together, their Sum acts upon the Body: But when we are to fubtract one of them from the other, we conceive that which is to be fubtracted to be a Power with an oppofite Direction, and if it be greater than the other, it will prevail by the Difference. This Change of Quality however only takes place where the Quantity is of fuch a Nature as to admit of fuch a Contrariety or Oppofition. We know nothing analogous to it in Quantity abstractly confidered; and cannot fubtract a greater Quantity of Matter from a leffer, or a greater Quantity of Light from a leffer. And the Application of this Doctrine to any Art or Science is to be derived from the known Principles of the Science. a= 0. § 7. A Quantity that is to be added is likewife called a Pofitive Quantity; and a Quantity to be fubtracted is faid to be Negative: They are equally real, but oppofite to each o ther, fo as to take away each other's Effect, in any Operation, when they are equal as to Quantity. Thus 330, and a But tho' and a are equal as to Quantity, we + do not fuppofe in Algebra that a = a; becaufe to infer Equality in this Science, they muit not only be equal as to Quantity, but of the fame Quality, that in every Operation the one may have the fame Effect as the other. A Decrement may be equal to an Increment, but but it has in all Operations a contrary Effect; a Motion downwards may be equal to à Motion upwards, and the Depreffion of a Star below the Horizon may be equal to the Elevation of a Star above it: But thofe Pofitions are oppofite, and the Distance of the Stars is greater than if one of them was at the Horizon fo as to have no Elevation above it, or Depreffion below it. It is on account of this Contrariety that a Negative Quantity is faid to be less than Nothing, because it is oppofite to the Pofitive, and diminishes it when joined to it, whereas the Addition of o has no Effect. But a Negative is to be confidered no lefs as a Real Quantity than the Pofitive. Quantities that have no Sign prefixed to them are understood to be Pofitive. § 8. The Number prefixed to a Letter is called the Numeral Coefficient, and fhews how often the Quantity reprefented by the Letter is to be taken. Thus 2 a imports that the Quantity represented by a is to be taken twice; 3 a that it is to be taken thrice; and fo on. When no Number is prefixed, Unit is understood to be the Coefficient. Thus I is the Coefficient of a or of b. Quantities are faid to be like or fimilar, that are reprefented by the fame Letter or Letters equally repeated. Thus + 3a and like; but a and b, or a and a a are unlike. 5 a are or A Quantity is faid to confift of as many Terms as there are Parts joined by the Signs + ; thus ab confifts 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 fimple Quantity confifts of one Term only, as+a, or +ab, or +abc. The other Symbols and Definitions neceffary in Algebra shall be explained in their proper Places, CHA P. II, Of ADDITION. 59. like and have like Signs. C ASE I. To add Quantities that are Rule, Add together the Coefficients, to their Sum prefix the common Sign, and fubjoin the common Letter or Letters. To 34- 4* Sum 84-12X Cafe II. To add Quantities that are like but Rule. Subtract the leffer Coefficient from the This Rule is eafily deduced from the Nature of Pofitive and Negative Quantities, If there are more than two Quantities to be added together, firft add the Pofitive together into one Sum, and then the Negative (by Cafe 1.) Then add thefe two Sums together (by Cafe II.) EX |