more than 6 per cent. ought to be given for the age of 20; 6 for the age of 25; 64 for that of 30; 74 for 40; 8 for 50; 10 at 56; 11 at 60; 163 at 70; 27 at 80; and 39 at 85.

It is therefore a great mistake to imagine, that on account of the great number of persons who sink money in these loans, on annuities, made by governments, the latter are soon freed from paying a part of the annuities by the death of a part of the annuitants. The slow increase of annuities in tontines is a sufficient proof of the falsity of this idea; besides, the greatness of the number of the persons is precisely the cause, why the extinction of the annuities takes place more in conformity to the laws of probability already explained. A lucky chance, at the end of a few years, may free a person from the payment of an annuity, established on the life of a man 30 years of age; but if this annuity were shared out on 300 different lives, the ages being nearly the same, it is certain that he would not be liberated from it before nearly 65 years; and after 32 or 33 nearly one half of the annuitants would be living. This Pariceux has clearly shewn, by examining the lists of the tontines.

111

MAGIC SQUARES.

A MAGIC square is a series of figures arranged in the cells of a square, in such a manner, that the figures in each band, whether vertical, horizontal, or diagonal, form exactly the same sum. They are divided into two kinds: odd and even.

These squares have been called magic, because the ancients ascribed to them great virtues, and because this arrangement of numbers formed the basis and principle of several of their talismans.

One square, containing unity, was, according to them, the symbol of the Deity, on account of his unity and immutability; for they observed, that this square, by its nature, was single and immutable; the product of unity by itself being always unity.

A square containing four divisions or cells, was the symbol of imperfect matter, on account of the impossibility of arranging figures in it so as to form a magic square.

The square, with 9 divisions, was consecrated to Saturn; that with 16 to Jupiter; that with 25 to Mars; that with 36 to the Sun; that with 49 to Venus; that with 64 to Mercury; and that with 81 to the Moon.

Those who can find any relation between the arrangement of numbers and the planets, must be indeed not a little visionary; but such was the spirit of the mysterious philosophy of Iamblichus and Porphyry, and of all their disciples, who were slaves to the most stupid superstition, and to all the absurdities of judicial astrology.

We shall here confine ourselves to the mechanical method of forming a magical square, either even or odd.

METHOD OF CONSTRUCTING AN ODD
SQUARE.

1st. Place unity below the middle cell.

2nd. Place the following number in the cells which descend diagonally from left to right.

3rd. When you come to the last diagonal cell, go up to the highest cell of the next following band.

4th. When the diagonal cell is filled up, carry the next figure to the most distant cell on the left of the lower band.

5th. In following the diagonal, if you meet with a cell already filled up, pass over that cell, and place the figure in the 'diagonal from right to left. See the following figures, one of which represents a square of 9 divisions, and the other one of 25.

We shall apply this method to a square of 16 cells, which is filled up in the following manner:

1st. Place 1 in the cell A (fig. M.) of the vertical band on the left; then pass the two next, and place 4 in the upper cell of the perpendicular band, on the right.

2nd. Omit 5, and place 6, 7, and the other figures, as seen in fig. M.

The remaining 8 divisions, which are left vacant, must be filled up after the manner of fig. N. Reckon

1 in the cell B without writing it down, and place 2 and 3 in the two next cells; then omit 4, and set down 5 in the first cell of the next band; omit 6 and 7, and write down 8, and so on. If you then fill up each of these squares from the other, you will have a square of 16 divisions. See the figures.

To arrange in a square, consisting of 9 cells, the 9 terms of a geometrical progression, in such a manner, that the product arising from the continued multiplication of the numbers in each band shall be always the same, and equal to the cube of the middle term.

Let the terms of the progression be 1:2:4:8; 16 32 64; 128: 256. If you arrange these 9 terms in a square of 9 cells, in the manner as you did the 9 terms of the arithmetical progression of the natural numbers, 1, 2, 3, &c. you will find that the product of them, in every direction, amounts to 4096; which is exactly the cube of the middle term 16, as may be seen in the annexed figure.

To make the knight pass over all the squares of the chess board, one after the other, without passing twice over the

same.

As the reader may perhaps be unacquainted with the movement of the knight in the game of chess, we shall here describe it. If the knight be placed in the square A, he cannot be moved into any of the squares immediately around him, as those marked 1, 2, 3, 4, 5, 6, 7, 8; nor into the squares 9, 10, 11, 12, which are directly above or below, or on one side; nor into the squares 13, 14, 15, 16, which are in the diagonals, but only into one of those which, in the figure, are left vacant. See fig. B.

Several celebrated men who amused themselves with this problem, have given solutions of it; but the following is the simplest of them all, and the easiest to be remembered.