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which represent these lines, they being estimated according to a linear unit taken at pleasure. The sense of this word being thus fixed, there is no difficulty in making use of it. The same is to be understood of the product of a surface by a line, of a surface by a solid, &c. It is sufficient to have established once for all that these products are or ought to be considered as the products of numbers, each of a kind that is adapted to it. Thus the product of a surface by a solid is nothing else than the product of a number of superficial units by a number of solid units.

We often use the word angle in common discourse to designate the point situated at its vertex; this expression is faulty. It would be more clear and more exact to denote by a particular name, as that of vertices, the points situated at the vertices of the angles of a polygon, or of a polyedron. In this sense is to be understood the expression vertices of a polyedron, which we have used.

We have followed the common definition of similar rectilineal figures; but we would observe, that it contains three superfluous conditions. For, in order to construct a polygon of which the number of sides is n, it is necessary in the first place to know a side, and then to have the position of the vertices of the angles situated without this side. Now the number of these angles is n 2, and the position of each vertex requires two data; whence it follows that the whole number of data necessary to construct a polygon of n sides is 1+ 2 n — 4, 3. But in the similar polygon there is one side to be taken at pleasure; thus the number of conditions, by which one polygon becomes similar to a given polygon, is 2 n 4. But the common definition requires, 1. that the angles should be equal, each to each, which makes n conditions; 2. that the homologous sides should be proportional, which makes n 1 conditions. There are then in all 2n-1 conditions, or three too many. In order to obviate this inconvenience we can resolve the definition into two others, in this

or 2 n

manner.

1. Two triangles are similar, when they have two angles equal, each to each.

2. Two polygons are similar, when there can be formed in the one and the other the same number of triangles similar, cach to each, and similarly disposed.

But, in order that this last definition should not itself contain superflous conditions, it is necessary that the number of triangles should be equal to the number of sides of the polygon minus two, which may take place in two ways. We can draw from two homologous angles diagonals to the opposite angles; then all the triangles formed in

each polygon will have a common vertex, and their sum will be equal to the polygon; or rather we can suppose that all the triangles formed in a polygon have for a common base a side of the polygon, and for vertices those of the different angles opposite to this base. In each case the number of triangles formed being n→→ 2, the conditions of their similitude will be equal to the number 2 n 4; and the definition will contain nothing superfluous. This new definition being adopted, the ancient one will become a theorem, which may be demonstrated immediately.

If the definition of similar rectilineal figures is imperfect in books of elements, that of similar solid polyedrons is still more so. In Euclid this definition depends upon a theorem not demonstrated; in other authors it has the inconvenience of being very redundant; we have, therefore, rejected these definitions of similar solids.t

The definition of a perpendicular to a plane may be regarded as a theorem; that of the inclination of two planes also requires to be supported by reasoning; the same may be said of several others. It is on this account that, while we have placed the definitions according to ancient usage, we have taken care to refer to propositions where they are demonstrated; sometimes we have merely added a brief explanation which appeared sufficient.

The angle formed by the meeting of two planes, and the solid angle formed by the meeting of several planes in the same point, are distinct kinds of magnitudes, to which it would be well perhaps to give particular names. Without this it is difficult to avoid obscurity and circumlocutions in speaking of the arrangement of planes which compose the surface of a polyedron; and as the theory of solids has been little cultivated hitherto, there is less inconvenience in introducing new expressions, where they are required by the nature of the subject.

I should propose to give the name of wedge to the angle formed by two planes; the edge or height of the wedge would be the common intersection of the two planes. The wedge would be designated by four letters, of which the two middle ones would answer to the edgc. A right wedge then would be the angle formed by two planes perpendicular to each other. Four right wedges would fill all the solid angular space about a given line. This new denomination would not prevent the wedge always having for its measure the angle formed by two lines drawn from the same point, the one in one of

The author here refers to a distinct note on the equality and similitude of polyedrons, not given in this translation.

the planes and the other in the other, perpendicularly to the edge or common intersection.

II.

By the Translator.

THE improvements referred to in the preceding note, so far as they have been adopted by the author, have been carefully preserv ed in the translation. Indeed it has been found necessary in a few instances to use English words in a sense somewhat different from their ordinary acceptation. The word polygon is generally restricted to figures of more than four sides. It is used in this work with the latitude of the original word polygone to stand for rectilineal figures generally; and polyedron is adopted in a similar manner for solids. Quadrilateral is employed as a general name for four-sided figures. The word losenge is rendered by rhombus, and trapéze by trapezoid, the English words, as they are commonly used, corresponding to the French. The perpendicular let fall from the centre of a regular polygon upon one of its sides is called in the original apotheme. It occurs but a few times, and as there is no English word answering to it, it is rendered by a periphrasis, or simply by the word perpendicular. The portion of the surface of a sphere comprehended between the semicircumferences of two great circles is denoted in the original by fuseau; Dr. Hutton uses the word lune in the same sense; others have employed lunary surface; as lune properly stands for the surface comprehended between two unequal circular curves, the latter denomination was thought the least exceptionable, and is adopted in the translation.

