Page:Cyclopaedia, Chambers - Volume 2.djvu/128

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with 3, the third Rank therefore mult commence with 5, the fourth with 7, the fifth with z, the fixth with 4, and the feventh with 6. The Commencement of the Ranks which follow the firft be- ing thus determined, the other Numbers, as we have already obferved, mull be written down in the Order wherein they ftand in the firlt, going on to 5, 6, and 2, fyc. till every Number in the 1 every Rank underneath, accord- ing to the Order arbitrarily pitched upon at firft. By this means, 'tis evident, no Number whatever can be repeated twice in the fame Rank, and by confequence that the feven Numbers 1. 2. 5. 4. 5.6". 7. being in each Rank, they muft of Neceflity make the fame Sum.

It appears, from this Example, that the Arrangement of the Numbers in the firft Rank being chofen at plea- fure, the other Ranks may be continued in four different Manners ; and fince the firft Rank may have 5040 diffe- rent Arrangements, there are no lefs than 2.0 1 60 different Manners of conftru&ing the Magic Stuart of feven Num- bers repeated.

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The Order of the Numbers in the firft Rank being determined ; if in beginning with the fecond Rank, the fccond Number 2, or the laft Number 7 /hould be pitched upon 5 in one of thofe Cafes one of the Diagonal Ranks would have the fame Number conftantly repeated 5 and, in the other Cafe, the other Diagonal would have it re- peated j of confequence therefore, either the one or the other Diagonal would be falfe, unlefs the Number re- peated feven times /hould happen to be 4, for four times feven is equal to the Sum of it. 2. 3. 4. 5. 6. 7. and, in general, in every Square confiding of an uneven Num- ber of Terms, in Arithmetical Progreflion, one of the Diagonals would be falfe according to thofe two Conftruc- tions, unlefs the Term, always repeated in that Diago- nal, were the middle Term of the Progreflion.

'Tis not however at all neceflary to take the Terms in an Arithmetical Progreflion ; for, according to this Method, one may conftrucl: a Magic Square of any Num- bers at pleafure, whether they be according to any cer- tain Progreflion or not. If they be in an Arithmetical Progreflion, 'twill be proper, out of the general Me- thod, to except thofe two Conftruftions, which produce a continual Repetition of the fame Term in one of the two Diagonals j and only take in the Cafe, wherein that Repetition would prevent the Diagonal from being juft. Which Cafe being abfolutely difregarded, when we computed that the Square of 7 might have 20160 dif- ferent Conftruclions j 'tis evident, that by taking that Cafe in, it mutt have vaftly more.

To begin the fecond Rank with any other Number befides the fecond and the laft, muft not however be looked on as an univerfal Rule. It holds good for the Square of 7, but if the Square of 9, for inftance, were to be conftrucled, and the fourth Figure of the firft Horizontal Rank were pitched on for the firft of the fe- 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 reft. The general Rule therefore muft be conceived thus: Let the Number in the firft Rank pitched on, for the Commencement of the fecond, have fuch an Exponent of its Quota, thatis, 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 Part of the Root of the Square ; thatis, cannot divide it equally. If, for Example, in the Square of 7, the third Number of the firft Horizontal Rank be pitched on for the firft of the fecond, fuch Conftruclion will be juft j becaufe the Exponent of the Place of that Number, viz. 3, fubttracYing 1, that is 2, cannot divide 7. Thus alfo might the fourth Number of the fame firft Rank be chofen, becaufe 4 — 1, mas, 3. cannot divide 7, and for the fame Reafon the fifth or fixth Number might be taken : Bat in the Square of 

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the firft Rank muft not be taken, becaufe 4 — r, •viz. $, does divide y. The Reafon of this Rule will appear ve- ry evidently, by confidering in what manner the Re- turns of the lame Numbers do or do not happen, taking them always in the fame manner in any given Se- ries. And hence it follows, that the fewer Divifions the Root of any Square to be conftructed has, the more dif- ferent Manners of constructing it there are, and that the prime Numbers, that is, thofe which have no Divifions, as 5. 7. 11. 13. &c are thofe whofe Squares will admit of the moft Variations in proportion to their Quantities.

The Squares conftrucled, 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 Diagonal muft neceffarily be fhorter, and have fewer Cells than the Diagonal itfelf, by adding to it the correfponding Parallel which has the Number of Cells, the other falls ftiort of the Diagonal 3 the Num- bers of thofe two Parallels, placed, as it were, end to end, ftill follow the fame Order with thofe of the Diagonal ; be- fides that, their Sums are likewife equal 5 fo that they are magical on another account.

Inftead of the Squares, which we have hitherto form'd by Ho- rizontal Ranks, one might alfo form them by Vertical Ones j the Cafe is the fame in both.

All we have hitherto fa id re- gards only the firft Primitive Square, whofe Numbers, in the propofed Example, were 1. 2. 3. 4.5.5.7? there ftill remains the fecond Primitive, whofe Numbers are 0.7. 14.21. 28. 35. 42. M. tie la Hire proceeds in the fame manner here as in the former ; and this may likewife be conftruded in 20160 diffe- rent Manners, as containing the feme Number of Terms with the firft. Its Conftruclion 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 firft of each, the two Numbers of the fecond, of the third, &c. and difpofe them in the 49 correfponding Cellsof a third Square j it will likewife be Magic, in regard its Ranks, formed by the Addition of equal Sums to equal Sums muft of neceflity be equal among themfelves. All that remains in doubt is, whether or no, by the Addition of the correfponding Cells of the two firft 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 1 1049, but alfo that this Number be different from that of any of the reft, which is the End and Defign of the whole Operation.

As to this, it muft be obferv'd, that if in the Conftruc- tion of the fecond Trimiwce Square, care has been taken in the Commencement of the fecond Horizontal Rank, to obferve an Order with regard to the firlt, different from what was obferv'd in the Conftrucfion of the firft Square ; for inftance, il the fecond Rank of the firft Square begun with the third Term of the firft Rank, and the fecond Rank of the fecond Square commence with the fourth of the firft Rank, as in the Example it actually does ; each Number of the firft Square may be combined once, and only once, by Addition with all theNumbersof the fecond. And as the Numbers of the firft are here 1. 2. 3. 4. 5- 6- 7. and thofe of the fecond 0.7. 14. 21 bining them in this manner. the Progreflion from 1 to 49, without having any of em repeated ; which is the TerfeB Magic Square propofed.

The Neceflity of conflrufting the two Primitive Squares in a different manner, does not at all hinder but that each of the 20160 Ccnftruclions of the one may be combined with all the 20160 Conductions of the orher : of confe- quence therefore 20160 multiplied by itfelf, which makes 406425(500, is the Number of different Constructions that . may be made of the PerfeB Sqmrej which here confifls of

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