onely And it is called a binomium that is hauing two names because it is made of two such lines as of his partes which are onely commensurable in power and not in length and therefore ech part or line or at the least the one of them as touching length is vncertaine and vnknowne Wherefore being ioyned together their quantitie cannot be expressed by any one number or name but ech part remayneth to be seuerally named in such sort as it may And of these binomiall lines there are sixe seuerall kindes the first binomiall the second the third the fourth the fifth and the sixt of what nature and condition ech of these is shal be knowne by their definitious which are afterward set in their due place ¶ The 25. Theoreme The 37. Proposition If two mediall lines commensurable in power onely containing a rationall superficies be added together the whole line is irrationall and is called a first bimediall line LEt these two mediall lines AB and BC being commensurable in power onely and contayning a rationall superficies the 27. of the tenth teacheth to finde out two such lines be composed Then I say that the whole line AC is irrationall For as ãâã sayd in the proposition next going before that which is composed of the squares of the ãâã AB and BC is incommeÌsurable to that which is contained vnder the lines AB and BC twisâ wherefore by the 16. of the tenth that which is composed of the squares of the lines AB and BC together with that which is contained vnder the lines AB and BC twise that is the square of the line AC is incommensurable to that which is contayned vnder the lines AB and BC twise But that which is contayned vnder the lines AB and BC twise iâ commensurable to that which is contayned vnder the lines AB and BC once by the 6. of the tenth wherefore the square of the whole line AC is by the 13â of the tenth incâmmensurable âo that which is contained vnder the lines AB and BC once But by supposition the lines AB and BC comprehend a rationall superâicies Wherefore the square of the whole line AC is irrationall wherefore also the line AC is irrationall And it is called a first bimediall line The third irrational line which is called a first bimediall line is shâwed by this proposition and the definition thereof is by it made manifest which is this A first bimediall line is an irrationall line which is composed of two mediall lines commensurable in power onely contayning a rationall parallelograme It is called a first bimediall line by cause the two mediall lines or partes whereof it is composed contayne a rationall superficies which is preferred before an irrationall ¶ The 26. Theoreme The 38. Proposition If two mediall lines commensurable in power onely contayning a mediall superficies be added together the whole line is irrationall and is called a second bimediall line LEt these two medial lines AB and BC being commensurable in power onely and contayning a mediall superficies the 28. of the tenth teacheth to findâ out two such lines be added together Then I say that the whole line AC is irrational Take a rationall line DE. And by the 44. of the first vpon the line DE apply the parallelograme DF equal to the square of the line AC whos 's other side let be the line DG And forasmuch as the square of the line AC is by the 4. of the second equall to that which is composed of
the squares of the lines AB and BC together with that which is contained vnder the lines AB and BC twise but the square of the line AC is equall to the parallelograme DF. Wherefore the parallelograme DF is equall to that which is composed of the squares of the lines AB and BC together with that which is contayned vnder the lines AB and BC twise Now then agayne by the 44 of the first vpon the line DE apply the parallelograme EH equall to the squares of the lines AB and BC. Wherefore the parallelograme remayning namely HF is equall to that which is contained vnder the lines AB and BC twise And forasmuch as either of these lines AB and BC is mediall therefore the squares of the lines AB and BC are also mediall And that which is contained vnder the lines AB and BC twise is by the corollary of the â4 of the tenth mediall For by the 6. of this booke it is commeÌsurable ââ that ãâã is contained vnder the lines AB and BC once which is by supposition medial ãâ¦ã squares of the lines AB and BC is equall the parallelograme EH and vnto that ãâã contayned vnder the lines AB and BC twise is equall the parallelograme HE ãâ¦ã either of these parallelogrames HE and AF is mediall and they are applyed vpon the rationall line ED. Wherefore by the 22. of the tenth either of these lines DH and HG is a rationall line and incâmmensurable in length vnto the line DE. And forasmuch as by supposition the line AB is incommensurable in length vnto the line BC. But as the line AB is to the line BC â so is the square of the line AB to the parallelograme which is contayned vnder the lines AB and BC by the first of the sixt Wherefore by the 10 of this bookâ the square of the line AB is incommensurable to the parallelograme contayned vnder the lines AB and BC. But to the square of the line AB is commensâable that which is composed of the squares of the lines AB and BC by the 15. of the tenth For the squares of the lines AB and BC are commensurable when as the lines AB and BC are put to be commensurable in power onely And to that which is contayned vnder the lines AB and BC is commensurable that which is contained vnder the lines AB and BC twise by the 6 of the tenth wherefore that which is composed of the squares of the lines AB and BC is incommensurable to that which is contayned vnder the lines AB and BC twise But to the squares of the lines AB and BC is equall the parallelograme EH And to that which is contayned vnder the lines AB and BC twise is equall the parallelograme FH Wherfore the parallelograme FH is incommensurable to the parallelograme HE. Wherfore the line DH is incommensurable in length to the line HG by the 1 of the sixt and 10 of this booke And it is proued that they are rationall lines Wherefore the lines DH HG are rationall commensurable in power onely Wherefore by the 36. of the tenth the whole line DG is irrationallâ And the line DE is rationall But a rectangle superâicies comprehended vnder a rationall line and an irrationall line is by the corollary added after the 21 of the tenth irrationall Wherefore the superâicies DF is irrationall And the line also which containeth it in power is irrational But the line AC containeth in power the superficies DF. Wherefore the line AC is irrationall And it is called a second bimediall line This Proposition sheweth the generation of the fourth irrationall line called a second bimediall line The definition wherof is euident by this Proposition which is thus A second bimediall line is an irrationall line which is made of two mediall lines commensurable in power onely ioyned together which comprehend a mediall superficies And it is called a second bimediall because the two mediall lines of which it is composed coÌtaine a mediall superficies and not a rationall Now a mediall is by nature in knowledge after a rationall ¶ The 27. Theoreme The 39. Proposition If two right lines incoÌmensurable in power be added together hauing that which is composed of the squares of them rationall and the parallelograÌme contayned vnder them mediall the whole right line is irrationall and is called a greater line LEt tâese two right lines AB and BC being incommensurable in power onely and making that which is required in the Proposition The 33. of the tenth teacheth to finde out two such lines be added together Then I say that the whole line AC is irrationall For forasmuch as by supposition the parallelogramme contained vnder the lines AB and BC is mediall therefore the parallelogramme contained twise vnder the lines AB and BC is mediall For that which is contained vnder AB and BC twise is commensurable to that which is coÌtained vnder AB and BC once by the 6. of the tenth Wherefore by the Corollary of the 23. of the tenth that which is contained vnder AB BC twise is mediall But by supposition that which iâ composed of the squares of the lines AB and BC is rationall Wherefore that which is contained vnder the lines AB and BC twise is incommensurable to that which is composed of the squares of the lines AB and BC. Wherfore by the 16. of the tenth that which is composed of the squares of the lines AB and BC together with that which is contayned vnder the lines AB BC twise which is by the 4. of the second equall to the square of the line AC is incommensurable to that which is composed of the squares of the lines AB and BC. But that which is composed of the squares of the lines AB and BC is rationall Wherefore the square of the whole line AC is irrationall Wherefore the line AC also is irrationall And is called a greater line And it is called a greater line for that that which is composed of the squares of the lines AB BC which are rationall is greater then that which is contayned vnder the lines AB and BC twise which are mediall Now it is meete that the name should be geuen according to the propertie of the rationall An Assumpt This Proposition teacheth the production of the fift irrationall line which is called a greater line which is by the sense of this Proposition thus defined A greater line is an irrationall line which is composed of two right lines which are incommensurable in power the squares of which added together make a rationall superficies and the parallelogramme which they containe is mediall It is therefore called a greater line as Theon sayth because the squares of the two lines of which it is composed added together being rationall are greater then the mediall superficies contained vnder them twise And it is conuenient that the denomination be taken of the proprietie of the
let the same be E. And as the number D is to the number AB so let the square of the line E be to the square of FG. Wherefore by the 6. of the tenth the line E is commensurable in power to the line FG the line E is rationall Wherfore also the line FG is rationall And for that the number D hath not âo the number AB that proportion that a square nuÌber hath to a square number therefore neither also shall the square of the line E haue to the square of the line FG that proportion that a square number hath to a square number Wherefore the line FG is incommensurable in length to the line E. Againe as the number BA is to the number AC so let the square of the line FG be to the square of the line GH Wherefore by the 6. of the tenth the square of the line FG is commensurable to the square of the line GH And the square of the line FG is rationall Wherefore the square of the line GH is also rationall Wherefore also the line GH is rationall And for that the number AB hath not to the number AC that proportion that a square number hath to a square number thereâore neither also hath the square of the line FG to the square of the line GH that proportion that a square number hath to a square number Wherefore the line FG is incommensurable in length to the line GH Wherefore the lines FG and GH are rationall commensurable in power onely Wherefore the whole line FH is a binomiall line I say moreouer that it is a sixt binomiall line For for that as the number D is to the number AB so is the square of the line E to the square of the line FG. And as the number BA is to the number AC so is the square of the line FG to the square of the line GH Wherefore of equalitie by the 22. of the fift as the number D is to the number AC so is the square of the line E to the square of the line GH But the number D hath not to the nuÌber AC that proportion that a square number hath to a square number Wherefore neither also hath the square of the line E to the square of the line GH that proportion that a square number hath to a square number Wherfore the line E is incommensurable in length to the line GH And it is already proued that the line FG is also incommensurable in length to the line E. Wherefore either of these lines FG and GH is incommensurable in length to the line E. And for that as the number âA is to the number AC so is the square of the line FG to the square of the line GH therfore the square of the line FG is greater then the square of the line GH Vnto the square of the line FG let the squares of the lines GH and K be equall Wherefore by euersion of proportion as the number AB is to the number BC so is the square of the line FG to the square of the line K. But the number AB hath not to the number BC that proportion that a square number hath to a square number Wherefore neither also hath the square of the line FG to the square of the line K that proportion that a square number hath to a square number Wherefore the line FG is incommensurable in length vnto the line K. Wherefore the line FG is in power more then the line GH by the square of a line incommensurable in length to it And the lines FG and GH are rationall commensurable in power onely And neither of the lines FG GH is commensurable in length to the rationall line geuen namely to E. Wherefore the line FH is a sixt binomiall line which was required to be found out ¶ A Corollary added out of Flussates By the 6. formâr Proposiâââââ it iâ manifest hoâ ãâã divide any right line geuen into the names of euery one of the sixâ foresayd binomiall lines For if it be required to deuide a right line geuen into a first binomiall line then by the 48â of this booke finde out a first binomiall line And this right line being so found out deuided into his names you may by the 10. of the sixt deuide the right line geuen in like sort And so in the other fiue following Although I here note vnto you this Corollary out of ãâ¦ã in very conscience and of gratefull âindeâ I am enforced to certifie you that iâ any yeareâ before the trauailes of Flussas vpoÌ Euâliâââ Geometricall Elementes were published the order how to deuide not onely the 6. Binomiall lines into their names but also to adde to the 6. Residââls their due partes ând fârthermore to deuide all the other irrationalâ lines of this tenth booke into the partes distinct of which they are composed with many other straunge conclusions Mathematicall to the better vnderstanding of this tenth booke and other Mathematicall bookes most necessary were by M. Iohn Dee inuented and demonstrated as in his booke whose title is Tyrocinium Mathematicum dedicated to Petruâ Nonnius An. 1559. may at large appeare Where also is one new arte with sundry particular pointes whereby the Mathematicall Sciences greatly may be enriched Which his booke I hope God will one day allowe him opportunitie to publishe with diuers other his Mathematicall and Metaphysicall labours and inuentions ¶ An Assumpt Is a right line be deuided into two partes how soeuer the rectangle parallelogramme contayned vnder both the partes is the meane proportionall betwene the squares of the same parts And the rectangle parallelogramme contained vnder the whole line and one of the partes is the meane proportionall betwene the square of the whole line and the square of the sayd part Suppose that there be two squares AB and BC and let the lines DB and BE so be put that they both make one right line Wherefore by the 14. of the first the lines FB and BG make also both one right line And make perfect the parallelogramme AC Then I say that the rectangle parallelogramme DG is the meane proportionall betwene the squares AB and BC and moreouer that the parallelogramme DC is the meane proportionall betwene the squares AC and CB. First the parallelogramme AG is a square For forasmuch as the line DB is equall to the line BF and the line BE vnto the line BG therfore the whole line DE is equall to the whole line FG. But the line DE is equall to either of these lines AH KC and the line FG is equall to either of these lines AK and HC by the 34. of the first Wherfore the parallelograÌme AC is equilater it is also rectangle by the 29. of the first Wherefore by the 46. of the first the parallelograÌme AC is a square Now for that as the line FB is to the line BG so is the line DB
the definition of a first binomiall line seâ before the 48. proposition of this booke the line DG is a first binomiall line which was required to be proued This proposition and the fiue following are the conuerses of the sixe former propositions ¶ The 43. Theoreme The 61. Proposition The square of a first bimediall line applied to a rationall line maketh the breadth or other side a second binomiall line SVppose that the line AB be a first bimediall line and let it be supposed to be deuided into his partes in the point C of which let AC be the greater part Take also a rationall line DE and by the 44. of the first apply to the line DE the parallelograÌme DF equall to the square of the line AB making in breadth the line DG Then I say that the line DG is a second binomiall line Let the same constructions be in this that were in the Proposition going before And forasmuch as the line AB is a first bimediall line and is deuided into his partes in the point C therefore by the 37. of the tenth the lines AC and CB are mediall commensurable in power onely coÌprehending a rationall superficies Wherfore also the squares of the lines AC and CB are mediall Wherefore the parallelogramme DL is mediall by the Corollary of the 23. of the tenth and it is applied vppon the rationall line DE. Wherefore by the 22. of the tenth the line MD is rationall and incommensurable in length to the line DE. Againe forasmuch as that which is coÌtayned vnder the lines AC and CB twise is rationall therefore also the parallelogramme MF is rationall and it is applied vnto the rationall line ML Wherefore the line MG is rationall and commensurable in length to the line ML that is to the line DE by the 20. of the tenth Wherefore the line DM is incommensurable in length to the line MG and they are both rationall Wherefore the lines DM and MG are rationall commensurable in power onely Wherefore the whole line DG is a binomiall line Now resteth to proue that it is a second binomiall line Forasmuch as the squares of the lines AC and CB are greater then that which is contayned vnder the lines AC and CB twise by the Assumpt before the 60. of this booke therefore the parallelogramme DL is greater then the parallelogrrmme MF Wherefore also by the first of the sixt the line DM is greater then the line MG And forasmuch as the square of the line AC is commensurable to the square of the line CB therefore the parallelogramme DH is commensurable to the parallelogramme KL Wherefore also the line DK is commensurable in length to the line KM And that which is contayned vnder the lines DK and KM is equall to the square of the line MN that is to the fourth part of the square of the line MG Wherefore by the 17. of the tenth the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM and the line MG is commensurable in length to the rationall line put namely to DE. Wherefore the line DG is a second binomiall line which was required to be proued ¶ The 44. Theoreme The 62. Proposition The square of a second bimediall line applied vnto a rationall line maketh the breadth or other side therof a third binomiall lyne SVppose that AB be a second bimediall line and let AB be supposed to be deuided into his partes in the point C so that let AC be the greater part And take a rationall line DE. And by the 44. of the first vnto the line DE apply the parallelogramme DF equall to the square of the line AB and making in breadth the line DG Then I say that the line DG is a third binomiall line Let the selfe same constructions be in this that were in the propositions next going before And forasmuch as the line AB is a second bimediall line and is deuided into his partes in the point C therfore by the 38. of the tenth the lines AC and CB are medials commensurable in power only compreheÌding a mediall superficies Wherfore that which is made of the squares of the lines AC and CB added together is mediall and it is equall to the parallelogramme DL by construction Wherefore the parallelogramme DL is mediall and is applied vnto the rationall line DE wherfore by the 22. of the tenth the line MD is rationall and incommensurable in length to the line DE. And by the lyke reason also the line MG is rationall and incommensurable in length to the line ML that is to the line DE. Wherfore either of these lines DM and MG is rational and incommensurable in length to the line DE. And forasmuch as the line AC is incommensurable in length to the line CB but as the line AC is to the line CB so by the assumpt going before the 22. of the tenth is the square of the line AC to that which is contained vnder the lines AC and CB. Wherfore the square of the line AC is incâmmmensurable to that which is contayned vnder the lines AC and CB. Wherfore that that which is made of the squares of the lines AC and CB added together is incommensurable to that which is contained vnder the lines AC and CB twise that is the parallelogramme DL to the parallelogramme MF Wherfore by the first of the sixt and 10. of the tenth the line DM is incommensurable in length to the line MG And they are proued both rationall wherfore the whole line DG is a binomiall line by the definition in the 36. of the tenth Now resteth to proue that it is a third binomiall line As in the former propositions so also in this may we conclude that the line DM is greater then the line MG and that the line DK is commensurable in length to the line KM And that that which is contained vnder the lines DK and KM is equall to the square of the line MN Wherfore the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM and neither of the lines DM nor MG is commensurable in length to the rational line DE. Wherfore by the definition of a third binomiâll line the line DG is a third binomiall line which was required to be proued ¶ Here follow certaine annotations by M. Dee made vpon three places in the demonstration which were not very euident to yong beginners â The squares of the lines AC and Câ are medials ãâã iâ taught after the 21â of this tenth and therâore forasmuch as they are by supposition commeÌsurable th' one to the other by the 15. of the teÌth the compound of them both is commensurable to ech part But the partes are medials therfore by the coâollary of the 23. of the tenth the compound shall be
fift binomiall line But if neither of the lines AE nor EB be commensurable in length to the rationall line geuen neither also of the lines CF nor FD shal be commensurable in length to the same and so either of the lines AB and CD shal be a sixt binomiall line A line therefore commensurable in length to a binomiall line is also a binomiall line of the selfe same order which was required to be proued ¶ The 49. Theoreme The 67. Proposition A line commensurable in length to a bimediall line is also a bimediall lyne and of the selfe same order SVppose that the line AB be a bimediall line And vnto the line AB let the lyne CD be commensurable in length Then I say that the line CD is a bimediall line and of the self order that the line AB is Deuide the line AB into his partes in the point E. And forasmuch as the line AB is a bimediall line and is deuided into his partes in the point E therfore by the 37. and 38. of the tenth the lines AE and EB are medials commensurable in power onely And by the 12. of the sixt as the line AB is to the line CD so let the line AE be to the line CF. Wherfore by the 19. of the fift the residue namely the line EB is to the residue namely to the line FD as the line AB is to the line CD But the line AB is commensurable in length to the lyne CD Wherfore the line AE is commensurable in length to the line CF and the line EB to the line FD. Now the lines AE and EB are mediall wherfore by the 23. of the tenth the lines CF and FD are also mediall And for that as the line AE is to the line EB so is the line CF to the line FD. But the lines AE and EB are commensurable in power onely wherfore the lines CF and FD are also commensurable in power onely And it is proued that they are mediall Wherfore the lyne CD is a bimediall line I say also that it is of the selfe same order that the line AB is For for that as the line AE is to the line EB so is the line CF to the line FD but as the line CF is to FD so is the square of the lyne CF to the parallelogramme contained vnder the lynes CF and FD by the first of the sixt Therfore as the line AE is to the line EB so by the 11. of the fift is the square of the line CF to the parallelogramme contained vnder the lines CF and FD but as AE is to EB so by the 1. of the sixt is the square of the line AE to the parallelogramme contained vnder the lines AE and EB therfore by the 11. of the fift as the square of the line AE is to that which is contained vnder the lines AE and EB so is the square of the line CF to that which is contained vnder the lines CF and FD. Wherfore alternately by the 16. of the fift as the square of the line AE is to the square of the line CF so is that which is contained vnder the lines AE and EB to that which is contained vnder the lines CF FD. But the square of the line AE is commensurable to the square of the line CF because AE and CF are commensurable in length Wherfore that which is contained vnder the lines AE and EB in commensurable to that which is contained vnder the lines CF and FD. If therfore that which is contained vnder the lines AE and EB be rationall that is if the line AB be a first bimediall line that also which is contained vnder the lines CF and FD is rationall Wherfore also the line CD is a first bimediall line But if that which is contained vnder the lines AE and EB be mediall that is if the line AB be a second bimediall line that also which is contayned vnder the lines CF and FD is mediall wherfore also the line CD is a second bimediall line Wherfore the lines AB and CD are both of one and the selfe same order which was required to be proued ¶ A Corollary added by Flussates but first noted by P. Montaâreus A line commensurable in power onely to a bimediall line is also a bimediall line and of the selfe same order Suppose that AB be a bimediall line either a first or a second wherunto let the line GD be coÌmensurable in power onely Take also a rationall line EZ vpon which by the 45. of the first apply a rectangle parallelogramme equall to the square of the line AB which let be EZFC and let the rectangle parallelogramme CFIH be equall to the square of the line GD And forasmuch as vpon the rationall line EZ is applyed a rectangle parallelogramme EF equall to the square of a first bimediall line therefore the other side therof namely EC is a second binomiall line by the 61. of this booke And forasmuch as by supposition the squares of the lines AB GD are commensurable therefore the parallelogrammes EF and CI which are equall vnto them are also commensurable And therefore by the 1. of the sixt the lines EC and CH are commensurable in length But the line EC is a second binomiall line Wherefore the line CH is also a second binomiall line by the 66. of this booke And forasmuch as the superficies CI is contayned vnder a rationall line EZ or CF and a second binomiall line CH therefore the line which contayneth it in power namely the line GD is a first bimediall line by the 55. of this booke And so is the line GD in the selfe same order of bimediall lines that the line AB is The like demonstration also will serue if the line AB be supposed to bâ a second bimediall line For so shall it make the breadth EC a third binomiall line whereunto the line CH shall be commensurable in length and therefore CH also shall be a third binomiall line by meanes whereof the line which contayneth in power the superficies CI namely the line GD shall also be a second bimediall line Wherefore a line commensurable either in length or in power onely to a bimediall line is also a bimediall line of the selfe same order But so is it not of necessitie in binomiall lines for if their powers onely be commensurable it followeth not of necessitie that they are binomialls of one and the selfe same order but they are eche binomialls eyther of the three first kindes or of the three last As for example Suppose that AB be a first binomiall line whose greater name let be AG and vnto AB let the DZ be coÌmeÌsurable in power onely Then I say that the line DZ is not of the selfe same order that the line AB is For if it be possible let the line DZ be of the selfe same order that the line AB is Wheâefore the line DZ may
