312 Froceedmgs of 'Philosophical Societies. [Avril^ 



ijsost advantageous. This cliapter is one of those of which the 

 applications are the most frequent and the most easy. Philoso- 

 phers, especiallj^ astronomers^ may almost make continual use 

 of it. They will find what is the probability that the sum of 

 errors shall be included between such and such a limit. This is 

 the most ordinary Cf.se in astronomy. Though it be almost cer» 

 tain that each observation is affected with an error, we know 

 almost always that this error cannot pass a very narrow limit* 

 To correct the tables, we compare them with a great number of 

 observatlonsj each of which gives a relation between the effects 

 of the errors of each of the elements of the tables. M. Laplace 

 determines by his analysis the methods which should lead to the 

 most probable results. He considers the case when there are 

 only two elements to correct, and those in which there are any 

 number whatever; and he always arrives at that method which 

 M. Legendre, who is the first known author of it, calls the 

 method of smaller squares. We must always suppose that the 

 number of observations is very great. It was according to this 

 theory that the tables ofM. Burckhardt were judged superior to 

 those of M. Burg, which already possessed so great a degree of 

 precision. 



" The same principles apply to the investigation of phenomena, 

 and their causes ; and what is very remarkable, we may ascertain 

 the very small effect of a cause always constant, by means of a 

 long series of observations, the errors of which may exceed the 

 effect itself. Thus we may ascertain that the diurnal variation 

 of the barometer depends entirely upon the sun, though these 

 heights are also affected by other inequalities which have not a 

 constant period. We may ascertain the small deviation to the 

 east which the rotation of the earth produces in a body that falls 

 freely from a considerable height. This remaik explains how 

 astronomers have been able to determine certain inequalities iu 

 the motion of the moon. It was thus that M. Laplace himself 

 was led (knowing the cause) to discover in the motion of th^_ 

 moon two very small inequalities, which depend upon the flatness 

 of the earth, and which they are capable of determining. From 

 the astronomical researches of MM. Burg and Burckhardt, M. 

 Laplace fixes the flatness at -^-i^. The degrees of France and 

 Peru, of France and the polar circle, have given it -j-Jj-^ to -^l--^. 

 It was likewise by the same method that Laplace was led to his 

 beautiful discoveries respecting the inequalities of Jupiter and 

 Saturn. He concludes from this, that we ought to be attentive 

 to the indications of nature, when they are the result of a great 

 number of observations, though they should be inexplicable by 

 the methods known. We are so far from knowing all the agents 

 of nature, that it would be unphilosophical to deny the existence 

 of phenomena solely because they are inexplicable in the actual 



