It is customary with Euclid, and various geometrical writers, to give the name equal triangles, to triangles which are equal only in surface; and of equal solids, to solids which are equal only in solidity. We have thought it more suitable to call such triangles or solids equivalent; reserving the denomination equal triangles or solids for such as coincide when applied to each other.

In solids and curve surfaces, it is farther necessary to distinguish two sorts of equality, which differ in some respects. Two solids, two solid angles, two spherical triangles or polygons, may be equal in all their constituent parts, and yet be incapable of coinciding when applied to each other, an observation which seems to have escaped the notice of elementary writers, as their inattention to it has vitiated certain demonstrations relying on the coincidence of figures, where no such coincidence can exist. Such are the demonstrations by which the equality of spherical triangles is sometimes imagined to be shewn, in the same manner as that of rectilineal triangles which are similarly related. A striking example of this oversight is exhibited by Robert Simson,* when this geometer impugns the demonstration of Euclid's Prop. 28. XI., yet falls himself into the error of grounding his own demonstration upon a coincidence which cannot take place. For these reasons, we have judged it necessary to assign a particular name to this kind of equality, which is not accompanied by coincidence; we have called it equality by symmetry, the figures to which it applies being called symmetrical.

Thus the terms equal figures, symmetrical figures, equivalent figures, refer to different objects, and should not be confounded in the same denomination.

In those propositions which relate to polygons, solid angles, and polyedrons, we have formally excluded all figures that have re-entrant angles. Our reason was, that, besides the propriety of limiting an elementary work to the simplest cases, if this exclusion had not taken place, several propositions either would not have been true, or, at least, would have required some modifications. We thought it better to restrict our reasoning to those lines and surfaces which we have named convex, and which are such that a straight line cannot cut them in more than two points.

We have frequently employed the expression, product of two or more lines; by which is meant, the product of the numbers representing those lines, when valued according to a linear unit, assumed at will. The signification of the phrase once fixed, there can be no objection against using it. In the same manner must be understood what is meant by the product of a surface by a line, of a solid by a surface, &c. It is enough to have settled, once for all, that such products are, or ought to be, considered as products of numbers, each of the kind proper to it. Thus, the product of a surface by a solid; is nothing but the product of a number of superficial units by a number of solid units.

In ordinary language, the word angle is often employed to designate the point situated at the vertex. This expression is inaccurate. It would be more correct and precise to use a particular name, such as that of vertices, for designating the points at the corners of a polygon or of a

* See his work, entitled: Euclidis Elementorum libri sex, &c. Glasguæ, 1756.

polyedron. The denomination vertices of a polyedron, as employed by us, is to be understood in this sense.

We have followed the common definition of similar rectilineal figures; we must observe, however, that it contains three superfluous conditions. In order to construct a polyedron, the number of whose sides is n, it is necessary first to know one side, and, next, the position of the vertices of all the angles situated out of this side. Now, the number of those angles is n-2, and the position of each vertex requires two data; hence, the whole number of data requisite for constructing a regular polygon of n sides, is 1 + 2 n 4, or 2 n 3. But in the similar polygon, one side may be assumed at will, therefore the number of conditions regulating the similarity of a polygon to a given polygon, is -4. Now the common definition requires, first, that the angles be equal each to each, which amounts to n conditions; secondly, that the homologous sides be proportional, which amounts to n— 1 conditions. Consequently, there are 2 n - 1 conditions in all, therefore three too many. To obviate this inconvenience, the definition might be subdivided into two, as follows:

2n

First. Two triangles are similar, when they have two angles in each respectively equal.

Second. Two polygons are similar, when both may be divided into the same number of triangles, similar each to each, and similarly placed.

But to prevent this latter definition itself from including any superfluous conditions, the number of triangles must be fixed equal to the number of the polygon's sides, minus two; which may be accomplished in either of the following ways: From the vertices of two homologous angles, diagonals may be drawn to all the opposite angular points; in which case, all the triangles formed in each polygon will have a common vertex, and their sum will be equal to the polygon: Or, let all the triangles formed in one polygon, have a side of it for their common base, and for vertices, the vertices of the different angles opposite to this base. In both cases, the number of triangles formed in the respective polygons being n-2, the conditions of their similarity will amount to n- 4; the definition will contain nothing superfluous whatever; and this being once settled, the old definition will become a theorem susceptible of immediate demonstration.

If the definition of similar rectilineal figures usually given in elementary works is imperfect, that of similar solid polyedrons is much more so. Euclid makes this definition to depend on a theorem which is not proved; in other treatises, it has the inconvenience of being very redundant. We have, therefore, rejected those definitions of similar solids, and substituted one in their place, which is founded on the principles just explained. But as many other observations upon this subject solicit our attention, we shall return to it in a separate Note.

Our definition of the perpendicular to a plane may be looked upon as a theorem; that of the inclination of two planes likewise requires to be sanctioned by a train of reasoning; several others do the same. Accordingly, while in conformity to custom, we have retained the old definitions, care has been taken to refer the reader to those Propositions where their accuracy is demonstrated; or, in other cases, to subjoin a brief explanation.

The angle formed by the meeting of two planes, and the solid angle formed by the meeting of several planes at the same point, are magnitudes, each of its own kind, to which it would perhaps be convenient to give separate names. As they stand at present, it is difficult to avoid obscurities or circumlocutions, when speaking of the arrangement of the planes which compose the surface of a polyedron. And as the theory of those solids has hitherto been little investigated, no great inconvenience could arise from introducing any new expressions which are called for by the nature of the objects.

