412 Historical Note on the Method of Least Squares. 



as Professor in Columbia College and in Pennsylvania Univ . 

 as well as by his correspondence, Dr. Adrain is well kuu 

 have contributed powerfully to the progress of Matbeuui; 

 studies in this his adopted country — (he was born andeducat^v. 

 in Dublin) — and his apparently independent demonstration ui 

 the method of least squares seems quite in accordance with the 

 originality shown in many other of the elegant solutions offered 

 by him to the different problems on which he busied himself. 

 A number of interesting and probably valuable mathematical 

 manuscripts still remain in the possession of his family at New- 

 Brunswick, New Jersey, which it is to be hoped may some day 

 see the light. At present I would offer toward the historv of 

 mathematics in America the following extracts from the Analyst 

 and other publications. 



The problem " to correct the distances and bearings of a 

 survey, so as to deduce the most probable area of the enclosed 

 field, '"* had been proposed by Professor Patterson in a previous 

 number of the Analyst, and after being a second time renewed 

 as a prize question, was at length in number IV, solved by a 

 course of special reasoning, by Dr. Bowditch, to whom Dr. 

 Adrain awarded the prize. Dr. Bowditch's results coincided 

 with what would have been deduced had the Gaussian method 

 been applied to this case. Immediately following Dr. Bowditch s 

 special solution, the editor adds his own solution of the more 

 general problem as follows: (The Analyst, pp. 93-95 inclusive). 

 " Research concerning the probabilities of the errors which happen 

 in making observations.^^ 



" The question which I propose to resolve is this ; supposing 

 AB to be the true value of any quantity of which the measure 

 by observation or experiment is AS, the error being BJ ; what 

 is the expression of the probabilitv that the error Bb happens 

 in measuring AB? ^ 



Let AB, BC, &c., be several successive distances o± whicti 

 the values by measure are Ai, 5c, &c., the whole error being 

 Cc ; now supposing the measures Ah, be, to be given and also 

 the whole error Cc, we assume as a self-evident principle, that 

 the most probable distances AB, BC are proportional to the 

 measures A6, be ; and therefore the errors belonging to AB, i>^ 

 are proportional to their lengths, or to their measured values 

 A6, he. If therefore we represent the values of AB, BC or ot 

 their measures A5, he by a, 5, the whole error Cc by C, and the 

 errors of the measures Ai, be by x, y, we must for the greatest 

 probability, have the equation |-= f- Let X and Y be simi- 

 lar functions of a, x, and of b, y, expressing the probabihti^ 

 that the errors x, y happen in the distances a, b ; and, by tne 



