■27 \- 



♦ KNOWi^EDGE • 



[Ja-v. 27, 1882. 



In tlip onjfrnvinft by Albert Dflrer, called " Mcloncholia I," the 

 Hquaro of 4 in rcprc-HontoH on thp wall of ii hoiiBC, iind Mr. 

 W. n. Scott, in his "Life of Albert Ufircr" (p. 09), when 

 iloscribinp the enfrrnvinj;, culls the miunres on the wall " the 

 mnffic qnnrlrnnt of miniemlfi of Cornolius Ajfrippn." This cnfrmviiifj 

 was executed by Albert Diirer probably between 1507 and 15M. 

 So mnrh for "the new <iame of fli," as it has been lately callcil. — 

 Faithfullv vonrs, "K. V. 1{. 



MAGIC SQUARES. 



(232] — In my former paper on this subject I pave examples of 

 Bachet'sand Poignard's method in squares of odd roots ; an<l I now 

 proceed to irivo a third rule, and to show how many different 

 arrangements may bo made of each square by these methods. 



Rule 3, for Odd Squares. 

 Example of a Square of 5. 



Fig. 1. 



In Fig. 1 place the mean 

 numbers of tbo series (in this 

 case 3) in the right-hand top- 

 corner cell, and the rest of the 

 numbers in the other cells of the 

 top row in any order at plea- 

 sure. Begin the second row with 

 the second number of the first, 

 and so on till the square is liUed 

 np. 



Fill npFip. 2 with the multiples 

 of the root, beginning with 0, by 

 placing the mean number in tlie 

 left-hand top corner cell, and the 

 others in any order. Then begin 

 the second row with the last 

 number of the first, and so on, 

 and it will be seen that the mean 

 numbers, 3 and 10, occupy the 

 diagonals of the squares in acon- 

 trarj- direction. In Fig. 3 place 

 the sums of the numbers in the 

 corresponding cells of Figs. 1 and 

 2, and the result is a magic square 

 wherein, in every case, the mean 

 number of the progression occu- 

 pies the centre square. 



By this method 576 different 

 arrangements can be made of 

 the square of 5. The square of 

 7 may be varied in the samo 

 way 518,.100 times, and the 

 square of 5) upwards of 20 

 million times. 



By Poignard's rule the square 

 of 5 may be varied 57,600 times 

 (exactly one hundred times as 

 often as by Knle 3), and the 

 square of 7 no less than 

 406,485,600 times !— all differing 

 from the results of Rule 3. 



And Poignard's squares have another siiiieriority over the others. 

 in the nnniber of ways the total eastings of the figures is 

 obtained from them. In Bachet's and the .squares by Rule 3 these 

 consist of the addition of each vertical and horizontal band, and 

 of the two diagonals- making twelve readings in the square of 5 

 and sixteen in the square of 7 — Poignard's give eight more 

 readings in the square of 5, and twelve more in the square of 7 

 — as shown in the annexed figure, whore, in addition to the sum 

 of 65 being made by the addi- 

 tion of the vertical and hori- , 

 zontal bands, and by the two " 

 main diagimals ; it is also 

 made by adding each partial b' 

 diagonals — a and the cell or 

 cells on the opposite side n', c' 

 Ac, making five cells, and b 

 and i)', (• and r' : &c., ioar only ,j' 

 are here shown, but four 

 more are found in the partial 

 diagonals at right angles 

 to those marked (twenty 

 readings in all). 



But this is not the limit of the number of readings, for Mr. Snart 

 has, by great perseverance and ingenuity, constructed a npiure of 7 

 ha\-ing no less than forty-two readings, which I will send to you. 



J. A. Mills. 



[233] — As a climax to odd Magic Squares, I send yon a scjiiare of 

 ■19 colls, which was conBtnictcd by a Mr. Snart, having gome 

 curious properties, not to be found in other squares : — 



/ 



In a Bachet's square of 7 there are sixteen readings of the total 

 175. In Poignard's there are twenty-eight ; but in the above 

 square, by Snart. there are forty-two readings. 

 li horizontal and vertical. 



2 diagonal. 



8 Right angles — a, c, d, — b, c, e, &c. 



8 Acute angles — a, c, b, — ?>, c, d, &c. 



S Obtuse angles — a, c, e,- b, c. f, &c. 



1 The centre square and the four corner squares. 



1 The centre square and the four squares a, d, f, and )i. 



Total 42 



It is seen that the highest number of the progression occupies 

 the centre cell, and is called into operation thirty times. 



Fakenham, Dec. 30, 1881. J. A. JIiles. 



VARIABLE MAGIC SQUARE. 



[234] — In the an-angements given in Knohleboe of numberSi 

 1 to 11 3 71 is always an odd number, and there is no mention of thSi 

 following arrangement of the numbers from 1 to 16 : — 



15 10 5 

 G 3 16 



in which 34 is made by the addition, not only of either diagonal" 

 or any horizontal or perpendicular line, but also by eierii four^- 

 adjacent fiijures, and by the four corner fij,'ures, as : — 



15 10 



3 16 



Jloroover, these conditions are still fulfilled if I shift the horii 

 zontnl linos from top to bottom, or vice rrrsi, or the )>er]x'jidicalar 

 linos from right to left, or vice versX. 1 may, in short, by 

 moving about the lines, bring any nun\ber 1 choose to any give 

 jiosition in the square. Florence E. Boyck. 



ABSTRACT TER>IS IN SCIENCE (Abstract). 



[235J — Words which are n\erely abstractions and conveuicntj 

 working terms to the scientist and mathematician are handled 

 such a w.iy that non-scientific persons are apt to give to the purelj 

 abstract conception a concrete meaning. 



