326 
NUMBER. 
cafe, is 9; the fecond row 
mull be the inverfe of 
the firft; the third mull 
be like the firft ; the 
fourth, like the fecond, 
and fo on, alternately, 
till the half of the fquare 
is filled-up ; after which, 
the other half may be 
formed by merely reverf- 
ing the firft,as maybe feen 
in the annexed figure. 
This will be the firft pri¬ 
mitive fquare. 
1 
6 
5 
2 
7 
4 
3 
8 
8 
3 
4 
7 
2 
s 
6 
I 
I 
6 
5 
2 
7 
4 
3 
8 
8 
3 
4 
7 
2 
5 
6 
I 
8 
3 
4 
7 
2 
5 
6 
I 
I 
6 
5 
2 
7 
4 
3 
8 
8 
3 
4 
| 7 
2 
5 
6 
I 
I 
6 
5 
I 2 
7 
4 
3 
8 
which means, we (hall have the fquare correfted, and ma¬ 
gically arranged. 
Second Primitive Square. 
Perfefl Square. 
48 
8 
481 8 
8 
48 
8 
48 
16 
40 
16 j4o 
40 
r 6 
40 
1 6 
32 
24 
32I24 
2 4 
32 
24 
3 2 
0 
5 6 
0 I56 
56 
0 
O 
56 
O 
5 6 i 0 
O 
5 6 
O 
56 
24 
32 
24 32 
24 
32 
24 
40 
.6 
40 1 16 
1 6 
40 
16 
40 
8 
48 
8 48 
48 
8 
48 
8 
49 
14 
53 
10 
1 ei 52. 
1 I 
56 
24 
43 
20 
47 
42I21 
46.(17 
33 
3 ° 
37 
26 
3 1 136 27(40 
8 
39 
4 
63 
5 «l 5 
62 
I 
64 
3 
60 
7 
2 |6 1 
6 
57 
2 5 
i£ 
29 
34 
39| 2 8 
11 
H 
48 
1 9 
44 
2 3 
I2 |45 
22 
1 
9 
54 
13 
5 ° 
55 1 2 
5 1 
l6 
Firft Primitive. 
Second Primitive. 
5 
6 
3 
4 
I 
2 
24 
6 
24 
24 
6 
24 
2 
I 
4 
3 
6 
5 
O 
30 
O 
O 
30 
O 
5 
6 
3 
4 
I 
2 
12 
18 
I 2 
12 
18 
I 2 
T 5 
6 
3 
4 
I 
2 
18 
12 
18 
18 
I 2 
18 
2 
I 
4 . 
3 
6 
5 
30 
O 
30 
30. 
0 
30 
5 
6 
3 
4 
I 
2 
6 
24 
6 
6 
24. 
6 
A 
Perfefl Square. 
29 
12 
2 7 
28 
7 
26 
2 
31 
4 
3 
36 
5 
17 
2 4. 
!5 
1 6 
19 
14 
23 
18 
21 
22 
13 
20 
32 
1 
34 
33 
6 
35 
11 
30 
9 
10 
2 5 
8 
29 
7 
28 
9 
12 
26 
32 
31 
3 
i 
36 
5 
23 
18 
15 
l 6 
J 9 
20 
2 4 
24. 
21 
22 
13 
17 
2 
1 
34 
33 
6 
35 
I I 
2 5 
TO 
27 
30 
8 
To form the fecond, fill it up, according to the fame 
principle, with the multiples of the root, beginning with 
o ; that is to fay, o, 8, 16, 24, 32, 40, 48, 56 ; taking care 
that the extremes (hall always make 56; but, inftead of 
arranging thefe numbers in a horizontal direction, they 
muft be arranged vertically,, as in the following figure. 
When this is done, add together the fimilar cells of the 
two fquares, and you will have a fquare of 8 on each fide, 
as in the laft figure below. 
Squares oddhj-even. —We fhall take, by way of example, 
the fquare of the root 6. To fill it up, inferib'e in it the 
firft fix numbers of the arithmetical progreffion, 1, 2, 3, 
&c. according to the above method ; which will give the 
firft primitive fquare, as in the annexed figure. 
The fecond muft be formed by filling-up the cells, in a 
vertical direction, according to the fame principle, with 
the multiples of the root, beginning at o ; viz. o, 6, 12, 
18, 24, 30. 
