tm 
N U M B E R. 
11 
T 
7 
T 
20 
8 
3 
4 
16 
1/ 
5 
13 
I 
2 1 
9 
IO 
18 
14 
22 
23 
6 
l 9 
2, 
15 
Mofcopulus places unity immediately below the central 
cell; then fets the following numbers, defcending from 
left to right,, and, when a number goes without the fquare', 
he carries it to the high eft cell of the vertical row to which 
it belongs; he then continues defcending obliquely from 
left to right, and, when,a number goes beyond the fquare 
on the right, he carries it to 
the moft diftant cel! on the 
left, from which he continues 
according to the firft rule ; if 
he meets with a cell filled up, 
he carries his figure two cells 
below that laft written ; and, 
when he arrives at the end of 
the diagonal, he carries the 
following number as high as 
pofiible in the fame vertical 
row. In the laft place, when a number, which ought to 
be carried two cells lower than the one laft placed, goes 
beyond the fquare, he carries it to the top of the lame 
row. This defcription of his method, together with an 
Example, will be fufticient to give a proper idea of it; but 
it is fomewhat more complex than the Indian method. 
Bachet’s rule is as 
follows: Raife upon 
each fide-or the given 
fquare, cells in the 
form of Heps, as feen 
in the figure ; and 
then, beginning at 
the higheft cell, in- 
fcribe alt the num¬ 
bers of the progref- 
fion, defcending dia¬ 
gonally, as feen from 
i to 5, and from 6 to 
io, &c. 
I 
6 
2 
11 
/ 
7 
m 
3 jj 
16 
h 
12 
b 
8 
g 1 4 
1 21 
17- 
C 
1 3 
(l 
9 1 
22 
k 
18 
a 
14 
*’ | JO 
: 2 3 
e 
19 
Jii 
|24_ 
20 
2 S 
When this is done, tranfpofe into the cell a, the next 
below the centre, the higheft number ; and, in like man- 
tranfpofe 25 into b, the 
J I 
24- 
7 
20 
3 
4 
12 
a 5 
8 
l6 
17 
5 
13 
2 I 
9 
10 
I 0 
I 
14 
22 
i 2 3 
6 
19 
2 
!5 
J 
3 
j 
a 
4 
5 
2 
4 
1 
3 
4 
1 
3 
5 
2 ! 
3 
5 
2 
4 
1 
2, 
4 
1 
v3 
5 
5 
O 
is 
IO 
20 
10 
20 
5 
O 
15 
o 
is 
IO 
20 
5 
20 
5 
O 
is 
IO 
!S 
10 
20 
5 
O 
next above the centre ; let 5, 
for the fame reafon, be tranf- 
pofed to c, and 21 to d; then 
tranfpofe 6 to c, and 24 to /’, 
20 to m > and 2 to l. See. By 
thefe means you will obtain 
the annexed magic fquare, and 
the fum of each row, whether 
vertical, or horizontal, or dia¬ 
gonal, will make 65. This 
rule, though different from that of Mofcopulus, gives ab- 
folutely the fame refult. 
But all thefe methods are inferior to the following, in¬ 
vented by M. Poignard, hinted at before, and improved 
and enlarged by M. de la Hire ; for the preceding are li¬ 
mited, whereas the one here alluded to is capable of giv¬ 
ing an almoft infinite number of combinations. 
Let it be required, for example, to fill up a fquare hav¬ 
ing an odd root, fuch as 5. Having conftru&ed the fquare 
.of ceils, place in the firft I10- 
3‘izontal row, at the top, the 
five firft numbers of the natu- 
•ral progreffion, in any order, 
at pleafure, which we fhallhere 
iuppofe to be, 1, 3, 5, 2,4; 
then make choice of a number 
which is prime to the root 5, 
and which, when diminifhed 
by unity, does not meafure it; 
let this number be 3 ; and, for 
that reafon, begin with the third figure of the feries, and 
count from it, to fill up the fecond horizontal row, 5, 2, 
4, 1, 3 ; then begin again by the next third figure, in¬ 
cluding the 5, that is to fay, by 4, which will give fot 
the third row, 4, i, 3, 5, 2; by following the fame pro- 
pels', weHiall thendiave the feries of numbers 3, 5, z, 4, i, 
to fill uj) the fourth row-; continue in this manner, al¬ 
ways beginning at the third figure, the preceding included, 
until the whole fquare is filled up. This fquare will be 
one of the components of the required fquare, and will be 
magic; for the fum of each row, whether horizontal, or 
vertical, or diagonal, is the fame, as the five numbers of 
the prOgreffion are contained in each, without the fame 
figure being ever repeated. 
