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• But the straight line AZ may be demonstrated to be Book XII." greater than AG otherwise, 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 bisected, and its half again bisected, and so on, there will at length be left a circumference less than the circumference which is subtended by a straight line equal to GU, inscribed in the circle BCDÉ: Let this be the circumference KB: Therefore the straight line KB is less than GU: And because the angle BZK is obtuse, 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 squares of AG, GU, are equal to the square of AB, that is, to the squares of AZ, ZB: but the square of BZ is less than the square of GU; therefore the square of AZ is greater than the square of AG, and the straight line AZ consequently greater than the straight line AG.

Cor. And if in the lesser sphere there be described a solid polyhedron, by drawing straight lines betwixt the points in which the straight lines from the centre of the sphere drawn to all the angles of the solid polyhedron in the greater sphere meet the superficies of the lesser; in the same order in which are joined the points in which the same lines from the centre meet the superficies of the greater sphere; the solid polyhedron 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 solids be divided into the same number of pyramids, and in the same order, the pyramids shall be similar to one another, each to each: Because they have the solid angles at their common vertex, the centre of the sphere, the same in each pyramid, and their other solid angle at the bases equal to one another, each to eacha, be- · B. 11. cause they are contained by three plane angles, each equal to each; and the pyramids are contained by the same number of similar planes; and are therefore similarb to one an-b11 Defill. other, each to each: But similar pyramids have to one another the triplicate ratio of their homologous sides. There--Cor. 8. 12 fore the pyramid of which the base is the quadrilateral KBOS, and vertex A, has to the pyramid in the other sphere of the same order, the triplicate ratio of their hom mologous sides, that is, of that ratio which AB from the centre of the greater sphere has to the straight line from

Book XII. the same centre to the superficies of the lesser sphere.

And in like manner, each pyramid in the greater sphere has to cach of the same order in the lesser, the triplicate ratio of that which AB has to the semidiameter of the lesser sphere. And as one antecedent is to its consequent, so are all the antecedents to all the consequents. Wherefore the whole solid polyhedron in the greater sphere has to the whole solid polyhedron in the other, the triplicate ratio of that which AB the semidiameter 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. SPHERES have to one another the triplicate ratio of that which their diameters have.

Let ABC, DEF, be two spheres, of which the diameters are BC, EF. The sphere 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 less or greater than DEF, the triplicate ratio of that which BC has to EF. First, let it have that ratio to a less, viz, to the sphere GHK; and let the sphere DEF have

the same centre with GHK: and in the greater sphere • 17.12. DEF describea a solid polyhedron, the superäcies of which

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Cor. 17.
12.

does not meet the lesser sphere GHK; and in the sphere ABC describe another similar to that in the sphere DEF: Therefore the solid polyhedron in the sphere ABC has to the solid polyhedron in the sphere DEF, the triplicate ratio b of that which BC has to EF. But the sphere ABC has to the sphere GHK, the triplicate ratio of that which BC has to EF; therefore as the sphere ABC to the sphere GHK, so is the said polyhedron in the sphere ABC to the solid polyhedron in the sphere DEF: But the sphere ABC is greater than the solid polyhedron in it: therefore also Book XIT, the sphere GHK is greater than the solid polyhedron in the sphere DEF: But it is also less, because it is contained within it, which is impossible: Therefore the sphere ABC has not to any sphere less than DEF, the triplicate ratio of that which BC has to EF. In the same manner, it may be demonstrated, that the sphere DEF has not to any sphere less 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 bas to EF: For, if it can, let it have that ratio to a greater sphere LMN: Therefore, by inversion, the splaere LMN has to the sphere ABC, the triplicate ratio of that which the diameter EF has to the diameter BC. But as the sphere LMN to ABC, so is the sphere DEF to some sphere, which must be less than the sphere ABC, because the sphere LMN is greater than the sphere DEF, therefore the sphere DEF has to a sphere less than ABC the triplicate ratio of that which EF has to BC; which was shown to be impossible: Therefore the sphere ABC has not to any sphere greater than DEF the triplicate ratio of that which BC has to EF: and it was demonstrated, that neither has it that ratio to any sphere less than DEF. Therefore the sphere ABC has to the sphere DEF, the triplicate ratio of that which BC has to EF. Q. E. D.

c 14. 5.

END OF THE ELEMENTS.

re

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 also Observations on some of the Propositions.

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By ROBERT SIMSON, M. D. Emeritus Professor of Mathematics in the University of

Glasgow.

LONDON:

Printed for F. WINGRAVE, and the rest of the Proprietors.

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