PROBLEM I.

To describe an hyperbola by finding points in the curve, having the diameter, or axis A B, its abscissa B C, and double ordinate D E.

1. Through B draw G F parallel to D E; from D and E draw D G and E F parallel to B C, cutting GF in F and G.

2. Divide C D and C E, each into any number of equal parts, as four through the points of division 1, 2, 3, draw lines to A.

3. Likewise divide D G and E F into the same number of equal parts, viz. four; from the divisions on D G and EF draw lines to B, and a curve being drawn through the intersections at B a b c E, will be the hyperbola required.

PROBLEM II.

Given the asymptotes A B, C D, and a point E in the curve, to describe the hyperbola.

1. Through the given point E draw any right line E F, cutting the asymptotes in the points i and I,

2. Make i F equal to I E; from F draw as many lines as you please, cutting the asymptotes in the points g, h, i, k, &c. and G, H, I, K, &c.

3. Make Gf, Hf, Kf, &c. respectively equal g F, h F, k F, &c. through the points f, f, f, describe a curve, and it is the hyperbola required.

In the same manner may the opposite hyperbola be described.

PROBLEM