The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

¶The fourtenth booke of Euclides Elementes. (Book 14)

IN this booke, which is commonly accompted the 14. booke of Euclide is more at large intreated of our prin∣cipal purpose:* 1.1 namely, of the comparison and propor∣tion 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 vndoub∣tedly for the woorthines and hardnes thereof (for thinges of most price are most hardest) were first sear∣ched, and found out of Philosophers, not of the inferi∣or 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 amōgst 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 Eu∣clides, 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 wōderful bookes writtē 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 witnes∣seth 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 dodecahe∣dron 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 the same preface seme to import.

The Preface of Hypsicles before the fourtenth booke.

FRend Protarchus, whē that Basilides of Tire came into Alexandria, hauing familiar frendship with my father by reason of his knowledge in the mathe∣maticall sciences, he remayned with him a long time, yea euen all the time of the pestilence. And sometime reasoning betwene themselues of that which A∣pollonius had written touching the comparison of a dodecahedron and of an Icosahedron inscribed in one and the selfe same sphere, what proportion such bodies haue the one to the other, they iudged that Apollonius had somewhat erred therein. Wherefore they (as my father declared vnto me) diligently weighing it, wrote it perfectly. Howbeit after∣ward I happened to finde an other booke written of Apollonius, which contayned in it the

right demonstration of that which they sought for: which when they saw, they much reioy∣sed. As for that which Apollonius wrote, may be sene of all men, for that it is in ••uery mans hand. And that which was of vs more diligently afterward written agayne, I thought good to send and dedicate vnto you, as to one whome I thought worthy commendation, both for that deepe knowledge which I know you haue in all kindes of learning, and chiefely in Geome∣trie, so that you are able redily to iudge of those thinges which are spoken, and also for the greate loue and good will which you beare towardes my father and me. Wherfore vouchsafe gently to accept this, which I send vnto you. But now is it time to end our preface, and to be∣gin the matter.

¶ The 1. Theoreme. The 1. Proposition. * 1.2A perpendicular line drawen from the centre of a circle to the side of a Pentagon described in the same circle: is the halfe of these two lines, name∣ly, of the side of an hexagon figure, and of the side of a decagon figure be∣ing both described in the selfe same circle.

SVppose that the circle be ABC.* 1.3 And let the side of an equilater Pentagon described in the circle ABC, be BC. And (by the 1. of the third) take the centre of the circle, and let the same be D. And (by the 12. of the first) from the point D draw vnto the line BC a perpendicular line DE. And extend the right line DE direct∣ly to the point F. Then I say, that the line DE (which is drawen from the centre to BC the side of the pentagon) is the halfe of the sides of an hexagon and of a decagon taken together and descri∣bed in the same circle. Draw these right lines DC and CF. And vnto the line EF put an equall line GE. And draw a right

[illustration]

line from the point G to the point C.* 1.4 Now forasmuch as the circumference of the whole circle is quintuple to the circū∣ference BFC (which is subtended of the side of the penta∣gon) and the circumference ACF is the halfe of the cir∣cumference of the whole circle, and the circumference CF (which is subtended of the side of the decagon) is the halfe of the circumference BCF: therefore the circumference ACF is quintuple to the circumference CF (by the 15. of the ••i••t). Wherefore the circumference AC is qradruple to the circumference FC. But as the circumference AC is to the circumference FC, so is the angle ADC to the angle FDC, by the last of the sixt. Wherefore the angle ADC is quadruple to the angle FDC. But the angle ADC is double to the angle EFC, by the 20. of the third: Wherefore the angle EFC is double to the angle GDC. But the angle EFC is equall to the angle EGC, by the 4. of the first. Wherfore the angle EGC is double to the an∣gle EDC. Wherefore the line DG is equall to the line GC (by the 32. and 6. of the first). But the line GC is equall to the line CF, by the 4. of the first. Wherfore the line DG is equall to the line CF. And the line GE is equall to the line EF (by construction). Wherefore the line DE is equall to the lines EF and FC added together. Vnto the lines EF and FC adde the line DE. Wherefore the lines DF and FC added together, are double to the line DE. But the line DF is equall to the side of the hexagon: and FC to the side of the decagon.

