consider carefully fig. 21, which shows that, when the root is a prime,
and not composite, number, as 7, eight letters a, b, . . h may proceed
from any, the same, cell, suppose that marked 0, each letter being
repeated in the cells along different paths. These eight paths are
called “ normal paths, " their number being one more than the root
Observe here that, excepting the cells from which any two letters
start, they do not occupy again the
same cell and that two letters startin
E B f

1 g

from any two different cells along different paths, will appear together in one and only one cell. Hence, if p, be placed in the cells of one of the n-l-I normal A dot is here placed on three faces of a cubelet at the corner, showing that this cubelet belongs to each of the faces AOB, BOC, COA, of the cube. Dots are placed on the cubelets of' some path of AOB (here the knight's path), beginning from O, also on the cubelets of a knight's path in BOC. Dots are now placed in the cubelets of similar paths to that on BOC in the other six sections parallel to BOC, starting from their dots in AOB, Forty-nine of the tl-ll-ee hundred and forty-three cubelets will now contain a dot; and it will be observed that the dots in sections perpendicular to BO have arranged themselves in similar paths. In this manner, p, , g, , 1, being placed in the corner cubelet paths, each of the remaining n normal paths will contain one, and only one, of these p, 's. If now we fill each row with pg, ps, .., p, , in the same order, commencing from the pl in that row, the p2's, p3's and p, ,'s will lie each in a path similar to that of p, , and each of the n normal paths will contain one, and only one, of the letters p, , p2, . .p, ,, whose sum will be Ep. Similarly, if gl -be placed along any of the normal paths, different from that of the p's, and each row filled as above with the letters gz, ga, gn, the sum of the g's along any normal path different from that of the gl will be Eg. The 112 cells of the square will now be found to contain all the combinations of the p's and g's; and if the g's be multiplied by n, the p's made equal to 1, 2, n, and the g's to 0, 1, 2, . . (n-1) in any order, the Nasik square of n will be obtained, and the summations along all the normal paths, except those traversed by the p's and g's, will be the constant Eng + Ep. When the root is an odd composite number, as 9, 15, &c., it will be found that in some paths, different from the two along which the pl and gl were placed, instead of having each of the p's and g's, some will be wanting, while some are repeated. Thus, in the case Of 9, the tf1Pl@t§ » P, i>..1>, , i>, P, P, ,, 1>, P.1>, , and q, q.q, , q2q, q, , q, q, q, , Ocwf, each triplet thrice, along paths whose summation should be-Ep 45 lfdn I

E uf E

IIII

IIIIEI

FIG. 21.

EEEIEIE

E ll!!

EIEEEEE

ll E I

I ll E

cd fg

1, /, h, 1,

and Er 36. But if we make p, , pz, . .p9, = 1, 3, 6, 5, 4, 7, 9, 8, 2, and the rl, 72, . .19 =0, 2, 5, 4, 3, 6, 8, 7, 1, thrice each o the above sets V V of triplets will equal Ep and Eg respectively. If now the

63 88 74 '3 8 M 53 48 g's are multiplied by 9, and added to the p's in their

several cells, we shall have a

52, 47 36 62 87 76 13 7 26 Nasik square, with a constant summation along eight of its

ten normal paths. In fig. 22

the numbers are in the nonary

scale; that in the centre is

the middle one of I to 92, and

the sum of pair of numbers

equidistant from and opposite

to the central 45 is twice 45;

and the sum of any number

and the 8 numbers 3 from it,

diagonally, and in its row and

column, is the constant Nasical summation, e.g. 72 and

27. The numbers in fig. 22 being kept 34

xxig:5 5: 49 35 6x 89

75

68-84731B4=3s€4433

19 5 1| 59 45 31 69 Us 71

57463a61867zx76zz

54 83 73 X4 3 =B 54 43 39

I5 I 29 55 4|'39 65 81 79

sd 43 37 66 8: 77 16 zi 27

Flo. 22.

32, 22, 76, 77, 26, 37, 36,

in the nonary scale, it is not necessary to add any nine of them together in order to test the Nasical summation; for, taking the first column, the figures in the place of units are seen at once to form the series, 1, 2, 3, .... 9, and those in the other place three triplets of 6, 1, 5. For the squares of 15 the p's and g's may be I1 2! 1018! 6? I4Y 157 II! 41 13V 917131 12157 O' 11 9179 5, 13, 14, Io, 3, 12, 8, 6, 2, 11, 4, where five times the sum of every third number and three times the sum of every fifth number makes Ep and Eg; then, if the g's are multiplied by I5, and added to the O, these letters are severally placed in the cubelets of three different paths of AOB, and again along

any similar paths in the seven

sections perpendicular to AO, starting from the letters' position in

AOB Next, p, g,1'2, p3g31'3,

p, g,1', are placed in the other cubelets of the edge AO, and dispersed

in the same manner as plglrl.

