SYMMETRY, SYMMETRICAL (Mathematics). These terms are now applied to order and regularity of any kind, but this is not their mathematical meaning. Euclid first used the word " summetros" (crinair spas) to signify commensurable, and this well-known Latin word is in fact merely the literal translation of the Greek : two magnitudes then were symmetrical which admitted of a common measure. In later times, and those comparatively recent, the word was adopted both in geometry and algebra in different senses.
Since symmetrical applies in its etymology to two magnitudes which can be measured together (by the same magnitude), the term would, as to space-mak,rnitudes, naturally apply to those which may be made to coincide. But the term equal had occupied this ground ; and when, in Euclid, the word equal, which was originally defined in the manner just expressed, had degenerated into signifying equality of area only, the term SIMILAR entered to express sameness of form, so that figures having perfect capability of coincidence, or the same both in size and form, were called equal and similar. The word symmetrical was therefore not wanted, and was finally introduced to signify that obvious relation of equal and similar figures which refers to their position merely, and consists in their corresponding portions being similarly placed on different aides of the same straight line ; so that coincidence cannot be procured without turning one Cgare round that straight line. Suppose, for instance, the front of a building to be symmetrical draw a vertical line through the middle of the elevation, and the two lateral portions are equal and similar, as Euclid uses those words. But they are more than equal and similar; they are sym metrical : the right-hand side stands in the right-hand portion of space, with respect to the dividing line, and in exactly tho same manner as the left-hand side stands in the left-hand portion of space. if the architect were to preserve equality and similarity, without symmetry, he would make two left sides, or two right sides, to his building, but not one right and one left. In the letter w there is a want of qui nictry, but not in o to make w symmetrical, both the inner lines should IT thin, and both the outer ones thick.
Euclid assumes the power of turning a plane round, so as to apply the faces of two figures to one another, in such manner that, after the application, the spectator must be supposed to see through the paper or other imaginary substance of which his plane is the surface. He has then no occasion to consider symmetry ; that is, figures being equal and similar, no cases can arise in which it makes any difference of demonstration whether they be symmetrical or not. When ho comes to solid figures, he assumes a postulate in the garb of a defini tion, which dispenses him from the consideration of symmetry ; namely, that solid figures consisting of the same number of equal planes, similarly placed, are equal. He seems to imagine that such solids must evidently be capable of being made to occupy the same space, which, though true as to quantity of space, is not true as to its disposition. Two.solids may be equal in every respect, and yet it may
be impossible (and precisely on account of their symmetry) to make one occupy the space previously occupied by the other. The two hands furnish an instance : they give the idea of equality (of size), similarity (of form), and symmetry (of disposition). Yet they cannot be made to occupy the same space, so as, for instance, to fit exactly the same glove ; and a sculptor who should cast both hands from the same mould, would be detected immediately as having given his figure two right hands or two left hands. Again, suppose two solids, irregular pyramids for instance, composed of planes similar and equal, each of one to one of the other. Let coincidence be attempted geometrically : the two bases must of course be made to coincide. If, then, the two vertices fall on the same sido of the common base, the figures will coincide altogether; but if the two vertices fall on opposite sides of the bases, absolute coincidence is impossible. Legendre proposed to call such solids by the name of symmetrical, in doing which he introduced the term of common life in an appropriate manner.
In algebra, a function is said to be symmetrical with respect to any two letters when it would undergo no change if these letters were interchanged, or if each were made to take the place of the other. Thus, a°x-s ab+ is symmetrical with respect to a and b; interchange would give L2x-Fta + avx, the same as before. But this expression is not symmetrical with respect to a and x, for interchange would here give An expression is symmetrical with respect to any number of letters when any two of them whatsoever may be interchanged without altera tion of the function. Thus + at° + + aca+ + is cal with respect to a, b, and c. It is not sufficient that certain conteno poraneous changes should be practicable without producing alteration : any two must be interchangeable, the rest remaining. Thus + + is unaltered if a become b, b become c, and c become a, at the same time, but it is not symmetrical ; for if a and b only be inter changed, it becomes b2a+ a2c+ or is altered. , Attention to symmetry is of the utmost consequence in mathematical notation. Here the word means that quantities which in any manner have a common relation should have something common in the symbols of notation ; and analogy is perhaps a better word than symmetry. Suppose, for instance, we bad taken, for the equation of a SURFACE OP THE SECOND DEGREE, + + (bey + cxz+ filz+ fix + +17 +1=0. Our formula) would have been confused masses of letters, no set of which would have presented any similarity, or have easily remained in the memory. But in the article cited there is no set of forninhe of which more than one need be remembered ; the others must be sug gested by it.