LIBRATION OF THE MOON. 337 



with a second instrument termed the mural circle. These 

 two inequalities of very different magnitudes connected 

 with the cause which produces them by analytical com- 

 binations of totally different kinds have, however, both 

 conducted to the same value of the ellipticity. It must 

 be borne in mind, however, that the ellipticity thus de- 

 duced from the movements of the moon, is not the ellip- 

 ticity corresponding to such or such a country, the 

 ellipticity observed in France, in England, in Italy, in 

 Lapland, in North America, in India, or in the region of 

 the Cape of Good Hope, for the earth's materials having 

 undergone considerable upheavings at different times and 

 in different places, the primitive regularity of its curva- 

 ture has been sensibly disturbed by this cause. The 

 moon, and it is this circumstance which renders the result 

 of such inestimable value, ought to assign, and has in 

 reality assigned the general ellipticity of the earth ; in 

 other words, it has indicated a sort of mean value of the 

 various determinations obtained at enormous expense, 

 and with infinite labour, as the result of long voyages 

 undertaken by astronomers of all the countries of Eu- 

 rope. 



I shall add a few brief remarks, for which I am mainly 

 indebted to the author of the Mecanique Celeste. They 

 seem to be eminently adapted for illustrating the pro- 

 found, the unexpected, and almost paradoxical char- 

 acter of the methods which I have just attempted to 

 sketch. 



What are the elements which it has been found neces- 

 sary to confront with each other in order to arrive at 

 results expressed even to the precision of the smallest 

 decimals ? 



On the one hand, mathematical formulas deduced from 



15 



