SlMAKt'BA BAUK. 



SIM1I.AK. SIMILAR FIGURES. 



b pound on the ghws, together with a little polishing-powder. When 

 wU Sorted, the solution U rolled with an india-rubber roll* to expel 

 all air- bubble*. Then more eolation U poured on, and more heat 

 applied, until a precipitate of pure aUrer U deposited on the glass, 

 which Ukea place in about a quarter of an hour. The surplus liquid 

 being poured off. and the silver washed, it U found to preeent a beauti- 

 ful reflecting surface. No mercury being here employed, the proceea 

 u leat harmful than that usually adopted. 



SIMAHUBA BARK U obtained from the root of the Simamba 

 ammra, Aublet S. ofnmalu, Dec. It U a native of Guyana, and alio 

 ,,f Jamaica (unless this latter be a distinct *peciei), and the more 

 wuthern of the United State*. 



It i intanaely bitter, and yielda an infusion more bitter than the 

 decoction. The infusion made with cold water ia beet It may be 

 need M an emetic or tonic, in fever and dysentery ; also, againit 

 wanna. If the wood be used in cooking, the viand* acquire an intense 

 bitterness. The Simarmba wmeotor (Aug. St. Hilaire), a native of 

 Braiil, he* similar propertie*, but it* internal use causes stupor and 

 other narcotic symptoms. Thia effect might be avoided by making the 

 infusion with tuld water, a* in the earn of quassia. [QUASSIA.] 



SIMILAR, SIMILAR FIGURES (Geometry). Similarity, resem- 

 blance, or likeness, means umeneM in some, if not in all, particulars. 

 In geometry, the word refers to a sameness of one particular kind. 

 The two most important notions which the view of a figure will give 

 are thuse of tat and tkofx, ideas which have no connection whatsoever 

 with each other. Figure* of different sices may have the same shape, 

 and figures of different shapes may have the same sise. In the latter 

 case they are called by Euclid equal, in the former rimilar (similar 

 figures, &JUM* axtnaral The first term [EQUAL ; RELATION], in Euclid's 

 first uae of it, includes united sameness both of site and shape ; but he 

 soon drops the former notion, and, reserving equal to signify samenees 

 of sice only, introduces the word similar to denote sameness of form : 

 so that the equality of the fundamental definition is the subsequent 

 combined equality and similarity of the sixth book. 



Similarity of form, or, as we ahall now technically say, similarity, is 

 a conception which is better defined by things than by words ; being 

 in fact one of our fundamental ideas of figure. A drawing, a map, a 

 model, severally appeal to a known idea of similarity, derived from, it 

 may be, or at least nourished by, the constant occurrence in nature 

 and art of object* which have a general, though not a perfectly mathe- 

 matical, similarity. The rudest nations understand a picture or a map 

 almost instantly. It is not necessary to do more in the way of detini- 

 iiul we must proceed to point out the mathematical testa of 

 ximilarity. We may observe indeed that errors or monstrosities of 

 size are always more bearable than those of form, so much more do 

 our conceptions of objects depend upon form than upon size. A 

 painter may be obliged to diminish the size of the minor parts of his 

 picture a little, to give room for the more important objects : but no 

 one ever thought of making a change of form, however slight, in one 

 object, for the sake of its effect on any other. The giant of Rabelais, 

 with whole nations carrying on the business of life inside his mouth, is 

 not so monstrous as it would have been to take the ground on which a 

 nation might dwell, England, France, or Spain, invest it with the 

 intellect and habit* of a human being, and make it move, talk, and 

 reason : the more tasteful fiction of Swift is not only bearable and 

 conceivable, but has actually made many a simple person think it was 

 meant to be taken as a true history. 



Granting then a perfect notion of similarity, we now ask in what 

 way it is to be ascertained whether two figures are similar or not To 

 simplify the question, let them be plane figures, say two maps ol 

 Kngland of different sizes, but made on the same projection. It is 

 obvious, in the first place, that the lines of one figure must not only 

 be related to one another in length in the same manner as in the 

 other, but also in position. Let us drop for the present all the curved 

 lines of the coast, &c., and consider only the dots which represent the 

 towns. Join every such pair of dots by straight lines : then it is plaii 

 that similarity of form requires that any two lines in the first shoulc 

 not only be in the same proportion, as to length, with the two corre 

 ponding lines in the second, but that the first pair should incline ai 

 the same angle to each other as the second. Thus, if I. v be the line 

 which join* London and York, and r c that which joins Falmouth anc 

 Chester, it is requisite that I. v should be to K u in the same proportion 

 in the one map that it is in the other ; and if r c produced meet L Y 

 produced in o, the angle o r in one map must be the same as in the 

 other. Hence, if there should be 100 towns, which are therefore 

 joined two and two by 4950 straight lines, giving about 1 2 millions 

 and a quarter of pairs of lines, it is clear that we must have the means 

 tfying 1-1 millions of proportions, and as many angular agree 

 ment*. But if it be only assumed that similarity is a possible thing 

 it U easily shown that this large number is reducible to twice ! 

