University of California Berkeley 
 
 THE THEODORE P. HILL COLLECTION 
 
 of 
 
 EARLY AMERICAN MATHEMATICS BOOKS 
 
I 
 
ELEMENTARY 
 
 ALGEBRA: 
 
 EMBRACING 
 
 THE FIRST PRINCIPLES 
 
 OF 
 
 THE SCIENCE 
 
 BY CHARLES DAVIES, LL.D. 
 
 ATTTHO B OF 
 
 ARITHMETIC, ELEMENTARY GEOMETRY, ELEMENTS OF SURVEYING, 
 ELEMENTS OF DESCRIPTIVE AND ANALYTICAL GEOMETRY, ELE- 
 MENTS OF DIFFERENTIAL AND INTEGRAL CALCULUS, 
 AND A TREATISE ON SHADES, SHADOWS, 
 AND PERSPECTIVE- 
 
 NEW YORK: 
 
 PUBLISHED BY A. S. BARNES & CO. 
 CINCINNATI :-H. W. DERBY & CO 
 1850. 
 
Entered according to the Act of Congress, in the year oae thousand 
 eight hundred and Forty-Five, by CHARLES DAVIES, in the Clerk's 
 Office of the District Court of the United States, for the Southern 
 District of New York 
 
 
 F. C. GUTIERREZ, 
 
 PRINTER, 
 
 Cor. John and Dutch-streets, N. Y. 
 
PIIEFACE. 
 
 ALTHOUGH Algebra naturally follows Arithmetic in a course 
 ef scientific studies, yet the change from numbers to a sys- 
 tem of reasoning entirely conducted by letters and signs is 
 rather abrupt and not unfrequently discourages the pupil. 
 
 In this work it has been the intention to form a connect- 
 ing link between Arithmetic and Algebra, to unite and blend, 
 as far as possible, the reasoning on numbers with the more 
 abstruse method of analysis. 
 
 The Algebra of M. Bourdon has been closely followed. 
 Indeed, it has been a part of the plan, to furnish an introduc- 
 tion to that admirable treatise, which is justly considered, 
 both in Europe and this country, as the best work on the 
 subject of which it treats, that has yet appeared. 
 
 This work, however, even in its abridged form, is too 
 voluminous for schools, and the reasoning is too elaborate 
 and metaphysical for beginners. 
 
 It has been thought that a work which should so far mo- 
 dify the system of M. Bourdon as to bring it within the 
 scope of our common schools, by giving to it a more prac- 
 tical and tangible form, could not fail to be useful. Such is 
 the object of the ELEMENTARY ALGEBRA. 
 
 Having within the past year carefully revised the Algebra 
 of M. Bourdon, and made therein many important changes 
 and alterations, both by the addition of new rules and in the 
 abridgment and simplification of those before given, it be- 
 came necessary to make corresponding changes in the in- 
 troductory work. The alterations before made, in the form 
 of an Introduction, the Treatise on Logarithms, and the 
 Supplement containing practical examples, with solutions 
 given in the Key, have all been retained ; and the work is 
 now presented to the public in a form which it is hoped will 
 not require alteration. 
 
 WBST POINT, January, 1845. ( iii ) 
 
DAVIES' 
 
 COURSE OF MATHEMATICS. 
 
 DAVIES' TABLE BOOK. 
 
 DAVIES' FIRST LESSONS IN ARITHMETIC 
 
 DAVIES' ARITHMETIC. Designed for the use of Academies and 
 Schools. 
 
 DAVIES' GRAMMAR OF ARITMETIC. 
 KEY TO DAVIES' ARITHMETIC. 
 
 DAVIES' UNIVERSITY ARITHMETIC Embracing the Science 
 of Numbers, and their numerous applications. 
 
 KEY TO DAVIES' UNIVERSITY ARITHMETIC. 
 
 DAVIES' ELEMENTARY ALGEBRA Being an Introduction to 
 the Science, and forming a connecting link between ARITHMETIC and 
 ALGEBRA. 
 
 KEY TO DAVIES' ELEMENTARY ALGEBRA. 
 
 DAVIES' ELEMENTARY GEOMETRY. This work embraces the 
 elementary principles of Geometry. The reasoning is plain and con- 
 cise, but at the same time strictly rigorous. 
 
 DAVIES' ELEMENTS OF DRAWING AND MENSURATION 
 
 Applied to the Mechanic Arts. 
 
 DAVIES' BOURDON'S ALGEBRA Including Sturms' Theorem, 
 Being an Abridgment of the work of M. Bourdon, with the addition of 
 practical examples 
 
 DAVIES' LEGENDRE'S GEOMETRY AND TRIGONOMETRY. 
 
 Being an Abridgment of the work of M. Legendre, with the addition 
 of a Treatise on MENSURATION OF PLANES AND SOLIDS, and a Table of 
 LOGARITHMS and LOGARITHMIC SINES. 
 
 DAVIES' SURVEYING With a description and plates of the THEOD- 
 OLITE, COMPASS, PLANE-TABLE, and LEVEL : also, Maps of the TOPO- 
 GRAPHICAL SIGNS adopted by the Engineer Department an explana- 
 tion of the method of surveying the Public Lands, and an Elementary 
 Treatise on NAVIGATION. 
 
 DAVIES' ANALYTICAL GEOMETRY Embracing the EQUA 
 
 TIONS OF THE PoiNT AND STRAIGHT LlNE of the CoNIC SECTIONS of 
 
 the LINE AND PLANE IN SPACE also, the discussion of the GENERAL 
 EQUATION of the second degree, and of SURFACES of the second order. 
 
 DAVIES' DESCRIPTIVE GEOMETRY, With its application U 
 SPHERICAL PROJECTIONS. 
 
 DAVIES' SHADOWS AND LINEAR PERSPECTIVE. 
 DAVIES' DIFFERENTIAL AND INTEGRAL CALCULUS 
 
CONTENTS. 
 
 CHAPTER I. 
 
 PRELIMINARY DEFINITIONS AND REMARKS. 
 
 ARTICLES. 
 
 Algebra Definitions Explanation of the Algebraic Signs, - 1 23 
 
 Similar Terms Reduction of Similar Terms, - 23 26 
 
 Addition Rule, 26 28 
 
 Subtraction Rule Remark, - - - - - - 28 33 
 
 Multiplication Rule for Monomials, 33 36 
 
 Rule for Polynomials and Signs, 36 38 
 
 Remarks Properties Proved, 3842 
 
 Division of Monomials Rule, ------ 42 45 
 
 Signification of the Symbol ao, - - - - - 45 46 
 
 Of the Signs in Divison, - - - - - - . - 46 47 
 
 Division of Polynomials, ---.-- 47 49 
 
 CHAPTER II. 
 
 ALGEBRAIC FRACTIONS. 
 