Wherefore the line DE is the halfe of the side of the hexagon, and of the side of the decagon being both added together and described in one and the selfe same circle.

It is manifest * 1.5 by the Propositions of the thirtenth booke, that a perpendicular line drawen from the centre of a circle to the side of an equilater triangle described in the same circle; is halfe of the semidiameter of the circle. Wherefore by this Proposition, a perpendicular dra∣wen from the c••ntre of a circle to the side of a Pentag••n, is equall to the perpendicular drawen from the centre to the side of the triangle, ••nd to halfe of the side of the decagon described in the same circle.

¶ The 2. Theoreme. The 2. Proposition. One and the selfe same circle comprehendeth both the Pentagon of a Do∣decahedron, and the triangle of an Icosahedron,* 1.6 described in one and the selfe same Sphere.

THis Theoreme is described of Aristeus in that booke whose title is, The comparison of the fiue figures, and is described of Apollonius in his se∣cond edition of the cōparison of a Dodecahedron to an Icosahedron, which is, * 1.7 that as the super••icies of a Dodecahedrō is to the superficies of an Icosahedron, so is the Dode∣cahedron to an Icosahedron, for that a perpendicular line drawen from the centre of a sphere to the pentagon of a dodecahedron and to the triangle of an Icosahedron is one and the selfe same. Now must we also proue that one and the selfe same circle comprehendeth both the pentagon of a Dodecahedron, and also the triangle of an Icosahedron described in one and the selfe same sphere, first this being proued:* 1.8

[illustration]

If in a circle be described an equilater pentagon, the squares which are made of the side of the pentagon, and of that right line which is sub∣tended vnder two sides of the pentagon, are quintuple to the square of the semidiameter o•• the circle. Suppose that ABC be a circle.* 1.9 And let the side of a pentagon in the circle ABC, be AC. And take (by the 1. of the third) the centre of the circle and let the same be D. And (by the 12. of the first) from the point D draw vnto the line AC a perpendicular line DF. And extend the line DF on either side to the pointes B and E. And draw a right line from the point A to the point B. Now I say, that the squares of the lines BA and AC are quint••ple to the square of the line DE. Drawe a right line from the point A to the point E. Wherefore the line AE is the side of a decagon figure. And forasmuch as the line BE is double to th•• line DE:* 1.10 therefore the square of the line BE is quadruple to the square of DE (by the 20. of the sixt). But vnto the square of the line BE, are equall the squares of the lines BA and AE (by the 47. of the first, for the angle BAE is a right angle, by the 31. of the third). Wherefore the squares of the lines BA and AE, are quadruple to the square of the line DE. Wherfore the squares of the lines AB, AE, and DE, are quintuple to the square of the line DE. But the squares of the lines DE and AE, are equall to the square of the line AC (by the 10. of the thirtenth). Wherefore the squares of the lines BA and AC, are quintuple to the square of the line DE.

This being thus proued, now is to be demonstrated that one and the selfe same circle cō∣prehendeth both the pentagon of a dodecahedron, & the triangle of an Icosahedron described in one & the self same cir••le.* 1.11 Take the diameter of the sphere, &•• let the same be AB. And in the same sphere describe a dodecahedron, & also an Icosahedron. And let one of the pētagons of the dodecahedron be CDEFG, & let one of the triangles of the Icosahedron be KLH. Now I say that the semidiameters of the circles which are described about them are equall,

that is, that one and the selfe same circle contayneth both the pentagon CDEFG, and the triangle KLH. Draw a right line from the point D to the point G. Wherfore the line DG is the side of a cube (by the corollary of the 17. of the thirtēth). Take a certayne right line MN. And let the square of the line AB be quintuple to the square of the line MN, by the assumpt put after the 6. propositiō of the tēth. But the diameter of a sphere is in power quintuple to the square of the semidiameter of the circle, on which is described the Icosahedrō (by the corolla∣ry of the 16. of the thirtenth). Wherefore the line MN is the semidiameter of the circle on which is described the Icosahedron. Diuide (by the 30. of the sixth) the line MN by an extreame and meane proportion in the poynt X. And

