610 LAPLACF. 



1044. Laplace's pages 464 — 484 are headed Additions; see 

 Arts. 916, 921. There are three subjects discussed. 



I. Laplace demonstrates Wallis's theorem, and he gives an 

 account of the curious way in which the theorem was discovered, 

 although it cannot be said to have been demonstrated by its dis- 

 coverer. 



II. Laplace demonstrates a formula for A"s' which he had 

 formerly obtained by a bold assumption ; see Arts. 916, 966. 



III. Laplace demonstrates the formula marked (p) on page 168 

 of the Theorie...des Prob.; see Art. 917. 



1045. The first Supplement to the Theorie...des Proh. is en- 

 titled Sur V application du Calcul des Prohabilites d la Philosophie 

 Naturelle ; it occupies 34 pages: see Art. 926. The title of the 

 Supplement does not seem adapted to give any notion of the 

 contents. 



1046. We have seen in Art. 1009 that in Laplace's theory of 

 the errors of observations a certain quantity occurs the value of 

 which is not known a priori, but which may be approximately 

 determined from the observations themselves. Laplace proposes 

 to illustrate this point, and to shew that this approximation is one 

 which we need not hesitate to adopt : see pages 7 — 11 of the first 

 Supplement. It does not appear to me however that much con- 

 viction could be gained from Laplace's investigation. 



A very remarkable theorem is enunciated by Laplace on page 8 

 of the first Supplement. He gives no demonstration, but says 

 in his characteristic way, L'analyse du n° 21 du seconde Livre 

 conduit a ce theoreme general.... The theorem is as follows: 

 SupjDose, as in Art. 1011, that certain quantities are to be deter- 

 mined by the aid of observations ; for simplicity we will assume 

 that there are three quantities x, y, z. Let values be found for 

 these quantities by the most advantageous method, and denote 

 these values by o;^, y^, z^, respectively. Put 



aj = a?j + f , y = y^ + V, z = z^+^. 

 Then Laplace's theorem asserts that the probability of the simul- 



