m2 N U M 
UJe of the Tulle .—Look clown the middle column of 
figures for that number whofe permutation you with to 
know, from one to fifty. On the right hand you will 
find the queftion anfwered, and on the left its progenitor, 
or the permutation of the preceding number. Thus, if 
you feek for the changes which may be rung upon twelve 
bells5 on the right hand of 12. Hands 479001600, the an- 
fwer, and on the left 39916800, which fum is the permu¬ 
tation of 11. In like manner proceed with any other 
fum. But in large produfts, divide the fums into periods 
of 6 figures each, and half-periods'of 3 figures; every 
whole period, in notation, being a million times the va¬ 
lue of its right-hand precurfdr, and the half-periods a 
thousand times that of theirs. 
Thus, the fum of the permutations of fifty, being the 
maximum of the Table, when fo divided off, will Hand as 
follows : 30,4i4'093,joi - 7i3,378'043,6i2 , 6o8,i66'o64,768 - 
844,377'64i,s68'o6o,5i2'ooo,ooo'ooo,ooo, and is to be nu¬ 
merated thus : 30 thoufands 414 decillions, 093 thoufands 
aoi novillions, 713 thoufands 378 oftiilions, 043 thou¬ 
fands 612 fcptillions, 608 thoufands 166 fextillions, 064 
thoufands 768 quintillions, 844thoufands 377 quadrillions, 
641 thoufands £.68 trillions, 960 thoufands 512 billions, 
there being no millions, no thoufands, no hundreds, no 
tens, no units. Thofe who read this maximum produft 
may eafily read any of the fmajlcr fums, they being all 
denominated in this one example. 
©f Magic Sotjares and Circles. 
A fquare figure, formed of a feries of numbers, in ma¬ 
thematical proportion, fo difpofed in parallel and equal 
ranks, as that the fums of each row, taken either perpen¬ 
dicularly, horizontally, or diagonally, are equal, is called 
a magic fquare. 
The feveral numbers which compofe any fquare num¬ 
ber (for inftance, 1, 2, 3, 4, 5, &c. to 25 inclufive, which 
compofe the fquare number 25), being difpofed after each 
other, in a fquare figure of 25 cells, each in its cell; if 
then you change the order of thefe numbers, and difpofe 
them in the cells in fuel) a manner as that the five num¬ 
bers which fill an horizontal rank of cells, being added 
together, finall make the fame fum with the five numbers 
in any other rank of cells, whether horizontal or vertical, 
and even the fame number with the five in each of the 
two diagonal ranks; this difpgfition of numbers is called 
a magic fquare ; in oppofition to the former difpofition, 
which is caiied a natural fquare. 
Natural Square. Magic Square. 
16 
14 
8 
2 
2 5 
3 
22 
20 
I I 
9 
15 
6 
4 
2 3 
17 
24 
18 
I 2 
IO 
I 
7 
5 ! 21 
19 
13 
1 
2 
3 
4 
5 
6 
7 
8 
9 
IO 
I I 
j 2 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
One would imagine that thefe magic fquares had that 
name given them, becaufe this property of all their ranks, 
which, taken any way, make always the fame fum, ap¬ 
peared extremely furprifing, efpecially in certain ignorant 
ages, when mathematics palled for magic. But there is 
reafon to fufpeft, that thefe fquares merited their name 
kill farther, by the fit peril itious operations they were em¬ 
ployed in, as the conllruftion of tali [mans. See. for, ac¬ 
cording to the childifh philofophy of thofe days, which 
attributed virtues to numbers, what virtues might not 
be expefted from numbers fo wonderful ? However, what 
was at firft the vain practice of makers of talifmans and 
conjurers, has (ince become the fubjeft ofrefearch among 
mathematicians ; not that they imagine it will lead to any 
thing of folid ufc or advantage, but only as it is a kind 
of play, where the difficulty makes the merit, and if may 
chance to produce fome new views of numbers, which 
mathematicians will not lofe the occafion of. 
