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3. Square. A square number, or a second power, is a name given to a number which is made by multiplying a number by itself. Thus 9 is a square number (3 times 3) ; 10 is not. The series of square numbers is

1 4 9 16 25 36 49 64 81 100 &c.

Questions. Why is it that a square number cannot end with 2, 3, or 7? Why is it that the square of an even number must be even, and of an odd number odd?

4. Cube. A cube number, or a third power, is a name given to a number which is made by multiplying a number twice by itself, or multiplying three equal numbers together; such as 64 (4 times 4 times 4; or 4, 4, and 4 multiplied together). The series of cubes is as follows:

1 8 27 64 125 216 343 512 729 1000 &c.

5. Fourth powers, Fifth powers, &c. These are names given to numbers which arise from multiplying together four equal numbers, five equal numbers, &c. Thus, 2, 2, 2, 2, 2, multiplied together, give 32, which is the fifth power of 2. The eighth power of 3 is 6561.

Questions. Why is it that every fourth power is a square; every sixth power a cube, and also a square?

III. We shall now proceed to notice some relations between numbers, such as may easily be verified.

1. Odd numbers added together, beginning from 1, and not leaving out any, always give a square number. For instance,

1 and 3 make 4, or the square of 2

1,3, and 5 . 9 . . 3

1,3,5, and 7 . 16 . . 4

and so on. If a person who knows this were to lay a bet with another that he would form the squares of all numbers under 100 sooner than the other, he would certainly win; for while the second would have to perform a multiplication at each step, the first would proceed as follows:

1

3

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and so on.

'2. An uninterrupted sum of cubes beginning from 1, is always a square number. The series of cubes is,

1 8 27 64 125 216 343 &c.

1 and 8 give 9 the square of 3

1,8, . 27 . 36 .... 6
1,8,27, . 64 . 100 .... 10

and so on.

8. Every number is either the sum of two, three, or four square numbers. The square numbers are,

1 4 9 16 25 36 49 &c.

and the following are instances of numbers in the first column, and the squares which compose them in the second.

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4. Every odd number can be made up of three square numbers at most, except those which, when divided by 8, leave a remainder 7. In the first column following, are odd numbers; in the second, the remainders of the same odd numbers divided by 8; in the third, the least number of squares of which the said number may be composed, which is never necessarily four, except when the remainder is 7.

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5. Every odd number whatsoever can be made up of squares not exceeding four in number, of which two at least shall be equal; as follows, in which the first column contains the odd number, the second the squares which compose it.

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6. Every even number greater than 2, is either the sum of two or three squares, or of four squares, two at least of which are equal. For instance,

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7. If we take any two numbers, one or other of the following three,

The sum of their squares,
The difference of their squares,
The product of the numbers,

will always be divisible by 5. For instance, 7 and 8 : the sum of the squares (49 and 64) is 118, the difference 15, the product (7 times 8) is 56. The second of these is divisible by 5.

8. If a square number be divided by 8, the remainder will be either 0, 1, or 4; if by 12 or 16, it will be either 0, 1, 4 or 9.

9. Every number which, divided by 8, leaves a remainder 6, is the sum of six odd squares.

14 is made up of 9 11111 62 do. 25 25 9 1 1 1 102 do. 81 9 9 1 1 1

also every number which, divided by 24, leaves a remainder 5, is the sum of five odd squares.

XLV.

THE MACE AS MUCH AS THE SPEAKER.

There are certain odd forms of proceeding connected with our legislative assemblies, which it may be presumed that very few but those acquainted with the details of Parliamentary business have any notion of. Many persons, for instance, may have seen, while standing in the lobby of the House of Commons, Mr. Speaker in his robes enter, preceded by a tall gentleman with a bag-wig and a sword by his side, carrying on his shoulder a heavy gilt club surmounted by a crown,—in short, a Mace: but few people are cognizant how important this toy is to the legislative duties of their representatives. Be it known then, that without it the House of Commons does not exist—and that it is as essential that the mace should be present at the deliberations of our senate, as that Mr. Speaker should be there himself:—without a Speaker the House never proceeds to business, and without his mace Mr. Speaker cannot take the chair. At the commencement of a session, and before the election of a Speaker, this valuable emblem of his dignity is hidden under the table of the House while the clerk of the table presides during the election; but no sooner is the Speaker elected, than it is drawn from its hiding-place, and deposited on the table, where it ever after remains during the sitting of the House: at its rising, Mr. Speaker carries it away with him, and never trusts it out of his keeping. This important question, of the Speaker's duty in retaining constant possession of this, which may be called his gilt walking-stick, was

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