 Definitions Entire Quantity Mixed Quantity, - - 49 52 
 
 To Reduce a Fraction to its Simplest Terms - 52 
 
 To Reduce a Mixed Quantity to a Fraction, ... 63 
 
 To Reduce a Fraction to an Entire or Mixed Quantity, - 64 
 
 To Reduce Fractions to a Common Denominator, - 55 
 
 To Add Fractions, ' - 56 
 
VI CONTENTS. 
 
 ARTICLES. 
 
 To Subtract Fractions, 57 
 
 To Multiply Fractions, 58 
 
 To Divide Fractions, -- 69 
 
 CHAPTER III 
 
 EQUATIONS OF THE FIRST DEGREE. 
 
 Definition of an Equation Properties of Equations, - 60 66 
 
 Transformation of Equations First and Second, - 66 70 
 
 Resolution of Equations of the First Degree Rule, - 70 
 
 Questions involving Equations of the First Degree, - '- 71 72 
 Equations of the First Degree involving Two Unknown 
 
 Quantities, - - 72 
 
 Elimination By Addition By Subtraction Bv Comparison, - 73 76 
 Resolution of Questions involving Two or more Unknown 
 
 Quantities, 76 79 
 
 CHAPTER IV. 
 
 OF POWERS. 
 
 Definition of Po .vers, ----..-. 79 
 
 To raise Monomials to any Power, - 80 
 
 To raise Polynomials to any Power, - 81 
 
 To raise a Fraction to any Power, - .... 82 83 
 
 Binomial Theorem, 8490 
 
 CHAPTER V. 
 
 Definition of Squares Of Square Roots And Perfect 
 
 Squares, 90 96 
 
 Rule for Extracting the Square Root of Numbers, - - 96 100 
 
 Square Roots of Fractions, - 100 103 
 
 Squan s Roots of Monomials, 103 107 
 
 Calculus of Radicals of the Second Degree, - - - 107109 
 
CONTENTS. 
 
 Vll 
 
 Addition of Radicals, .... 
 
 Subtraction of Radicals, - 
 
 Multiplication of Radicals, 
 
 Division of Radicals, .... 
 
 Extraction of the Square Root of Polynomials, 
 
 ARTICLES. 
 
 109 
 110 
 111 
 112 
 113116 
 
 CHAPTER VI. 
 
 Equations of the Second Degree, ... 
 Definition and Form of Equations, - 
 Incomplete Equations, - 
 Complete Equations, ..... 
 
 Four Forms, ------- 
 
 Resolution of Equations of the Second Degree, - 
 Properties of the Roots, - - - 
 
 116 
 
 116118 
 118122 
 
 122 
 
 123127 
 127128 
 128134 
 
 CHAPTER VII. 
 
 Of Progressions, ........ 135 
 
 Progressions by Differences, ...... 136 138 
 
 Last Term, 138140 
 
 Sum of the Extremes Sum of the Series, ... 140 141 
 
 The Five Numbers To Find any Number of Means, - 141 144 
 
 Geometrical Proportion and Progression, - 144 
 
 Various Kinds of Proportion, - - - - 144 166 
 
 Geometrical Progression, ------- 166 
 
 Last Term Sum of the Series, 167 171 
 
 Progressions having an Infinite Number of Terms, - - 171 172 
 
 The Five Numbers To Find One Mean - 172173 
 
 CHAPTER VIII. 
 
 Theory of Logarithms 174 179 
 
INTRODUCTION. 
 
 LESSON I. 
 
 1. JOHN and Charles have twelve apples between them, 
 and each has as many as the other : How many has each ? 
 
 If we suppose the apples divided into two equal parts, it is 
 plain that John will have one part and Charles the other : 
 hence, they will each have six apples. 
 
 In Algebra, we often represent numbers by the letters of 
 the alphabet ; that is, we take a letter to stand for a number. 
 Thus, let x stand for the apples which John has. Then, as 
 Charles has an equal number, x will also stand for the apples 
 which he has. But together, they have twelve apples ; hence, 
 twice x must be equal to 12. This we write thus : 
 
 and if twice x is equal to 12, it follows that once a?, or a?, will 
 be equal to 6. This we write thus : 
 
 a?= =6 
 
 ~~2 ~~ 
 
 When we write x by itself, we mean one #, or the same as 
 la?. If we write 2a;, we mean that x is twice taken ; if 3a?, 
 that it is taken three times, &,c. 
 
 QUEST. 1. In the first question, how many apples has each boy? 
 By what are numbers represented in Algebra ? If a: stands by itself, how 
 many times x are expressed? What does Zx denote? What 3x1 
 What4#, &c. If \vc have x-\-x, to how many times x is it equal ? ff 
 we have the value of 2or, how do we iiud the value of x I 
 
 w 
 
ELEMENTARY ALGEBRA 
 
 3. James and John together have 24 penches, and one has 
 as many as the other : How many has each ? 
 
 Let x stand for the number of peaches which James has : 
 then x will also be equal to the number of peaches which 
 John has ; and since they have 24 between them, 
 
 that is, 2a?=24 and a?==12. 
 
 Therefore, each has twelve peaches. 
 
 3. William and John have 36 pears, and one has as many 
 as the other : How many has each ? 
 
 Let the number which each has be denoted by x. 
 then a;+;r=36 ; 
 
 that is, 2*=36 and *=-??==* 18. 
 
 4:. What number is that which added to itself will give a 
 sum equal to 20 ? 
 
 Let the number be denoted by x : then, as the number is 
 to be added to itself, we have 
 
 that is, 2z=20 or x=^ U =10. 
 
 Hence, 10 is the number. 
 
 5. What number is that which added to itself will give 9 
 sum equal to 30 ? 
 
 QUEST. 2. In the second question, what does x stand for 1 What 
 is twice x equal to ? How then do you find the value of x 1 3. In the 
 third question, what does x stand for 1 What is x equal to 7 How do 
 you find the value of x 1 4. In the fourth question, what does x stand 
 for 7 What is twice x equal to 1 How do you then find x 1 5. In the 
 fifth question, what does x stand for 1 How do you find its value 1 
 
 
INTRODUCTION. 3 
 
 6 What number is that which added to itself will give a 
 sum equal to 60 ? 
 
 7. What number is that which added to itself will give a 
 sum equal to 100 ? 
 
 8. What number added to itself will give a sum equal to 80. 
 
 9. What number added to itself will give a sum equal to 25. 
 10.. What number added to itself will give a sum equal 
 
 to 37J. 
 
 LESSON II. 
 
 1. JOHN and Charles together have 12 apples, and Charles 
 has twice as many as John : How many has each ? 
 
 If we now suppose the apples to be divided into three equal 
 parts, it is evident that John will have one of the parts and 
 Charles two. 
 
