NUMBER. 
327 
ranged any how in the four empty cells of the firft row, 
and their complements below, the fecond. muft be ar¬ 
ranged in the cells of the firft: vertical row, and theircom- 
plements each at the extremity of the fame horizontal 
row ; by which means we fhall have the new fquare with 
a border, as feen above. 
If it were required to form a bordered fquare of the 
root 8, it would be neceflary to referve for the interior 
fquare of 36 cells, the 36 mean numbers of the pro- 
greffion ; and they might be formed into a bordered fquare 
around the magic fquare of 16 cells ; with the 28 remain¬ 
ing numbers we might then form a border to the fquare 
of 36 cells, &c. Hence it appears in what manner we 
might form a magic fquare, which, when fuccefiively lef- 
fened by one, two, or three, rows, fhall Hill remain magic. 
Magic Squares in Compartments. —Another property, of 
which molt magic fquares are fufceptible, is, that they are 
not only magic when entire, but that, when divided into 
thofe fquares into which they can be refolved, thefe por¬ 
tions of the original fquare are themfelves magic. A 
fquare of 8 cells on a fide, for example, formed of four 
fquares, each having 4 for its root, being propofed, it is 
required, that not only the fquare of 64 fhall be difpofed 
magically, but each of thofe of 16 ; and that the latter 
even, however arranged, fhall flill conrpofe a magic fquare. 
To conftruft a fquare of 64 in this manner, take the 
firft 8 numbers of the natural progreftion from 1 to 64, 
and the 8 laft, and arrange them, magically, in a fquare of 
j6 cells; do the fame thing with the 8 terms which fol¬ 
low the firft 8, and the 8 which precede the laft 8, and by 
thefe means you will have a fecond magic fquare; form a 
fimilar fquare of the 8 
following numbers with 
their correfponding ones, 
anti another w’ith the 16 
means: the refult will be 
four fquares of 16 cells, 
tie numbers in which 
will be equal when added 
together, either in rows 
or diagonally; for they 
will every-where be 130. 
It is therefore evident 
that, if thefe fquares be 
arranged fide by fide, in any order whatever, the fquare 
refulting from them will be magic, and the fum in every 
direction will be 260. 
To arrange the fquare of 9 in this manner, divide the 
progrefiion, from 1 to 81 inclufively, into nine others, 
as j, 10, 19 .... 73; 2, 11, 20 .... 74; 3, 12, 21- 75 ; 
See. and arrange each of thefe progrefiions magically in a 
fquare of 9 cells, marking the firft I, the fecond II, See. 
But it will beobferved, that in thefe different fquares, the 
fums of the rows, and thole of the diagonals, will be them¬ 
felves in arithmetical progreffion ; viz, in the fquare I the 
fum will be 111, in the fquare II it will be 114, and fo on. 
If thefe 9 fquares be arranged magically, it may be readily 
feen, that the total will ftill be magical; but the partial 
fquares cannot be tranfpofed,as in the preceding one of 64. 
The fquare of 15 may be refolved into 25 fquares of 9 
cells. If 25 fquares therefore of 9 cells be arranged 
magically, filling them up with the 25 progrefiions, which 
may be formed in this manner, 1, 26, 51 .... 201; 2, 25, 
52 ... . 202 ; 3, 28, 53 ... . 203; See. thefe fquares will 
have fucceffively, and in order, for the fums of their row's 
and their diagonals, 303, 306, 309, Sec. to the laft, which 
will make 375 in each of its rows and diagonals. By ar^ 
ranging thefe 25 fquares, magically, in this manner, call¬ 
ing the firft I, the fecond II, the third III, and the laft 
XXV, you will obtain a magic fquare ; and, whatever 
number of variations the fquare of 25 cells may be fuf¬ 
ceptible of, the fquare of 15 will be capable of receiving 
as many, being at the fame time magic, as well as all the 
fquares of which it is compofed. 
