tions. Fundamen- Rule I. The terms of the dividend are to be ranged tal operaaccording to the powers of fome one of its letters; and thofe of the divifor, according to the powers of the fame letter. Thus, if a'+2ab+b is the dividend, and a +h the divifor, they are ranged according to the powers of a. 2. The first term of the dividend is to be divided by the first term of the divifor (obferving the general rule for the figns); and this quotient being fet down as a part of the quotient wanted, is to be multiplied by the whole divifor, and the product fubtracted from the dividend. If nothing remain, the divifion is finished the remainder, when there is any, is a new dividend. Thus, in the preceding example, a divided by a, gives a, which is the first part of the quotient wanted: and the product of this part by the whole divifor a+b, viz. a+ab being fubtracted from the given dividend, there remains in this example ab+b2. 3. Divide the first term of this new dividend by the first term of the divifor as before, and join the quotient to the part already found, with its proper fign: then multiply the whole divifor by this part of the quotient, and subtract the product from the new dividend; and thus the operation is to be continued till no remainder is left, or till it appear that there will always be a remainder. Thus, in the preceding example, tab, the firft term of the new dividend divided by a, gives b; the product of which, multiplied by a+b, being fubtracted from ab+b, nothing remains, and a+b is the true quotient. The entire operation is as follows. tal operations. is ftill a remainder from which the operation may be Fundamen 403 continued without end. This expreffion of a quotient is called an infinite feries; the nature of which shall be confidered afterwards. By comparing a few of the first terms, the law of the feries may be difcovered, by which, without any more divifion, it may be continued to any number of terms wanted. Of the General Rule. The reason of the different parts of this rule is evident; for, in the course of the operation, all the terms of the quotient obtained by it are multiplied by all the terms of the divifor, and the products are fucceffively fubtracted from the dividend till nothing remain: that, therefore, from the nature of division, must be the true quotient. Note. The fign÷is sometimes used to exprefs the quotient of two quantities between which it is placed: Thus, a+x+a+x, expreffes the quotient of a'+x' divided by a+x. $2. OF FRACTIONS. Definitions. 1. WHEN a quotient is expreffed by a fraction, the divifor above the line is called the numerator; and the divifor below it is called the denominator. 2. If the numerator is less than the denominator, it is called a proper fraction. 3. If the numerator is not lefs than the denominator, it is called an improper fraction. 4. If one part of a quantity is an integer, and the other a fraction, it is called a mixt quantity. 5. The reciprocal of a fraction, is a fraction whofe numerator is the denominator of the other; and whose denominator is the numerator of the other. The reciprocal of an integer is the quotient of divided by that integer. Thus, b - a of m. The diftinctions in Def. 2, 3, 4, properly belong to common arithmetic, from which they are borrowed, and are scarcely used in algebra. The operations concerning fractions are founded on the following propofition: If the divifor and dividend be either both multiplied or both divided by the fame quantity, the quotient is the fame; or, if both the numerator and denominator of the fraction be either multiplied or divided by the fame quantity, the value of that fraction is the famine. of divifion, if the quotient (c) be multiplied by b the divifor b, the product muft be the dividend a. Hence bca, and likewise ma=mbc, and dividing (xb=) bc= both by mb, c. Conversely, ifc, then alfo a It often happens, as in the laft example, that there c. 40+ Fundamen tal opera tions. 2. The fimple divifors of each of the quantities are Fundamento be taken out, the remainders in the feveral opera- tal operations are alfo to be divided by their fimple divisors, and the quantities are always to be ranged according to the powers of the fame letter. The fimple divifors in the given quantities, or in the quantity, Befides the fimple divifors in the remainders From the nature of this operation, it is plain that it may always be continued till there be no remainder. The rule depends on the two following principles: 1. A quantity which measures both divifor and remainder muft measure the dividend. 