 Let us denote by x the apples which John has. Then, 2x 
 will be equal to what Charles has, and x+2x will be equal 
 to all the apples. This equality is thus expressed : 
 
 that is, 3#=>12 or ;r==4 
 
 o 
 
 therefore, John has 4 apples, and Charles 8. 
 
 QUEST. 6. How do you find the value of x in the 6th question t 
 8. How in the 8th 1 9. How in the 9th ? 1O. How in the 10th 1 
 
 QUESTIONS ox LESSON IT. 1. Into how many parts may we suppose 
 the 12 apples to be divided 1 How many of the parts will John have! 
 What is the value of each part 1 If a? stands for one of the parts, what 
 will stand for two parts'? What for three parts'? If you have the value 
 of 3ar, how will you find the value of x 1 
 
4 ELEMENTARY ALGEBRA 
 
 2. James and John have 30 pears, and John has twice as 
 many as James : How many has each ? 
 
 Here, again, let us suppose the whole number to be divided 
 into three equal parts, of which James must have one part, 
 and John two. 
 
 Let iis then denote by #, the number of pears which James 
 has: then 2x will be equal to the number of pears, which * 
 John has, and x-\-2x will be equal to the whole number of 
 pears : and we shall have 
 
 on 
 
 that is, 3a? 30 or #= = 10. 
 
 3 
 
 3. William and John have 48 quills between them, and 
 John has twice as many as William : How many has each ? 
 
 Let the number of quills which William has be denoted by 
 ./" : then, since John has twice as many, his will be denoted 
 by 2#, and the quills possessed by both of them, by x-{-2x f 
 Hence, we shall have 
 
 t hat is, 3^=48 or #==16. 
 
 O 
 
 [Tence, William has 16 quills, and John 32. 
 
 4. What number is that which added to twice itself, will 
 jive a sum equal to 60 ? 
 
 Let the number sought be denoted by #, then twice the 
 number will be denoted by 2#, and we shall have 
 
 fin 
 
 that is, 3#=60 or a=_=20; 
 
 o 
 
 and we see that 20 added to twice itself will give 60. 
 
 QUEST. 2. In question second, what is the value of one of the parts? 
 3- What in question 3rd 1 4. How do you state question 4th 1 
 
INTRODUCTION. 5 
 
 5. John says to Charles, " give me your marbles and I shall 
 nave three times as many as I have now." " No," says 
 Charles, but give me yours, and I shall have just 51." How 
 many had each ? 
 
 Let the number of marbles which John has be denoted by 
 x : then, 2x will denote the number which Charles has, and 
 since they have 51 in all, we write 
 
 that is, 3x=5l or #=-=17. 
 
 o 
 
 6. What number is that which added to twice itself will 
 give a sum equal to 75 ? 
 
 Let the number be denoted by x : then, twice the number 
 will be expressed by 2#, and 
 
 that is, 3#=75 and ar==25. 
 
 o 
 
 7. What number added to twice itself will give a sum 
 equal to 90 ? 
 
 8. What number added to twice itself will give a sum 
 equal to 57 ? 
 
 9. What number added to twice itself will give a sum 
 equal to 39 ? 
 
 10. What number added to twice itself will give a sum 
 equal to 21 ? _ 
 
 LESSON III. 
 
 1. If James and John together have 24 quills, and John 
 has three times as many as James, how many will each have ? 
 
 QUEST. 5- How do you state question 5th 1 ' 6- Explain the 6th ques- 
 tion! % Also the 7th. 8. What is the required number in the 8th 1 
 9. What in the 9th f 1O. What in the 10th 1 
 
 2 
 
5 ELEMENTARY ALGEBRA. 
 
 It is plain that if we suppose the twenty-four quills to be 
 divided in four equal parts, that James will have one of the 
 parts, and John three. 
 
 Let us now designate by x the number of quills which 
 James has : then 3x will denote the number of quills which 
 John has, and we shall have 
 
 24 
 that is, 4#i=24 and a?= =6. 
 
 2. What number is that which added to three times itself 
 will give a sum equal to 48 ? 
 
 If we denote the number by a?, we shall have 
 
 that is, 4#=48 and #= = 12. 
 
 3. John and Charles have 60 apples between them, and 
 Charles has three times as many as John : How many has 
 each ? 
 
 If we suppose the number of apples to be divided into four 
 equal parts, it is evident that John will have one of those 
 parts, and Charles three. 
 
 Let x= the apples which John has : then 3x will stand for 
 the apples which Charles has, and we shall have 
 
 60 
 that is, 4#=60 and #==15. 
 
 Hence, John will have 15 and Charles 45. 
 
 QUEST. 1. If the 24 quills be divided into four equal parts, how 
 many parts will John have 1 How many will James have ] What is 
 each part equal to 1 . If three times a number be added to the number, 
 how many times will the number be taken 1 If 4.r is equal to 48, what 
 is the value of x 1 3. Explain the third question ! If 4.r is equal to 
 60, how do you find the value of a: 1 
 
INTRODUCTION. 
 
 4. What number is that which being added to three times 
 itself will give a sum equal to 100 ? 
 Let the number be denoted by x : then 
 
 inn 
 
 that is, 4c=100 and a?=_=25. 
 
 4 
 
 5. What number is that which if added to four times itself, 
 the sum will be equal to 60 ? 
 
 Let a? denote the number. Then, 
 
 that is, 5z=60 and *==12. 
 
 
 
 6. What number is that which being multiplied by 3, and the 
 product added to twice the number will give a sum equal to 75 ? 
 
 Let the number be denoted by x. 
 Then, 3x= the product of the number by 3 ; 
 and 2x twice the number; 
 and 3x+2x=5x=75- y 
 
 and a?= = 15, the required number. 
 
 5 
 
 7. What number is that which being added to three times 
 itself will give a sum equal to 140 ? 
 
 8. What number is that which being multiplied by 5, and 
 the product added to the number, will give a sum equal to 240 ? 
 
 9. What number is that which being multiplied by 2, and 
 then by 3, and the products added together, will give a sum 
 equal to 125 ? 
 
 QUEST. 5. If a number be added to four times itself, how many times 
 will the number be taken ! 6. If x stands for any number, what will 
 stand for three times that number 1 What for twice the number] T. 
 Explain the 7th question 1 How do you state it 1 What is 4x equal to 1 
 Why 1 How then do you find x 1 8. How do you state the 8th question ? 
 What is Qx equal to 1 How then do you find xl 9. If x denotes a number, 
 what will stand for twice the number 1 What for three times the number ? 
 
ELEMENTARY ALGEBRA. 
 
 LESSON IV. 
 
 1. John and Charles together have 80 apples, and Charles 
 has four times as many as John : How many has each ? 
 
 If we suppose the 80 apples to be divided into 5 equal 
 parts, it is evident that John will have one of the parts, and 
 Charles four. 
 