Every cubelet will then be found to contain a different combination of the p's, g's and 1"s. If therefore the p's are made equal to I, 2, 7, and the g's and 1's to 0,

1, 2, . 6, in any order, and the g's multiplied by 7, and the TYS as

o%¢¢

C o§ ¢;¢;og¢

!~ &=:i$°%

~~ *f°¢*a¢»

Q* *ra E%*n*n*¢

u u u un 4 n n

- 20:52, 23 January 2015 (UTC)»¢»*n*a

u$$§ ¢595I5

FIG. 25-Nasik Cube.

MMMM.

P391 A72 15:93 A74

ME@M

MEMM

p's, the Nasik square of I5

is obtained. When the root

is the multiple of 4, the same

process gives us, for the

square o 4, fig. 23. Here

the columns give Ep, but

alternately 2g, , 2g3, and 2q2, 24142 and the rows give Eg,

Elll

by 72, then, as in the case of the squares, the 73 cubelets will contain the numbers from I to 73, and the Nasical summations will be E721-l-E7g -l- p. If 2, 4, 5 be values of 1', p, g, the number for that cubelet is written 245 in the septenary scale, and if all the cubelet numbers are kept thus, the paths along which summations are found can be seen without adding, as the seven numbers would contain I, 2, 3, . . .7 in the unit place, and o, 1, 2, 6 in each of the other places. In all Nasik cubes, if such values are given to the letters on the central cubelet that the number is the middle one of the series I to na, the sum of all the pairs of numbers opposite 1 8 ag al xx 14 23 18

3|;6 3 6 l IQ 17 xx I3

FIG. 26. V

to and equidistant from the middle number is the double of it. Also, if around a Nasik cube the twenty-six surrounding equal cubes be placed with their cells filled with the same numbers, and their corresponding faces looking the same way, -and if the surrounding space be conceived thus filled with similar cubes, and a straight line of unlimited length be drawn through any two cubelet centres, one in each of any two cubes, -the numbers along that line will be found to recur in groups of seven, which (except in the three cases where the same p, g or 1' recur in the group) together make the Nasical summation of the cube. Further, if we take n similarly filled Nasik cubes of n, n new letters, s, , S2, . .sm can be so placed, one in each of the n* cubelets of this group of n cubes, that each shall contain a different combination of the p's, g's, 1's and s's. This is done by placing sl on each of the n2 cubelets of the first cube that EBEBBE

IEEEBH

EBBEEH

EEHEBI

BEIIEE

IEEE!!

EBI°B

EEE"E

ll

but alternately 2pl, 2pl,

and 2p, , 2p, ; the diagonals

If pl! P27 P37 P4 qll q27

FIG. 23. FIG. 24.

11, , Q, be I, 2, 4, 3,

and 0, 1, 3, 2, we have the Nasik square of fig. 24. A square like this is engraved in the Sanskrit character on the gate of the fort of Gwalior, in India. The squares of higher multiples of 4 are readily obtained by a similar adjustment. Nasik Cubes.-A Nasik cube is composed of ni small equal cubes, here called cubelets, in the centres of which the natural numbers from 1 to ni* are so placed that every section of the cube by planes perpendicular to an edge has the properties of a Nasik square; also sections by planes perpendicular to a face, and passing through the cubelet centres of any path of Nasical summation in that face. Fig. 25 shows by dots the way in which these cubes are constructed. giving Ep and Eg.

FIG. 27.

Fig. 28.

contain pl, and on the n2 cubelets or the 2d, 3d, and nth cube that contain p.2, pa, . pn respectively. This process is repeated with 52, beginning with the cube at which we ended, and so on with the other s's; the n4 cubelets, after multiplying the g's, r's, and s's by n, n2, and ni' respectively, will now be filled with the numbers from 1 to n4, and the constant summation will be En3s -- 21127 + Eng -1- Ep. This process may be carried on without limit; for, if the n cubes are placed in a row with their faces resting on each other, and the corresponding faces looking the same way, n such parallelepiped's might be put side by side, and the n5 cubelets of this solid square be Nasically filled by the introduction of a new letter t; while, by introducing another letter, the 77/G cubelets of the compound cube of ni* Nasi