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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 482

( 4 -
aiagonal Ranks, this Difpofition of Numbers is called
a Magic £juare, in oppofition to the former Difpofition,
which is called a Natural Square.  See the Figures- ad-
Natural Square.   Magic Square.  One would ima-
gine  that   Magic
S2 3 4 _      t6 14 81 2e.5 Squares had    that
~6 _8 _9lo     322 xz GI r _Name given them,
_ _51642.3        ____Hin regard this Pro-
. I t2 I13I4 1 15  I 6 4 423 17 prty of all their
16 17 8 15 2C   4 4 18  H IC 1 I  Ranks, which, ta-
- - - - -      ken any way, make
2.1 2. 2.3  4 e5  _    5  911 9I  always the fame
Sum, appeared ex-
tremely furprizing, efpecially in certain ignorant Ages,
*hen Mathematics paffed for Magic: But there is a great
deal of Reafon to fufpel, that thefe Squares merited
their Name fill further by the fuperflitious Operations
they were imployed in, as the Confirutioo of Talifmans,
&c. for according to the childilh Philofophy of thofe Days,
which attributed Virtues to Numbers, what Virtues might
not be expe~ied from Numbers fo wonderful?
However, what was at firft the vain Praclice of Ma-
kers of Talifmans, and Conjurers, has fince become the
Subjet of a ferious Refearch among the Mathemati-
Cians; not that they imagine it will lead them to any
thing of folid Ufe or Advantage. MagicSquares favour too
much of their Original to be of any Ufe. But only as
Itis a kind of Play, where the Difficulty makes the Me-
rit ; and as it may chance to produce fome new Views of
Numbers which Mathematicians will not lofe the Occa-
fion of.
Eman. Mefcbopulus, a Greek Author of no great Anti-
quity, is the firfi that appears to have fpoke of Magic
Squares; and by the Age wherein he lived, there is Rea-
fan to imagine he 'did not look on them merely as a
Mathematician.  However, he has left us fome Rules
for their Conflrulion. In the Treatife of Cor. Agrippa,
fo much accufed of Magic, we find the Squares of fe-
ven Numbers, viz. from three to nine inclufive, dif-
pofed magically; and it muff not be fuppofed that thofe
Seven Numbers were preferred to all the others without
a. very good Reafon.  In effect, 'tis, becaufe their
Squares, according to the Syflem of Agrippa and his Fol-
lowers, are planetary.  The Square of 3, for Inflance,
belongs to Saturn, that of 4 to 5fupiter, that of 5 to Mars,
that of 6 to the San, that of 7 to Venus, that of 8 to Mer-
cury, and that of 9 to the Moon. M. sachet applied him-
felf to the Study of Magic Squares, on the Hint he had
taken from the Planetary Squares of A4grippa; as being un-
acquainted with the Work of Mofcbopulus, which is only
in Manufcript in the French King's Library; and, with-
out the Affiflance of any other Author, found out a new
Method for thofe Squares whofe Root is uneven, for in-
fiance 25, 49, ZOc. but could not make any thing of
thofe whofe Root is even.
After him came Mr. Frenicle, who took the fame Sub-
je& in hand. A great Algebraift was of opinion, that
whereas the fixteen N umbers, which compofe the Square,
might be difpofed 2092.789888000 different Ways in a
natural Square (as from the Rules of Combination 'tis
certain they may) could not be difpofed in a Magic
Square above fixteen different Ways.  But M. Frenicle
Thewed, that they might be difpofed 878 different Ways ;
whence it appears how much his Method exceeds the
former, which only yielded the 55th Part of Magic
Squares of that of Mr. Frenkcle.  To this Enquiry he
thought fit to add a Difficulty, that had not yet been
confidered: The Magic Square of 7, for inflance, being
conflructed, and its 49 Cells filled, if the two Horizon.
tal Ranks of Cells, and at the fame time the two Ver-
tical ones, the moal remote from the middle, be re-
trenched, that is, if the whole Border or Circumference
of the Square be taken away; there will remain a
Square, whofe Root will be 5, and which will only
confifof a5 Cells. Now 'tis not at all furprizing that
the Square Ihould be no longer Magic, in regard the Ranks
of the large one were not intended to make the fame
Sum, excepting when taken entire with all the 7 Num-
bers that fill their feven Cells; fo that being mutilated
ach of two Cells, and having loll two of their Num-
bers, it may be well expected that their Remainders
will not any longer make the fame Sum. But Mr. Fre-
tuicle would not be fatisfied, unlefs when the Circumfe-
rence or Border of the Magic Square was taken away, and
even any Circumference at pleafure, or in fine feveral
Circumferences at once, the remaining Square were flill
Agic : which laf Condition, no doubt, made thefe
Squares vafil y more magical than ever.
