ALLUDED TO BY EEEMAT. 
639 
There remains one other matter to be mentioned, viz. a certain remarkable 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 1 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 between 
that and the next term are found by the processes mentioned in the former part of this 
paper. The roots of the even numbers are obtained by an analogous process, and these 
are used as bases or roots of the polygonal numbers, which are placed in columns, 
with their sums, as appears in the Table. See Diagram No. 4 for the mode in which 
the polygonal numbers are formed. 
It will be observed that the sum of the roots or bases is 17, but if they be used to 
form trigonal numbers, the increment of the sum of the resulting trigonal numbers, 
above the sum of the roots or bases, is 28; and so on of the rest, each successive column 
increasing by the same number, viz. 28. If the roots or bases be n, n, n, w+1, that is, 
a term in the gradation series, the increment of the sums of the successive columns will 
be 2w 2= p^, a trigonal number. 
Again, in the trigonal numbers the difference between the sums of the first and second 
term is 0 ; in the square numbers it is 1, in the pentagonal numbers 2, in the hex- 
agonal numbers 3, in the heptagonal numbers 4; but in all of them the difference 
between the second and third terms isl, and this continues throughout. The difference 
between the 3rd and 4th, the 5th and 6th, the 7th and 8th, &c. increases by 1 in each 
column, but the difference between the 2nd and 3rd, the 4th and 5th, the 6th and 7th, 
&c. is always 1 in each column ; and the result is, that by adding 1 in the pentagonal 
column, by adding 1, or 1, 1 in the hexagonal, by adding 1, or 1, 1, or 1, 1, 1 in the 
heptagonal, every number, odd or even, can be made by not exceeding four square 
numbers, or five pentagonal numbers, or &c., as clearly appears by the Table. 
This corresponds with what was discovered by Cauchy, published at the end of 
Legendre’s “ Theorie des Nombres,” viz. that four only of each class of numbers is 
necessary, the rest may be supplied by 1, repeated as often as necessary. But I must 
not omit to say that, although all the odd numbers are sufficiently obedient, there is one 
class of even numbers quite refractory, viz. the powers of 2. They may easily be ex- 
pressed in squares, pentagonal numbers, &c., but they cannot be brought within the 
rule that otherwise prevails. 
