immense, though scattered, literature of its own. In their simple form
of consecutive whole numbers arranged in a square so that every column,
every row, and each of the two long diagonals shall add up alike, these
magic squares offer three main lines of investigation: Construction,
Enumeration, and Classification. Of recent years many ingenious methods
have been devised for the construction of magics, and the law of their
formation is so well understood that all the ancient mystery has
evaporated and there is no longer any difficulty in making squares of
any dimensions. Almost the last word has been said on this subject. The
question of the enumeration of all the possible squares of a given order
stands just where it did over two hundred years ago. Everybody knows
that there is only one solution for the third order, three cells by
three; and Frénicle published in 1693 diagrams of all the arrangements
of the fourth order--880 in number--and his results have been verified
over and over again. I may here refer to the general solution for this
order, for numbers not necessarily consecutive, by E. Bergholt in
_Nature_, May 26, 1910, as it is of the greatest importance to students
of this subject. The enumeration of the examples of any higher order is
a completely unsolved problem.

As to classification, it is largely a matter of individual
taste--perhaps an æsthetic question, for there is beauty in the law and
order of numbers. A man once said that he divided the human race into
two great classes: those who take snuff and those who do not. I am not
sure that some of our classifications of magic squares are not almost as
valueless. However, lovers of these things seem somewhat agreed that
Nasik magic squares (so named by Mr. Frost, a student of them, after the
town in India where he lived, and also called Diabolique and
Pandiagonal) and Associated magic squares are of special interest, so I
will just explain what these are for the benefit of the novice.