590 LAPLACE. 



a^ 



Now, as in Art. 1004, we may take -^ as the greatest value 



c \/3 

 of 1c y so that the least value of r is — 77^ ; also a = 4, c = 400, 



a \l{^s) ' ' 



5\/ 3 

 s = 400 : thus the least value of r is —j^ , that is \/(S^'5). 



1 r 



Hence 1 j- \ e~^'^dt is found to be very nearly equal to 



unity. We may therefore regard it as nearly certain that the 

 sum of the excesses would fall below 400 if there were no constant 

 cause : that is we have a very high probability for the existence of 

 a constant cause. 



1019. Laplace states that in like manner he had been led 

 by the theory of probabilities to recognise the existence of con- 

 stant causes of various results in physical astronomy obtained by 

 observation ; and then he had proceeded to verify the existence 

 of these constant causes by mathematical investigations. The 

 remarks on this subject are given more fully in the Introduction, 

 pages LVII — LXX ; see Art. 938. 



1020. Laplace on his pages 359 — 362 solves BufFon's problem, 

 which we have explained in Art. 650. 



Suppose that there is one set of parallel lines ; let a be the 

 distance of two consecutive straight lines of the system, and 2r 

 the length of the rod : then the chance that the rod will fall 



across aline is — . Hence, bv Art. 993, if the rod be thrown 



down a very large number of times we may be certain that the 

 ratio of the number of times in which the rod crosses a line 



to the whole number of trials will be very nearly — : we might 



therefore determine by experiment an approximate value of tt. 



8r 

 Laplace adds . . . et il est facile de voir que le rapport — qui, 



pour un nombre donn^ de projections, rend I'erreur a craindre la 

 plus petite, est I'unite... Laplace seems to have proceeded thus. 

 Suppose^ the chance of the event in one trial; then, by Art. 993, 



