16 
NEW METHOD OF FINDIXG THE 
in sucli a case, the possible squares must end in either 1, 4, 
0, 9, or 0. 0 we have dealt with above. If the last figure 
but one be odd, then to be a multiple of four, the number 
must end with either 2 or 6, so that in this case we can only 
have 6 as a possible square, and we may reject all numbers 
ending with any other figure. 
If the last figure be a 5, the last figure but one will be 2 
if the number is a square, for every square number ending 
with 5 is divisible by 25, and we have already seen that if 
the last number is 5, the last figure but one must be an even 
number, so that the last two figures cannot be 75. If the 
last figure therefore be 5, we may reject it unless the last but 
one is 2. 
No square number has its last two figures the same, 
unless they be 00 or 44 ; and if the number ends with a 0, 
it cannot be a square unless the number of cyphers is even. 
We are further helped by another criterion suggested by 
Professor Michie Smith depending on the last figure. 
Suppose the last figure of a is 9, then 9 can be pro- 
duced as a last figure by 1 x 9, 3 x 3, or 7 x 7, but by no 
others. So that the factors must have one or other of these 
pairs of figures as their last figures, and it is easy to see 
whether any possible root of a number can end in a figure 
such that, when added to and subtracted fi-om the corre- 
sponding value of X, it will produce numbers having either 
of these paii's of figures as their final figures. In this way 
we may generally reject many of the differences which are 
possible squares, and only go through the process of extract- 
ing the square root in those cases in which the necessary 
conditions are present. 
By using this method a trained computor would discover 
the two factors of the number given by Jevous in half an 
hour instead of many weeks. 
The number given by Jevons is 8,616,460,799. "We first 
take the square root of the given number. This is found to 