 Let x stand for the apples which John has : then 4# wilJ 
 stand for the apples which Charles has ; and 
 
 on 
 
 that is, 5x=SO and x==16. 
 
 5 
 
 2. What number added to four times itself will give a sum 
 equal to 90 ? 
 
 3. What number added to five times itself will give a sum 
 equal to 120 ? 
 
 4. What number added to six times itself will give a sum 
 equal to 245 ? 
 
 5. What number added to seven times itself will give a 
 sum equal to 360 ? 
 
 N. B. The questions in the preceding Lessons will give 
 some idea of the questions to which Algebra may be applied. 
 
 QUEST. 1. If x stands for John's apples, what will denote Charles' 1 
 What will stand for the apples which they both have 1 If nx is equal to 
 80, what will x be equal to 1 2. If a number be added to 4 times itself, how 
 many times will the number be taken 1 If 5 times a number is equal to 
 90, what is the value of the number] 3- Explain example 3rd. 4. 
 Explain question 4th 1 What does x stand for ? 5. Explain the 5th 
 question ? 
 
ELEMENTARY 
 
 ALGEBRA. 
 
 CHAPTER I. 
 Preliminary Definitions and Remarks. * 
 
 1. QUANTITY is a general term applied to every thing 
 which can be estimated or measured. 
 
 2. MATHEMATICS is the science of quantity. 
 
 3. ALGEBRA is that branch of mathematics in which the 
 quantities considered are represented by letters, and the ope- 
 rations to be performed upon them are indicated by signs. 
 These letters and signs are. called symbols. 
 
 4. The sign + , is called plus ; and indicates the addition 
 of two or more quantities. Thus, 9+5, is read, 9 plus 5, 
 or 9 augmented by 5. 
 
 If we represent the number nine, by the letter a, and 
 the number 5 by the letter &, we shall have a+b, which is 
 read, a plus b ; and denotes that the number represented by 
 a is to be added to the number represented by b. 
 
 5. The sign , is called minus ; and indicates that one 
 
 QUEST. 1,. What is quantity 1 2. What is Mathematics 1 3. What 
 is Algebra 1 What are these letters and signs called 1 4. What does the 
 sign plus indicate 1 5. What does the sign minus indicate 7 
 2* 
 
10 ELEMENTARY ALGEBRA. 
 
 quantity is to be subtracted from another. Thus, 9 5 is 
 read, 9 minus 5, or 9 diminished by 5. 
 
 In like manner, a 6, is read, a minus 6, or a diminished 
 by b. 
 
 6. The sign x , is called the sign of multiplication ; and 
 when placed between two quantities, it denotes that they 
 are to be multiplied together. The multiplication of two 
 quantities is also frequently indicated by simply placing a 
 point between them. Thus, 36 x 25, or 36.25, is read, 36 
 multiplied by 25, or the product 36 by 25. 
 
 7. The multiplication of quantities, which are represented 
 by letters, is indicated by simply writing them one after the 
 other, without interposing any sign. 
 
 Thus, ab signifies the same thing as a x b, or as a.b ; 
 and abc the same as a x b x c, or as a.b.c. Thus, if we 
 suppose o = 36, and 6=25, we have 
 
 Again, if we suppose a =2, 6=3 and c=4, we have 
 abc=2x 3x4=24. 
 
 It is most convenient to arrange the letters of a product 
 in alphabetical order. 
 
 8. In the product of several letters, as abc, the single let- 
 ters, a, b, and c, are called factors of the product. Thus, 
 in the product 06, there are two factors, a and b ; in the 
 product obey there are three, a, 6, and c. 
 
 QUEST. 6. What is the sign of multiplication 1 "What does the sign 
 of multiplication indicate 1 In how many ways may multiplication be 
 expressed 1 7. If letters only are used, how may their multiplication be 
 expressed? 8. In the product of several letters, what is each letter 
 called 1 How many farror* in uh ?--Tn abr. ? In ahrd J In abcdfJ 
 
DEFINITION OF TERMS. 11 
 
 9. There are three signs used to denote division. Thus 
 a-r- b denotes that a is to be divided by b. 
 j- denotes that a is to be divided by b. 
 a\b denotes that a is to be divided by b. 
 
 1 0. The sign , is called the sign of equality, and is 
 read, is equal to. When placed between two quantities, it 
 denotes that they are equal to each other. Thus, 9 5=4 : 
 that is, 9 minus 5 is equal to 4 : Also, a-\-bc, denotes that 
 the sum of the quantities a and b is equal to c. 
 
 If we suppose a = 10, and b = 5, we have 
 
 a+b = c, and 10 + 5=c = 15. 
 
 1 1. The sign >, is called the sign of inequality, and is 
 used to express that one quantity is greater or less than 
 another. 
 
 Thus, a > b is read, a greater than b ; and a < b is 
 read, a less than b ; that is, the opening of the sign is turned 
 towards the greater quantity. Thus, if a =9, and b=4, we 
 write, 9>4. 
 
 12. If a quantity is added to itself several times as 
 c+a-f a+a-fa+a, we generally write it but once, and 
 then place a number before it to show how many times it 
 is taken. Thus, 
 
 51 fc 4 H 
 
 QUEST. 9. How many signs are used in division "? What are they ? 
 10. What is the sign equality 1 When placed between two quantities, 
 what does it indicate 1 11. For what is the sign of inequality usedl 
 Which quantity is placed on the side of the opening 1 12. What is a co- 
 efficient 1 How many times is ab taken in the expression ab 1 In Bab 1 
 In 4ab 1 In bob 1 In Gab 1 If no co-efficient is written, what co efficient 
 is undorstood ' 
 
12 ELEMENTARY ALGEBRA. 
 
 The number 5 is called the co-efficient of a, and denotes 
 that a is taken 5 times. 
 
 When the co-efficient is 1 it is generally omitted. Thus, 
 a and la are the same, each being equal to a, or to one a. 
 
 13. If a quantity be multiplied continually by itself, as 
 axaXaXaXa, we generally express the product by writing 
 the letter once, and placing a number to the right of, and a 
 little above it : thus, 
 
 aXaXaXaXa=a 5 . 
 
 The number 5 is called the exponent of a, and denotes 
 the number of times which a enters into the product as a 
 factor. For example, if we have a 3 , and suppose a =3, 
 we write, 
 
 If a=4, 3 =4 3 
 
 and for a=5, a 3 =5 3 = 5x5 X 5 = 125. 
 
 If the exponent is 1 it is generally omitted. Thus, a 1 is 
 the same as a, each expressing a to the first power. 
 
 1 4. The power of a quantity is the product which results 
 from multiplying the quantity by itself. Thus, in the example 
 a 3_ 4 3_4 X 4 X 4_ 64j 
 
 64 is the third power of 4, and the exponent 3 shows the 
 degree of the power. 
 
