ON DIRECT PROBABILITIES. 



1 13341)59049(4=4 



53364 



2 5685)13341(2X4+1=M 



11370 



2x2+1=5 1971)5685(2x9+4=22 



39*2 



5x1+2=7 1743)1971(1 x 2+9=31 



1743 



7x7+5=54. 228)1743(7x314-22=239 



1596 



54x1+7=61 147)228(1x239+31=270 



147 



81)147(1 

 81 



66 &c. 



Opposite to the first quotient write 1, and proceed tc 

 form columns out of the several quotients, in the fol- 

 lowing manner : 



1st Quotient = 1st Denomina- 

 tor. 



2d Qu. x 1st Qu. + 1 = 2d 

 Den. 



3d Qu. x2d Den. + 1st Den. 

 = 3d Den. 



4th Qu. x 3d Den. + 2d Den. 



1 = 1 st Numerator. 

 2nd Qu. = 2d Num. 



2d Num. x 3d Qu. + 1st Num. 



= 3d Num. 

 3d Num. x 4th Qu. + 2d Num. 



= 4th Num. 



4th Num. x 5th Qu. + 3d Num. 

 = 5th Num. 



Any numerator multiplied by 

 the next quotient, and pro- 

 duct increased by preceding 

 numerator, gives succeeding 

 numerator. 



= 4th Den. 



5th Qu. x 4th Den. + 3d Den. 

 = 5th Den. 



Any denominator multiplied 

 by the next quotient, and 

 product increased by pre- 

 ceding denominator, gives 

 succeeding denominator. 



Thus, 1, -|, -%, J_, -^-g, -f^ } &c. &c., is a set of 

 fractions which approach nearer and nearer to -Jilo^-g- 

 The first is always too great, the second too small, the 

 third too great, the fourth too small : every odd one too 

 great, every even one too small. 



Test of correctness. Take any two successive numer- 

 ators and denominators, multiplied crosswise ; they give 

 products which differ by unity. 