in like sort be deuided as the line AB is by that which hath bene demonstrated in the 66. Proposition of this bookeâ let it be so deuided in the poynt E. Wherefore it can not be so deuided in any other poynt by the 42â of this booke And for that the line AB ââ to the line DZ as the line AG is to the line DE but the lines AG DE namely the greater names are commensurable in length the one to the other by the 10. of this booke for that they are commensurable in length to ãâã and the selfe same rationall line by the first definition of binomiall lines Wherefore the lines AB and DZ are commensurable in length by the 13. of this booke But by supposition they are commensurable in power onely which is impossible The selfe same demonstration also will serue if we suppose the line AB to be a second binomial line for the lesse names GB and EZ being commensurable in length to one and the selfe same rationall line shall also be commensurable in length the one to the other And therefore the lines AB and DZ which are in the selfe same proportion with them shall also be commensurable in length the one to the other which is contrary to the supposition Farther if the squares of the lines AB and DZ be applyed vnto the rationall line CF namely the parallelogrammes CT and HL they shall make the breadthes CH and HK first binomiall lines of what order soeuer the lines AB DZ whose squares were applyed vnto the rational line are by the 60. of this booke Wherefore it is manifest that vnder a rationall line and a first binomiall line are confusedly contayned all the powers of binomiall lines by the 54. of this booke Wherfore the onely commensuration of the powers doth not of necessitie bryng forth one and the selfe same order of binomiall lines The selfe same thyng also may be proued if the lines AB and DZ be supposed to be a fourth or fifth binomiall line whose powers onely are conmmensurable namely that they shall as the first bring forth binomiall lines of diuers orders Now forasmuch as the powers of the lines AG and GB and DE and EZ are commensurable proportionall it is manifest that if the line AG be in power more then the line GB by the square of a line commensurable in length vnto AG the line DE also shall be in power more then the line EZ by the square of a line commensurable in length vnto the line DE by the 16. of this booke And so shall the two lines AB and DZ be eche of the three first binomiall lines But if the line AG be in power more then the line GB by the square of a line incommensurable in length vnto the line AG the line DE shall also be in powâr ãâã then the line EZ by the square of a line incomensurable in length vnto the line DE by the selfâ same Pâoposition And so shall eche of the lines AB and DZ be of the three last binomiall lines But why it is not so in the third and sixt binomiall lines the reason is For that in them neither of the nameâ is commensurable in length to the rationall line put FC ¶ The 50. Theoreme The 68. Proposition A line commensurable to a greater line is also a greater line SVppose that the line AB be a greater line And vnto the line AB let the line CD be commensurable Then I say that the line CD also is a greater line Deuide the line AB into his partes in the point E. Wherfore by the 39. of the tenth the lines AE and EB are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contained vnder theÌ mediall And let the rest of the construction be in this as it was in the former And for that as the line AB is to the line CD so is the line AE to the line CF thâ line EB to the line FD but the line AB is commensurable to the line CD by suppositioÌ Wherfore the line AE is commensurable to the line CF and the line EB to the line FD. And for that as the line AE is to the line CF so is the line EB to the line FD. Therfore alternately by the 16. of the fift as the line AE is to the line EB so is the line CF to the line FD. Wherfore by composition also by the 18. of the fift as the line AB is to the line EB so is the line CD to the line FD. Wherefore by the 22. of the sixt as the square of the line AB is to the square of the line EB so is the square of the line CD to the square of the line FD. And in like sort may we proue that as the square of the line AB is to the square of the line AE so is the square of the line CD to the square of the line CF. Wherfore by the 11. of the fift as the square of the lyne AB is to the squares of the lines AE and EB so is the square of the line CD to the squares of the lines CF and FD. Wherfore alternately by the 16. of the fift as the square of the line AB is to the square of the line CD so are the squares of the lines AE and EB to the squares of the lines CF and FD. But the square of the line AB is commensurable to the square of the line CD for the line AB is commensurable to the line CD by suppositioÌ Wherfore also the squares of the lines AE and EB are commensurable to the squares of the lines CF and FD. But the squares of the lines AE and EB are incommensurable and being added together are rationall Wherfore the squares of the lines CF and FD are incommensurable being added together are also rationall And in like sort may we proue that that which is contained vnder the lines AE and EB twise is commensurable to that which is contained vnder the lines CF and FD twise But that which is contained vnder the lines AE and EB twise is mediall wherfore also that which is contained vnder the lines CF and FD twise is medial Wherfore the lines CF and FD are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contained vnder theÌ mediall Wherfore by the 39. of the tenth the whole line CD is irrationall is called a greater line A line therfore commensurable to a greater line is also a greater line An other demonstration of Peter Montaureus to proue the same Suppose that the line AB be a greater line and vnto it let the line CD be commensurable any way that is either both in length and in power or els in power onely Then I say that the line CD also is a greater
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 parallelograÌme 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 coÌmeÌsurable in leÌgth 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 coÌmeÌsurable 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 parallelograÌme 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 incommânsurable 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 incommensârable 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 coÌtayning 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 coÌtayneth in power the superficies CI namely the line GD is a line contayning in power two medialls by the 59. of
two other propositions going next before it so farre misplaced that where they are word for word before duâly placed being the 105. and 106. yet here after the booke ended they are repeated with the numbers of 116. and 117. proposition Zambert therein was more faythfull to follow as he found in his greke example than he was skilfull or carefull to doe what was necessary Yea and some greke written auncient copyes haue them not so Though in deede they be well demonstrated yet truth disorded is halfe disgracedâ especially where the patterne of good order by profession is auouched to be But through ignoraunce arrogancy and âemerltie of vnskilfull Methode Masters many thinges remayne yet in these Geometricall Elementes vnduely tumbled in though true yet with disgrace which by helpe of so many wittes and habilitie of such as now may haue good cause to be skilfull herein will I hope ere long be taken away and thinges of importance wanting supplied The end of the tenth booke of Euclides Elementes ¶ The eleuenth booke of Euclides Elementes HITHERTO HATH âVCLIDâ IN THâSâ former bookes with a wonderfull Methode and order entreated of such kindes of figures superficial which are or may be described in a superficies or plaine And hath taught and set forth their properties natures generations and productions euen from the first roote ground and beginning of them namely from a point which although it be indiuisible yet is it the beginning of all quantitie and of it and of the motion and slowing therof is produced a line and consequently all quantitie coÌtinuall as all figures playne and solide what so euer Euclide therefore in his first booke began with it and from thence went he to a line as to a thing most simple next vnto a point then to a superficies and to angles and so through the whole first booke he intreated of these most simple and plaine groundes In the second booke he entreated further and went vnto more harder matter and taught of diuisions of lines and of the multiplication of lines and of their partes and of their passions and properties And for that rightlined âigures are far distant in nature and propertie from round and circular figures in the third booke he instructeth the reader of the nature and conditioÌ of circles In the fourth booke he compareth figures of right lines and circles together and teacheth how to describe a figure of right lines with in or about a circle and contraâiwiâe a circle with in or about a rectiline figure In the fifth booke he searcheth out the nature of proportion a matter of wonderfull vse and deepe consideration for that otherwise he could not compare âigure with figure or the sides of figures together For whatsoeuer is compared to any other thing is compared vnto it vndoubtedly vnder some kinde of proportion Wherefore in the sixth booke he compareth figures together one to an other likewise their sides And for that the nature of proportion can not be fully and clearely sene without the knowledge of number wherein it is first and chiefely found in the seuenth eight and ninth bookes he entreatâth of number of the kindes and properties thereof And because that the sides of solide bodyes for the most part are of such sort that compared together they haue such proportion the one to the other which can not be expresâed by any number certayne and therefore are called irrational lines he in the teÌth boke hath writteÌ taught which lineâ are coÌmeÌsurable or incoÌmeÌsurable the one to the other and of the diuersitie of kindes of irrationall lines with all the conditions proprieties of them And thus hath Euclide in these ten foresayd bokes fully most pleÌteously in a meruelous order taught whatsoeuer semed necessary and requisite to the knowledge of all superficiall figures of what sort forme so euer they be Now in these bookes following he entreateth of figures of an other kinde namely of bodely figures as of Cubes Piramids Cones Columnes Cilinders Parallelipipedons Spheres and such othersâ and sheweth the diuersitie of theÌ the generation and production of them and demonstrateth with great and wonderfull art their proprieties and passions with all their natures and conditions He also compareth one oâ them to an other whereby to know the reason and proportion of the one to the other chiefely of the fiue bodyes which are called regular bodyes And these are the thinges of all other entreated of in Geometrie most worthy and of greatest dignitie and as it were the end and finall entent of the whole are of Geometrie and for whose cause hath bene written and spoken whatsoeuer hath hitherto in the former bookes bene sayd or written As the first booke was a ground and a necessary entrye to all the râst âollowing so is this eleuenth booke a necessary entrie and ground to the rest which follow And as that contayned the declaration of wordes and definitions of thingeâ requisite to the knowledge of superficiall figures and entreated of lines and of their diuisions and sections which are the termes and limites of superficiall figures so in this booke is set forth the declaration of wordes and definitions of thinges pertayning to solide and corporall figures and also of superficieces which are the termes limites of solides moreouer of the diuision and intersection of them and diuers other thinges without which the knowledge of bodely and solide formes can not be attayned vnto And first is set the definitions as followâth Definitions A solide or body is that which hath length breadth and thicknes and the terme or limite of a solide is a superficies There are three kindes of continuall quantitie a line a superficies and a solide or body the beginning of all which as before hath bene sayd is a poynt which is indiuisible Two of these quantities namely a line and a superficies were defined of Euclide before in his first booke But the third kinde namely a solide or body he there defined not as a thing which pertayned not then to his purpose but here in this place he setteth the definitioÌ therof as that which chiefely now pertayneth to his purpose and without which nothing in these thinges can profitably be taught A solide sayth he is that which hath leÌgth breadth and thicknes or depth There are as before hath bene taught three reasons or meanes of measuring which are called coÌmonly dimensions namely lângth breadth and thicknes These dimensions are ascribed vnto quantities onely By these are all kindes of quantitie deâined ââ are counted perfect or imperfect according as they are pertaker of fewer or more of them As Euclide defined a line ascribing vnto it onely one of these dimensions namely length Wherefore a line is the imperfectest kinde of quantitie In defining of a superficies he ascribed vnto it two dimensions namely length and breadth whereby a superficies is a quantitie of
if the angle LMK be an acute angle then is that angle the inclination of the superficies ABCD vnto the superficies EFGH by this definition because it is contained of perpendicular lines drawen in either of the superficieces to one and the self same point being the common section of them both 5 Plaine superficieces are in like sort inclined the onâ ãâ¦ã her when the sayd angles of inclination are equall the one to the o ãâ¦ã This definition needeth no declaration at all but is most manifest by the definition last going before For in considering the inclinations of diuers superficieces to others if the acute angles contayned vnder the perpendicular lines drawen in them from one point assigned in ech of their common sections be equall as if to the angle LMK in the former example be geuen an other angle in the inclination of two other superficieces equall then is the inclination of these superficieces like and are by this definition sayd in like sort to incline the one to the other Now also let there be an other ground plaine superficies namely the superficies MNOP vnto whom also let leane and incline the superficies QââT and let the common section or segment of them be the line QR And draw in the superficies MNOP to some one point of the coÌmon section as to the point X the line VX making with the common section right angles namely the angle VXR or the angle VXQ also in the superficies STQR draw the right line YX to the same point X in the common section making therwith right angles as the angle YXâ or the angle YXQ. Now as sayth the definition if the angles contayned vnder the right lines drawen in these superficieces making right angles with the common section be in the pointes that is in the pointes of their meting in the common section equall then is the inclination of the superficieces equall As in this example if the angle LGH contayned vnder the line LG being in the inclining superficies âKEF and vnder the line HG being in the ground superficies ABCD beÌ equall to the angle YXV contayned vnder the line VX being in the ground superficies MNOP and vnder the line YX being in the inclining superficies STQR then is the inclination of the superâicies IKEF vnto the superficies ABCD like vnto the inclination of the superficies STQR vnto the superficies MNOP And so by this definition these two superficieces are sayd to be in like sort inclined 6 Parallell plaine superficieces are those which being produced or extended any way neuer touch or concurre together Neither needeth this definition any declaration but is very easie to be vnderstanded by the definition of parallell lines âor as they being drawen on any part neuer touch or come together so parallel plaine superâicieces are such which admitte no touch that is being produced any way infinitely neuer meete or come together 7 Like solide or bodily figures are such which are contained vnder like plaine superficieces and equall in multitude What plaine superâicieces are called like hath in the beginning of the sixth booke bene sufficiently declared Now when solide figures or bodies be contained vnder such like plaine superficieces as there are defined and equall in number that is that the one solide haue as many in number as the other in their sides and limites they are called like solide figures or like bodies 8 Equall and like solide or bodely figures are those which are contained vnder like superficieces and equall both in multitude and in magnitude In like solide figures it is suâficient that the superficieces which containe them be like and equall in number onely but in like solide figures and equall it is necessary that the like superficieces contaynyng them be also equal in magnitude So that besides the likenes betwene them they be eche being compared to his correspondent superâicies oâ one greatnes and that their areas or fieldes be equal When such superâicieces contayne bodies or solides then are such bodies equall and like solides or bodies 9 A solide or bodily angle is an inclination of moe then two lines to all the lines which touch themselues mutually and are not in one and the selfe same superâicies Or els thus A solide or bodily angle is that which is contayned vnder mo then two playne angles not being in one and the selfe same plaine superficies but consisting all at one point Of a solide angle doth Euclide here geue two seueâall definitioÌs The first is geuen by the concurse and touch of many lines The second by the touch concurse of many superficiall angles And both these definitions tende to one and are not much different for that lynes are the limittes and termes of superficieces But the second geuen by superâiciall angles is the more naturall definition because that supeâficieces aâe the next and immediate limites of bodies and so are not lines An example of a solide angle cannot wel and at âully be geueÌ or described in a plaââe superficies But touchyng this first definitioÌ lay before you a cube or a die and coÌsider any of the corners or angles therof so shal ye see that at eueây angle there concurre thre lines for two lines coÌcurring cannot make a solide angle namely the line or edge of his breadth of his leÌgth and of his thicknes which their so inclining coÌcurring touether make a solide angle and so of others And now coÌcârning the second definitioÌ what superâicial or plaine angles be hath bene taught before in the first bokâ namely that it is the touch of two right lines And as a superâiciall or playne angle is caused coÌtained of right lines so si a solide angle caused coÌtayned of plaine superficiall angles Two right lines touching together make a plaine angle but two plaine angles ioyned together can not make a solide angle but according to the definitioÌ they must be moe theÌ two as three âoure âiue or moâ which also must not be in one the selfe same superficiâs but must be in diuers superficieces âeeting at one point This definition is not hard but may easily be coÌceiued in a cube or a die where ye see three angles of any three superficieces or sides of the die concurre and meete together in one point which three playne angles so ioyned together make a solide angle Likewise in a Pyramiâ or a spiâe of a steple or any other such thing all the sides therof teÌding vpward narower and narower at length ende their angles at the heigââ or toppe therof in one point So all their angles there ioyned together make a solide angle And for the better âigât thereof I haue set here a figure wherby ye shall more easily conceiue ââ the base of the figure is a triangle namely ABC if on euery side of the triangle ABC ye rayse vp a triangle as vpon the side AB ye raise vp the triangle AFB and vpon the side AC the
