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II.

O 47. I.

Book XII angles to all the right lines drawn in that plain "; join BZ, ZK, then AZ will be perpendicular to BZ, ZK. But the n def. 3. fquares of AK, AB, are equal, and the fquares of AZ, ZB, are equal to the fquare of AB °, and the fquares of AZ, ZK, are equal to the fquare of AK"; therefore the fquares of AZ, ZB, are equal to the fquares of AZ, ZK. Take the common square of AZ from both, then the fquares of EZ, ZK, are equal; that is, BZ equal to ZK. In like manner, lines drawn from Z to the points O, S, may be proved equal to BZ, ZK; therefore a circle defcribed about the center Z, with either of thefe diftances, will pafs through the points O, S, K, B; and, because KBSO is a quadrilateral figure infcribed in a circle, and OB, K, KS, are equal, and OS lefs than BK, the angle BZK will be obtufe; therefore BK is greater than BZ; but GL is greater than KB, and therefore much greater than BZ; and the fquares of AG, GL, are equal to the fquare of AL, AB, or AK; therefore the fquares of BZ, ZA, are equal to the fquare of AL; and the fquares of AZ, ZB, are equal to the fquares of AG, GL; but the fquare of GL was proved greater than the fquare of BZ; therefore the fquare of AZ is greater than the fquare of AG; that is, the right line AZ greater than AG; but AZ is perpen dicular to one of the bafes of the polyhedron; and AG reaches the fuperficies of the leffer fphere; therefore the polyhedron does not touch the fuperficies of the leffer fphere. Wherefore, &c.

b 12.

P 15. 5.

r 12. 5.

COR. If a folid polyhedron is infcribed in another sphere, fimilar to that in BCDE, they fhall be to one another in the tripli cate ratio of the fquares of their diameters; for, the folids being divided into pyramids, equal in number, and of the fame order, they will be fimilar; and therefore to one another in the triplicate ratio of their homologous fides P; that is, as AB drawn from the center of the fphere BCDE, to the femidiameter of the other fphere; but the femidiameters of spheres are as their diameters; and one of the antecedents is to one of the confequents as all the antecedents to all the confequents ; therefore, the polyhedron in the one fphere is to the fimilar polyhedron in the other fphere, in the triplicate ratio of their diameters.

PRO P. XVIII. THE OR.

S PHERES are to one another in the triplicate ratio of their

diameters.

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Let ABC, DEF, be two spheres, and BC, EF, their diameters, Book XII the fphere ABC is to the sphere DEF, in the triplicate ratio of BC to EF. If not, let the fphere ABC be to a sphere GHK lefs than the sphere DEF, in the triplicate ratio of BC to EF. Let this sphere GHK be infcribed within the fphere DEF; likewise, in DEF, inscribe a polyhedron, which shall not touch the superficies of the leffer fphere GHK. In the fphere ABC infcribe a a 17. polyhedron, fimilar and alike fituate to that in DEF; then these fimilar polyhedrons are to one another in the triplicate ratio of their dian.eters BC, EF; but the sphere ABC, to the fphere b cor. 17. GHK, hath a triplicate ratio of BC to EF; therefore the sphere c hyp. ABC is to the sphere GHK, as the polyhedron ABC to the fimilar polyhedron in DEF; but the fphere ABC is greater than the polyhedron in it; therefore the fphere GHK is likewife greater than the polyhedron in DEF; but it is lefs, as contained in it; which is abfurd; therefore the fphere ABC, to the fphere lefs than DEF, has not a triplicate ratio of BC to EF. For the fame reason, the sphere DEF, to a sphere less than ABC, has not a triplicate ratio of EF to BC. Again, the fphere ABC, to a fphere of LMN, greater than 'DEF, has not a triplicate ratio of BC to EF. If it can, then, by inverf. the fphere LMN, to the sphere ABC, fhall have a triplicate ratio of the diameters EF to the diameter BC; but the fphere LMN is to the sphere ABC as the fphere DEF to fome fphere lefs than ABC, because the fphere LMN is greater than DEF; therefore the fphere DEF, to a sphere lefs than ABC, has a triplicate ratio of what EF has to BC; which is proved abfurd; therefore the fphere ABC, to a fphere greater or lefs than DEF, has not a triplicate ratio of what BC has to EF. Therefore ABC has to the fphere DEF a triplicate ratio of what BC has to EF: Which was to be demonftrated.

THE

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