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Diagonal Ranks, this Difpofition of Numbers is called a Magic Square, in oppofition to the former Difpofirion, which is called a Natural Square. See the Figures ■ ad- joining.
Natural Square. Magic Square. One would ima-
gine that Magic Squares had that Name given them, in regard this Pro- perty of all their Ranks, which, ta- ken any way, make always the fame Sum, appeared ex- tremely furprizing, efpeci.illy in certain ignorant Ages, when Mathematics paffed for Magic : But there is a great deal of Reafon to fufpect, that thefe Squares merited their Name ftill further by the fuperftitious Operations they were imploycd in, as the Conftruction of Tali/mans, &c. for according to thechildifh Philofophy of thofeDays, which attributed Virtues to Numbers, what Virtues might not be expected from Numbers fo wonderful ?
However, what was at firft the vain Practice of Ma- kers of Talifmans, and Conjurers, has fince become the Subject of a ferious Refearch among the Mathemati- cians ; not that they imagine it will lead them to any thing of folidUfe or Advantage. Magic Squares favour too much of their Original to be of any Ufe. But only as 'tis a kind of Play, where the Difficulty makes the Me- rit j and as it may chance to produce fome new Views of Numbers which Mathematicians will not lofe the Occa- fion of.
Email. Mefcb'fulus, a Greek Author of no great Anti- quity, is the firft that appears to have fpoke of Magic Squares 5 and by the Age wherein he lived, there is Rea- fon to imagine he did not look on them merely as a Mathematician. However, he has left us fome Rules for their Construction. In the Treatife of Cor. Jgrifja, fo much accufed of Magic, we find the Squares of fe- ven Numbers, viz. from three to nine inclufive, dif- pofed magically 5 and it muft not be fuppofedthat thofe Fevcn Numbers were preferred to all the others without a very good Reafon. In effect, 'tis- becaufe their Squares, according to the Syftem of Agrippa and his Fol- lowers, are planetary. The Square of ;, for Inftance, belongs to Saturn, that of 4. to Jupiter, that of 5 to Mars, that of 6 to the&iw, that of 7 to Venus, that of 8 to Mer- cury, and that of 9 to the Moon. M. Backet applied him- felf to the Study of Magic Squares, on the Hint he had taken from the Planetary Squares of Agrif fa ; as being un- acquainted with the Work of Mofchopalus, which is only in Manufcript in the trench King's Library 5 and, with- out the Affiftance of any other Author, found out a new Method for thofe Squares whofe Root is uneven, for in- ftance 25, 49, &c. but could not make any thing of thofe whole Root is even.
After him came Mr. Frenicle, who took the fame Sub- ject in hand. A great Algebraift was of opinion, that whereas the fixteen Numbers, which compofe the Square, might be difpofed 20922789888000 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 /hewed, that they might be difpofed 87S different Ways ; whence it appears how much his Method exceeds the former, which only yielded the 55 th Part of Magic Squares of that of Mr. Frenicle. To this Enquiry he thought fit to add a Difficulty, that had not yet been confidered: The Magic Square of 7, for inftance, being conflicted, and its 49 Cells filled, if the two Horizon- tal Ranks of Cells, and at the fame time the two Ver- tical ones, the moil 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 confift of 2 5 Cells. Now 'tis not at all furprizing that the Square fhould 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 each of two Cells, and having loft two of their Num- bers it may be well expected that their Remainders will'not any longer make the fame Sum. But Mr. Fre- nicle 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 Mill Magic : which laft Condition, no doubt, made thefe Squares vaftly more magical than ever.
Again, he inverted that Condition, and required that any Circumference taken at pleafure, or even feveral Cir-
cumferences fhould be infeparable from the Square ; that is, it fhould ceafe to be Magic when they were re- moved, and yet continue Magic after the Removal of a- ny of the reft. Mr. Frenicle, however, gives no general Dcmonflration of his Methods, and frequently fe'ems to have no other Guide but his groping. "lis true, his Book was not publilhed by himfelf, nor did it appear till after his Death, visa: in KS93.
In 1705, Mr. Foignard, Canon of T.nJJels, publilhed a Treatife of Sublime Magic Squares. Before him there had been no Magic Squares made but for Series's of na- tural Numbers that formed a Square; but M. Foig- nard made two very confiderablc Improvements : (1.) In- ftead of taking all the Numbers that fill a Square, for Inftance, the ;6 fucceffive Numbers, which would fill all the Cells of a natural Square, "whofe Side is 6, he only takes as many fucceffive Numbers as there are 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 s.(f Cells, that none of them are repeated twice in the fame Rank, whether it be horizontal, ver- tical, or diagonal : whence it follows, that all the Ranks, taken all the Ways poffible, mull always make the Tame Sum, which Mr. Foignard calls repeated Frogteffion. (2.) Inftead of being confined to take thefe Numbers according to the Series and Succcffi in of the . natural Numbers, that is, in an Arithmetical Progreffion, he takes them likewife in a Geometrical Progreffion, .and even an Harmonical Progreffion. But with thefe two laft Progrcffions the Magic mull neceffarily be different from what it was. In the Squares, filled with Numbers in Geometrical Progreffion, it confills in this, that the Products of all the Ranks are equal, and in the Har- monical Progreffion, the Numbers of all the Ranks con- tinually folic w that Progreffion : he makes Squares of each of thefe three Progreffions repeated.
This Book of M. Foignard gave occafion to M. de let 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 firft confiders uneven Squares : all his Pre'deceffors on the Sub- ject having found the Conftruction of even ones by much the moft difficult ; for which Reafon M. de laHirere- ferves thofe for the laft. 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 inftance, being multiplied by the Number of Terms in the Pro- greffion, the Product is equal to the Sum of all the Terms.
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 confills in decompounding Motions, and relolving them into others more fimple, fo does M. dela-Hire's Method confift in refolving the Square, that is to be conftructed, into two fimple and primitive Squares. It mull be owned, however, 'tis not quite fo eafy to con- ceive thofe two fimple and primitive Squares in the com- pound or perfelt Square, as in an oblique Motion to ima- ginea Parallel and a Perpendicular one.
Suppcfe a Square of Cells, whofe Root is uneven ; for Inftance 7, and that its 49 Cells are to be filled ma- gically with Numbers, for inftance, the firft 7. M. de la. Hire, on the one fide, takes the firft feven Numbers, be- ginning with Unity, and ending with the Root 7, and on the other 7, and all its Multiples to 49 exclufively ; and as thefeonly make fix Numbers, he addso, which makes this an Aritmetical Progreffion of feven Terms as well as the other, o. 7. 14.21. 28. 35. 42.
This done, with the firft Progreffion repeated, he fills the Squate of the Root 7 magically. In order to this, he writes in the firft fevcn Cells of the firft Horizontal Rank the feven Numbers propofed, in what Order he pleafes, for that is absolutely 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 firll Horizon- tal Rank, be it what it will, is that which determines their Order in all the reft. For the fecond Horizontal Rank, he places in its firft Cell, either the third, the fourth, the fifth, or the fixth Number from the firft Number of the firft 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 heobferved in the fecond with regard to the firft, and fo of the reft. For inftance, fuppofe the firft Horizontal Rank filled with the feven Num- bers in their natural Order, 1. 2. 3. 4. 5.C. 7. the fecond Horizontal Rank may either commence with 2, with 4, with 5, or with 6 j but in this Inftance it commences
with