in the middest wherof is a point from which all lines drawen to the circumference therof are equall this definition is essentiall and formall and declareth the very nature of a circle And vnto this definition of a circle is correspondent the deâinition of a Sphere geueÌ by Theodosius saying that it is a solide oâ body in the middest whereof there is a point from which all the lines drawen to the circumference are equall So see you the affinitie betwene a circle and a Sphere For what a circle is in a plaine that is a Sphere in a Solide The fulnes and content of a circle is described by the motion of a line moued about but the circumference therof which is the limite and border thereof is described of the end and point of the same line moued about So the fulnes content and body of a Sphere or Globe is described of a semicircle moued about But the Sphericall superficies which is the limite and border of a Sphere is described of the circumference of the same semicircle moued about And this is the superficies ment in the definition when it is sayd that it is contained vnder one superficies which superficies is called of Iohannes de âacro Busco others the circumference of the Sphere Galene in his booke de diffinitionibus mediciâ â geueth yet an other definitioÌ of a Sphere by his propertie or coÌmon accideÌce of mouing which is thus A Sphere is a figure most apt to all motion as hauing no base whereon th stay This is a very plaine and witty deâinition declaring the dignitie thereof aboue all figures generally All other bodyes or solides as Cubes Pyramids and others haue sides bases and angles all which are stayes to rest vpon or impedimentes and lets to motion But the Sphere hauing no side or base to stay one nor angle to let the course thereof but onely in a poynt touching the playne wherein ãâã standeth moueth freely and fully with out let And for the dignity and worthines thereof this circular and Sphericall motion is attributed to the heauens which are the most worthy bodyes Wherefore there is ascribed vnto them this chiefe kinde of motion This solide or bodely figure is also commonly called a Globe 13 The axe of a Sphere is that right line which abideth fixed about which the semicircle was moued As in the example before geuen in the definition of a Sphere the line AB about which his endes being fixed the semicircle was moued which line also yet remayneth after the motion ended is the axe of the Sphere described of that semicircle Theodosius defineth the axe of a Sphere after this maner The axe of a Sphere is a certayne right line drawen by the centre ending on either side in the superficies of the Sphere about which being fixed the Sphere is turned As the line AB in the former example There nedeth to this definition no other declaration but onely to consider that the whole Sphere turneth vpon that line AB which passeth by the centre D and is extended one either side to the superficies of the Sphere wherefore by this definition of Theodosius it is the axe of the Sphere 14 The centre of a Sphere is that poynt which is also the centre of the semicircle This definition of the centre of a Sphere is geuen as was the other definition of the axe namely hauing a relation to the definition of a Sphere here geuen of Euclide where it was sayd that a Sphere is made by the reuolution of a semicircle whose diameter abideth fixed The diameter of a circle and of a semicrcle is all one And in the diameter either of a circle or of a semicircle is contayned the center of either of them for that they diameter of eche euer passeth by the centre Now sayth Euclide the poynt which is the center of the semicircle by whose motion the Sphere was described is also the centre of the Sphere As in the example there geuen the poynt D is the centre both of the semicircle also of the Sphere Theodosius geueth as other definition of the centre of a Sphere which is thus The centre of a Sphere is a poynt with in the Sphere from which all lines drawen to the superficies of the Sphere are equall As in a circle being a playne figure there is a poynt in the middest from which all lines drawen to the circumfrence are equall which is the centre of the circle so in like maner with in a Sphere which is a solide and bodely figure there must be conceaued a poynt in the middest thereof from which all lines drawen to the superficies thereof are equall And this poynt is the centre of the Sphere by this definition of Theodosius Flussas in defining the centre of a Sphere comprehendeth both those definitions in one after this sort The centre of a Sphere is a poynt assigned in a Sphere from which all the lines drawen to the superficies are equall and it is the same which was also the centre of the semicircle which described the Sphere This definition is superfluous and contayneth more theÌ nedeth For either part thereof is a full and sufficient diffinition as before hath bene shewed Or ells had Euclide bene insufficient for leauing out the one part or Theodosius for leauing out the other Paraduenture Flussas did it for the more explication of either that the one part might open the other 15 The diameter of a Sphere is a certayne right line drawen by the ceÌtre and one eche side ending at the superficies of the same Sphere This definitioÌ also is not hard but may easely be couceaued by the definitioÌ of the diameter of a circle For as the diameter of a circle is a right line drawne froÌ one side of the circuÌfrence of a circle to the other passing by the centre of the circle so imagine you a right line to be drawen from one side of the superficies of a Sphere to the other passing by the center of the Sphere and that line is the diameter of the Sphere So it is not all one to say the axe of a Sphere and the diameter of a Sphere Any line in a Sphere drawen from side to side by the centre is a diameter But not euery line so drawen by the centre is the axe of the Sphere but onely one right line about which the Sphere is imagined to be mouedâ So that the name of a diameter of a Sphere is more general then is the name of an axe For euery axe in a Sphere is a diameter of the same but not euery diameter of a Sphere is an axe of the same And therefore Flussas setteth a diameter in the definition of an axe as a more generall word ân this maner The axe of a Sphere is that fixed diameter aboue which the Sphere is moued A Sphere as also a circle may haue infinite diameters but it can haue but
by his motion described the round Conical superficies about the Cone And as the circuÌfereÌce of the semicircle described the round sphericall superficies about the Sphere In this example it is the superficies described of the line DC By this definition it is playne that the two circles or bases of a cilinder are euer equall and parallels for that the lines moued which produced them remayned alwayes equall and parallels Also the axe of a cilinder is euer an erected line vnto either of the bases For with all the lines described in the bases and touching it it maketh right angles Campane Vitellâo with other later writers call this solide or body a round Columnâ or piller And Campane addeth vnto this definition this as a corrollary That of a round Columne of a Sphere and of a circle the ceÌtre is one and the selfe same That is as he him selfe declareth it proueth the same where the Columne the Sphere and the circle haue one diameter 20 Like cones and cilinders are those whose axes and diameters of their bases are proportionall The similitude of cones and cilinders standeth in the proportion of those right lines of which they haue their originall and spring For by the diameters of their bases is had their length and breadth and by their axe is had their heigth or deepenes Wherefore to see whether they be like or vnlike ye must compare their axes together which is their depth and also their diameters together which is thier length breadth As if the axe âG of the cone ABC be to to the axe EI of the cone DEF as the diameter AC of the cone ABC is to the diameter DF of the cone DEF then aâe the cones ABC and DEF like cones Likewise in the cilinders If the axe LN of the cilinder LHMN haue that proportion to the axe OQ of the cilinder ROPQ which the diameter HM hath to the diameter RP then are the cilinders HLMN and ROPQ like cilinders and so of all others 21 A Cube is a solide or bodely figure contayned vnder sixe equall squares As is a dye which hath sixe sides and eche of them is a full and perfect square as limites or borders vnder which it is contayned And as ye may conceiue in a piece of timber contayning a foote square euery way or in any such like So that a Cube is such a solide whose three dimensions are equall the length is equall to the breadth thereof and eche of them equall to the depth Here is as it may be in a playne superficies set an image therof in these two figures wherof the first is as it is commonly described in a playne the second which is in the beginning of the other side of this leafe is drawn as it is described by arte vpoÌ a playne superficies to shew somwhat bodilike And in deede the latter descriptioÌ is for the sight better theÌ the first But the first for the demoÌstrations of Euclides propositions in the fiue bookes following is of more vse for that in it may be considered and sene all the fixe sides of the Cube And so any lines or sections drawen in any one of the sixe sides Which can not be so wel sene in the other figure described vpon a playnd And as touching the first figure which is set at the ende of the other side of this leafe ye see that there are sixe parallelogrammes which ye must conceyue to be both equilater and rectangle although in dede there can be in this description onely two of them rectangle they may in dede be described al equilater Now if ye imagine one of the sixe parallelogrammes as in this example the parallelogramme ABCD to be the base lieng vpon a ground playne superfices And so conceiue the parallelogramme EFGH to be in the toppe ouer it in such sort that the lines AE CG DH BF may be erected perpendicularly from the pointes A C B D to the ground playne superficies or square ABCD. For by this imagination this figure wil shew vnto you bodilike And this imagination perfectly had wil make many of the propositions in these fiue bookes following in which are required to be described such like solides although not all cubes to be more plainly and easily conceiued In many examples of the Greeke and also of the Latin there is in this place set the diffinition of a Tetrahedron which is thus 22 A Tetrahedron is a solide which is contained vnder fower triangles equall and equilater A forme of this solide ye may see in these two examples here set whereof one is as it is commonly described in a playne Neither is it hard to conceaue For as we before taught in a Pyramis if ye imagine the triangle BCD to lie vpon a ground plaine superficies and the point A to be pulled vp together with the lines AB AC and AD ye shall perceaue the forme of the Tetrahedron to be contayned vnder 4. triangles which ye must imagine to be al fower equilater and equiangle though they can not so be drawen in a plaine And a Tetrahedron thus described is of more vse in these fiue bookes following then is the other although the other appeare in forme to the eye more bodilike Why this definition is here left out both of Campane and of Flussas I can not but maruell considering that a Tetrahedron is of all Philosophers counted one of the fiue chiefe solides which are here defined of Euclide which are called coÌmonly regular bodies without mencion of which the entreatie of these should seeme much maimed vnlesse they thought it sufficiently defined vnder the definition of a Pyramis which plainly and generally taken includeth in deede a Tetrahedron although a Tetrahedron properly much differeâh from a Pyramis as a thing speciall or a particular from a more generall For so taking it euery Tetrahedron is a Pyramis but not euery Pyramis is a Tetrahedron By the generall definition of a Pyramis the superficieces of the sides may be as many in number as ye list as 3.4 5.6 or moe according to the forme of the base whereon it is set whereof before in the definition of a Pyramis were examples geuen But in a Tetrahedron the superficieces erected can be but three in number according to the base therof which is euer a triangle Againe by the generall definition of a Pyramiâ the superficieces erected may ascend as high as ye list but in a Tetrahedron they must all be equall to the base Wherefore a Pyramis may seeme to be more generall then a Tetrahedron as before a Prisme seemed to be more generall then a Parallelipipedon or a sided Columne so that euery Parallelipipedon is a Prisme but not euery Prisme is a Parallelipipedon And euery axe in a Sphere is a diameter but not euery diameter of a Sphere is the axe therof So also noting well the definition of a Pyramis euery Tetrahedron may be called a Pyramis
âignifieth Last of all a Dodecahedron for that it is made of Pântagoâ whose angles are more ample and large then the angles of the other bodies and by that âeaâââ draw more ââ rounânes ãâã to the forme and nature of a sphere they assigned to a sphere namely ãâ¦ã Who so will ãâ¦ã in his Tineus shall âead of these figures and of their mutuall proportionâââraunge maâterâ which hâre are not to be entreated of this which is sayd shall be sufficient for the ãâã of them and for thâ declaration of their diffinitions After all these diffinitions here set of Euclide Flussas hath added an other diffinition which ãâã 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 womeÌs 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 hadââtore of that pasted paper by you for so shal yoâ vpon it ãâ¦ã the formes of other bodies as Prismes and Parallelipopedons ãâ¦ã set forth in these fiue bookes following and see the very ãâã of thâse bodies there meÌcioned 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 consistâth of twâluââquilââââ and âquianglâ Pântâââââ vpoâ the foresaid mattâr and finely cut as before was ââught tâââlâuân lines containâd within thâ figurâ and bow and folde the Penââgonâ accordingly And they will so close toâethââ thaâ thây will ââkâ 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 ninâtâne 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 âake 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 sâraunge vnto theÌ I haue for their more âase 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 ãâã pointes then one vâlesse 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 Flâsââs 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 ãâ¦ã the outward angââ ABâ is eqââll 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 Whârefore 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 playâer declaration of Euclides demonstration iâ will not be hard for you to coââââe this figure which âlussâs putteth for his demonstââtion â wherein
two Prismes vnder equall altitudes the one haue to his base a parallelogramme and the other a triangle and if the parallelogramme be double to the triangle those Prismes are equall the one to the other SVppose that these two Prismes ABCDEF GHKMON be vnder equall altitudes and let the one haue to his base the parallelogramme AC and the other the triangle GHK and let the parallelogramme AC be double to the triangle GHK Then I say that the Prisme ABCDEF is equall to the Prisme GHKMON Make perfecte the Parallelipipedons AX GO And forasmuch as the parallelogramme AC is double to the triangle GHK but the parallelogramme GH is also by the 41. of the first double to the triangle GHK wherefore the parallelogramme AC is equall to the parallelogramme GH But Parallelipipedons consisting vpon equall bases and vnder one and the selfe same altitude are equall the one to the other by the 31. of the eleuenth Wherefore the solide AX is equall to the solide GO But the halfe of the solide AX is the Prisme ABCDEF and the halfe of the solide GO is the Prisme GHKMON Wherfore the Prisme ABCDEF is equall to the Prisme GHKMON If therefore there be two Prismes vnder equall altitudes and the one haue to his base a parallelogramme the other a triangle and if the parallelogramme be double to the triangle those Prismes are equall the one to the other which was required to be proued This Proposition and the demonstration thereof are not hard to conceaue by the former figures but ye may for your fuller vnderstanding of theÌ take two equall Parallelipipedons equilateâ and equiangle the one to the other described of pasted paper or such like matter and in the base of the one Parallelipipedon draw a diagonall line and draw an other diagonall line in the vpper superficies opposite vnto the said diagonall line drawen in the base And in one of the parallelogrammes which are set vpon the base of the other Parallelipipedon draw a diagonall line and drawe an other diagonall line in the parallelogramme opposite to the same For so if ye extend plaine superficieces by those diagonall lines there will be made two Prismes in ech body Ye must take heede that ye put for the bases of eche of these Parallelipipedons equall parallelograÌmes And then note theÌ with letters according to the letters of the figures before described in the plaine And coÌpare theÌ with the demonstration and they will make both it and the Proposition very clere vnto you They will also geue great light to the Corollary following added by Flussas A Corollary added by Flussas By this and the former propoâitions it is manifest that Prismes and solides contayned vnder two poligoâon figures equall like and parallels and the rest parallelogrammes may be compared the one to the other after the selfe same maner that parallelipipedons are For forasmuch as by this proposition and by the second Corollary of the 2â of this booke it is manifest that euery parallelipipedon may be resolued into two like and equal Prismes of one and the same altitude whose base shal be one and the selfe same with the base of the parallelipipedon or the halfe thereof which Prisâes also shal be contâyned vnder the selfe same sideâ with the parallelipipedoÌ the sayde sideâ beyng also sides of like proportion I say that Prismeâ may be compared together after the like maner that their Parallelipipedonâ areâ For if we would deuide a Prisme like vnto his foliâe by the 25. of this booke ye shall finde in the Corollaryes of the 25. propoââtioÌ that that which is set forth touching a parallelipipedon followeth not onely in a Prisme but also in any sided columne whose opposite bases are equall and like and his sides parallelogrammes If it be required by the 27. proposition vpon a right line geuen to describe a Prisme like and in like sorte situate to a Prisme geuen describe âââst the whole parallelipipedon whereof the prisme geuen is the halfe which thing ye see by this 40. proposition may be done And vnto that parallelipipedoÌ describe vpon the right line geuen by the sayd 27. proposition an other parallelipipedon like and the halfe thereof shal be the prisme which ye seeke for namely shal be a prisme described vpon the right line geuen and like vnto the prisme geuen In deede Prismes can not be cut according to the 28. proposition For that in their opposite sides can be drawen no diagonall lines howbeit by that 28. proposition those Prismes are manifestly confirmed to be equall and like which are the halues of one and the selfe same parallelipipedon And as touching the 29. proposition and the three following it which proueth that parallelipipedons vnder one and the selfe same altitude and vpon equall bases or the selfe same bases are equal or if they be vnder one and the selfe same altiââdâ they are in proportion the one to the other as their bases areâ to apply these comparisons vnto ãâã it is to ãâã required that the bases of the Prismes compared together be either all parallelogrammes or all triaâgles For so one and the selfe altitude remayning the comparison of thinges equall ãâã one and thââ selfe same and the halfes of the bases are euer the one to the other in the same proportion that their wholes are Wherfore Prismes which are the halues of the parallelipipedons and which haue the same proportion the one to the other that the whole parallelipipedons haue which are vnder one and thâ selfââame altitude must needes cause that their bases being the halues of the baseâ of the parallelipâpââââââe in the same proportioÌ the one to the other that their whole parallelipipâdonâ are If thereâoââ the wâole parallelipipedons be in the proportion of the whole bases their hâlâââ also which are Prismes shal be in the proportion either of the wholes if their bases be parallelâgrâmmâââ or of the halââââf they be triangles which is euer all one by the 15. of the fiueth And forasmuch as by the 33. proposition like parallelipipedons which are the doubles of their Prismes are in treble proportion the one to the other that their sides of like proportion are it is manifest that Prismes being their halues which haue the one to the other the same proportion that their wholes haue by the 15 of the fiueth and hauing the selfe same sides that theiâ parallelipipedons haue are the one to the other in treble proportion of that which the sides of like proportion are And for that Prismes are the one to the other in the same proportion that their parallelipipedons are and the bases of the Prismes being all either triangles or parallelograÌmes are the one to the other in the same proportion that the bases of the parallelipipedons are whose altitudes also are alwayes equall we may by the 34. proposition conclude that the bases of the prismes and the bases of the parallelipipedons their doubles being ech the one to the
last proposition in the second booke and also of the 31. in the sixth booke ¶ A Probleme 5. Two vnequall circles being geuen to finde a circle equall to the excesse of the greater to the lesse Suppose the two vnequal circles geueÌ to be ABC DEF let ABC be the greater whose diameter suppose to be AC the diameter of DEF suppose to be DF. I say a circle must be found equal to that excesse in magnitude by which ABC is greater thâ DEF By the first of the fourth in the circle ABC Apply a right line equall to DF whose one end let be at C and the other let be at B. FroÌ B to A draw a right line By the 30. of the third it may appeare that ABC is a right angle and thereby ABC the triangle is rectangled wherfore by the first of the two corollaries here before the circle ABC is equall to the circle DEF For BC by construction is equall to DF and more ouer to the circle whose diameter is AB That circle therefore whose diameter is AB is the circle conteyning the magnitude by which ABC is greater then DEF Wherefore two vnequal circles being geuen we haue found a circle equall to the excesse of the greater to the lesse which ought to be doone A Probleme 6. A Circle being geuen to finde two Circles equall to the same which found Circles shall haue the one to the other any proportion geuen in two right lines Suppose ABC a circle geuen and the proportion geuen let it be that which is betwene the two right lines D and E. I say that two circles are to be found equall to ABC and with al one to the other in the proportioÌ of D to E. Let the diameter of ABC be AC As D is to E so let AC be deuided by the 10. of the sixth in the poynt F. At F to the line AC let a perpeÌdicular be drawne FB and let it mete the circuÌfereÌce at the poynt B. From the poynt B to the points A and C let right lines be drawne BA and BC. I say that the circles whose diameteâ are the lines BA and BC are equall to the circle ABC and that those circles hauing to their diameters the lines BA and BC are one to the other in the proportion of the line D to the line E. For first that they are equal it is euident by reason that ABC is a triangle rectangle wherfore by the 47. of the first the squares of BA and BC are equall to the square of AC And so by this second it is maniâest the two circles to be equall to the circle ABC Secondly as D is to â so is AF to FC by construction And as the line AF is to the line FC so is the square of the line âA to the square of the line BC. Which thing we will briefely pâoue thus The parallelogramme contayned vnder AC and AF is equall to the square of BA by the Lemma after the 32. of the tenth booke and by the same Lemma or Assumpt the parallelogramme contayned vnder AC and âC is equall to the square of the line BC. Wherfore as the first parallelogramme hath it selfe to the secondâ so hath the square of BA equall to the first parallelogramme it selfe to the square of BC equall to the second parallelogramme But both the parallelograÌmes haue one heigth namely the line AC and bases the lines AF and FC wherefore as AF is to FC so is the parallelogâamme contayned vnder AC AF to the parallelogramme contayned vnder AC FC by the fiâst of the sixth And therefore as AF is to FC so is the square of BA to the square of BC. And as the square of BA is to the square of BC so is the circle whose diameter is BA to the circle whose diameter is BC by this second of the twelfth Wherefore the circle whose diameter is BA is to the circle whose diameter is BC as D is to E. And before we proued them equall to the circle ABC Wherâfore a circle being geuen we haue found two circles equall to the same which haue the one to the other any proportion geuen in two right lines Which ought to be done Note Heâe may you perâeiue an other way how to execute my first probleme for if you make a right angle conteyned of the diameters geueÌ as in this figure suppose them BA and BC and then subtend the right angle with the line AC and from the right angle let fall a line perpendicular to the base AC that perpândicular at the point of his fall deuideth AC into AF and FC of the proportion required A Corollary It followeth of thinges manifestly proued in the demonstration of this probleme that in a triangle rectangle if from the right angle to the base a perpendicular be let fall the same perpendicular cutteth the base into two partes in that proportion one to the other that the squares of the righâ lines conteyning the right angle are in one to the other those on the one side the perpendicular being compared to those on the other both square and segment A Probleme 7. Betwene two circles geuen to finde a circle middell proportionall Let the two circles geuen be ACD and BEF I say that a circle is to be fouÌd which betwene ACD and BEF is middell proportionall Let the diameter of ACD be AD and of BEF let Bâ be the diameter betwene AD and BF finde a line middell proportionall by the 13. of the sixth which let be HK I say that a circle whose diameter is HK is middell proportionall betwene ACD and BEF To AD HK and BF three right lines in continuall proportion by construction let a fourth line be found to which BF shal haue that proportion that AD hath to HK by the 12. of the sixth let that line be â It is manifest that the âower lines AD HK BF and L are in continuall proportion For by coÌstruction as AD is to HK so is Bâ to L. And by construction on as AD is to HK so is HK to BF wherefore HK is to BF as BF is to L by the 11. of the fifth wherfore the 4. lines are in continuall proportion Wherefore as the first is to the third that is AD to BF so is the square of the first to the square of the second that is the square of AD to the square of HK by the corollary of the 20. of the sixth And by the same corollary as HK is to L so is the square of HK to the square of BF But by alternate proportion the line AD is to BF as HK is to L wherefore the square of AD is to the square of HK as the square of HK is to the square of BF Wherefore the square of HK is middell proportionall betwene the square of AD and the square of BF But as the squares are