The angle formed by two planes, I should therefore propose to denominate a corner; the edge or ridge of the corner might designate the common intersection of the two planes. The corner might be named by means of four letters, the middle two corresponding to the edge. A right corner would be the angle formed by two planes perpendicular to each other. Four right corners would fill up all the solid angular space about a given line. Under this new denomination, the corner would still, not the less, have for its measure, the angle formed by the two perpendiculars, drawn each in its own plane, at the same point, to the edge or common intersection.

NOTE II.

On the Demonstration of Prop. 20. Book I, and of some other fundamental Propositions in Geometry.

PROP. 20. I. is only a particular case of the famous postulate, on which Euclid has founded the doctrine of parallels, and likewise the theorem concerning the sum of the three angles of a triangle. This postulate has never hitherto been demonstrated in a way strictly geometrical, and independent of all considerations about infinity,-a circumstance attributable, doubtless, to the imperfection of our common definition of a straight line, on which the whole of geometry hinges. But viewing the matter in a more abstract light, we are furnished by analysis with a very simple method of rigorously proving both this and the other fundamental propositions of geometry. We here propose to explain this method, with all requisite minuteness, beginning with the theorem concerning the sum of the three angles of a triangle. By superposition, it can be shewn immediately, and without any preliminary propositions, that two triangles are equal when they have two angles and an interjacent side in each equal. Let us call this side P, the two adjacent angles A and B, the third angle C. This third angle C, therefore, is entirely determined, when the angles A and B, with the side p, are known; for if several different angles C might correspond to the three given magnitudes A, B, p, there would be several different triangles, each having two angles and the interjacent side equal, which is impossible; hence the angle C must be a determinate function of the three quantities A, B, p, which I shall express thus, C=?: (A, B, p). Let the right to express than (A, B, Let the right angle be equal to unity, then the angles A, B, C,

will be numbers included between o and 2; and since C=ø: (A, B, p), I assert, that the line p cannot enter into the funtion . For we have already seen that C must be entirely determined by the given quantities A, B, p alone, without any other line or angle whatever. But the line p is heterogeneous with the numbers A, B, C; and if there existed any equation between A, B, C, p, the value of p might be found from it in terms of A, B, C; whence it would follow, that p is equal to a number; which is absurd: hence p cannot enter into the function, and we have simply C=9: (A, B).*

This formula already proves, that if two angles of one triangle are equal to two angles of another, the third angle of the former must also be equal to the third of the latter; and this granted, it is easy to arrive at the theorem we have in view.

First, let ABC be a triangle right-angled at A; from the point A, draw AD perpendicular to the hypotenuse. The angles B and D of the triangle ABD are equal to the angles B and A of the triangle B BAC; hence, from what has just been proved, the

third angle BAD is equal to the third C. For a like reason, the angle DAC= B, hence BAD + DAC, or BAC B+C; but the angle BAC is right; hence the two acute angles of a right-angled triangle are together equal to a right angle.

Now, let BAC be any triangle, and BC a side of it not less than either of the other sides; if from the opposite angle A, the perpendicular AD is let fall on BC, this perpendicular will fall B within the triangle ABC, and divide it into two

right-angled triangles BAD, DAC. But 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 DAC, ACD are also equal to a right angle; hence all the four taken together, or, which amounts to the same thing, all the three BAC, ABC, ACB are together equal to two right angles; hence in every triangle, the sum of its three angles is equal to two right angles.

It thus appears, that the theorem in question does not depend, when considered a priori, upon any series of propositions, but may be deduced immediately from the principle of homogeneity; a principle which must

• Against this demonstration it has been objected, that if it were applied word for word to spherical triangles, we should find that two angles being known, are sufficient to determine the third, which is not the case in that species of triangles. The answer is, that in spherical triangles, there exists one element more than in plane triangles, the radius of the sphere, namely, which must not be omitted in our reasoning. Let r be the radius ; instead of C = 9 (A, B, p), we shall now have C4 (A, B, p, r), or by the law of homogeneity, simply C =

(A, B, 2). But since the ratio is a number, as well as A, B, C, there is

r

nothing to hinder from entering the function 4, and consequently we have no right to infer from it, that C = 4 (A, B).

=? «

display itself in a relations subsisting between all quantities of whatever sort. Let us continue the investigation, and shew that from the same source, the other fundamental theorems of geometry may likewise be derived.

Retaining the same denominations as above, let us farther call the side opposite the angle A by the name of m, and the side opposite B by that of n. The quantity m must be entirely determined by the quan

tities A, B, p alone; hence m is a function of A, B, p, and

m

m

m

is one

also; so that we may put = 4 : (A, B, p). But is a number,

as well as A and B ; hence the function

m

cannot contain the line p,

and we shall have simply = : (A, B), or m = p↓ : (A, B).

Hence, also, in like manner, n = p 4 (B, A).

Now, let another triangle be formed with the same angles A, B, C, and with sides m', n', p', respectively opposite to them. Since A and B are not changed, we shall still, in this new triangle, have m' = p'y: (A, B), and n'p'y: (A, B). Hence m: m'::n: n':: p: p'. Hence in equiangular triangles, the sides opposite the equal angles are proportional.

From this general proposition, we can deduce, as a particular case, the property assumed in the text for demonstrating Prop. 20. I. For the triangles AFG, AML have each two angles respectively equal, namely, the angle A common, and a right angle: hence they are equiangular, hence we have the proportion AF: AL:: AG: Am, by means of which Prop. 20 is completely proved.

The proposition concerning the square of the hypotenuse, we already know, is a consequence of that concerning equiangular triangles. Here then are three fundamental propositions of geometry,

that concerning the three angles of a triangle, that concerning equiangular triangles, and that concerning the square of the hypotenuse, which may be very simply and directly deduced from the consideration of functions. In the same way, the propositions relating to similar figures and similar solids may be demonstrated with great ease.