234 



♦ KNOWLEDGE ♦ 



[Oct. 12, 1883. 



or cells, be cut in a paper, tlu'ough wliich any of the sixteen little 

 cells may be seen, and the paper laid on the great square, the sum 

 of all the sixteen numbers seen through the hole is always equal to 

 205G. 



In short, there is scarcely an end to the variety of ways by ivhich 

 2056 may be made, ard to use the constructor's own words, " it is 

 the most magical of all magic squares ever made by any magician." 



Clavis. 



Copied from The Mirror of Oct. 28, 1824. 



MAGIC .SQUARES. 

 [959] — It may interest some of the readers of Knowledge to be 

 shown that a " Magic Square " can be made by a very simple me- 

 chanical process. Take anj- square number (witli an odd root) of 

 terms in continued arithmetical progression, say 1 to 121, and 



/" 



write them down in the form of a square, as in the appended 

 diagram. The next step is to inscribe a second square (a h c d) in 

 the first. This second square is, of course, just half the size of the 

 first, and will contain one more than one-half of all the terms in 

 the series, and all in their right places in the proposed square. 

 Hence it only remains to remove the other terms from the four 

 outside triangles to their respective positions. For convenience 

 draw the diagonals x x, xx, &c., y y, y y, &c., and then cut off the 

 four triangles from the inscribed square. Take now one of the 

 triangular pieces, say alf, and place it on the opposite side of the 

 inscribed square, with the side ab on d e, and the vertex 8 on the 

 centre, that is on the term "61." Dispose of the other thi-ee 

 triangular pieces along the corresponding sides; and supposing 

 transparent paper to have been used, ar.d the figures to have been 

 accurately arranged, every term will fall into its proper place, 

 thereby completing the " Magic Square." James Macfadzean. 



A VERY COMPLETE MAGIC SQUARE. 



[560]. — "G. S." sends you, and you publish under the above title 

 (888, p. 61), a magic square of nine, which he thinks it would be 

 difficult to surpass. I do not clearly understand what " G. S." 

 means when he says with reference to his square that the number 

 369 is produced " by the angles, whether right, acute, or obtuse, 

 from the corners or the middle from the centre," but I imagine that 

 he refers to some of the ways of making 369 with four pairs of 

 numbers, each making 82, and the central number, -41, added. If 

 this is so, his square possesses only two properties which do not 

 belong to all magic squares ; the first, that each of the nine small 

 squares makes 369 ; and the second, that each number with the 

 number opposite to it makes 82. The latter property gives a choice 

 of four out of forty pairs of numbers which, with 41, make 369. 

 This choice can be made in 91,390 ways ; and yet this property is 

 not at all a rare one, or one which makes a magic square parti- 

 cularly curious. It is found in the magic squares made by the very 

 simple rules of Moschopulus and others, and also in squares 

 not magic, made by writing the numbers down in their natural 

 order. 



Let me now offer you a magic square of nine, which contains all 

 the properties that I could discover in " G. S.'s" square and some 

 others. 



Additional properties of this square ; — 



1. If one or more rows be transferred from top to bottom, or 

 from one side to the other, or both, the square will still be magic. 

 In other words, all the broVen diagonals make 369; a broken dia- 

 gonal being a set of numbers beginning with one in the top, and 

 running diagonally to one side, then entering on the opposite side 

 in the next lower line and running diagonally to the bottom. 



2. In any square, whether that given above or one formed by 

 transfer of rows, any nine numbers forming a square make 369. 



3. In the small squares into which the above square is divided, 

 each horizontal or vertical row of three numbers makes 123, one- 

 third of 3C9. A. 



[For the present magic squares are done with. — R.P.] 



LETTERS RECEIVED AND SHORT ANSWERS. 

 An I.nquirek. There is no evidence showing that " the seasons 

 are not what they used to be twenty or thirty years ago ; " but 

 when you hear any one say that, you may safely conclude that he 

 (or she) is not what he (or she) used to be twenty or thirty years 

 ago. — G. JiNMAX. A happy shot, but made pretty safely. — A.x Old 

 WoMAX. Sorry to hear Mr. Jago mistaken in thinking all Liver- 

 pool bread good. — Geologist. Know of no such work. — F. C. 

 Dixon. Know nothing about blow-pipe, but believe some blow- 

 trumpet articles are appearing in the weekly you mention. — Clash. 

 Newcomb's explanation of the tides unsound. The usual theory is 

 statically correct, bnt the true theory is the dynamical one, for 

 sketch of which see Sir E. Beckett's "Astronomy without Mathe- 

 matics." Do not know where you can get photographs of the sun. 

 Gall's projection not known to me. — Jas. K. Thokxetox, Glasgow. 

 My dear (but atrabilious) sir, you are the solitary survivor 

 (I fancy) of the foohsh folk among Kxowledge readers who 

 mistake fun for fury. I thought we had got rid of the last 

 of you. Tour weekly twopence would be far better bestowed 

 on PiUd. Antib. (Cockl.). — F. Cowley. I don't think I promised 

 explanation, only comment. No one can conceive space of 

 four dimensions or more. Would not mind some pretending they 

 can if they would not waste so much time and thought about it. 

 — Vega. Hitherto I have foimd no place for poetry, of which a 

 good deal has been sent me. If I make an exception in your 

 case it is because of the bearing of your ideas on my own views, 

 with which (by the way) they are not so inconsistent as many 

 might imagine. The lines shall appear ; but the case must not be 

 regarded as a precedent. If readers should approve of the lines, 

 however, the others of which you speak may perhaps find a place. 

 I do not consider myself a judge — for a reason yon will understand. 

 — A Stargazek. IIow odd that your letter should come next! 

 Your stanzas (as you call them), beginning " The stars so bright ! 

 The stars so bright ! How beautiful they shine ! " &c., would 

 scarcely suit these columns, even if poetry were in general admis- 

 sible here, which is not the case. Yet the last lines are profoundly 

 touching, and quite merit your notes of admiration : — 

 With folded hands ! Aye, folded hands ! and likewise down-bowed 



head ! 

 I gazed upon the glowing orbs that shone high overhead ! 

 I mused upon tiiose nii/sh'c orbs till day came and they fled! 

 I can understand the folded hands (you should have kept them so) 

 but a down-bowed head hardly suits one who is gazing on stars 

 overhead. Your rhyming also is a little shaky ; but the same fault 

 may perhaps be found with the following little poem which has 



