112 LECTURES 



in coming to the meridian than she would have been if she had re- 

 tained the same mean motion as she had in the time of the earliest 

 Chaldean observations." If we adopt the new coefficient of Hansen, 

 13", it will increase the difference by about half an hour. Hence, in 

 assigning the moon's place in any remote period in the past by the 

 use of the modern tables, the assigned place will be less advanced in 

 longitude than the actual place. The accurate determination of this 

 coefficient of acceleration is of the utmost importance in verifying the 

 ancient eclipses. 



The cause of this acceleration for a long time baffled the severest 

 scrutiny. The French Academy offered its astronomical prize of 1770, 

 1772, and 1774, for an investigation which should have for its object 

 to show whether it could be produced by the law of gravitation. The 

 first prize was taken by Euler, the second was shared by Euler and 

 Legrange, the third was awarded to Legrange alone. Both of these 

 eminent mathematicians concurred in the conclusion that it could not 

 be produced by the action of gravity. The language of Euler in the 

 memoir which bore off the prize is very explicit, as follows: "There 

 is not one of the equations about which any uncertainty prevails, and 

 now it appears to be established by indisputable evidence that the 

 secular inequality in the moon's mean motion cannot be produced by 

 the forces of gravitation." The failure of Euler and Legrange in at- 

 tempting to trace this inequality to the effect of gravity, and the 

 great weight of their authority, may have deterred others from re- 

 peating the investigation, and turned their attention to some other 

 quarter for a solution. 



Newton was inclined to refer it to the resistance of the medium in 

 which the moon moved, and his conclusion — the only one, indeed, 

 which his premises permitted — was, that the moon was winding in 

 toward the earth, and must ultimately fall upon it. 



But the profound analysis of Laplace, in 1787, revealed what had 

 escaped the sagacity of Newton and Legrange. He traced it to a very 

 slow and almost imperceptible change in the form of the earth's orbit, 

 technically called the secular diminution of the eccentricity of the 

 earth's orbit. 



The general manner in which this inequality is produced is not 

 difficult to be understood. One of the consequences of gravity deduced 

 by Newton was, that the disturbing force of the sun's attraction upon 

 the moon had the effect to diminish the gravity of the moon to the 

 earth by about .^ „- part of it. If the earth and moon were removed 

 to a greater distance from the sun the consequence would be that the 

 moon would be drawn down nearer to the earth, and would make her 

 revolution in a shorter time, because, with an unchanged actual 

 volocity, she would revolve in a smaller orbit. Now, this is precisely 

 the effect of the diminution of the eccentricity of the earth's orbit. 

 Thus, in fig. 9, let A C B D represent the earth's orbit, S the 

 place of the sun, A B the major axis, and C D the minor. The form 

 of the orbit has been for many ages undergoing a gradual change, 

 almost imperceptible, but still continuous. This change is due to the 

 disturbing action of the planets upon the earth, and is one of those 