Now conftruit a fecond geometrical fquare, of 25 cells, 
in the firft row of which inferibe the multiples of the 
root 5, beginning with a ci¬ 
pher, viz. o, 5, 10, 15, 20, and 
in any order at pleafure, fuch 
for example as 5, o, 1 5,10, 20 ; 
then fillup the fquare,accord¬ 
ing to the fame principle as 
before, taking care not to af- 
fume the lame number in the 
feries always to begin with. 
Thus, for example, as in the 
former fquare the third figure 
in the feries was taken, in the prefent one the fourth 
mult be affumed; and, thus we filial 1 have a fquaj'e of the 
multiples, as feen in the annexed figure. This is the fe¬ 
cond component of the required magic fquare; and is it- 
felf magic, iince the fum of each row, in.every direction, 
is the fame. 
Now to obtain the magic fquare required, nothing is 
neceffary but to inferibe in a third fquare, of 25 cells, the 
fum of the numbers found in 
the correfponding cells of the 
preceding two; for example, 
5-[-i, or 6, in the firft on the 
left, at the top of the required 
fquare; 0+3, or 3, in the fe¬ 
cond, and lo on ; by thefe 
means we filial 1 have the an¬ 
nexed fquare of 25 cells, 
which will neceffarily be ma¬ 
gic. 
By thefe means, any of the numbers may be'made to 
fall in any cells at pleafure; for example, 1 in the cen¬ 
tral cell. Nothing is neceffary for this purpofe, but to 
fill up the middle row with the feries of numbers in fuch 
a manner that 1 may be in the centre, as feen in the firft: 
fquare below; and then to fill up the reft of the fquare ac¬ 
cording to the above principles, beginning at the higheft 
row, when the lowed has been filled up. To form the fecond 
fquare, place a cipher in the centre, as feen in the annexed 
figure, and fill up the remaining cells in the fame man¬ 
ner as before, taking care not to afl’ume the fame quanti¬ 
ties as in the former for beginning the rows. In the laft 
place, form a third fquare, by adding together the num¬ 
bers in the fimilar cells, and you will have the annexed 
fquare, where 1 will neceffarily occupy the centre. 
6 _ 
3 
20 | 12 
a 4 
1 5 
22 
9 1 1 
18 
4 
16 
13 25 
7 
.23 
10 
2 | 1 9 
I I 
17 
14 
21 1 s 
5 
2 
3 
I 
4 
5 
A \JL 
2 11 
_5_ 
2 
1 
314 
I 
j_ 
4 
51 2 
4 
5 
2 
113 
120, 5 
10 
O 
1 5 
O In 
20 
5 _ 
IO 
5 |io 
O 
■5 
20 
1 5I20 
5 
IO 
O 
10! 0 
IS 
20 
s 
22 
6 
Jii + 
20 
_ 3 _ 
11 
vs 
7 
I I 
IO 
I 2 
I 
18 
24 
16 
11 
9 
15 
2 
24 
5 
17 
2 I 
8 
Wemuft here obferve, that when the number of the 
root is not prime, that is, if it be 9, 15, 21, &e. it is im- 
poftlble to avoid a repetition of fome of the numbers at 
leaft, in one of the diagonals; but, in that cafe, it nnift be 
arranged in fuch a manner, that the number repeated in 
that diagonal (hall be the middle one of the progreffion; 
for example, 5, if the root of the fquare be 9 ; 8, if it be 
15 ; and, as the fquare formed of the multiples will be 
liable to the fame accident, care mull: he taken, in filling 
them up, that the oppofite diagonal fhall contain the mean 
3 multiple 
