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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

(1728)

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

Page 483

M A C (483 ) MAG - 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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