Imperial Institute of France. 243 



Boscovich long since had it in contemplation to make 

 the sum of the positive errors equal with the negative ones ; 

 and all astronomers have had this object in view in the con- 

 struction of their tables. He also maintained that the sum 

 of errors without the distinction of signs was the least pos- 

 sible ; which is likewise the opinion of all astronomers : 

 but to be more certain, he gave, according to his custom, a 

 graphical construction of the problem, to which a calcula- 

 tioQ may be applied when greater nicety is required. It 

 is to be observed, that he introduces the centre of gravity 

 of all the extreme points of the abscissa, which in his con- 

 struction represent the degrees measured : because it was 

 also on account of the figure of the earth that he under- 

 took his researches. 



Count Laplace, in adopting the principal opinions of 

 Boscovich, treated the same problem in a more analytical 

 and more rigorous manner in the 2d vol. of his Mecanique 

 Celeste; and he was led to a flattening of -j^, almost as 

 great as that of M. Legendre: his 45th degree differed a 

 little less from the arc adopted, and the eiTors of their lati- 

 tudes were nearly the same. Thus two methods, absolutely 

 different, led to results almost identical. 



M. Gauss, in his Theory of the Motions of Celestial 

 Bodies, published 1809, endeavours to determine the degree 

 of probability of a system of elements for a planet from a 

 considerable number of observations. He soon meets with 

 an insoluble equation, which forces him to take another 

 course. He inquires upon what function, taken tacitly as 

 the base, the principle vulgarly adopted is supported, and 

 what is the value of the mean result between several obser- 

 vations equally well made, which value is not rigorously 

 exact, but only probably so ? By this inverted course his 

 demonstration is very analogous to that of M. Legendre. 



Setting out from an elegant theorem of Count Laplace, 

 he arrives at a function which gives expressly the sum of 

 the squares, which ought to be a minimum. Hence he 

 concludes, that the principle of the small squares has the 

 same certainty with the common principle which allows the 

 greatest probability to the arithmetical method. But he 

 remarks that this consequence cannot be correct but on the 

 supposition that all the observations are entitled to the same 

 confidence; and in order to render the principle more 

 general, he multiplies each of the squares by a co-efficient, 

 which expresses the probability of the observation to which 

 he refers; and it is the su.n thus modified which ought to 

 be a minimum. He afterwards examines whether the eli- 

 Q 3 mination 