[illustration]

let the greater segment thereof be MX. Wherefore the line MX is the side of a decagon described in the same circle (by the corollary of the 9. of the thirtenth).* 1.12 And forasmuch as the square of the line AB is quintuple to the square of the line MN: But the square of the line BA is treble to the square of the line DG (by the co∣rollary of the 15, of the thirtenth). Wherfore three squares of the line DG are equall to fiue squares of the line MN.* 1.13 But as thre squares of the line DG are to ••iue squares of the line MN, so are three squares of the line CG to fiue squares of the line MX. Wherfore three squares of y^e line CG are equll to fiue squares of the line MX. But fiue squares of the line CG are equal to ••iue squares of the line MN & to fiue squares of the MX. For (by the 10. of the thirtenth) one square of the line CG is equall to one square of the line MN & to one square of the line MX. Wherfore fiue squares of the line CG are equall to thre squares of the line DG and to three squares of the line CG (as it is not hard to proue, marking what hath before bene proued). But three squares of the line DG, together with three squares of the line CG, are equall to fiftene squares of the semidiameter of the circle described about the pentagon CDEFG (for it was before proued in the assumpt put in this proposition) that the square•• of DG and G C taken once, are quintuple to the square of the semidiameter of the circle d••scribed about the pentagon CDEFG). And fiue squares of the line KL are equall to fiftene squares of the semidiameter of the circle described about the triangle KLH. (For by the 12. of the thirtenth, one square of the line LK is triple to one square of the line drawne from the centre to the circumference). Wherefore fiftene squares of the line drawne from the centre to the circumference (of the circle which contayneth the pentagon CDEFG) are equall to fiftene squares of the line drawne from the centre to the circumference of the circle which contayneth the triangle KLH): wherefore one of the squares which is drawne from the centre to the circumference of the one circle, is equall to one of the squares which is drawne from the centre to the circumference of the other circle. Wherefore the diameter is equall to the diameter, wherefore one and the selfe same circle comprehendeth both the pen∣tagon of a dodecahedron and the triangle of an Icosahedron described in one and the selfe same circle: which was required to be proued.

¶ The 3. Theoreme. The .3 Proposition. * 1.14If there be an equilater and equiangle pētagon, aud about it be described a circle, and from the centre to one of the sides be drawne a perpendicular line, that which is contayned vnder one of the sides and the perpendicular

line thirty times, is equall to the superficies of the dodecahedron.

SVppose that ABCD be an equilater and equiangle pentagon. And about the same pentagon, describe (by the 14. of the fourth) a circle.* 1.15 And let the centre ther∣of be the poynt F. And from the poynt F draw (by the 12. of the first) vnto the line CD a perpendicular line FG. Now I say that that which is contayned vn∣der the lines CD and GF thirty times, is equall to 12. pentagons of the same quantitie that the pentagon ABCD is. Draw these right lines CF and FD.* 1.16 Now forasmuch as that which is contayned vnder the lines CD and FG is double to

[illustration]

the triangle CDF (by the 41. of the first) therefore that which is contayned vnder the lines CD and FG fiue times is equall to ten of those triangles. But ten of those triangles are two pentagōs, and sixe times ten of those triangles are all the pentagons. Wherefore that which is contayned vnder the lines CD and FG thirty times is equall to 12. pentagons But 12. pentagons are the superficies of dodecahedron. Wher¦fore that which is contayned vnder the lines CD and FG thirty times is equall to the superficies of the dodecahedron.

In like sort also may we proue that if there be an equilater triangle, as for example,* 1.17 the triangle ABC, and about it be described a circle, and the centre of the circle be the point D, and the perpendicular line be the line DE: that which is contay∣ned

[illustration]

vnder the lines BC and DE thirty times, is equall to the su∣perficies of the Icosahedron.* 1.18 For agayne forasmuch as that which is contayned vnder the lines DE and BC is double to the trian∣gle DBC (by the 41. of the first): therefore two triangles are e∣quall to that which is contayned vnder the lines DE and BC, and three of those triangles contayne the whole triangle. Where∣fore sixe such triangles as DBC is, are equall to that which is contayned vnder the lines DE and BC thrise. But sixe s••ch tri∣angles as DBC is, are equall to two such triangles as ABC is. Wherefore that which is contained vnder the lines DE and BC thrise, is equall to two such triangles as ABC is. But two of those triangles takē ten times contayneth the whole Icosahe∣dron. Wherfore that which is contayned vnder the lines DE & BC thirty times, is equall to twenty such triangles as the triangle ABC is, that is, to y^e whole superficies of the Icosahedrō. * 1.19 Wherefore as the superficies of the dodecahedron is to the supe••ficies of the Icosahedron, so is that which is con∣tayned vnder the lines CD and FG to that which is contayned vnder the lines BC and DE.

