## Problem XXXII

To describe a parabola, by finding points in the curve; the axis A B, or any diameter being given, and a double ordinate C D.

Through A draw E F. Fig. 33, parallel to C D; through C and D draw D F and C E parallel to A B, cutting E F at E and F. Divide B C and B D, each into any number of equal parts, as four; likewise divide C E and D F into the same number of equal parts. Through the points 1, 2, 3, etc. in C D, draw the lines 1 a, 2 b, 3 c, etc. parallel to A B; also through the points, 1, 2, 3, etc. in C E and D F, draw the lines 1 A, 2 A, 3 A, cutting the parallel lines at the points a, b, c; then the points a, b, c, are in the curve of the parabola.

## Problem XXXIII

To describe an hyperbola.

If B and C, Fig. 34, be two fixed points, and a ruler A B be made movable about the point B, a string ADC being tied |to the other end of the ride, and to the point C; and if the point A be moved round the centre B, towards G, the angle D of the string ADC, by keeping it always tight, and close to the edge of the ruler A B, will describe a curve D H G, called an hyperbola.

Fig. 33. Fig. 34. Fig. 35. If the end of the ruler at B were made movable about the point C, the string being tied from the end of the rule A to B, and a curve being described after the same manner, is called an opposite hyperbola.

The foci are the two points B and C, about which the ruler and string revolve.

The transverse axis is the line l H terminated by the two curves passing through the foci, if continued.

The centre is the point M, in the middle of the transverse axis l H.

The conjugate axis is the line N O, passing through the centre M, and terminated by a circle from H, whose radius is M C, at N and O.

A diameter is any line VW, drawn through the centre M, and terminated by the opposite curves.

Conjugate diameter to another, is a line drawn through the centre, parallel to a tangent with either of the curves, at the extremity of the other diameter terminated by the curves.

Abscissa is when any diameter is continued within the curve, terminated by a double ordinate and the curve; then the part within is called the abscissa.

Double ordinate is a line drawn through any diameter parallel to its conjugate, and terminated by the curve.

Parameter, or latus rectum, is a line drawn through the focus, perpendicular to the transverse axis, and terminated by the curve.

## Problem XXXIV

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

Through G draw E F, Fig. 35, parallel to C D; from C and D draw C E and D F, parallel to B G, cutting E F in E and F. Divide C B and B D, each into any number of equal parts, as four; through the points of division, 1, 2, 3, draw lines to A. Likewise divide E C and D F into the same number of equal parts, viz. four; from the divisions on C E and D F, draw lines to G; a curve being drawn through the intersections at G, a, b, etc. will be the hyperbola required.