147. In the preceding pages we have adopted several processes, e. g., bisecting and trisecting finite lines, bisecting rectilineal angles and dividing them into other equal parts, drawing perpendiculars to a given line, etc. Let us now examine the theory of these processes.

148. The general principle is that of congruence. Figures and straight lines are said to be congruent, if they are identically equal, or equal in all respects.

In doubling a piece of paper upon itself, we obtain the straight edges of two planes coinciding with each other. This line may also be regarded as the intersection of two planes if we consider their position during the process of folding.

In dividing a finite straight line, or an angle into a number of equal parts, we obtain a number of congruent parts. Equal lines or equal angles are congruent.

149. Let X'X be a given finite line, divided into any two parts by A'. Take O the mid-point by doubling the line on itself. Then OA' is half the difference between A'X and X'A'. Fold X'X over O, and take A in OX corresponding to A'. Then A A' is the difference between A'X and X'A' and it is bisected in 0.

Fig. 52.

As A' is taken nearer O, A'O diminishes, and at the same time A'A diminishes at twice the rate. This property is made use of in finding the mid-point of a line by means of the compasses.

150. The above observations apply also to an angle. The line of bisection is found easily by the compasses by taking the point of intersection of two circles.

151. In the line X'X, segments to the right of

O may be considered positive and segments to the left of O may be considered negative. That is, a point moving from O to A moves positively, and a point moving in the opposite direction OA' moves negatively.

AX= OX - OA.

OA' = OX' - A'X', both members of the equation being negative.*

152. If OA, one arm of an angle A OP, be fixed and OP be considered to revolve round O, the angles which it makes with OA are of different magnitudes.

*See Beman and Smith's New Plane and Solid Geometry, p. 56.

All such angles formed by OP revolving in the direction opposite to that of the hands of a watch are regarded positive. The angles formed by OP revolving in an opposite direction are regarded negative.*

153. After one revolution, OP coincides with OA. Then the angle described is called a perigon, which evidently equals four right angles. When OP has completed half the revolution, it is in a line with OAB. Then the angle described is called a straight angle, which evidently equals two right angles.† When OP has completed quarter of a revolution, it is perpendicular to OA. All right angles are equal in magnitude. So are all straight angles and all perigons.

154. Two lines at right angles to each other form four congruent quadrants. Two lines otherwise inclined form four angles, of which those vertically opposite are congruent.

155. The position of a point in a plane is determined by its distance from each of two lines taken as above. The distance from one line is measured parallel to the other. In analytic geometry the properties of plane figures are investigated by this method. The two lines are called axes; the distances of the point from the axes are called co-ordinates, and the intersection of the axes is called the origin. This method was invented by Descartes in 1637 A. D.* It has greatly helped modern research.

* See Beman and Smith's New Plane and Solid Geometry, p. 56. †Ib., p.5.

156. If X'X, YY' be two axes intersecting at O, distances measured in the direction of OX, i. e., to the right of 0 are positive, while distances measured to the left of 0 are negative. Similarly with reference to YY', distances measured in the direction of O Y are positive, while distances measured in the direction of OY' are negative.

157. Axial symmetry is defined thus : If two figures in the same plane can be made to coincide by turning the one about a fixed line in the plane through a straight angle, the two figures are said to be symmetric with regard to that line as axis of symmetry.†

158. Central symmetry is thus defined : If two figures in the same plane can be made to coincide by turning the one about a fixed point in that plane through a straight angle, the two figures are said to be symmetric with regard to that point as center of symmetry. ‡

In the first case the revolution is outside the given plane, while in the second it is in the same plane.

If in the above two cases, the two figures are halves of one figure, the whole figure is said to be symmetric with regard to the axis or center - these are called axis or center of symmetry or simply axis or center.

*Beman and Smith's translation of Fink's History of Mathematics, p. 230. †Beman and Smith's New Plane and Solid Geometry, p. 26. ‡Ib., p. 183.

159. Now, in the quadrant XOY make a triangle PQR. Obtain its image in the quadrant YOX' by folding on the axis YY' and pricking through the paper at the vertices. Again obtain images of the two triangles in the fourth and third quadrants. It is seen that the triangles in adjacent quadrants posses axial symmetry, while the triangles in alternate quadrants possess central symmetry.

Fig. 53.

160. Regular polygons of an odd number of sides possess axial symmetry, and regular polygons of an even number of sides possess central symmetry as well.

161. If a figure has two axes of symmetry at right angles to each other, the point of intersection of the axes is a center of symmetry. This obtains in regular polygons of an even number of sides and certain curves, such as the circle, ellipse, hyperbola, and the lemniscate; regular polygons of an odd number of sides may have more axes than one, but no two of them will be at right angles to each other. If a sheet of paper is folded double and cut, we obtain a piece which has axial symmetry, and if it is cut fourfold, we obtain a piece which has central symmetry as well, as in Fig. 54.