1. Simple quantities are multiplied together by multiplying the coefficients one into the other, and to the product annexing the quantity which, according to the method of notation, expreffes the product of the fpecies; prefixing the fignor, according as the figns of the given quanti ties are like or unlike. Thus 2a. Alfo bab And 11adf makes bab. makes 30abc. makes makes 77aabdf. Thus, by way of illuftration, abcde will appear to be abced, &c. For, the former of these being equal to every other product of the clafs, or terminatione (by hypothefis and equal multiplication), and the latter equal to every other Product of the clafs, or termination d; it is evident, therefore, that all the Products of different claffes, as well as of the fame class, are mutually equal to each other. So far relates to the firft general obfervation: It remains to prove that abcd × pqrst is a xbx c x d x p x q x r x s xt. In order to which, let abcd be denoted by x, then will abcd × pqrst be denoted by x × pqrst, or pqrst xx (by cafe 1), that is, by p x q x r x s x t xx; which is equal to xxpxqxrxsxt, or axbx cx d x pxqxrxsxt, by the preceding Demonftration. The Reafon of Rule 1 depends on thefe two general Obfervations for it is evident from hence, that 24 x 36 (in the first example) is = 2 x a x 3 x b = 2 x 3 xa x b = 6 × a x b = 6ab: And, in the faine manner, 11adf x 7ab (in the third example) appears to be II xa x d x ƒ x 7 x ax b = 11 x 7 x a x a x b x d x f = 77 × aabdf = 77aabdf. But the grounds of the method of proceeding may be otherwife explained, thus: It has been obferved that ab (according to the method of notation) defines the product of the Species a, b (in the first example), therefore the product of a by 3b, which must be three times as great (because the multiplier is here three times as great), In the preceding examples all the products are affir mative, the quantities given to be multiplied being fo; but, in thofe that follow, fome are affirmative, and others negative, according to the different cafes fpecified in the latter part of the rule; whereof the reafons will be explained hereafter. 5a 6b Prod. + 30ab. Mult. + 5a Mult. Mult. by Mult. 7av aa +xx 6bv aa-yy Prod.-35×Vax XV.cy.Prod. +42abx√ aa+xx ×√ aa-yy In the two laft examples, and all others, where radical quantities are concerned, every fuch quantity may be confidered, and treated in all refpects as a fimple quantity, expreffed by a fingle letter; fince it is not the Form of the expreffion, but the value of the quantity that is here regarded. 2o. A Fraction is multiplied, by multiplying the nume rator thereof by the given multiplier, and making the product a numerator to the given denominator. will be truly defined by gab, or ab taken three times: but, fince the product of a by 3b appears to be 3ab, it is plain that the product of 2a by 3b muft be twice as great as that of a by 36, and therefore will be truly expreffed by 6ab. Thus alfo, the product of the Species ab and c (in the fecond example) being abc (by bare notation) it is evident that the product of 6ab by c will be truly defined by babe, or abe fix times taken, and confequently the product of 6ab and 5c, by 30abc, or 6abc taken five times, the multiplier here being five times as great. The Reason of Rule 2o may be thus demonftrated: Let the numerator of any propofed fraction be denoted by A, the 3. Fractions are multiplied into one another, by multiplying the numerators together for a new numerator, and the denominators together for a new denominator. the denominator by B, and the given multiplicator by C: AC then, I fay, that is equal to A denotes the quantity which arifes by dividing AC by B, A and the quantity which arifes by dividing A by B, it is B evident that the former of these two quantities must be C times as great as the latter (because the dividual is C times as great in the one cafe as in the other) and therefore must be equal to the latter C times taken, that is, muft be equal to × C, as was to be fhewn. AC B A B The Reason of Rule 3° will appear evident from the preceding demonftration of Rule 2°. For, it be ing there proved that C is equal to A B AC B' it is ob 4. Surd quantities under the fame radical fign are multiplied like rational quantities, only the product must stand under the fame radical fign. Thus, 7 x 5 5 = √35; Va x ✓ 7bc × 5ad =✔ 35abcd; 3√ ab × 5 2aV 2cy × 36V 5ax (= 6ab × √2cy × √ 5ax) 2aV2cy √b = Nab; c = 15 √ abc; = 6ab√10acxy; and 7ab 8x 13d 9d 26 C caufe, the multiplier here, is but the D part of the D' AC former multiplier C: But is alfo equal to the D AC part of the fame B BD because its divifor is D times as great as that of AC: therefore these two quanti B A C AC ties, X and being the fame part of one and the fame quantity, they must neceffarily be equal to each other; which was to be proved. As to Rule 4° for the multiplication of fimilar radical quantities, it may be explained thus: Suppofe VA and B to reprefent the two given quantities to be multiplied together; let the former of them be denoted by a, and the latter by b, that is, let the quantities reprefented by a and b be fuch, that aa may be = A, and bb = B; then the product of VA by VB, or of a by b, will be expreffed by ab, and its fquare by ab xab: but ab x ab is axbx axb=aax bb (by the general obfervations premifed at the beginning of this fection); whence the fquare of the product is likewife truly expreffed by aa x bb, or its equal A×B; and confequently the product itself, by VAXB, that is, by the quantity which, being multiplied into itfelf, produces A x B. In 5°. Powers, or roots of the fame quantity are multiplied together, by adding their exponents: But the exponents here understood are thofe defined in p. 5, where roots are reprefented as fractional powers. In the fame manner the product of Ã× VB will ' appear to be AB: for, if VA be denoted by a, and VB by b; or, which is the fame, if aaa A, and bbb B; then will VA × VB = axb (or ab) and its cube ab× ab × ab = aaa × bbb = AB (by the aforefaid obfervations) whence the product itself will evidently be expreffed by AB. I * The Grounds of thefe Operations may be thus explained. First, when the exponents are whole numbers, as in example 1, the demonftration is obvious, from the general obfervations premised at the beginning of the section: For, by what is there fhewn, x2 × ×3, or xx × xxx is = × × × × × × × × × = x5 (by Notation.) But in the laft example, where the exponents are fractions, let + be reprefented by x; c y that is, let the quantity x be fuch, that x x x × × × xxxxx, or x may be equal to c + y; so shall c + y be expreffed by x3; because, by what has been already fhewn, 3 x3 isx: and, in the fame manner, will c+y be expressed by x2; because 2 × x2 × x2 is |3 likewifex. Therefore c+ y X I I 1 2 c+ y 3 is = 3⁄43 × x2 = x2 = the fifth power of c + y +y, by Notation. C 2 6 |