: 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 straight lines AB, AK, it may be demonstrated that TP is parallel to KB in the very same way that SO was shewn 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 same reason, the quadrilateral TPRY is in one plane: And the figure YRX is also in one planeb. a 9. 11. ba. 11. Therefore, if from the points, O, S, P, T, R, Y there be drawn straight lines to the point A, there shall be formed a folid po. lyhedron between the circumferences BX, KX compofed of pyramids the bases of which are the quadrilaterals KBOS, SOPT, TPRY, and the triangle YRX, and of which the common vertex is the point A: And if the fame construction be made upon each of the fides KL, LM, ME, as has been done upon BK, and the like be done also in the other three quadrants, and in the other hemisphere; there shall be formed a folid polyhedron described in the sphere, compo. fed t 2 CRITICAL AND GEOMETRICAL; CONTAINING An account of those Things in which this Edition differs from the Greek Text; and the Reafons of the Alterations which have been made. As also Observations on fome of the Propofitions. BY ROBERT SIMSON, M. D. Emeritus Professor of Mathematics in the University of Glafgow 1 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 sphere GHK, so is the folid polyhedron in the sphere ABC to the folid polyhedron in the sphere DEF: But the sphere ABC is greater than the solid polyhedron in it; therefore & al-c 14. 5. fo the sphere GHK is greater than the folid polyhedron in the sphere DEF: But it is also less, because it is contained within it, which is impoffible: 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 demonftrated, 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 fphere 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 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 fphere, which must be less than the sphere ABC, because the sphere LMN is greater than the sphere DEE: Therefore the sphere DEF has to a sphere less than ABC the triplicate ratio of that which EF has to BC; which was shewn to be impoffible: 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 then DEF. Therefore the sphere ABC has to the sphere DEF, the triplicate ratio of that which BC has to EF. Q. E. D. FINIS T3 |