¶ Corollary.

By this it is manifest, that as the superficies of the Dodecahedron is to the superficies of the Icosahedron,* 1.20 so is that which is contained vnder the side of the Pentagon, and the perpēdicular line which is drawen from the cen∣tre of the circle described about the Pentagon to the same side, to that which is contained vnder the side of the Icosahedron and the perpendicular line which is drawen from the centre of the circle described about the tri∣angle to the same side: so that the Icosahedron and Dodecahedron be both described in one and the selfe same Sphere.

¶ The 4. Theoreme. The 4. Proposition. * 1.21This being done, now is to be proued, that as the superficies of the Do∣decahedron is to the superficies of the Icosahedron, so is the side of the cube to the side of the Icosahedron.

* 1.22TAke (by the 2. Theoreme of this booke) a circle containing both the penta∣gon of a Dodecahedron, and the triangle of an Icosahedron, being both de∣scribed in one and the selfe same sphere, and let the same circle be DBC. And in the circle DBC describe the side of an equilater triangle, namely, CD, and the side of an equilater pentagon, namely, AC. And take (by the 1. of the third) the centre of the circle, and let the same be E. And from the point E drawe vnto the lines DC and AC, perpendicular lines EF and EG. And extend the line EG directly to the point B. And drawe a right line from the point B to the point C. And let the side of the cube be the line H. Now I say, that as the superficies of the Dodecahedron is to the superficies of the Icosahedron, so is the line H to the line CD.* 1.23 Forasmuch as the line made of the lines EB and BC added together (namely, of the side of the hexagon, and of the side of a decagon) is (by the 9. of the thirtenth) diuided by an extreme and meane proportion, and his greater segment is the line BE: and the line EG is also (by the 1. of the fo••retenth) the halfe of the same line, and the line EF is the halfe of the line BE (by the Corollary of the 12. of the thirtenth). Wherefore the line EG being diuided by

[illustration]

an extreme and meane proportion,* 1.24 his greater seg∣ment shall be the line EF. And the line H also being diuided by an extreme & meane proportion, his grea∣ter segment is the line CA, as it was proued† 1.25 in the Dodecahedron. * 1.26 Wherefore as the line H is to the line CA, so is the line EG to the line EF. Wherefore (by the 16. of the sixt) that which is contained vnder the lines H and EF, is equall to that which is contained vnder the lines CA and EG. And for that as the line H is to the line CD, so is that which is contained vn∣der the lines H and EF, to that which is contained vnder the lines CD and EF (by the 1. of the sixt). But vnto that which is contained vnder the lines H and EF, is equall that which is contained vnder the lines CA and EG. Wherefore (by the 11. of the fift) as the line H is to the line CD, so is that which is contained vnder the lines CA and EG, to that which is contained vnder the lines CD and EF, that is (by the Corollary next going before) as the superficies of the Dodeca∣hedron is to the superficies of the Icosahedron, so is the line H to the line CD.

An other demonstration to proue that as the superficies of the Dodecahe∣dron is to the superficies of the Icosahedron, so is the side of the cube to the side of the Icosahedron. † 1.27

LEt there be a circle ABC. And in it describe two sides of an equilater pen∣tagon (by the 11. of the fift) namely, AB and AC: and draw a right line from the point B to the point C. And (by the 1. of the third) take the centre of the circle, and let the same be D. And draw a right line from the point A to the point D, and extend it directly to the point E, and let it cut the line BC in the point G. And let the line DF be halfe to the line DA, and let

the line GC be treble to the line HC, by the 9. of the sixt.* 1.28 Now I say, that that which is contained vnder the lines AF and BH, is equall to the pentagon inscribed in the circle ABC. Draw a right line from the point B to the point D. Now forasmuch as the line AD is double to the line DF, there∣fore the line AF is sesquialter to the line AD.