Eman. Mofcopulus, 5 Greek author of no great'anti- 
quity, is the firft that appears to have fpoken of magic 
fquares ; and, by the age in which he lived, there is rea¬ 
fon to imagine he did not look on them merely as a ma¬ 
thematician. However, he has left ns fome rules for 
their conllruftion. In the treatife of Cornelius Agrippa, fo 
much accufe'd of magic, we find the fquares of feven num¬ 
bers difpofed magically; and it mult not be fuppofed that 
thofe feven numbers were preferred to all the others with¬ 
out fome very good reafon ; in effect, it is becaufe their 
fquares, according to the fyftem of Agrippa and his fol¬ 
lowers, are-planetary. According to this idea, a fquare 
of one cell, filled up with unity, was the fymbol of the 
Deity, on account of the unity and immutability of God ; 
for they remarked that this fquare was by its nature 
unique and immutable; the produft of unity by itfelf 
being always unity. The fquare of the root 2, was the 
fymbol of imperfeft matter, both on account of the four 
elements, and of the impoffibility of arranging the fquare 
magically. The fquare of 3, as it was called, or a fquare 
of 9 cells, was aiiigned or confecrated to Saturn ; one of 
16, to Jupiter; of 25, to Mars; of 36, to the Sun ; of 
49, to Venus; of 64, to Mercury; and of 81, or nine 011 
each fide, to the Moon. 
M. Bacbet applied himfelf to the fludy of magic fquares, 
on the hint he had taken from the planetary fquares of 
Agrippa, as being unacquainted with the work of Mof- 
copulus, which is only in manufeript in the French king’s 
library; and, without the aflirtance of any author, he 
found out a new method for thofe fquares whofe root is 
uneven ; for inftance, 25, 49, &c. but he could not make 
any thing of thofe whofe root is even. After him came 
M. Frenicle, who took the fame fubjedt in hand. A 
certain great algebraift was of opinion, that, whereas the 
16 numbers which compofe a fquare might be difpofed 
20922789888000 different ways in a natural fquare (as 
from the rules of combination it is certain they may), 
they could not be difpofed in a magic fquare above 16 
different ways; but M. Frenicle fnowed, that they 
might be thus difpofed 878 different ways: whence it 
appears how much his method exceeds the former, which 
only yielded the 55th part of magic fquares of that of M. 
Frenicle. 
To this enquiry lie thought fit to add a difficulty, that' 
had not yet been confidered ; the magic fquare of 7, for 
inftance, being conftrufted, and its 49 cells filled, if the 
two horizontal ranks of cells, and, at the fame time, the 
two vertical ones, the moft remote from the middle, be 
retrenched, that is, if the whole border or circumference 
of the fquare be taken atvay, there will remain a fquare, 
whofe root will be 5, and which will only confift of 25 
cells. Now, it is not at all furprifing, that the fquare 
ftiouid be no longer magical, becaufe the ranks of the 
large ones were not intended to make the fame fum, ex¬ 
cepting when taken entire with all the feven numbers that 
fii! their feven cells; fo that, being mutilated each of two 
cells, and having loft two of their numbers, it maybe well 
expefted that their remainders' will not any longer make 
the fame fum. But M. Frenicle would not be fatisffed 
unlefs, when the circumference or border of the magic 
fquare was taken away, and even any circumference at 
pleafure, or, in fine, feveral circumferences at once, the 
remaining fquares were .ftill magical; which laft condi¬ 
tion, no doubt, made thefe fquares vaftly more magical 
than ever. Again, he inverted that condition, and re¬ 
quired that any circumference taken at pleafure, or even 
feveral circumferences, ihould be inseparable from the 
fquare ; that is, that it ihould ceafe to be magical when 
they were removed, and yet continue magical, alter the 
removal of any of the reft. M. Frenicle, however, gives 
no general demonilration of his methods, and frequently 
feems to have no other guide but chance. It is true, his 
book 
