﻿180 ON THE USE or EQUIVALENT FACTORS 



" When the number of functions or equations proposed for solution is considerable, 

 the computations become laborious, the more so from the circumstance that the co- 

 efficients by which the primitive equations are to be multiplied are almost ahvays com- 

 plicated decimal fractions. If it is not thought worth the trouble in such a case to 

 calculate the products with exactness by means of logarithms, it will generally be suf- 

 ficient to substitute for them (i. e. for the multiplying factors) more simple numbers 

 diftering but slightly from them." * 



In his subsequent researches, it does not appear that Gauss has given any further 

 development to the suggestion here put forth. Indeed, the introduction of modifications 

 of a like nature, however desirable in a practical point of view, would have deprived 

 a purely theoretical discussion of much of its elegance and symmetry. Yet the passage 

 above quoted lends the support of the highest authority to the leading proposition 

 which we shall have occasion most frequently to insist upon ; namely, the propriety of 

 allowing some relaxation of theory in appljing the calculus of probabilities to the dis- 

 cussion of data affected by ordinary errors of observation, whenever the modification 

 conduces to convenience and the saving of labor at the sacrifice of no appreciable ad- 

 A^antages. 



Even an iniqualified admission of the superior probability of results which exactly 

 fulfil the criterion proposed in the method of least squares, does not relieve us from the 

 necessity of restricting it to examples which never actually occur, that is, if the ques- 

 tion be made a rigorous one ; I — to such, for instance, as involve the discussion of obser- 

 vations which are entirely free from unknown constant errors, or errors following any 

 law of facility which does not imply the assimiption that the mean error is proportional 

 to the square root of the mean of the squares of the individual errors. But we know 

 that this proposition, which lies at the foundation of the whole subject, is not suscepti- 

 ble of absolute demonstration by any process of mathematical reasoning. Further than 

 this, we know from constant experience that the law of distribution of errors recognized 

 in the method of least squares practically fails, in extreme cases, both for very large and 

 for very small errors. If any illustration of the failure of the assumed law be needed, 

 it will be found in the familiar instance of computing by it the probable error of the 

 arithmetical mean of a very large number of observations, where common sense assures 

 us that the theoretical probable errors of the result are invariably smaller than they 

 should be. 



Why, then, should an implicit adherence to its minutest details be requii-ed as essen- 



* Theoria Motus, § 185. t Theor. Comb., § 17. 