[illustration]

Againe,* 1.29 forasmuch as the line GC is treble to the line CH, therefore the line GH is double to the line CH. Wherefore the line GC is sesquialter to the line HG. Wherefore as the line FA is to the line AD, so is the line GC to the line GH. Wherefore (by the 16. of the sixt) that which is contained vnder the lines AF & HG, is equall to that which is contained vnder the lines DA and GC. But the line GC is equall to the line BG (by the 3. of the third). Wherfore that which is contained vnder the lines AD and BG, is e∣quall to that which is contained vnder the lines AF and GH. But that which is contained vn∣der the lines AD and BG, is equall to two such triangles as the triangle ABD is (by the 41. of the first). Wherefore that which is contained vnder the lines AF and GH, is equall to two such triangles as the triangle ABD is. Wherefore that which is contained vnder the lines AF and GH ••iue times, is equall to ten triangles. But ten triangles are two pentagons. Wherefore that which is contained vnder the lines AF and GH fiue times, is equall to two pentagons. And forasmuch as the line GH is double to the line HC, therefore that which is contained vnder the lines AF and GH, is double to that which is contained vnder the lines AF and HC (by the 1. of the sixt). Wherefore that which is contained vnder the lines AF and CH twise, is equall to that which is contained vnder the lines AF and GH once. Take eche of those parallelogrammes fiue times. Wherefore that which is contained vnder the lines AF and HC ten times, is equall to that which is contained vnder the lines AF & GH fiue times, that is, to two pentagons. Wherefore that which is contained vnder the lines AF and HC fiue times, is equall to one pentagon. But that which is contained vnder the lines AF and HC fiue times, is equall (by the 1. of the sixt) to that which is contained vnder the lines AF and HB, for the line HB is quintuple to the line HC (as it is easie to see by the con∣struction) and they are both vnder one & the selfe same altitude, namely, vnder AF. Wher∣fore that which is contained vnder the lines AF and BH, is equall to one pentagon.

This being proued, now let there be drawne a Circle comprehending both the Pentagon of a Dodecahedron, and the triangle of an Icosahedron, being both described in one and the selfe same Sphere.

* 1.30LEt the circle be ABC. And in it describe as before, two sides of an equilater pentagon, namely BA and AC•• and draw a right line from the point B to the point C: and take the centre of the circle and let the same be E. And from the point A to the point E draw a right line AE: and extend the line AE to the point F. And let it cut the line BC in the point K. And let the line AE be do∣ble to the line EG, & let the line CK be treble to the line CH, by the .9. of the sixth. And frō the point G raise vp (by the .11. of the first) vnto the line AF a perpendicular line GM: and extend the line GM directly to the point D. Wherfore the line MD is the side of an equi∣liter triāgle, by the corollary of the .1••. of the thirtenth: draw these right lines AD and AM. Wherfore ADM is an equilater triangle.* 1.31 And for as much as that which is contained vn∣der the lines AG and BH is equal to the pentagon (by the former assump••) and that which

is cōtained vnder the lines AG and GD is equal

[illustration]

to the triangle ADM: therefore as that which is contained vnder the lines AG and HB is to that which is contained vnder the lines AG and GD, so is the pentagon to the triāgle. But as that which is contained vnder the lines BH & AG is to that which is contained vnder the lines AG and GD, so is the line BH to the line DG (by the .1. of the sixth) wherefore (by the .15. of the fifth) as 12. such lines as BH is, are to .20 such lines as DG is, so are 12. pentagons to 20. triangles, that is the su∣perficies of the Dodecahedron, to the superficies of the Icosahedron. And 12. suche lines as BH is, are equall to tenne suche lines as BC is (for the line HB is quintuple to the line HC): and the line BC is sextuple to the line CH•• Wherfore six such lines as BH is, are equal to fiue such lines as BC are: and in the same pro∣portion are their doubles: and 20. such lines as the line DG is, are equal to .10. such lines as the line DM is: for the line DM is double to the line DG. Wherfore as 10. such lines as BC is, are to 10. such lines as DM is, that is, as the line BC is to the line DM, so is the superficies of the Dodecahedron to the superficies of the Icosahedron. But the line BC is the side of the cube, and the line DM the side of the Icosahedron: wherefore (by the 11. of the fifth) as the superficies of the Dodecahedron is to the superficies of the Icosahedron, so is the line BC to the line DM, that is, the side of the cube to the side of the Icosahedron.

