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Chambers, Ephraim, 1680 (ca.)-1740 / Cyclopædia, or, An universal dictionary of arts and sciences : containing the definitions of the terms, and accounts of the things signify'd thereby, in the several arts, both liberal and mechanical, and the several sciences, human and divine : the figures, kinds, properties, productions, preparations, and uses, of things natural and artificial : the rise, progress, and state of things ecclesiastical, civil, military, and commercial : with the several systems, sects, opinions, &c : among philosophers, divines, mathematicians, physicians, antiquaries, criticks, &c : the whole intended as a course of antient and modern learning

M - mapparius,   pp. 478-497 PDF (19.4 MB)

Page 483

(483 )
-   with 3, the third junktheretore
7  muAl commence with 5,the fourth
i   with 7, the fifth with a, the fixth
with 4, and the feventh with 5.
4   The Commencement of the
6   Ranks which follow the firfl be-
ing thus determined, the other
_ Numbers, as we have already
obferved, mufl be written down
in the Order wherein they fland
in the hrit, going on to 5, 6, and
7, and returning to T, A, &c. till every Number in the
firet Rank be found in every Rank underneath, accord-
ing to the Order arbitrarily pitched upon at firfl.  By
this means, tis evident, no Number whatever can be
repeated twice in the fame Rank, and by confequence
that the feven Numbers l. '. 3- 4. 5. 6. 7. being in each
Rank, they mu{} of Neceffity make the fame Sum.
It appears, from this Example, that the Arrangement
of the Numbers in the firfm Rank being chofen at plea-
fure, the other Ranks may be continued in four different
Manners; and fince the firfm Rank may have 5040 diffe-
rent Arrangements, there Ire no lefi than zoi6o different
Manners of confiruaing the Magic Square of even Num-
bers repeated.
1     3 45    6 7      I 2  3 4   5  6 7
2 34   5 67I                  3     4  56
4  5 6   7    2. 3     5 6 7         3 4
S 5           4    I45        67 7    24
6 7 _ 2
7  l 2 34      56       2.34    561711
The Order of the Numbers in the fird Rank being
determined ; if in beginning with the fecond Rank, the
fecond Number 2, or the laft Number 7 mhould be pitched
upon ; in one of thofe Cafes one of the Diagonal Ranks
would have the fame Number conflantly repeated ; and,
in the other Cafe, the other Diagonal would have it re-
peated ; of confequence therefore, either the one or the
other Diagonal would be falfe, unlefs the Number re-
peated feven times mhould happen to be 4, for four times
feven is equal to the Sum of i. 2.. 3. 4. 5. 6. 7. and, in
general, in every Square confifling of an uneven Num-
ber of Terms, in Arithmetical Progreffion, one of the
Diagonals would be falfe according to thofe two Confiruc-
tions, unlefs the Term, always repeated in that Diago-
nal, were the middle Term of the Progreffion.
'Tis not however at all neceffary to take the Terms
in an Arithmetical Progrefflon ; for, according to this
Method, one may confirud a Magic Square of any Num-
bers at pleafure, whether they be according to any cer-
tain Progreffion or not. If they be in an Arithmetical
Progreffion, 'twill be proper, out of the general Me-
thod, to eicept thofe two Conffruffions, which produce
a continual Repetition of the fame Term in one of the
two Diagonals ; and only take in the Cafe, wherein that
Repetition would prevent the Diagonal from being jull.
Vhich Care being absolutely difregarded, when we
computed that the Square of 7 might have 2oi60 dif-
ferent Conflru&ions; 'tis evident, that by taking that Cafe
in, it mufl have vaflly more.
To begin the fecond Rank with any other Number
befides the feccond and the lal, mull not however be
looked on as an univerfal Rule.  It holds good for the
Square of 7, but if the Square of 9, for inflance, were
to be confiructed, and the fourth Figure of the firfm
Horizontal Rank were pitched on for the firfi of the fe-
P cond, the Confequence would be, that the fifth and
eighth Horizontal Ranks would likewife commence with
the fame Number, which would therefore be repeated
three times in the fame vertical Rank, and occafion
other Repetitions in all the rell.  The general Rule
therefore mull be conceived thus: Let the Number in
the firfi Rank pitched on, for the Commencement of the
fecond, have fuch an Exponent of its Quota, that is, let
the Order of its Place be fuch, as that if an Unit be
taken from it, the Remainder will not be any juft Quota
Vart of the Root of the Square; that is, cannot divide it
equally. If, for Example, in the Square of 7, the
third Number of the firfl Horizontal Rank be pitched on
for the firf{ of the fecond, fuch Conflrudion will be
jual; becaufe the Exponent of the Place of that Number,
viz. 3, fubilraffing i, that is a, cannot divide 7. Thus
alfo might the fourth Number of the fame firfl Rank be
