of the one polyedron, is similar to the triangle pnm which joins the three homologous vertices of the other.

Again, let Q and q be two homologous vertices; and the triangle PQN will be similar to pqn. We assert farther, that the inclination of the planes PQN, PMN is equal to that of the planes pqn, pmn.

For, joining QM and qm, we shall still have the triangle QNM similar to qnm, and therefore the angle QNM equal to qnm. Imagine a solid angle to be formed at N, by the three plane angles QNM, QNP, PNM; and another solid angle to be formed at n, by the three plane angles qnm, qnp, pmm. Since these plane angles are respectively equal, the solid angles must be equal also. Hence the inclination of the two planes PNQ, PNM is equal to that of their homologous planes pnq, pnm (359.); hence, if the two triangles PNQ, PNM were in the same plane, in which case, the angle QNM would be equal to QNP+PNM, we should likewise have the angle qnm equal to qnp+pnm, and the two triangles qnp, pnm, would also be in the same plane.

All that we have now proved is true, whatever be the value of the angles M, N, P, Q, compared with their corresponding ones m, n, p, q•

Let us next suppose the surface of one of the polyedrons to be divided into triangles ABC, ACD, MNP, NPQ, &c.; the surface of the other polyedrons will evidently contain an equal number of triangles abc, acd, mnp, npq, &c., similar and similarly placed; and if several triangles, as MNP, NPQ, &c. belong to one face, and lie in the same plane, their corresponding triangles mnp, npq, &c. will likewise lie in one plane. Hence every polygonal face in the one polyedron will correspond to a similar polygonal face in the other polyedron; hence the two polyedrons will be terminated by the same number of planes similar, and similarly placed. We assert farther, that their homologous solid angles will be equal.

For if the solid angle N, for example, is formed by the plane angles QNP, PNM, MNR, QNR, the corresponding solid angle n will be formed by the plane angles qnp, pnm, mnr, qnr. But these plane angles are equal each to each; and the inclination of any two adjacent planes is equal to that of the two which correspond to them: hence the two solid angles are equal, because they would exactly coincide.

Hence, finally, two similar polyedrons have their homologous faces similar and their homologous solid angles equal.

432. Cor. From the precoding demonstration, it follows, that if a triangular pyramid were formed with four vertices of

one polyedron, and a second with the four corresponding vertices of a similar polyedron, these two pyramids (430.) would be similar, because their homologous edges would be proportional.

It is also evident that two homologous diagonals (157.), as AN, an, are to each other as two homologous sides AB, ab.

PROBLEM.

433. Two similar polyedrons may be divided into the same number of triangular pyramids, similar each to each, and similarly placed.

For, it has already been shewn, that the surfaces of two polyedrons may be divided into the same number of triangles similar each to each, and similarly placed. Consider all the triangles of the one polyedron, except those which form the solid angle A, as the bases of so many triangular pyramids having their common vertex in A; those pyramids taken together will compose the whole polyedron. Divide the other polyedron, in the same manner, into pyramids having for their common vertex the angle a, homologous to A: the pyramid which joins four vertices of the one polyedron, will evidently be similar to the pyramid which joins the four homologous vertices of the other polyedron. Hence two similar polyedrons, &c.

THEOREM.

434. Two similar pyramids are to each other as the cubes of their homologous sides.

For two pyramids being similar, the smaller may be placed within the greater, so that the solid angle S shall be common to both. In that position, the bases ABCDE, abcde will be parallel; because, since (423.) the homologous faces are similar, the angle Sab is equal to SAB, and Sbc to SBC; hence (344.) the plane ABC is parallel to the plane abc. This granted, let SO be the perpendi- A cular drawn from the vertex S to the plane ABC, and o the point where this perpendi

E

Ο

C

B

cular meets the plane abc: from what has already been shewn (406.), we shall have SO : So :: SA : Sa :: AB : ab; and consequently,

SOS0: AB: ab.

But the bases ABCDE, abcde being similar figures, we have ABCDE: abcde: AB2: ab2. (221.)

Multiply the corresponding terms of these two proportions; there results the proportion,

ABCDEX SO : abcde×3So :: AB3 : ab3.

Now ABCDE×SO is the solidity of the pyramid SABCDE and abcdex So is that of the pyramid Sabcde (413.); hence two similar pyramids are to each other as the cubes of their homologous sides.

THEOREM.

435. Two similar polyedrons are to each other as the cubes of their homologous sides.

logous sides AB, ab. The same ratio exists between every other pair of homologous pyramids hence the sum of all the pyramids which compose the one polyedron, or that polyedron itself, is to the other polyedron, as the cube of any one side in the first is to the cube of the homologous side in the second.

General Scholium.

436. The chief propositions of this Book relating to the solidity of polyedrons, may be exhibited in algebraical terms, and so recapitulated in the briefest manner possible.

Let B represent the base of a prism; Ĥ its altitude: the solidity of the prism will be BXH, or BH.

Let B represent the base of a pyramid; H its altitude; the solidity of the pyramid will be B×3H, or H×B, or BH.

Let H represent the altitude of the frustum of a pyramid, having parallel bases A and B; VAB will be the mean proportional between those bases; and the solidity of the frustum will be H×(A+B+√AB).

Let B represent the base of the frustum of a triangular prism; H, H', H" the altitudes of its three upper vertices: the solidity of the truncated prism will be B×(H+H+H′′).

In fine, let P and p represent the solidities of two similar polyedrons; A and a two homologous edges or diagonals of these polyedrons: we shall have P : p :: A3 : a3.

BOOK VII.

THE SPHERE.

Definitions.

437. The sphere is a solid terminated by a curve surface, all the points of which are equally distant from a point within, called the centre.

438. The radius of a sphere is a straight line drawn from the centre to any point of the surface; the diameter or axis is a line passing through this centre, and terminated on both sides by the surface.⚫

All the radii of a sphere are equal; all the diameters are equal, and each double of the radius.

439. It will be shewn (452.) that every section of the sphere, made by a plane, is a circle: this granted, a great circle is a section which passes through the centre; a small circle one which does not pass through the centre.

440. A plane is tangent to a sphere, when their surfaces have but one point in common.

441. The pole of a circle of a sphere is a point in the surface equally distant from all the points in the circumference of this circle. It will be shown (464.) that every circle, great or small, has always two poles.

442. A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.

Those arcs, named the sides of the triangle, are always supposed to be each less than a semicircumference. The angles, which their planes form with each other, are the angles of the triangle.

443. A spherical triangle takes the name of right-angled, isosecles, equilateral, in the same cases as a rectilineal triangle.

444. A spherical polygon is a portion of the surface of a sphere terminated by several arcs of great circles.

445. A lune is that portion of the surface of a sphere, which is included between two great semicircles meeting in a common diameter.

446. A spherical wedge or ungula is that portion of the solid sphere which is included between the same great semicircles, and has the lune for its base.

447. A spherical pyramid is a portion of the solid sphere, included between the planes of a solid angle whose vertex is the centre. The base of the pyramid is the spherical polygon intercepted by the same planes.

448. A zone is the portion of the surface of the sphere, included between two parallel planes, which form its bases. One of those planes may be tangent to the sphere; in which case the zone has only a single base.

449. A spherical segment is the portion of the solid sphere, included between two parallel planes which form its bases.