2. A quantity which measures both divifor and dividend muft alfo measure the remainder. For a quantity which meafures two other quantities, muft alfa meafure both their fum and difference; and, from the nature of divifion, the dividend confifts of the divifor repeated a certain number of times, together with the remainder. By the first it appears, that the number found by this rule is a common measure; and, by the fecond, it is plain there can be no greater common measure; for, if there were, it muft neceffarily measure the quantity already found lefs than itself, which is abfurd. a2 —b2)a2 —zab+b2 ( 1. tions. -2ab+2b Remainder, which If the quantities given are 8a'b'—10ab3 +2ba, and 9a1b—9a3b2+3a2b3—3ab4. The fimple divifors being taken out, viz. 26 out of the firft, it becomes 4a2-5ab+b2, and 3ab out of the fecond, it is. 3a3-3a2b+ab2-b3. As the latter is to be divided by the former, it must be multiplied by 4, to make the operation fucceed, and then it is as follows: When the greatest common measure of algebraical quantities is required, if either of them be fimple, any cominon fimple divifor is eafily found by infpection. If they are both compound, any common fimple divifor may also be found by infpection. But, when the greatest compound divifor is wanted, the preceding rule is to be applied; only, 4a2-5ab2) 12a3-12a3b+4ab2—4b3 (3a. 3a2b+ab2-4b3 This remainder is to be divided by b, and the new dividend multiplied by 3, to make the divifion proceed. Thus, 3a2+ab—4b2) 12a2—15ab+3b2(4 —19ab19b2. and this remainder, divided by -19b, gives a-b, which. being made a divifor, divides 3a+ab-4b2 without a remainder, and therefore a-b is the greatest compound. divifor: but there is a fimple divifor b, and therefore a-bXb is the greatest common measure required. PROB. II. To reduce a Fraction to its lowest Terms. Rule. Divide both numerator and denominator by their greateft common measure, which may be found by Prob. 1. ALG Fundamen- PROB. III. To reduce an Integer to the Form of a Fraction: tal operations. Rule. Multiply the given integer by any quantity for a numerator, and fet that quantity under the pro duct for a denominator. Cor. Hence, in the following operations concerning fractions, an integer may be introduced; for, by this problem, it may be reduced to the form of a fraction. The denominator of an integer is generally made 1. PROB. IV. To reduce Fractions with different Denominators to Fractions of equal Value, that shall have the fame Denominator. Rule. Multiply each numerator, feparately taken, into all the denominators but its own, and the products fhall give the new numerators. Then multiply all the denominators into one another, and the product shall give the common denominator. Example. Let the fractions be a, c, ba refpectively equal to bdf bdf bdf adf, bef, bde. The reafon of the operation appears from the preceding propofition; for the numerator and denominator of each fraction are multiplied by the fame quantities; and the value of fractions therefore is the fame. PROB. V. To add and fubtract Fractions. Rule. Reduce them to a common denominator, then add or fubtract the numerators; and the fum or difference fet over the common denominator is the fum or remainder required.. ace, b Ex. Add together dJ C From fubt. the difference is ad d From the nature of divifion it is evident, that, when feveral quantities are to be divided by the fame divifor, the fum of the quotients is the fame with the quotient of the fum of the quantities divided by that common divifor. In like manner, the difference of two fractions having the fame denominator, is equal to the difference of the numerators divided by that common denominator. Cor. 1. By Cor. Prob. 3. integers may be reduced to the form of fractions, and hence integers and frac tions may be added and subtracted by this rule. Hence also what is called a mixed quantity may be reduced into the form of a fraction by bringing the integral part into the form of a fraction, with the fame denominator as the fractional part, and adding or fubtracting the numerators according as the two parts are connected by the figns or —. C bd+c; d Thus, b+2= 2a2—a2+b2 _a2+b2 Or, let m, and=n. Then a=bm, and c=dn,, and ac=bdmn, and (mn=) —×2=6d. a ac but is bc da. a Or, let =m, and=n; then a=bm, and c=dn; alfo ad=bdm and be=bdn; therefore (bdm Scholium. n m bc -= aa By these problems, the four fundamental operations may be performed, when any terms of the original. quantities, or of those which arife in the courfe of the operation, are fractional Ex. 406 Of Propor tion. Cor. 1. Of four arithmetical proportionals, any three Of Propor being given, the fourth may be found. Thus, let a, b, c, be the 1ft, 2d, and 4th terms, and let x be the 3d which is fought. By the preceding operations quantities of the fame kind may be compared together. Then by def. a+c=b+x, and x=a+c—b. Cor. 2. If three quantities be arithmetical proportionals, the fum of the extremes is double of the middle term; and hence, of three fuch proportionals, any two being given, the third may be found. 