BUFFON. 3-i7 



them ; they were communicated to the Academy in that year ; 

 see Art. 354. 



650. Buffon then proceeds to a more difficult example which 

 requires the aid of the Integi'al Calculus. A large plane area is 

 ruled with equidistant parallel straight lines ; a slender rod is 

 thrown down : required the probability that the rod will fall across 

 a line. Bufifon solves this correctly. He then proceeds to con- 

 sider what he says might have appeared more difficult, namely to 

 determine the probability when the area is ruled with a second 

 set of equidistant parallel straight lines, at right angles to the 

 former and at the same distances. He merely gives the result, 

 but it is wrong. 



Laplace, without any reference to Buffi^n, gives the problem in 

 the T]ieorie..,des Proh., pages 359 — 362. 



The problem involves a compound probability ; for the centre 

 of the rod may be supposed to fall at any point within one of 

 the figures, and the rod to take all possible positions by turning 

 round its centre : it is sufficient to consider one figure. Bufifon and 

 Laplace take the two elements of the problem in the less simple 

 order ; we mil take the other order. 



Suppose a the distance of two consecutive straight lines of one 

 system, h the distance of two consecutive straight lines of the 

 other system ; let 2r be the length of the rod and assume that 

 2r is less than a and also less than h. 



Suppose the rod to have an inclination 6 to the line of length 

 a ; or rather suppose that the inclination Hes between 6 and 

 6 + dd. Then in order that the rod may cross a line its centre 

 must fall somewhere on the area 



ah — {a — 2r cos 6) {h — 2r sin 6), 



that is on the area 



2r (a sin ^ + Z* cos 0) — h-^ sin 6 cos 0. 



Hence the whole probability of crossing the lines is 

 1 2r {a sinO + b cos 0) - 4r^ sin 6 co&6\ dO 



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