The fimilar cells of the two fquares, if then added, will 
form a third fquare, A, B, C, D, which will require only 
a few corredtions to be magic. This third fquare is as 
below. 
To render the fquare magic, leaving the corners fixed, 
tranfpofe the other numbers of the upper horizontal row, 
•and of the firft vertical one, on the left, by reverfing all 
the remainder of the row; writing 7, 28, 27, 12, inftead 
of 12, 27, &c. and in the vertical one, 32, 23, 17, and 2, 
from the top downwards, inftead of 2, 17, &c. It will be 
neceflary, alfo, to exchange the numbers in the two cells 
of the middle of the fecond horizontal row at the top; of 
the lowed of the fecond vertical row, on the left; and of 
the laft, on the right. The numbers in the cells A and B 
muft alfo be exchanged, as well as thofe in C and D; by 
B 
Of Magic Squares with Borders. —Modern arithmeti¬ 
cians have added a new difficulty to the fubjedl of magic 
fquares, by propofing not only to arrange magically, in a 
fquare, a progreffion of numbers, but by requiring that 
this fquare, when leftened by a row on each Side, or two 
or three rows, &c. (hall Hill remain magic ; or, a magic 
fquare being given, to add to it a border, of one or more 
rows, in fuch a manner, that the enlarged fquare, thence 
refulting, (hall be ftiil magic. 
To give an example of this conftrudlion, let it be re¬ 
quired to form a magic fquare of the root 6, and to fill it 
up with the natural numbers, from 1 to 36. The firlt 
even magic fquare poffible being that of 4 on each fide, 
we lhall firft arrange it magically, filling it up with the 
mean terms of the progreffion, to the number 16, and re- 
ferving the firft and the laft 10 for the border. For the 
interior fquare, therefore, we lhall take the numbers, 11, 
12, &c. as far as 26 inclufively, and lhall give them any 
magic difpofition whatever ; there will then remain the 
numbers, 1, 2, &c. as far as 10, and 27 as far as 36, for the 
border. 
To difpofe thefe numbers in the border, firft place the 
numbers 1,6, 31, 36, in thefour 
corners, and in fuch a manner, 
that diagonally they lhall make 
37. As each row muft make 111, 
it will be necelfary to place in 
the firft row four fuch numbers, 
that theirfum lhall be 104; and, 
as their complements to 37 
muft be found in the lowed, 
where there is already 67, it will 
be necelfary that they fhould to¬ 
gether make 44: there are leve- 
ral combinations of thefe numbers, four and four, which 
can make 104, and their complements 44; but it is ne¬ 
celfary, at the fame time, that four of thole remaining fhould 
make 79, to fill-up the firft vertical row, while their com¬ 
plements make 69, to complete the laft. This double con¬ 
dition limits the combination to 35, 34, 30, 5, which may¬ 
be placed in the firlt row in any order whatever, provided 
their complements be placed below each of them in the laft 
row; and the four numbers requifite to fill-up the firft ver¬ 
tical row will be 33, 28, 10, 8, which may be arranged any 
how at plealure, provided the complement of each be 
placed oppofite to it, in the correfponding cell on the 
other fide. 
It is not abfolutely neceflary that 1, 6, 31, 36, fhould 
be placed in the four corners 
of the fquare ; if we fuppofe 
them to be filled-up, in the fame 
order, with 2, 7, 30, 35, it would 
be then necelfary that the four 
firft numbers Ihould make 102, 
and theircomplementS46, while 
the four laft make 79, and their 
complements 69; but it is found, 
that the four firft numbers are 
36, 31, 27, 8 ; and the fecond, 
34, 32, 9,4. The firlt being ar- 
1 ranged 
I 
35 
34 
51 
30 
6 I 
33 
I I 
2 5 
24. 
14 
4 | 
28 
22 
16 
17 
*9 
9 | 
8 
18 
20 
21 
1 5 
29 
IO 
23 
13 
12 
’ 26 
27 
3 i 
2 
3 
32 
7 
36 
2 |36 | 31|27 
8 
7 
34 | 11 
2 5 | 2 4 
14 
3 
32 I 22 
16 j 17 
J 9 
5 
9 118 
20 | 21 
15 
28 
4 1 2 3 
1 3 | 21 
62 
33 
3° | 1 
6 I 10 
2 9 
35 
