The Magic Squares of Manuel Moschopoulos - The Sum of the Side

It is necessary first to speak of the Side produced by the numbers from one to the [area of the] square we seek. This value we find as follows: We add the numbers from one to the square and then divide the total of this addition by the number which when multiplied by itself gives the square. This quotient¹¹ we consider to be the Side of the numbers from one to the square we seek. For example, suppose we seek the Side of the numbers from one to 9. We therefore add 2 to one, giving 3; then add 3 to 3 which makes 6, then 4 to 6, which makes 10, and 5 to 10, which makes 15 and so on as far as 9. The total then becomes 45. This we divide by 3, since that number, multiplied by itself, gives 9. The quotient then is 15. These are the Sides of the numbers from one to 9. The method is the same in regard to other squares.

But lest, if the number happens to be very large, we grow weary of adding the numbers from one, having made inquiry, we found a method where we can easily arrive at the total of the addition of the numbers from one to as far as we wish. This method is as follows: We keep in mind the number to which the addition will proceed, and we multiply it by itself. We divide the number obtained from the multiplication into two equal parts, and then to one of these parts we add half of the number we multiplied by itself. It is a fact¹² that the result of the addition of half of the number obtained from the multiplication and [half of]¹³ the number being multiplied by itself, is equal to that obtained from the addition of the numbers from one to the number which was multiplied by itself.

This would become clearer when applied to some specific numbers, for example: Suppose again that the result of the sum of the numbers from one to 9 is sought. We multiply this number then by itself and it becomes 81. This we divide into two equal parts, so there corresponds 40 and a half to each part. Then we divide 9 into two equal parts, and there corresponds to each part 4 and a half. We add this to half of the result of the multiplication, that is to say, to 40 and a half, and together they make 45. The result of the addition of the numbers from 1 to 9 was in fact also 45. The same thing happens also on all other occasions.