Again, he inverted that Condition, and required that
sny Circumference taken at pleafure, or even feveral Cir,
cumferences fhould be infeparable from  the Square;
that is, it Thould ceafe to be Magic when they were re-
moved, and yet continue Magic after the Removal of a-
ny of the relt. Mr. Frenicle, however, gives no general
Demonrfration of his Methods, and frequently feems to
have no other Guide but his groping.  'Tis true, his
Book was not publilhed by him felf, nor did it appear
till after his Death, viz. in 1693.
In 1703, Mr. Poignard, Canon of Brtffels, publifhed
a Treatife of Sudblime Magic Squares. Before him there
had been no MagicSquares made but for Series's of na-
tural Numbers that formed a Square; but M. Poig-
nard made two very confiderable Improvements: (i.) In-
flead of taking all the Nu mbers that fill a Square,
for Inflance, the 36 fucceflive Numbers, which would
fill all the Cells of a natural Square, _whufe Side is 6,
he only takes as many fucceilive Numbers as there art,
Units in the Side of the Square, which in this Cafe are
6; and thefe fix Numbers alone he difpofes in fuch man.
ner, in the 36 Cells, that none of them are repeated
twice in the fame Rank, whether it be horizonta l, ver-
tical, or diagonal: whence it follows, that all the Ranks,
taken all the Ways pofible, mull alwa)s make the fame
Sum, which Mr. Poignard calls repeated Progreffion.
(a.) Inflead of being confined to take thefe Nlumbers
according to the Series and Succeffion of the .natural
Numbers, that is, in an Arithmetical Progreffion, he
takes them likewife in a Geometrical Progrefflon, -and
even an Harmonical Progreflion.  But with thefe two
lafl Progressions the Magic mull neceffarily be different
from what it was. In the Squares, filled with Numbers
in Geometrical Progreflion, it confifis in this, that the
Produ61s of all the Ranks are equal, and in the Har-
monical Progreffion, the Numbers of all the Ranks con-
tinually follkw that Progreflion: he makes Squares of
each of thefe three Progreffions repeated.
This Book of M. Poignard gave occafion to M. de la
Hire to turn his Thoughts the fame way, which he did
with good Succefs, infomuch that he feems to have well-
nigh compleated the Theory of Magic Squares. He firfi
confiders uneven Squarer: all his Predeceffirs on the Sub-
jed having found the Confirudion of even ones by much
the mofa difficult ; for which Reafon M. de la Hire -re-
ferves thofe for the lafb.  This Excefs of Difficulty may
arife partly from hence, that the Numbers are taken in
an Arithmetical Progreffion. Now in that Progreffion, if
the Number of Terms be uneven, that in the middle has
fome Properties, which may be of Service ; for inflance,
being multiplied by the Number of Terms in the Pro-
greflion, the Produ&  is equal to the Sum of all the
M. de la Hire propofes a general Method for uneven
Squares, which has fome Similitude with the Theory of
compound Motions, fo ufeful and fertile in Mechanics. As
that confifds in decompounding Motions, and refolving
them into others more fimple, fo does M. dela-Hir's
Method confifd in refolving the Square, that is to be
conflrutled, into two fimple and primitive Squares. It
muff be owned, however, 'tis not quite fo eafy to con-
ceive thofe two fimple and primitive Squares in the com-
pound or perfef Square, as in an oblique Motion to ima-
gine a Parallel and a Perpendicular one.
Suppofe a Square of Cells, whofe Root is uneven;
for Inflance 7, and thatits49 Cells are to be filled ma-
gically with Numbers, for infiance, the firfl 7.  M. de la
Hire, on the one fide, takes the firff feven Numbers, be-
ginning with Unity, and ending with the Root 7, and on
the other 7, and all its Multiples to 49 exclufi vely; and
as thefeonly make fix Numbers, he addso, which makes
this an Aritmetical Progreffion of feven Terms as well a3
the other. c. 7. 14.2I. .28 3  42 .
This done, with the firil Progreflion repeated, he fills
the Square of the Root 7 magically.  In order to this,
he writes in the firfi feven Cells of the firil Horizontal
Rank the feven Numbers propofed, in what Order he
pleafes, for that is abfolutely indifferent; and 'tis pro-
per to obferve here, that thofe feven Numbers may be
ranged in 504c different Manners in the fame Rank.
The Order in which they are placed in the firdl Horizon-
tal Rank, be it what it will, is that which determines
their Order in all the ref. For the fecond Horizontal
Rank, he places in its firfi Cell, either the third, the
fourth, the fifth, or the fixth Number from the firft
Number of the firfi Rank, and after that writes the fix
others in the Order as they follow. For the third Hori-
zontal Rank, he obferves the fame Method with regard
to the fecond, that he obferved in the fecond with regard
to the firff, and fo of the redi. For inflance, fuppofe
the firfm Horizontal Rank filled with the feven Num-
bers in their natural Order, I a. 3. 4. 5. 6. 7. the fec ond
Horizontal Rank may either commence with 3, with 4,
with 5, or with 6; but in this Inflance it commences

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