SECT. 3.] ANY NUMBER OF OBSERVATIONS. 269 



185. 



The subject we have just treated might give rise to several elegant analytical 

 investigations, upon which, however, we will not dwell, that we may not be too 

 much diverted from our object. For the same reason we must reserve for another 

 occasion the explanation of the devices by means of which the numerical calcu 

 lation can be rendered more expeditious. I will add only a single remark. 

 When the number of the proposed functions or equations is considerable, the 

 computation becomes a little more troublesome, on this account chiefly, that the 

 coefficients, by which the original equations are to be multiplied in order to ob 

 tain P, Q, R, S, etc., often involve inconvenient decimal fractions. If in such 

 a case it does not seem worth while to perform these multiplications in the most 

 accurate manner by means of logarithmic tables, it will generally be sufficient 

 to employ in place of these multipliers others more convenient for calculation, 

 and differing but little from them. This change can produce sensible errors in 

 that case only in which the measure of precision in the determination of the 

 unknown quantities proves to be much less than the precision of the original 

 observations. 



186. 



In conclusion, the principle that the sum of the squares of the differences 

 between the observed and computed quantities must be a minimum may, in the 

 following manner, be considered independently of the calculus of probabilities. 



When the number of unknown quantities is equal to the number of the ob 

 served quantities depending on them, the former may be so determined as exactly 

 to satisfy the latter. But when the number of the former is less than that of the 

 latter, an absolutely exact agreement cannot be obtained, unless the observations 

 possess absolute accuracy. In this case care must be taken to establish the best 

 possible agreement, or to diminish as far as practicable the differences. This idea, 

 however, from its nature, involves something vague. For, although a system of 

 values for the unknown quantities which makes all the differences respectively 



