SCIENCE. 



Esq. Glasgow, and made by the dissolution of caout- ed in the same way except the middle 

 Curiosities chouc, or thenaptha obtained from coal-tar, is peculiar- column, which contains the same num- 

 ._,_'"'...._. ly fitted for rendering balloons strong arid air tight. ber 13. 



Fig. 2. magical. Science, 

 Curiosities 



Arithmetic. 



"Magic 



squares. 



Magic 

 s-quares of 

 wild num- 

 bers. 



PLATE 

 .rcccLxxxiv, 



ARITHMETIC. 



As we do not mean to occupy our pages with the 

 numerous arithmetical tricks which are now to be 

 found in every popular work, we propose to confine 

 our attention to the subject of magic squares and cir- 

 cles. The following treatise on this subject prepared 

 for this work by an able correspondent, contains many 

 new original views and constructions which cannot but 

 prove interesting to the curious reader. 



1. Magic Square*. 



Magic squares are of two kinds; the roots of the one 

 being even numbers, of the other odd numbers. 



The rules for their .construction are peculiar to each 

 kind, and we shall begin with giving those for odd 

 numbers. 



Magic Squares of odd numbers. 



The lowest square of this kind has 3 for its root, 

 but as it is incapable of any variation in its arrange- 

 ment, we shall elucidate the rules we give chiefly from 

 the square of 5. 



Having divided the square A, B, C, D, into 25 cells, 

 fill them up with the numbers 1 to 25 in their natural 

 order, as in Plate CC.CCLXXX1V, Fig. 4. 



In this square inscribe another square, E, F, G, H, 

 and divide it likewise into 25 cells ; 13 of which will 

 now appear filled with numbers. The remaining 12, 

 which are crossed by the subdivisions of the exterior 

 square, being empty. To fill them up proceed as fol- 

 lows : 



Transfer the numbers in the upper triangle E, A, 

 F, viz. 1, 6, 2, to the three empty cells immediately 

 below the centre (13), and in the same order. Trans- 

 fer the numbers 24, 20, 25, in the lower triangle G, 

 D, H, to the empty cells above the centre ; the num- 

 bers 21, 16, 22, in the triangle C, E, G, on the left to 

 the empty cells on the right of the centre, and the 

 numbers 4, 10, 5, in the triangle F, H, B, on the 

 right to the empty cells on the left of the centre. 



The figures in the interior square being now made 

 permanent with ink, and the pencil marks rubbed out, 

 the magic square E, F, G, H, will remain. The 

 amount of each column, horizontal or vertical, and also 

 of each of the diagonals, being all the same or 65. 



This is a very simple and easy method of making a 

 magic square of odd numbers, and is applicable to 

 every one of the kind, whatever may be its dimensions. 

 It is said to be the invention of M. Bachet, and some 

 of the rules commonly given to make these squares are 

 evidently derived from it. It would appear that it was 

 thought incapable of being varied in the arrangement; 

 as no mention is made of this property in any treatise 

 on the subject we have seen, we shall therefore show 

 how this can be done with little trouble. 



Fig. 1. natural. 



The natural arrangement of the num- 

 bers in the exterior square A, B, C, D, 

 may be varied in two ways ; 1st, In the 

 vertical columns, any one of which may 

 be shifted from its situation except the 

 middle column, which contains the 

 central number 13; 2d, In the hori- 

 zontal- columns, which may be shifu 

 5 



Tn this way, no less than 576 differ- 

 ent arrangements may be given to the 

 square of 5. The square of 7 may be 

 varied 518,400 different ways, and that 

 of 9 upwards of twenty millions of 

 ways. 



Fig. 1. and 2. show the vertical co- 

 lumns altered, and the magical square 

 derived from it. Fig. 3. and 4. show 

 the horizontal columns altered, with 

 its magical square. 



Fig. 5. and 6. show them both al- 

 tered, and the magical square resulting 

 from it. It is remarkable that all these 

 variations can be made without shifting 

 the central number of the square. 



Fig. 3. natural. 



Fig 4. magical. 



Fig. 5. natural. Fig. 6. magical. 



7 8J 6 10 



24'22!23J21 ! 25 



4 2 



3 li 5 



_! _i_ 



19;i7;18, ! 16!20 



If a still greater variety is wanted, the following very 

 ingenious method, invented by Poignard, and improv- 

 ed by De la Hire, will, we have no doubt, give ample 

 satisfaction. 



Poignard's Method. 

 Example in the Square of 5. 



In the square A. B, C, D, Fig. 1. divided into 25 Poi s nard> * 

 cells, place in the first horizontal column at top, the method ' 

 five first numbers of the natural progression in any 

 order at pleasure, which we shall here suppose to be 

 1, 3, 5, 2, 4-. Then make choice of a number which 

 is prime to the root 5, and which, when diminished by 

 unity, does not measure it. Let this number be 3, 

 and for that reason begin with the third 

 figure of the series, and count from it 

 to fill up the second horizontal column 

 5, 2, 4, 1, 3. Then begin again with 

 the next third figure, including the 5, 

 that is to say by 4, which will give for 

 the third column 4, 1, 3, 5, 2. By fol- 

 lowing the same process, we shall have 

 the series of numbers 3, 5, 2, 4, 1, to 

 fill up the fourth column, and 2, 4, 1, 



