Proof. Since the triangle 0 C D is an equilateral triangle by construction, the angle C 0 D is one-third of two right. angles and one-sixth of four right angles. Hence arc C D is one-sixth of the circumference and the chord is a side of a regular hexagon.

Problem 17. To draw a line tangent to a circle at a given 'point on the circumference.

With 0 as a center and a radius of about 1 1/4 inches draw the given circle. Assume some point P on the circumference and join the point P with the center 0. By the method given in Problem 6, Plate IV, construct a perpendicular to P 0, which perpendicular will be the desired tangent to the circle at the point P.

Proof. A line perpendicular to a radius at its extremity is tangent to the circle.

Problem 18. To draw a line tangent to a circle from a point outside the circle.

With 0 as a center and a radius of about 1 inch draw the given circle. Assume P some point outside of the circle about 2 1/2 inches From the center. Draw a straight line passing through P and O. Bisect P O and with the middle point F as a center describe the circle passing through P and O. Draw a line from P through the intersection of the two circumferences C. The line P C is tangent to the given circle. Similarly P E is tangent to the circle. JANUARY 29, /9/0. HERBERT CHANDLER, CH/CAGO, /LL . PLATE VII. Proof. The angle P C 0 is inscribed in a semicircle and hence is a right angle. Since P C O is a right angle, P C is perpendicular to C 0. The perpendicular to a radius at its extremity is tangent to the circumference.

Inking. In inking Plate VI, the same method should be followed as in previous plates.

## Plate VII

Penciling. Lay out this plate in the same manner as the preceding plates.

Problems 19 and 20. To draw an ellipse when the axes are given.

First Method. Draw the lines L M and C D about 3 1/4 and 2 1/4 inches long respectively, making C D perpendicular to L M at its middle point P and having C P = P D. The two lines, L M and C D, are the axes. With C as a center and a radius L P equal to one-half the major axis, draw the arc, cutting the major axis at E and F. These two points are the foci.

Now locate several points on P M, such as A, B, and G. With E as a center and a radius equal to L A, draw arcs above and below L M. With F as a center and a radius equal to A M describe short arcs cutting those already drawn as shown at N. With E as a center and a radius equal to L B draw arcs above and below L M as before. With F as a center and a radius equal to B M, draw arcs intersecting those already drawn as shown at 0. The point P and others are found by repeating the process. The student is advised to find at least 12 points on the curve - 6 above and 6 below L M. These 12 points with L, C, M, and D will enable him to draw the curve.

After locating these points, draw a free-hand curve passing through them.

Second Method. Draw the two axes A B and P Q in the same manner as in the first method. With 0 as a center and a radius equal to one-half the major axis, describe a circle. Similarly with the same center and a radius equal to one-half the minor axis, describe another circle. Draw any radii such as 0 C, 0 D, 0 E, 0 F, etc., cutting both circumferences. These radii may be drawn with the 60 and 45 degree triangles. From C, D, E, and F, the points of intersection of the radii with the large circle, draw vertical lines and from C', D', E', and F', the points of intersection of the radii with the small circle, draw horizontal lines The intersections of these lines are points on the ellipse.

Draw a free-hand curve passing through these points; about five points in each quadrant will be sufficient.

Problem 21. To draw an ellipse by means of a trammel.

As in Problems 19 and 20, draw the major and minor axes, U V and X Y. Take a slip of paper having a straight edge and mark off C B equal to one-half the major axis, and D B equal to one-half the minor axis. Place the slip of paper in various positions keeping the point D on the major axis and the point C'on the minor axis. If this is done, the point B will mark various points on the curve. Find as many points as necessary and sketch the ellipse.

Problem 22. To draw a spiral of one turn in a circle.

Draw a circle with the center at 0 and a radius of 1 1/2 inches. Locate twelve points, 1/8 inch apart on the radius 0 A and draw circles through these points. Now, by means of the 30-degree triangle, draw radii O B, O C, O D, etc., 30 degrees apart, thus dividing the circle into 12 equal parts.

The points on the spiral are now located; the first is at the center 0; the next is at the intersection of the line 0 B and the first circle; the third is at the intersection of 0 C and the second circle; the other points are located in the same way. Sketch in pencil a smooth curve passing through these points.

Problem 23. To draw a parabola when the abscissa and ordinate arc given.

Draw the straight line A B - about three inches long - as the axis, or abscissa of the parabola. At A and B draw the lines C D and E F perpendicular to A B, and with the T-square draw E C and F D, 1 1/2 inches above and below A B, respectively. Let A be the vertex of the parabola. Divide A E and E C into the same number of equal parts. Through R, S, T, U, and V, draw horizontal lines and connect L, M, N, 0, and P, with A. The intersections of the horizontal lines with the oblique lines are points on the curve. For instance, the intersection of A L and the line V is one point and the intersection of A M and the line U is another.

The lower part of the curve A D is drawn in a similar manner.

Problem 24. To draw a hyperbola when the abscissa E X, the ordinate A E, and the diameter X Y are given.