NUMBER. 
323 
book was not publifhed by liamfelf, nor did it appear till 
after his death, fiz. in 1693. 
In 1703, M. Poignard, canon of Brufiels, publiflied a 
treatife of fublime magic .fquares. Before him there had 
been no magic fquar-es made but for feriefes of natural 
numbers that formed a fquare ; but M. Poignard made 
two very confiderable improvements. i°. Inftead ot tak¬ 
ing all the numbers that fill a fquare, for inftance, the 
thirty-fix fuccefiive numbers, which would fill all the 
cells of a natural fquare whole fide is fix, he only takes 
as many fuccefiive numbers as there are units in the fide 
of the fquare, which, in this cafe, are fix ; and thefe fix 
numbers alone he difpofes in fuch manner in the thirty-fix 
cells, that none of them is repeated twice in the fame 
rank, whether it be horizontal, vertical, or diagonal ; 
whence it follows, that all the ranks, taken all the ways 
poflible, mull always make the fame fum, which M. Poig¬ 
nard calls repeated progrefflou. a 0 . Inftead of being con¬ 
fined to take thefe numbers according to the feries and 
fucceflion of the natural numbers, that is, in an arithme¬ 
tical progreftion, he takes them likewife in a geometrical 
progrefiion, and even in an harmonical progreftion. But 
with thefe two lall progrefllons the magic mu ft necefiarily 
be different to what it was : in the fquares filled with 
numbers in geometrical progreftion, it confifts in this, 
that the produCis of all the ranks are equal; and, in the 
harmonical progreftion, the numbers of all the ranks con¬ 
tinually follow that progreftion ; he makes fquares of each 
of thefe three progreftions repeated. 
This book of M. Poignard gave occafion to M. de la 
Mire to turn his thoughts the fame way, which he did with 
fuch fuccefs, that he feems to have well-nigh completed 
the theory of-magic fquares. He firft conliders uneven 
fquares ; all his predeceflors on the fubjeCt having found 
the conftru&ion of even ones by much the molt difficult; 
for which reafon M. de la Hire referves thofe for the laft. 
This excefs of difficulty may arife partly from hence, that 
the numbers are taken in arithmetical progrefiion. Now 
in that progreftion, if the number of terms be uneven, that 
in the middle has fome properties which may be of fer- 
vice; for inftance, being multiplied by the number of 
terms in the progrefiion, the product is equal to the fum 
of all the terms. 
Of Odd Magic Squares. 
There are feveral rules for the conftruCtion of thefe 
Squares; but, in our opinion, the fimpleft and moft con¬ 
venient, is that which, according to M. de la Loubere, is 
employed by the Indians of Surat, among whom magic 
fquares feem to be held in as much eftimation as they were 
formerly among the ancient vifionaries before mentioned. 
We fnall here fuppofe an odd fquare, the root of which 
is 5, and that it is required to fill it up with the firft 25 of 
the natural numbers. In this cafe, begin by placing 
unity in the middle cell of the 
horizontal row at the top; 
then proceed from left to right, 
afeending diagonally, and, 
when you go beyond the 
fquare, transport the next 
number, 2, to the loweft cell of 
that vertical row to which it 
belongs; fet 3 in the next cell, 
afeending diagonally from left 
to right, and, as 4 would go 
beyond the fquare, tranfport it to the moft diftant cell of 
the horizontal row to which it belongs; fet 5 in,the next 
cell, afeending diagonally from left to right; and, as the 
following cell, where 6 would fall, is already occupied 
"by 1, place 6 immediately below 5 ; place 7 and 3 in the 
two next cells, afeending diagonally, as feen in the figure ; 
and then, in confequence of the firft rule of tranfpofition, 
let 9 at the bottom of the laft vertical row; then 10, in 
confequence of the fecond, in the laft cell of the left of the 
fecond horizontal row; then ii below it, according to the 
17 
24 
I 
8 
23 
5 
7 
14 
l6 
4 
6 
13 
20 
22 
IO 
12 
*9 
21 
3 
i 1, 
18 
25 1 2 
9 
third rule ; after which, continue to fill up the diagonal 
with the numbers 12, 13, .14, 15 ; and, as you can afcenct 
no farther, place the following number, 16, below 15 ; 
if you then proceed in the fame manner, the remaining 
cells of the fquare may be filled up without any diffi¬ 
culty, as feen in the above figure ; and any line or row 
of them will make 65, as in the firft example. 
