SYMMETRICAL, in botany (of the parts of a flower), related to each other in number, the same in number, or one a multiple of the other, as in Saxifraga, which has five divisions of the calyx, five petals, and five stamens; or Epilobiunn, which has a four-parted calyx, four pet als, and eight stamens.
In mathematics, possessing the at tribute of symmetry; having correspond ing parts or relations. In geometry, two points are symmetrically disposed with respect to a straight line, when they are on opposite sides of the line and equally distant from it, so that a straight line joining them intersects the given line, and is at right angles to it. A curve is symmetrical with respect to a straight line, when for each point on one side of the line there is a corresponding point on the other side, and equally distant from it. The line is called an axis of symmetry. In conic sections, the axes are the only true axes of symmetry. Two plane figures are symmetrically sit uated with respect to a straight line, when each point of one has a correspond ing point in the other on the opposite side of the axis, and equally distant from it. A line or surface is symmetrical with
respect to a plane, when for each point on one side of the plane there is a sec ond point on the other side, equally dis tant from it. The plane is called the plane of symmetry, and is, in conical sec tions, a principal plane. Symmetrical lines and surfaces in space cannot, in general, be made to coincide with each other. Spherical triangles are symmet rical when their sides and angles are equal each to each, but not similarly sit uated. In analysis, an expression is symmetrical with respect to two letters, when the places of these letters may be changed without changing the expres sion. Thus the expression x' ab b'x is symmetrical with respect to a and b; for, if we change the place of a and b, we have x' b'x ba a'x, the same expression. An expression is symmetrical with respect to several let ters, when any two of them may change places without affecting the expression; thus, the expression ab ba' c'a ± be' is symmetrical with re spect to the three letters, a, b, c.