14. 5. is greater than the solid polyhedron in it: therefore also Book XII. 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 has to EF: For, if it can, let it have that ratio to a greater sphere LMN: Therefore, by inversion, 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 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. END OF THE ELEMENTS. 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. 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. 1814. NOTES, &c. 289 DEFINITION I. BOOK I. IT is necessary to consider a solid, that is, a magnitude Boox I. which has length, breadth, and thickness, in order to un- H G F N D B M of each of them. It cannot be a The boundary of a superficies is called a line, or a line is the common boundary of two superficies that are contiguous, or which divides one superficies into two contiguous parts: Thus if BC be one of the boundaries which contain the superficies ABCD, or which is the common boundary of this superficies, and of the superficies KBCL which is contiguous to it, this boundary BC is called a line, and has no breadth: For if it have any, this must be part either of the breadth of the superficies ABCD, or of the superficies KBCL, or part of each of them. It is not part of the breadth of the superficies KBCL: for, if this superficies be U |