M A G
358
M A G
middle. If it be required that 2 be placed in the cenlre of thethird square, nothing is necessary in this case but to place 2 in thecentre of the tir=t square, and 0 in the centre of the second ; andm like manner we must proceed for any other number which isrequired to be placed in any particular cell of the third square;bv taking care so to dispose the numbers of the two first squares,that the sum of the numbers in the two corresponding cells of thefirst and second square shall amount to the number proposed tobe placed in the corresponding cell of the third square.
22
38
54
59
28
15
75
8
70
10
7S
3
71
25
40
47
63
32
43
49
56
36
14
73
6
66
26
77
1
69
21
44
52
58
29
18
53
61
31
11
81
5
64
24
89
9
68
19
42
48
62
-'A
04
13
74
57
35
16
76
2
72
23
37
51
65
27
41
46
60
30
17
79
4
33
12
80
7
67
20
45
50
55
r [ he construction of these squares is attended with considerablymore difficulty than that of the odd squares; and the degree ofdifficulty is different according as they are evenly even or oddlyeven: tor this reason we must divide them into two classes.Squares evenly even are those the root of which when halved iseven, or can be divided by 4 without a remainder; of this kindare the squares of 4, 8, 12, 16, 20, 24, 28, &c. The oddly evenare those whose root when divided by 2 gives an odd number; asthose of 6, 10, 14, 18, 22, 26, 30, &c. In order to construct themagic squaie of 4, which is evenly even, we must begin by filling upfirst the two diagonals; and for that purpose count the naturalnumbers in order 1, 2, 3, &c. on the cells of the first horizontalband, from left to right; and then proceed to the second band,and when you come to the cells belonging to the diagonals, in-scribe the numbers counted as you fall upon them, by whichmeans you have tiie arrangement as represented in the first of thefollowing figures. When the diagonals have been thus filled, tofill up the cells which remain vacant begin to count the same num-bers, proceeding from the last cell of the last horizontal band tothe right hand, and going from right to left, and in like manner inthe next above it, and when any cells are found empty, fill them upwith the numbers that belong to them ; in this manner you willhave the square filled up magically, as seen in the last of the fol-lowing figures and the sum of each band and diagonal willbe 34.
i
4
6
7
10
11
13
16
1
15
14
4
12
6
7
9
8
10 1! j 5
13
3 i 2
1
16
Obs. 1. It is not necessary, in order to form a magic square olcells by the preceding rule, that the arithmetical ratio of the nun'"hers occupying the cells should be only 1 ; the rule, will h'dgood, (as in the case of the odd squares,) if any other arithmetic* 1ratio be adopted; as in the first of the six subjoined square®’where 2 is the common ratio ; and the series of the numbers, 1, *?’5, 7,9,11, 13, 15, 17, 19, 21, 23, 25,27, 29, 31. Obs. 2. U ! salso nut necessary that the arithmetical progression be continued; 1may be disjunct, provided the numbers in the vertical bands p' 1 "serve an arithmetical progression, as for example, the numbers Jj10, 11,12,13, 14,15, 16, 49, 50, 51,52,53, 54, 55,56, inthesecon"of the following figures. The third figure exhibits these numb eI 'magically arranged in a square of 16 cells. The fourth fig lll _is the magic arrangement of the continued series of i' 11 ' 1 !1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 37, 40, 43, 46. The fi'h 1figure is the magic arrangement of the disjunct series of numb® 1 *’11 2, 3, 4, 5, 7, 8, 57, 58, 59, 60, 61, 62, 63, 64. The last fig«£is a magic arrangement of the disjunct series 17 to 24 and 85 to -■*'
1
29
27
7
23
11
13
17
15
19
21
9
25
5
3
31
9
10
11
12
13
14
15
16
49
50
51
52
53
54
55
56
i
43 i
40
10
34
16 i
19
25
22
28
31
13
37
7
4
46
1
63
62
4
60
6
7
57
8
58
59
5
|61
3
o
64
9
55
54
52
14
15
16
50
51
53
11
10
17
91
90
88
22
23
24
86
87
89
19
18
43
20
2l
cal band reversed ; let the third veitical band be like the first<fourth like the second, See. &c. and complete this square insame manner as the former by making the latter half of the vccal bands exactly the reverse of the former. The third sq"*' fwhich is the one required, is formed by the addition of the $cells of the two first. We here give the magic squares of 4 a' 1in illustration of the preceding rules.
4
3
o
*•
i
! 12
0
0
12
16
3
2
-
—
1
a
3
4
8
4
4
8
9
6
7
_
.- *
1
•)
3
4
4
8
8
4
5
10
11
.
- —
4
3
2
I
0
12
12
0
4
15
14_
13
General Method for Constructing MagjoSquarE sEvenly Even.
Fill the top horizontal band of the first square with the nui TI,,t fof the root of the required square in such a manner that the ct ||equally distant from the central one of t he first horizontal bandbe occupied by numbers the sum of which will amount to the f°of the square + 1, as, e. gr. 5 in a square of 16 cells, 9 in a sq 11 * 1of 64 cells, &c. &c. The second horizontal band must be *'reverse of the first; the third, like the first, the fourth like‘ssecond, &c. till you come to the two middle-hands, the lattC rwhich must be occupied with figures exactly in the same ° r< cwith its immediately preceding band ; and so complete the Pl l u . 3 i ieas that the latter half of the horizontal bands shall be exactly *^reverse of the former. In the constructing of the second s< l ll ^q|jdispose the numbers of the multiples of the root, (beginning ''Ijy0, and omitting the last multiple,) so that the sum of the cells cq ua tdistant from the centre of the first vertical band shall an’ 0 'to the number of cells in the square minus the root of the sq 11 * 1 ,j,Fill the second vertical band with the numbers of the first ve
th*
th«