388 mids are given, having equal lJi1SlS and equal altitudes, aml if each be divided as above, then the two triangular prisms in the one are equal to those in the other, and each of the remaining pyramids in the one has its base and altitude equal to the base and altitude of the remaining pyramids in the other. Hence to these pyramids the same process is again applicable. We are. thus enabled to cut out of the two given pyramids equal parts, each greater than half the original pyramid. Of the remainder vc can; again cut out equal parts greater than half these remainders, and so on as far as we like. This process may be continued till the last remainder is smaller than any assignable quantity, however small. It follows, so we should conclude at present, that the two volumes must be. equal, for they cannot differ by any assignable quantity. To Greek mathematicians this conclusion offers far greater diIli- eulties. They prove elaborately, by a rcductio ad absurdum, that the volumes cannot be unequal. This proof must be read in the Elements. A further discussion of this method of exhaustion, as it is called, would belong to a treatise on the history of geometry. We refer readers to ll-ankel, Gcschichte dcr Jlathcmatik (p. 115 sq). We must, however, state that we have in the above not proved l~'.nelid's Prop. 5, but only a special case of it. Euclid does not suppose that the bases of the two pymmids to be compared are equal, and hence he proves that the volumes are as the bases. The reasoning of the proof becomes clearer in the special case, from which the general one may be easily deduced. § 56. Prop. 6 extends the result to pyramids with polygonal bases. From these results follow again the rules at present given for the mensuratiou of solids, viz., a pyramid is the third part of a triangular prism having the same base and the same altitude. But a triangular prism is equal in volume to a parallelepiped which has the same base and altitude. Hence if B is the base and h the altitude, we have Volume of prism = Bh, Volume of pyramid = -13- Bh, statements which have to be taken in the sense that B means the number of square units in the base, h the number of units of length in the altitude, or that B and 1; denote the numerical values of base and altitude. § 87. A method similar to that used in proving Prop. 5 leads to the following results relating to solids bounded by simple curved surfaces :— Prop. 10. Eccry cone is the third part of a cylinder which has the .5-imc base, and is of an equal altitude with it. Prop. 11. Cones and cylinders of the same altitude arc to one an- other as their bases. Prop. 12. Similar cones and cylinders lmrc to one another the triplicate ratio of that which the diameters of their bases hm-c. Prop. 13. If a cylinder bc cut by a plane parallel to its opposite planes or bases, it dicidcs the cylinder into two cylinders, one of 1(‘]l.l(.'h is to the other as the axis of the first to the axis of thc othcr; which m:i_v also be stated tlms:— Cylinders on the same base arc proportional to their altitudes. Prop. 14. Cones and cyli-ndcrs upon equal bases are to one another as their altitudes. Prop. 15. The buses and altitudes of equal cones and cylinders are reciprocally proportional, and if the bases and altitudes be rcciprocally proportional, the cones and cylinders are cqu-at to one another. These theorems again lead to formulae in mcnsuration, if we com- pare a cylinder with a prism having its base and altitude equal to the base and altitude of the Cylinder. This may be done by the method of exhaustion. We get, then, the result that their bases are equal, and have, if B denotes the numerical value of the base, and Ii that of the altitude, Volume of cylinder Bh. ,1 Bh. .5 Volume of conc = §88. The remaining propositions relate to circles and spheres. Of the sphere only one property is proved, viz. :— Prop. 18. Spheres have to one another the triplicate ratio qf that which their diameters have. The mensuration of the sphere, like that of the circle, the cylinder, and the cone, had not been settled in the time of Euclid. It was done by Archimedes. BOOK XIII. § 89. The 13th and last book of Euclid's Elements is devoted to the regular solids. It is shown that there are five of them, viz. :— 1. The regular tetrahedron, with 4 triangular faces and 4 vertiees; 2. The cube, with 8 vertiees and 6 square faces ; The octahedron, with 6 vertiees and 8 triangular faces; 4. The dodccahcdron, with 12 pentagonal faces, 3 at each of the ‘Z0 vertices; 5. The icosahcdron, with '20 triangular faces, 5 12 vertiees. at each of the G E O M 1*} T R Y [rr.o.11scT1vE. It is shown ho' to inscribe these solids in a given sphere, and hov to determine the lengths of their edges. These results are—if 1- denotes the radius of the circumscribed sphere, and a the side of the regular solid- ‘2 For tetrahedron (I n ,, octahedron (r = ,, ln-_‘-ahedron or cube u'-' = — 1"‘, U . .. F’; ,, icosahedron a-= 2 1 — __‘ r‘-, 5 /7' ,, (lodeeahedron a‘-' == :2(I ~ I‘ )1"-’. ‘J § 90. The 13th book, and therefore the Elements. conclude with the scholium, “that no other regular solid e.ists lusides the tire oncs enumerated." The proof is very simple. Each face is a regular polygon. hence the angles of the faces at any vertex must be angles in equal regular polygons, must be together less than four right angles (X1. '.31, and must be three or more in number. Each angle in a regular triangle equals two-thirds of one right angle. Hence it is possible to form a solid angle with three, four, or five regular triangles or faces. These give the solid angles of the tetrahedron, the octohedron, and the icosahedron. The angle in a square (the regular quadrilateral) equals one right angle. llcm-e three. will form a solid angle, that of the cube, aml four will not. The angle in the regular pentagon equals of a right angle. llence three of them equal if (i.c.. less than 4) right angles, and form the solid angle of the dodeeahcdron. Three regular polygons of six or more sides cannot foriu a solid angle. Therefore no other regular solids are possible. Secriox II.—II 1011131: on l’no.1Ecr1vJ-_' (:1-;oM1~;Tnr. It is difficult, at the outset, to characterize l’1-ojective Geometry as compared with Euclidian. lint a few c.;amples will at least indicate the diIl'erence between the two. In lSuclid's Elements almost all propositions refer to the 7m_(yu[twlc of lines, angles, areas, or volumes, and therefore to measurement. The statement that an angle is right-, or that two straight lines are parallel, refers to measurement. On the other hand, the fact that a straight line does or does not cut a circle is independent of measure- ment, it being dependent only upon the mutual “ posi- tion” of the line and the circle. This difference becomes clearer if we project any figure from one plane to another. liy this the length of lines, the magnitude of angles and areas, is altered, so that the projection, or shadow, of a square on a plane will not be a square ; it will, however, be some quadrilateral. Again, the projection of a circle will not be a circle, but some other curve more or less resembling a circle. llut one property may he stated at once,—no straight line can cut the projection of a circle in more than two points, because no straight line can cut a circle in more than two points. There are, then, some properties of figures which do not alter by projection, whilst others do. To the latter belong nearly all properties relating to measurement, at least in the form in which they are generally given. The others are said to be projective properties, and their investigation forms the subject of Projective Geometry. Different. as are the kinds of properties investigated in the old and the new sciences, the methods followed differ in a still greater degree. In "l-Euclid each pro- position stands by itself; its connexion with others is never indicated; the leading ideas contained in its proof are not stated; general principles do not exist. In the modern methods, on the other hand, the greatest import- ance is attached to the leading thoughts which pervade the whole; and general principles, which bring whole groups of theorems under one aspect, are given rather than separate propositions. The whole tendency is towards generalization. A straight line is considered as given in
its entirety, extending both ways to infinity, while Euclid