Book XII. in which these parallels are, and the quadrilateral figure KBOS is in one plane: And if PB, TK be joined, and perpendiculars be drawn from the points P, T to the ftraight lines AB, AK, it may be demonftrated that TP is parallel to KB in the very fame way that SO was fhewn to be parallel to the fame KB; wherefore a TP is parallel to SO, and the quadrilateral figure SOPT is in one plane: For the fame reason, the quadrilateral TPRY is in one plane: And the figure YRX is alfo in one plancb.

II.

b 2. II.

Therefore, if from the points, O, S, P, T, R, Y there be drawn ftraight lines to the point A, there fhall be formed a folid po lyhedron between the circumferences BX, KX compofed of pyramids the bafes of which are the quadrilaterals KBOS, SOPT, TPRY, and the triangle YRX, and of which the com mon vertex is the point A: And if the fame conftruction be made upon each of the fides KL, LM, ME, as has been done upon BK, and the like be done alfo in the other three qua drants, and in the other hemifphere; there fhall be for med a folid polyhedron defcribed in the fphere, compo fed

2

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 alfa Obfervations on fome of the Propofitions.

BY

ROBERT SIMSON, M. D.

Emeritus Profeffor of Mathematics in the University of Glafgow

EDINBURGH:

Printed for J. NOURSE, London; and J. BALFOUR, Edinburgh M,DCC,LXXV.

tio of that which BC has to EF; therefore as the sphere ABC Book XII. to the fphere GHK, fo is the folid polyhedron in the sphere ABC to the folid polyhedron in the fphere DEF: But the sphere ABC is greater than the folid polyhedron in it; therefore al- c 14. 5. fo the fphere GHK is greater than the folid polyhedron in the Sphere DEF: But it is alfo lefs, because it is contained within it, which is impoffible: Therefore the fphere 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 demonftrated, that the fphere DEF has not to any fphere lefs than ABC, the triplicate ratio of that which EF has to BC. Nor can the fphere ABC have to any fphere 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 fphere LMN: Therefore, by inverfion, the fphere LMN has to the fphere ABC, the triplicate ratio of that which the diameter EF has to the diameter BC. But, as the fphere LMN to ABC, fo is the fphere DEF to fome fphere, which must be less than the fphere ABC, because the fphere LMN is greater than the fphere DEE: Therefore the fphere DEF has to a fphere lefs than ABC the triplicate ratio of that which EF has to BC; which was fhewn to be impoffible: Therefore the sphere ABC has not to any fphere 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 lefs then DEF. Therefore the fphere ABC has to the sphere DEF, the triplicate ratio of that which BC has to EF. Q. E, D.

FINIS

T3