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risks his human doors, and suffering the ethereal tides to roll and circulate through him."

What all this may imply about the ultimate metaphysical nature of God is, no doubt, worth discussing, and conceivably of great import. But men are coming in these latter days to a humbler sense of their intellectual limitations; we are realizing that we know nothing of the inner nature of anything, save of our own conscious life as it passes. What is matter? What is electricity? What is God? Perhaps we cannot know. But what is practically important is to understand and utilize the experiences out of which these concepts have grown. If we can use electricity in our telephones and dynamos and trolley cars, we can be content to confess our ignorance of its inner nature. So if we can comprehend and repeat the religious experiences out of which the concept of God has arisen, it matters less if our knowledge of God is limited to that experience

contact.

Souls of different types and needs will naturally formulate their experience in different terms; there is no need that any one to whom the generalization is not personally useful should express his Godidea in a trinitarian formula. Trinitarianism should never be a dogma. And with the arguments and disputes of the Greek doctors of the third and fourth centuries, through whom that dogma took shape, we may have scant patience. Certainly all that sort of speculation is very alien to our modern scientific world-view. But on the other hand, the arguments of the rationalists of to-day for a God-idea divorced from those experiences in which it has its natural roots, are equally alien to the outlook and spirit of science. To believe in God is a mere act of credulity, except as we see the meaning of the God-idea in human life. When we do thus turn to experience, we find ourselves led to the God-conception from the three sorts of experience mentioned. So, as an embodiment of the profound truth of the threefold basis of our human conception of God, the Trinitarianism of the saints should command our sympathy and respect.

Trinitarianism, Unitarianism-as mutually exclusive dogmas, both are cramping and arrogant. What is important is to keep alive the experience that each term enshrines. The essential oneness of all God-experiences, and of the God-idea which they unite in producing, is important, no doubt. But the bare insistence upon unity has, now that the extravagances of polytheism are forever

past, little religious value, and tends to a contentment with less than the full gamut of religious experience. No one of the three forms of God-experience can be dispensed with in a rich and fruitful spiritual life; and it is no wonder that the orthodox have generally felt a merely negative Unitarianism to have an impoverishing tendency. However crude the creedal affirmations of Trinitarianism may be, the fulness of the Christian life has by it been fostered and preserved. So, however loath we may be to seem to accept the description of a quasi-human Being who is somehow Three Persons and yet One, if we take the doctrine (as we must take all religious doctrines) in its inner and spiritual sense, which is its empirical foundation-sense, we shall see it as a more or less blind expression of a great truth-that Christians can attain to the vision of God in three ways, through contemplation of the outer world, through faith in their Master Christ, and through obedience to the Holy Spirit in their hearts.

VASSAR COLLEGE.

DURANT DRAKE.

GENERAL NOTES ON THE CONSTRUCTION OF MAGIC SQUARES AND CUBES WITH PRIME NUMBERS.

The series of numbers generally used in the construction of magic squares are in arithmetical progression. The progression of the prime number series is very irregular, and therefore cannot be used as freely as an arithmetical series. This naturally leads to the investigation of the possible irregularities in groups or series of numbers which may be formed into magic squares. It is also necessary to find means of discovering these groups of numbers in the prime number series.

It is the writer's aim to describe here simple rules for constructing prime number squares, methods of finding the numbers to be used, and to point the way to the solution of a few of the problems not yet mastered.

THE SQUARE OF THE 3d ORDER.

There is only one possible construction of this square and there is only one rule governing the series, and that is, when the series is written in tabular form, as in Fig. 1, the differences between all

vertically adjacent cells must be equal, and the differences between all horizontally adjacent cells must be equal, but the vertical and horizontal differences must be unequal to avoid duplicate numbers. These differences are indicated by numbers at the sides of the lattice and it will be by these that we will identify the nature of the series

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used in the following magic squares.

71 5 101

89 59 29

17 113 47

Fig. 3.

We will represent these differences by letters, using the letters of the fore part of the alphabet for one set of differences and those of the other end of the alphabet for the other set, as is shown in Fig. 2, like letters indicating the necessity of like differences.

Fig. 2 is arranged into the magic by using the middle column and middle line as diagonals, the position of the remaining numbers then being easily found. The resulting magic is shown in Fig. 3.

THE SQUARE OF THE 4th ORDER.

Any series or set of 16 numbers, when written in the tabular form previously mentioned, wihch gives the differences a, b, c and x, y, z, may be formed into a magic square by the Jaina method as follows. Fig. 4 shows a table of prime numbers with irregular differences. Four sets of the upper line of numbers of this table are arranged

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in a subsidiary square, as shown in Fig. 5, so that each line, column, and the two diagonals contain each of the four different numbers. Subtracting the initial number of the table (in this case 1) from each of the numbers in the left-hand column of the table, will give the numbers, 0, 30, 60, 126, which are to be arranged in a second subsidiary square with the same arrangement as in Fig. 5, only that

the pattern is turned 90 degrees. The two subsidiary squares are then added together, cell to cell, to produce the magic square. A resulting square is shown in Fig. 6.

In selecting numbers from the tables for the subsidiary squares, the column and line containing the lowest numbers should be chosen, but it makes no difference which set the initial number is subtracted from.

A balanced series of numbers whose tabular differences are

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a, b, a, and x, y, x, may be arranged into an associated square, or a pandiagonal square. Such a series is shown in Fig. 7. By revolving the two diagonals of Fig. 7, 180 degrees, it will produce the associated magic square shown in Fig. 8. To produce a pandiagonal square, we select, as before mentioned, two subsidiary sets of numbers from which are formed two subsidiary squares of the pattern shown in Fig. 5. The numbers in the upper line should be so arranged that the sum of the left-hand pair equals the sum of the right-hand pair. One of these subsidiary squares is revolved 90 degrees and added to the other to produce the magic. A pandiagonal square resulting from such a construction is shown in Fig. 9.

Another form of subsidiary square which may be used to produce a pandiagonal square from a balanced series is shown in Fig. 10, which is exemplified with arbitrary numbers. The numbers

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must be so arranged that the pairs indicated by dotted lines will have like summations. One subsidiary square is revolved 90 degrees from the other and the two added together to produce the final

square. Fig. 11 is a pandiagonal square produced from the series in Fig. 7, by the method last described.

THE SQUARE OF THE 5th ORDER.

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A series of 25 numbers whose tabular differences are a, b, C, and w, x, y, z is shown in Fig. 12. Such a series may be formed

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into a pandiagonal square as follows. Two sets of subsidiary numbers are selected and arranged in two subsidiary squares according to the pattern shown in Fig. 13; the pattern of one subsidiary square should, however, be in a reversed or reflected order from the other. The two squares are then united to form the final square. Fig. 14 shows one example resulting from the series in Fig. 12.

To produce an associated pandiagonal square of the 5th order, it requires a series whose tabular differences are a, b, b, a and x, y, y, x. The writer, at present, knows of only one series which suits the above requirements. Its initial number is 41 and the tabular differences are 60, 390, 390, 60 and 72, 138, 138, 72 respec

1013 251 449 911 881

839 1301 941 | 113 |311

41 173 701 1229 1361|

1091 1289 461 | 101 | 563|

521 491 953 1151 389
Fig. 15.

tively. The subsidiary squares are arranged in associated formation and according to the pattern in Fig. 13. The solution of this difficult problem was accomplished by Mr. Chas. D. Shuldham, and his resulting magic is shown in Fig. 15.

Mr. Shuldham has succeeded by other methods in constructing

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