N U M B E R. 
325 
multiple between o and the greateft; for example, 36 if 
the root be 9, 105 if it be 15, &c. 
The fame thing may bedonealfo in fquares which have 
a prime number for their root. By way of example, we 
fhall here form a magic fquare of the firft two of the fol¬ 
lowing ones : in the firft of which, 3 is reprefented in the 
diagonal defcending from right to left; and in the fecond, 
10 is repeated in the diagonal defcending from left to 
right. Thi?, however, does not prevent the third fquare, 
formed by their addition, from being magical. 
_5 
4 _ 
4-1 3_ 
1 
± _ 3 . 
U_L 
1 I 2 
lI_l 
514 
10 , 0 
_ 5 _ 
F 5 
20 
20JI 0 
O 
5 
'5 
> 5^0 
IO 
O 
_ 5 _ 
5 15 
20 
I 0 
O 
O 
15 
20 
IO 
11 
2 
1 O 
1 9 
23 
22 
>5 
4 
8 
16 
20 
24 
'3 
I 
7 
9 
18 
2 I 
I 2 
5 
3 
6 
17I25 
14 
1 
4 
6 
7 
13 
I O 
I I 
l6 
Of Even Magic Squares. 
The conftru&ion of thefe fquares is attended with more 
difficulty than that of the odd fquares, and the degree of 
difficulty is different, according as they are evenly-even, 
oroddly-even ; for this reafon, we muft divide them into 
two daffies. 
Squares evenly-cven, are thofe the root of which, when 
halved, is even, or can be divided by 4 without remain¬ 
der ; of this kind are the fquares of 4, 8, 12, &c. The 
oddly-even, are thofe the root of which, when halved, gives 
an odd number; as thofe of 6, 10, 14, &c. 
As the ancients have left us no general rule 011 this 
fubjeft, but only fome examples of even fquares magically 
arranged, we fhall here give the beft methods invented by 
the moderns ; and fhall begin with fquares evenly-even. 
Let us fuppofe then, that the annexed fquare, ABCD, is 
to be filled-up magically with the firft 16 of the natural 
numbers: fill-up firft the two diagonals ; and, for that 
purpofe, begin to count the natural ^ g 
numbers, in order, 1, 2, 3, 4, See. on 
the cells of the firft horizontal row, 
from left to right; then proceed to 
the fecond row, and, when you come 
to the cells belonging to the diago¬ 
nals, inferibe the numbers counted, 
as you fall upon them; by which 
means you will have the arrangement 
reprefented in the annexed figure. C D 
When the diagonals have been thus filled, to fill-up the 
cells which remain vacant, begin to count the fame num¬ 
bers, proceeding from the angle D in the cells of the 
lower row, going from right to left, and 
then in the next above it; and, when 
any cells are found empty, fill them up 
with the numbers that belong to them : 
in this manner you will have the fquare, 
16, filled-uprmagically, as feen in the 
annexed figure; and the fum of each 
row, and each diagonal, will be 34. 
The cafe is here the fame as in regard to the odd 
fquares : every progreffion of numbers which, when ar¬ 
ranged in order in a geometrical fquare, exhibits in every 
direction, horizontally and vertically, an arithmetical 
progreffion, will be fufceptible of being arranged magi¬ 
cally in the fame fquare. 
It is not even necefiary, in the vertical direction, that 
the arithmetical proportion fhould 
be continued: it may disjunct; for 
example, if the numbers, 1, 2, 3, 4, 
5> 6, 7. 8; 57, 58, 59, 60, 6 1, 62, 63, 
64, when arranged according to their 
natural order, in a fquare of 16 cells, 
exhibit the arithmetical proportions, 
J> 5> 573. 61 ; 6, 58, 62, &c. only in- 
the vertical direction, they may be ar¬ 
ranged magically in the fame fquare, as 
feen in the annexed figure, where the 
fum of each diagonal row is 130. 
Vol. XVII. No. 1180. 
13 
1 5 
J 4 | 4 
16 
5 
57 
61 
2 3 4 
678 
58 59 60 
62 63 64 
I 
63 
62 
4 
60 
6 
7 
57 
O 
58 
59 
5 
6l 
3 
2 
64 
We (hall fuppofe that a fquare of 8 on a fide, or 64 
cells, is to be filled-up with the firft 64 numbers of the 
natural progreffion. 
