Ex. 28.—To describe an octagon about a circle, [Fig. 133]. Describe a square about the given circle A B, draw perpendiculars H and K, to the diagonals, touching the circle to form the octagon. Or, the points H, K, etc., may be found by cutting the sides from the corners, by lines parallel to the diagonals.
Fig. 134.
Ex. 29.—To describe an ellipse when the length and breadth are given, [Fig. 134]. On the center C, with A E as radius, cut the axis A B at F and G, the foci, fix a couple of pins into the axis at F and G, and loop on a thread or cord upon them equal in length to the axis A B, so as when stretched to reach the extremity C of the conjugate axis, as shown in dot-lining. Place a pencil or drawpoint inside the cord, as at H, and guiding the pencil in this way, keeping the cord equally in tension, carry the pencil round the pins F, G, and so describe the ellipse.
Note.—The ellipse is an oval figure, like a circle in perspective. The line that divides it equally in the direction of its great length is the transverse axis, and the line which divides the opposite way is the conjugate axis.
Second Method. Along the straight edge of a piece of stiff paper mark off a distance a c equal to A C, half the transverse axis; and from the same point a distance a b equal to C D, half the conjugate axis. Place the slip so as to bring the point b on the line A B of the transverse axis, and the point c on the line D E; and set off on the drawing the position of the point a. Shifting the slip, so that the point travels on the transverse axis, and the point c on the conjugate axis, any number of points in the curve may be found, through which the curve may be traced. See [fig. 135].
Fig. 135.
Trigonometry.
Trigonometry is that portion of geometry which has for its object the measurement of triangles. When it treats of plane triangles, it is called Plane Trigonometry; and as the engineer will continually meet in his studies of higher mathematics the terms used in plane trigonometry, it is advantageous for him to become familiar with some of the principles and definitions relating to this branch of mathematics.