Magic Squares - Introduction

MAGIC SQUARES

by Manuel Moschopoulos - 13th century

An exposition by the most learned and most blessed master Manuel Moschopoulos with regard to the invention of number squares, which he has made at the insistence of Nicolaos Artabasdos of Smyrna, arithmetician and geometer, the Rhabdas¹.

Introduction

Among the set of numbers, some are odd and some are even. Furthermore, among the even numbers, some are evenly-even² - those capable of repeated division into two equal parts, finally resulting in the number one³; and some are evenly-odd⁴  - those which are not capable of repeated division into two equal parts resulting in the number one.

Every number when multiplied by itself, makes a square of equal sides⁵. For example, the number⁶ 3 , multiplied by itself makes 9, and 9 is a square, its side⁷ being the number 3. For the side of each square is the number that when multiplied by itself, produces that square. It is possible for [the sum of the numbers along] this side⁸ to be the same in all directions even along the diagonals, but so that this may become clearer by a diagram, let a square be drawn (Fig. 1) and in it let the cells⁹ of the number square be contained by lines, thus:\[\begin{array} {| c | c | c |} \hline 1 & 1 & 1 \\ \hline 1 & 1 & 1 \\ \hline 1 & 1 & 1 \\ \hline \end{array}\quad{Side}=3\]

Figure 1

Further, let a one be placed in each of the cells, then it is altogether obvious that the addition of all these ones amounts to 9, but that the addition of (the numbers on) each of the sides in any direction gives 3, even along the diagonals. This is easy to understand. But if a square were drawn, and cells are drawn in it equal to the (side) number squared, and furthermore, there were placed not ones, but the number one, and those numbers one after another from the number one, then the Sides will no longer be equal in all directions, since the numbers in succession have been placed in the cells. Now if an arrangement were sought which could make the Sides equal in all directions, including the diagonals, this would not be found very easily. Even if after some difficulty such an arrangement were found for one square, there is no reason to expect one might be found in a different (size) square. However, if one were guided by a method, one would easily have the arrangement which produces this effect, in whatever square one likes.

There is no single method for this, but one method which works for squares arising from odd numbers, another for those arising from evenly-even numbers, and yet another for those arising from evenly-odd numbers. Concerning these things¹⁰ we now propose to speak.