been imagined for repreſenting their affections, relations, and dependencies.
The relation of equality is expreſſed by the ſign =; thus, to expreſs that the quantity repreſented by a is equal to that which is repreſented by b, we write a = b. But if we would expreſs that a is greater than b, we write a > b; and if we would expreſs algebraically that a is leſs than b, we write a < b.
Quantity is what is made up of parts, or is capable of being greater or leſs. It is increaſed by addition, and diminished by ſubtraction; which are therefore two primary operations that relate to quantity. Hence it is, that any quantity may be ſupposed 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 ſubtracted. The ſign + (plus) is the mark of addition, and the ſign − (minus) of ſubtraction. Thus the quantity being repreſented by a, + a imports that a is to be added, or repreſents an increment; but − a imports that a is to be ſubtracted, and repreſents a decrement. When ſeveral ſuch quantities are joined, the ſigns ſerve to ſshow which are to be added and which are to be ſubtracted. Thus + a + b denotes the quality that ariſes when a and b are both conſidered as increments, and therefore expreſſes the ſum of a and b. But + a − b denotes the quantity that ariſes, when from the quantity a the quantity b is ſubtracted; and expreſſes the exceſs of a above b. When a is greater than b, then a − b is itſelf an increment; when a = b, then a − b = 0; and when a is leſs than b, then a − b is itſelf a decrement.
As addition and ſubtraction are oppoſite, or an increment is oppoſite to a decrement, there is an analogous oppoſition between the affections of quantities that are conſidered in the mathematical ſciences; as, between exceſs and defects; between the values of effects or money due to a man, and money due by him. When two quantities, equal in reſpect and magnitude, but of thoſe opposite kinds, are joined together, and conceived to take place in the ſame ſubject, they deſtroy 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 eſtimating his ſtock may be both neglected. When two unequal quantities of thoſe oppoſite quantities are joined in the ſame ſubject, the greater prevails by their difference. And, when a greater quantity is taken from the leſſer of the ſame kind, the remainder becomes that of the opposite effect.
A quantity that is to be added is likewiſe called a poſitive quantity; and a quantity to be ſubtracted is ſaid to be negative: They are equally real, but oppoſite to each other, ſo 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 ſuppoſe in algebra that +a = −a; becauſe, to infer equality in this ſcience, they muſt not only be equal as to quantity, but of the ſame 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 depreſſion of a ſtar below the horiſon may be equal to the elevation of a ſtar above it: But thoſe poſitions are oppoſite, and the diſtance of the ſtars is greater than if one of them was at the hiron, ſo as to have no elevation above it, or depreſſion below it. It is on account of this contrariety, that a negative quantity is ſaid to be leſs than nothing, becauſe it is oppoſite to the positive, and diminiſhes when joined to it; whereas the addition of 0 has no effect. But a negative is conſidered no leſs as a real quantity than the poſitive. Quantities that have no ſign prefixed to them are understood to be poſitive.
The number prefixed to a letter is called the numeral coefficient, and ſhows how often the quantity repreſented by the letter is to be taken. Thus 2a imports that the quantity repreſented by a is to be taken twice; 3a that it is to be taken thrice; and ſo on. When no number is prefixed, unit is underſtood to be the coefficient. Thus 1 is the coefficient of a or of b.
Quantities are ſaid to be like or ſimilar, that are repreſented by the ſame 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 ſaid to conſiſt of as many terms as there are parts joined by the ſigns + or −; thus a + b conſiſts of two terms, and is called a binomial; a + b + c conſiſts of three terms, and is called a trinomial. Theſe are called compound quantities: A ſimple quantity conſiſts of one term only, as +a, +ab, or +abc.
Chap. I. Of Addition.
| Example. | To | a | to | −6b | to | a + b |
| Add | +4a | add | −2b | add | 3a + 5b | |
| Sum | +9a | Sum | −8b | Sum | 4a+6b | |
Rule. Subtract the leſſer coefficient from the greater, prefix the ſign of the greater to the remainder, and ſubjoin the common letter or lettters.
| Example. | To | −4a | + | 5b − 6c |
| Add | +7a | − | 3b + 8c | |
| Sum | +3a | 2b + 2c | ||
This rule is eaſily deduced from the nature of poſitive and negative quantities.
If there are more than two quantities to be adddd together, firſt add the poſitive number together into one ſum, and then the negative (by Caſe I.); then add theſe two ſums together (by Caſe II.)