a triangle and if the parallelogramme be double to the triangle those Prismes are by the 40. of the eleuenth equall the one to the other therefore the Prisme contained vnder the two triangles BKF and EHG and vnder the three parallelogrammes EBFG EBKH and KHFG is equall to the Prisme contained vnder the two triangles GFC and HKL and vnder the three parallelogrammes KFCL LCGH and HKFG And it is manifest that both these Prismes of which the base of one is the parallelogramme EBFG and the opposiâe vnto it the line KH and the base of the other is the triangle GFC and the opposite side vnto it the triangle KLH are greater then both these Pyramids whose bases are the triangles AGE and HKL and toppes the pointes H D. For if we drawe these right lines EF and EK the Prisme whose base is the parallelogramme EBFG and the opposite vnto it the right line HK is greater then the Pyramis whose base is the triangle EBF toppe the point K. But the Pyramis whose base is the triangle EBF and toppe the point K is equall to the Pyramis whose base is the triangle AEG and toppe the point H for they are contained vnder equall and like plaine superficieces Wherefore also the Prisme whose base is the parallelogramme EBFG and the opposite vnto it the right line HK is greater then the Pyramis whose base is the triangle AEG and toppe the point H. But the prisme whose base is the parallelogramme EBFG and the opposite vnto it the right line HK is equall to the prisme whose base is the triangle GFC and the opposite side vnto it the triangle HKL And the Pyramis whose base is the triangle AEG and toppe the point H is equall to the Pyramis whose base is the triangle HKL and toppe the point D. Wherefore the foresaid two prismes are greater then the foresaid two Pyramids whose bases are the triangles AEG HKL and toppes the pointes H and D. Wherefore the whole Pyramis whose base is the triangle ABC and toppe the point D is deuided into two Pyramids equall and like the one to the other and like also vnto the whole Pyramis hauing also triangles to their bases and into two equall prismes and the two prismes are greater then halfe of the whole Pyramis which was required to be demonstrated If ye will with diligence reade these fower bookes following of Euclide which concerne bodyes and clearely see the demonstrations in them conteyned it shall be requisite for you when you come to any proposition which concerneth a body or bodies whether they be regular or not first to describe of pâsâed paper according as I taught you in the end of the definitions of the eleuenth booke such a body or bodyes as are there required and hauing your body or bodyes thus described when you haue noted it with letters according to the figure set forth vpoÌ a plaine in the propositioÌ follow the construction required in the proposition As for example in this third propositioÌ it is sayd that Euery pyramis hauing a triangle to âis base may be deuided into two pyramids c. Here first describe a pyramis of pasted paper haâing his base triangled it skilleth not whether it be equilater or equiangled or not only in this proposition is required that the base be a triangle Then for that the proposition supposeth the base of the pyramis to be the triangle ABC note the base of your pyramis which you haue described with the letters ABC and the toppe of your pyramis with the letter D For so is required in the proposition And thus haue you your body ordered ready to the construction Now in the construction it is required that ye deuide the lines AB BC CA. c namely the sixe lines which are the sids of the fower triangles contayning the piramis into two equall partes in the poyntet â F G c. That is ye must deuide the line AB of your pyramis into two equall partes and note the poynt of the deuision with the letter E and so the line BC being deuided into two equall partes note the poynt of the deuision with the letter F. And so the rest and this order follow ye as touching the rest of the construction there put and when ye haue finished the construction compare your body thus described with the demonstration and it will make it very playne and easy to be vnderstaÌded Whereas without such a body described of matter it is hard for young beginners vnlesse they haue a very deepe imagination fully to conceaue the demonstration by the sigâe as it is described in a plaine Here for the better declaration of that which I haue sayd haue I set a figure whose forme if ye describe vpon pasted paper noted with the like letters and cut the lines âA DA DC and folde it accordingly it will make a Pyramis described according to the construction required in the proposition And this order follow ye as touching all other propositions which concerne bodyes ¶ An other demonstration after Campane of the 3. proposition Suppose that there be a Pyramis ABCD hauing to his base the triangle BCD and let his toppe be the solide angle A from which let there be drawne three subtended lines AB AC and AD to the three angles of the base and deuide all the sides of the base into two equall partes in the three poyntes E F G deuide also the three subteÌded lines AB AC and AD in two equall partes in the three points H K L. And draw in the base these two lines EF and EG So shall the base of the pyramis be deuided into three superficieces whereof two are the two triangles BEF and EGD which are like both the one to the other and also to the whole base by the 2 part of the secoÌd of the sixth by the definitioÌ of like superâiciecâs they are equal the one to the other by the 8. of the first the third superficies is a quadrangled parallelogramme namely EFGC which is double to the triangle EGD by the 40. and 41. of the first Now then agayne from the poynt H draw vnto the points E and F these two subtendent lines HE and HF draw also a subtended line from the poynt K to the poynt G. And draw these lines HK KL and LH Wherefore the whole pyramis ABCD is deuided into two pyramids which are HBEF and AHKL and into two prismes of which the one is EHFGKC and is set vpon the quadrangled base CFGE the other is EGDHKL and hath to his base the triangle EGD Now as touching the two pyramids HBEF and AHKL that they are equall the one to the other and also that they are like both the one to the other and also to the whole it is manifest by the definition of equall and like bodyes and by the 10. of the eleuenth and by 2. part of the second of the sixth And that the two Prismes are equall it
the â of this booke Wherefore a cylinder inclined shall be triple to euery cone although also the cone be erected set vpon one and the same base with it and being vnder the same altitude But the cilinder erected was the triple of the same cone by the tenth of this booke Wherefore the cilinder inclined is equall to the cilinder erected being both set vpon one and the selfe same base and hauing one and the selfe same altitude The same also cometh to passe in cones which are the third partes of equall cilinders therefore are equall the one to the other Wherefore according to the eleuenth of this booke it followeth that cylinders and cones inclined or erected being vnder one and the selfe same altitude are in proportion the one to the other as their bases are For forasmuch as the erected are in proportion as their bases are and to the erected cilinders the inclined are equall therefore they also shall be in proportion as their bases are And therefore by the 12. of this booke like cones and cylinders being inclined are in triple proportion of that in which the diameters of the bases are For forasmuch as they are equall to the erected which haue the proportion by the 12. of this booke and their bases also are equall with the bases of the erected therefore they also shall haue the same proportion Wherefore it followeth by the 13. of this booke thaâ cylinder inclined being cut by a playne superficies parallel to the opposite playne superficieces therof shall be cut according to the proportions of the axes For suppose that vpon one and the selfe same base âe set an erected cylinder and an inclined cylinder being both vnder one and the selfe same altitude which ãâ¦ã a playne superficies parallel to the opposite bases Now it is manifest that the sections of the one cylinder are equall to the section of the other cylinder for they are set vpon equall bases and vnder one and the selfe same altitude namely betwene the parallel playne superficieces And their axes also are by those parallel playne superficiâ ãâã proportionally by the 1ââ of âhe âleuenâh Wherefore the inclined cylinders being equall to the erected cylinders shall haue the proportion of theiâ axes aâ also haue the erected For in echâ the proportion of the axes is one and the same Wherefore inclined Cones and Cylinders being set vpon equall bases shall by the 14. of this booke be in ãâã as their altitudeâ ãâ¦ã forasmuch aâ the iâclined are equall to the erected which haue the selfe same bases and altitude and the erected are iâ proportion as their altitudes therfore the inclined shall be in proportion the one to the other as âhe selfe same altiâudes which are common to ech namely to the inclined and to the erected And therefore in equall cones and cylinders whether they be inclined or erected the bases shall be reciprokally proportionall with the altitudes and contrariwise by the 15. of this booke For forasmuch as we haue oftentimes shewed that the inclined cones and cylinders are equall to the erected hauing the selfâ same bases and altitudes with them and the erected vnto whome the inclined are equall haâe their bases rââiproâall proportionally with their altitudes therefore it followeth that the inclined being equall to the erected haue also their bases and altitudes which are common to eche reciprokally proportionall Likewise if theiâ altitudes bases be reciprokally proportionall they theÌselues also shall be equall for that they are equall to the erected cylinders and cones set vpon the same bases and being vnder the same altitudeâ which erected cylinders are equall the one to the other by the same 15. of this booke Wherefore we may conclude that those passions proprieties which in this twelfth booke haue bene proued to be in cones and cylinders whose altitudes are erected perpendicularly to the ãâ¦ã set obliquely vpoâ their bases Howbeit this is to be noted that such inclined cones or cylinders are not perfect rouâd as are the erected so that if they be cut by a playne superficies passing at right angles with their altitude this section is a Conicall section which is called Ellipsis and shall not describe in their superficies a circle as it doth in erected cylinders cones but a certaine figure whose lesse diameter is in cylinders equall to the dimetient of the base that is is one and the same with it And the same thing happeneth also in cones inclined being cut after the same maner The 1. Probleme The 16. Proposition Two circles hauing both one and the selfe same centre being geuen to inscribe in the greater circle a poligonon figure which shall consist of equall and euen sides and shall not touch the superficies of the lesse circle SVppose that there be two circles ABCD and EFGH hauing one the selfe same centre namely K. It is required in the greater circle which let be ABCD to inscribe a poligonon figure which shal be of equal and euen sides and not touch the circle EFGH Drawe by the centre K a right line BD. And by the 11. of the first from the point G rayse vp vnto the right line BD a perpendicular line AG and extend it to the point C. Wherefore the line AC toucheth the circle EFGH by the 15. of the third Now therfore if by the 30. of the third we diuide the circumference BAD into two equall partes and againe the halfe of that into two equal partes and thus do coÌtinually we shall by the corollary of the 1. of the tenth at the length leaue a certayne circumference lesse then the circumference AD. Let the circumference left be LD And from the point L. Drawe by the 12. of the first vnto the line BD a perpendiculare line LM and extende it to the point N. And draw these right lines LD and DN And forasmuch as the angles DML and DMN are right angles therfore by the 3. of the third the right line BD diuideth the right line LN into two equall parts in the pointe M. Wherfore by the 4. of the first the rest of the sides of the triangles DML and DMN namely the lines DL and DN shal be equall And forasmuch as the line AC is a parallell to the LN by the 28. of the first But AC toucheth the circle EFGH wherfore the line LM toucheth not the circle EFGH and much lesse do the lines LD and DN touch the circle EFGH If therefore there be applied right lines equall to the line LD continually into the circle ABCD by the 1. of the fourth there shal be described in the circle ABCD a poligonon figure which shal be of equall and euen sides and shall not touch the lesse circle namely EFGH by the 14. of the third or by the 29. which was required to be done ¶ Corollary Hereby it is manifest that a perpendicular line drawen from the poynt L to the line BD toucheth not
line which subtendeth the angle ZOB to the third line which subtendeth the angle ZKB But by construction BO is equall to BK therefore OZ is equall to KZ And the third alâo is equall to the third Wherefore the point Z in respecte of the two triangles rectangles OZB and KZB determineth one and the same magnitude iâ the line BZ Which can not be if any other point in the line BZ were assigned nearer or farther of from the point B. One onely poynt therefore is that at which the two perpendiculars KZ and OZ fall But by construction OZ falleth at Z the point and therefore at the same Z doth the perpendicular drawen from K fall likewyse Which was required to be demonstrated Although a briefe monition mought herein haue serued for the pregnant or the humble learner yet for them that are well pleased to haue thinges made plaine with many wordes and for the stiffenecked busie body it was necessary with my controlment of other to annexe the cause reason therof both inuincible and also euident A Corollary 1. Hereby it is manifest that two equall circles cutting one the other by the whole diameter if from one and the same end of their common diameter equall portions of their circumferences be taken and from the pointes ending those equall portions two perpendiculars be let downe to their common diameter those perpendiculars shall fall vpon one and the same point of their common diameter 2. Secondly it followeth that those perpendiculars are equall ¶ Note From circles in our first supposition eche to other perpendicularly erected we procede and inferre now these Corollaries whether they be perpendicularly erected or no by reasou the demonstration hath a like force vpon our suppositions here vsed ¶ The 16. Theoreme The 18. Proposition Spheres are in treble proportion the one to the other of that in which their diameters are SVppose that there be two spheres ABC and DEF and let their diameters be BC and EF. Then I say that the sphere ABC is to the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF. For if not then the sphere ABC is in treble proportion of that in which BC is to EF either to some sphere lesse then the sphere DEF or to some sphere greater First let it be vnto a lesse namely to GHK And imagine that the spheres DEF and GHK be both about one and the selfe same centre And by the proposition next going before describe in the greater sphere DEF a polihedron or a solide of many sides not touching the superficies of the lesse sphere GHK And suppose also that in the sphere ABC be inscribed a polihedron like to the polihedron which is in the sphere DEF Wherefore by the corollary of the same the polihedron which is in the sphere ABC is to the polihedron which is in the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF. But by supposition the sphere ABC is to the sphere GHK in treble proportion of that in which the diameter BC is to the diameter EF. Wherefore as the sphere ABC is to the sphere GHK so is the polihedroÌ which is described in the sphere ABC to the polihedroÌ which is described in the sphere DEF by the 11. of the fift Wherfore alternately by the 16. of the fift as the sphere ABC is to the polihedron which is described in it so is the sphere GHK to the polihedron which is in the sphere DEF But the sphere ABC is greater then the polihedroÌ which is described in it Wherfore also the sphere GHK is greater then the polihedroÌ which is in the sphere DEF by the 14. of the fift But it is also lesse for it is contayned in it which impossible Wherefore the sphere ABC is not in treble proportioÌ of that in which the diameter BC is to the diameter EF to any sphere lesse then the sphere DEF In like sort also may we proue that the sphere DEF is not in treble proportion of that in which the diameter EF is to the diameter BC to any sphere lesse then the sphere ABC Now I say that the sphere ABC is not in treble proportioÌ of that in which the diameter BC is to the diameter EF to any sphere greater theÌ the sphere DEF For if it be possible let it be to a greater namely to LMN Wherfore by conuersion the sphere LMN is to the sphere ABC in treble proportion of that in which the diameter EF is to the diameter BC. But as the sphere LMN is to the sphere ABC so is the sphere DEF to some sphere lesse theÌ the sphere ABC as it hath before bene proued for the sphere LMN is greater then the sphere DEF Wherfore the sphere DEF is in treble proportioÌ of that in which the diameter EF is to the diameter BC to some sphere lesse theÌ the sphere ABC which is proued to be impossible Wherefore the sphere ABC is not in treble proportion of that in which BE is to EF to any sphere greater theÌ the sphere DEF And it is also proued that it is not to any lesse Wherefore the sphere ABC is to the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF which was required to be demonstrated A Corrollary added by Flussas Hereby it is manifest that spheres are the one to the other as like Polihedrons and in like sort described in them are namely eche are in triple proportion of that in which the diameters A Corollary added by Mâ Dee It is then euident how to geue two right lines hauing that proportion betwene them which any two spheres geuen haue the one to the other For if to their diameters as to the first and second lines of fower in continuall proportion you adioyne a third and a fourth line in continuâll proportion as I haue taught before The first and fourth lines shall aunswere the Pâobleme How generall this rule is in any two like solides with their correspondent or Omologall lines I neede not with more wordes declare ¶ Certaine Theoremes and Problemes whose vse is manifolde in Spheres Cones Cylinders and other solides added by Ioh. Dee A Theoreme 1. The whole superficies of any Sphere is quadrupla to the greatest circle in the same sphere contayned It is needeles to bring Archimedes demonstration hereof into this place seing his boke of the Sphere and Cylinder with other his woâkes are euery where to be had and the demoÌstration therof easie A Theoreme 2. Euery sphere is quadruplâ to that Cone whose base is the greatest circle height the semidiameter of the same sphere This is the 32. Proposition of Archimedes fiâst booke of the Sphere and Cylinder A Probleme 1. A Sphere being geuen to make an vpright Cone equall to the same or in any other proportioâ geuen betwene two right lines And as concerning the other part of
also deuided into two equall parts being cylinders which two equall cylinders let be IG and FK the axe of IG suppose to be HN and of FK the axe to be NM And for that FG is an vpright cylinder and at the poynt N cut by a playne Superficies parallell to his opposite bases the common section of that playne superficies and the cylindâr FG must be a circle equall to his base FLB and haue his center the point N. Which circle let be IOK And seing that FLB is by supposition equall to the greatest circle in A IOK also shall be equall to the greatest circle in A contained Also by reason MH is by supposition equal to the diameter of A and NH by constructioÌ half of MH it is manifest that NH is equall to the semidiameter of A. If therefore you suppose a cone to haue the circle IOK to hiâ base and NH to his heith the sphere A shall be to that Cone quadrupla by the 2. Theoreme Let that cone be HIOK Wherefore A is quadrupla to HIOK And the Cylinder IG hauing the same base with HIOK the circle IOK and the same heith the right line NH is triple to the cone HIOK by the 10. of this twelfth booke But to IG the whole cylinder FG is double as is proued Wherefore FG is triple and triple to the cone HIOK that is sextuple And A is proued quadrupla to the same HIOK Wherefore FG is to HIOK as 6. to 1 and A is to HIOK as 4. to 1 Therfore FG is to A as 6 to 4 which in the least termes is as 3 to 2. but 3 to 2 is the termes of sesquialtera proportion Wherefore the cylinder FG is to A sesquialtera in proportion Secondly forasmuch as the superficies of a cylinder his two opposite bases excepted is equall to that circle whose semidiameter is middell proportionall betwene the side of the cylinder and the diameter of his base as vnto the 10. of this booke I haue added But of FG the side BG being parallell and equall to the axe MH must also be equall to the diameter of A. And the base FLB being by supposition equall to the greatest circle in A contained must haue his diameter FB equal to the sayd diameter of A. The middle proportional therfore betwene BG and FB being equall eche to other shalâ be a line equall to either of them As iâ ãâã set BG and FB together as one line and vpon that line composed as a diameter make a semicirclâ and from the center to the circumference draw a linâ perpendicular to the sayd diameter by the ãâã of the sixth that perpendicular is middel proportional betwene FB and BG the semidiametersâ and he him selfe also a semidiameter and therfore by the definition of a circle equall to FB and likewise to BG And a circle hauing his semidiameter equall to the diameter FB is quadruple to the circle FLB For the square of euery whole line is quadruple to the squâre of his halfe line as may be proued by the 4. of the second and by the second of this twelfth circles are one to the other as the squares of their diameters are Wherfore the superficies cylindricall of FG alone is quadrupla to his base FLB But if a certayne quantity be dupla to one thing and an other quadrupla to the same one thing those two quantities together are sextupla to the same one thing Therefore seing the base opposite to FLB being equall to to FLB added to FLB maketh that coÌpound double to FLB that double added to the cylindricall superâicies of FG doth make a superficies sextupla to FLB And the superficies of A is quadrupla to the same FLB by the first Theoreme Therefore the cylindricall superficies of FG with the superficieces of his two bases is to the superficies FLB as 6 to 1 and the superficies of A to FLB is as 4 to 1. Wherfore the cylindricall superficies of FG his two bases together are to the superficies of A as 6 to 4 that is in the smallest termes as 3 to 2. Which is proper to sesâuialtera proportion Thirdly it is already made euident that the superficies cylindrical of FG onely by it self is quadrupla to FLB And also it is proued that the superficies of the sphere A is quadrupla to the same FLB Wherefore by the 7. of the fifth the cylindricall superficies of FG is equall to the superficies of A. Therfore euery cylinder which hath his base the greatest circle in a sphere and heith equal to the diameter of that sphere is sesquialtera to that spere Also the superficies of that cylinder with his two bases is sesquialtera to the superficies of the sphere and without his two bases is equall to the superficies of the sphere which was to be demonstrated The Lemma If A be to C as 6 to 1 and B to C as 4 to 1 A is to B as 6 to 4. For seing B is to C as 4 to 1 by supposition therefore backward by the 4. of the fifth C is to B as 1 to 4. Imagine now two orders of qnantities the first A C and B the second 6 1 and 4. Forasmuch as A is to C as 6 to 1 by supposition and C is to B as 1 to 4 as we haue proued wherfore A is to B as 6 to 4 by the 22 of the fift Therfore if A be to C as 6 to 1 and B to C as 4 to 1 A is to B as 6 to 4. which was to be proued Note Sleight things some times lacking euideÌt proufe brede doubt or ignorance And I nede not warnâ you how genârall this demonstration is for if you put in the place of 6 and 4 any other numbers the like manner of conclusion will follow So likewise in place of 1. any other one number may be as if A be to C as â to 5 and B vnto C be as 7 to 5 A shall be to B as 6 to 7. c. A Probleme 4. To a Sphere geuen to make a cylinder equall or in any proportion geuen betwene two right lines Suppose the geuen Sphere to be A and the proportion geuen to be that betwene X and Y. I say that a cylinder is to be made equall to Aâ or els in the same proportion to A that is betwene X to Y. Let a cylinder be made such one as the Theoreme next before supposed that shall haue his base equall to the greatest circle in A and height equall to the diameter of A Let that cylinder bâ the vpright cylinder BC. Leâ the one side of BC be the right line QC Deuide QC into three equal partâ of which let QE containe two and let the third part be CE. By the point E suppose a plaine parallel to the bases of BC to passe through the cylinder BC cutting the same by the circle DE. I say that the cylinder BE is equall to the Sphere A. For seing BC being an
