chapter xii : Euler, Commentationes Arithmeticae Collectae, St Petersburg, 1849,

vol. II, pp. 503—602 : Violle, Traiti Complet des Carris Magiques, 3 vols, Paris, 1837-8.
140
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[CH. VII
♦ De la Louhhe*s Method*. If the reader will look at figure iii he will see one way in which such a square containing 25 cells can be constructed. The middle cell in the top row is occupied by 1. The successive numbers are placed in their natural order in a diagonal line which slopes upwards to the right, except that (i) when the top row is reached the next number is written in the bottom row as if it came immediately above the top rov/ ; (ii) when the right-hand column is reached, the next number is written in the left-hand column, as if it immediately succeeded the right-hand column ; and (iii) when a cell which has been filled up already, or when the top right-hand square is reached, the path of the series drops to the row vertically below it and then continues to mount again. Probably a glance at the diagram in figure iii will make this clear.
17^
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0t3
10 fl
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Figure iii.
i
- V Vo Figure iv.
The reason why such a square is magic can be explained best by expressing the numbers in the scale of notation whose radix is 5 (or n, if the magic square is of the order n), except that 5 is allowed to appear as a unit-digit and 0 is not allowed to appear as a unit-digit. The result is shown in figure iv. From that figure it will be seen that the method of construc- tion ensures that every row and every column shall contain one and only one of each of the unit-digits 1, 2, 3, 4, 5, the sum of which is 15 ; and also one and only one of each of the radix- digits 0, 5, 10, 15, 20, the sum of which is 50. Hence, as
* De la Loubere, Du Royaume de Siam (Eng. Trans.), London, 1693, vol. n, pp. 227 — 247. De la Loubere was the envoy of Louis XiV to Siam in 1687-8, and there learnt this method.
CH. VIl] MAGIC SQUARES 141
far as rows and columns are concerned, the square is magic. Moreover if the square is odd, each of the diagonals will contain one and only one of each of the unit-digits 1, 2, 3, 4, 5. Also the leading diagonal will contain one and only one of the radix-digits 0, 5, 10, 15, 20, the sum of which is 50; and if, as is the case in the square drawn above, the number 10 is the radix-digit to be added to the unit-digits in the right diagonal, then the sum of the radix-digits in that diagonal is also 50. Hence the two diagonals also possess the magical property.
And generally if a magic square of an odd order n is constructed by De la Loubere's method, every row and every column must contain one and only one of each of the unit- digits 1, 2, 3, ..., 72; and also one and only one of each of the radix-digits 0, n, 2n, ..., n (n — 1). Hence, as far as rows and columns are concerned, the square is magic. Moreover each diagonal will either contain one and only one of the unit-digits or will contain n unit-digits each equal to ^ (n + 1). It will also either contain one and only one of the radix-digits or will contain n radix-digits each equal to ^n(n — l). Hence the two diagonals will also possess the magical property. Thus the square will be magic.
I may notice here that, if we place 1 in any cell and fill up the square by De la Loubere's rule, we shall obtain a square that is magic in rows and in columns, but it will not in general be magic in its diagonals.
It is evident that other squares can be derived from De la Loubere's square by permuting the symbols properly. For instance, in figure iv, we may permute the symbols 1, 2, 3, 4, 5 in 5 ! ways, and we may permute the symbols 0, 5, 15, 20 in 4 ! ways. Any one of these 5 ! arrangements combined with any one of these 4 ! arrangements will give a magic square. Hence we can obtain 2880 inagic squares of the fifth order of this kind, though only 720 of them are really distinct. Other squares can however be deduced, for it may be noted that from any magic square, \xiiether even or odd, other magic squares of the same order can be formed by the mere inter-
14j2 magic squares [CH. VII
chano-e of the row and the column which intersect in a cell on a diagonal with the row and the column which intersect m the complementary cell of the same diagonal.
*; Bachet proposed* a similar rule. In this, we begin by placing 1 in the cell above the middle one, and then we write the successive numbers in a diagonal line sloping upwards to the right, subject to the condition that when the cases (i) and (ii) mentioned under De la Loubere's method occur the rules there given are followed, but when the case (iii) occurs the path of the series, instead of going on to the cell already occupied, is continued from one cell to the cell next but one vertically above it. If this cell is above the top row the path continues from the corresponding cell in one of the bottom two rows following the analogy of rule (i) in De la Loubere's method. Bachet's method leads ultimately to this arrangement ; except that the rules are altered so as to make the line slope down- wards. This method also gives 720 magic squares of the fifth order.
In the notation given later (see pp. 157, 158), De la Loubere's rule is equivalent to taking steps a = — 1, 6 = 1, and cross-steps x = l, y = 0. Bachet's form of it, as here enunciated, is equiva- lent to. taking steps a = - 1, 6 = 1, and cross-steps a? = - 2,
2/ = 0.
- De la Hires Method^. I shall now give another rule for the formation of odd magic squares. To form an odd magic square of the order n by this method, we begin by constructing two subsidiary squares, one of the unit-digits, 1, 2, ..., n, and the other of multiples of the radix, namely, 0, /i, 2/i, . . . , (n — 1 ) n. We then form the magic square by adding together the numbers in the corresponding cells in the two subsidiary squares.
De la Hire gave several ways of constructing such sub- sidiary squares. I select the following method (props, x and xiv of his memoir) as being the simplest, but I shall apply it to form a square of only the fifth order. It leads to the same results as the second of the two rules given by Moschopulus.
* Bachet, Problem xxi, p. 161.
t Meinoires de I'Academie des Sciences for 1705, part i, pp. 127—171.
CH. VIl]
MAGIC SQUARES
143
The first of the subsidiary squares (figure v, below) is constructed thus. First, 3 is put in the top left-hand corner, and then the numbers 1, 2, 4, 5 are written in the other cells of the top line (in any order). Next, the number in each cell of the top line is repeated in all the cells which lie in a diagonal line sloping downwards to the right (see figure v) according to the rule (ii) in De la Loubere's method. The cells filled by the same number form a broken diagonal. It follows that every row and every column contains one and only one 1, one and only one 2, and so on. Hence the sum of the numbers in every row and m every column is equal to 15; also, since we placed 3, which is the average of these numbers, in the top left-hand corner, the sum of the numbers in the left diagonal is 15;
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First Subsidiary Square. Figure v.
