points o, s, P, T, R, V right lines be conceived to be drawn to A, there will be made folid polyhedrons compofed of pyramids, between the circumferences BX, KX whose bases are the quadrilateral figures K BOS, SOPT, TPRV, and the triangle VRX, and vertex the point A. Now if upon each of the fides KL, LM, ME, we make the fame conftruction as on K B, as alfo in the rest of the three quadrants, and in the remaining hemisphere; there will be defcribed a folid polyhedron in the fphere compofed of pyramids, whofe bafes are quadrilateral figures of the fame order with those and the triangle already mentioned, and vertex the point A: I fay the faid polyhedron does not touch the fuperficies of the leffer sphere, in which the circle FGH is. Draw [by 11. 11.] AY from the point A perpendicular to the plane of the quadrilateral figure KBOs, meeting it in the point y, and join BY, YK. Then because AY is perpendicular to the plane of the quadrilateral figure K BOS, [by 3. def. 11.) it will be perpendicular to all right lines that touch it, and are in the fame plane: Therefore AY is at right angles to B Y, and to Y K. And because A B is equal to a K ; the square of A B will be equal to the fquare of AK. But [by 47. 1.] the fquares of A Y, Y B are equal to the fquare of A B, for [by conftr.] the angle at y is a right angle; and the squares of AY, YK are equal to the fquare of AK: Therefore the squares of A Y, Y B are equal to the squares of A Y, YK: Take away the common fquare of AY, and the fquare of BY remaining is equal to the fquare of YK remaining; and fo the right line BY is equal to the right line YK. After the fame manner we demonftrate that each of the right lines drawn from the point to o, s, is equal to BY, or Y K. Therefore a circle de fcribed about the centre y with either of the diftances YB, or YK, will pafs thro' the points o, s; and fo KBOS will be a quadrilateral figure in a circle. And because KB is greater than zq, and zo is equal to so; KB will be greater than so. But KB is equal to Ks of BO Therefore KS or BO are each greater than so. There fore fince KBOs is a quadrilateral figure in a circle, and KB, BO, KS are equal, and os leffer, than either of them, and BY is a line drawn from the centre; the fquare of KB will be greater than twice the fquare of BY, Draw the perpendicular Kw from the point K to BD. Dd 2 Then

Book XII. Then because BD is lefs than twice DW, and [by 1. 6.] as DB is to DW, fo is the rectangle contained under DB, Bw to the rectangle contained under DW, WB; wherefore the fquare of BW being defcribed, and the parallelogram upon WD being compleated, the rectangle under DB, BW is less than twice the rectangle under DW, WB. And moreover joining KD, the rectangle under DB, BW [by 8. 6.] is equal to the fquare of KB, and the rectangle under DW, WB equal to the fquare of Kw. Therefore the fquare of KB is lefs than twice the fquare of Kw. But the fquare of KB is greater than twice the fquare of By: Therefore the fquare of Kw is greater than the fquare of BY. And becaufe BA is equal to KA, the fquare of BA will be equal to the fquare of KA. And [by 47. 1.] the fquares of BY, YA are equal to the fquare of BA, and the fquares of KW, WA equal to the fquare of KA: Therefore the fquares of BY, YA are equal to the fquares of KW, WA; but the fquare of Kw is greater than the fquare of By: Therefore the remaining fquare of wa is lefs than the fquare of YA; and accordingly the right line AY is greater than the right line AW: Therefore AV is much greater than AG. But AY is at one bafe of the polyhedron, and AG at the fuperficies of the leffer fphere: Wherefore the polyhedron does not touch the fuperficies of the leffer sphere.

Otherwife.

It may be demonftrated more eafily and expeditiously that AY is greater than AG. From the point G draw GL at right angles to AG, and join AL. Then the circumference EB being bifected, and one half of it alfo bifected, and this being continually done, there will at last [by 1. 10.] be left a part of the circumference less than the arch of the circumference of the circle BCD, whose *fubtenfe is equal to GL. Let the part of the circumference KB be this ultimate remainder. Then the right line KB is less than GL. And because BKSO is a quadrilateral figure in a circle, and OB, BK, KS are equal, and os lefs than either; the angle BYK will be obtufe; and fo BK is greater than BY. But GL [by conftr.] is greater than BK: Wherefore GL is much greater than BY, and the fquare of GL is greater than the fquare of BY. And because AL is equal to AB, the fquare of AL *A right line drawn from one end of it to the other.

will be equal to the fquare of AB. But the fquares of AG, GL are equal to the fquare of AL, and the fquares of BY, YA equal to the fquare of AB: Therefore the fiquares of AG, GL are equal to the fquares of BY, YA : But the fquare of By is lefs than the fquare of GL: Wherefore the remaining iquare of YA is greater than the remaining fquare of AG; and fo the right line AY is greater than the right line AG.

Therefore two fpheres having the fame centre, being given, a folid polyhedron is defcribed in the greater fphere that does not touch the fuperficies of the leffer fphere. Which was to be demonftrated.