 15. The sign >/ , is called the radical sign, and when 
 
 QUEST. 13. What does the exponent of a letter show? How many 
 times is a a factor in 02 ] In s ] In a4 1 In afi ? If no exponent is 
 written, what exponent is understood 1 14. What is the power of a 
 quantity 1 What is the third power of 2 1 Express the 4th power of a. 
 15. Express the square root of a quantity. Also the cube root. Also 
 the 4th root. 
 
DEFINITION OF TERMS. 13 
 
 prefixed to a quantity, indicates that its root is to be ex- 
 tracted Thus, 
 
 y/o~or simply -/^"denotes the square root of a. 
 ^/a denotes the cube root of a. 
 -y/a" denotes the fourth root of a. 
 
 The number placed over the radical sign is called the in- 
 dex of the root. Thus, 2 is the index of the square root, 3 
 of the cube root, 4 of the fourth root, &c. 
 
 If we suppose a = 64, we have 
 
 16. Every quantity written in algebraic language, that 
 is, with the aid of letters and signs, is called an algebraic 
 quantity, or the algebraic expression of a quantity. Thus, 
 
 o f is the algebraic expression of three times 
 
 I the number a ; 
 
 2 ( is the algebraic expression of five times 
 < the square of a ; 
 
 ^ is the algebraic expression of seven times 
 7a 3 b 2 < the product of the cube of a by the square 
 
 * of 6; 
 
 _ , ( is the algebraic expression of the difference 
 \ between three times a and five times b ; 
 
 {is the algebraic expression of twice the 
 square of a, diminished by three times 
 the product of a by b, augmented by four 
 times the square of b. 
 
 1. Write three times the square of a multiplied by the 
 cube of b. Ans. 
 
 QUEST. 16. What is an algebraic quantity 1 Is 5ab an 
 quantity 1 Is 9a ! Is 4y 1 Is 3b xl 
 
ELEMENTARY ALGEBRA. 
 
 2. Write nine times the cube of a multiplied by b, dimin- 
 ished by the square of c multiplied by d. Ans. 9a 3 b c 2 d. 
 
 3. If a=2, b = 3, and c = 5, what will be the value of 
 3a 2 multiplied by b 2 diminished by a multiplied by b multi- 
 Dlied by c. We have 
 
 3a 2 b 2 abc = 3 X 2 2 X 3 2 2 X 3 X 5 = 78. 
 
 4. If a = 4, b = 6, c=7, d=8, what is the value of 
 9a 2 +fo ad\ Ans. 154. 
 
 5. If a = 7, b = 3, c = 7, d=l, what is the value of 
 6ad+3b 2 c4d 2 i Ans. 227 
 
 6. If a=5, b = 6, c=6, d=5, what is the value of 
 9abc 8<zd-|-46c? Ans. 1564. 
 
 7. Write ten times the square of a into the cube of b into 
 c square into d 3 . 
 
 1 7 . When an algebraic quantity is not connected with 
 any other, by the sign of addition or subtraction, it is called 
 a monomial, or a quantity composed of a single term, or sim- 
 ply, a term. Thus, 
 
 3a, 5a 2 , 7a 3 b 2 , 
 
 are monomials, or single terms. 
 
 18. An algebraic expression composed of two or more 
 parts, separated by the sign + or , is called a polynomial, 
 or quantity involving two or more terms. For example, 
 
 3a 5b and 2a 2 3cb + 4b 2 
 are polynomials. 
 
 19. A polynomial composed of two terms, is called a bi- 
 nomial ; and a polynomial of three terms is called a trinomial. 
 
 QUEST. 17. What is a monomial"? Is 3ai a monomial 1 18. What 
 is a polynomial? Is 3a b a polynomial! 19. What is a binomial! 
 What is a trinomial 1 
 
DEFINITION OF TERMS. 15 
 
 20. Eacl of the literal factors which compose a term is 
 called a dimension of this term : and the degree of a term is 
 the number of these factors or dimensions. Thus, 
 
 S is a term of one dimension, or of the first 
 3a < 
 
 f degree. 
 
 _ , ( is a term of two dimensions, or of the 
 ( second degree. 
 
 is of six dimensions, or of the sixth 
 degree. 
 
 2 1 . A polynomial is said to be homogeneous, when all 
 its terms are of the same degree. The polynomial 
 
 3 a 2Z-j-c is of the first degree and homogeneous. 
 
 4ab-\-b z is of the second degree and homogeneous. 
 
 5 a 2 c 4c 3 -|-2c 2 d is of the third degree and homogeneous. 
 
 Sa 3 -\-4ab-\-c is not homogeneous. 
 
 22. A vineulum or bar , or a parenthesis ( ), 
 
 is used to express that all the terms of a polynomial are to 
 be considered together. Thus, 
 
 a+b+cxb, or (o+5+c)x&, 
 
 denotes that the trinomial a+b-\-c is to be multiplied by b ; 
 also, a+b+cxc+d+f, or (a+6+c) x (c+d+/), 
 
 denotes that the trinomial a+b+c is to be multiplied by 
 the trinomial c-\-d-\-f. 
 
 When the parenthesis is used, the sign of multiplication 
 is usually omitted. Thus, 
 
 (a-\-b+c)xb is the same as (a-\-b + c)b. 
 
 QWEST. 2O. What is the dimension of a term? What is the degree 
 of a term 1 How many factors in 3aic 1 Which are they! What 
 is its degree 1 21. When is a polynomial homogeneous 1 Is the polyno- 
 mial 2a3i4-3o2&2 homogeneous 1 Is 2a*6 is] 22. For what, is tb 
 vineulum or bar used 1 Can you express the same with the parenthesis 7 
 
1 6 ELEMENTARY ALGEBRA 
 
 23. The terms of a polynomial which are composed of 
 the same letters, the same letters in each being affected 
 with like exponents, are called similar term?. 
 
 Thus, in the polynomial 
 
 the terms 7ab, and Sab, are similar : and so also are the 
 terms 4a 3 b 2 and 50 3 Z> 2 , the letters and exponents in both 
 being the same. But in the binomial 8a 2 b -\-7ab 2 , the 
 terms are not similar ; for, although they are composed of 
 the same letters, yet the same letters are not affected with 
 like exponents. 
 
 24. When an algebraic expression contains similar 
 terms, it may be reduced to a simpler form. 
 
 1. Take the expression 3ab+2ab, which is evidently 
 equal to 5ab. 
 
 2. Reduce the expression Sac -\-9ac -\-2ac to its simplest 
 form. Ans. I4ac. 
 
 3. Reduce the expression abc-\-4abc-\-5abc to its s'm- 
 plest form. 
 
 In adding similar terms together we abc 
 
 take the sum of the coefficients and 4abr 
 
 annex the literal part. The first term, 5abc 
 
 abc, has a coefficient 1 understood, lOabr. 
 (Art. 12). 
 
