42. _The Ellipse._--An _ellipse_ is a closed curve which has two points within it, the sum of whose distances from every point on the curve is the same. These two points are called the _foci_ of the ellipse. [Illustration: Fig. 53.] One method of describing an ellipse is shown in Fig. 53. Two tacks, _F_ and _F'_, are stuck into a piece of paper, and to these are fastened the two ends of a string which is longer than the distance between the tacks. A pencil is then placed against the string, and carried around, as shown in the figure. The curve described by the pencil is an ellipse. The two points _F_ and _F'_ are the foci of the ellipse: the sum of the distances of these two points from every point on the curve is equal to the length of the string. When half of the ellipse has been described, the pencil must be held against the other side of the string in the same way, and carried around as before. The point _O_, half way between _F_ and _F'_, is called the _centre_ of the ellipse; _AA'_ is the _major axis_ of the ellipse, and _CD_ is the _minor axis_. 43. _The Eccentricity of the Ellipse._--The ratio of the distance between the two foci to the major axis of the ellipse is called the _eccentricity_ of the ellipse. The greater the distance between the two foci, compared with the major axis of the ellipse, the greater is the eccentricity of the ellipse; and the less the distance between the foci, compared with the length of the major axis, the less the eccentricity of the ellipse. The ellipse of Fig. 54 has an eccentricity of 1/8. This ellipse scarcely differs in appearance from a circle. The ellipse of Fig. 55 has an eccentricity of 1/2, and that of Fig. 56 an eccentricity of 7/8. [Illustration: Fig. 54.] [Illustration: Fig. 55.] [Illustration: Fig. 56.] 44. _Kepler's First Law._--Kepler first discovered that _all the planets move from west to east in ellipses which have the sun as a common focus_. This law of planetary motion is known as _Kepler's First Law_. The planets appear to describe loops, because we view them from a moving point. The ellipses described by the planets differ in eccentricity; and, though they all have one focus at the sun, their major axes have different directions. The eccentricity of the planetary orbits is comparatively small. The ellipse of Fig. 54 has seven times the eccentricity of the earth's orbit, and twice that of the orbit of any of the larger planets except Mercury; and its eccentricity is more than half of that of the orbit of Mercury. Owing to their small eccentricity, the orbits of the planets are usually represented by circles in astronomical diagrams. [Illustration: Fig. 57.]