Of the Variations of Magic Squares .—The fquare having 
3 for its root is fufceptible of no variation; whatever 
I 
63 
62 
4 
9 
JL 
54 
12 
60 
6 
7 
57 
5 2 
14 
15 
49 
8 
5 * 
59 
5 
1 6 
5 ° 
51 
13 
61 
3 
2 
6 4 { 5 3 
10 
5 ° 
■7 
47 
46 
20 
! 25 
39 
38 
28 
44 
22 
2 3 
41 
36 
30 
31 
33 
24 
42 
43 
2 I 
3 2 
34 
35 
2 5 
4-5 
!9 
18 
48 
37 
27 
26 
40 
method may be employed, or whatever arrangement mgy 
be given, to the numbers of the progreffion from 1 to 9, 
the fame fquare will always arife, except that it will be in¬ 
verted, or turned from left to right, which is not a varia¬ 
tion. But this is not the cafe with the fquare having 4 
for its root, or that of 16 cells; this being fufceptible of 
878 variations. 
The fquare of 5 is fufceptible of at leaft 57600 diffe¬ 
rent combinations ; for, according to the procefs of M. 
de la Hire, the 5 firft numbers may be arranged 120 dif¬ 
ferent ways in the firft row of the firft primitive fquare ; 
and, as they may be afterwards arranged in the lower 
rows, beginning again by two different quantities, this 
will make 240 variations, at leaft, in the primitive fquare ; 
which, combined with the 240 of the fecond, form 57600 
variations in the fquare of 5. But there are doubtlefs a 
great many more; fora bordered fquare of 5 cannot be 
reduced to the method of M. de la Hire; but one bordered 
fquare of 5, the corners remaining fixed, as well as the in¬ 
terior fquare of 3, may experience 36 variations. What 
a number, therefore, of other variations muft be produced 
by changing the interior fquare and the angles ! 
A bordered fquare of 6, when once conftrufted, the 
corners remaining fixed, and the interior fquare being 
compofed of the fame numbers, may be varied 4055040 
different ways; for the interior fquare may be varied and 
differently tranfpofed in the centre 7040 ways ; each of the 
horizontal rows at top and at bottom, the extremities re¬ 
maining fixed, may be varied 24 ways; for there are four 
pairs of numbers fufceptible of change in their place, 
which may be combined 24 ways; and there are alfo four 
pairs in the vertical rows between the corners. The num¬ 
ber of the combinations, therefore, is the produCl of 7040 
by 576, the fquare of 24, which gives 4055040 variations. 
But the corners may be varied, as well as the numbers af- 
fumed to form the interior fquare; and it hence follows, 
that the whole number of the variations of a fquare of 6, 
while it ftill remains bordered, is equal to feverul millions 
of times the former. 
The fquare of 7, by M. de la Hire’s method alone, 
may be varied 406425600 different ways. Thefe varia¬ 
tions, however numerous, ought to excite no furprife ; 
for the number of difpolitions, magic or not magic, of 
49 numbers, for example, forms one of fixty-two figures, 
of which the preceding is, as we may fay, but a part infi¬ 
nitely fmail. 
Of Geometrical Magic Squares. —We have already ob¬ 
ferved, that numbers in geometrical, as well as in arith¬ 
metical, progrefiion, might be arranged in the cells of a 
fquare, and in fuch a manner, that the product of thefe 
numbers, in each row, whether vertical, or horizontal, or 
diagonal, fhall always be the fame. To conftruft a fquare 
of this kind, the fame principles muft be followed as in the 
conftruCfion of other magic fquares ; and this may be ea- 
fily demonftrated from the property of logarithms. With¬ 
out enlarging further, therefore, on this 
fubjeft, we fhall confine ourfelves to 
giving one example; it is that of the 9 
firft terms of the double geometric pro¬ 
greftion, 1, 2, 4, 8, &c. arranged in a 
f quare of 3 cells on each fide. The pro¬ 
duct is evidently the fame in every di¬ 
rection, viz. 4096. 
The ingenious Dr. Franklin carried this curious fpecu- 
lation further than any of his predeceffors in the fame 
way. He conftrufted both a magic fquare of fquares, and 
a magic circle of circles, the defeription of which is as 
follows: The magic fquare of fquares is formed by di¬ 
viding the great fquare as in fig. 5 of the Engraving. The 
great fquare is divided into 256 little fquares, in which all 
the numbers from 1 to 256, or the fquare of 16, are placed 
in 16 columns, which may betaken either horizontally or 
vertically. Their chief properties are as follow: 1. 
The fum of the 16 numbers in each column or row, 
vertical or horizontal, is 2056. 2. Every half-column, 
vertical and horizontal, makes 1028, or juft one-half 
of 
128 
I 
32 
4 
l6 
64 
8 
256 
2 