 25. Of the different terms which compose a polynomial, 
 some are preceded by the sign -f- , and the others by the 
 sign . The first are called additive terms, the others 
 subtractive terms. 
 
 QUEST. 23. What are similar terms of a polynomial"? Are 
 and 6a2i2 similar 1 Are 2a2#2 and 2a3i 2 1 24. If the terms are positive 
 and similar, may they be reduced to a simpler form 7 In what wa.v v 
 
DEFINITION OF TERMS. 11 
 
 The first term of a polynomial is commonly not preceded 
 by any sign, but then it is understood to be affected with the 
 sign +. 
 
 1. John has 20 apples and gives 5 to William: how 
 many has he left ? 
 
 Now, let us represent the number of apples which John 
 has by , and the number given away by b : the number he 
 would have left would then be represented by a b. 
 
 2. A merchant goes into trade with a certain sum of 
 money, say a dollars ; at the end of a certain time he has 
 gained b dollars : how much will he then have ? 
 
 Ans. a+b dollars. 
 
 If instead of gaining he had lost b dollars, how much 
 would he have had ? Ans. a b dollars. 
 
 Now, if the losses exceed the amount with which he 
 began business, that is, if b were greater than a, we must 
 prefix the minus sign to the remainder to show that the 
 quantity to be subtracted was the greatest. 
 
 Thus, if he commenced business with $2000, and lost 
 $3000, the true difference would be 1000: that is, the 
 subtractive quantity exceeds the additive by $1000. 
 
 3. Let a merchant call the debts due him additive, and 
 the debts he owes subtractive. Now, if he has due him 
 $600 from one man, $800 from another, $300 from another, 
 and owes $500 to one, $200 to a second, and $50 to a 
 third, how will the account stand ? Ans. $950 due him. 
 
 4. Reduce to its simplest form the expression 
 
 QUEST. 25. What are the terms called which are preceded by the 
 sign + ? What are the terms called which are preceded by the sign . 
 If no sign is prefixed to a term, what sign is understood ? If some of the 
 terms are additive and some subtractive. rnay they be reduced if similar ? 
 Give the rule for reducing them. Does the reduction affeet the expo- 
 nents, or only the coefficients ? 
 
 3 
 
18 ELEMENTARY ALGEBRA. 
 
 Additive terms. Subtractive terms. 
 
 -f 
 -f 
 
 
 _ 
 
 Sum +12a^ Sum IQatb. 
 
 But, , I2a z b Wa 2 b = ( . 
 
 Hence, for the reduction of the similar terms of a polyno- 
 mial we have the following 
 
 RULE. 
 
 I. Add together the coefficients of all the additive terms, 
 and annex to their sum the literal part ; and form a single 
 subtractive term in the same manner. 
 
 II. Then, subtract the less coefficient from the gr eater * 
 and to the remainder prefix the sign of the greater co- 
 efficient, to which annex the literal part. 
 
 REMARK. It should be observed that the reduction afle'cts 
 only coefficients, and not the exponents. 
 
 EXAMPLES. 
 I. Reduce to its simplest form the polynomial 
 
 Find the sum of the additive and subtractive terms sepa- 
 rately, and take their difference : thus, 
 
 Additive terms. Subtractive terms. 
 
 8a 3 bc* 
 Sum 
 Sum -f-19a 3 6c 2 
 
 IJence, the given polynomial reduces to 
 
ADDITION. 
 
 2. Reduce the polynomial 4a 2 6 Sa z b 9a 2 6+lla 2 6 to 
 its simplest form. Ans. 2a 2 b. 
 
 3. Reduce the polynomial 7.abc 2 abc 2 7abc 2 + 8abc 2 
 f 6ac 2 to its simplest form. Ans. I3abc 2 . 
 
 4. Reduce the polynomial 9cb 3 8ac z +15cb 3 +8ca + 9ac* 
 24cb 3 to its simplest form. Ans. ac 2 -\-8ca. 
 
 The reduction of similar terms is an operation peculiar to 
 algebra. Such reductions are constantly made in Algebraic 
 Addition, Subtraction, Multiplication, and Division. 
 
 ADDITION. 
 
 26. Addition in Algebra, consists in finding the simplest 
 equivalent expression for several algebraic quantities. Such 
 equivalent expression is called their sum. 
 
 1. What is the sum of 
 
 3ax+2ab and 2ax+ab. 
 
 3ax+2ab 
 
 We reduce the terms as in Art. 25, 2ax+ ab 
 and find for the sum ax+3ab 
 
 ( 3a 
 
 2. Let it be required to add together \ ^ 
 
 the expressions : );' 
 
 The result is 3<z+54-2< 
 
 an expression which cannot be reduced to a more simple 
 form. 
 
 QUEST. 26. What is addition in Algebra ? What is such simplest 
 and equivalent expression called ? 
 
20 ELEMENTARY ALGEBRA. 
 
 Again, add together the monomials J 2a 2 b 3 
 
 The result after reducing (Art. 25), is . . I3a 2 b 3 
 
 r 2a 2 4ab 
 3. Let it be required to find the sum \ 
 
 of the expressions , 
 
 2ab-5b 2 
 
 Their sum, after reducing (Art. 25) is . 5a 2 Sab 4b 2 
 
 27. As a course of reasoning similar to the above 
 would apply to all polynomials, we deduce for the addition 
 of algebraic quantities the following general 
 
 RULE. 
 
 I. Write down the quantities to be added so that the similar 
 terms shall fall under each other, and give to each term its 
 proper sign. 
 
 II. Reduce the similar terms, and annex to the results the 
 terms which cannot be reduced, giving to each term its respec- 
 tive sign. 
 
 EXAMPLES. 
 
 1. What is the sum of Sax, 5ax, 2ax and I3ax. ? 
 
 Ans. I9ax. 
 
 2. What is the sum of 4ab+8ac and 2a 7ac+d. ? 
 
 Ans. Gab -\-ac-\-d. 
 
 3. Add together the polynomials, 
 
 3a 2 2b z 4ab, 5a 2 b 2 +2ab, and Sab 3c 2 25 2 . 
 The term 3a 2 being similar to 
 5a 2 , we write 8a z for the result 
 
 - , 
 
 of the reduction of these two terms, 
 
 ,. , , . 
 
 at the same time slightly crossing 
 
 ^ . -i /. 
 
 them, as in the first term. 
 
 _ 2 , , 2 
 
 -- O 
 
 , 2 _ 
 
 2b* 3c z 
 
 =-5 - 
 
 o<2 -\- CLO OO oC 
 
 QUEST. 27. Give the rule for the addition of Algebraic quantities. 
 