Second Subsidiary Sqxiare. Resulting Magic Square. Figure vi. Figure vii.
and, since the right diagonal contains one and only one of each of the numbers 1, 2, 3, 4, and 5, the sum of the numbers in that diagonal also is 15.
The second of the subsidiary squares (figure vi) is con- structed in a similar way with the numbers 0, 5, 10, 15, and 20, except that the mean number 10 is placed in the top right- hand corner ; and the broken diagonals formed of the -same numbers all slope downwards to the left. It follows that every row and every column in figure vi contains one and only oneL 0, one and only one 5, and so on ; hence the sum of the numbers in every row and every column is equal to 50. Also the sum of the numbers in each diagonal is equal to 50.
If now we add together the numbers in the corre6ponding cells of these two squares, we shall obtain 25 numbers such
144 MAGIC SQUARES [CH. VII
that the sum of the numbers in every row, every column, and each diagonal is equal to 15 + 50, that is, to 65. This is represented in figure vii. Moreover, no two cells in that figure contain the same number. For instance, the numbers 21 to 25 can occur only in those five cells which in figure vi are occu- pied by the number 20, but the corresponding cells in figure v contain respectively the numbers 1, 2, 3, 4, and 5; and thus in figure vii each of the numbers from 21 to 25 occurs once and only once. De la Hire preferred to have the cells in the subsidiary squares which are filled by the same number con- nected by a knight's move and not by a bishop's move ; and usually his rule is enunciated in that form.
By permuting the numbers 1, 2, 4, 5 in figure v we get 4 ! arrangements, each of which combined with that in figure vi would give a magic square. Similarly by permuting the numbers 0, 5, 15, 20 in figure vi we obtain 4! squares, each of which might be combined with any of the 4! arrangements deduced from figure V. Hence altogether we can obtain in this way 144 magic squares of the fifth order.
There are various other methods by which odd magic squares of any order can be constructed, but most or all of them depend on the form of n. I content myself here with the two methods described above; later, when discussing pandiagonal squares, I shall mention another rule for odd magic squares whose order is higher than three, which permits us to place a selected number in any cell we like.
Magic squares of an even order. The above methods are inapplicable to squares of an even order. I proceed to give two methods for constructing any even magic square of an order higher than two.
It will be convenient to use the following terms. Two rows which are equidistant, the one from the top, the other from the bottom, are said to be complementary. Two columns which are equidistant, the one from the left-hand side, the other from the right-hand side, are said to be complementary. Two cells in the same row, but in complementary columns, are said to be horizontally related. Two cells in the same column,
CH. VIl] MAGIC SQUARES 145
but in complementary rows, are said to be vertically related, Two cells in complementary rows and columns are said to be skewly related ; thus, if the cell h is horizontally related to the cell a, and the cell d is vertically related to the cell a, then the cells h and d are skewly related; in such a case if the cell c is vertically related to the cell 6, it will be horizontally related to the cell d, and the cells a and c are skewly related: the cells a, h, c, d constitute an associated group, and if the square is divided into four equal quarters, one cell of an associated group is in each quarter.
A horizontal interchange consists in the interchange of the numbers in two horizontally related cells. A vertical inter- change consists in the interchange of the numbers in two vertically related cells. A skew interchange consists in the interchange of the numbers in two skewly related cells. A cross interchange consists in the change of the numbers in any ^eTland in its horizontally related cell with the numbers in the cells skewly related to them ; hence, it is equivalent to two vertical interchanges and two horizontal interchanges.
First Method*, This method is the simplest with which I am acquainted. Begin by filling the cells of the square with the numbers 1, 2, ...,7i^in their natural order commencing (say) with the top left-hand corner, writing the numbers in each row from left to right, and taking the rows in succession fi:om the top. I will commence by proving that a certain number of horizontal and vertical interchanges in such a square must make it magic, and will then give a rule by which the cells whose numbers are to be interchanged can be at once picked out.
First we may notice that the sum of the numbers in each diagonal is equal to N^ where iV = ?i(?i^-|- l)/2; hence the diagonals are already magic, and will remain so if the numbers therein are not altered.
Next, consider the rows. The sum of the numbers in the
* It seems to have been first enunciated in 1889 by W. Firth, but later was independently discovered by various writers: see the Messenger of Mathematics, Cambridge, September, 1893, vol. xxm, pp. 65 — 69, and the Monist, Chicago, 1912, vol. XXII, pp. 53 — 81. I leave my account as originally written, though perhaps the procedure used by C. Planck in the latter paper is somewhat simpler.
B. R. 10
146 MAGIC SQUARES [CH. VII
a^tli row fi'om the top is N — n^ (n — 1x 4- 1)/2. The sum of the numbers in the complementary row, that is, the a;th row from the bottom, is iV + 71^ (n — 2a; + l)/2. Also the number in any cell in the a?th row is less than the number in the cell vertically related to it by 71 (?i — 2^7 + 1). Hence, if in these two rows we make 7?/2 interchanges of the numbers which are situated in vertically related cells, then we increase the sum of the numbers in the icth row by ti x ?i {n — 2a) + l)/2, and therefore make that row magic ; while we decrease the sum of the numbers in the complementary row by the same number, and therefore make that row magic. Hence, if in every pair of complementary rows we make ?i/2 interchanges of the numbers situated in vertically related cells, the square will be made magic in rows. But, in order that the diagonals may remain magic, either we must leave both the diagonal numbers in any row unaltered, or we must change both of them with those in the cells vertically related to them.
The square is now magic in diagonals and in rows, and it remains to make it magic in columns. Taking the original arrangement of the numbers (in their natural order) we might have made the square magic in columns in a similar way to that in which we made it magic in rows. The sum of the numbers originally in the yth column from the left-hand side is N — n{n — 2y + 1)/2. The sum of the numbers originally in the complementary column, that is, the yih column from the right- hand side, is H + n(n— 2y + l)/2. Also the number originally in any cell in the yth column was less than the number in the cell horizontally related to it by n — 2y + 1. Hence, if in these two columns we had made n/'2 interchanges of the numbers situated in horizontally related cells, we should have made the sum of the numbers in each column equal to N. If we had done this in succession for every pair of complementary columns, we should have made the square magic in columns. But, as before, in order that the diagonals might remain magic, either we must have left both the diagonal numbers in any column unaltered, or we must have changed both of them with those in the cells horizontally related to them.
