The answer was provided by Newton. He shewed that when a small body is allowed to move freely under the gravitational pull of a big body, it will run away altogether if its speed exceeds a certain critical amount, in which case its orbit is the curve called a hyperbola. But if its speed is less than this critical amount, its orbit will always be an ellipse—a sort of pulled out circle or oval curve[5] (fig. 4, p. 47). Previous to this Kepler had found that the actual paths of the planets round the sun were not exact circles but ellipses, although for the most part ellipses which did not differ greatly from circles; they are what the mathematician calls “ellipses of small eccentricity.” This provides still further confirmation of Newton’s law of gravitation, for it can be proved that if the force of gravitation falls off in any way other than according to Newton’s law of the inverse square of the distance, the orbits of the planets will not be elliptical. [5] 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. [Illustration: Fig. 4. The oval curve is an ellipse; the points _S_, _T_ are its “foci.”]