another, each to each, because they are contained by three Book XII. plane angles, equal each to each; and the pyramids are con tained by the same number of similar planes; and are therefore a B. 11. similar to one another, each to each: But similar pyramids b 11.def.11. have to one another the triplicate ratio of their homologous c Cor.8.12. sides. Therefore 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 homologous sides; that is, of that ratio which AB from the centre of the greater sphere has to the straight line from the same centre to the superficies of the less sphere. And, in like manner, each pyramid in the greater sphere has to each of the same order in the less, the triplicate ratio of that which AB has to the semidiameter of the less 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 DEF describe a 17. 12. a solid polyhedron, the superficies of which does not meet the less 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 of that which BC b Cor. 17. 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 12. B c 14. 5. Book XII. sphere DEF: But the sphere ABC is greater than the solid polyhedron in it; therefore also 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 tripli cate 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. NOTES, CRITICAL AND GEOMETRICAL. DEFINITION I. BOOK I. It is necessary to consider a solid, that is, a magnitude which Book I. has length, breadth, and thickness, in order to understand aright the definitions of a point, line and superficies; for these all arise from a solid, and exist in it: The boundary or boundaries which contain a solid are called superficies, or the boundary which is common to two solids which are contiguous, or which divides one solid into two contiguous parts, is called a superficies: Thus, if BCGF be one of the boundaries which contain the solid ABCDEFGH, or which is the common boundary of this solid, and the solid BKLCFNMG, and is therefore in the one as well as in the other solid, called a superficies, and has no thickness: For, if it has any, this thickness must either be a part of the thickness of the solid AG, or of the solid BM, or a part of the thickness of each of them. It cannot be a part of the thickness E of the solid BM; because if this H G M F N C Ꭰ L B K solid AG; because, if this be removed from the solid BM, the superficies BCGF, the boundary of the solid BM does nevertheless remain; therefore, the superficies BCGF has no thickness, but only length and breadth. 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 has any, this must be part either of the breadth of the super Book I. ficies 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 removed from the superficies ABCD, the line BC which is the boundary of the superficies ABCD, remains the same as it was: Nor can the breadth that BC is supposed to have, be a part of the breadth of the superficies ABCD; because if this be removed from the superficies KBCL, the line BC, which is the boundary of the superficies KBCL, does nevertheless remain: Therefore the line BC has no breadth And because the line BC is in a superficies, and a superficies has no thickness, as was shown; therefore a line has neither breadth nor thickness, but only length. The boundary of a line is called a point, or a point is the common boundary or extremity of two lines that are contiguous: E H G M F N C D L or of the line KB. It is not part A. B K line KB be removed from AB, the point B, which is the extremity of the line AB, remains the same as it was: Nor is it part of the length of the line AB; for, if AB be removed from the line KB, the point B which is the extremity of the line KB, does nevertheless remain: therefore the point B has no length And because a point is in a line, and a line has neither breadth nor thickness, therefore a point has no length, breadth nor thickness. And in this manner the definitions of a point, line, and superficies, are to be understood. DEF. VII. B. I. INSTEAD of this definition as it is in the Greek copies, a more distinct one is given from a property of a plane superficies, which is manifestly supposed in the Elements, viz. that a straight line drawn from any point in a plane to any other point in it, is wholly in that plane. DEF. VIII. B. I. It seems that he who made this definition designed that it should comprehend not only a plane angle contained by two straight lines, but likewise the angle which some conceive to Book I. be made by a straight line and a curve, or by two curve lines, which meet one another in a plane: But, though the meaning of the words i' susies, that is, in a straight line, or in the same direction, be plain, when two straight lines are said to be in a straight line, it does not appear what ought to be understood by these words, when a straight line and a curve, or two curve lines, are said to be in the same direction; at least it cannot be explained in this place; which makes it probable that this definition, and that of the angle of a segment, and what is said of the angle of a semicircle, and the angles of segments, in the 16th and 31st Proposition of Book 3, are the additions of some less skilful editor: On which account, especially since they are quite useless, these definitions are distinguished from the rest by inverted double commas. DEF. XVII. B. I. THE words, "which also divides the circle into two equal "parts," are added at the end of this definition in all the copies, but are now left out, as not belonging to the definition, being only a corollary from it. Proclus demonstrates it by conceiving one of the parts into which the diameter divides the circle to be applied to the other; for it is plain that they must coincide, else the straight lines from the centre to the circumference would not be all equal. The same thing is easily deduced from the 31st Proposition of Book 3, and the 24th of the same; from the first of which it follows, that semicircles are similar segments of a circle; and from the other, that they are equal to one another. DEF. XXXIII. B. I. THIS definition has one condition more than is necessary; because every quadrilateral figure which has its opposite sides equal to one another, has likewise its opposite angles equal; and on the contrary. Let ABCD be a quadrilateral figure, of which the opposite sides AB, CD are equal to one another; as also AD and BC: Join BD; A the two sides AD, DB, are equal to the two CB, BD, and the base AB is equal to the base CD; therefore by Prop. 8. of Book 1, the angle B |