Firft, write down thefe 64 numbers, as feen in the two 
lower lines of the four following periods: 
I 
■f 
• 1 
2 
3 
4 
4 
3 
2 
£ 
1 
2 
3 
4 
5 
6 
7 
8 
.64 
63 
62 
61 
60 
59 
58 
57 
4 
1 
2 
3 
3 
2 
1 
4 
9 
10 
11 
12 
13 
14 
15 
16 
5<5 
55 
54 
53 
5 2 
5 1 
5 ° 
49 
3 
4 
i 
2 
2 
1 
4 
3 
17 
18 
>9 
20 
21 
22 
23 
24 
48 
47 
46 
45 
44 
43 
42 
4 i 
2 
3 
4 
1 
1 
4 
3 
2 
2 5 
26 
27 
28 
29 
30 
31 
32 
40 
39 
' 38 
37 
36 
35 
34 
33 
lower lines in the preceding feries 
h of which forms 65. 
make 
32 cou- 
IV, 
Then form the arithmetical progreffion, 1, 2, 3, See. 
which muft be continued as far as the number that ex¬ 
press the half of the root; in this cafe it will be 1,2, 3, 
4 ; after which form the following three: 4, 1, 2, 3 ; 3, 4, 
I > 2 j 2 > 3 , 4 > 1 > inferibe, in order, each of thefe feries of 
numbers over the firft terms of each of the above periods 
of numbers ; and, as thefe figures will extend only to the 
firft four, and as there is twice that number, they muft be 
written in an inverfe order over the remaining four. 
When this preparation has been made, nothing will be 
necefiary but to write down in order all thefe numbers 
in the cells of the fquare; taking care, ift. When the cou¬ 
ple of numbers have over them an odd number, to write 
down the upper number: of this kind are the numbers 
3 ? 3 > ft? 8, Jos 
but, when the 
couple have over 
them an even 
number, the low¬ 
er number muft 
be written down, 
ad. When you 
continue by 33, 
34> 3 5j &c. after 
the firft . feries 
has been exhauf- 
ted, the cafe will 
be the contrary. 
Thus, the num¬ 
bers to be fuc- 
celfively inferib- 
ed in the fquare, are, 1,63,3,61,60,6, 58,8; which will form 
the firft row : the fecond will be found by continuing 56, 
10,54, 12, 13, 51, &c. and, by proceeding in this manner, 
you will obtain a fquare of 8 cells on each fide, as here 
annexed. 
If the fpirit of this method has been properly compre¬ 
hended, it muft be feen, that, in confequence of it, the 
firft and laft rows are neceflarily filled-up with the 16 
numbers of the firft period, and in fuch a manner, that 
the cells centrally oppofite form always 65. The cafe 
is the fame with the fecond row, and the laft but one, 
being filled-up with the numbers of the fecond period, 
and in the fame manner. The fame may be faid of the 
third and fixth rows, and the fourth and fifth -. it thence 
follows that the diagonals alfo muft be exadft. 
The following rule will equally ferve for evenly-even, 
and oddly-even, magic fquares. 
Let it be required, to fill-up magically a fquare of 8 
cells on each fide. 
For this purpofe, arrange, in the firft horizontal row, in 
a fquare of that kind, the firft eight numbers of the arith¬ 
metical progreffion; but in fuch a manner, that thofe 
equally diftant from the middle fhall form the fame fum; 
viz. that of the root augmented by unity, which, in this 
4 O cafe. 
I 1 63 
3 
61 
60 
6 
00 
00 
56 
IO 
54 
12 
3 3 
5 1 
15 
49 
37 
47 ' 
'9 
45 
9 
44 
22 
42 
24. 
40 
26 
38 
28 
29 
35 
31 
33 
3 1 
34 
30 
36 
37 
27 
39 
2 5 
4 i 
23 
42 
2 I 
20 
46 
18 
48 
16 
5 ° 
14 
5 2 
53 
I I 
55 | 9 
57 
7 
59 
5 
4 
62 
2 1 64 
