(In. 77.) and tax+b-ax-b (In. 389.) therefore Tab=a+b (In. 190.) confequently tab-ab COROLLARY XI. a. 392. Whence we have this general Rule for Species Multiplication and Divifion, that Co-affected Quantities give a Pofitive Product or Quotient, and Contra-affected Quantities a Defective one; i. e. give+Product Ex.S+ab=+ax+b=~ax--b +ab -ab in the Quotientgr.+a=. To and +x= {and { −x+2give-Product Ex. COROLLARY XII. = دات 393. Hence every Power, whofe Exponent is an even Number, i. e. every Square, Biquadrate, Sixth Power, &c. must be Pofitive, whether the Root be Pofitive or Defective (In. 386.) COROLLARY XIII. 394. Therefore the Root of every Defective Square, Biquadrate, Sixth Power, &c. is impoffible; becaufe it fuppofes two Co-affected Factors to give a Defective Product: And the Root of every Defective Power, whofe Exponent is an odd Number, (i. e. of every Defective Cube, Fifth, Seventh, Ninth, &c. Power) is Defective. DEFINITION VIII. 395. Quantities defigned by one Species are ftiled Monomes, and thofe defigned by more connected with the Signs and, Polynomes. SCHOLIUM I. 396. Every Quantity is fuppofed to be affected with one or other of thefe Signs (In. 377.) and whenever a Quantity has no Sign prefixed to it, as generally the leading Member in every Polynome has not, then is that Member understood to be affected with the Sign. Thus ab is a+b-c2 bis +b, dd, &c. DEFINITION IX. 397. Subftitution is the putting of one Expreffion in the room of another, and generally the more fimple Expreffion in the room of a lefs fimple one. B EN. Ex. gr. in thẹ Polynome a†-6b+2xz3 + z—x2 + by fubftituting p = ¿b+2x23 + z—x2 +2, we have this more fimple Expression a-+p, which -X is equivalent to the former, and therefore may be used in its ftead. COROLLARY XIV. 398. Whence every Polynome by fubftituting a Letter for all the Members except the firft, may be taken as a Binomial. DEFINITION X. 399. And the Ufe of this Species Arithmetic in the Investigation of Theorems, and Effection of Problems, is called Algebra. 400. TO multiply one Polynome into another. Effection. The Effection of this Problem in Species is directly as in Numbers, obferving In. 392. for the Signs. As follows. a2-3ab+bd x3-x2+x1 6xz|a— 32 Effection. 1. Let that Term in the Divifor which is contained in moft Terms, and the oftneft in each Term, of the Dividend be put in the first Place, that which is the next often in the fecond Place, and fo on. 2. Let the Terms in the Dividend be ordered accordingly; viz. Let thofe Terms which contain the firft Term in the Divifor ftand in the firft Place, thofe which contain the fecond, in the fecond Place, and fo on; obferving in the first Place to fet thofe Terms, which contain the firft dividing Term oftneft, before those which contain it lefs often. 3. Set down the Divifor and the Dividend according to the Directions (In. 321) obferving (In. 392) for the Signs. Examples follow. |