### Penciling

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

## Problems 19 And 20. How 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 LA, 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 R 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. How 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.