2. Of Geometrical Proportion. Definition. If of four quantities, the quotient of the firft and fecond is equal to the quotient of the third and fourth, thefe quantities are faid to be in geometrical proportion. They are alfo called proportionals. Thus, if a, b, c, d, are the four quantities, then and their ratio is thus denoted a: b::c: The relation arifing from this comparifon is called ratio or proportion, and is of two kinds. If we confider the difference of the two quantities, it is called arithmetical proportion; and if we confider their quotient, it is called geometrical proportion. This laft being moft generally ufeful, is commonly called fimply proportionals. a C d. Cor. Three quantities may be geometrical propor tionals, viz. by fuppofing the two middle terms of the four to be equal. If the quantities are a, b, c, then a 1. Of Arithmetical Proportion. b - , and the proportion is expreffed thus, a: b: c. Prop. I. The product of the extremes of four quantities geometrically proportional is equal to the product of the means; and converfely. Cor. Three quantities may be arithmetically proportional, by fuppofing the two middle terms of the four to be equal. Prop. In four quantities arithmetically proportional, the sum of the extremes is equal to the fum of the means. Let the four be a, b, c, d. Therefore from Def. 4-b-c-d; to thefe add b+d and a+d=b+c. Let a:b::c: d. and multiplying both by bd, ad=bc. If ad=bc, then dividing by bd,=, that is, a:b::c:d. Cor. 1. The product of the extremes of three quantities, geometrically proportional, is equal to the fquare of the middle term. Cor. 2. Of four quantities geometrically proportional, any three being given, the fourth may be found. Ex. Let a, b, c, be the three firft; to find the 4th. Let it be x, then a:b::c:x, and by this propofition, ax=bc tion. This coincides with the Rule of Three in arithmetic, and may be confidered as a demonftration of it. In applying the rule to any particular cafe, it is only to be obferved, that the quantities muft be fo connected and so arranged, that they be proportional, according to the preceding definition. 3. Cor. Of three geometrical proportionals, any two being given, the third may be found. Prop. II. If four quantities be geometrically proportional, then if any equimultiples whatever be taken of the firft and third, and alfo any equimultiples whatever of the second and fourth; if the multiple of the firft be greater than that of the second, the multiple of the third will be greater than that of the fourth; and if equal, equal; and if lefs, lefs. For, let a, b, c, d, be the four proportionals. Of the tions. Of Equa- the first and third, ma and me may represent any equimultiples whatever, and alfo nb, nd, may represent any equimultiples of the fecond and fourth. Since a: b::c:d, ad=bc; and hence multiplying by mn, innad mnbc, and therefore (Conv. Prop. 1.) ma: nb:: me: nd; and from the definition of proportionals, it is plain, that if ma is greater than nb, e must be greater than nd; and if equal, equal; and if lefs, lefs. For fince ad bc, it is obvious, that in each of thefe cafes the product of the extremes is equal to the product of the means; the quantities are therefore proportionals. (Prop. 1.) Prop. IV. If four numbers be proportionals, according to Def. 5. V. B. of Euclid, they will be trically proportional, according to the preceding defi nition. geome 1, Let the four numbers be integers, and let them be a, b, c, d. Then if b times a and b times c be taken, and alfo a times b and a times d, fince ba the multiple of the firft is equal to ab the multiple of the fecond, bc, the multiple of the third, must be equal to ad the multiple of the fouth. And fince bead, by Prop. 1. a, b, c, and d, must be geometrical proportionals. •5. 2dly, If any of the numbers be fractional, all the four being multiplied by the denominator of the frac tions, they continue proportionals, according to Def. 5 B. V. Euclid (by Prop. 4. of that book); and the four integer quantities produced being fuch proportionals, they will be geometrical proportionals, by the first part of this prop.; and therefore being reduced by divifion to their original form, they manifeftly will remain proportionals, according to the algebraical definition. CHAP III. SECT. I. Of Equations in general, and of the Solution of fimple Equations. Definitions. 1. AN Equation may in general be defined to be a propofition afferting the equality of two quantities; and is expreffed by placing the fign between Of Equa 407 2. When a quantity ftands alone upon one fide of an of it. 3. When an unknown quantity is made to ftand alone 4. Equations containing only one unknown quantity If the higheft power of Ift, The E-Simple, C =12, treated of in algebra, are almoft univerfally either that is in the investigation of quantities that are unknown, The folution of the different orders of equations will 1. Of fimple Equations, and their Refolution. Rule 1. Any quantity may be transposed from one fide By tranfp. 3x=a-b. This rule is obvious from prob. 1. and 2.; for it is the equation, or to fubtracting equal quantities from equivalent to adding equal quantities to both fides of both fides. Cor. tions. |