EVOLUTWN. 153 



or the ncpct less than, the first period, a^d put the root of 

 it to the right hand o£ the given number, after the manner 

 of a quotient figure in division, and it virill be the first fig- 

 ure of the root required. 



■ 1. Subtract the assumed square from the first period, 

 and. to the remainder bring dovrn the next period for a 

 dividend. 



3. Place- 



viz, looooo ; but icocoo has one place less than 1000 and-too 

 together have : and since 1000 and 100 were taken the least pos- 

 sible, the product of any other tv/o numbers, of the same number 

 of plaqes, will be greater than i 00000 ;. consequently the product 

 of any two numbers can have, at least, but one place less than 

 both, the factors. ... 



Again, take two numbers of any number of places, that shall 

 be the greatest of those places possible, as 999 and 99. Now 

 999 X 99 ^s ^^55 ^ban 999 x 100 ; but 999 X 100 (==99900) con- 

 tains only as many places of figures, as are in 999 and 99; there- 

 fore 999 X 99 or the product of any other two numbers, consisting 

 of the same number of places, cannot have more places of figure^^^ 

 than are in both its factors. 



Corollary i. A square number cannot have more praces of 

 figures, than doable the places of the root, and, at least, but one 

 less. ., '; .u 



Cor. z. 'A cube number cannot have more places of figures 

 than triple the places of the root, and, at least, but two less. 



The truth of the rule may be shev/n algebraically thus ; 



Let iVrr the number, whose square root is to be found. 



Now, it appears from the leinma, -that there will be always g^ 

 many places of figures in the root, as there are points or periods 

 in the given number, and therefore the figures of these places may 

 be represented by letters. • ' 



u 



