value of b, being supposed equal to 3, we shall have 263 54.

12. By the root of a quantity, is meant a number which being multiplied one or more times by itself, will give that quantity. There is therefore a square root cube root, &c.

The different roots of quantities may be expressed by the following signs:,,, in this manner, /a, /a, /a, 2/16, 3/27.

13. We shall now shew how to extract the square root of any quantity; that is to say, how to find a number, which being multiplied by itself, will give that quantity, if it be a complete square, or at least the greatest square which it contains.

EXAMPLE I.

Let the number, the square root of which is required be 1156. First divide this number from right to left into periods of two figures, and then proceed as follows:

4/256

Find the greatest square contained in 11, 56 (34 11, which is 9, and write down its root 9 3, as seen in the annexed example. Square 64/256 3, which gives 9, subtract it from 11, and the remainder will be 2. Then bring down the next period, which is 56, that it may serve as a dividend along with the figure 2 on its left. Take 6 as a divisor, that is to say, the double of the root 3 already found; place it on the left, and find how often it is contained in 25; the quotient will be 4, which must be written down in the root after 3; and also after 6, the divisor, which will give 64. Then multiply the last number by the second root 4, and the product will be 256. As there is no remainder, it is a proof that 1155 is a perfect square, the root of which is 34. Had the last product been too large to be subtracted, it would have been necessary to diminish the last figure in the root, in order to make it small enough for that purpose.

EXAMPLE II.

What is the Square Root of the Number 214369?

As in the preceding examples, we must first divide the number into periods of two, from right to left, and there will be as many figures in the root as there are periods.

Then, as the greatest square contained in 21 is 16, the square root of which is 4, 86)

write down the 4 in the root; square 4, which will give 16,

1

21, 43, 69 (463

16

543

6 516

and having subtracted it 923) 2769 from 21, the remainder will

3 2769

be 5. Bring down the following period 43, which, with the preceding figure, must be divided by the double of the root already found, that is to say, by 8. The quotient of 54 divided by 8 is 6, which must be placed after the first root 4, and also after 8 the divisor; then multiply 86 by 6, and subtract the product 516 from 543. Place the remainder 27 under 516, and bring down the next period 69. Take, as the divisor of 276, the double of the two roots already found, which is 92. Divide 27 by 9, and place the quotient 3 in the root after 46, and also after 92. Then multiply 923 by 3, and if the product 2769 be subtracted from the number 2769, there will be no remainder. The truth of this operation may be proved by squaring 463; that is to say, by multiplying it by itself.

After the last subtraction, if any thing remains, it is a proof that, though the root found is not exactly the real root, it does not want unity to be so; but if it were required to approach still nearer to the real root, nothing would be necessary but to reduce the remainder to decimals, and to continue the operation, taking care to separate the whole numbers in the root from the decimals. However, as we propose here only to give a few amusing problems, there will be no necessity for carrying the extraction of the square root beyond whole numbers.

OF EQUATIONS.

14. As certain questions cannot be easily resolved without some knowledge of analysis and equations, we shall here give a short explanation of them, and such as may be easily understood.

By equations is meant the application of numerical and algebraic rules, to the solution of different questions, which may be proposed respecting quantity.

The first and most difficult thing in analysis, is to comprehend properly the state of the question, and the relation which the known quantities bear to the unknown, in order that they may be clearly expressed in an equation.

Every equation is composed of two members, separated by the sign; and each member may consist of several terms. An example of the whole may be seen in the following equations:

73+4; 8-52 + 1; 3 x 4 = 12; 2=3.

There may be equations also consisting of algebraic quantities alone, or in which arithmetical quantities are mixed with algebraic ones, as in the following: x + b = a; x -b4c2x.

- y = a + b ; 3a

GENERAL RULES IN REGARD TO EQUATIONS.

RULE I.

15. Any quantity may be transposed from one member of an equation to another, without deranging the equation, provided that the signs be changed.

Thus, as 12 39, we may write 12 For the same reason if a x +3b= d have a + yd = x

36.

9 + 3.

y, we shall

This method of operation, in regard to equations, is called transposition, and is employed when it is necessary to free one member of an equation from any quantity connected with it either by addition or subtrac

RULE II.

16. When an unknown quantity is involved in an equation either by multiplication or division, it may be disengaged from it, in the first case, by division; and in the second by multiplication. For example,

b

If 3x b, then = and if

This method of disengaging an unknown quantity will be more easily comprehended, if we give determinate values to the quantities a and b. If we suppose, for example, that b = 12, and a = 8, the two equations above mentioned will be reduced to the following, x = 12 = 4, ÿ = 24.

17. It appears, therefore, that the whole art of analysis consists, first, in comparing in the equations the unknown with the known quantities, and disengaging them from each other by the means already pointed out, in such a manner, that the known quantity may remain` alone in one member of the equation, and the unknown in the other.

To facilitate the solution of algebraic questions, the unknown quantities are generally denoted by some of the last letters of the alphabet, v, y, z; and the known quantities by some of the first, as a, b, c, &c.

OF RATIOS AND PROPORTIONS.

18. Relation or Ratio is what results from the comparison of two quantities. As two quantities may be compared with each other two ways, ratio is distinguished into two kinds, arithmetical and geometrical.

Arithmetical relation, is that of two quantities compared with each other by subtraction.

Geometrical relation, is that of two quantities compared with each other by division.

Thus, for example, the arithmetical

ratio of 12 to 4 is 8; and the geometrical ratio of the same quantities, is 3; for 12 48, and 2 = 3.

19. Proportion is an equality of ratios. As there are two kinds of ratio, there are also two kinds of proportion, arithmetical and geometrical: the first consists in an equality of differences, and the second in an equality of quotients.

Every ratio is expressed by two terms; the first of which is called the antecedent, and the second the consequent.

Two equal ratios form a proportion; which is either arithmetical or geometrical, according as they contain either the same difference or the same quotient. Thus 3.5.7.9, expresses an arithmetical proportion; the meaning of which is, that 3 is arithmetically to 5, as 7 is to 9; and 6: 3 :: 16 8, expresses a geometrical proportion; the meaning of which is, that 6 is geometrically to 3 as 16 is to 8.

The first and last terms of each of these proportions are called the extremes; and the other two the means.

20. Proportion is continued when the same term is the consequent of that which precedes it, and the antecedent of that which follows it. Thus the two following proportions, one of which is arithmetical, and the other geometrical, are continued, viz. ÷ 3 : 16. The meaning of which is,

3

5

7;4:8 5 5

7;

When a continued proportion has more than three terms, it is called a progression. Thus 1 9 is an arithmetical progression, and 32 64 is a geometrical progression.

3.5.7. 4:8; 16:

Properties of Arithmetical Proportion and Progression.

THEOREM I.

21. In every arithmetical proposition, the sum of the extremes is equal to that of the means.

Then 3 +95 +7, and 9 +56 +8.

It hence follows, that when three terms of an arithmetical proportion are known, we may easily find the