216 M A G M A G 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 ^7, 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, .!, s.j, . . . s n , can be so placed, one in each of the ?i 4 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 s 1 on each of the ?i 2 cubelets of the first cube that contain p lt and on the ?t 2 cubelets of the 2d, 3d, . . . and nth cube that contain jt;. 2 , p.,, . . . p n respect ively. This process is repeated with s 2 , beginning with the cube at which we ended, and so on with the other s s ; the n 4 cubelets, after multiplying the q s, r s, and s s by ?i, ?i 2 , and ri* respectively, 1 8 29 28 11 14 23 18 30 27 2 7 21 20 9 16
X.
4
5
32
25
10
15
22
19
31
26
3
6
24
17
12
13
will now be filled with the numbers from 1 to n 4 , and the constant
summation will be 2/i 3 s + 2. 2 r + 2>ig + 22>. 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 n 5 cubelets of this solid square be Nasically filled by
the introduction of a new letter t ; while, by introducing another
letter, the ?i 6 cubelets of the compound cube of ?i 3 Nasik cubes
Z.
30
21
6
15
28
19
7
16
29
20
5
14
22
31
8
35
18
27
9
3(!
17
26
13
4
32
23
2
11
34
25
1
10
33
24
3
12
might be filled by the numbers from 1 to n e , and so ad infinitum.
When the root is an odd composite number the values of the three
groups of letters have to be adjusted as in squares, also in cubes of
an even root. A similar process enables us to place successive
numbers in the cells of several equal squares in which the Nasical
summations are the same in each, as in diagrams X.
Among the many ingenious squares given by various writers, this
article may justly close with two by Euler, in the 77/sfozYc dc TAca-
demie Royale des Sciences, Berlin, 1759. In diagram Y the natural
numbers show the path of a knight that moves within an odd square
in such a manner that the sum of pairs of numbers opposite to
and equidistant from the middle figure is its double. In diagram
Z the knight returns to its starting cell in a square of 6, and the
difference between the pairs of numbers opposite to and equidistant
from the middle point is 18.
A model consisting of seven Nasik cubes, constructed by
Mr Frost, can be seen in the South Kensington Museum. The
centres of the cubes are placed at equal distances in a straight
line, the similar faces looking the same way in a plane parallel to
that line. Each of the cubes lias seven parallel glass plates, to
which, on one side, the seven numbers in the septenary scale are
fixed, and behind each, on the other side, its value in the common
scale. 1201, the middle number from 1 to 7 4 , occupies the central
cubelet of the middle cube. Besides each cube having separately
the same Nasical summation, this is also obtained by adding the
numbers in any seven similarly situated cubelets, one in each
cube. Also, the sum of all pairs of numbers, in a straight
line through the central cube of the system, equidistant from it,
in whatever cubes they are, is twice 1201.
A very complete bibliographical index of writers on (his subject is given in
IVofessor Lucas s Recreations Matheinatiques, Paris, 1882. (A. II. F.)
MAGISTRATE. The term magistrate, derived from
the Latin magistratus, is one of more general and compre
hensive meaning than JUSTICE OF THE PEACE, which has
already been treated of (vol. xiii. p. 789), and is of far
higher antiquity. In its full significance it indicates one
side of the universal public relation by which men are
connected together as governors and governed in other
words, as magistrates and people. Of magistrates some
are supreme, in whom the sovereign power and executive
government of the state reside, as the king or queen
regnant, or the president of a republic, as of the United
States; and such a functionary would formally be designated
the first magistrate of the realm or state. Speaking gene
rally, a magistrate may be described as a public civil officer
invested with legal or other authority ; but the term is
more particularly applied to subordinate officers, as justices
of the peace and the like, deriving their authority solely
from the chief of the state or in virtue of legislative enact
ment. During the Roman republic the offices of magistrate
and judge were distinct and separate. A magistrate was
appointed cumjurisdictione et imperio ; to a judge belonged
only mida notio sine jurisdictione et imperio. The office of
the magistrate was to inquire into matters of law ; and
whatever business was transacted before him was said to
be done in jure. The office of the judge was to inquire
into matters of fact ; and whatever was transacted before
him was said to be done in judicio. This distinction is
thus clearly defined by Cicero in his well-known oration for
Cluentius : " Legum ministri, magistratus; legum inter-
pretes, judices." When the magistrate took cognizance both
of the law and the fact he was said to administer justice
extra ordinem ; and the judgment so administered was
called extraordinary. The magistrate, when he decided on
matters of law, was assisted by a council of ten, called
decemviri litibus judicandis. To these was added in
important cases another council of one hundred and five
persons, selected from each tribe, whose judgment was final;
this was called judicium centumvirale. After the decline
of the Roman republic the offices of magistrate and judge
were united, by which means all judgments became extra
ordinary, and the distinction of what was done injure and
in judicio was abolished. The magistrates were chosen
only from the patricians in the early republic, but in the
course of time the plebeians shared in these honours. The
chief magistrates of Athens were designated a.rchons.
They were nine in number, and none were chosen but
such as were descended from ancestors who had been free
citizens of the republic for three generations. They took
an oath that they would observe the laws, administer
