328 N U M 
of the fame fum, 2.055. 5. Half a diagonal afcending, 
added to half a diagonal defcending, makes alfo the fame 
{11111,2056; taking thefe half-diagonals from the ends of 
any fide of the fquare to the middle of it; and fo reckon¬ 
ing them either upward or downward, or fideways from 
right to l?ft, or from left to right. 4. The fame with all 
the parallels to the half-diagonals, as many as can be 
drawn in the great fquare : for, any two of them being 
directed upward and downward, from the place where 
they begin to that where they end, their films ftill make 
the fame 2056. Alfo, the fame holds true downward and 
upward; as well as if taken fideways to the middle, and 
hack to the fame fide again. Only one fet of thefe half¬ 
diagonals and their parallels, is drawn in the fame fquare 
upward and downward; but another fet may be drawn 
from any of the other three fides. 5. The four corner- 
numbers in the great fquare added to the four central 
numbers in it, make 1028, the half-fum of any vertical 
or horizontal column which contains 16 numbers; and 
alfo equal to half a diagonal, or its parallel. 6. If a fquare 
hole, equal in breadth to four of the little fquares or cells, 
be cut in a paper, through which any of the 16 little 
cells in the great fquare may be feen, and the paper be laid 
upon the great fquare ; the fum of all the 16 numbers feen 
through the hole is always equal to 2056, the fum of the 
16 numbers in any horizontal or vertical column. 
The magic circle of circles, fig. 6. by the fame au¬ 
thor, is compofed of a feries of numbers, from 12 to 
75 inclufive, divided into concentric circular fpaces, 
and arranged in 8 radii of numbers, with the number 
12 in the centre; which number, like the centre, is 
common to all thefe circular fpaces, and to all the radii. 
The numbers are fo placed, that, ill. The fum of all thofe 
in either of the concentric circular fpaces above-mention¬ 
ed, together with the central number, 12, amount to 360, 
the fame as the number of degrees in a circle. 2. The 
numbers in each radius alfo, together with the central 
number 12, make juft 360. 3. The numbers in half of 
any of the above circular fpaces, taken either above or 
below the double horizontal line, with half the central 
number, 12, make juft 180, or half the degrees in acircle. 
4. If any four adjoining numbers be taken, as if in a 
fquare, in the radial divifions of thefe circular fpaces, the 
fum of thefe, with half the central number, make alfo 
the fame, 180. 5. There are alfo included four fets of 
other circular fpaces, bounded by circles that are eccen¬ 
tric with regard to the common centre ; each of thefe fets 
containing five fpaces, and the centres of them being at 
A, B, C, D. For diftindtion, thefe circles are drawn with 
different marks, fome dotted, others by Ihort uncon¬ 
nected lines, &c. or, ftill better, with inks of divers co¬ 
lours, as blue, red, green, yellow. Thefe fets of eccen¬ 
tric circular fpaces interfedt thofe of the concentric, and 
each other; and yet, the numbers contained in each of 
the eccentric fpaces, taken all around through any of the 
20 which are eccentric, make the fame fum as thofe in 
the concentric, namely 360, when the central number, 12, 
is added. Their halves alfo, taken above or below the 
double horizontal line, with half the central number, 
make up 180. It is obfervable, that there is not one of 
the numbers but what belongs at leaft to two of the 
circular fpaces ; fome to three, fome to four, fome to 
five ; and yet they are all fo placed, as never to break the 
required number, 360, in any of the 28 circular fpaces 
within the primitive circle. 
Thelaft of thefe curious combi nations or arrangements of 
fio ureswefliall take notice of, and which, in the frequency of 
producingthe number fought, tranfcends all the former, is 
that invented and conftrudted by Mr. Snart, (fee fig. 7.) 
founded upon the fquare of feven, con tain ingall the natural 
numbers thereof to be once u fed from 1 to 49, both inclufive. 
Which numbers are fo difpofed in the 49 cells, that, when¬ 
ever any feven of them are added together, they uniform¬ 
ly produce the defired fum of 175; no matter whether 
thefe columns of numbers are taken vertically, horizon- 
B E R. 
tally, or obliquely; making thus 16 readings, which, 
heretofore, have been confidered the greateft number 
that could be produced in a fquare of 49 numbers. 
