To Draw the Tractrix and Catenary

Draw a base–line across the foot of the paper and mark points on it at equal intervals, beginning at the left–hand margin and continuing as far as the middle of the paper. Number these points 0, 1, 2, 3, …, from left to right. Through points 1, 3, 5,…, drawn lines at right angles to the base–line. (These will be called ‘vertical lines 1, 3, 5, … ’.) With points 0, 2, 4, …, as centres draw quadrants of circles of fixed radius c, as shown in Fig. 85. (These will be called ‘quadrants 0, 2, 4, … ’.) On the vertical line 1 choose a point P1 and from it draw tangents P1>T0 and P1>T2> to quadrants 0 and 2 respectively. With P1> as centre, draw an arc from T>0 to T>2>. Let P>1>T>2> cut the vertical line 3 at P3. With centre P>3> draw an arc from T>2> to meet quadrant 4 at T>4>. Joint P>3>T>4>, cutting vertical line 5 at P>5>; and so on.

The arcs joining T>o>, T>2>, T>4>, …, will form a curve approximating to a tractrix, and the points P1, P3, P5, …, will lie approximately on a catenary.

Suitable Dimensions

The paper may be placed either way. The intervals between the points 0, 1,2, …, may conveniently be 0·2 in. or 0·5 cm.