Part 16

EX. 19.--To describe a circle about a square, and to inscribe a square in a circle, Fig. 124.

First. To describe the circle. Draw the diagonals A B, C D of the square, cutting at E; on the center E with the radius E A describe the circle.

Second. To inscribe the square. Draw the two diameters A B, C D at right angles and join the points A B, C D to form the square.

In the same way a circle may be described about a triangle.

EX. 20.--To inscribe a circle on a square, and to describe a square about a circle, Fig. 125.

First. To inscribe the circle. Draw the diagonals A B, C D of the square, cutting at E; draw the perpendicular E F to one side, and with the radius E F describe the circle.

Second. To describe the square. Draw two diameters A B, C D at right angles, and produce them; bisect the angle D E B at the center by the diameter F G, and through F and G draw perpendiculars A C, B D, and join the points A D and B C where they cut the diagonals to complete the square.

EX. 21.--To inscribe a circle in a triangle, Fig. 126. Bisect two of the angles A C of the triangle by lines cutting at D; from D draw a perpendicular D E to any side, and with D E as radius describe a circle.

EX. 22.--To inscribe a pentagon in a circle, Fig. 127. Draw two diameters A C, B D at right angles cutting at O; bisect A O at E, and from B with radius B E cut the circumference at G H and with the same radius step round the circle to I and K; join the points to form the pentagon.

EX. 23.--To construct a hexagon upon a given straight line, Fig. 128. From A and B, the ends of the given line, describe arcs cutting at G; from G with the radius G A describe a circle. With the same radius set off the arcs A C, C F and B D, D E; join the points so found to form the hexagon.

EX. 24.--To inscribe a hexagon in a circle, Fig. 129. Draw a diameter A C B; from A and B as centers, with the radius of the circle A C cut the circumference at D, E, F, G, and draw A D, D E, etc., to form the hexagon. The points D E, etc., may be found by stepping the radius (with the dividers) six times round the circle.

EX. 25.--To describe an octagon on a given straight line, Fig. 130. Produce the given line A B both ways and draw perpendiculars A E, B F; bisect the external angles A and B by the lines A H, B C, which make equal to A B. Draw C D and H G parallel to A E and equal to A B; from the center G D, with the radius A B, cut the perpendiculars at E F, and draw E F to complete the hexagon.

EX. 26.--To convert a square into an octagon, Fig. 131.--Draw the diagonals of the square cutting at E; from the corners A, B, C, D, with A E as radius, describe arcs cutting the sides at G, H, etc., and join the points so found to complete the octagon.

EX. 27.--To inscribe an octagon in a circle, Fig. 132. Draw two diameters A C, B D, at right angles; bisect the arcs A B, B C, at E, F, etc., to form the octagon.

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.

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.

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.

The circumferences of all circles contain the same number of degrees, but the greater the radius the greater is the absolute measures of a degree. The circumference of a fly wheel or the circumference of the earth have the same number of degrees; yet the same number of degrees in each and every circumference is the measure of precisely the same angle.

The circumference of a circle is supposed to be divided into 360 degrees or divisions, and as the total angularity about the center is equal to four right angles, each right angle contains 90 degrees, or 90°, and half a right angle contains 45°. Each degree is divided into 60 minutes, or 60′; and for the sake of still further minuteness of measurement, each minute is divided into 60″. In a whole circle there are, therefore, 360 × 60 × 60 = 1,296,000 seconds. The annexed diagram, fig. 136, exemplifies the relative positions of the

Sine, Co-sine, Versed Sine,

Tangent, Co-Tangent, Secant and Co-secant

of an angle.

These may be defined thus:

DEFINITIONS.

1. The Complement of an arc is 90° minus the arc.