The following are the fquares of 3 and 7 filled up by 
the fame method ; and, as thefe examples will be fuffi- 
cient to exercife'fuch of our readers as have a tafte for 
amufements of this kind, we lhall proceed to a few ge¬ 
neral remarks on the properties of a fquare arranged ac¬ 
cording to this principle. 
iff. According to this difpolition, the moft regular of 
all, the middle number of the progreftion occupies the 
centre, as 5 in the fquare of 9 cells, 13 in that of 25, and 
25 in that of 49 ; but this is not needfary in the arrange¬ 
ment of all magic fquares. 
2d. In each of the diagonals, the numbers which oc¬ 
cupy the cells equally diftant from the centre, are double 
that in the centre: thus 30+20—47 + 3=284-22—244-26, 
&c. are double the central number, 25. The cafe is the 
fame with the cells centrally oppolite, that is to fay, thole 
fimilarly ft United in regard to the centre, but in oppofite 
directions, both laterally and perpendicularly ; thus, 31 
and 19 are cells centrally oppolite, and the cafe is the fame 
in regard 1048 and 2, 13 and 37, 14 and 36, 32 and 18. 
3d. It may be readily feen, that it is not necefiary that 
the progrefiion, to be arranged magically, ftiould be that 
of the natural numbers 1,2, 3, 4," &c. any arithmetical 
progreftion whatever, 3, 6, 9, 12, See. or 4, 7, 10, 13, 16, 
Sec. may be arranged in the fame manner. 
4th. Nor is it necefiary that the progrefiion ftiould be 
continued ; it may be disjunct, and the rule is as follows; 
If the numbers of the progreftion, arranged according to 
their natural order in the cells of the fquare, exhibit in 
every direction, vertical and horizontal, an arithmetical 
progrefiion, they are fufceptible of being arranged magi¬ 
cally in the fame fquare, and by the fame procefs. Let 
us take for example the feries 
of numbers 1, 2, 3, 4, 5 ; 7, 8, 
9, 10,11 ; 13, 14, 15, 16, 17 ; 
19, 20, 21, 22, 23 ; 25, 26, 27, 
28, 29 ; as thefe, when ar¬ 
ranged in the cells of a fquare, 
every-where exhibit an arith¬ 
metical progreftion, they may 
be arranged magically; and 
indeed, according to the above 
rule, they may be formed into the annexed magic fquare. 
5th. In like manner, and 
forthe'fame reafon, the num¬ 
bers 1, 6,11,16, 21 ; 2, 7, 12, 
17, 22 ; 3, 8, 13, 18, 23 ; 4, 
9, 14, 19, 24; 5, 10, 15, 20, 
25 ; may be arranged magi¬ 
cally by the fame procefs, as 
in the annexed figure, and 
give a fquare of 25. Of the 
variations of the fame fquare 
we fiiall fpeak hereafter. 
9 
20 
I 
12 J 23 
15 
21 
7 
18 
4 
l 6 
2 
13 
24 
IO 
22 
8 
19 
5 
II 
3 
14 
25 
6 ! 17 
20 | 28 
J 
I 
9 
17 
27 1 5 
4 i 7 
1 
8 
16 
*9 
J S 
23 
26 
11 j 14 
'Ti 
Oj 
3 
14 ! 21 
29 
2 
IO 
8 
1 
6 
3 
5 
7 
4 
9 
2 
30 
39 
48 
1 
IO 
19 
28 
47 
7 
9 
18 
27 
29 
46 
6 
8 
*7 
26 
35 
37 
s 
- 14 
16 
2 5 
34 
36 
45 
13 
24 
33 
42 
44 
4 
21 
23 
32 
41 
45 
3 
12 
r ' 
122 
31 
40 
49 
2 
I I 
20 
Mofcopulus 
