But the ftraight line AZ may be demonftrated to be greater Book XII. than AG otherwife, and in a shorter manner, without the help of Prop. 16. as follows. From the point G draw GU at right angles to AG and join AU. If then the circumference BE be bifected, and its half again bifected, and so on, there will at length be left a circumference less than the circumference which is fubtended by a straight line equal to GU infcribed in the circle BCDE: Let this be the circumference KB: Therefore the straight line KB is less than GU: And because the angle BZK is obtufe, as was proved in the preceding, therefore BK is greater than BZ: But GU is greater than BK; much more then is GU greater than BZ, and the square of GU than the square of BZ; and AU is equal to AB; therefore the square of AU, that is, the fquares of AG, GU are equal to the fquare of AB, that is, to the fquares of AZ, ZB; but the square of BZ is lefs than the fquare of GU; therefore the square of AZ is greater than the square of AG, and the straight line AZ confequently greater than the ftraight line AG.

COR. And if in the leffer sphere there be described a folid polyhedron by drawing ftraight lines betwixt the points in. which the straight lines from the centre of the sphere drawn to all the angles of the folid polyhedron in the greater sphere meet the fuperficies of the leffer; in the fame order in which are joined the points in which the fame lines from the centre meet the fuperficies of the greater sphere; the folid polyhe dron in the sphere BCDE has to this other solid polyhedron the triplicate ratio of that which the diameter of the sphere BCDE has to the diameter of the other sphere: For if these two folids be divided into the fame number of pyramids, and in the fame order; the pyramids fhall be fimilar to one another, each to each: Because they have the folid angles at their common vertex, the centre of the fphere, the fame in each pyramid, and their other folid angle at the bafes equal to one another, each to each a, because they are contained by three a B. 11. plane angles equal each to each; and the pyramids are contained by the fame number of similar planes; and are therefore fimilar b b 11. Def. to one another, each to each: But fimilar pyramids have to one another the triplicate ratio of their homologous fides. e Gor. 8, Therefore the pyramid of which the bafe is the quadrilateralKBOS, and vertex A, has to the pyramid in the other sphere of the fame order, the triplicate ratio of their homologous fides; that is, of that ratio, which AB from the centre of the greater sphere has to the straight line from the fame centre to

11.

12.

Book XII the fuperficies of the leffer sphere. And in like manner, each pyramid in the greater iphere has to each of the fame order in the leffer, the triplicate ratio of that which AB has to the femidiameter of the leffer fphere. And as one antecedent is to its confequent, fo are all the antecedents to all the confequents. Wherefore the whole folid polyhedron in the greater fphere has to the whole folid polyhedron in the other, the triplicate ratio of that which AB the femidiameter of the first has to the semidiameter of the other; that is, which the diameter BD of the greater has to the diameter of the other sphere.

PROP. XVIII. THEOR.

PHERES have to one another the triplicate ratio of that which their diameters have.

Let ABC, DEF be two fpheres of which the diameters are BC, EF. The fphere ABC has to the sphere DEF the triplicate ratio of that which BC has to EF.

For, if it has not, the sphere ABC shall have to a sphere either lefs or greater than DEF, the triplicate ratio of that which BC has to EF. First, let it have that rátio to a lefs, viz. to the fphere GHK; and let the sphere DEF have the fame

a 17. 12. centre with GHK; and in the greater sphere DEF defcribe a

a folid polyl edron, the fuperficies of which does not meet the leffer fphere GHK and in the sphere ABC describe another fimilar to that in the fphere DEF: Therefore the folid polyhedron in the 1phere ABC has to the folid polyhedron in the Cor. 17. fphere DEF, the trip'icate ratio b of that which BC has to EF. But the fphere ABC has to the sphere GHK, the triplicate ra

I 2.

tio

N

tio of that which BC has to EF; therefore, as the sphere ABC Book XII. to the sphere GHK, fo is the faid polyhedron in the sphere ABC to the folid polyhedron in the sphere DEF: But the sphere ABC is greater than the folid polyhedron in it; therefore c al- c 14. 5. fo the sphere GHK is greater than the folid polyhedron in the sphere DEF: But it is also lefs, because it is contained within it, which is impoffible: Therefore the sphere ABC has not to any fphere lefs than DEF, the triplicate ratio of that which BC has to EF. In the fame manner, it may be demonstrated, that the sphere DEF has not to any sphere lefs than ABC, the triplicate ratio of that which EF has to BC. Nor can the sphere ABC have to any sphere greater than DEF, the triplicate ratio of that which BC has to EF: For, if it can, let it have that ratio to a greater sphere LMN: Therefore, by inverfion, the sphere LMN has to the fphere ABC, the triplicate ratio of that which the diameter EF has to the diameter BC. But as the sphere LMN to ABC, fo is the fphere DEF to fome sphere, which must be less than the fphere ABC, because the fphere LMN is greater than the sphere DEF: Therefore the fphere DEF has to a sphere lefs than ABC the triplicate ratio of that which EF has to BC; which was fhewn to be impoffible: Therefore the fphere ABC has not to any sphere greater than DEF the triplicate ratio of that which BC has to EF: And it was demonftrated, that neither has it that ratio to any sphere less than DEF. Therefore the sphere ABC has to the fphere DEF, the triplicate ratio of that which BC has to EF. Q. E. D.

FINI S.

T3

NOTE S

CRITICAL AND GEOMETRICAL

CONTAINING

An Account of those things in which this Edition differs from the Greek text; and the Reasons of the Alterations which have been made. As alfo Obfervations on fome of the Propofitions.

BY ROBERT SIMSON, M. D.
Emeritus Profeffor of Mathematics in the University of Glasgow.

EDINBURGH:

Printed for F. WINGRAVE, London; and E. BALFOUR, Edinburgh.

M,DCC, XCIV.