RULES FOR ASCERTAINING SQUARE NUMBERS. $$% 



Prop. 4. If a square number terminate with 5 it will If with 5 the 



. , * preceding fi- 



. terminate with 25. guie m ust be 



Demonstration. For such a square number would have a 2. 

 its root ending in 5, that is, would have its root of the form 

 10 m + 5, and consequently the square number itself would 

 be of the form 100 m* -f- 100 m -f 25; where it is evident, 

 that, whatever value be given to iw, the sum of the two first 

 terms will end with two ciphers, and therefore that the whole 

 sum will terminate with 25. Q. E. D. 



Prop. 5. If a square number terminate in an odd num- If with an oaA 



ber. the last figure but one will be an even number, but jf number, the 

 O last figure but 



it terminate in any even number, except 4, the last figure one will be 



but one will be an odd number. V^lXZSfr 



it will be odd. 



Demonstration. By prop. 1 , if a square number end in 

 an odd number, it must be in 1, 5, or 9 ; and by the last 

 prop, when it ends in 5, the last figure but one will always 

 be 2, which is an even number; and when it ends in 1, or 

 9, its root must end in 1 or 3, that is, must be of the form 

 10 m -h 1, or 10 m + 3 ; and therefore, the square number 

 itself of the form 100 m* +20m+l, or 100 m* -f 60 m 

 + 9; where, whatever be the value of m, it is evident, that ' 

 the sum of the two first terms in either expression will give 

 an even number of tens; and an even number of tens plus 

 1 or 9 will have the last figure but one an even number. 

 Again, if a square number end in an even number except 

 4, by prop. 1 it can only be in 6, and its root must end in 4 

 or 6; that is, it must be of the form 10 m + 4, and conse- 

 quently the square number itself of the form 100 m* + 80 m 

 -f- l(j; where it is evident, that, whatever be the value of m, 

 the sum or difference of the two first terms will always give an 

 even number of tens, and an even number of tens plus 16 

 must have the last figure but one an odd number. Q. E. D. 



Corollary. Hence no square number can terminate with A square num- 

 two equal figures, except two fours or two ciphers. cannot ter- 



Vide cor. to prop. 6. two similar 



Prop. 6. A square number cannot terminate with more ^oro's U " 

 than three fours. and not with 



Demonstration. For, if it could end in four fours, such JJJ^ l 4s an 

 a number might be expressed by a . 10 4 -f- 4 . 10 3 -f 4 . 10* 



+ 4 