CH. VIl]
MAGIC SQUARES
147
It remains to show that the vertical and horizontal inter- changes, which have been considered in the last two paragraphs, can be made independently, that is, that we can make these interchanges of the numbers in complementary columns in such a manner as will not affect the numbers already interchanged in complementary rows. This will require that in every column there shall be exactly n/2 interchanges of the numbers in vertically related cells, and that in every row there shall be exactly n/2 interchanges of the numbers in horizontally related cells. I proceed to show how we can always ensure this, if n is greater than 2. I continue to suppose that the cells are initially filled with the numbers 1, 2, ..., ti^ in their natural order, and that we work from that arrangement.
A doubly-even square is one where n is of the form 4m. If the square is divided into four equal quarters, the first quarter will contain 2m columns and 2m rows. In each of these columns take m cells so arranged that there are also m cells in each row^j and change the numbers in these 2??i^ cells and the 6m^ cells associated with them by a cross interchange. The result is equivalent to 27n interchanges in every row and in every column, and therefore renders the square magic.
One way of selecting the 2m^ cells in the first quarter is to divide the whole square into sixteen subsidiary squares each
a
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♦ a
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Figure viii.
containing m' cells, which we may represent by the diagram above, and then we may take either the cells in the a squares or those in the b squares ; thus, if every number in the eight a squares is interchanged with the number skewly related to it the resulting square is magic. A magic square of the eighth order, constructed in this way, is shown in figure xxiii on page 164.
10—2
148 MAGIC SQUARES [CH. VII
Another way of selecting the 2m^ cells in the first quarter would be to take the first m cells in the first column, the cells 2 to m+1 in the second column, and so on, the cells m+1 to 2/?i in the (m + l)th column, the cells m + 2 to 2m and the first cell in the (m + 2)th column, and so on, and finally the 2mth cell and the cells 1 to m — 1 in the 2mth column.
A singly-even square is one where n is of the form 2 (2m + 1). If the square is divided into four equal quarters, the first quarter will contain 2m + 1 columns and 2m + 1 rows. In each of these columns take m cells so arranged that there are also m cells in each row: as, for instance, by taking the first m cells in the first column, the cells 2 to m + 1 in the second column, and so on, the cells m + 2 to 2m + 1 in the {m + 2)th column, the cells m + 3 to 2m + 1 and the first cell in the (m + 3)th column, and so on, and finally the (2m + l)th cell and the cells 1 to m — 1 in the (2m+l)th column. Next change the numbers in these m (2m + 1) cells and the 3m {2m + 1) cells associated with them by cross interchanges. The result is equivalent to 2m inter- changes in every row and in every column. In order to make the square magic we must have n/2, that is, 2m + 1, such inter- changes in every row and in every column, that is, we must have one more interchange in every row and in every column. This presents no difficulty; for instance, in the arrangement indicated above the numbers in the (2m + l)th cell of the first column, in the first cell of the second column, in the second cell of the third column, and so on, to the 2mth cell in the {2m + l)th column may be interchanged with the numbers in their vertically related cells ; this will make all the rows magic. Next, the numbers in the 2mth cell of the first column, in the (2m + l)th cell of the second column, in the first cell of the third column, in the second cell of the fourth column, and so on, to the (2m — l)th cell of the (2m + l)th column may be interchanged with those in the cells horizontally related to them; and this will make the columns magic without affecting the magical properties of the rows.
It will be observed that we have implicitly assumed that m is not zero, that is, that n is greater than 2; also it would seem
CH. VIlJ MAGIC SQUARES 149
that, if w = 1 and therefore n= 6, then the numbers in the diagonal cells must be included in those to which the cross interchange is applied, but, if n is greater than 6, this is not necessary, though it may be convenient.
The construction of odd magic squares and of doubly-even magic squares is very easy. But though the rule given above for singly-even squares is not difficult, it is tedious of applica- tion. It is unfortunate that no more obvious rule — such, for instance, as one for bordering a doubly-even square — can be suggested for writing down instantly and without thought singly-even magic squares.
De la Hires Method*. I now proceed to give another way, due to De la Hire, of constructing any even magic square of an order higher than two.
In the same manner as in his rule for making odd magic squares, we begin by constructing two subsidiary squares, one of the unit-digits 1,2,3, ..., n, and the other of the radix- digits 0, n, 2n, ..., (n— l)n. We then form the magic square by adding together the numbers in the corresponding cells in the two subsidiary squares. Following the analogy of the notation used above, two numbers which are equidistant from the ends of the series 1, 2, 3, ..., n are said to be comple- mentary. Similarly numbers which are equidistant from the ends of the series 0, n, 2n, ..., (n — l)n are said to be comple- mentary.
For simplicity I shall apply this method to construct a magic square of only the sixth order, though an exactly similar method will apply to any even square of an order higher than the second.
The first of the subsidiary squares (figure ix) is constructed
as follows. First, the cells in the leading diagonal are filled
with the numbers 1, 2, 3, 4, 5, 6 placed in any order whatever
that puts complementary numbers in complementary positions
* Tlie rule is due to De la Hire (part 2 of his memoir) and is given by Montucla in his edition of Ozanam's work : I have used the modified enunci- ation of it inserted in Labosne's edition of Bachet's Problemes, as it saves the introduction of a third subsidiary square. I do not know to whom the modifi- cation is due.
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{ex. gr. in the order 2, 6, 3, 4, 1, 5, or in their natural order 1, 2, 3, 4, 5, 6). Second, the cells vertically related to these are filled respectively with the same numbers. Third, each of the remaining cells in the first vertical column is filled either with the same number as that already in two of them or with the complementary number {ex. gr. in figure ix with a "1" or a "6") in any way, provided that there are an equal number of each of these numbers in the column, and subject also to proviso (ii) mentioned in the next paragraph. Fourth, the cells hori- zontally related to those in the first column are filled with the complementary numbers. Fifth, the remaining cells in the second and third columns are filled in an analogous way
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First Subsidiary Square. Figure ix.
Second Subsidiary Square. Figure x.
Resulting Magic Square. Figure xi.
to that in which those in the first column were filled : and then the cells horizontally related to them are filled with the comple- mentary numbers. The square so formed is necessarily magic in rows, columns, and diagonals.
