Elements of Algebra

140 ELEMENTS SECT. III. other integer, leads to the theory of polygonal numbers, which are formed by adding together the terms of any such progression. 426. Suppose the difference to be 1; then, since the first term is 1 also, we shall have the arithmetical progression, $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, \ldots$ and if in this progression we take the sum of one, of two, of three, $\ldots$ terms, the following series of numbers will arise: $1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, \ldots$ for $1=1, 1+2=3, 1+2+3=6, 1+2+3+4=10, \ldots$ Which numbers are called triangular, or trigonal numbers, because we may always arrange as many points in the form of a triangle as they contain units, thus: $1 \quad 3 \quad 6 \quad 10 \quad 15$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427. In all these triangles, we see how many points each side contains. In the first triangle there is only one point; in the second there are two; in the third there are three; in the fourth there are four, $\ldots$: so that the triangular numbers, or the number of points, which is simply called the triangle, are arranged according to the number of points which the side contains, which number is called the side; that is, the third triangular number, or the third triangle, is that whose side has three points; the fourth, that whose side has four; and so on; which may be represented thus: which Fermat considered as very interesting, (see his notes on Diophantus, page 16), is this: that if from the $n$th term of any series the $(n-1)$ term of the same series be subtracted, the remainder will be the $n$th term of the preceding series. Thus, in the third series above given, the $n$th term is $\frac{n.(n+1).(n+2)}{2.3}$; consequently, the $(n-1)$ term, by substituting $(n-1)$ instead of $n$, is $\frac{(n-1).n.(n+1)}{2.3}$; and if the latter be subtracted from the former, the remainder is $\frac{n.(n-1)}{2}$, which is the $n$th term of the preceding order of numbers. The same law will be observed between two consecutive terms of any one of these sums.