III.

On the Demonstration of the Proposition of Article 58.

THE proposition of art. 58 is only a particular case of the celebrated postulate upon which Euclid has established the theory of parallel lines, as well as the theorem upon the sum of the three angles of a triangle. This postulate has not yet been demonstrated in a manner entirely geometrical, and independent of the consideration of infinity, which is undoubtedly to be attributed to the imperfection of the definition of a straight line, which serves as the basis of the elements. But, if we consider this subject in a point of view more abstract, analysis offers a very simple method of demonstrating the proposition rigorously.

We show immediately by superposition, and without any prelimi nary proposition, that two triangles are equal, when a side and the two adjacent angles of the one are equal to a side and the two adjacent angles of the other, each to each. Let us call p the side in question, A and B the two adjacent angles, C the third angle. The angle C then must be entirely determinate, when the angles A and B are known with the side p; for, if several angles C could correspond to the three given things A, B, p, there would be as many different triangles, which would have a side and the two adjacent angles of the one equal to a side and the two adjacent angles of the other, which is impossible; therefore the angle C must be a determinate function of the three quantities A, B, p; which may be expressed

thus

C: (A, B, p).

Let the right angle be equal to unity, then the angles A, B, C, will be numbers comprehended between 0 and 2; and, since

C = 4 (A, B, p),

we say that the line p does not enter into the function . Indeed we have seen that C must be entirely determined by the data A, B, p, merely, without any other angle or line whatever; but the line p is of a nature heterogeneous to the numbers A, B, C; and if, having any equation whatever among A, B, C, p, we could deduce the value of p, in A, B, C, it would follow that p is equal to a number, which is absurd; therefore Р cannot enter into the function p, and we have simply C: (A, B).....*.

This formula proves already that, if two angles of a triangle are equal to two angles of another triangle, the third must be equal to the third; and, this being supposed, it is easy to arrive at the theorem we have in view.

*It has been objected to this demonstration that if it were applied, word for word, to spherical triangles, it would follow that two known angles would be sufficient to determine the third, which would not be true in this kind of triangles. The answer is, that in spherical triangles there is one element more than in plane triangles, and this element is the radius of the sphere which must not be omitted. Accordingly, let r be the radius; then, instead of having

€ = 9 (A, B, p), we shall have C = 9 (A, B, p, r), or simply C=9 (A, B,2),

by the law of homogenous quantities. Now, since the ratio is a number, as well as A, B, C, there is nothing to prevent being found in the fraction ¶, and then we can no longer conclude that

C = ¶ (A, B).

In the first place let ABC (fig. 274) be a triangle right-angled at Figs 274. A; from the point A let fall upon the hypothenuse the perpendicular AD. The angles B and D of the triangle ABD are equal to the angles B and A of the triangle BAC; therefore, according to what has just been demonstrated, the third angle BAD is equal to the third C; for the same reason the angle DAC = B; consequently BAD +DAC, or BAC = B + C; but the angle BAC is a right angle; therefore the two acute angles of a right-angled triangle, taken together, are equal to a right angle.

Again, let BAC (fig. 275) be any triangle, and BC a side which is Fig. 273. not less than each of the two others; if from the opposite angle A the perpendicular AD be let fall upon BC, this perpendicular will fall within the triangle ABC, and will divide it into two right-angled triangles BAD, DAC. Now in the right-angled triangle BAD the two angles BAD, ABD, are together equal to a right angle; in the right-angled triangle DAC the two angles DAC, ACD, are also equal to a right angle. Consequently the four united, or the three BAC, ABC, ACB, are together equal to two right angles; therefore, in every triangle the sum of the three angles is equal to two right angles.

We see by this that the theorem, considered a priori, does not depend upon a series of propositions, but is deduced immediately from the principle of homogeneity, a principle which exists in every relation among quantities of whatever kind. But we proceed to show that another fundamental theorem of geometry may be deduced from the same source.

The above denominations being preserved, and the side opposite to the angle A being called m, and the side opposite the angle B being called n; the quantity m must be entirely determined by the quantities A, B, p; consequently m is a function of A, B, p, as also

m

m

m

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that we can make = (A, B, p). But is a number, as well

Р

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as A and B; therefore the function y must not contain the line

m

p, and we have simply w: (A, B), or m=py: (A, B). We

Р

have also in a similar manner n = p w : (A, B).

Let there be another triangle formed with the same angles A, B, C, and having for the opposite sides m', n', p', respectively. Since A and B do not change, we have in this new triangle

m'p' y (A, B),

and n'p'y: (B, A). Therefore m: m' :: n' :: p: p'. Therefore in equiangular triangles the sides opposite to the equal angles are proportional.

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