The second of the subsidiary squares (figure x) is con- structed as follows. First, the cells in the leading diagonal are filled with the numbers 0, 6, 12, 18, 24, 30 placed in any order whatever that puts complementary numbers in comple- mentary positions. Second, the cells horizontally related to them are filled respectively with the same numbers. Third, each of the remaining cells in the first horizontal row is filled either with the same number as that already in two of them or with the complementary number {ex. gr. in figure x with a "0" or a "30") in any way, provided (i) that there are an equal
CH. VIl] MAGIC SQUARES 151
number of each of these numbers in the row, and (ii) that if any cell in the first row of figure ix and its vertically related cell are filled with complementary numbers, then the corre- sponding cell in the first row of figure x and its horizontally relat/ed cell must be occupied by the same number*. Fourth, the cells vertically related to those in the first row are filled with the complementary numbers. Fifth, the remaining cells in the second and the third rows are filled in an analogous way to that in which those in the first row were filled: and then the cells vertically related to them are filled with the complementary numbers. The square so formed is necessarily magic in rows, columns, and diagonals.
It remains to show that proviso (ii) in the third step de- scribed in the last paragraph can be satisfied always. In a doubly-even square, that is, one in which n is divisible by 4, we need not have any complementary numbers in vertically related cells in the first subsidiary square unless we please, but even if we like to insert them they will not interfere with the satisfac- tion of this proviso. In the case of a singly-even square, that is, one in which n is divisible by 2, but not by 4, we cannot satisfy the proviso if any horizontal row in the first square has all its vertically related squares, other than the two squares in the diagonals, filled with complementary numbers. Thus in the case of a singly-even square it will be necessary in con- structing the first square to take care in the third step that in every row at least one cell which is not in a diagonal shall have its vertically related cell filled with the same number as itself: this is always possible if n is greater than 2.
The required magic square will be constructed if in each cell we place the sum of the numbers in the corresponding cells of the subsidiary squares, figures ix and x. The result of this is given in figure xi. The square is evidently magic. Also every number from 1 to 36 occurs once and only once, for the numbers from 1 to 6 and from 31 to 36 can occur only in the top or the bottom rows, and the method of construction ensures that the
* The insertion of this step evades the necessity of constructing (as Montucla did) a third subsidiary square.
J
152 MAGIC SQUARES [CH. VII
same number cannot occur twice. Similarly the numbers from 7 to 12 and from 25 to 30 occupy two other rows, and no number can occur twice ; and so on. The square in figure i on page 137 may be constructed by the above rules; and the reader will have no difficulty in applying them to any other even square.
Other Methods for Constructing any Magic Square. The above methods appear to me to be the simplest which have been proposed. There are however two other methods, of less generality, to which I will allude briefly in passing. Both depend on the principle that, if every number in a magic square is multiplied by some constant, and a constant is added to the product, the square will remain magic.
The first method applies only to such squares as can be divided into smaller magic squares of some order higher than two. It depends on the fact that, if we know how to construct magic squares of the mth and nth. orders, we can construct one y of the mnth order. For example, a square of 81 cells may be V considered as composed of 9 smaller squares each containing 9 cells, and by filling the cells in each of these small squares in the same relative order and taking the small squares themselves in the same order, the square can be constructed easily. Such squares are called Composite Magic Squares.
The second method, which was introduced by Frenicle, con- sists in surrounding a magic square with a Border, Thus in figure xii on page 153 the inner square is magic, and it is surrounded with a border in such a way that the whole square is also magic. In this manner from the magic square of the 3rd order we can build up successively squares of the orders 5, 7, 9, &c., that is, any odd magic square. Similarly from the magic square of the 4th order we can build up successively any higher even magic square.
If we construct a magic square of the first n^ numbers by bordering a magic square of (n — 2)^ numbers, the usual process is to reserve for the 4(7i— 1) numbers in the border the first 2 (n - 1) natural numbers and the last 2 (n — 1) numbers. Now the sum of the numbers in each line of a square of the order
CH. VIl]
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(n - 2) is i (?? - 2) {(n - 2)»+ 1}, and the averasre is J{(n-2)Hll. Similarly the average number in a square of the t?th order is J (n^ + 1). The difference of these is 2 (n — 1). We begin then by taking any magic square of the order (n — 2), and we add to every number in it 2(n — 1); this makes the average number
The numbers reserved for the border occur in pairs, n^ and 1, n* — 1 and 2, n^ — 2 and 3, &c., such that the average of each pair is ^ (n^ + 1), and they must be bordered on the square so that these numbers are opposite to one another. Thus the bordered square will be necessarily magic, provided that the sum of the numbers in two adjacent sides of the external border is correct. The arrangement of the numbers in the borders will be somewhat facilitated if the number n^-\-l—p (which has to be placed opposite to the number p) is denoted by p, but it is not worth while going into further details here.
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Bordered Magic Square. Figure xii.
Since a doubly-even square can be formed with ease, this method would provide a simple way of constructing a singly- even square if definite rules for bordering a square could be laid down. I am however unable to formulate precise rules for this purpose. It will illustrate sufficiently the general method if I explain how the square in figure xii is constructed. A magic square of the third order is formed by De la Loubere's rule, and to every number in it 8 is added: the result is the inner square in figure xii. The numbers not used are 25 and 1, 24 and 2, 23 and 3, 22 and 4, 21 and 5, 20 and 6, 19 and 7, 18 and 8. The sum of each pair is 26, and obviously they must be placed at opposite ends of any line.
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[CH. VII
I believe that with a little patience a magic square of any order can be thus built up, and of course it will have the property that, if each border is successively stripped off, the square will still remain magic. This is the method of con- struction commonly adopted by self-taught mathematicians, some of whom seem to think that the empirical formation of such squares is a valuable discovery.
I may add here (figure xiii) the following general solution of a magic square of the fourth order in which any numbers (not necessarily consecutive) are used*. There are eight indepen- dent quantities.
V~v
Y+v + y
X+x-y
Z^x
Z + v-z
X
Y
V-v+z
Y-x + z
V
Z
X+x - z
X+x
Z-v-y
V-x + y
Y+v
32768
4
2
4096
16
512
1024
128
256
32
64
2048
8
16384
8192
1
Figure xiii.
Figure xiv.
I may also mention that Montucla suggested the construc- tion of squares whose cells are occupied by numbers such that the product of the numbers in each row, column, and diagonal is constant. The formation of such figures is immediately deducible from that of magic squares, for if the consecutive numbers in a magic square are replaced by consecutive powers of any number m the products of the numbers in every line will be magic. This is obvious, for if the numbers in any line of a square are a, a\ a" , &c., such that %a is constant for every line in the square, then TimP' is also constant. For instance, figure xiv represents such a magic figure in which the product of the numbers in each line is 1,073,741,824 ; it is constructed from the magic square given in figure i on page 137.