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 âFG 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 ceÌter 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
superficies or soliditie in the hole or in partâ such certaine knowledge demonstratiue may arise and such mechanical exercise thereby be deuised that sure I am to the sincere true student great light ayde and comfortable courage farther to wade will enter into his hart and to the Mechanicall witty and industrous deuiser new maner of inuentions executions in his workes will with small trauayle for fete application come to his perceiueraunce and vnderstanding Therefore euen a manifolde speculations practises may be had with the circle his quantitie being not knowne in any kinde of smallest certayne measure So likewise of the sphere many Problemes may be executed and his precise quantitie in certaine measure not determined or knowne yet because both one of the first humane occasioÌs of inuenting and stablishing this Arte was measuring of the earth and therfore called Geometria that is Earthmeasuring and also the chiefe and generall end in deede is measure and measure requireth a determination of quantitie in a certayne measure by nuÌber expressed It was nedefull for Mechanicall earthmeasures not to be ignorant of the measure and contents of the circle neither of the sphere his measure and quantitie as neere as sense can imagine or wish And in very deede the quantitie and measure of the circle being knowne maketh not onely the cone and cylinder but also the sphere his quantitie to be as precisely knowne and certayne Therefore seing in respect of the circles quantitie by Archimedes specified this Theoreme is noted vnto you I wil by order vpon that as a supposition inferre the conclusion of this our Theoremes Note 1. Wherfore if you deuide the one side as TQ of the cube TX into 21. equall partes and where 11. partes do end reckening from T suppose the point P and by that point P imagine a plaine passing parallel to the opposite bases to cut the cube TX and therby the cube TX to be deuided into two rectangle parallelipipedons namely TN and PX It is manifest TN to be equall to the Sphere A by construction and the 7. of the fift Note 2. Secondly the whole quantitie of the Sphere A being coÌtayned in the rectangle parallelipipedon TN you may easilie transforme the same quantitie into other parallelipipedons rectangles of what height and of what parallelogramme base you list by my first and second Problemes vpon the 34. of this booke And the like may you do to any assigned part of the Sphere A by the like meanes deuiding the parallelipipedon TN as the part assigned doth require As if a third fourth fifth or sixth part of the Sphere A were to be had in a parallelipipedon of any parallelograââe base assigned or of any heith assigned then deuiding TP into so many partes as into 4. if a fourth part be to be transformed or into fiue if a fifth part be to be transformed c. and then proceede âs you did with cutting of TN from TX And that I say of parallelipipedons may in like sort by my ââyd two problemes added to the 34. of this booke be done in any sided columnes pyramids and prismeâ so thââ in pyramids and some prismes you vse the cautions necessary in respect of their quan ãâ¦ã odyes hauing parallel equall and opposite bases whose partes ãâ¦ã re in their propositions is by Euclide demonstrated And finally ãâ¦ã additions you haue the wayes and orders how to geue to a Sphere or any segmeââ oâ the same Cones or Cylinders equall or in any proportion betwene two right lines geuen with many other most necessary speculations and practises about the Sphere I trust that I haue sufficiently âraughted your imagination for your honest and profitable studie herein and also geuen you reaââ ââtter wheââ with to sââp the mouthes of the malycious ignorant and arrogant despisers of the most excellent discourses trauayles and inuentions mathematicall Sting aswel the heauenly spheres sterres their sphericall soliditie with their conueâe spherical superficies to the earth at all times respecting and their distances from the earth as also the whole earthly Sphere and globe it selfe and infinite other cases concerning Spheres or globes may hereby with as much ease and certainety be determined of as of the quantitie of any bowle ball or bullet which we may gripe in our handes reason and experience being our witnesses and without these aydes such thinges of importance neuer hable of vs certainely to be knowne or attayned vnto Here ende M. Iohn d ee his additions vpon the last proposition of the twelfth booke A proposition added by Flussas If a Sphere touche a playne superficiesâ a right line drawne from the center to the touche shall be erected perpendicularly to the playne superficies Suppose that there be a Sphere BCDL whose centre let be the poynt A. And let the playne superficies GCI touch the Spere in the poynt C and extend a right line from the centre A to the poynt C. Then I say that the line AC is erected perpendicularly to tâe playne GIC. Let the sphere be cutte by playne superficieces passing by the right line LAC which playnes let be ABCDL and ACEL which let cut the playne GCI by the right lines GCH and KCI Now it is manifest by the assumpt put before the 17. of this booke that the two sections of the sphere shall be circles hauing to their diameter the line LAC which is also the diameter of the sphere Wherefore the right lines GCH and KCI which are drawne in the playne GCI do at the poynt C fall without the circles BCDL and ECL. Wherefore they touch the circles in the poynt C by the second definition of the third Wherefore the right line LAC maketh right angles with the lines GCH and KCI by the 16. of the third Wherefore by the 4. of the eleuenth the right line AC is erected perpendicularly to to the playne superficies GCI wherein are drawne the lines GCH and KCI If therefore a Sphere touch a playne superficies a right line drawne from the centre to the touche shall be erected perpendicularly to the playne superficies which was required to be proued The ende of the twelfth booke of Euclides Elementes ¶ The thirtenth booke of Euclides Elementes IN THIS THIRTENTH BOOKE are set forth certayne most wonderfull and excellent passions of a lyne deuided by an extreme and meane proportion a matter vndoubtedly of great and infinite vse in Geometry as ye shall both in thys booke and in the other bookes following most euidently perceaue It teacheth moreouer the composition of the fiue regular solides and how to inscribe them in a Sphere geuen and also setteth forth certayne comparisons of the sayd bodyes both the one to the other and also to the Sphere wherein they are described The 1. Theoreme The 1. Proposition If a right line be deuided by an extreme and meane proportion and to the greater segment be added the halfe of the whole line the square made of those two
lines added together shal be quintuple to the square made of the halfe of the whole lyne SVppose that the right line AB be deuided by an extreme and meane proportioÌ in the point C. And let the greater segment therof be AC And vnto AC adde directly a ryght line AD and let AD be equall to the halfe of the line AB Then I say that the square of the line CD is quintuple to the square of the line DA. Describe by the 46. of the first vpon the lines AB and DC squares namely AE DF. And in the square DF describe and make complete the figure And extend the line FC to the point G. And forasmuch as the line AB is deuided by an extreme and meane proportion in the point C therefore that which is contayned vnder the lines AB and BC is equall to the square of the line AC But that which is contayned vnder the lines AB and BC is the parallelogramme CE and the square of the line AC is the square HF. Wherefore the parallelogramme CE is equall to the square HF. And forasmuch as the line BA is double to the line AD by constructâon ãâã the lyne BA is equall to the line KA and the line AD to the lyne AH therefore also the lyne KA is double to the line AH But as the lyne KA is to the line AH so is the parallelogramme CK to the parallelogramme CH Wherefore the parallelogramme CK is double to the parallelogramme CH. And the parallelogrammes LH and CH are double to the parallelogramme CH for supplementes of parallelogrammes are bâ the 4â of the first equall the one to the other Wherefore the parallelogramme CK is equall to the parallelogrammes LH CH. And it is proued that the parallelogramme CE is equall to the square FH Wherefore the whole square AE is equall to the gnâmon MXN And forasmuch as the line BA iâ double to the line AD therefore the square of the line BA is by the 20. of the sixth quadruple to the square of the line DA that is the square AE to the square DH But the square AE is equall to the gnomoÌ MXN wherefore the gnomoÌ MXN is also quadruple to the square DH Wherefore the whole square DF is quintuple to the square DH But the square DF iâ the square of the line CD and the square DH is the square of the line DA. Wherefore the square of the line CD is quintuple to the square of the line DA. If therefore a right line be deuided by an extreame and meane proportion and to the greater segment be added the halfe of the whole line the square made of those two lines added together shal be quintuple to the square made of the halfe of the whole line Which was required to be demonstrated Thys proposition is an other way demonstrated after the fiueth proposition of this booke The 2. Theoreme The â Proposition If a right line be in power quintuple to a segment of the same line the double of the sayd segment is deuided by an extreame and meane proportion and the greater segment thereof is the other part of the line geuen at the beginning Now that the double of the line AD that is AB is greater then the line AC may thus be proued For if not then if if it be possible let the line AC be double to the line AD wherefore the square of the line AC is quadruple to the square of the line AD. Wherefore the squares of the lines AC and AD are quintuple to the squares of the line AD. And it is supposed that the square of the line DC is quintuple to the square of the line AD wherefore the square of the line DC is equall to the square of the lines AC and AD which is impossible by the 4. of the second Wherefore the line AC is not double to the line AD. In like sorte also may we proue that the double of the line AD is not lesse then the line AC for this is much more absurd wherefore the double of the line AD is greater theÌ the line ACâ which was required to be proued This proposition also is an other way demonstrated after the fiueth proposition of this booke Two Theoremes in Euclides Method necessary added by M. Dee A Theoreme 1. A right line can be deuided by an extreame and meane proportion but in one onely poynt Suppose a line diuided by extreame and meane proportion to be AB And let the greater segment be AC I say that AB can not be deuided by the sayd proportion in any other point then in the point C. If an aduersary woulde contend that it may in like sort be deuided in an other point let his other point be supposed to be D making AD the greater segment of his imagined diuision Which AD also let be lesse then our AC for the first discourse Now forasmuch as by our aduersaries opinion AD is the greater segment of his diuided lineâ the parallelogramme conteyned vnder AB and DB is equall to the square of AD by the third definition and 17. proposition of the sixth Booke And by the same definition and proposition the parallelogramme vnder AB and CB conteyned is equall to the square of our greater segment AC Wherefore as the parallelogramme vnder AB and Dâ is to the square of AD so iâ ãâã parallelogramme vnder AB and CB to the square of AC For proportion of equality is concluded in them both But forasmuch as Dââ iâ byâ supposition greater theÌ CB the parallelograÌme vnder AB and DB is greater then the parallelogramme vnder AC and CB by the first of the sixth for AB is their equall heith Wherefore the square of AD shal be greater then the square of AC by the 14. of the fifth But the line AD is lesse then the line AC by supposition wherefore the square of AD is lesse then the square of AC And it is concluded also to be greater then the square of AC Wherefore the square of AD is both greater then the square of ACâ and also lesse Which is a thing impossible The square therefore of AD is not equall to the parallelogramme vnder AB and DB. And therefore by the third definition of the sixth AB is not deuided by an extreame and meane proportion in the point D as our aduersary imagined And Secondly in like sort will the inconueniency fall out if we assigne AD our aduersaries greater segment to be greater then our AC Therefore seing neither on the one side of our point C neither on the other side of the same point C any point can be had at which the line AB can be deuided by an extreame and meane proportion it followeth of necâssitie that AB can be deuided by an extreame and meane proportion in the point C onely Therefore a right line can be deuided by an extreame
line AI. Wherefore the square of the line BW is equall to the square of the line AI wherefore also the line BT is equall to the line AI. And by the same reason are the lines ID and WC equall to the same lines Now forasmuch as the lines AI and ID and the lines AL and LD are equall and the base IL is common to them both the angles ALI and DLI shal be equall by the 8. of the first and therefore they are right angles by the 10. diffinition of the first And by the same reason are the angles WHB and WHC right angles And forasmuch as the two lines HT and TW are equall to the two lines Lct and ctI and they contayne equall angles that is right angles by supposition therefore the angles WHT and ILct are equall by the 4. of the first Wherefore the playne superficies AID is in like sort inclined to the playne superficies ABCD as the playne superficies BWC is inclined to the same playne ABCD by the 4. diffinition of the eleuenth In like sort may we proue that the playne WCDI is in like sort inclined to the playne ABCD as the playne BVZC is to the playne EBCF. For that in the triangles YOH and ctPK which consist of equall sides eche to his correspondent side the angles YHO and ctKP which are the angles of the inclination are equall And now if the right line ctK be extended to the point a and the pentagon CWIDa be made perfect we may by the same reason proue that that playne is equiangle and equilater that we proued the pentagon BVZCW to be equaliter and equiangle And likewise if the other playnes BWIA and AID be made perfect they may be proued to be equall and like pentagons and in like sort situate and they are set vpon these common right lines BW WC WI AI and ID And obseruing this methode there shall vpon euery one of the 12. âidâs of the cube be set euery one of the 12. pentagons which compose the dodecahedron ¶ Certayne Corollaryes added by Flussas First Corollary The side of a cube is equall to the right line which subtendeth the angle of the pentagon of a dodecahedron contayned in one and the selfe same sphere with the cube For the angles BWC and AID are subtended of the lines BC and AD. Which are sides of the Cubeâ ¶ Second Corollary In a dodecahedron there are sixe sides euery two of which are parallels and opposite whose sections into two equall partes are coupled by three right lines which in the center of the sphere which contayneth the dodecahedron deuide into two equall partes and perpendicularly both them selues and also the sides For vpon the sixe bases of the cube are set sixe sides of the dodecahedron as it hath bene proued by the lines ZV WI c. which are cutte into two equall partes by right lines which ioyne together the centers of the bases of the cube as the line YO produced and the other like Which lines coupling together the centers of the bases are three in number cutting the one the other perpendicularly for they are parallels to the sides of the cube and they cutte the one the other into two equall partes in the center of the sphere which contayneth the cube by that which was demonstrated in the 15. of this booke And vnto these equall lines ioyning together the centers of the bases of the cube are without the bases added equall partes OY P ct and the other like which by supposition are equall to halfe of the side of the dodecahedron Wherefore the whole lines which ioyne together the sectoins of the opposite sides of the dodecahedron are equall and they cut those sides into two equall partes and perpendicularly Third Corollary A right line ioyning together the poynts of the sections of the opposite sides of the dodecahedron into two equall partes being diuided by an extreame and meane proportion the greater segment thereof shal be the side of the cube and the lesse segment the side of the dodecahedron contayned in the selfe same sphere For it was proued that the right line YQ is diuided by an extreame and meane proportion in the poynt O and that his greater segment OQ is halfe the side of the cube and his lesse segment OY is halfe of the side VZ which is the side of the dodecahedron Wherefore it followeth by the 15. of the fifth that their doubles are in the same proportion Wherefore the double of the line YQ which ioyneth the poynt opposite vnto the line Y is the whole and the greater segment is the double of the line OQ which is the side of the cube the lesse segment is the double of the line YO which is equall to the side of the dodecahedron namely to the side VZ ¶ The 6. Probleme The 18. Proposition To finde out the sides of the foresayd fiue bodies and to compare them together TAke the diameter of the Sphere geuen and let the same be AB and diuide it in the point C so that let the line AC be equall to CB by the 10. of the first and in the point D so that let AD be double to DB by the 9. of the sixt And vpon the line AB describe a semicircle AEB And from the pointes C and D raise vp by the 11. of the first vnto the line AB perpendicular lines CE and DF. And draw these right lines AF FB and BE. Now forasmuch as the line AD is double to the line DB therefore the line AB is treble to the line DB. Wherefore the line BA is sesquialter to the line AD for it is as 3. to 2. But as the line BA is to the line AD so is the square of the line BA to the square of the line AF by the 6. of the sixt or by the Corollary of the same and by the Corollary of the 20. of the same for the triangle AFB is equiangle to the triangle AFD Wherefore the square of the line BA is sesquialter to the square of the line AF. But the diameter of a sphere is in power sesquialter to the sidâ of the pyramis by the 13. of this booke and the line AB is the diameter of the sphere Wherefore the line AF is equall to the side of the pyramis Againe forasmuch as the line AB is treble to the line BD but as the line AB is to the line BD so is the square of the line AB to the square of the line FB by the Corollaries of the 8. and 20. of the sixt Whereâore the square of the line AB is treble to the square of the line FB But the diameter oâ a sphere is in power treble to the side of the cube by the 15. of this booke and the diameter of the sphere is the line AB Wherefore the line BF is the side of the cube And forasmuch as the line FB is the
double to the side of the Octohedron the side is in power sequitertia to the perpeÌdiclar line by the 12. of this booke wherfore the diameter thereof is in power duple superbipartiens tertias to the perpendicular line Wherfore also the diameter and the perpeÌdicular line are rationall and commensuâable by the 6. of the tenth As touching an Icosahedron it was proued in the 16. of this booke that the side thereof is a lesse line when the diameter of the sphere is rationall And forasmuch as the angle of the inclination of the bases thereof is contayned of the perpendicular lines of the triangles and subtended of the right line which subtendeth the angle of the Pentagon which contayneth fiue sides of the Icosahedron and vnto the perpendicular lines the side is commensurable namely is in power sesquitertia vnto them by the Corollary of the 12. of this booke therefore the perpendicular lines which contayne the angles are irrationall lines namely lesse lines by the 105. of the tenth booke And forasmuch as the diameter contayneth in power both the side of the Icosahedron and the line which subtendeth the foresayd angle if from the power of the diameter which is rationall be taken away the power of the side of the Icosahedron which is irrationall it is manifest that the residue which is the power of the subtending line shal be irrationall For if it shoulde be rationall the number which measureth the whole power of the diameter and the part taken away of the subtending line should also by the 4. common sentence of the seuenth measure the residue namely the power of the side which is irrationall for that it is a lesse line which were absurd Wherefore it is manifest that the right lines which compose the angle of the inclination of the bases of the Icosahedron are Irrationall lines For the subtending line hath to the line contayninge a greater proportion then the whole hath to the greater segment The angle of the inclination of the bases of a dodecahedron is contayned vnder two perpendiculars of the bases of the dodecahedron and is subtended of that right line whose greater segment is the side of a Cube inscribed in the dodecahedron which right line is equall to the line which coupleth the sections into two equal parts of the opposite sides of the dodecahedron And this coupling line we say is an irrationall line for that the diameter of the sphere contayneth in power both the coupling line and the side of the dodecahedron but the side of the dodecahedron is an irrationall line namely a residuall line by the 17. of this booke Wherefore the residue namely the coupling line is an irrationall line as it is âasy to proue by the 4. coÌmon sentence of the seueÌth And that the perpeÌdicular lines which contayne the angle of the inclination are irrationall is thus proued By the proportion of the subtending line of the foresayd angles of inclination to the lines which containe the angle is found out the obliquitie of the angle For if the subtending line be in power double to the line which contayneth the angle then is the angle a right angle by the 48. of the first But if it be in power lesse then the double it is an acute angle by the 23. of the second But if it be in power more then the double or haue a greater proportion then the whole hath to the greater segmeÌtâ the angle shal be an obtuse angle by the 12. of the second and 4. of the thirtenth By which may be proued that the square of the whole is greater then the double of the square of the greater segment This is to be noted that that which Flussas hath here taught touching the inclinations of the bases of the âiue regular bodies Hypsicles teacheth after the 5 proposition of the 15. booke Where he confesseth that he receiued it of one Isidorus and seking to make the mater more cleare he endeuored himselfe to declare that the angles of the inclination of the solides are geuen and that they are either acute or obtuse according to the nature of the solide although âuclidâ in all his 15. bookes hath not yet shewed what a thing geuen is Wherefore Flussas framing his demoÌstration vpon an other ground procedeth after an other maner which semeth more playne and more aptly hereto be placed then there Albeit the reader in that place shal not be frustrate of his also The ende of the thirtenth Booke of Euclides Elementes ¶ The fourtenth booke of Euclides Elementes IN this booke which is commonly accompted the 14. booke of Euclide is more at large intreated of our principal purpose namely of the comparison and proportion of the fiue regular bodies customably called the 5. figures or formes of Pythagoras the one to the other and also of their sides together eche to other which thinges are of most secret vse and inestimable pleasure and commoditie to such as diligently search for them and attayne vnto them Which thinges also vndoubtedly for the woorthines and hardnes thereof for thinges of most price are most hardest were first searched and found out of Philosophers not of the inferior or meane sort but of the depest and most grounded Philosophers and best exercised in Geometry And albeit this booke with the booke following namely the 15. booke hath bene hetherto of all men for the most part and is also at this day numbred and accompted amoÌgst Euclides bookes and supposed to be two of his namely the 14. and 15. in order as all exemplars not onely new and lately set abroade but also old monumentes written by hand doo manifestly witnes yet it is thought by the best learned in these dayes that these two bookes are none of Euclides but of some other author no lesse worthy nor of lesse estimation and authoritie notwithstanding then Euclide Apollonius a man of deepe knowledge a great Philosopher and in Geometrie maruelous whose woÌderful bookes writteÌ of the sections of cones which exercise occupy thewittes of the wisest and best learned are yet remayning is thought and that not without iust cause to be the author of them or as some thinke Hypsicles him selfe For what can be more playnely then that which he him selfe witnesseth in the preface of this booke Basilides of Tire sayth Hypsicles and my father together scanning and peysing a writing or books of Apollonius which was of the comparison of a dodecahedron to an Icosahedron inscribed in one and the selfe same sphere and what proportion these figures had the one to the other found that Apollonius had fayled in this matter But afterward sayth he I found an other copy or booke of Apollonius wherein the demonstration of that matter was full and perfect and shewed it vnto them whereat they much reioysed By which wordes it semeth to be manifest that Apollonius was the first author of this booke which was afterward set forth by Hypsicles For so his owne wordes after in
