SIMILARITY (from similar, from Lat. simi lis, similar, like; connected with simul, together, Gk. Ziaa, Kama, together, Skt. sama, like, equal, same, and ultimately with Eng. same). In geom etry, the theory of similar systems and similar figures. Two systems of points A„ B„ and are said to be similar when they can be so placed that all lines, joining corresponding points form a pencil whose vertex, 0, divides each line into segments having a constant ratio r.
In the figures OA, : OA, = OB, : OB, — r. Two figures are said to be similar when their systems of points are similar. The symbol ct. for similarity, is due to Leibnitz and is de rived from the letter S.
When two similar figures are so placed that lines through their corresponding points form a pencil, they are said to be in perspective, and the vertex of the pencil is called their centre of similitude. The above figures are placed in per spective, and in each case 0 is the centre of similitude. In similar figures, if the ratio, r, known as the ratio of similitude, is 1, the figures are evidently symmetric with respect to a centre.
Hence, central symmetry is a special case of similar figures in perspective. The term centre of similitude is clue to Euler. (See SYMMETRY.) Some of the principal propositions of Similarity are: Two triangles are similar if they have two angles of one equal to two angles of the other, respectively. Mutually equiangular triangles are similar. if two triangles have the sides of the one respectively parallel or perpendicular to the sides of the other, they are similar. If two triangles have one angle of the one equal to one angle of the other, and the including sides pro portional, the triangles are similar. If two tri angles have their sides proportional, they are similar. If two polygons are mutually equi angular and have their corresponding sides pro portional, they are similar. Areas of similar polygons are proportional to the squares of the corresponding sides. Volumes of similar solids are proportional to the cubes of their like di mensions. Consult Beman and Smith, Seen• Plane and Solid Geometry (Boston, 1899), pp. 182, 364.