We have hitherto supposed that both circumstanced timid be looked to; proper lengths and proper angles; truth of linear proportion and truth of relative direction. But it is one of the first things which the student of geometry learns (in reference to this subject), that the attainment of correctness In either secures that of the other. If the smaller map be made true in all its relative lengths, it must be true In all ite directions ; If it be made true in all its directions, it must be true in all its relative lengths. The foundation of this simplifying theorem reats on three propositions of the sixth book of Euclid, as follows : 1. The angles of a triangle (any two, of course) alone are enough to determine its form : or, as Euclid would express it, two triangles which have two angles of the one equal to two angles of the other, each each, have the third angles equal, and all the sides of one in the mine proportion to the corresponding sides of the other.
2. The proportions of the sides of a triangle (those of two of them to the third) are alone enough to determine its form, or if two triangles have the ratios of two sides to the third in one the same as the corresponding ratios in the other, the angles of the one are seve rally tho same as those of the other.
8. One angle and the proportion of the containing sides are sufficient to determine the form of a triangle : or, if two triangles have one angle of the firs equal to one of the second, and the sides about those angles proportional, the remaining angles are equal, each to each, and the aides about equal angles are proportional.
From these propositions it is easy to show the truth of all that has been asserted about the conditions of similarity, and the result is, that any number of points are placed similarly with any other number of points, when, any two being taken in the first, and the corresponding two in the second, say A, D, and a, b, any third point c of the first gives a triangle A B c, which is related to the corresponding triangle a b c of the second, in the manner described in either of the three pre ceding propositions. For instance, let there be five points in each figure : In the triangles BA E and bee, let the angles A a is and E B A be seve rally equal to aeb and eba. In the triangles ADD and adb let DA : : :da:ab, and on:BA : :db:ba. In tho triangles A c B and a cb let the angles A H C and abc be equal, and An:De:: ab: he. These conditions being fulfilled, it can be shown that the figures are similar in form. There is no angle in one but is equal to its corre sponding angle in the other : no proportion of any two lines in one but is the same as that of the corresponding lines in the other. Every con
ception necessary to the complete notion of similarity is formed, and the one figure, in common language, is the same as the other in figure, but perhaps on a different scale.
The number of ways in which the conditions of similarity can be expressed might be varied almost without limit ; if there he n points, they are twice (e-2) in number. It would be most natural to take either a sufficient number of ratios, or else of angles : perhaps the latter would be best. Euclid confines himself to neither, in which he is guided by the following consideration :-11e uses only salient or convex figures, and his lengths, or sides, are only those lines which form the external contour. The internal lines or diagonals ho rarely considers, except in the four-sided figure. He lays it down as the definition of similarity, that all the angles of the one figure (meaning only angles made by the sides of the contour) are equal to those of the other, each to each, and that the sides about those angles are pro portional. This gives 2 a conditions in an ti-sided figure, and con sequently four redundancies, two of which are easily detected. In the above pentagons, for instance, if the angles at A, E, D, 0, be severally equal to these at a, e, d, c, there is no occasion to say that that at B must be equal to that at b, for it is a necessary consequence: also, if BA: AE:: bet: ae, and so on up to ne:cB: :dc:cb, there is no occasion to lay it down as a condition that cis : BA : cb :ba, for it is again a consequence. These points being noted, the definition of Euclid is admirably adapted for his object, which is, in this as in every other case, to proceed straight to the establishment of his propositions, without casting one thought upon the connection of his preliminaries with natural geometry.
Let us now euppose two similar curvilinear figures, and to simplify the question, take two arcs A TI and a b. Having already detected the test of similarity of position with reference to any number of points, it will be easy to settle the conditions under which the arc A II is altogether similar to a b. By hypothesis, a and n are the points curre spending to a and b. Join A, B, and a, 6; and in the arc A u take any point P. Make the angle bap equal to is A r, and abp equal to A s r; and let ap and by meet in p. Then, if the curves be similar, p must be on the arc a b ; for every point on A is is to have a corresponding point on ob. Hence the definition of similarity is as follows :—Two curves are similar when for every polygon which can be inscribed in the first, a similar polygon can be inscribed in the second.