no doubt true; but it has the inconvenience of containing several superfluous conditions. If the condition of the solid angles being equal were suppressed, we should again fall into the statement of Euclid, which is defective, because it presupposes the demonstration of a theorem concerning equal polyedrons. To avoid all embarrassment, we thought it best to divide the definition of similar triangular pyramids into two parts: first, we gave the definition of similar triangular pyramids, then we defined similar solids as those which have similar bases, and whose homologous vertices out of those bases are determined by triangular pyramids, similar each to each.

This definition requires two conditions for the bases, supposing them to be triangular; and three conditions for each vertex out of the base; so that, if S represent the number of solid angles in each of the two polyedrons, the similarity of these polyedrons will require 2+3 (S-3) equal angles on both sides, or 3 S7 conditions; and no one of these For, in this deficonditions is superfluous or contained in the others. nition, we regard two polyedrons as having simply the same number of vertices or solid angles; in which case, all the 3 S-7 conditions without one exception are required to make the two solids similar: but if we suppose them before-hand to be of the same species, in other words, to have the same number of faces, these faces compared with each other having also the same number of sides; then such a supposition, in the case where some faces have more than three sides, would include some conditions, and these would diminish the number 3 S7 to the same extent; so that in place of 3 S-7 conditions, we should then require only A-1; (upon which see Note VIII). We now perceive what gives rise to the difficulty of settling a good definition of similar solids; it is that they may either be regarded as being of the same species, or only as having an equal number of solid angles. In the latter case, the whole difficulty vanishes; the 3 S-7 conditions implied in the definition must all be fulfilled before the solids can be similar; and from their similarity we conclude, a fortiori, that they are of the same species. As our definition was perfectly complete, we have deduced Simson's from it as a theorem.

It appears then, that in the Elements we might altogether omit the theorem concerning the equality of polyedrons; but as this is interesting of itself, the reader will not be displeased to find a demonstration of it here, and thus to have the theory of polyedrons complete in all respects.*

The question is, to know whether by varying the inclinations of the planes which compose the surface of a given convex polyedron, a second convex polyedron may be formed that shall be included under the same polygonal planes combined with each other in the same order.

We may observe, in the first place, that if there be a second polyedron satisfying the conditions of the problem, it cannot be the polyedron

• The demonstration here given, with the exception of some developements, is the same as that which M. Cauchy lately communicated to the Institute. He has founded it on some ideas, which had been proposed for the same purpose, in the first edition of these Elements, page 327 et seq.

which is symmetrical to the given one, because in these two polyedrons the equal polyedrons are combined in an inverse order about the corresponding solid angles. Hence the consideration of symmetrical polyedrons must be entirely kept out of view in the object we are engaged with.

*

We may observe, in the second place, that if the given polyedron contains one or more triple solid angles, these angles are in their nature invariable, the knowledge of three plane angles being sufficient to determine the mutual inclination of their planes when joined into a solid angle. In the proposed solid, we are therefore at liberty to suppress all triangular pyramids which form the triple solid angles; and if the new polyedron resulting from this suppression has still any triple solid angles, they two may be suppressed, and so on till we arrive at a polyedron every one of whose solid angles combines not less than four plane angles. In fact, if the proposed solid may alter its figure by any change in the inclinations of its planes, this alteration cannot take place in the triangular pyramids which are cut off; it must be wholly confined to the polyedron what remains after the suppression of all the triangular pyramids. In what follows, therefore, we shall only contemplate such polyedrons as have solid angles uniting at least four plane angles in each.

This being fixed, let S (see the next figure) be any solid angle in the polyedron; and from the vertex S as a centre, let a spherical surface be described whose intersections with the planes of the solid angle may form the spherical polygon ABCDEF. The sides of this polygon, AB, BC, &c. serve as measures to the plane angles ASB, BSC, &c., and are consequently invariable: as for the angles A, B, C, &c. of the polygon, each of them measures the inclination of two adjacent planes in the solid angle; thus the angle B measures the inclination of the planes ASB, BSC, which inclination, for the sake of brevity, we shall style the inclination to the edge SB; in like manner, the angle C measures the inclination to the edge SC; and so on.

We shall now be able to judge respecting the changes of figure in each solid angle S, by the changes of the spherical polygon ABCDEF, whose sides are constant, and whose angles may vary in any way provided the polygon do not cease to be convex. Now, in such polygons, the signs of variations in the angles present some remarkable laws, which we shall explain in the following Lemmas.

LEMMA I.

All the sides of a spherical polygon AB, BC, CD, DE, with the exception of one AF, being given, if any one of the angles B, C, D, E, opposite the side AF, be made to vary, all the rest remaining constant, then will the side AF augment as the angle augments, and diminish as

* If the same edge were common to two triple solid angles, we could only, in the first operation, suppress one of these angles.

it diminishes. In all cases the polygon is supposed to be convex both before and after its change of figure.

Suppose, first, that the angle B is made to vary, the other three C, D, E remaining constant; if BF is joined, the figure BCDEF will undergo no change, and BF will be constant. Hence we shall have a spherical triangle ABF, whose sides AB, BF are constant, while its angle ABF varies by the same quantity as the angle ABC of the polygon, since the part FBC remains constant. Now, by properties which are admitted, we know that the side AF will increase if the angle ABF increases, and diminish if the angle ABF diminishes.

B

ཝ

Next, suppose the angle C varies, the other three B, D, E being constant; if the diagonals AC, FC are drawn, these diagonals will evidently remain constant, and likewise the angles ACB, FCD: hence we shall again have a spherical triangle ACF, whose sides AC, CF are constant, while its angle ACF varies by the same quantity as the angle C of the polygon; whence we shall, in like manner, infer that the side AF will augment if the angle C augments, and diminish if that angle diminishes.