Magic Stars. Some elegant magic constructions on star- shaped figures (pentagons, hexagons, &c.) may be noticed in passing, though I will not go into details. One instance will suffice. Suppose a re-entrant octagon is constructed by the
• E. Bergholt, Nature, London, vol. Lxxxm, Maj 26, 1910, pp. 368—369.
CH. Vllj
MAGIC SQUARES
155
intersecting sides of two equal concentric squares. It is required to place the first 16 natural numbers on the corners and points of intersection of the sides so that the sum of the numbers on the corner of each square and the sum of the numbers on eycry
15
Magic Star. Figure xv.
side of each square is equal to 34. Eighteen fundamental solu- tions exist. One of these is given above*.
There are magic circles, rectangles, crosses, diamonds, and other figures : also magic cubes, cylinders, and spheres. The theory of the construction of such figures is of no value, and I cannot spare the space to describe rules for forming them.
In the above sketch, two questions remain unsolved. One is the determination of a definite rule for bordering a square ; such a rule might lead to a simpler method than that given above for forming oddly-even squares. The other is the determination of the number of magic squares of the fifth (or any higher) order. There is, in effect, only one magic square of the third order though by reflexions and rotations it can be presented in 9 forms. There are 880 magic squares of the fourth order, but by reflexions and rotations these v/ can be presented in 7040 forms. De la Hire showed that,
* Communicated to me by Mr K. Strachey.
156 MAGIC SQUARES [CH. VII
apart from mere reflexions and rotations, there were 57600 magic squares of the fifth order which could be formed by the methods he enumerated. Taking account of other methods, it would seem that the total number of magic squares of the fifth order is very large, perhaps exceeding half a million. Notwith- ' standing these two unsolved problems we may fairly say that the theory of the construction of magic squares as defined above has been worked out in sufficient detail — though not exhaustively, since methods other than those given above may be expounded. Accordingly attention has of late been chiefly directed to the construction of squares which, in addition to being magic, satisfy other conditions. I shall term such squares hyper-magic.
Hypee-Magic Squares. Of hyper-magic squares, I will deal only with the theory of Pan-Diagonal and of Symmetrical Squares, though I will describe without going into details what are meant by Doubly and Trebly Magic Squares.
Pandiagonal Squares. One of the earliest additional
I conditions to be suggested was that the square should be magic
) along the broken diagonals as well as along the two ordinary
diagonals*. Such squares are called Pandiagonal. They are
also known as Nasik, or perfect, or diabolic squares.
For instance, a magic pandiagonal square of the fourth order is represented in figure ii on page 137. In it the sum of the numbers in each row, column, and in the two diagonals is 34, as also is the sum of the numbers in the six broken diagonals formed by the numbers 15, 9, 2, 8, the numbers 10, 4, 7, 13, the numbers 3, 5, 14, 12, the numbers 6, 4, 11, 13, the numbers 3, 9, 14, 8, and the numbers 10, 16, 7, 1.
It follows from the definition that if a pandiagonal square be cut into two pieces along a line between any two roAvs or
* Squares of this type were mentioned by De la Hire, Sauveur, and Euler. Attention was again called to tbem by A. H. Frost in the Quarterly Journal of Mathematics, London, 1878, vol. xv, pp. 34—49, and subsequently their properties have been discussed by several writers. Besides Frost's papers I have made considerable use of a paper by E. McClintock in the American Journal of Mathe- matic^, vol. xix, 1897, pp. 99—120.
CH. VIl]
MAGIC SQUARES
157
any two columns, and the two pieces be interchanged, the new square so formed will be also pandiagoiially magic. Hence it is obvious that by one vertical and one horizontal transj)osition of this kind any number can be made to occupy any specified cell.
Pandiagonal magic squares of an odd order can be con- structed by a rule somewhat analogous to that given by De la Loubere, and described above. I proceed to give an outline of the method.
If we write the numbers in the scale of notation whose radix is w, with the understanding that the unit-digits run from 1 to n, it is evident, as in the corresponding explanation of why De la Loubere's rule gives a magic square, that all we have to do is to ensure that each row, column, and diagonal (whether broken or not) shall contain one and only one of each of the unit-digits, as also one and only one of each of the radix-digits.
7
20
3
11
24
13
21
9
17
5
19
2
15
23
6
25
8
16
4
12
1
14
22
10
18
5 + 2
15 + 5
0+3
10 \-\
20 + 4
10 + 3
20+1
5 + 4
15 + 2
0 + 5
15 + 4
042
10 f 5
20 + 3
5 + 1
20 + 5
5 + 3
15+1
0 + 4
10 + 2
0 + 1
10 + 4
20 f 2
5 + 5
15 + 3
Figure xvi.
Figure xvii.
This is seen to be the case in the square of the fifth order delineated above, figures xvi and xvii.
Let us suppose that we write the numbers consecutively, and proceed from cell to cell by steps, using the term step (a, h) to denote going a cells to the right and 6 cells up. Thus a step (a, h) will take us from any cell to the ath column to the right of it, to the ^th row above it, to the (6 -I- a)th diagonal above it sloping down to the right, and to the (6 — a)th diagonal above it sloping down to the left. In all cases we have the convention, as in De la Loubere's rule, that the move- ments along lines are taken cyclically ; thus a step n + a is equivalent to a step a. Of course, also, if a means going a cells
158 MAGIC SQUARES [CH. VII
to the right, then — a will mean going a cells to the left ; thus if the 6th upper line is outside the square we take it as equivalent to the (n — 6)th lower line.
It is clear that a and b cannot be zero, and that if a and b are prime to n (that is, if each has no divisor other than unity which also divides n) we can make n — 1 steps from any cell from which we start, before we come to a cell already occupied. Thus the first n numbers form a path which will give a different unit-digit in every row, column, and in one set of n diagonals ; of the other diagonals, n — 1 are empty, and one contains every unit-digit — thus they are constructed on magical lines. We must take some other step (h, k) from the cell n to get to an unoccupied cell in which we place the number w -h 1. Con- tinuing the process with n — \ more steps (a, b) we get another series of n numbers in various cells. If h and h are properly selected this second series will not interfere with the first series, and the rows, columns, and diagonals, as thus built up, will continue to be constructed on magical lines provided h and k are chosen so that the same unit-digit does not appear more than once in any row, column, and diagonal. We will suppose that this can be done, and that another cross-step Qi, k) of the same form as before enables us to continue filling in the numbers in compliance with the conditions, and that this process can be continued until the square is filled. If this is possible, the whole process will consist of n series of n steps, each series consisting oi n — \ uniform steps (a, b) followed by one cross-step (A, k). The numbers inscribed after the n cross-steps will be n + 1, 2w -♦- 1, 3n + l, ..., and these will be themselves connected by uniform steps {u, v\ where u = {n — \)a-\-h^k— a, mod. n, and v={n — \)b-{-k=k — b.