ADDITION. 21 
 
 Passing then to the term 4ab, which is similar to -f 2ab 
 and +3ab, the three reduce to -\~ab, which is placed after 
 8a 2 , and the terms crossed like the first term Passing 
 then to the terms involving b 2 , we find their sum to be 
 5& 2 , after which we write 3c 2 . 
 
 The marks are drawn across the terms, that none of them 
 may be overlooked and omitted. v 
 
 (4) (5) (6) 
 
 a 5a 
 
 a 5a 5b 
 
 J2a lla 5a+5b 
 
 (9) (10) 
 
 7abc+9ax 8007+ 3b 
 
 3abc 3ax 5ax 9b 
 
 4abc-\-ax j.3ax6b 
 
 NOTE. If a=5, b=4, c=2, a?=l, what are the values 
 of the several sums above found. 
 
 (12) (13) (14) 
 
 9a-\-f 6ax 8ac 3af-\- g -\-m 
 
 6a-}-g 7ax 9ac ag 3af m 
 
 2af . ax+\7ac ab 
 
 a-\-g ab+4g 
 
 (15) (16) 
 
 7a?+3a6+ 3c 
 
 -3x3ab 5c 
 
 5x 9ab 9c 4x 2 4- 4acx 
 
 9x 9ab lie 3# 2 -f- 
 
 (17) (18) 
 
 22A 3c ' 
 
 2ah 2 6a 3 b*+ ax 3 - 
 3* 
 
22 ELEMENTARY ALGEBRA. 
 
 (19)'' (20) 
 
 7x 9y+5*+3 g 8a+ b 
 
 a 3y 8 # 2a 6+ c 
 
 a?+ y-3^+1 + 7^ 3a+ fc 
 
 2x+6y+3z 1 ^ 66 3c-f3d 
 
 5a +7c 8<f 
 
 4a?H-3y+0 +4 + 5^ 2a 5b+5c 3d 
 
 21. Add together 6+3c ^ 115e+6/-5^, 36-2c 
 3rf__ C 4_27/, 5c 8rf+3/ 7^, 7i 6c+17<Z+9e 5/ 
 , 3J 5d 2e+6/ 
 
 22. Add together the polynomials 7a 2 b 3abc 8b 2 c 9c 3 
 
 and 4a 2 5 8c 3 -f95 2 c 
 35 2 c 14c 3 -f 2cd 2 3d 3 . 
 
 23. What is the sum of 5a 2 5c+65a: 4af, 3a 2 bc6bx 
 af, af+9bx+2a 2 bc, +6af8bx+6a?bc. 
 
 Ans. Watbc+bx+lSaf. 
 
 24. What is the sum of 2 n 2 +3a 3 m-f b, 6a 2 w 2 6a 3 m b, 
 
 25. What is the sum of 4a 3 b 2 c16a*x9ax 3 d, +6a 3 b 2 c 
 Gaxtd+Wtfx, -\-l6ax 3 da*x9a 3 b 2 c. 
 
 Ans. a 3 b z c-\-ax 3 d. 
 
 26. What is the sum of 7^+3^4-4^2^ +3# 
 
 3b-i 2b. Ans. 0. 
 
 27. What is the sum of ab-\-3xy m n, 6xy 3m 
 + lln+cd, +3iry + 4m Wn+fg. Ans. ab + cd+fg. 
 
 28. What is the sum of 4xy+n -\-6ax-\-9am, 6xy+6n 
 
 6ax8am, 2xy7n+axam. Ans. +ax. 
 
SUBTRACTION. 23 
 
 29. Add the polynomials IQaWb I2a 3 cb, 
 
 Wax, 2a 2 x 3 bl2a*cb, and - I8a 2 x 3 b 12a 3 cb+9ax. 
 
 Ans. 4a?x 3 b22a 3 cbax. 
 
 30. Add together 3a-f-+c, 5a+2b + 3ac, a+c+ac, 
 and 3a 9ac 8b. Ans. 6a 55+2c 5ac. 
 
 31. Add together . 5a?b + 6cx+9bc*, 7cx8a?b, and 
 
 15cx 9bc 2 +2a 2 b. Ans. a 2 b 2cx. 
 
 32. Add together 8a#+5a&-f 3a 2 5 2 c 2 , 18aa;-f6a 2 -f-10a& 
 and 10a# I5ab 6a 2 b z c 2 . Ans. 3a 2 6 2 c 2 -f 6a 2 . 
 
 33. Add together 3a 2 +5a 2 J 2 c 2 9a 3 #, 7a 2 8a 2 5 2 c 2 10a 3 
 and 
 
 SUBTRACTION. 
 
 28. Subtraction, in Algebra, consists in finding the sim- 
 plest expression for the difference between two algebraic 
 quantities. 
 
 Thus, the difference between 6a and 3<z is expressed by 
 
 60 3=3a; 
 and the difference between 7a?b and 3a 3 & by 
 
 In like manner the difference between 4a and 3b is 
 expressed by 4a 3b. 
 
 Hence, If the quantities are positive and similar, subtract 
 the coefficients, and to their difference annex the literal part. 
 If they are not similar, place the minus sign before the quantity 
 to be subtracted. 
 
 QUEST. 28. In what does subtraction in Algebra consist ? How do 
 you find this difference when the quantities are positive and similar ? 
 When they are not similar, how do you express the difference ? 
 
ELEMENTARY ALGEBRA. 
 
 From 
 
 take 
 
 Rem. 
 
 From 
 
 take 
 
 Rem. 
 
 From 
 
 take 
 
 Rem. 
 
 (1) 
 
 3ab 
 
 2ab 
 
 ab 
 
 (4) 
 
 (3) 
 
 9abc 
 Tabc 
 
 (6) 
 
 (9) 
 2am 
 
 ax 
 
 3ax8c 
 
 4abx 9ac 
 
 2am ax. 
 
 29. Let it be required to subtract from 4a 
 the binomial . 2b 3c 
 
 The difference may be put under the form 4a (2b 3c). 
 We must now remark that it is the difference between 2b 
 and 3c which is to be taken from 4a. 
 
 If then, we write 4a 2b, 
 
 we shall have taken away too much by the units in 3c ; 
 hence, 3c must be added to give the true remainder, .which 
 is 4a 26 + 3c. 
 
 To illustrate this example by figures, suppose a =5, 
 b=5 t and c=3. 
 
 We shall then have 4a=20 
 
 and . 2b 3c -109 1 
 
 which may be written . 4a (2b 3c)20 1"= 19. 
 
 QUEST. 29. If 2i 3c is to be taken from 4a, what is proposed to 
 be done ? If you subtract 26 from 4a, have you taken too much ? /low 
 then must you supply the deficiency ? 
 