The same reasoning would evidently apply to the variation of either of the angles D and E; it would likewise evidently hold good for any other spherical polygon having more than three sides. Hence, in all cases, our conclusion will agree with the enunciation of the Proposition, still however supposing, that before as well as after its change of figure, the polygon is convex. This restriction is necessary; for if the angle E, for example, should diminish till the point F fell on the diagonal AE, then AF would be a minimum; and if reckoning from this point, the angle E continued to diminish, the side AF would evidently augment instead of diminishing; but in this latter case, the angle AFE would become re-entrant, and the polygon would cease to be convex,

Cor. The same things being admitted, if several of the angles opposite the undetermined side AF augment, and none of them diminish, the side AF will of necessity augment by means of all the joint variations. The contrary effect will take place, if several of the angles opposite the side AF diminish and none of them augment.

For, if by reason of the simultaneous increase or diminution, the angles A, B, C, &c., of the polygon are to be changed into A', B', C', &c., we may pass successively from the proposed polygon to that which contains only one varied angle A'; from this latter, to the polygon which contains only two varied angles A' and B'; and so on. Now, in each of these transitions, the application of the Theorem above demonstrated is manifest, and always leads to the same conclusion.

* This proposition is demonstrated in the same manner as Prop. 10. Book I. for rectilineal triangles.

LEMMA II.

Given a spherical polygon having more than three sides, and all of them constant, if the angles are made to vary in any way, so the polygon do not cease to be convex; if, farther, the sign + is put at the vertex of each angle that augments, the sign at the vertex of each angle that diminishes, no sign whatever being put at the angles that remain constant; then I assert that, in going quite round the polygon, we shall find at least four changes of sign from one vertex to the next

vertex.

First. If n is the number of angles in the polygon, there cannot be 2 consecutive angles, all of which increase at once, or of which some increase and the others remain constant; for if either of these results took place, it would follow by the Corollary of the preceding Lemma, that the side of the polygon, lying opposite to those n 2 angles, would increase; which contradicts the hypothesis that all the sides are constant. For a like reason, there cannot be n- -2 consecutive angles, all of which diminish at once, or of which some diminish, the others remaining constant. Hence, in the series of ncutive angles, there must occur at least one change of sign; and a fortiori, must this change be observed in the series of n consecutive angles, in going quite round the polygon.

2 conse

Secondly. The variations in the angles of the polygon cannot be such as to offer only one series of + signs, and one series of signs, thus offering only two changes of sign in the whole circuit of the polygon.

F

E

B

For, let A, B, C be the three angles marked with the sign+, and D, E, F, G the angles, marked with the sign; an hypothesis which includes all such as refer to a less number of signs in each series, when some of the angles are invariable. If the figure represents the initial state of the polygon, the diagonal GD must increase when the three angles A, B, C, or only some of them increase; but the same diagonal GD, as belonging to the polygon GFED, of which the other sides are constant, must diminish along with the angles F and E, or at least remain constant, if of the four angles D, E, F, G, there be only D and G, or merely one of them which diminishes: hence the hypothesis in question cannot be correct; hence the variation of the angles cannot be such as to offer only two series, one of signs +, another of signs

-;

Thirdly. It is farther impossible that in going round the polygon we should not find three alternate series of signs +, and of signsfor on this hypothesis the first and the third series would be of the same sign, and following each other immediately, would form only a single series; from which it appears, that in going round the polygon there would in reality be only two series, the one of signs +, the other of signs; which we have already shewn to be impossible.

Hence, finally, the changes of sign to be found in going quite round the polygon will at least amount to four.

S

Cor. What we have just proved with regard to spherical polygons is immediately applicable to the solid angles, of which these polygons are the measure. Thus in a convex solid, an angle being given which unites more than three plane angles, if the inclinations to the edges are made to vary in any way, provided the solid angle do not cease to be convex; if farther the sign + or the sign — is put on each edge according as the inclination to it augments or diminishes, no sign whatever being marked upon the edges whereof the inclination continues constant, I assert that in going quite round the solid angle, we shall find at least four changes of sign from one edge to the next.

By means of this Proposition, and of Euler's theorem concerning polyedrons (Prop. 25. VII.) we are now in a condition to demonstrate the following theorem under its most general form.

THEOREM.

Given a convex polyedron, each of whose solid angles unites more than three plane angles, it is impossible to vary the inclinations of the planes in this solid, so as to produce a second polyedron formed with the same planes arranged with each other in the same order as in the given polyedron.

In demonstrating this Proposition, two cases must be distinguished, according as the inclination to all the edges is made to vary, or only to some of them.

First Case.

Suppose the inclinations on all the edges are made to vary at once; and let N be the total number of changes of sign that can be found from one edge to another, in going round each solid angle.

In Lemma II. we saw, that for each solid angle, the number of changes of sign cannot be less than four.

Hence if we call S the number of solid angles, we shall have N➖ 4 S, the sign not excluding equality.

Now it is plain that two consecutive edges of a solid angle always belong to one face in a polyedron, and belong only to one; hence the total number of changes of sign observed in the consecutive edges of each solid angle, must be equal to the total number of changes of sign observed in the consecutive sides of each face; for there is no change of sign in the one system, which does not correspond to a similar change in the other.

Now for each triangular face, the number of changes of sign cannot be greater than two; since by making the series + + or the series +return into itself, we obtain only two changes of sign. For each quadrangular face, the number of changes of sign is evidently four at most.

In general, if the number of sides belonging to a face is even 2 n, the greatest number of changes of sign that can be found in going round the sides is 2 n, which will occur when the sides have the signs + and alternately.

But if the number of sides belonging to a face is odd=2n+1, the greatest number of changes of sign will be 2 n only; because by