22. of the sixt Wherefore by coÌposition by the 18. of the fifth as both the lines AB BC added the one to the other together with the line AC that is as two such lines as AB is are to the line AC so are both the lines DE and EF added the one to the other together with the line DF that is two such lines as DE is to the line DF. And in the same proportion are the halues of the antecedents by the 15. of the fifth Wherefore as the line AB is to the line AC so is the line DE to the line DF. And therefore by the 19. of the fifth as the line AB is to the line BC so is the line DF to the line FE Wherefore also by diuision by the 17. of the fifth as the line AC is to the line CB so is the line DF to the line DE Now that we haue proued that any right line whatsoeuer being diuided by an extreame and meane proportion what proportion the line contayning in power the squares made of the whole line and of the greater segment added together hath to the line contayning in power the squares made of the whole line and of the lesse segment added together the same proportion hath the side of the cube to the side of the Icosahedron Now also that we haue proued that as the side of the cube is to the side of the Icosahedron so is the superficies of the Dodecahedron to the superficies of the Icosahedron being both described in one and the selfe same sphere and moreouer seing that we haue proued that as the superficies of the Dodecahedron is to the superficies of the IcosahedroÌ so is the DodecahedroÌ to the Icosahedron for that both the pentagon of the Dodecahedron and the triangle of the Icosahedron are comprehended in one and the selfe same circle All these thinges I say being proued it is manifest that if in one and the selfe same sphere be described a Dodecahedron and an Icosahedron they shall be in proportion the one to the other as a right line whatsoeuer being diuided by an extreame and meane proportion the line contayning in power the squares of the whole line and of the greater segment added together is to the line containing in power the squares of the whole line and of the lesse segment added together For for that as the Dodecahedron is to the Icosahedron so is the superficies of the Dodecahedron to the superficies of the Icosahedron that is the side of the cube to the side of the Icosahedron but as the side of the cube is to the side of the Icosahedron so any right line what so euer being diuided by an extreame and meane proportion is the line contayning in power the squares of the whole line and of the greater segment added together to the line contayning in power the squares of the whole line and of the lesse segment added together Wherefore as a Dodecahedron is to an Icosahedron described in one and the selfe same sphere so any right line what so euer being diuided by an extreame and meane proportion is the line contayning in power the squares of the whole line of the greater segment added together to the line contayning in power the squares of the whole line and of the lesse segment added together The ende of the fourtenth Booke of Euclides Elementes after Hypsicles ¶ The fourtenth booke of Euclides Elementes after Flussas FOr that the fouretenth Booke as it is set forth by Flussas containeth in it moe Propositions then are found in Hypsicles also some of those Propositions which Hypsicles hath are by him somewhat otherwise demonstrated I thought my labour well bestowed for the readers sake to turne it also all whole notwithstanding my trauaile before taken in turning the same booke after Hypsicles Where note ye that here in this 14. booke after Flussas and in the other bookes following namely the 15. and 16. I haue in alleadging of the Propositions of the same 14. booke followed the order and number of the Propositions as Flussas hath placed them ¶ The first Proposition A perpendicular line drawen from the centre of a circle to the side of a Pentagon inscribed in the same circle is the halfe of these two lines taken together namely of the side of the hexagon and of the side of the decagon inscribed in the same circle TAke a circle ABC and inscribe in it the side of a pentagon which let be BC and take the centre of the circle which let be the point D and froÌ it draw vnto the side BC a perpendicular line DE which produce to the point â And vnto the line E F put the line EG equall And draw these right lines CG CD and CF. Then I say that the right line DE which is drawen from the centre to BC the side of the pentagon is the halfe of âhe sideâ of the decagon and hexagon taken together Forasmuch as the line DE is a perpendicular ânto the line BC therefore the sections BE and EC shall be equall by the 3. of the third and the line EF is common vnto them both and the angles FEC and FEB are right angles by supposition Wherefore the bases BF and FC are equall by the 4. of the first But the line BC is the side of a pentagon by construction Wherefore FC which subtendeth the halfe of the side of the pentagon is the side of the decagon inscribed in the circle ABC But vnto the line FC is by the 4. of the first equall the line CG for they subtend right angles CEG and CEF which are contained vnder equall sides Wherefore also the angles CGE and CFE of the triangle CFG are equall by the 5. of the first And forasmuch as the arke FC is subtended of the side of a decagon the arke CA shall be quadruple to the arke CF Wherefore also the angle CDA shall be quadruple to the angle CDF by the last of the sixâ And forasmuch as the same angle CDA which is set at the center is double to the angle CFA which is set at the circumference by the 20. of the third therefore the angle CFA or CFD is double to the angle CDâ namely the halfe of quadruple But vnto the angle CFD or CFG is proued equall the angle CGF Wherefore the outward angle CGF is double to the angle CDF Wherefore the angles CDG and DCG shall be equall For vnto those two angles the angle CGF is equall by the 32. of the first Wherefore the sides GC and GD are equall by the 6. of the first Wherefore also the line GD is equall to the line FC which is the side of the decagon But vnto the right line FE is equall the line EG by construction Wherefore the whole line DE is equall to the two lines Câ and FE Wherefore those lines taken together namely the lines DF and FC shall
spherâ containeth the Dodecahedron of this pentagon and the Icosahedron of this triangle by the 4. of this booke â and the line CL falleth perpendiculaâly 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 âuperficies of that solide vpon whose side the perpendiculâr 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 timâs 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 sâdes 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 âG iâ to the liââ GD For forasmuche as the line EI beinâ diuided by an extreme and meane proportion the greater segment thârof 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 ârâ cut proportionally by the second of this bâoke Whârâfore as the line CG is to the line AG so is the line EI to the line EZ Wherâfore that which it contained vnder the extreames CG and EZ is âquall to that which iâ contaynâd vnder the meanes AG and EI. by the 16. of the sixth But as that which iâ contained vnder the linââ CG and âZ 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 oââ 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 liââ 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 ââ 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 AâCIG 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 AâH 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 âases 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 AâCâG For forasmuch as the line âZ is sesqâialter to the line AD for the line Dâ is diuided into two equall partes in the poynt Z by the corollary of the â2â 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 ãâã âT Wherefore that which is contayned vndeâ the ãâã 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 âK âroued 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 contaynethâ ãâ¦ã equall âo 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 ãâ¦ã 1. of the sixth that which is coÌteyned 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 ââTâ therefore that which is contayned vnder the lines AZ
together And a perpendicular line drawne from the centre of the sphere to any base of the cube is equall to halfe the side of the cube which was required to be prouâd ¶ A Corollary If two thirds of the power of the diameter of the sphere be multiplyed into the perpendicular line equall to halfe the side of the cube there shall be produced a solide equall to the solide of the cube For it is before manifest that two third partes of the power of the diameter of the sphere are equall to two bases of the cube If therefore vnto eche of those two thirds be applyed halfe the altitude of the cube they shall make eche of those solides equall to halfe of the cube by the 31. of the eleuenth for they haue equall bases Wherefore two of those solides are equall to the whole cube You shall vnderstand gentle reader that Campane in his 14. booke of Euclides Elementes hath 18. propositioÌs with diuers corollaries following of them Some of which propositions and corollaries I haue before in the twelfth and thirtenth bookes added out of Flussas as corollaries which thing also I haue noted on the side of those corollaries namely with what proposition or corollary of Campanes 14. booke they doo agree The rest of his 18. propositions and corollaries are contained in the twelue former propositions and corollaries of this 14. booke after Flussas where ye may see on the side of eche proposition and corollary with what proposition and corollary of Campanes they agree But the eight propositions following together with their corollaries Flussas hath added of him selfe as he him selfe affirmeth The 13. Proposition One and the self same circle containeth both the square of a cube and the triangle of an Octohedron described in one and the selfe same sphere SVppose that there be a cube ABG and an Octohedron DEF described in one and the selfe same sphere whose diameter let be AB or DH And let the lines drawne from the ceÌtres that is the semidiameters of the circles which ctoÌaine the bases of those solides â be CA and ID Then I say that the lines CA and ID are equal Forasmuch as AB the diameter of the sphere which containeth the cube is in power triple to BG the side of the cube by the 15. of the thirtenth vnto which side AG the diameter of the base of the cube is in power double by the 47. of the first which line AG is also the diameter of the circle which coÌtaineth the base by the 9. of the fourth therfore AB the diameter of the sphere is in power sesquialter to the line AG namely of what partes the line AB containeth in power 12. of the same the line AG shal containe in power 8. And therfore the right line AC whiche is drawn from the ceÌtre of the circle to the circumference conteineth in power of the same partes 2. Wherefore the diameter of the sphere is in power sextuple to the lyne which is drawne from the centre to the circumference of the circle whiche containeth the square of the cube But the Diameter of the selfe same Sphere whych containeth the Octohedron is one and the selfe same with the diameter of the cube namely DH is equall to AB and the same diameter is also the diameter of the square which is made of the sides of the Octohedron wherefore the saide diameter is in power double to the side of the same Octohedron by the 14. of the thirtenth But the side DF is in power triple to the line drawne from the centre to the circumference of the circle which containeth the triangle of the octohedron namely to the line ID by the 12. of the thirtenth Wherfore the selfe same diameter AB or DH which was in power sextuple to the line drawne from the centre to the circumference of the circle which containeth the square of the cube is also sextuple to the line ID drawne from the centre to the circumference of the circle which containeth the triangle of the Octohedron Wherefore the lines drawne from the centres of the circles to the circumferences which containe the bases of the cube and of the octohedron are equal And therfore the circles are equal by the first diffinition of the third Wherfore one and the selfe same circle containeth c. as in the proposition which was required to be proued A Corollary Hereby it is manifest that perpendiculars coupling together in a sphere the centres of the circles which containe the opposite bases of the cube and of the Octohedron are equal For the circles are equal by the second corollary of the assumpt of the 16. of the twelfth and the lines which passing by the centre of the sphere couple together the centres of the bases are also equal by the first corollary of the same Wherfore the perpendicular which coupleth together the opposite bases of the Octohedron is equal to the side of the cube For either of them is the altitude erected The 14. Proposition An Octohedron is to the triple of a Tetrahedron contained in one and the selfe same sphere in that proportion that their sides are SVppose that there be an octohedron ABCD and a Tetrahedron EFGH vpon whose base FGH erect a Prisme which is done by erecting from the angles of the base perpendicular lines equal to the altitude of the Tetrahedron which prisme shal be triple to the Tetrahedron EFGH by the first corollary of the 7. of the twelfth Then I say that the octohedron ABCD is to the prisme which is triple to the Tetrahedron EFGH as the side BC is to the side FG. For forasmuch as the sides of the opposite bases of the octohedron are right lines touching the one the other and are parellels to other right lines touching the one the other for the sides of the squares which are coÌposed of the sides of the octohedroÌ are opposite Wherfore the opposite plaine triangles namely ABC KID shal be parallels and so the rest by the 15. of the eleuenth Let the diameter of the Octohedron be the line AD. Now then the whole Octohedron is cut into foure equal and like pyramids set vpon the bases of the octohedron and hauing the same altitude with it being about the Diameter AD namely the pyramis set vpon the base BID and hauing his toppe the poynt A and also the pyramis set vppon the base BCD hauing his top the same poynt A. Likewise the pyramis set vpoÌ the base IKD hauing his toppe the same poynt A and moreouer the pyramis set vpon the base CKD and hauing his toppe the former poynt A which pyramids shal be equal by the 8. diffinition of the eleuenth for they eche consist of two bases of the octohedron and of two triangles contained vnder the diameter AD and two sides of the octohedroÌ Wherfore the prisme which is set vpon the base of the Octohedron
DEF whose side let be DE and let the right line subtending the angle of the pentagon made of the sides of the Icosahedron be the line EF. Then I say that the side ED is in power double to the line H the lesse of those segmentes Forasmuch as by that which was demonstrated in the 15. of this booke it was manifest that ED the side of the Icosahedron is the greatâr segment of the line EFâ and that the diameter DF containeth in power the two lines ED and EF namely the whole and the greater segment but by suppoâition the side AB coÌtaineth in power the two lines C H ioined together in the self same proportioÌ Wherefore the line EF is to the line ED as the line C is to the line H by the â oâ this bokeâ And altârnaâây by the 16. of the fiueth the line EF is to the line C as the line ED is to the line H. And forasmuche as the line DF containeth in power the two lines ED and EF and the line AB containeth in power the two lines C and H therefore the squares of the lines EF and ED are to the square of the line DF as the squares of the lines C and H to the square AB And alternately the squares of the lines EF and âD are to the squares of the lines C and H as the square of the line DF is to the square of the line ABâ But DF the diameter is by the 14. of the thirtenâh iâ power double to AB the side of the octohedron inscribed by supposition in the same sphere Wherefore the squares of the lines EF and ED are double to the squares of the lines C and H. And therfore one square of the line ED is double to one square of the line H by the 12. of the fifth Wherfore ED the side of the Icosahedron is in power duple to the line H which is the lesse segment If therfore the poweâ of the side of an octohedron be expressed by two right lines ioyned together by an extreme and meane proportion the side of the Icosahedron contained in the same sphere shal be duple to the lesse segment The 17. Proposition If the side of a dodecahedron and the right line of whome the said side is the lesse segment be so set that they make a right angle the right line which containeth in power halfe the line subtending the angle is the side of an Octohedron contained in the selfe same sphere SVppose that AB be the side of a Dodecahedron and let the right line of which that side is the lesse segment be AG namely which coupleth the opposite sides of the Dodecahedron by the 4. corollary of the 17. of the thirtenth and let those lines be so set that they make a right angle at the point A. And draw the right line BG And let the line D containe in power halfe the line BG by the first proposition added by Flussas after the laste of the sixth Then I say that the line D is the side of an Octohedron contayned in the same sphere Forasmuche as the line AG maketh the greater segment GC the side of the cube contained in the same sphere by the same 4. corollary of the 17. of the thirtenth and the squares of the whole line AG. and of the lesse segment AB are triple to the square of the greater segment GC by the 4. of the thirtenth Moreouer the diameter of the sphere is in power triple to the same line GC the side of the cube by the 15. of the thirtenth Wherfore the line BG is equal to the ãâã For it conââineth in power the two lines AB and AG by the 47. of the first and therefore it containeth in power the triple of the line GC But the side of the Octohedron contained in the same sphere is in power triple to halfe the diameter of the sphere by the 14. of the thirtenth And by suppoââtion the line D contaiââââ in powââ the halfe of the line BG Wherefore the line D containing in power the halfe of the same diameter is the side of an octohedron If therfore the side of a Dodecahâdron and the right line of whome the said side is the lesse segment be so set that they make a right angle the right line which containeth in power halfe the line subtending the angle is the side of an Ocâââedron contained in the selfe same sphere Which was required to be proued A Corollary Vnto what right line the side of the Octoâedron is in power sesquialter vnto the same line the side of the Dodecahedron inscribed in the same sphere is the greater segment For the side of the Dodecahedron is the greater segment of the segment CG vnto which D the side of the Octohedron is in power sesquiâlter that is is halfe of the power of the line BG which was triple vnto the line CG ¶ The 18. Proposition If the side of a Tetrahedron containe in power two right lines ioyned together by an extreme and meane proportion the side of an Icosahedron described in the selfe same Sphere is in power sesquialter to the lesse right line SVppose that ABC be a Tetrahedron and let his side be AB whose power let be diuided into the lines AG and GB ioyned together by an extreme and meane proportion namely let it be diuided into AG the whole line and GB the greater seâment by the Corollary of the first Proposition added by Flussas after the last of the sixth And let ED be the side of the Icosahedron EDF contained in the selfe same Sphere And let the line which subtendeth the angle of the Pentagon described of the sides of the Icosahedron be EF. Then I say that ED the side of the Icosahedron is in power sesquialter to the lesse line GB Forasmuch as by that which was demonstrated in the 15. of this booke the side ED is the greâter segment of the line EF which subtendeth the angle of the Pentagon But as the whole line EF is to the greater segment ED so is the same grââter segment to the lesse by the 30. of the sixth and by supposition AG was the whole line and Gâ the greater segment Wherefore as EF is to ED so is AG to Gâ by the second of the fouretenth And alternately the line EF is to the line AG as the line ED is to the line GB And forasmuch as by supposition the line AB containeth in power the two lines AG and GB therefore by the 4â of the first the angle AGB is a right angle But the angle DEF is a right angle by that which was demonstrated in the 15. of this booke Wherefore the triangles AGâ and FED are equiangle by the â of the sixth Wherefore their sides are proportionall namely as the line ED is to the line GB so is the line FD to the line AB by the 4. of the sixth But by that which hath before
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 segmeÌt 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 Wheâfore 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 coÌtayneth 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 incoÌmeÌsurable 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 studeÌt 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
equall partes by euery one of the three equall squares which diuide the Octohedron into two equall partes and perpendicularly For the three diameters of those squares do in the centre cut the one the other into two equall partes and perpendicularly by the third Corollary of the 1â of the thirtenth which squares as for example the square EKLI do diuide in sunder the pyramids and the prismes namely the pyramis KLTD and the prisme KLTEIA from the pyramis EKZB and the prisme EKZILG which pyramids are equall the one to the other and so also are the prismes equall the one to the other by the 3. of the twelfth And in like sort do the rest of the squares namely KZIT and ZLTE which squares by the second Corollary of the 14. of the thirtenth do diuide the Octohedron into two equall partes ¶ The 3. Proposition The 3. Probleme In a cube geuen to describe an Octohedron TAke a Cube namely ABCDEFGH And diuide euery one of the sides thereof into two equall partes And drawe right lines coupling together the sections as for example these right lines PQ and RS which shall be equall vnto the side of the cube by the 33. of the first and shall diuide the one the other into two equall parts in the middest of the diameter AG in the point I by the Corollary of the 34. of the first Wherefore the point I is the centre of the base of the cube And by the same reason may be found out the centres of the rest of the bases which let be the pointes K L O N M. And drawe these right lines LI IM MO OL KI KL KM KO NI NL NM NO And now forasmuch as the angle IPL is a right angle by the 10. of the eleuenth for the lines IP and PL are parallels to the lines RA and AB And the right line IL subtendeth the right angle IPL namely it subtendeth the halfe sides of the cube which containe the right angle IPL and likewise the right line IM subtendeth the angle IQM which is equall to the same angle IPL and is contained vnder right lines equall to the right lines which containe the angle IPL Wherefore the right line IM is equall to the right line IL by the 4. of the first And by the same reason may we proue that euery one of the right lines MO OL KI KL KM KO NI NL NM and NO which subtend angles equall to the selfe same angle IPL and are coÌtained vnder sides equall to the sides which containe the angle IPL are equall to the right line IL. Wherefore the triangles KLI KLO KMI KMO and NLI NLO NMI NMO are equilater and equall and they containe the solide IKLONM Wherefore IKLONM is an Octohedron by the 23. definition of the eleuenth And forasmuch as the angles thereof do altogether in the pointes I K L O N M touch the bases of the cube which containeth it it followeth that the Octohedron is inscribed in the cube by the first definition of this booke Wherefore in the cube geuen is described an Octohedron which was required to be done ¶ A Coâollary aâded by âlussââ Hereby it is manifest that right lines ioyning together the âentres of the opposite bases of the cube do cut the one the other into two equall parts and perpendicularly in the centre of the cube or in the centre of the Sphere which containeth the cube For forasmâch as the right lines LM and IO which knââ together the centres of the opposite bases of the cube do also knit together the opposite anglâs of the Octâhedron inscribed in the cube it followeth by the 3. Corollary of the 14. of the thirtenth that those lines LM and IO do cut the one the other into two equall partes in a point But the diameters of the cube do also cut the one the other into two equall partes by the 39. of the eleuenth Wherfore that point shall be the centre of the sphere which containeth the câââ For making that point the centre and the space some one of the semidiameters describe a sphere and it shall passe by the angles of the cube and likewise making the same point the centre and the space halfe of the line LM describe a sphere and it shall also passe by the angles of the Octohedron ¶ The 4. Proposition The 4. Probleme In an Octohedron geuen to describe a Cube SVppose that the Octohedron geuen be ABGDEZ And let the two pyramids thereof be ABGDE and ZBGDE And take the centres of the triangles of the pyramis ABGDE that is take the centres of the circles which containe those triangles and let those centres be the pointâs T I K L. And by these centres let there be drawen parallâl lines âo the sides of the square BGDE which parallel âigââ linââ let be MTN NLX XKO OIM. And forasmuch as thâse parallel right lines do by the 2. of the sixth cut the equall right lines AB AG AD and AE proportionally therfore they concurre in the pointes M N X O. Wherefore the right lines MN NX XO and OM which subtend equall angles set at the point A contained vndâr âquall right lines are equall by the 4. of the first And moreouer seing that they are parallels vnto the lines BG GD DE Eâ which make a square therefore MNXO is also a square by the 10. of the eleuenth Wherefore also by the 15. of the âame the square MNXO is parallel to the squarâ BGDE For all tâe right lines touch the one the other in the pointes of their sections From the centres T I K L drawe these right lines TI IK KL LTâ And drawe the right line AIC And forasmuch as I is the centre of the equilater triangle ABE therefore the right line AI being extended cutteth the right line BE into two equall partes by the Corollary of the 12. of the thirtenth And forasmuch as MO is a parallel to BE therefore the triangle AIO is like to the whole triangle ACE by the Corollary of the 2. of the sixth And the right line MO is diuided into two equall partes in the point I by the 4. of the sixth And by the same reason may we proue that the right lines MN NX XO are diuided into two equall partes in the pointes T L K. Wherefore also againe the bases TI IK KL LT which subtend the angles set at the pointes M O X N which angles are right angles and are contained vnder equall sides those bases I say are equall And forasmuch as TIM is an Isosceles triangle therefore the angles set at the base namely the angles MTI and MIT are equal by the ââ of the first But the angle M is a right angle wherefore eche of the angles MIT and MTI is the halfe of a right angle And by the same reason the angles OIK OKI are equall Wherefore the angle remayning namely TIK is a right angle