But, on infpedting the plate, it will be feen, that, be- 
fides the 16 columnar readings, or lineal columns, there 
are 24 angular ones, all of which are obvious in the 
engraving, and may be diftindtly read, without de¬ 
ranging thefe columns at all; that is, there are eight 
acute angles of 45 degrees each, eight redtangles of 
'90 degrees, and eight obtufe of 135 degrees, as ftiown 
within the circumfcribing circle on each fide the fquare. 
All thefe, however, to the amount of 24 more readings, 
he has, by infinite labour and ingenuity", made fubfervient 
to his firft purpofe of producing the fum fought, without 
interfering at all with the lineal columns. To accomplifh 
this increafe or reiteration of readings, he has, contrary 
to the former cuftom, placed his higheft number, 49, in 
the centre, conllantly making that the angular point, and 
which is called into adtion no lefs than 30 times ; that is, 
once in the vertical readings, once in the horizontal, once 
in the oblique dexter, and once in the oblique finifter; 
24 times in the angular; and twice in the detached read¬ 
ings of five numbers each, i. e. the four extreme corners 
and the centre 49=175. And, laftly, the four middle ex¬ 
teriors marked 175, and the centre, amount to the fame 
fum, 175, as lhown by the rings at the corners, and the 
loops on the fides, containing each 175; making a total 
of 42 readings, or nearly triple thofe fquares w'hich were 
called the ne plus ultra of artificial arrangement. How¬ 
ever, (ufing his own words,) even this is but a bagatelle 
to what, if it were worth while, might be produced; be- 
caufe the permutations of 49 numbers, were they iingle 
and dillindt digits, in groups of feven each, amount to 
5040, as ftiown by his Table; but, when taken feparately, 
to 6082818640342675608722521633212953768S755283137- 
9210240000000000, whofe quotient, by feven divifor, muft 
convince any one, that even this fquare is not the maxi¬ 
mum of perfedtion, however ingenious it may be. It will 
be obvious to the attentive reader, that the total fumtna- 
tion of the 49 numbers, is — 1225-^-7=175, from which 
fource all is derived; but yet, from the nature of the de- 
fign of making the angular columns, &c. read, it will be 
equally plain, that the primary fquares can have no re¬ 
ference at all to the conitrudtion of this, becaufe the order 
of thefe fquares, w'hich confines the number of readings 
within the limits of 16, is inevitably broken in the firft 
inftance, to produce the more numerous angular read¬ 
ings, without facrificing one of the lineal. However, the 
numbers are equally balanced, or elfe the objedl could 
not have been produced, which now is completely ac- 
complifhed, as well as that of preferving a bold uniformity 
on each corner, which would naturally have been broken 
by placing a fingle digit in either of the corner cells. 
It was our intention to have introduced, at the clofe of 
this article, fome entertaining problems upon the powers 
and arrangement of numbers, with their folutions, by 
way of iliuftration of what has preceded. But we have 
fo far exceeded our limits, that we {hall fuddenly, though 
reludtantly, conclude with the following jeu d'ejprit, as 
defcribing a kind of magic fquare. 
Prob. In what manner can counters be difpofed in the eight 
external cells of a fquare, fo that there map be ahvays 9 in each 
roiv, and yet the whole number Jhall vary from 20 to 32 ?— 
Ozanam propofed this problem in the following manner, 
with a view, no doubt, to excite the curiofity of his 
readers: A certain convent confifted of nine cells, of 
which the centre one was occupied by a blind abbefs, and 
the reft by her nuns. The good abbefs, to allure herfelf 
that the nuns did not violate their vows, vifited all the 
cells; and, finding 3 nuns in each, which made 9 in every 
row, retired to reft. Four nuns, however, went out; and 
the abbefs, returning at midnight to count them, ftill 
found 9 in each row, and therefore retired as before. 
The four nuns then came back, each with a gallant; 
and the abbefs, on paying them another vifit, having 
again 