chofen, becaufe 4 -1, viz. 3 cannot divide 7, and for
the fame Reafon the fifth or fixth Number might be
taken: But in the Square of 9, the fourth Number of
the firft Rank mull not be taken, becaufe 4 - r, VZZ. 3,
does divide 9. The Reafon of this Rule will appear ve-
ry evidently, by confidering in what manner the Re-
turns of the fame Numbers do or do not happen,
taking them always in the famne manner in any given Se-
ries. And hence it follows, that the fewer Divifions the
Root of any Square to be conflruaed has, the more dif-
ferent Manners of conftruaing it there are, and that the
prime Numbers, that is, tho?'e which have no Divifions,
as 5. 7.tI. 13. Ec. are thofe whole Squares will admit
of the moff Variations in proportion to their Quantities.
The Squares confirua ed, according to this Method,
have fome particular Property not required in the Pro-
blem: For the Numbers that compofe any Rank pa-
rallel to one of the two Diagonals, are ranged in the
fame Order with the Numbers that compofe the Dia
gonal, to which they are parallel. And as any Rank pa.
rallel to a Diaaonal mutt neceffarily be Ihorter, and
have fewer Cells than the Diagonal itfelf, by adding to
it the correfponding Parallel which has the Number of
Cells, the other falls fhort of the Diagonal; the Num-
bers of thofe two Parallels,
placed, as it were, end to end,  .Frjf Primitive.
frili follow the fame Order
with thofe of the Diagonal; be-  -- 2  451-     7
fides that, their Sums are     3 4 5 6 7 1 2
likewife equal ; fo that they are  5 6 7  2 3 4
magical on another account.    -  __     -
Inflead of the Squares, which  7 1 2 3 4 5 6
we have hitherto form'd by Ho-  2. 34 5 67 1
rizontal Ranks, one might alfo  4 _  6 7_I 2; 3
form them by Vertical Ones;   6 -
the Cafe is the fame in both.  6 7 1 2 3 4' 5
All we havehitherto faidre-
gards only the firf 'Primitive   Second Prirmitve.
Square, whofe Numbers, in the   '    I z |
propofed Example, were I. 2. 3.  C -
4.5.6. 7; there fill remains    2.1 835 42 C 7Z14
the fecond Primitive, whofe    42 o 71 21 z8'35
Numbersareo.7. 142.2. 28. 35.   _ _    _ _ _1_
42. M. de la Hire proceeds in   14 21 . 3542 0 7
the fame manner here as in the  3542.C 7 14 2Ij8
former; and this may likewife r   42   8    3542.0
be conflruaed in 2o06b diffe-   _ _        35 _  __
rent Manners, as containing the  2 835142 G 7 14 21I:
fame Number of Terms with
the firff.  Its Confiruffion being made, and of confe-
quence all its Ranks making the fame Sum, 'tis evi-
dent, that if we bring the two into one, by adding to-
gether the Numbers of the two correfponding Cells of
the two Squares, that is, the two Numbers of the firil of
each, the two Numbers of the fecond, of the third, & c.
and difpofe them in the 49 correfponding Cells of a third
Square; it will likewife be Magic, in regard its Ranks,
formed by the Addition of equal Sums to equal Sums
mufl of necessity be equal among themrfelves. All that
remains in doubt is, whether or no, by the Addition
of the correfponding Cells of the two'firil Squares, all
the Cells of the third will be filled in fuch manner, as
that each not only contain one of the Numbers of the
Progreflion from I to49, but alfo that this Number be
different from that of any of the reff, which is the End
and Defign of the whole Operation.
As to this, it mull be obferv'd, that if in the Conflruc-
tion of the fecond Przmitite Square, care has been taken
in the Commencement of the fecond Horizontal Rank,
to obferve an Order with regard to the firfl, different
from what was obferv'd in the Conflruftion of the firit
Square; for inflance, if the fecond
Rank of the firA Square begun      Perfeg Square.
with the third Term of the firil  1 9 17 z 533 41 49
Rank, and the fecond Rank of the  4 _ 0 8 7_7
fecond Square commence with the     - 3
fourth of the firfi Rank, as in the 47 6 14 15 231O31 39
Example it adtually does; each   2.122131 38 4615 13
Number of the firfi Square may be             -
combined once, and only once, by  ;7L14 12 2.0z29
Addition with all theNumbersof the  1 t 9 27 3 5 36146 3
fecond. And as the Numbers of 6           re'8 z 6
the firil are here i.a. 2  5. )6. 7. "
and thofe of the fecond 0. 7.14- 2.1 28- 35. 42. by com-
bining them in this manner, we have all the Numbers in
the Progreflion from I to 49, without having anyof'em
repeated ; which is the Perfeaf Magic Square propofed.
The Neceffity of confiruding the two Primitive Squares
in a different manner, does not at all hinder but that each
of the 20160 Conflruclions of the one may be combined
with all the 2or6o Confiruaions of the other: of confe-
quence therefore z0160 multiplied by itfelf, which makes
40642 5600, is the Number of different Conflruflions that
may be made of the Perfe~f Square, which here confifis of
.. .-   -   .  I

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