Corollary. Alfo if a folid polyhedron be infcribed in the leffer fphere fimilar to that defcribed in the greater fphere BCDE: the folid polyhedron in the leffer fphere will be to the folid polyhedron in the greater sphere BCDE in the triplicate ratio of the diameter of the greater sphere BCDE to the diameter of the leffer fphere. For the folids being divided into equal numbers of pyramids of the fame order, these pyramids will be fimilar. And fimilar pyramids [by cor. 8. 12.] are to one another in the triplicate ratio of their homologous fides: Therefore the pyramid whose base is the quadrilateral figure KBOS, and vertex the point A, is to the pyramid in the other sphere of the fame order in the triplicate ratio of the homologous fide of the one to the homologous fide of the other; that is, of the femidiameter AB of one of the spheres, to the femidiameter of the other. In like manner each of the pyramids in one of the spheres to the correspondent pyramid in of the other, will be in the triplicate ratio of the femidiameter of one of the spheres to the femidiameter of the other. But [by 12. 5.] as one of the antecedents is to one of the confequents, fo are all the antecedents to all the confequents: Wherefore the whole folid polyhedron which is in one of the spheres about the centre A, is to the other in the triplicate ratio of the femidiameter of one of the spheres, to the femidiameter of the other; that is, of the diameter B D of the one, to the diameter of the other.

8 This problem, being a fort of unlimited one, is generally ftumbled at by young beginners, who feldom or never underftand it till they have read the demonftration. It is of no other ufe in thefe Elements but to demonstrate the next theorem; and, to tell the truth, is not very inviting in itself.

Dd 3

PROP

PROP. XVIII.

THEOR.

.

Spheres are to one another in the triplicate ratio of their diametersh.

Conceive ABC, DEF to be two spheres, whofe diameters are BC, EF: I fay the sphere ABC is to the sphere DEF in the triplicate ratio of BC to EF.

For if not, the fphere ARC to fome sphere either greater or lefs than DEF, will be in the triplicate ratio of BC to EF. Firft let it be fo to a sphere which is lefs, viz. to GHK; and conceive the fphere DEF to have the fame centre as the fphere GHK; and [by 17. 12.] in the

greater of these spheres DEF defcribe a folid polyhedron that does not touch the fuperficies of the leffer fphere GHK; and in the fphere ABC defcribe a folid polyhedron fimilar to that defcribed in DEF: Then the folid polyhedron in the fphere ABC is to the folid polyhedron in the fphere DEF, [by cor. 17. 12.] in the triplicate ratio of BC to EF. But [by fuppofition] the fphere ABC is to the fphere GHK in the triplicate ratio of BC to EF: Therefore [by 11. 5.] as the fphere ABC is to the sphere GHK, fo is the folid polyhedron in the fphere ABC to the folid polyhedron in the fphere DEF. And alternately as the sphere ABC is to the polyhedron which is in it, fo is the fphere GHK to the polyhedron in the fphere DEF. But the fphere ABC is greater than the folid polyhedron which is in it; and therefore the fphere GHK is greater than the polyhedron in the fphere DEF: But it is lefs too, for it is contained in it; which is impoffible: Therefore the sphere ABC is not to a sphere less than DEF in the triplicate ratio of BC to EF. In like manner we demonftrate that the sphere DEF is not to fome fphere lefs then ABC in the triplicate ratio of EF to BC. I fay also

that

that the sphere ABC to fome fphere greater than DEF, is not in the triplicate ratio of BC to EF. For if poffible, let it be so to fome greater fphere LMN: Then inversely the sphere LMN is to the fphere ABC in the triplicate ratio of the diameter EF to the diameter BC. And as the sphere LMN is to the sphere ABC, fo is the fphere DEF, to fome sphere lefs than ABC, as has been already demonftrated; because the fphere LMN is greater than DEF; therefore the sphere DEF is to a sphere less than ABC in the triplicate ratio of EF to BC; which has been already proved to be impoffible. Wherefore the sphere ABC is not to some sphere greater than DEF in the triplicate ratio of BC to EF; and it has been demonftrated not to be fo to some sphere lefs: Therefore the sphere ABC will be to the sphere DEF in the triplicate ratio of BC to EF. Which was to be demonftrated.

h This theorem may be demonftrated fhorter, by fuppofing fpheres to be fimilar folids having infinite equal numbers of fmall equal fquare faces, the four angles of each square being in the fuperficies of the fpheres; for then a fphere will confift of an infinite number of fmall equal pyramids, whofe vertexes will all be at the centre of the sphere, and small square bases at the fuperficies of the fphere: So also may any other sphere confift of the like number of fmall fquare pyramids, and each of these pyramids in each of the spheres, will be fimilar; and fo will be to one another in the triplicate ratio of the correfpondent fides of the pyramid; that is, of the femidiameters of the spheres; and fo will all fuch pyramids in one sphere, be to all in the other; that is, one sphere to the other.