* 1.32Nowe will we proue that a right line being deuided by an extreme and meane proportiō, what proportiō the line cōtaining in power the squares of the whole line and of the greater segment, hath to the line containing in power the squares of the whole line and of the lesse segment, the same proportion hath the side of the cube to the side of the Icosahedron, being both described in one and the selfe same sphere.

* 1.33SVppose that AB be a circle conta••ning both the pentagon of a Dodecahedron & the triangle of an Icosahedron described bothe in one and the selfe same sphere. Take the centre of the circle, and let the same be C. And from the point C extend to the circumference a right line at all auentures, and let the same BC. And (by the 30. of the sixth) deuide the line BC by an extreme and meane proportion in the point D, and let the greater segment therof be CD. Wherfore the line CD is the side of a Decagō de∣scribed in the same circle (by the corollary of the 9. of the thirtenth). Take the side of an Ico∣sahedron, and let the same be the line E, and the side of a Dodecahedron, and let the same be the line F, and the side of a cube & let the same be the line G.* 1.34 Wherfore the line E is the side of an equilater triangle, and F of an equaliter pentagon described in one and the selfe same circle. And the line G being deuided by an extreme and meane proportion, his greater seg∣ment is the line F, by the corollary of the 17. of the thirtēth. Now forasmuch as the line E is the side of an equilater triangle, but (by the 12. of the thirtenth) the side of an equilater triangle is in power treble to the line BC, (which is drawne from the center to the circum∣ference) therefore the square of the line E is treble to the square of the line BC: but the squares of the line BC and BD are (by the 4. of the thirtenth) treble to the square of the line CD. Wherfore as the square of the line E is to the square of the line CB, so are the squares of the lines CB and BD to the square of the line CD. Wherefore alternately (by the 16. of

the fifth) as the square of the line E is to the squares

[illustration]

of the lines CB and BD, so is the square of the line CB to the square of the line CD. * 1.35 But as the square of the line BC is to the square of the line CD, so is the square of the line G (the side of the cube) to the square of the line F, the side of Dodecahedron. For the line F is the greater segmēt of the line G (as was before proued.) Wherfore (by the .11. of the fift) as the square of the line E is to the squares CB and BD, so is the square of the line G, to the square of the line F. Wherefore alternately (by the 16. of the fifth) & also by conuersion (by the corollary of the 4. of the fift) as the square of the line G, is to the square of the line E, so is the square of the line F, to the squares of the lines CB & BD. But vnto the square of the line F are equal the squares of the lines BC & CD, for the side of a pentagon cōtaineth in power both the side of a sixe angled figure, and the side of a ten angled figure (by the 10. of the thirtenth.) Wherfore as the square of the line G, is to the square of the line E, so are the squares of the lines BC and CD to the squares of the lines CB and BD. But as the squares of the lines CB and CD are to the squares of the lines CB & BD,† 1.36 so (any right line what so euer it be, being diuided by an extreme and meane proportion) is the line containing in power the squares of the whole line, and of the greater segmēt, to the line containing in power the squares of the whole line, and of the lesse segment: wherfore (by the 11. of the fifth) as the square of the line G (the side of the cube) is to the square of the line E, so (any right line being deuided by an extreme and meane proportion) is the line containing in power the squares made of the whole line, and of the greater segmēt, to the line containing in power the squares made of the whole line, and of the lesse segment: but the line G is the side of the Cube, and the line E of the Icosahedron (by supposition.) If therfore a right line be deuided by an extreeme and meane proportion, as the line cōtaining in power the squares of the whole line, and of the greater segment, is to the line containing in power the squares of the whole line and of the lesse segment: so is the side of the cube to the side of the Icosahedron, being both described in one and the selfe same sphere.

Now will we proue that as the side of the Cube is to the side of the Ico∣sahedron,* 1.37 so is the solide of the Dodecahedron to the solide of the Icosa∣hedron.

