Page:Popular Science Monthly Volume 67.djvu/668

${\begin{aligned}B&=\;\;7{,}460{,}514,\qquad \qquad &b&=4{,}893{,}246,\\W&=10{,}366{,}482,\qquad \qquad &w&=7{,}206{,}360,\\D&=\;\;7{,}358{,}060,\qquad \qquad &d&=3{,}515{,}820,\\Y&=\;\;4{,}149{,}387,\qquad \qquad &y&=5{,}439{,}213,\\\end{aligned}}$

are the least numbers satisfying the conditions of the first problem. The total number of cattle is 50,389,082, not too great to graze upon the island Sicily, the area of which is about 7,000,000 acres. If the above numbers are multiplied by 80 they give the results stated in the appendix to the Wolfenbüttel manuscript.

The second or complete problem includes the determination of numbers which satisfy not only equations (1) to (7), but also

W	$+$	B	$=$	a square number,	(8)
D	$+$	Y	$=$	a triangular number,	(9)

and this is to be done by finding an integer N to multiply into each of the results of the first problem. Since W + B is 17,826,996 and D + Y is 11,507,447, these equations become

17,826,996	N	=	a square number,
11,507,447	N	=	a triangular number.

A number N that will satisfy one of these conditions can be found without difficulty, but to determine one that will satisfy both is a task requiring an enormous amount of labor and patience. In fact, this required number N has never been completely computed.

It has been claimed by some critics that the ninth condition should be rejected altogether, for they asserted that there is no evidence that Archimedes or the Greek mathematicians had the idea of a triangular number. On this hypothesis the solution is easy. Since W + B is 17,826,966 N or 4 $\times$ 4,456,749 N, and since 4,456,749 contains no number that is a perfect square, it is plain that N must be 4,456,749. Accordingly, each of the numbers found in the first solution must be multiplied by 4,456,749 in order to satisfy equations (1) to (8) inclusive; the number W + B is then 79,450,446,596,004, which is a perfect square, but the number D + Y is 51,285,802,909,803, which is not a triangular number. This solution is identical with that of Leiste as published by Lessing in 1773.

For the benefit of those who are neither novices nor of high skill in numbers, it is now time to explain what is meant by a triangular number. The number 10 is triangular because ten dots can be arranged in rows in the form of a triangle, there being one dot in the first row, two in the second, three in the third and four in the fourth. The next higher triangular number is 15 and the next 21, and in general ¹⁄₂ n (n + 1) is a triangular number whenever n is an integer, n being the number of rows parallel to one side of the triangle. The proof that 51,285,802,909,803 is not a triangular number consists in