I proceed to investigate the conditions that a, 6, A, and k must satisfy in order that the square can be constructed as above described with uniform steps (a, b) and {h, k). We notice at once that in order to secure the magic property in the rows and columns, we must have a and b prime to n ; and to secure it in the diagonals, we must have a and b unequal and b-¥ a and 6 — a prime to n. The leading numbers of the
CH. VII] MAGIC SQUARES 159
n sequences of n numbers, namely 1, n-\-l, 2n+ 1, ..., are con- nected by steps (u, v), where « = h — a and v = k — b. Hence, if these are to fit in their places, we must also have u and v unequal, and u, v, n + v, and u — v prime to n. Also a, b, u, and V cannot be zero. Lastly the cross-steps (h, k) must be so chosen that in no case shall a cross-step lead to a cell already occupied. This would happen, and therefore the rule would fail, if p steps (a, h) from any cell and q steps {u, v) from it, w^here p and q are each less than n, should lead to the same cell. Thus, to the modulus n, we cannot have pa = qit, = q(h — a), and at the same time pb = qv = q {k — b).
It is impossible to satisfy these conditions if n is equal to 3 or to a multiple of 3. For a and b are to be unequal, not zero, and less than n, and a + 6 is to be less than n and prime to n. Thus we cannot construct a pandiagonal square of the third order.
Next I will show that, if n is not a multiple of 3, these conditions are satisfied when a = l, 6 = 2, h=0, k= — l, and therefore that in this case these values provide a particular solution of the general problem. It is at once obvious that in this case a and b are unequal, not zero, and prime to n, that 6 -f a and b — a are prime to n, and that the correspond- ing relations for u and v are true. The remaining condition for the validity of a rule based on these particular steps is that it shall be impossible to find integral values of p and q each less than n, which will simultaneously make p = ^q, and 2p = — dq. This condition is satisfied. Hence, any odd pan- diagonal square of an order which is not a multiple of 3 can be constructed by this rule. Thus, to form a pandiagonal square of the fifth order we may put 1 in any cell; proceed by four successive steps, like a knight's move, of one cell to the right and two cells up, writing consecutively numbers 2, 3, 4, 5 in each cell, until we come to a cell already occupied ; then take one step, like a rook's move, one cell down, and so on until the square is filled. This is illustrated by the square delineated in figure xvi.
Further discussion of the general case depends on whether or not n is prime ; here I will confine myself to the simpler
160
MAGIC SQUARES
[CH. VII
alternative, and assume that n is prime : this will sufficiently illustrate the theory. From the above relations it follows that we cannot have pqa (k — h) = pqb (h — a), that is, ^iq (ak — bh) = 0. Therefore ak — bh cannot be a multiple of n, that is, it must be prime to n. If this condition is fulfilled, as well as the other conditions given above, each cross-step {h, k) can be made in due sequence, and the square can be constructed. The result that ak — bh is prime to n shows that the cross-step (/i, k) must be chosen so as to take us to an unoccupied ce'^ not in the same row, column, or diagonal (broken or not) as the initial number. By noting this fact we can in general place any two given numbers in two assigned cells.
There are some advantages in having the cross-steps uniform with the other steps, since, as we shall see later, the square can then be written in a form symmetrical about the centre. This will be effected if we take h — — h,k = a. If n is prime our conditions are then satisfied if h be any number jfrom 2 to {n — 1)/2, if a be positive and less than b, and if a^ + 6^ be prime to n. We can, if we prefer, take h = h, k = — a; but it is not possible to take h = a and k = ^b, or h = — a and k = b, since they make u=iO or v = 0.
For instance, if we use a knight's move, we may take a—1, 6 = 2. The square of the seventh order given below (figure xviii)
35
23
18
13
1
45
40
4
48
36
31
26
21
9
22
17
12
7
44
39
34
47
42
30
25
20
8
3
16
11
6
43
38
33
28
41
29
24
19
14
2
46
10
5
49
37
32
27
15
A Pandiagonal Symmetrical Square. Figure xviii.
is constructed by this rule. But in the case of a square of the fifth order we cannot use a knight's move, since, if a= 1, and 6 = 2, we have a^ + b^ = 5. Hence the use of a knight's move is not applicable when n is 5, or a multiple of 5.
CH. VII]
MAGIC SQUARES
101
The construction of singly-even pandiagonal sqnares (that is those whose order is 4m + 2) is impossible, but that of doubly- even squares (that is, those whose order is 4m) is possible
Here is one way of constructing a doubly-even square. Suppose the order of the square is 4m. and as before let us write the number in a cell in the scale 4m, that is, as 4mp + r so that p and r are the radix and unit-digits, with the conven- tion that r- cannot be zero. Place p„p„p„ ..„ p^^ ;„ ^rder m the cells m the bottom row. Proceeding from p. by steps (^m, 1) fill up 2m cells with it. And proceed similarly with the other radix-digits. Next place r„ n, ..., r,„ in order in the cells m the first column. Proceeding from r, by steps (1, 2m) fill up 2m cells with it. And proceed similarly with the other unit-digits. Then if we take for r„ r„ ..., r,„, the
the values 0, 1, .... 2m -1, 4m -1, .... 2m, the square will be pand.agonally magic. I leave the demonstration to my readers The resulting square in the case when m = 1, « = 4, and the p and r subsidiary squares are shown below. This is the square represented in figure ii on page 137.
3
2
0
l\
0
1
3
2
7j2
0
1
0 1
3
2
3
2
3
2
4
1
4
1
2
3
2
3
[T 4
1
4
12 + 3
8 + 2
0 + 3
4 + 2
0 + 4
4 + 1
12 + 4
8 + 1
12 + 2
— — —
8 + 3
0 + 2
4+3
1 0 + 1
4 + 4
12 + 1 8 + 4|
Subsidiary p Square. Figure xix.