FOrasmuche as equal circles comprehend both the pentagon of a Dodecahe∣dron, and the triangle of an Icosahedron, being both described in one and the selfe same sphere, by the 2. of this booke: but in a sphere equal circles are equally distant from the centre (for the perpendicular lines drawn from the centre of the sphere to the plaine superficieces of the circles are equal, and do fall vpon the centres of the circles. † 1.38Wherfore perpendicular lines drawne from the centre of the sphere, to the centre of the circle, comprehending bothe the triangle of an Icosahedron, and the pentagon of a Dodecahedron are equal: wherefore the pyramides, whose bases are the pentagons of the Dodecahedron, are of equal altitude with the piramides whose bases are the triangles of the Icosahedron. But piramids of equal altitude, are in that proportion the one to the other, that their bases are (by the 5. of the twelfth) wherefore as the pentagon is to the triangle, so is the pyramis whose base is the pentagon of the Dodecahe∣dron and toppe the centre of the sphere, to the pyramis whose base is the triangle and top the

centre also of the sphere. Wherfore (by the 15. of the fifth) as 12. pētagons are to 20. triangles, so are 12. pyramids hauing pentagons to theyr bases to 20. pyramids hauing triāgles to their bases. But 12. pentagons are the superficies of the D••decahedron, and 20. triangles are the superficies of the Icosahedron. Wherefore as the superficies of the Dodecahedron is to the su∣perficies of the Icosahedron, so are 12. pyramids hauing pentagons to their bases to 20. pyra∣mids hauing triangles to their bases. But 12. pyramids hauing pentagons to their bases, are the solide of the Dodecahedron, and 20. pyramids hauing triangles to their bases are the so∣lide o•• the Icosahedron.* 1.39 Wherfore (by the 11. of the fifthe) as the superficies of the Dodecahedron is to the superficies of the Icosahedron ••o is the solide of the Dodecahedron to the solide of the Icosahedron. But as the superficies of the Dodecahedron, is to the superficies of the Icosahedron, so haue we proued that the side o•• the cube is to the side of the Icosahedron. Wherfore, by the 11. of the fifth, as the side of the cube is to the side of the Icosahedron, so is the solide of the Dodecahedron to the solide of the Icosahedron.

* 1.40If two right lines be diuided by an extreame and meane proportion, they shall euery way be in like proportion: which thing is thus demonstrated.

LEt the line AB be (by the 30. of the sixth) diuided by an extreame and meane pro∣portion in the poynt C, and let the greater segment thereof be the line CA. And likewise also let the line DE be diuided by an extreame and meane proportion in the poynt F, and let the greater segment thereof be the line DF. Then I say that as the whole line AB is to the greater segment thereof AC, so is the whole line DE to the grea∣ter segment thereof DF. For forasmuch as that which is contayned vnder the lines AB and BC is equall to the square of the line AC (by the diffinition of a line diuided be an ex∣treame and meane proportion):* 1.41 and that which is contayned vnder the lines DE and EF is also equall to the square of the line DF (by the same diffinition): therefore as that which is contayned vnder the lines AB and BC is to the square of the line AC, so is that which is contayned vnder the lines DE and EF to the square of the line DF. For in eche is the pro∣portion of equalitie. Wherfore as that which is contayned vnder the lines AB and BC fower times, is to the square of the line

[illustration]

AC, so is that which is contaynd vnder the lines DE and EF fower times to the square of the line DF (by the 15. of the fifth). Wherfore by composition (by the 18. of the ••i••th) as that which is contayned vnder the lines AB and BC fower times, together with the square of the line AC, is to the square of the line AC, so is that which is contayned vnder the lines DE and EF fower times, together with the square of the line DF, to the square of the line DF. Wherefore as the square which is made of the lines AB and BC ad∣ded together and made one line (which square by the 8. of the second is equall to that which is contayned vnder the lines AB and BC fower times together with square of the line AC) is to the square of the line AC, so is the square made of the lines DE & EF added together and made one line (which square is also, by the same, equal to that which is contayned vnder the lines DE and EF fower times together with the square of the line DF) to the square of the line DF. Wherefore also as the lines AB & BC added together are to the line AC, so are the lines DE & EF added together to the line DF (by the 22. of the sixt). Wherefore by cō∣position (by the 18. of the fifth) as both the lines AB & BC added the one to the other, toge∣ther 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, * 1.42 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 propor∣tion hath the side of the cube to the side of the Icosahedron: Now also that we haue proued, † 1.43 that as the side of the cube is to the side of the Icosahedron, so is the superficies of the Dode∣cahedron to the superficies of the Icosahedron, being both described in one and the selfe same sphere: and moreouer seing that we haue proued, † 1.44that as the superficies of the Dodecahedron is to the superficies of the Icosahedrō, so is the Dodecahedrō 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:* 1.45 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 pro∣portion, 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 pow∣er 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.

