LAPLACE. 611 



taneoiis existence of f, 77, f, as values of the errors of the quantities 

 to be determined, is 23roportional to e~'^, where 



I am compelled to omit the demonstration of this theorem for want 

 of space ; but I shall endeavour to publish it on some other 

 occasion. 



1047. Laplace next supposes that six elements are to be 

 determined from a large number of observations by the most ad- 

 vantageous method. He arranges the algebraical work in what 

 he considers a convenient form, supposing that we wish to de- 

 termine for each variable the mean value of the error to be appre- 

 hended, or to determine the probability that the error wdll lie 

 within assigned limits ; see pages 11 — 19 of the first Supplement. 

 He then, on his pages 21 — 26, makes a numerical application, and 

 arrives at the result to which we have alreadv referred in Art. 939. 



1048. Laplace observes that all his analysis rests on the as- 

 sumption that positive and negative errors are equally likely, and 

 he now proposes to shew that this limitation does not practically 

 affect the value of his results: see his pages 19 — 21. Here again 

 however it does not appear to me that much conviction would be 

 gained from Laplace's investigation. 



1049. The first Supplement closes with a section on the Pro- 

 bability of judgments; it is connected with the eleventh ChajDter : 

 see Art. 1043. 



1050. The second Supplement is entitled Application du 

 Calcul des Prohahilites aux operations geodesiques : it occupies 50 

 pages: see Art. 927. This Supplement is dated February 1818. 



This Sup23lement is very interesting, and considering the sub- 

 ject and the author it cannot be called difficult. Laplace shews 

 how the knowledge obtained from measuring a base of verification 

 may be used to correct the values of the elements of the triangles 

 of a survey. He speaks favourably of the use of repeating circles; 

 see his pages 5, 8, 20. He devotes more space than the subject 

 seems to deserve to discuss an arbitrary method proj^osed by 



39—2 



