13. To Draw an Approximate Ellipse with the Compasses to Given Dimensions, Using Two Sets of Centers. - First Method. - In Fig. 217, let A B represent the length of the required figure and D E its width. Draw A B and D E at right angles to each other, and intersecting at their middle points. At the point A erect the perpendicular A F, and in length make it equal to C D. Bisect A F, obtaining the point N. Draw N D. From F draw a line to E, as shown, cutting N D in the point G. Bisect the line G D by the line H I, perpendicular to G D and meeting D E in the point I. In the same manner draw lines corresponding to G I, as shown by E I, MO and R O. From I and O as centers, and with I G as radius, strike the arcs G D L and M B R, and from K and P as centers. with K G as radius, strike the ares G A M and L B R, thus completing the figure.

74. To Draw an Approximate Ellipse with the Compasses to Given Dimensions, Using; two Sets of Centers. -Second Method. - In Fig. 218, let C D represent the length of a required ellipse and A B the width. Lav off these two dimensions at right angles to each other, as shown. On C D lay off a space equal to the width of the required figure, as shown by D E. Divide the remainder of D O, or the space E C, into three equal parts, as shown in the cut. With a radius equal to two of these parts, and from R as center, strike the circle G S F T. Then with F and G as centers, and F G as radius, strike the ares, as shown, intersecting upon A B prolonged at O and P. From O, through the points G and F, draw O L and 0 M, and likewise from P, through the same points, draw P K and P N. From O as center, with O A as radius, strike the arc L M, and with the same radius, and P as center, strike the arc K N. From F and G as centers, and with F D and G C as radii, strike the arcs N M and K L respectively, thus completing the figure.

Fig. 217.-First Method.

Fig. 218. - Second Method.

Fig. 219. - Third Method.

To Draw an Approximate Ellipse with the Compasses, Using Two Sets of Centers.

75. To Draw an Approximate Ellipse with the Compasses to Given Dimensions, Using: Two Sets of Centers. - Third Method. - In Fig. 219, let B C represent the length of the required figure and D E its width. B C and D E are drawn at right angles to each other, intersecting at their middle points at F. The next step in describing the figure is to obtain the difference in length between the axes F D and F B, which can be done as indicated by the arc D G. This difference, G B, is to be set off from the center F on F B and F D, as shown by F H, F J, then draw H J and set off half of H J to L, as indicated by the are K L. The object of the operation so far has been to secure the point L. From F as center, with F L as radius, describe a circle, as shown, thus establishing the points M, N and O, which, with L, are the centers from which the ellipse is to be struck. From M, draw M L Q and MNS indefinitely, and in a similar manner O L P and O N R. With O as center, and O O as radius, strike the arc P D R, cutting O P and O R, as shown. In a similar manner, and with the same radius (or which is the same, with M F as radius) and M as center, describe the arc Q E S. With L and N as centers, and with L B or N C as radius, strike the arcs Q B P and R C S, thus completing the figure.

The above methods of drawing approximate ellipses are only available within certain limits of proportion, as will be discovered if an attempt is made to draw them very much elongated, the limit being reached when the long diameter is about equal to two times the shorter diameter. Beyond this limit in the first two methods, if the final arc be drawn with the radius G K (Figs. 217 and 218), it will not reach the end of the long diameter, but will strike it at a point inside of A or C. By the third method, if the long diameter be increased until it is about 2f times the shorter, the point L (Fig. 219) will fall at the extreme limit of the long diameter (B), thus completely cutting out the small arc P Q. It must, therefore, in extreme cases be left to the judgment of the draftsman to adjust or vary the lengths of the radii of the two arcs so as to produce the result which will look the best.

76. To Draw an Approximate Ellipse with the Compasses to Given Dimensions, Using Three Sets of Centers. - In Fig. 220, let A B represent the length of the required figure and D E the width. Draw A B and D E at right angles to each other, intersecting at their middle points, as shown at C. From the point A draw

A F perpendicular to A B, and in length equal to C

D. Join the points F and D, as shown. Divide A F into three equal parts, thus obtaining the points Z and I, and draw the lines Z D and I D. Divide A C into three equal parts, as shown by Y and G, and draw E G and E Y, prolonging them until they intersect with Z D and I D respectively, in the points J and H. Bisect J D, and draw K L perpendicular to its central point, intersecting D E prolonged in the point L. Draw J L and H J. Bisect H J, and draw M N perpendicular to its central point, meeting J L in N. Draw N H, cutting A B in the point O. L then is the center of the arc J D P, N is the center of the arc H J, and O is the center of the arc H A R. The points S and U, corresponding to N and O, from which to strike the remainder of the upper part of the figure, may be obtained by measurement, as indicated. Having drawn so much of the figure as can be struck from these centers, set the dividers to the distance L P or L J, and placing one point at E, the remaining center will be found at the other point of the dividers, in the line E D prolonged, as shown by X.