Notes

• * 1.1

  The argument of the fourtenth booke.
• * 1.2

  First proposi∣tion after Flussas.
• * 1.3

  Construction.
• * 1.4

  Demonstra∣tion.
• * 1.5

  This is mani∣fest by the 12. propositiō of the thirtenh booke as Campane well gathereth in a Corollary of the same.
• * 1.6

  The 4. p••pos••tiō after Flussas.
• * 1.7

  This is after∣ward proued in the 4. proposi∣tion.
• * 1.8

  This Assumpt is the 3. proposi∣tion after Flussas.
• * 1.9

  Construction of the Assumpt.
• * 1.10

  Demonstration of the Assumpt.
• * 1.11

  Construction of the proposition.
• * 1.12

  Demonstra∣tion of the ••••op••sition.
• * 1.13

  〈1 paragraph〉〈1 paragraph〉
• * 1.14

  Th•• 5. propo∣sition a••t••r 〈◊〉〈◊〉.
• * 1.15

  Construction.
• * 1.16

  Demo••stra∣tion.
• * 1.17

  The 5. propo∣sition a••ter F••ussas.
• * 1.18

  Demonstra∣tion.
• * 1.19

  This is the reason of the Corollary following.
• * 1.20

  A Corollary which also Flussas put∣teth as a Co∣rollary after the 5. proposi∣tion in his order.
• * 1.21

  The 6. p••••positiō ••••ter Flussas.
• * 1.22

  Co••struction.
• * 1.23

  Demonstra∣tion.
• * 1.24

  This is not hard to proue by the 15. 16. and 19. of the ••••••eth.
• † 1.25

  In the Corol∣lary of the 17. of the t••irtēth.
• * 1.26

  〈◊〉〈◊〉 againe is required the Assumpt which is afterward proued in this 4 proposition.
• † 1.27

  But first the Assumpt follow∣ing, the con∣struction wh••re••f here beginne••h, is to be proued.
• * 1.28

  The Assumpt, which also Flus∣sas putteth as an Assumpt a••ter the 6. propositiō.
• * 1.29

  Demonstration of the Assumpt.
• * 1.30

  Construction pertaining to the second d••∣monstration of the 4. propositiō.
• * 1.31

  Second demon∣stration o•• the 4. proposition.
• * 1.32

  The 7•• propo∣sition after Flussas.
• * 1.33

  Construction.
• * 1.34

  Demonstra∣tion.
• * 1.35

  Here againe is required the Assumpt af∣terward pro∣ued in this 4. proposition.
• † 1.36

  As may by the Assumpt afterward in this propositiō be plainely proued.
• * 1.37

  The 8. pro••i∣tion a••ter Flussas.
• † 1.38

  By the Co∣rollary added by Flussas af∣ter has As∣sumpt put af∣ter the 17. pro¦position of the 12. booke.
• * 1.39

  Corollary of the 8. after Flussas.
• * 1.40

  This Assumpt is the 3. propositiō a••ter ••lussas, and is it which 〈◊〉〈◊〉 times hath bene taken a•• g••aunted in this booke, and o••ce also in the last proposition of the 13. booke: as we haue be∣••ore noted.
• * 1.41

  Demonstra∣tion.
• * 1.42

  In the 4. sec∣tion ••f this proposition.
• † 1.43

  In the 1. and 3 section of the same propositiō.
• † 1.44

  In the 5. sectiō of the same proposition.
• * 1.45

  A Corollary.