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 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 e∣quall 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 sesquial••er to FM the square of the cube. From the line AE cut of a third part AN, and frō the line MG cut of like∣wise 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 tri∣ple to the pyramis ABKDL which is set vpon the same base, and is vnder the same altitude (by the se∣cond Corollary of the 7. of the twelfth). Wherefore the pyramis ABKDL is equall to the parallelipi∣pedon, which is set vpon the base BKDL, and

[illustration]

hath to his altitude the line AN. But vnto that pa∣rallelipipedō, 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 pyra∣mis is double the Octohedron ABKLDC, by the 2. Corollary of the 14. of the thirtenth. Where∣fore the Octohedron ABKDLC is equall to the parallelipipedon set vpon the base BKLD, & ha∣uing 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 ele∣uenth. (For it was before proued that the base BKDL is sesquialter to the base FM). Now then these two parallelipipedons, namely, the paralleli∣pipedon which is set vpō 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 paral∣lelipipedon is proued equall to the cube, and the parallelipipedon set vpon the same base BKDL, and hauing his altitude the line EN (which paral∣lelipipedon 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 alti∣tude EN, so is the cube FGHIM, to the Octo∣hedron ABKDLC, by the 7. of the fifth. But as the line MO is to the line EN, so is 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 in∣to 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 segmēt is the side of the Icosahedron. And the dodecahedron into two squares, whose proportion is quadru∣ple to the proportion of a line diuided

[illustration]

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 C∣D two equall sides of the octohedron. But the diameter contayneth in power the whole line AE and the greater seg∣ment thereof ED, which is the side of the Icosahedron, by the 15. of this booke. Whe••fore their powers being in duple proportiō 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 proportiō. Farther the diameter cōtayneth 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, •• 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 quadru∣ple 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 ge∣uen the side of euery one of the bodies inscribed. And forasmuch as three of those bo∣dies haue their sides commensurable in power onely, and not in length, vnto the dia∣meter 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 incom∣mensurabe 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 other•• which powers are con∣tained 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 Tetra∣hedron: 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 dedocahe∣dron is a residuall line, by the 17. of the same) therfore those sides are vnto the diame∣ter which is a rationall line set, incommensurable both in length and in power. Where∣fore 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 incōmēsurable 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 proportiō, 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 propor∣tion 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.

Notes

• Construction.
• Demonstra∣tion.