DANIEL BERNOULLI. 237 



the best result from a number of observations will be that 

 which makes the product of the probabilities of all the errors 

 a maximum. Thus, suppose that observations have given the 

 values a,h, c, ... for an element ; denote the true value bv x ; 

 then we have to find x so that the following product may be a 

 maximum : 



^y- _ (^ _ ay] sjy -{x- hy] s/y -{x- cy] . . . 



Daniel Bernoulli gives directions as to the value to be assigned 

 to the constant ?\ 



426. Thus Daniel Bernoulli agrees in some respects with 

 modern theory. The chief difference is that modern theory takes 

 for the curve of probability that defined by the equation 



while Daniel Bernoulli takes a circle. 



Daniel Bernoulli gives some good remarks on the subject ; 

 and he illustrates his memoir by various numerical examples, 

 which however are of little interest, because they are not derived 

 from real observations. It is a fatal objection to his method, even 

 if no other existed, that as soon as the number of observations 

 surpasses two, the equation from which the unknown quantity is 

 to be found rises to an unmanageable degree. This objection he 

 himself recognises. 



427. Daniel Bernoulli's memoir is followed by some remarks 

 by Euler, entitled Ohservationes in pj'aecedeiitem dissertationem ; 

 these occupy pages 24 — 33 of the volume. 



Euler considers that Daniel Bernoulli was quite arbitrary in 

 proposing to make the product of the probabilities of the errors 

 a maximum. Euler proposes another method, which amounts to 

 making the sum of the fourth powers of the probabilities a 

 maximum, that is, with the notation of Art. 425, 



y _ (a: - ayY + [r' - {x - ly]' 4- 17-^ -{x- c)^ + . . . 

 is to be a maximum. Euler sa3^s it is to be a maximum, but 



