It is easily shown that if on two lines, A and a, be described a first pair of polygons, P and p, and a second pair, q and q, the proportion of the first and second pairs is the same, or P : p : : q : q. The simplest similar polygons are squares; consequently, any similar polygons described on A and a are to one another in the proportion of the squares on A and a. This is also true if for the polygons we sub stitute similar curves; and it must be proved by the method of ex haustions [GEOMETRY], or by the theory of limits applied to the proposition, that any curve may be approached in magnitude by a polygon within any degree of nearness.
The theory of similar solids resembles that of similar polygons, but it is necessary to commence with three points instead of two. Let A, B, C, and a, b, c, be two seta of three points each, and let the triangles A B C and a be be similar: let them also be placed so that the sides of one are parallel to those of the other. If then any number of similar pyramids be described on A n c and abr., the vertices of these pyramids will be the corners of similar solids. If P and p be the vertices of one pair, then the pyramids Pane and p a be are similar if the vertices P and p be on the same side of A B C and ab c [SYMMETRY], and one of the triangles, say P A B, be similar to its corresponding triangle p ab, and so placed that the angle of the planes PAS and e AB is the same as that of the planesp a b and The simplest similar solids are cubes ; and any similar solids described on two straight lines are in the same proportion as the cubes on those lines. Similar curve surfaces are those which are such that every solid which can be inscribed in one has another similar to it, capable of being inscribed in the other.
It is worthy of notice that the great contested element of geometry [PARALLELS] would lose that character if it were agreed that the notion of form being independent of size is as necessary as that of two straight lines being incapable of enclosing a space ; so that whatever form can exist of any one size, a similar form must exist of every other. There can be no question that this universal idea of similarity involves as much as this, and no more ; that in the passage from one size to another, all lines alter their lengths in the same proportion, and all angles remain the same. It is the subsequent mathematical treat ment of these conditions which first points out that either of them follows from the other. If the whole of this notion be admissible, so in any thing leas; that is, the admission implies it to be granted that whatever figure may be described upon any one line, another figure having the same angles may be described upon any other line. If then we take a triangle A B c, and any other line ab, there can be drawn upon ab a triangle having angles equal to those of AB C. This can only be done by drawing two lines from a and b, making angles with a6 equal to B • 0 and A B C. These two lines must then meet in some point c, and the angle a cb will be equal to A C B. If then two triangles have two angles of one equal to two angles of the other, each to each, the third angle of the one must be equal to the third angle of the other; and this much being established, It is well knowu that the ordinary theory of parallels follows. The preceding assumption is not without resemblance to that required in the methods of Legendre.