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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.f
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 edge. 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.
The improvements referred to in the preceding note, so far as they have been adopted by the author, have been carefully preserved 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 apothéme. 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.
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 preliminary 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 musi 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= 9:(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 y. 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 p cannot enter into the function y, and we have simply C=4: (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 trian. gles. 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, letr be the radius ; then, instead of having c=9 (A, B,p), we shall have c=" (A, B, p,r), or simply c=" (A, B,”), 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 ,
P and then we can no longer conclude that
C=Y (A, B).
In the first place let ABC ( fig. 274) be a triangle right-angled at Fig. 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. 275. 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 me and the side opposite the angle B being called n; the quantity m must be entirely determined by the quantities A, B, p; consequently in is a function of A, B, p, as also
m that we can make = 4: (A, B, p). But is a number, as well р
P as A and B; therefore the function y must not contain the line P, and we have simply = 4:(A, B), or m = py: (A, B). We
P have also in a similar manner n=py:(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' pW (A,B), and n =
p' y: (B, 1). Therefore m: m' :: n' ::p:p'. Theres fore in equiangular triangles the sides opposite to the equal angles are proportional.
From this general proposition we deduce, as a particular case,
that which we have supposed in the text for the demonstration of the Fig. 35. proposition of art. 58. Indeed the triangles, AFG, AML (fig. 35),
have two angles equal, each to each, namely, the angle A common, and a right angle. Consequently these triangles are equiangular ; therefore we have the proportion AF : AL :: AG : AM, by means of which the proposition is fully demonstratedt.
+ In the note, of which the above is only a part, the author undertakes to demonstrate in a similar manner other fundamental propositions of geometry, For remarks upon the kind of reasoning here employed, the reader is referred to Leslie's Geometry, third edition, page 292.