SIMILAR
, in Arithmetic and Geometry, the same with like. Similar things have the same disposition or conformation of parts, and differ in nothing but as to their quantity or magnitude; as two squares, or two circles, &c.
In Mathematics, Similar parts, as A, a, have the same ratio to their wholes B, b; and if the wholes have the same ratio to the parts, the parts are Similar.
Similar angles, are also equal angles.
Similar ares, of circles, are such as are like parts of their whole peripheries. And, in general, similar arcs of any like curves, are the like parts of the wholes.
Similar bodies, in Natural Philosophy, are such as have their particles of the same kind and nature one with another.
Similar Curves. Two segments of two curves are said to be Similar when, any right-lined figure being inscribed within one of them, we can inscribe always a Similar rectilineal figure in the other.
Similar Conic Sections, are such as are of the same kind, and have their principal axes and parameters proportional. So, two ellipses are figures of the same kind, but they are not Similar unless the axes of the one have the same ratio as the axes of the other. And the same of two hyperbolas, or two parabolas. And generally, those curves are Similar, that are of the same kind, and have their corresponding dimensions in the same ratio.—All circles are Similar figures.
Similar Diameters of Conic Sections, are such as make equal angles with their ordinates.
Similar Figures, or plane figures, are such as have all their angles equal respectively, each to each, and | their sides about the equal angles proportional. And the same of Similar polygons.—Similar plane figures have their areas or contents, in the duplicate ratio of their like sides, or as the squares of those sides.
Similar Plane Numbers, are such as may be ranged into the form of Similar rectangles; that is, into rectangles whose sides are proportional. Such are 12 and 48; for the sides of 12 are 6 and 2, and the sides of 48 are 12 and 4, which are in the same proportion, viz, 6 : 2 :: 12 : 4.
Similar Polygons, are polygons of the same number of angles, and the angles in the one equal severally to the angles in the other, also the sides about those angles proportional.
Similar Rectangles, are those that have their sides about the like angles proportional.—All squares are Similar.
Similar Segments of circles, are such as contain equal angles.
Similar Solids, are such as are contained under the same number of Similar planes, alike situated.—Similar solids are to each other as the cubes of their like linear dimensions.
Similar Solid Numbers, are those whose little cubes may be so ranged, as to form Similar parallelopipedons.
Similar Triangles, are such as are equiangular ones, or have all their three angles respectively equal in each triangle. For it is sufficient for triangles to be similar, that they be equiangular, because that being equiangular, they necessarily have their sides proportional, which is a condition of Similarity in all figures. As to other figures, having more sides than three, they may be equiangular, without having their sides proportional, and therefore without being similar.—Similar triangles are as the squares of their like sides.