Page:EB1911 - Volume 17.djvu/327

 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, starting from any two different cells along different paths, will appear together in one and only one cell. Hence, if p1 be placed in the cells of one of the n + 1 normal paths, each of the remaining n normal paths will contain one, and only one, of these p1’s. If now we fill each row with p2, p3, pn in the same order, commencing from the p1 in that row, the p2’s, p3’s and pn’s will lie each in a path similar to that of p1, and each of the n normal paths will contain one, and only one, of the letters p1, p2, pn, whose sum will be p. Similarly, if q1 be placed along any of the normal paths, different from that of the p’s, and each row filled as above with the letters q2, q3, qn, the sum of the q’s along any normal path different from that of the q1 will be q. The n2 cells of the square will now be found to contain all the combinations of the p’s and q’s; and if the q’s be multiplied by n, the p’s made equal to 1, 2, n, and the q’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 q’s, will be the constant nq + p. 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 p1 and q1 were placed, instead of having each of the p’s and q’s, some will be wanting, while some are repeated. Thus, in the case of 9, the triplets, p1p4p7, p2p5p8, p3p6p9, and q1q4q7, q2q5q8, q3q6q9 occur, each triplet thrice, along paths whose summation should be—p 45 and r 36. But if we make p1, p2, p9, = 1, 3, 6, 5, 4, 7, 9, 8, 2, and the r1, r2, r9 = 0, 2, 5, 4, 3, 6, 8, 7, 1, thrice each of the above sets of triplets will equal p and q respectively. If now the q’s are multiplied by 9, and added to the p’s in their several cells, we shall have a 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 1 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 32, 22, 76, 77, 26, 37, 36, 27. The numbers in fig. 22 being kept 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 q’s may be respectively 1, 2, 10, 8, 6, 14, 15, 11, 4, 13, 9, 7, 3, 12, 5, and 0, 1, 9, 7, 5, 13, 14, 10, 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 p and q; then, if the q’s are multiplied by 15, and added to the p’s, the Nasik square of 15 is obtained. When the root is the multiple of 4, the same process gives us, for the square of 4, fig. 23. Here the columns give p, but alternately 2q1, 2q3, and 2q2, 2q4; and the rows give q, but alternately 2p1, 2p3, and 2p2, 2p4; the diagonals giving p and q. If p1, p2, p3, p4 and q1, q2, q3, q4 be 1, 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 n3 small equal cubes, here called cubelets, in the centres of which the natural numbers from 1 to n3 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. 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 three 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, p1, q1, r1 being placed in the corner cubelet 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, p2q2r2, p3q3r3, p7q7r7 are placed in the other cubelets of the edge AO, and dispersed in the same manner as p1q1r1. Every cubelet will then be found to contain a different combination of the p’s, q’s and r’s. If therefore the p’s are made equal to 1, 2, 7, and the q’s and r’s to 0, 1, 2, 6, in any order, and the q’s multiplied by 7, and the r’s by 72, then, as in the case of the squares, the 73 cubelets will contain the numbers from 1 to 73, and the Nasical summations will be 72r + 7q + p. If 2, 4, 5 be values of r, p, q, 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 1, 2, 3, 7 in the unit place, and 0, 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 1 to n3, the sum of all the pairs of numbers opposite 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, q or r 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, s1, s2, sn, can be so placed, one in each of the n4 cubelets of this group of n cubes, that each shall contain a different combination of the p’s, q’s, r’s and s’s.

This is done by placing s1 on each of the n2 cubelets of the first cube that contain p1, and on the n2 cubelets of the 2d, 3d, and nth cube that contain p2, p3, pn respectively. This process is repeated with s2, beginning with the cube at which we ended, and so on with the other s’s; the n4 cubelets, after multiplying the q’s, r’s, and s’s by n, n2, and n3 respectively, will now be filled with the numbers from 1 to n4, and the constant summation will be n3s + n2r + nq + p. 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 parallelepipeds 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 n6 cubelets of the compound cube of n3 Nasik