Note 5 — Chapter I · Exploring the Sky

The simplest definition of an ellipse is that it is the curve drawn by a moving point P which moves in such a way that the sum of its distances PS, PT from two fixed points S, T remains always the same. In practice we can most easily draw an ellipse by slipping an endless string SPTS round two drawing pins S, T stuck into a drawing board. Stretch the string tight with a pencil at P, and on letting the pencil move round, keeping the string always tight, we shall draw an ellipse. If the pins S, T in the drawing board are placed near to one another the curve described by the pencil P is nearly circular. The ratio of the distance ST to the length of the remainder of the string SP + PT is called the “eccentricity” of the ellipse; it is necessarily less than unity, because two sides of a triangle are together greater than the third side.

In the limiting case in which the eccentricity is made zero, the ellipse becomes a circle. If the eccentricity is nearly as large as unity, the ellipse is very elongated. All the different shapes of ellipses are obtained by letting the eccentricity change from 0 to 1, and these represent all the different shapes of orbit that a small body can describe around a heavy gravitating mass. The points S, T are called the foci of the ellipse, and the big attracting body always occupies one or other of the two foci of the ellipse.