to the Method of Least ^Squares. 163 



tnethode des moindres quarres, gerathe ich aiif die Vei'iniithung, dass 

 eiii Gruudsatz, desseii ich mieh schoii scit zwolf Jahreii bey mancherley 

 Reclnuingen bedient habe, und den ich auch in meinem Werke rait 

 gebrauchen werde, ob er wol zu meiner Methode eben nicht wesent- 

 lich gehort, — dass dieser Gruudsatz aiich von Legendre benutzt ist," 

 The work of Legendre here alluded to is the Noiwelles m'ethodes 

 ....1805, and that of Gauss the Theoria niotus ...1809, then in 

 preparation. Gauss mentions some of the advantages of his method 

 for computing orbits but gives no hint of the principle of Least 

 Squares. 



1808 BowDiTCH. 'Solution of Mr. Patterson's Prize Quastion for 

 correcting a survey, proposed in No. IL page 42, No. IIL page 68, 

 by Nathaniel Bowuitch, to whom the Editor has awarded the 

 prize of ten dollars.' :77te Anali/st or Math. Museum., Vol. I, pp. 

 88-92. 



The Prize Question was : " In order to find the content of a piece 

 of ground, . . .1 measured, with a common circumferentor and chain, 



the bearings and lengths of its several sides, But upon casting 



up the diiference of latitude and departure, I discovered .... that some 

 error had been contracted in taking the dimensions. Now it is re- 

 quired to compute the area of this enclosure, on the most prohaMe 

 supposition of this error." 



Bowditch's solution depends on several "principles" or hypotheses, 

 the cliief of which is " that in measuring the lengths of any lines 

 the errors would probably be in proportion to their lengths." No 

 principles of the Theory of Probability are employed except such as 

 are by common sense implied. His solution coincides Avith that given 

 by the ^lethod of Least Squares. 



This Prize Question undoubtedly led to the folloAving ' Research ' 

 by AuRAiN, the Editor of The Analyst. 



1808 Aurain. 'Research concerning the probabilities of the errors 

 which happen in making observations.' Th,e Analyst or Math. Mu- 

 seum, Vol. I, pp. 93-109. 



This paper seems to have been unknown to mathematicians until 

 1871 when |it was partly reprinted in Amer. Jour. Sel., see 1871 

 Abbe. It is of great historical interest as containing the first deduc- 

 tion of the law of facility of error 



(p (cc) =: ce"'*'^*^ 

 (p{x) l>eiiig the probability of any error x, and e and h constants 

 flepending upon the precision of the measurement. The term "Least 

 Squares" is not used, and Adrain seems to have been entii'ely unac- 

 quainted with Lkgendre's writings. 



Adrain gives two deductions of this law. The first, occupying 

 pages 93-' '5 has been reprinted as noted above and need not here be 

 repeated. It depends i;pon the "self-evident principle" that the ti*ue 

 errors of measured quantities are proportional to the quantities them- 



