bills enhanced; that is, the course of exchange will be raised in favour of the place to which the bills are to be remitted. In this way the fluctuation of the course of exchange is proportioned in general to the balance of debts, or remittances between two places, which again is produced by the balance of trade, or the difference between the values of the exports and imports of both places with respect to each other. Thus, if London export goods to Lisbon to the value of £1000, and import from Lisbon to the value of £ 1500, the difference or balance of trade is evidently against London, and in favour of Lisbon: and if London remit this balance in bills, their price in the money market must be raised, and consequently more money must be given in London, to procure the payment of a certain sum in Lisbon, than would be expressed by valuing these suins at the par, or just value of the coins of each country; and, on this account, when the course of exchange runs high against any country, remittances from thence ought, if possible, to be made in specie instead of bills, which will tend to reduce the rate of exchange nearer to par.

Exchanges are calculated by the rule of Practice: thus, if it be required to find the value of £ 420 sterling in French money at par of 25 livres for the pound sterling, the product of 420 by 25, or 10500 livres, is the answer; and the same sum of French money, at the course of 24 livres 2 sous, would be equal to £435.. 13.. 84. sterling.

THE

MODERN PRECEPTOR.

CHAPTER IV.

OF ALGEBRA.

ALGEBRA is an Arabic term of uncertain etymology, but generally supposed to signify the arts of restitution, comparison, resolution, and equation; meanings sufficiently denoting the nature of the art. By algebra we discover a general form of expressing the results of all questions comprehending similar circumstances, relating to magnitude, quantity, or number; or, in other words, by algebra we perform the several operations of addition, subtraction, multiplication, and division, employing certain characters or symbols of no real intrinsic value in themselves, but qualified to represent magnitudes, quantities, and numbers of every description. For example, let us suppose any number, as 3, to be represented by the symbol or character a; 5 to be represented by b; and their sum 8 by the symbol c; then, in algebraic language, a and b added together will be equal to c, or thus, a+b=c, that is, in this example 3+5=8. But the values of the arithmetical symbols 3, 5, and 8, having by long and unvaried usage become determinate, they are not susceptable of any change; whereas the values attributable to the symbols a, b, and c, may be varied indifinitely, and operations by them still give correct results; thus a may represent 12, b 15, and c 27, then a+b=c, for 12+

15=27.

Although

Although any letter of the alphabet may be employed to represent quantities in algebraic operations, yet it has been found convenient to use the first letters as a, b, c, d, e, &c. for quantities whose values are known or given, and the last letters, as v, x, y, z, for quantities neither given nor known; hence, as in the former example, where the values of a and b are given, and the value of their sum is required, we would say a+b=x.

Algebraic quantities are connected by means of certain signs, as(+) Plus, denoting that the quantities before and after the sign are to be added together, as 3+5 equal to 8. The sign(-) Minus, denotes that one of the quantities is to be subtracted from the other, as 8-5 equal to 3.

The sign(x) denotes that the quantities between which it stands are to be multiplied together, as 3 x 5 equal to 15; or axb-z. The product is also expressed by writing the symbols close together, as the letters in a word; thus axbab, and axbxc=abc.

Division is expressed by writing the dividend above a small line, and the divisor below it, as will signify that b is to be divided by a.

a

When quantities to be multiplied or divided are compound, a line, called a vinculum, is drawn over them thus ; axb-c, signifying that a is to be multiplied into the difference between b and c, and axb+m+c-d+e, will signify that a is to be multiplied by the difference between the sum of b, m, and c, and the sum of d and e; but, instead of this vinculum, or tye, over the compound quantities, many algebraists inclose these quantities as within a parenthesis; thus ax (b+m+c)−(d+e).

When an arithmetical figure stands before an algebraic symbol, it is called the numeral coefficient, and shows how often the algebraic quantity is to be repeated; thus 3 a will signify three times the value of a.

Equality

Equality is represented by the sign (=), as a+b=c, or the sum of a and b is equal to c.

Quantities are said to be like when they consist of the same characters; thus, 3 am and 5 am are like quantities, but 3 am and 5 am m would be unlike quantities.

Quantities having the same signs, whether or, are said to have like signs; but one having +, and another having-, have unlike signs.

Quantities having the sign + before them, are termed positive quantities; and those having before them, are negative quantities. It is true, that in the nature of things there can be no such thing as a negative quantity, that is, a quantity less than nothing; but the term is used in algebra to express such quantities whose value must be deducted from that of others with which they are connected; for example, the amount of a person's estate may be considered as a positive quantity, and that of his debts as a negative quantity, which being deducted from the former quantity, will show how much the person's real property is.

When no sign is prefixed to an algebraic quantity, it is always considered to be plus, or that the quantity is positive

Addition of algebraic quantities is performed in three different ways, according to the nature of the quantities.

1st. If the quantities be like, and have like signs, the rule is, to add together the co-efficients, (reckoning every character without a co-efficient for one,) annexing the common letter or letters, and prefixing the common sign.

2d. When the quantities are like, but the signs unlike, add all the positive quantities together, and all the négative quantities together, and subtracting the one sum from the other, the remainder will be the total sum required, having the sign belonging to the greatest sum.

To understand these two last examples, let us suppose all the positive quantities to represent the several articles of a person's effects, and the negative quantities to represent his debts; it will then be evident, that, to know the real value of his property, we must subtract the debts from the effects, and the remainder will correspond to the value of the whole positive and negative articles taken together. Hence, in the first example of this case, we have +4 m and + 9 m = 13 m, for the effects, and -Sm for the debts; consequently this sum being taken away from 13 m, will leave 5 m for the value of the property remaining.

- 3 m and

- 5 m =

3d. When the quantities to be added are all unlike, they are to be written down in succession, with their respective signs and co-efficients, in one line, as in the fol lowing examples.