Subsidiary r Sqtiare. Figure xx.
Resulting Square. Figure xxi.
The rows, columns, and all diagonals of pandiagonal squares possess the magic property. So also do a group of any n numbers connected cyclically by steps (c, d) provided the first two numbers of the group are such that when divided by n they have either different quotients or different remainders. Such groups include rows, columns, and diagonals as particular cases. Thus m the square delineated in figure ii on page 137 the numbers 1, 7, 10, 16 form a magic group whose sum is 34, connected
11
162 MAGIC SQUARES [CH. VII
cyclically by steps (1, 3). Again in the square delineated in figure xviii on page 160, 10, 30, 1, 28, 48, 19, 39 form a magic group connected cyclically by steps (2, 3).
Symmetrical Squares. It has been suggested that we might impose on the construction of a magic square of the order n the condition that the sum of any two numbers in cells geometrically symmetrical to the centre (eoo. gr. 22 and 28 in figure xviii) shall be constant and equal to n' + 1. Such squares are called Symmetrical.
The construction of odd symmetrical squares of the order n, when n is prime to 3 and 5, involves no difficulty. We can begin by placing the mean number in the middle cell and work from that, either in both directions or forwards, making the number 1 follow after n^ ; we can also effect the same result by constructing any pandiagonal square of the order n and then transposing a certain number of rows and columns. If the rule given above on page 160, where a = 1, 6 = 2, A = 2, A; = — 1, be followed, this will lead to placing the number 1 in the (w+3)/2th cell of the top row : see, for instance, figure xviii.
Such a square must be symmetrical, for if we begin with the middle number (n^ -\- 1)/2, which I will denote by m, in the middle cell, and work from it forwards with the numbers ni + l, m + 2, ..., and backwards with the numbers m—1, m — 2, ..., the pairs of cells filled by the numbers m + 1 and m — 1, m + 2 and VI — 2, &c., are necessarily situated symmetrically to the middle cell and the sum of each pair is 2m. I believe this was first pointed out by McClintock.
The construction of doubly-even symmetrical pandiagonal squares is also possible, but the analysis is too lengthy for me to find room for it here.
In a symmetrical square any n such pairs of numbers together with the number in the middle cell will form a magic group. For instance in figure xviii, the group 32, 18, 36, 14, 47, 3, and 25 is magic. So also is the group 47, 3, 35, 15, 13, 37, and 25. Thus in a symmetrical pandiagonal square, even of a low order, there are hundreds of magic groups of n numbers whose sum is constant.
CH. VIl]
MAGIC SQUARES
163
Bouhly-Magic Squares. In another species of hyper-magic squares the problem is to construct a magic square of the «th order in such a way that if the number in each cell is replaced by its mth power the resulting square shall also be magic. Here for example (see figure xxii) is a magic square* of the eighth order, the sum of the numbers in each line being equal
5
31
35
60
57
34
8
30
19
9
53
46
47
56
18
12
16
22
42
39
52
61
27
1
63
37
25
24
3
14
44
50
26
4
64
49
38
43
13
23
41
51
15
2
21
28
62
40
54
48
20
11
10
17
55
45
36
58
6
29
32
7
33
59
A Doubly-Magic Square, Figure xxii.
to 260, so constructed that if the number in each cell is re- placed by its square the resulting square is also magic (the sum of the numbers in each line being equal to 11180).
Trebly-Magic Squares. The construction of squares which shall be magic for the original numbers, for their squares, and for their cubes has also been studied. I know of no square of this kind which is of a lower order than 128.
Magic Pencils. Hitherto I have concerned myself with numbers arranged in lines. By reciprocating the figures com- posed of the points on which the numbers are placed we obtain a collection of lines forming pencils, and, if these lines be numbered to correspond with the points, the pencils will be magic f. Thus, in a magic square of the nth order, we arrange n^ consecutive numbers to form 2/i + 2 lines, each containing
* See M. Coccoz in L' Illustration, May 29, 1897. The subject has been studied by Messieurs G. Tarry, B. Portier, M. Coccoz and A. Rilly. More than 200 such squares have been given by the last-named in bis itude sur les Triangles et les Carres Magiqu£s aux deux premiers degres, Troyes, 1901.
t See Magic Reciprocals by G. Frankenstein, Cincinnati, 1875.
11—2
164
MAGIC SQUARES
[CH. VII
n numbers so that the sum of the numbers in each line is the same. Reciprocally we can arrange n^ lines, numbered con- secutively to form 2n + 2 pencils, each containing n lines, so that in each pencil the sum of the numbers designating the lines is the same.
For instance, figure xxiii represents a magic square of 64
1
2
02
Gl
GO
59
7
8
9
10
54
53
52
51
15
IG
48
47
19
20
21
22
42
41
40
39
27
28
29
30
34
33
32
31
35
36
37
38
26
25
24
23
43
44
45
46
18
17
49
60
14
13
12
11
55
5C
57
58
6
5
4
3
G3
64
Figure xxiii.
consecutive numbers arranged to form 18 lines, each of 8 numbers. Reciprocally, figure xxiv represents 64 lines arranged to form 18 pencils, each of 8 lines. The method of construc- tion is fairly obvious. The eight-rayed pencil, vertex 0, is cut by two parallels perpendicular to the axis of the pencil, and all the points of intersection are joined cross- wise. This gives 8 pencils, with vertices A,B, ... H ;S pencils, with vertices A\ ... H'\ one pencil with its vertex at 0 ; and one pencil with its vertex on the axis of the last-named pencil.
The sum of the numbers in each of the 18 lines in figure xxiii is the same. To make figure xxiv correspond to this we must number the lines in the pencil A from left to right, 1, 9, ..., 57, following the order of the numbers in the first column of the square: the lines in pencil B must be numbered similarly to correspond to the numbers in the second column of the square, and so on. To prevent confusion in the figure I have not in- serted the numbers, but it will be seen that the method of construction ensures that the sum of the 8 numbers which designate the lines in each of these 18 pencils is the same.
CH. VIl]
MAGIC SQUARES
165
We can proceed a step further, if the resulting figure is cut by two other parallel lines perpendicular to the axis, and if all the points of their intersection with the cross-joins be joined cross-wise, these new cross-joins will intersect on the
Figure xxiv.
axis of the original pencil or on lines perpendicular to it. The whole figure will now give 8^ lines, arranged in 244 pencils each of 8 rays, and will be the reciprocal of a magic cube of the 8th order. If we reciprocate back again we obtain a representation in a plane of a magic cube.
