SECT. VIII. ASTRONOMICAL TABLES. 59 



sun are given in terms of the time, by general analytical formula}. 

 These formulae will consequently give the exact place of the body 

 in the heavens, for any time assumed at pleasure, provided they 

 can be reduced to numbers. But before the calculator begins 

 his task the observer must furnish the necessary data, which 

 are, obviously, the forms of the orbits, and their positions with 

 regard to the plane of the ecliptic (N. 57). It is therefore neces- 

 sary to determine by observation, for each planet, the length of 

 the major axis of its orbit, the excentricity, the inclination of the 

 orbit to the plane of the ecliptic, the longitudes of its perihelion 

 and ascending node at a given time, the periodic time of the 

 planet, and its longitude at any instant arbitrarily assumed, as 

 an origin from whence all its subsequent and antecedent longi- 

 tudes are estimated. Each of these quantities is determined 

 from that position of the planet on which it has most influence. 

 For example, the sum of the greatest and least distances of the 

 planet from the sun is equal to the major axis of the orbit, and 

 their difference is equal to twice the excentricity. The longi- 

 tude of the planet, when at its least distance from the sun, is the 

 same with the longitude of the perihelion ; the greatest latitude 

 of the planet is equal to the inclination of the orbit : the longitude 

 of the planet, when in the plane of the ecliptic in passing towards 

 the north, is the longitude of the ascending node, and the 

 periodic time is the interval between two consecutive passages of 

 the planet through the same node, a small correction being made 

 for the precession of the node during the revolution of the planet 

 (1ST. 137). Notwithstanding the excellence of instruments and 

 the accuracy of modern observers, unavoidable errors of observa- 

 tion can only be compensated by finding the value of each ele- 

 ment from the mean of a thousand, or even many thousands of 

 observations. For as it is probable that the errors are not all in 

 one direction, but that some are in excess and others in defect, 

 they will compensate each other when combined. 



However, the values of the elements determined separately 

 can only be regarded as approximate, because they are so con- 

 nected that the estimation of any one independently will induce 

 errors in the others. The excentricity depends upon the longi- 

 tude of the perihelion, the mean motion depends upon the major 

 axis, the longitude of the node upon the inclination of the orbit, 

 and vice versa. Consequently, the place of a planet computed 



