662 . POPULAR SCIENCE MONTHLY. 



B= 7,460,514, 6 = 4,893,246, 



W = 10,366,482, w = 7,206,360, 



D= 7,358,060, d = 3,515,820, 



Y— 4,149,387, y = 5,439,213, 



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 Wolfenbiittel 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 -f- B is 17,826,996 and 

 D -4- Y is 11,507,447, these equations become 



17,826,996 N = a square number, 

 11,507,447 A' = 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 -4- B is 

 17,826,966 A 7 or 4 X 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) in- 

 clusive; 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 ar- 

 ranged 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 -| n{n -4- 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 



