$ 233] D^Alembert : Lunar Theory 297 



Great of Prussia, and preferred to keep his independence, 

 though he retained the friendship of both sovereigns and 

 accepted a small pension from the latter. He lived ex- 

 tremely simply, and notwithstanding his poverty was very 

 generous to his foster-mother, to various young students, 

 and to many others with whom he came into contact. 



233. Euler, Clairaut, and D'Alembert all succeeded in 

 obtaining independently and nearly simultaneously solutions 

 of the problem of three bodies in a form suitable for lunar 

 theory. Euler published in 1746 some rather imperfect 

 Tables of the Moon, which shewed that he must have 

 already obtained his solution. Both Clairaut and D'Alembert 

 presented to the Academy in 1747 memoirs containing 

 their respective solutions, with applications to the moon 

 as well as to some planetary problems. In each of these 

 memoirs occurred the same difficulty which Newton had 

 met with : the calculated motion of the moon's apogee was 

 only about half the observed result. Clairaut at first met 

 this difficulty by assuming an alteration in the law of gravi- 

 tation, and got a result which seemed to him satisfactory 

 by assuming gravitation to vary partly as the inverse square 

 and partly as the inverse cube of the distance.* Euler also 

 had doubts as to the correctness of the inverse square. 

 Two years later, however (1749), on going through his 

 original calculation again, Clairaut discovered that certain 

 terms, which had appeared unimportant at the beginning of 

 the calculation and had therefore been omitted, became 

 important later on. When these were taken into account, 

 the motion of the apogee as deduced from theory agreed 

 very nearly with that observed. This was the first of several 

 cases in which a serious discrepancy between theory and 

 observation has at first discredited the law of gravitation, 

 but has subsequently been explained away, and has thereby 

 given a new verification of its accuracy. When Clairaut 

 had announced his discovery, Euler arrived by a fresh 

 calculation at substantially the same result, while D'Alembert 

 by carrying the approximation further obtained one that 

 was slightly more accurate. A fresh calculation of the 

 motion of the moon by Clairaut won the prize on the 

 subject offered by the St. Petersburg Academy, and was 



* I.e. he assumed a law of attraction represented by fj./r* + v/r*. 



