252 



Sir F. Pollock on the Mysteries of Numbers [Feb. 13, 



A similar property belongs to all polygonal numbers ; in the trigonal 



the increase is w+ 1, 



in the square it is 2n-\-2 ^ 



in the pentagonal 821 + 3, 



and in the »2-gonal mn-\-m. 



"When the reverse operation takes place and the sum of the squares is 

 diminished, the + (plus) in the above expressions becomes — (minus). 

 There are some other modes mentioned also in dealing with the roots so as 

 to increase the sum of the squares by 2, although there be not two of the 

 roots which are equak A proof is offered, by means of a supplemental 

 square decreasing as the other increases, that if every number up to 2w-f- 1 

 has the properties of odd numbers above enumerated, then the number 

 2?i-f 3 will also possess them ; and if this be so, then every subsequent odd 

 number will likewise possess them. This is a mode of proof not un- 

 frequent in mathematical^investigations : it cannot be abbreviated ; but it 

 may be useful to state that the proof chiefly arises from this, that if one 

 term of a series corresponds with the law of it, then every term will do so, 

 and in all the series but two there will be one term obedient to the law 

 which renders all the rest so ; the other two series are treated differently. 



It is shown that if a term, in the series 1,3, 7, &c., whose terms (repre- 

 sented in the roots of the 4 squares of which they are the sum) will be 

 n, n, n {n + ^) be increased by 2, the roots being altered in the manner 

 described above, the operation may be carried on till one of the terms be- 

 comes zero (0) ; but the next term in the series will be reached before that 

 occurs. Then the next term may be taken as the beginning of another 

 similar operation, and may go on till another term is reached, and so on 

 without end. In this way the 4 squares into which any odd numbers may 

 be divided will be obtained ; and if every odd number is divisible into 4 

 squares, every even number will be so likewise. 



The next subject is considered the most material and important in the 

 paper, because it connects Lagrange's proof of the square numbers with 

 The Square (the subject of the last paper). Euler thought that no assistance 

 could be derived from the proof of Lagrange as to the other branches of 

 the theorem (see Euler, 'Opuscula Analytica,' vol. ii. p. 4). But if every 

 odd number is composed of 4 squares or less, then a number of the form 

 An-\-2 must be composed of 2, 3, or 4 squares, and in any of these cases 

 n (any number) will be equal to 4 trigonal numbers, which is shown in the 

 'paper. The expression + a + has been proved in a former paper of 

 the author to be a general expression for any 2 trigonal numbers ; and if 

 any number is composed of 4 trigonal numbers or less, a^-^a + h^-^-m^ 

 + will represent any number whatever, odd or even, and 2«^-|-2« 

 -f 25" -f 2m^ -f 2m 2n^ will represent any even number. This connects La- 

 grange's proof of the squares with The Square, which is the subject of the 

 last paper ; and if a series be composed of squares and double trigonal 

 numbers beginning with nothing, and having differences 2, 2, 4, 4, 6, 6, 



