1868.] 



alluded to by Fermat, 



253 



8, 8, &c., the series will be 2, 4, 8, 12, 18, &c., and any even number will 

 be made with some 4 terms of the series. Now The Square, the subject of 

 the last paper, has a property not noticed in the former paper, viz. that the 

 first term of The Square, supposing it to be of the form 4n-\-3, will be in- 

 creased in descending down the principal diagonal into the sum of the 

 squares of the roots n, n+l, n+ I, n + l, into which the number itself may- 

 be divided ; and if the form of the number be 4n-\-l, a term which is the 

 sum of the roots n, n, n, n-\-l (into which 4w+l may be divided) would 

 appear in the diagonal next below the principal diagonal ; and as every odd 

 number is of the form of either 4?z + 3, or 4?z + l, this applies to every 

 possible odd number, and each of these numbers is a term in the series 

 already mentioned, 1, 3, 7, 13, &c., and which may be increased by any 

 even number by means of the series 2b^, 2a^-\-2a, and so on. This, it is 

 shown in the paper, may be so altered as to correspond with the index of 

 some number in the principal diagonal of the square, or the one below it, 

 and will therefore ascend to the first term in The Square, and give the sum 

 of its roots equal to 1 ; and therefore (4?z + 3 — 1) divided by 2 will be com- 

 posed of 3 trigonal numbers, and in the other case (4?i + l— 1) is divided 

 by 2 ; that is, every odd and every even number is composed of 3 trigonal 

 numbers. If this be so, Fermat's theorem of the trigonal numbers is 

 proved from the case of the squares, which (it is believed) has not been 

 done before ; but this leads to other conclusions, which are shown in the 

 paper. If 1 be the first term of The Square, every term in it will have its 

 roots of the squares that compose it of the form -\-la,a, h, h, and the term 

 itself will be composed of two trigonal numbers ; but if each of these be 

 made the first term of a square, every odd number will be found in some 

 of the resulting squares ; and it is shown that every odd number not only 

 is of the form 1 4-2a^ + 2a + 26^ + 2m^ + 2?w4-2>z^, but also of the form 

 l + 2a' + 2fl + 25' + 2mH2^w, or H-2a' + 2o! + 26'-H2n2 ; so that, with re- 

 respect to every odd number, two of the squares that compose it may be 

 equal, and also two may have their roots differing by 1 . 



There remains one other matter to be mentioned, viz. a certain remark- 

 able relation which all the polygonal numbers bear to each other, and 

 which forms a connexion that runs through them all, from which it would 

 seem to follow that a solution of the theorem as to one would be a solution 

 as to all the rest (except the first). 



This relation arises in the square numbers by a property of the gradation 

 series, already in part alluded to, viz., as to the odd numbers, by which the 

 interval between any two terms can be filled up, all the terms having, as 

 to the odd numbers, the sum of the roots of the squares that compose them 

 equal to the sum of the roots of the first term ; but the intervals, as to the 

 even numbers, may be also filled up by making the sum of the roots one less 

 than that of the roots of the odd numbers (see the Table in Diagram No. 3, 

 which is thus constructed). A term in the gradation series is assumed (in 

 this case 73) ; its roots are 4, 4, 4, 5 ; and the roots of all the odd numbers 