wherfore the line Ce is to the line eg as the greater segment to the lesse and therefore their proportion is as the whole line IC is to the greater segment Ce and as the greater segment Ce is to the lesse segment eg wherefore the whole line Ceg which maketh the greater segment and the lesse is equall to the whole line IC or IE And forasmuch as two parallel plaine superficieces namely that which is extended by IOB and that which is extended by the line ag are cutte by the playne of the triangle BCE which passeth by the lines ag and IB their common sections ag and IB shall be parallels by the 16. of the eleuenth But the angle BIE or BIC is a right angle wherefore the angle agC is also a right angle by the 29. of the first and those right angles are contayned vnder equall sides namely the line gC is equall to the line CI and the line ag to the line BI by the 33. of the first wherfore the bases Ca and CB are equall by the 4. of the first But of the line CB the line CE was proued to be the greater segment wherefore the same line CE is also the greater segment of the line Ca but cn was also the greater segment of the same line Ca. Wherefore vnto the line CE the line cn which is the side of the dodecahedron and is set at the diameter is equall And by the same reason the rest of the sides which are set at the diameter may be proued eâuall to lines equall to the line CE. Wherfore the pentagon inscribed in the circle where in is contained the triangle BCE is by the 11. of the fourth equiangle and equilater And forasmch as two pentagons set vpon euery one of the bases of the cube doo make a dodecahedron and sixe bases of the cube doo receaue twelue angles of the dodecahedron and the 8. semidiameters doo in the pointes where they are cutte by an extreame and meane proportion receaue the rest therefore the 12. pentagon bases contayning 20. solide angles doo inscribe the dodecahedron in the cube by the 1. diffinition of this booke Wherefore in a cube geuen is inscribed a dodecahedron which was required to be done First Corollary The diameter of the sphere which containeth the dodecahedron containeth in power these two sides namely the side of the Dodecahedron and the side of the cube wherein the Dodecahedron is inscribed For in the first figure a line drawne from the centre O to the poynt B the angle of the Dodecahedron namely the line OB containeth in power these two lines OV the halfe side of the cube and VB the halfe side of the dodecahedron by the 47. of the first Wherefore by the 15. of the fiueth the double of the line OB which is the diameter of the sphere containing the Dodecahedron containeth in power the double of the other lines OV and VB which are the sides of the cube and of the dodecahedron ¶ Second Corollary The side of a cube diuided by an extreme and meane proportion maketh the lesse segment the side of the dodecahedron inscribed in it and the greater segment the side of the cube inscribed in the same Dodecahedron For it was before proued that the side of the dodecahedron is the greater segment of BE the side of the triangle BECâ but the side BE which is equall to the lineâ GB and SF is the greater segmeÌt of GF the side of the cube which line âE subtending thâ angle of the pentagon was by the â of this booke the side of the cube inscribed in the dodecahedron Third Corollary The side of a cube is equal to the sides of a Dodecahedron inscribed in it and circumscribed about it For it was manifest by this proposition that the side of a cube maketh the lesse segment the side of a Dodecahedron inscribed in it namely as in the first figure the line BS the side of the Dodecahedron inscribed is the lesse segmeÌt of the line GF the side of the cube And it was proued in the 17. of the thirtenth that the same side of the cube subteÌdeth the angle of the pentagon of the Dodecahedron circumscribed and therefore it maketh the greater segment the side of the Dodecahedron or of the pentagon by the first corollary of the same Wherefore it is equal to bothe those segments The 14. Probleme The 14. Proposition In a cube geuen to inscribe an Icosahedron SVppose that the cube geuen be ABC the Centres of whose bases let be the points D E G H I K by whiche poyntes draw in the bases vnto the other sids parallels not touching the one the other And deuide the lines drawn from the centres as the line DT c. by an extreme and meane proportion in the poyntes A F L M N B P Q R S C O by the 30. of the sixth and let the greater segmentes be about the ceÌtres And draw these right lines AL AG AM and TG And forasmuch as the lines cut are parallels to the sides of the cube they shall make right angles the one with the other by the 29. of the first and forasmuche as they are equal their sections shall be equal for that the sections are like by the 2. of the fourtenth Wherfore the line TG is equal to the line DT for they are eche halfe sides of the cube Wherfore the square of the whole line TG and of the lesse segment TA is triple to the square of the line AD the greater segment by the 4. of the thirteÌth But the line AG containeth in power the lines AT TG for the angle ATG is a right angle Wherefore the square of the line AG is triple to the square of the line AD. And forasmuch as the line MGL is erected perpendicularly to the plain passing by the lines AT which is parallel to the bases of the cube by the corollary of the 14. of the eleueÌth therfore the angle AGL is a right angle But the line LG is equal to the line AD for they are the greater segments of equal lines Wherfore the line AG which is in power triple to the line AD is in power triple to the line LG Wherefore adding vnto the same square of the line AG the square of the line LG the square of the line AL which by the 47. of the first containeth in power the two lines AG and GL shal be quadruple to the line AD or LG Wherefore the line AL is double to the line AD by the 20. of the sixth and therfore is equal to the line AF or to the line LM And by the same reason may we proue that euery one of the other lines which couple the next sections of the lines cut as the lines AM PF PM MQ and the rest are equal Wherfore the triangles ALM APF AMP PMQ and the rest such like are equal equiangle and equilater
by the 4. and eigth of the first And forasmuch as vpon euery one of the lines cut of the cube are set two triangles as the triangles ALM and BLMâ there shal be made 12. triaÌgles And forasmuch as vnder euery one of the â angles of the cube are subtended the other 8. triangles as the triangle AMP. c. of 1â and 8. triangles shall be produced 20. triangles equal and equilater coÌtaining the solidâ of an Icosahedron by the 25. diffinition of the eleuenth which shal be inscribed in the cube geuen ABC by the first diffinition of this booke The inuention of the demonstration of this dependeth of the ground of the former Wherfore in a cube geuen we haue described an Icosahedron which was required to be done First Corollary The diameter of a sphere which containeth an Icosahedron containeth two sides namely the side of the Icosahedron and the side of the cube which containeth the Icosahedron For if we drawe the line AB it shall make the angles at the poynt A right angles for that it is a parallel to the sides of the cube wherfore the linâ which coupleth the opposite angleâ of the Icosahedron at the poynts F and B coÌtaineth in power the line AB the sidâ of thâ cube and the line AF the side of the Icosahedron by the 47. of the first Which line FB is equâl to the âiameter of the sphere which contâineth the Icosahedron by the demonstration of the ââ of the thirteÌth Second Corollary The six opposite sides of the Icosahedron deuided into two equal parts their sections are coupled by three equal right lines cutting the one the other into two equal partes and perpendicularly in the centre of the sphere which containeth the Icosahedron For those three lines are the three lines which couple the centres of the bases of the cube which do in suche sort in the centre of the cube cut the one the other by the corollary of the third of this booke and therfore are equal to the sides of the cube But right lines drawne from the ceÌtre of the cube to the angles of the Icosahedron euery one of them shall subtend the halfe side of the cube and the halfe side of the Icosahedron which halfe sides containe a right angle wherefore those lines are equal Wherby it is manifest that the foresaid centre is the centre of the sphere which containeth the Icosahedron Third Corollary The side of a cube deuided by an extreme and meane proportion maketh the greater segment the side of an Icosahedron described in it For the half side of the cube maketh the halfe of the side of the Icosahâdron the greater segment wherefore also the whole side of the cube maketh the whole side of the Icosahedron the greater segment by the 15. of the fifthe for the sections are like by the â of the fourtenth ¶ Fourth Corollary The sides and bases of the Icosahedron which are opposite the one to the other are parallels Forasmuch as euery one of the opposite sides of the Icosahedron may be in the parallel lines of the cube namely in those parallels which are opposite in the cube and the triangles which are made of parallel lines are parallels by the 15. of the eleuenth therfore the opposite triângleâ of the Icosahedron as also the sides are pârallels the one to the other ¶ The 15. Probleme The 15. Proposition In an Icosahedron geuen to inscribe an Octohedron SVppose that the Icosahedron geuen be ACDF and by the former second Corollary let there be takeÌ the three right lines which cut the one the other into two equall partes perpendicularly and which couple the sections into two equall partes of the sides of the Icosahedron which let be BE GH and KL cutting the one the other in the point I. And drawâ these riâht lines âG GE EH and HB And forasmuch as the angles at the point I are by construction right angles ând are conââined under equâll linesâ the ãâã Gâ and ââ shall ãâ¦ã squâre by the â of the âirst Likewyse ânto thoâe ãâã shall be âquall the lines drâwân from ãâã pointes K and â to euery one of the poinââs â G. â H And therefore the triangles which ãâã the ââramis DGENK shall be equall ãâ¦ã And by ãâ¦ã ¶ The 16. Probleme The 16. Proposition In an Octohedron geuen to inscribe an Icosahedron LEt there be taken an Octohedron whose 6. angles let be A B C F P L. And draw the lines AC BF PL cutting the one the other perpendicularly in the point R by the 2. Corollary of the 14. of the thirtenth And let euery one of the 12. sides of the Octohedron be diuided by an extreme and meane proportion in the pointes H X M K D S N G V E Q T. And let the greater segmentes be the lines BH BX FM FK AD AQ CS CT PN PG LV LE And drawe these lines HK XM GE NV DS QT Now forasmuch as in the triangle ABF the sides are cut proportionally namely as the line BH is to the line HA so is the line FK to the line KA by the 2. of the fouretenth therefore the line HK shall be a parallel to the line BF by the 2. of the sixth And forasmuch as the line AC cutteth the line HK in the point Z and the line ZK is a parallel vnto the line RF the line RA shall be cut by an extreme and meane proportion in the point Z by the 2. of the sixth namely shall be cut like vnto the line FA and the greater segmeÌt therof shall be the line ZR Vnto the line ZK put the line RO equall by the 3. of the first and drawe the line KO now then the line KO shall be equall to the line ZR by the 33. of the âirst Draw the lines KG KE and KI And forasmuch as the triangles ARF and AZK are equiangle by the 6. of the sixth the sides AZ and ZK shall be equall the one to the other by the 4. of the sixth for the sides AR and RF are equall Wherfore the line ZK shall be the lesse segment of the line RA. But if the greater segment RZ be diuided by an extreme meane proportion the greater segment therof shall be the line ZK which was the lesse segment of the whole line RA by the 5. of the thirâenth And forasmuch as the two lines FE and FG are equall to the two lines AH and AK namely ech are lesse segmentes of equall sides of the Octohedron and the angles HAK and EFG are equall namely are right angles by the 14. of the thirtenth the bases HK and GF shall be equall by the 4. of the first And by the same reason vnto them may be proued equall the lines XM NV DS and QT And forasmuch as the lines AC BF and PL do cut the one the other into two equall parts and perpendicularly by construction the lines HK and GE which
subtend angles of triangles like vnto the triangles whose angles the lines AC BF and PL subtend are cut into two equall partes in the pointes Z and I by the 4. of the sixth so also are the other lines NV XM DS QT which are equall vnto the lines HK GE cut in like sort and they shall cut the lines AC BF and PL like Wherefore the line KO which is equall to RZ shall make the greater segment the line RO which is equall to the line ZK for the greater segment of the RZ was the line ZK and therefore the line OI shall be the lesse segment when as the whole line RI is equall to the whole line RZ Wherefore the squares of the whole line KO and of the lesse segment OI are triple to the square of the greater segment RO by the 4. of the thirtenth Wherfore the line KI which containeth in power the two lines KO and OI is in power triple to the line RO by the 47. of the first for the angle KOI is a right angle And forasmuch as the lines FE and FG which are the lesse segmentes of the sides of the Octohedron are equall and the line FK is coÌmon to them both and the angles KFG and KFE of the triangles of the Octohedron are equall the bases KG and KE shall by the 4 of the first be equall and therefore the angles KIE and KIG which they subtend are equall by the 8. of the first Wherefore they are right angles by the 13. of the first Wherefore the right line KE which containeth in power the two lines KI and âE by the 47. of the first is in power quadruple to the line RO or IE for the line RI is proued to be in power triple to the same line RO But the line GE is double to the line IE Wherfore the line GE is also in power ãâ¦ã PF And by the same reason may be proued that the âest of the eleuen solide angles of the ãâã are ãâ¦ã the sections of euery one of the sides of the Octohedron namely in the pointes E N V H â M â D S Q T. Wherefore there are 12. angles of the Icosahedron Moreouer forasmuch as euery one of the bases of the Octohedron do eche containe triangles of the Icosahedron ãâ¦ã pyramiââ ABCâFP which is the halfe of the Octohedron the triangle FCP receaueth in thâ section of his sides the â triangle GMS and the triangle CPB containeth the triangle NXS and thâ triangle âAP contayneth the triangle HND and moreouer the triangle APF containeth the triangle âDG and the same may be proued in the opposite pyramis ABCFL Wherefore there shall be eight triangleâ And forasmuch as besides these triangles to euery one of the solide angles of the Octohedron ãâã subtended two triangles as the triangles KEG amd MEG to the angle F and the triangles HNV and XNV to the angle B also the triangles NDS and âDS to the angle P likewise the triangleâ DHK and QHK to the angle A Moreouer the triangles EQT and VQT to the angle L and finally the triangles SXM and TXM to the angle C these 12. triangles being added to thââ for ãâã triangles shall produce â0 triangles equall and equilâter coupled together which shall male an Icosahedron by the 25. definition of the eleuenth and it shall be inscribed in the Octohedron geuen ABCââL by the first definition of this booke for the 1â angles thereof are set in 1â like sections of the sides of the Octohedron Wherefore in an Octohedron geuen is inscribed an Icosahedron ¶ First Corollary The side of an equilater triangle being diuided by an extreme and meane proportion a right line subtending within the triangle the angle which is contained vnder the greater segment and the lesse is in power duple to the lesse segment of the same side For the line KE which subtendeth the angle KFE of the triangle AFL which angle KFE is contained vnder the two segmentes KF FE was proued equall ãâã the line HK which containeth in power the two lesse segmentes HA and AK by the 47. of the âârst foâ ãâã angle HAK is ãâ¦ã Second Corollary The bases of the Icosahedron are concentricall that is haue one and the selfe same centre with the bases of the Octohedron which contayneth it For suppose that ãâ¦ã Octohedron ãâã ECD the base of an Icosahedron and let the centre of the base ABG be the point F. And drawe these right lines FA FB FC and FE Now then the ãâ¦ã to the two lines FB and BC for they are lines drawen from the centre and are also lesse segmentes and they contayne the ãâ¦ã ¶ The 17. Probleme The 17. Proposition In an Octohedron geuen to inscribe a Dodecahedron SVppose that the Octohedron geuen be ABGDEC whose 12. âides let be cut by an extreme and meane proportion as in the former Proposition It was manifest that of the right lines which couple thâse sections are made 20. triangles of which 8. are concentricall with the bases of the Octohedron by the second Corollary of the former Proposition If therefore in euery one of the centres of the 20. triangles be inscribed by the 1. of this booke euery one of the ââ ââgles of the Dodecahedron we shall finde that â angles of the Dodecahedron are set in the 8. centres of the bases of the Octohedron namely these angles I u ct O M a P and X and of the other 12. solide angles there are two in the centres of the two triangles which haue one side common vnder euery one of the solide angles of the Octohedron namely vnder the solide angle A the two solide angles K Z vnder the solide angle B the two solide angles H T vnder the solide angle G the two solide angles Y V vnder the solide angle D the two solide angles F L vnder the solide angle E the two solide angles S N vnder the solide angle C the two solide angles Q R and forasmuch as in the Octohedron are sixe solide angles vnder them shall be subtended 12. solide angles of the Dodâcahedron and so are mâde 20. solide angles composed of 12. equall and âquilâter superficiall pentagons as it was ãâã by the 5. of this booke which therefore containe a Dodecahedron by the 24. definition of the eleuenth And it is inscribed in the Octohedron by the 1. definition of this booke for that euery one of the bases of the Octohedron do receaue angles therof Wherefore in an Octohedron geuen is inscribed a Dodecahedron ¶ The 18. Probleme The 18. Proposition In a trilater and equilater Pyramis to inscribe a Cube SVppose that there be a trilater equilater Pyramis whose base let be ABC and âoppe the point D. And let it be comprehended in a Sphereâ by the 13. of the ãâã And lââ the centre of that Sphere be the point E. And from the solide angles A B C D draw right lines passing by the centre E vnto the opposite bases of
Cube FOr forasmuch as the side of the pyramis inscribed in the cube subteÌdeth two sides of the cube which containe a right angle by the 1. of the fiuetenth it is manifest by the 47. of the first that the side of the pyramis subteÌding the said sides is in power duple to the side of the cube Wherefore also the square of the side of the cube is the halfe of the square of the side of the pyramis The side therefore of a cube containeth in power halfe the side of an equilater triangular pyramis inscribed in the said cube ¶ The 7. Proposition The side of a Pyramis is duple to the side of an Octohedron inscribed in it FOrasmuch as by the 2. of the fiuetenth it was proued that the side of the Octohedron inscribed in a pyramis coupleth the midle sections of the sides of the pyramis Wherefore the sides of the pyramis and of the Octohedron are parallels by the Corollary of the 39. of the first and therefore by the Corollary of the 2. of the sixth they subtend like triangles Wherfore by the 4. of the sixth the side of the pyramis is double to the side of the Octohedron namely in the proportion of the sides The side therefore of a pyramis is duple to the side of an Octohedron inscribed in it ¶ The 8. Proposition The side of a Cube is in power duple to the side of an Octohedron inscribed in it IT was proued in the 3. of the fiuetenth that the diameter of the Octohedron inscribed in the cube coupleth the centres of the opposite bases of the cube Wherefore the said diameter is equall to the side of the cube But the same is also the diameter of the square made of the sides of the Octohedron namely is the diameter of the Sphere which containeth it by the 14. of the thirtenth Wherefore that diameter being equall to the side of the cube is in power double to the side of that square or to the side of the Octohedron inscribed in it by the 47. of the first The side therefore of a Cube is in power duple to the side of an Octohedron inscribed in it which was required to be proued ¶ The 9. Proposition The side of a Dodecahedron is the greater segment of the line which containeth in power halfe the side of the Pyramis inscribed in the sayd Dodecahedron SVppose that of the Dodecahedron ABGD the side be AB and let the base of the cube inscribed in the Dodecahedron be ECFH by the ââ of the fiuetenth And let the side of the pyramis inscribed in the cube be CH by the 1. of the fiuetenth Wherefore the same pyramis is inscribed in the Dodecahedron by the 10. of the fiuetenth Then I say that AB the side of the Dodecahedron is the greater segment of the line which containeth in power halfe the line CH which is the side of the pyramis inscribed in the Dodecahedron For forasmuch as EC the side of the cube being diuided by an extreme and meane proportion maketh the greater segment the line AB the side of the Dodecahedron by the âârst Corollary of the 17. of the thirtenth For they are contâined in one and the selfe same Sphere by the first of this booke and the line EC the side of the cube contayneth in power the halfe of the side CH by the 6. of this booke Wherefore AB the side of the Dodecahedron is the greater segment of the line EC which containeth in power the halfe of the line CH which is the side of the Dodecahedron inscribed in the pyramis The side therefore of a Dodecahedron is the greater segment of the line which containeth in power halfe the side of the Pyramis inscribed in the said Dodecahedron ¶ The 10. Proposition The side of an Icosahedron is the meane proportionall betwene the side of the Cube circumscribed about the Icosahedron and the side of the Dodecahedron inscribed in the same Cube SVppose that there be a cube ABFD in which let there be inscribed an icosahedron CLIGOR by the 14. of the fiuetenth Let also the Dodecahedron inscribed in the same be EDMNPS by the 13. of the same Now forasmuch as CL the side of the Icosahedron is the greater segmeÌt of AB the side of the cube circumscribed about it by the 3. Corollary of the 14. of the fiuetenth and the side ED of the DodecahedroÌ inscribed in thesame cube is the lesse segmeÌt of the same side AB of the cube by the 2. Corollary of the 13. of the fiuetenth it followeth that AB the side of the cube being diuided by an extreme and meane proportion maketh the greater segment CL the side of the Icosahedron inscribed in it and the lesse segment ED the side of the Dodecahedron likewise inscribâd in it Wherefore as the whole line AB the side of the cube is to the greater segment CL the side of the Icosahedron so is the greater segment CL the side of the Icosahedron to the lesse segment EDâ the side of the Dodecahedron by the third definition of the sixth Wherefore the side of an Icosahedron is the meane proportionall betwene the side of the cube circumscribed about the Icosahedron and the side of the Dodecahedron inscribed in the same cube ¶ The 11. Proposition The side of a Pyramis is in power Octodecuple to the side of the cube inscribed in it FOr by that which was demonstrated in the 18. of the fiuetenth the side of the pyramis is triple to the diameter of the base of the cube inscribed in it and therefore it is in power nonecuple to the same diameter by the 20. of the sixth But the diamer is in power double to the side of the cube by the 47. of the first And the double of nonecuple maketh Octodecuple Wherefore the side of the pyramis is in power Octodecuple to the side of the cube inscribed in it ¶ The 12. Proposition The side of a Pyramis is in power Octodecuple to that right line whose greater segment is the side of the Dodecahedron inscribed in the Pyramis FOrasmuch as the Dodecahedron and the cube inscribed in it are set in one and the sâlfâ same pyramis by the Corollary of the first of this booke and the side of the pyramis circumscribed about the cube is in power octodecuple to the side of the cube inscribed by the former Proposition but the greater segment of the selfe same side of the cube is the side of the Dodecahedron which containeth the cube by the Corollary of the 17. of the thirtenth Wherfore the side of the pyramis is in power octodecuple to that right line namely to the side of the cube whose greater segment is the side of the Dodecahedron inscribed in the pyramis ¶ The 13. Proposition The side of an Icosahedron inscribed in an Octohedron is in power duple to the lesse segment of the side of the same
