[ 3§7 ] 
¥b is the ratio of fb to AB. But ex hypothefi ac- 
cording as the ball W falls upon F b or not the 
point o will lie between f and b or not, and there- 
fore the probability the point o will lie between f and 
b is the ratio of f b to A B. 
Again j if the rectangles Cf F b, LA are not 
commenfurable, yet the laft mentioned probability 
can be neither greater nor lefs than the ratio of j b to 
A B 3 for, if it be lefs, let it be the ratio of fc to AB, 
and upon the line fb take the points p and t, fo 
that p t fhall be greater than f c, and the three lines 
Bp, pt , t A commenfurable (which it is evident may 
be always done by dividing A B into equal parts lefs 
than half cb , and taking p and t the neareft points 
of divilion to /'and c that lie upon fb). Then 
becaufe Bp, pt, t A are commenfurable, fo are the 
rectangles C p, D t, and that upon pt compleating 
the fquare AB. Wherefore, by what has been faid, 
the probability that the point o will lie between p and 
t is the ratio of p t to A B. But if it lies between p 
and t it muft lie between f and b. Wherefore, the 
probability it fhould lie between f and b cannot be 
lefs than the ratio of pt to A B, and therefore mufl 
be greater than the ratio of /r to AB (fince pt is 
greater than fc). And after the fame manner you 
may prove that the forementioned probability cannot 
be greater than the ratio of fb to AB, it mufl there- 
fore be the fame. 
Lem. 2. The ball W having been thrown, and 
the line o s drawn, the probability of the event M 
in a fingle trial is the ratio of A o to A B. 
For, in the fame manner as in the foregoing lem- 
ma, the probability that the ball o being thrown fhall 
reft 
