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80 been imagined for repreenting their affections, relations, and dependencies.

The relation of equality is expreed by the ign =; thus, to expres that the quantity repreented by a is equal to that which is repreented by b, we write a = b. But if we would expres that a is greater than b, we write a > b; and if we would expres algebraically that a is les than b, we write a < b.

is what is made up of parts, or is capable of being greater or les. It is increaed by addition, and diminished by ubtraction; which are therefore two primary operations that relate to quantity. Hence it is, that any quantity may be upposed to enter into algebraic computations two different ways, which have contrary effects; either as an increment, or as a decrement; that is, as a quantity to be added, or as a quantity to be ubtracted. The ign + (plus) is the mark of addition, and the ign − (minus) of ubtraction. Thus the quantity being repreented by a, + a imports that a is to be added, or repreents an increment; but − a imports that a is to be ubtracted, and repreents a decrement. When everal uch quantities are joined, the igns erve to show which are to be added and which are to be ubtracted. Thus + a + b denotes the quality that aries when a and b are both conidered as increments, and therefore exprees the um of a and b. But + a − b denotes the quantity that aries, when from the quantity a the quantity b is ubtracted; and exprees the exces of a above b. When a is greater than b, then a − b is itelf an increment; when a = b, then a − b = 0; and when a is les than b, then a − b is itelf a decrement.

As addition and ubtraction are oppoite, or an increment is oppoite to a decrement, there is an analogous oppoition between the affections of quantities that are conidered in the mathematical ciences; as, between exces and defects; between the values of effects or money due to a man, and money due by him. When two quantities, equal in repect and magnitude, but of thoe opposite kinds, are joined together, and conceived to take place in the ame ubject, they detroy each other's effect, and their amount is nothing. Thus, 100 L due to a man and 100 L due to him balance each other, and in etimating his tock may be both neglected. When two unequal quantities of thoe oppoite quantities are joined in the ame ubject, the greater prevails by their difference. And, when a greater quantity is taken from the leer of the ame kind, the remainder becomes that of the opposite effect.

A quantity that is to be added is likewie called a poitive quantity; and a quantity to be ubtracted is aid to be negative: They are equally real, but oppoite to each other, o as to take away each other's effect, in any operation, when they are equal as to quantity. Thus, 3 − 3 = 0, and a − a = 0. But though +a and −a are equal as to to quantity, we do not uppoe in algebra that +a = −a; becaue, to infer equality in this cience, they mut not only be equal as to quantity, but of the ame quality, that in ever operation the one may have the same effect as the other. A decrement may be equal to an increment, but it has in all operations a contrary effect; a motion downwards may be equal to a motion upwards, and the depreion of a tar below the horion may be equal to the elevation of a tar above it: But thoe poitions are oppoite, and the ditance of the tars is greater than if one of them was at the hiron, o as to have no elevation above it, or depreion below it. It is on account of this contrariety, that a negative quantity is aid to be les than nothing, becaue it is oppoite to the positive, and diminihes when joined to it; whereas the addition of 0 has no effect. But a negative is conidered no les as a real quantity than the poitive. Quantities that have no ign prefixed to them are understood to be poitive.

The number prefixed to a letter is called the numeral coefficient, and hows how often the quantity repreented by the letter is to be taken. Thus 2a imports that the quantity repreented by a is to be taken twice; 3a that it is to be taken thrice; and o on. When no number is prefixed, unit is undertood to be the coefficient. Thus 1 is the coefficient of a or of b.

Quantities are aid to be like or imilar, that are repreented by the ame letter or letters equally repeated. Thus +3a and −5a are like; but a and b, or a and a a are unlike.

A quantity is aid to conit of as many terms as there are parts joined by the igns + or −; thus a + b conits of two terms, and is called a binomial; a + b + c conits of three terms, and is called a trinomial. Thee are called compound quantities: A imple quantity conits of one term only, as +a, +ab, or +abc.

ASE I.

Rh Subtract the leer coefficient from the greater, prefix the ign of the greater to the remainder, and ubjoin the common letter or lettters.

This rule is eaily deduced from the nature of poitive and negative quantities.

If there are more than two quantities to be adddd together, firt add the poitive number together into one um, and then the negative (by Cae I.); then add thee two ums together (by Cae II.)