 Let it be granted that I y on the smaller map is to represent L T on 

 the larger. Lay down / and r in their proper place* on the smalle 

 map, each with reference to I and .>/, by comparison with the large 

 map : then / and c are in their proper place* with reference to eacl 

 other. For if not, one of them at least must be altered, which wouli 

 disturb the correctness of it with respect to I and y. Hither then there 

 is no such thing a* perfect similarity, or else it may be entirely ob- 

 tained by companion with / and y only. 



We hare hitherto supposed that both circumstance* must be looked 

 to; broper lengths and proper angle*; truth of linear pr..i 

 truth of relative direction. Hut it is one of the first thing* win. 1, ihe 



tudrnt of geometry learns (in reference to this subject), th 

 attainment of correctness in either secures that of the other. It th.- 

 smaller map be made true in all its relative lengths, it must be true 



n all its directions ; if it be made true in all iti directions, it must be 

 true in all its relative lengths. The foundation of thix Amplifying 



heorem rests on three propositions of the sixth book of Euclid, as 



ollows : 



1. The angle* of a triangle (any two, of course) alone are enough to 

 determine iU form : or, as Euclid would express it, two triangles whieh 



lave two angles of the one equal to two angles of the other, each to 

 each, hare the third angles equal, and all the sides of one in the same 

 iroportion to the corresponding sides of the other. 



2. The proportions of the sides of a triangle (those of two . i them 

 to the third) are alone enough to determine its form, or if two 



riangles 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 the same a* 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 tw.. triangles have one angle 

 of the first equal to one of the second, and the sides about those angles 



iroportional, the remaining angles are equal, each to each, .iii.l the 



tide* about equal angles are proportional. 

 From these propositions it is easy to show the truth of all that lias 



Men asserted about the conditions of similarity, and the result is, that 

 any number of points are placed similarly with any other number of 



mints, when, any two being taken in the first, and the corresponding 



;wo in the second, say A, B, and a, b, any third point c of the first 

 gives a triangle ABC, which is related to the corresponding triangle 

 ode 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 B A and bac, let the angles A K B and E B A be seve- 

 rally equal to a e b and e b a. In the triangles A D B and 

 A B : : d a : a b, and D B : B A : : d li : >i a. In the triangles A i 

 acb let the angles ABC and a b c be equal, and AB:BC::al:6e. 

 These conditions being fulfilled, it can be shown that the figut. 

 similar in form. There is no angle in one but is equal to its 

 ttponding 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 Jiyure, 

 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 be n points, 

 they are twice (M 2) in number. It would be most natural to take 

 either a sufficient number of ratios, or else of angles : perha|>s the 

 latter would be best Euclid confines himself to neither, in which he 

 is guided by the following consideration : He 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 he 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 conditions in an ?i-si<leil figure, ami eon 

 sequently four redundancies, two of which are easily detected. In the 

 above pentagons, for instance, if the angles at A, E, D, c, be severally 

 equal to these at a, e, d, e, there is no occasion to say that that at B 

 must be equal to that at h, for it is a necessary consequence : also, if 

 n A : A K : : Aa : c, and so on up to D c : C B : : d c : c b, there is in > 

 occasion to lay it down as a condition that c B : B A : : cb : IKI, fur it is 

 again a consequence. These points being noted, the defini' 

 Euclid is admit ably adapted for his object, which is, in this a.-, in every 

 other case, to proceed straight to the establishment of his \ 

 without casting one thought upon the connection of his | 

 with natural geometry. 



Let us now suppose two similar curvilinear figure mplify 



the question, take two arcs A B and ab. Having already detected the 

 test of similarity of position with reference to any number of points, it 



will be easy to settle the conditions \tnder which the arc A n is 

 altogether similar to ab. By hypothesis, A and n arc the points corre- 

 sponding to a and 6. Join A, B, and a, 6; and in the arc A B take any 



