62 A ‘Treatise of Algebra.

The Truth of this Rule is grounded upon self-evidentPrinciples; for if the equal Quantities, on each Side ofan Equation, be either multiply’d or divided, by oneand the fame Quantity, what results must necessarily beequal: Or, which is the same, if two Quantities areequal, their Halves, Thirds, £sV. or any equal Multi-ples of them must also be equal.

3°. Having ordered your Equation according to the pre-ceding Directions , if in it there he neither irreducible Surdsnor Fractions, let all the Terms on the right-hand Side ,in which the unknown Quantity is found , be carry'd overwith their Signs changed , to the left-hand Side , and let ,at the fame time , all the Terms on the left-hand whereinthe unknown Quantity is not found , be transposed, withcontrary Signs , to the right-hand Side ; contracting , asyou go along (by Rule i.) all such Terms as either destroyone another , or may be joined in one , and observing to placethe Terms according to the Order of the Powers of the un-known Quantity ; and you will thence get an Equation hav-ing all the unknown Terms on one Side , and all the knownones on the other ; of which all the Signs are to be changedwhen that of the first Term , or highest Power of the un-known Quantity comes out negative. Thus; let the Equa-tion $ax 2 — b 2 x d -j- a 2 b — — *3 — ax 2 be pro-

posed, where x is the unknown, or required Quantity.By looking over the Equation, it may be perceived that*3 is the highest Power of the unknown Quantity there-in concerned, and that it stands on the left-hand Side;therefore (in the next Step) *3 is to be transposed, andwrote down first, with the Sign -f- prefix’d to it, or,which is the same thing, without any Sign at all, forthen the Sign -f- is understood ; this being supposed done,look over the Equation again, and 3 ax 2 and — ax 2 willappear to be the Terms in which (x 2 ) the next inferiorPower of the unknown Quantity is concerned 3 and these,when — ax 2 is brought over, with it’s Sign changed,will give -f- qax 2 , to be next put down; to the twoTerms already wrote down annex— b 2 x, and then theSign of Equality ; look over the Equation, once again,and you will find c 3 -j- a 2 b to be transported to the right-

hand