Construction Two cases in this Proposition First caseâ Second case Demonstration Construction Demonstration Construction Demonstration An other way after Peliâarius Construction Demonstration Construction Demonstration Demonstration leading to an impossibilitie Three cases in this Propositiân 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 Demonstrâtion Demonstration leading to an absurditie A Corollary Construction Demonstration Demonstration An âther way to do the same after OroÌtius 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 aâthor of this booke Eudoxus The first definition A part taken two maner of wayes The fiâst 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 multiplicatiâa Pars aliquota This kinde of part coÌmonly vsed in Arithmetique The other kinde of part Pars constitâens or componens Pars aliquanta The second definition Numbers very necessary for the vnderstanding of this booke and the other bookes following The tâird definition Rational proportion deuided ââto 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 superquadâipartiens 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 âefinition 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 fâurth Example thereof The seuenth definition 9 12 3 4 Proportionality of two sortes contiâuall 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 propârtionalitie Example of discontinual proportionality in numbers In discoÌtinual 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 ân 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 twelfâh definition Example of this deâinition 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 fiâtâne definition This is the coÌuerse 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 seueÌtenth 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 defiâition 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 differâcâ of the first part Demonstratiâ of tâe same first differeÌce Second diffeâence Third diââerence The second part âf 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 Constrâction Demonstrationâ Construction Demonstration An addition of Campane Demonstration Construction Demonstration DemoÌstration of alternate proportion Construction Demonstration DemonstratioÌ of proportion by diuision Constrâctionâ Demonsâration Demonstration of proportion by composition This proposition is the conuerse of the former DemonstratioÌâeâaing to an âmpossibility 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 Thâr 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 AâCDEâGH 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 deâinition 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 Demonstâation 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 Demonstratiân 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 propoâition may a right line be deuided into what partes soeuer you will. Construction Demonstration An other way after Pelitarius An otâeâ way after Pelitarius Construction Demonstration An other way after Campane Construction Demonstratioâ A proposition added by Pelitarius The
the square of the line B wherefore the parallelograme EC is coÌmensurable vnto the parallelograme CF. But as the parallelograme EC is to the parallelograme CF so is the line ED to the line DF by the first of the sixt Wherefore by the 10. of the tenth the line ED is commensurable in length vnto the line DF. But the line ED is rationall and incoÌmensurable in length vnto the line DC wherefore the line DF is rationall and incommensurable in length vnto the line DC by the 13. of the tenth Wherefore the lines CD and DF are rationall commensurable in power onely But a rectangle figure comprehended vnder rationall right lines commensurable in power onely is by the â1 of the tenth irrationall and the line that containeth it in power is irrationall and is called a mediall line Wherefore the line that containeth in power that which is comprehended vnder the lines CD and DF is a mediall line But the line B containeth in power the parallelograme which is comprehended vnder the lines CD and DF Wherefore the line B is a mediall line A right line therfore commensurable to a mediall line is also a mediall line which was required to be proued ¶ Corollary Hereby it is manifest that a superficies commensurable vnto a mediall superficies is also a mediall superficies For the lines which containâ in power those superficieces are commensurable in power of which the one is a mediall line by the definitioÌ of a mediall line in the 21. of this tenth wherefore the other also is a mediall line by this 23. propositioÌ And as it was sayd of rationall lines so also is it to be sayd oâ mediall lines namely that a liâe commensurable to a mediall line is also a mediall line a line I say which is commensurable vnto a mediall line whether it be commensurable in length and also in power or ells in power onely For vniuersally it is true that lines commensurable in length are also commensurable in power Now if vnto a mediall line there be a line commensurable in power if it be commensurable in length theÌ are those lines called mediall lines commensurable in length in power But if they be commensurable in power onely thây are called mediall lines commensurable in power onely There are also other right lines incommensurable in length to the mediall line and commensurable in power onely to the same and these lines are also called mediall for that they are commensuâable in power to the mediall line And in aâ muâh as they are mediall lines they are commensurable in power the one to the other But being compared the one to the other they may be commensurable either in length and theâefoâe in power or ells in power onely And then if they be commensurable in length they are called also mediall lines commensuâable in length and so consequently they are vnderstanded to be commensurable in power But iâ they be commensurable in power onely yet notwithstanding they also are called mediall lines commensurable in power onely Flussates after this proposition teacheth how to come to the vnderstanding of mediall superficieces and lines by surd numbers after this maner Namely to expresse the mediall superficieces by the rootes of numbers which are not square numbers and the lines coÌtaining in power such medial superficieces by the rootes of rootes of numbers not square Mediall lines also commensurable are expressed by the rootes of rootes of like sâperficial numbers but yet not square but such as haue that proportion that the squares of square numbers haue For the rootes of those numbers and the rootes of rootes are in proportion as numbers are namely if the squares be proportionall the sides also shal be proportionall by the 22. of the sixt But mediall lines incommensurable in power are the rootes of rootes of numbers which haue not that proportion that square numbers haue For their rootes are the powers of mediall lines which are incommensurable by the 9. of the tenth But mediall lines commensurable in power onely are the rootes of rootes of numbers which haue that proportion that simple square numbers haue and not which the squares of squares haue For the rootes which are the powers of the mediall lines are commeÌsurable but the rootes of rootes which expresse the sayd mediall lines are incommensurable Wherefore there may be found out infinite mediall lines incommensurable in powâr by comparing infinite vnlike playne numbers the one to the other For vnlike playne numbers which haue not the proportion of square numbers doo make the rootes which expresse the superficieces of mediall lines incoÌmensurable by the 9. of the tenth And therefore the mediall lines containing in power those superficieces are incoÌmensurable in length For lines incommensurable in power are alwayes incommensurable in length by the corrollary of the 9. of the tenth ¶ The 21. Theoreme The 24. Proposition A rectangle parallelogramme comprehended vnder mediall lines coÌmensurable in length is a mediall rectangle parallelogramme SVppose that the rectangle parallelogramme AG be comprehended vnder these mediall right lines AB and BC which let be commensurable in length Then I say that AC is a mediall rectangle parallelogramme Describe by the 46. of the first vpon the line AB a square AD. Wherefore the square AD is a mediall superficies And âorasmuch as the line AB is commensurablâ in length vnto the line BC and the line AB is equall vnto the line BD therefore the line BD is commensurable in length vnto the line BC. But ãâã the line DB is to the line BC so is the square DA to the parallelogramme AC by the first of the sixt Wherfore by the 10. of the tenth the square DA is commensurable vnto the parallelogramme AC But the square DA is mediall for that it is described vpon a mediall line Wherefore AC also is a mediall parallelograÌme by the former Corollary A rectangleâ c which was required to be proued ¶ The 22â Theoreme The 25. Proposition A rectangle parallelogramme comprehended vnder mediall right lines commensurable in power onely is either rationall or mediall And now if the line HK be commensurable in length vnto the line HM that is vnto the line FG which is equall to the line HM then by the 19. of the tenth the parallelogramme NH is rationall But if it be incommensurable in length vnto the line FG then the lines HK and HM are rationall commensurable in power onely And so shall the parallelograÌme HN be mediall Wherefore the parallelogramme HN is either rationall or mediall But the parallelogramme HN is equall to the parallelogramme AG. Wherefore the parallelogramme AC is either rationall or mediall A rectangle parallelogramme therefore comprehended vnder mediall right lines commensurable in power onely is either rationall or mediall which was required to be demonstrated How to finde mediall lines commensurable in power onely contayning a rationall parallelogramme and also other mediall lines commensurable in power contayning a mediall
this booke Wherefore a line commensurable to a line contayning in power two medialls c. An Annotation If other to hath bene spoken of sixe Senarys of which the first Senary contayneth the prâduction of irrationall lines by composition the second the diuision of them namely that those lines are in one poinâ onely deuideâ the third the finding out of binomiall lines of the first I say the second the third the fourth the fift and the sixt after that beginneth the âourth Senary containing the difference of irrationall lines betwene them selues For by the nature of euery one of the binomiall lines are demonstrated the differences of irrational lines The fiueth entreâteth of the applications of the squares of euery irrational line namely what irrationall lines are the breadthes of euery superficies so applied In the sixt Senary is proued that any line commensurable to any irrationall line is also an irrationall line of the same nature And now shall be spoken of the seuenth Senary wherein againe are plainly set forth the rest of the differences of the said lines betwene them selues And theâe is euen in those irrationall lines an arithmeticall proportionalitie And that line which is the arithmeticall meane proportionall betwene the partes of any irrationall line is also an irrationall line of the selfe same kinde First it is certaine that there is an arithmeticall proportion betwene those partes For suppose that the line AB be any of the foresaid irrationall lines as for example let it be a binomiall line let it be deuided into his names in the point C. And let AC be the greater name from which take away the line AD equall to the lesse name namely to CB. And deuide the line CD into two equall partes in the point E. It is manifest that the line AE is equall to the line EB Let the line FG be equall to either of them It is plaine that how much the line AC differeth froÌ the line FG so much the same line FG diââereth from the line CB for in eche is the difference of the line DE or EC which is the propertie of arithmeticall proportionalitie And it is manifest that the line FG is commensurable in length to the line AB for it is the halfe thereof Wherefore by the 66. of the tenth the line FG is a binomiall line And after the selfe same maner may it be proued touching the rest of the irrationall lines ¶ The 53. Theoreme The 71. Proposition If two superficieces namely a rationall and a mediall superficies be coÌposed together the line which contayneth in power the whole superficies is one of these foure irrationall lines either a binomial line or a first bimediall lyne or a greater lyne or a lyne contayning in power a rationall and a mediall superficies But now let the lyne EH be in power more then the line HK by the square of a line incommensurable in length to the line EH now the greater name that is EH is commensurable in length to the rationall line geuen EF. Wherfore the line EK is afourth binomiall line And the line EF is rationall But if a superficies be contained vnder a rationall line and afourth binomiall line the line that containeth in power the same superficies is by the 57. of the tenth irrational and is a greater line Wherfore the line which containeth in power the parallelogramme EI is a greater line Wherefore also the line containing in power the superficies AD is a greater lyne But now suppose that the superficies AB which is rationall be lesse then the superficies CD which is mediall Wherfore also the parallelogramme EG is lesse then the parallelograÌme HI Wherfore also the line EH is lesse then the line HK Now the line HK is in power more then the lyne EH either by the square of a line coÌmensurable in length to the line HK or by the square of a lyne incommensurable in length vnto the lyne HK First let it be in power more by the square of a line commensurable in length vnto HK now the lesse name that is EH is commensurable in length to the rationall line geuen EF as it was before proued Wherfore the whole line EK is a second binomiall line And the line EF is a rationall line But if a superficies be contained vnder a rationall line and a second binomiall lyne the lyne that contayneth in power the same superficies is by the 55. of the tenth a first bimediall line Wherfore the line which contayneth in power the parallelograme EI is a first bimediall line Wherfore also the line that containeth in power the superficies AD is a first bimediall lyne But now let the line HK be in power more then the line EH by the square of a line incoÌmensurable in length to the lyne HK now the lesse name that is EH is coÌmensurable in length to the rationall lyne geuen EF. Wherfore the whole line EK is a fift binomiall lyne And the lyne EF is rationall But if a superficies be contayned vnder a rationall lyne and a fift binomiall lyne the line that contayneth in power the same superficies is by the 58. of the tenth a line containing in power a rationall and a mediall Wherefore the lyne that contayneth in power the parallelogramme EI is a line contayning in power a rationall and a mediall Wherfore also the lyne that containeth in power the superficies AD is a lyne contayning in power a rationall and a mediall If therfore a rationall and a mediall superficies be added together the lyne which contayneth in power the whole superficies is one of these foure irrationall lines namely either a binomiall line or a first bimediall line or a greater lyne or a lyne contayning in power a rationall and a mediall which was required to be demonstrated ¶ The 54. Theoreme The 72. Proposition If two mediall superficieces incommensurable the one to the other be composed together the line contayning in power the whole superficies is one of the two irrationall lines remayning namely either a second bimediall line or a line contayning in power two medialls LEt these two mediall superficieces AB and CD being incommensurable the one to the other be added together Then I say that the line which contayneth in power the superficies AD is either a second bimediall line or a line contayning in power two medialls For the superficies AB is either greater or lesse then the superficies CD for they can by no meanes be equall when as they are incommensurable First let the superficies AB be greater then the superficies CD And take a rationall line EF. And by the 44. of the first vnto the line EF apply the parallelogramme EG equall to the superficies AB and making in breadth the line EH and vnto the same line EF that is to the line HG apply the parallelogramme HI equall to the superficies CD making in breadth the line HK And forasmuch as
the residue or of this excesse But a pyramis is to the same cube inscribed in it nonecuple by the 30. of this booke Wherefore the Dodecahedron inscribed in the pyramis and containing the same cube twise taking away the selfe same third of the lesse segment and moreouer the lesse segment of the lesse segment of halfe the residue shall containe two ninth partes of the solide of the pyramis of which ninth partes eche is equall vnto the cube taking away this selfe same excesse The solide therefore of a Dodecahedron containeth of a Pyramis circumscribed about it two ninth partes taking away a third part of one ninth part of the lesse segment of a line diuided by an extmere and meane proportion and moreouer the lesse segment of the lesse segment of halfe the residue ¶ The 36. Proposition An Octohedron exceedeth an Icosahedron inscribed in it by a parallelipipedon set vpon the square of the side of the Icosahedron and hauing to his altitude the line which is the greater segment of halfe the semidiameter of the Octohedron SVppose that there be an Octohedron ABCFPL in which let there be inscribed an Icosahedron HKEGMXNVDSQTâ by the â6 of the fiuetenth And draw the diameters AZRCBROIF and the perpendicular KO âarallel to the line AZR Then I say that the Octohedron ABCFPL is greater thân the Icosahedron inscribed in it by a parallelipipedon set vpon the square of the side HK or GE and hauing to his altitude the line KO or RZ which is the greater segment of the semidiameter AR. Forasmuch as in the same 16. it hath bene proued that the triangles KDG and KEQ are described in the bases APF and ALF of the Octohedron therefore about the solide angle there remaine vppon the base FEG three triangles KEG KFE and KFG which containe a pyramis KEFG Vnto which pyramis shall be equall and like the opposite pyramis MEFG set vpon the same base FEG by the 8. definition of the eleuenth And by the âame reason shall there at euery solide angle of the Octohedron remayne two pyramids equall and like namely two vpon the base AHK two vpon the base BNV two vpon the base DPS and moreouer two vpon the base QLT. Now theÌ there shal be made twelue pyramids set vpon a base contained of the side of the Icosahedron and vnder two leââe segmentes of the side of the Octohedron containing a right angle as for example the base GEF And forasmuch as the side GE subteÌding a right angle is by the 47. of the âirst in power duple to either of the lines EF and FG and so the ââdeâ KH is in power duple to either of the sides AH and AK and either of the lines AH AK or EF FG is in power duple to eyther of the lines AZ or ZK which coÌtayne a right angle made in the triangle or base AHK by the perpendicular AZ Wherfore it followeth that the side GE or HK is in power quadruple to the triangle EFG or AHK But the pyramis KEFG hauing his base EFG in the plaine FLBP of the Octohedron shall haue to his altitude the perpendicular KO by the 4. definition of the sixth which is the greater segment of the semidiameter of the Octohedron by the 16. of the fiuetenth Wherfore three pyramids set vnder the same altitude and vpon equall bases shall be equall to one prisme set vpon the same base and vnder the same altitude by the 1. Corollary of the 7. of the twelfth Wherefore 4. prismes set vpon the base GEF quadrupled which is equall to the square of the side GE and vnder the altitude KO or RZ the greater segment which is equall to KO shall containe a solide equall to the twelue pyramids which twelue pyramids make the excesse of the Octohedron aboue the Icosahedron inscribed in it An Octohedron therefore excedeth an Icosahedron inscribed in it by a parallelipipedon set vpon the square of the side of the Icosahedron and hauing to his altitude the line which is the greater segment of halfe the semidiameter of the Octohedron ¶ A Corollary A Pyramis exceedeth the double of an Icosahedron inscribed in it by a solide set vpon the square of the side of the Icosahedron inscribed in it and hauing to his altitude that whole line of which the side of the Icosahedron is the greater segmeÌt For it is manifest by the 19. of the fiueteÌth that an octohedroÌ an IcosahedroÌ inscribed in it are inscribed in one the self same pyramis It hath moreouer bene proued in the 26. of this boke that a pyramis is double to an octohedroÌ inscribed in it Wherfore the two excesses of the two octohedrons vnto which the pyramis is equal aboue the two Icosahedrons inscribed in the said two octohedrons being brought into an solide the said solide shal be set vpon the selfe same square of the side of the Icosahedron and shall haue to his altitude the perpendicular KO doubled whose double coupling the opposite sides HK and XM maketh the greater segment the same side of the Icosahedron by the first and second corollary of the 14. of the fiuââenâh The 37. Proposition If in a triangle hauing to his base a rational line set the sides be commensurable in power to the base and from the toppe be drawn to the base a perpendicular line cutting the base The sections of the base shall be commensurable in length to the whole base and the perpendicular shall be commensurable in power to the said whole base And now that the perpendicular AP is commensurable in power to the base BG iâ thus proued Forasmuch as the square of AB is by supposition commensurable to the square of BG and vnto the rational square of AB is commensurable the rational square of BP by the 12. of the eleuenth Wherfore the residue namely the square of PA is commensurable to the same square of BP by the 2. part of the 15. of the eleuenth Wherefore by the 12. of the tenth the square of PA is commensurable to the whole square of BG Wherefore the perpendicular AP is commensurable in power to the base BG by the 3. diffinition of the tenth which was required to be proued In demonstrating of this we made no mention at all of the length of the sides AB and AG but only of the length of the base BG for that the line BG is the rational line first set and the other lines AB and AG are supposed to be commensurable in power only to the line BG Wherefore if that be plainely demonstrated when the sides are commensurable in power only to the base much more easily wil it follow if the same sides be supposed to be commensurable both in length and in power to the base that is if their lengthes be expressed by the rootes of square nombers ¶ A Corollary 1. By the former things demonstrated it is manifest that if from the powers of the base and of one of the sides be taken away the