166 MAGIC SQUARES [CH. VII
Magic Square Puzzles. Many empirical problems, closely related to magic squares, will suggest themselves; but most of them are more correctly described as ingenious puzzles than as mathematical recreations. The following will serve as specimens.
Magic Card Square*. The first of these is the familiar problem of placing the sixteen court cards (taken out of a pack) in the form of a square so that no row, no column, and neither of the diagonals shall contain more than one card of each suit and one card of each rank. The solution presents no difficulty, and is indicated in figure xxvi below. There are 72 fundamental solutions, each of which by reflexions and reversals produces 7 others.
Euler's Officers Problemf. A similar problem, proposed by Euler in 1779, consists in arranging, if it be possible, thirty- six officers taken fi:om six regiments — the officers being in six groups, each consisting of six officers of equal rank, one drawn from each regiment ; say officers of rank a, h, c, d, e, f, drawn fi:om the 1st, 2nd, 3rd, 4th, 5th, and 6th regiments — in a solid square formation of six by six, so that each row and each file shall contain one and only one officer of each rank and one and only one officer from each regiment. The problem is insoluble.
Extension of Eider^s Problem. More generally we may investigate the arrangement on a chess-board, containing n^ cells, of n^ counters (the counters being divided into n groups, each group consisting of n counters of the same colour and numbered consecutively 1, 2, . . . , ti), so that each row and each column shall contain no two counters of the same colour or marked with the same number. Such arrangements are termed Eulerian Squares.
* Ozanam, 1723 edition, vol. iv, p. 434.
t Euler's Comvientationes Arithmeticae, St Petersburg, 1849, vol. n, pp. 802— 361. See also a paper by G. Tarry in the Comptes rendus of the French Associ- ation for the Advancement of Science, Paris, 1900, vol. ii, pp. 170—203 ; and various notes in L' Inter mediaire des Mathematiciens, Paris, vol. m, 1896, pp. 17, 90 ; vol. V, 1898, pp. 83, 176, 252 ; vol. vi, 1899, p. 251 ; vol. vn, 1900, pp. 14, 311.
CH. VIl]
MAGIC SQUARES
1G7
For instance, if n = 3, with three red counters a^, a^, a^, three white counters h^, b^, 63, and three black counters Ci, C2, Cg, we can satisfy the conditions by arranging them as in figure xxv below. If ?i = 4, then with counters a^, a^, a^, a^) ^1, ^2, ^3> K; Ci, C3, C3, C4; di, di, di, di, we can arrange them as in figure XX vi below. A solution when n ^ 5 is indicated in figure xxvii.
->
a.
b.
C3
h
Ci
a._.

«3
^
«i
^a
C3
d.
C4
^3
'^2
b.
d,
Cl
^
63
f*4

Ci
^h
f^-2
^3

^5
K
Cl
d.

"4
'-^
d.
^\
«,
^
d.
^*
«&
^
^■i
«3
^4
^6
Figure xxv. Figure xxvi. Figure xxvii.
The problem is soluble if w is odd ; it is insoluble if n is of the form 2 (2m + 1). If solutions when n=a and when n = b are known, a solution when n= ab can be written down at once. The theory is closely connected with that of magic squares and need not be here discussed further.
Reversible Magic Squares. The digits 0, 1, 2, 6, and 8, when turned upside down, can be read as 0, 1, 7, 9, and 8. This property has been utilized in constructing magic squares
//
ZZ
62
29
69
22
IZ
Zl
2Z
61
19
12
Z2
19
21
6Z
Meversible Magic Square. Figure xxviii.
of non-consecutive numbers, which remain magic when the paper on which they are written is turned upside down. Here, for instance, figure xxviii*, is such a square in which only the digits 1, 2, 6 and their reversals are used.
* Sec The Tribune, April 29, 1907.
168
MAGIC SQUARES
[CH. VII
Magic Domino Squares. Analogous problems can be made with dominoes. An ordinary set of dominoes, ranging from double zero to double six, contains 28 dominoes. Each domino is a rectangle formed by fixing two small square blocks together side by side: of these 56 blocks, eight are
• • •• •
• • • • •
• • • •• • jV»
• • • • • • • •
• •
•
• • •
• • •
• • •
• • •
• •
• • •
• •
• •
• • •
• •
• •
• •
• •
• •
• •
• •
•
• • •
• • •
• •
e • •
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•
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•
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• • •
•
•
• • •
• • •
•
9 «
e
• •
• • •
• •
• •
• • •
• •
•
e • •
• • •
• • •
•
Magic Domino Square. Figure xxix.
blank, on each of eight of them is on.e pip, on each of another eight of them are two pips, and so on. It is required to arrange the dominoes so that the 56 blocks form a square of 7 by 7 bordered by one line of 7 blank squares and so that the sum of the pips in each row, each column, and in the two diagonals of the square is equal to 24. A solution* is given above.
If we select certain dominoes out of the set and reject the others we can use them to make various magic puzzles. As instance, I give on the next page magic squares of this kind due to Mr Escott and Mr Dudeney. Numerous squares of this kind can be formed.
* See V Illustration, July 10, 1897.
CH. VIl]
MAGIC SQUARES
169
• • •
• « •
• 1 •
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Magic Domino Squares. Figure xxx.
Magic Coin Squares*. There are somewhat similar ques- tions concerned with coins. Here is one applicable to a square of the third order divided into nine cells, as in figure xxv above. If a five-shilling piece is placed in the middle cell Cj and a florin in the cell below it, namely, in 0.3, it is required to place the fewest possible current English coins in the remaining seven cells so that in each cell there is at least one coin, so that the total value of the coins in every cell is different, and so that the sum of the values of the coins in each row, column, and diagonal is fifteen shillings : it will be found that thirteen additional coins will suffice. Similar problems can be proposed with postage stamps.
ADDENDUM.
Note. Page 169. Magic Coin Squares. Taking the notation of figure xiv we must put a double-florin and a sixpence in cell Oj, two double-florins in cell 62) a half-crown in cell C3, a florin and a shilling in cell 63, a crown and a florin in cell a2> a crown and a half-crown in cell C2, a crown and a sixpence in cell 61.
See The Strand Magazine, Londou, December, 1896, pp. 720, 721.
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