160 AN ESSAY ON THE ROOTS OF INTEGERS, 



is increased both by the increase of r and z, so evidently its maximum is 



produced when r and z are both maxima. Now by last paragraph, since 



r is not greater than 2a, so 2a is maximum of r. And since z is any 



integer at pleasure, so Infinity is the maximum of z. Substitute these 



4 a 2 co • 

 values of r and z, and this expression becomes . 



4 a 2 co 2 -j- 4 a co -J-l 



2 a 2 a 



. Now since 2 a co -J- 1 is infinitely great, so when 



2aco -f- 1 2aco -f- 1 



a is finite, becomes infinitely small, and vanishes. And the quantity 



4 a co -(- 1 being an infinite of the first order, vanishes before 4 a" co 2 an 



4a 2 co 2 

 infinite of the second order, and the expression is reduced to zz I 



4 a 2 co ~ 



as before. But it is to be considered whether it be correct Logic to 

 ascribe positive properties to the negative idea Infinity. 



69). By this it is evident not much advantage is gained, for by 

 Par. 56) the error on one side may be -f-, and here it may be 1 on the other. 

 To correct this, and to render the error of excess as small as possible, the 

 Arabian Arithmeticians direct that z should not be taken greater than 2, 



2r 



and hence the assumed Root of a 2 -}- r is a -| ■ and the error ex- 



4a-|- 1 



(2 a z -j- 1) r — r~ z* (4 a + 1) r — 4 r 2 



pressed by becomes . If in this 



(2«H I) 2 (4 a + l) 2 



case, the deficiency is positive, then by Par. 64), it cannot be greater 



1 1 1 

 than or or . But if this expression is negative, it is evident 



4 s 2 4-2 2 16. 



that it can only become negative by the increase of r. Now as before 

 maximum of r is 2 «. Substitute this value of r, and the expression 



