302 A Short History of Astronomy [Ca. XE. 



in subsequent researches, it may be worth while to attempt 

 to give a sketch of it. 



If perturbations are ignored, a planet can be regarded as 

 moving in an ellipse with the sun in one focus. The size and 

 shape of the ellipse can be defined by the length of its axis 

 and by the eccentricity ; the plane in which the ellipse is 

 situated is determined by the position of the line, called the 

 line of nodes, in which it cuts a fixed plane, usually taken 

 to be the ecliptic, and by the inclination of the two planes. 

 When these four quantities are fixed, the ellipse may still 

 turn about its focus in its own plane, but if the direction 

 of the apse line is also fixed the ellipse is completely 

 determined. If, further, the position of the planet in its 

 ellipse at any one time is known, the motion is completely 

 determined and its position at any other time can be 

 calculated. There are thus six quantities known as elements 

 which completely determine the motion of a planet not 

 subject to perturbation. 



When perturbations are taken into account, the path 

 described by a planet in any one revolution is no longer 

 an ellipse, though it differs very slightly from one ; while in 

 the case of the moon the deviations are a good deal greater. 

 But if the motions of a planet at two widely different 

 epochs are compared, though on each occasion the path 

 described is very nearly an ellipse, the ellipses differ in 

 some respects. For example, between the time of Ptolemy 

 (A.D. 150) and that of Euler the direction of the apse line 

 of the earth's orbit altered by about 5, and some of the 

 other elements also varied slightly. Hence in dealing with 

 the motion of a planet through a long period of time it is 

 convenient to introduce the idea of an elliptic path which 

 is gradually changing its position and possibly also its size 

 and shape. One consequence is that the actual path 

 described in the course of a considerable number of 

 revolutions is a curve no longer bearing much resemblance 

 to an ellipse. If, for example, the apse line turns round 

 uniformly while the other elements remain unchanged, the 

 path described is like that shewn in the figure. 



Euler extended this idea so as to represent any per- 

 turbation of a planet, whether experienced in the course 

 of one revolution or in a longer time, by means of changes 