SUBTRACTION. 25 
 
 Here it is required to subtract 1 from 20. If, then, we 
 subtract 2b=lO, from 4a = 20, it is plain that we shall 
 have taken too much by 3c=9, which must therefore be 
 added to give the true remainder. 
 
 3O. Hence, for the subtraction of algebraic quantities 
 re have the following general 
 
 RULE. 
 
 I. Write the quantity to be subtracted under that from which 
 it is to be taken, placing the similar terms, if there are any, 
 under each other. 
 
 II. Change the signs of all the terms of the polynomial to 
 be subtracted, or conceive them to be changed, and then reduce 
 the polynomial result to its simplest form. 
 
 EXAMPLES. 
 
 From 
 Take 
 Rem. 
 
 From 
 Take 
 Rem. 
 
 From 
 Take 
 Rem. 
 
 (i) 
 
 6ac 5ab-{- c 2 
 3ac-\-3ab-\-7c 
 
 The same with 
 the signs of the 
 lower line chan- 
 
 (1) 
 
 6ac 5ab+ c 2 
 Sac 3ab 7c 
 
 3ac8ab+ c 2 7c. 
 
 1 Sac 8ab+ c z 7c. 
 
 (2) 
 6ax a-f S5 2 
 9ax x+ b 2 
 
 (3) 
 6yx 3x 2 +5b 
 ^yx-T-3 -{- a 
 
 3axa-\-x-\-2b 2 . 
 
 5yx 3 2 +3 + 5& a. 
 
 (4) 
 
 5 a 3_ 4a 2_|_ 3 2 C 
 
 2a 3 +3a 2 Sb 2 c 
 
 (M 
 
 4ab cd-\-3a 2 
 5ab4cd+3a 2 +5b 2 
 
 7a*7a 2 b+nb 2 c. 
 
 ab+3cd5b 2 . 
 
 
 
 QUKST. 30. Give the rule for the subtraction of algebraic quantities. 
 
26 ELEMENTARY ALGEBRA. 
 
 6. From 6am-\-y take Sam x. Ans. 3am-\-x+y. 
 
 7. From Sax take 3>ax y. Ans. -j-y 
 
 8. From 7aWx* take I8a?b z +x z . 
 
 Ans. llaW2x 2 . 
 
 9. From 7f+3m8x take 6/ 5m 2x+3d+8. 
 
 Ans. f+8m6x3d8 
 
 10. From a 5b+7cd take 45 c + 2d+2A:. 
 
 Ans. a9b + 8c3d2k 
 
 11. From . . 3a+b 8c+7e 5/+3A 7x 13y take 
 *+2a 9c+8e 7a?-f-7/ y 3Z A. 
 
 ^.n^. 5a-f-f c e 12/+3A 12y+3/. 
 
 12. From a+b take a b. Ans. 2b. 
 
 13. From 2x 4a 2^+5 take 8 5&-fa-f 6#. 
 
 ^.n^. 4a? 5a+35 3 
 
 14. From 3a+J+c rf10 take c-f 2a d. 
 
 ^.n^. a-\-b 10 
 
 15. From 3a+5+c d 10 take 5 19+3a. 
 
 Aw5. c 
 
 16. From 2a&+5 2 4c+5c b take 3a 2 
 
 ^.ns. 2ai 3a 2 
 
 17. From a*+3b z c+ab z abc take b 3 +ab 2 abc. 
 
 Ans. a 3 +3 2 c R 
 
 18. From 12a?+6a 4&+40 take 4b 3a+4x + 6d 10. 
 
 Ans. 8;r-f9a 8b 6c?+50. 
 
 19. From 2x 3a+454-6c 50 take 9a+x+6b 6c 40. 
 
 Ans. x 12a 2-f-12c 10. 
 
 20. From 6a4i 12c+l2x take 2o; 8a+4* 6c. 
 
 ^.n^. 14a 8b 6c+10a?. 
 
 21. From 85c 12b 3 a+6cx 7xy take 7c xy13b 3 a. 
 
 A ns. 8abc -f- b*a ex 6xy 
 
SUBTRACTION. 27 
 
 31. By the rule for subtraction, polynomials may be 
 subjected to certain transformations. 
 
 For example, . . 6a 2 Sab -f 25 2 2bc, 
 
 becomes 6a 2 (3ab 2 2 
 
 En like manner . . 7a 3 8a z b 
 
 becomes .... 7a 3 (8a 2 6 + 46 2 c 6& 2 ), 
 
 or, again, .... 7a 3 8a 2 (46 2 c G6 2 ). 
 
 Also, 8a 3 7 3 + c d, 
 
 becomes .... 8<z 3 (7Z> 2 c +d). 
 
 Also, 96 3 a + 3a 2 d, 
 
 becomes .... 9b 3 (a 3a 2 -f d). 
 
 32. REMARK. From what has been shown in addition 
 and subtraction, we deduce the following principles. 
 
 1st. In algebra, the words add and sum do not always, as 
 m arithmetic, convey the idea of augmentation. For, if to a 
 we add 6, we have a &, which is, properly speaking, a 
 difference between the number of units expressed by , and 
 the number of units expressed by b. Consequently, this 
 result is numerically less than a. To distinguish this sum 
 from in arithmetical sum, it is called the algebraic sum. p27 
 
 Thus, the polynomial 2a 2 3a z b-\-3b 2 c ' is an algebraic 
 sum, so long as it is considered as the result of the union 
 
 QUEST. 31. How may you change the /orm of a polynomial ? 32. In 
 algebra do the words add and sum convey the same idea a"s in arithme- 
 tic ? What is the algebraic sum of 9 and/ 4? Of 8 and 2? 
 May an algebraic sum ever be negative ? What is the sum of 4 and 
 8"? Do the words subtraction and difference in algebra always con- 
 vey the idea of diminution ? What is the algebraic difference between 
 8 and 4? Between a and bl 
 
28 ELEMENTARY ALGEBRA 
 
 of the monomials 2a 2 , 3a 2 b, -f-35 2 c, with their respec 
 tive signs ; and, in its proper acceptation, it is the arithmeti- 
 cal difference between the sum of jthe units contained in the 
 additive terms, and the sum of the units contained in the 
 subtractive terms. 
 
 It follows from this, that an algebraic sum may, in the 
 numerical applications, be reduced to a negative number, 01 
 a number affected with the sign . 
 
 2nd. The words subtraction and difference do not always 
 convey the idea of diminution ; for, the numerical differenca 
 between -{-a and b being a-{-b, exceeds a. This result 
 is an algebraic difference, and can be put under the form of 
 
 a _ (b)= a+b 
 
 MULTIPLICATION. 
 
 33. If a man earns a dollars in one day, how much 
 will he earn in 6 days ? Here it is simply required to re- 
 peat the number a, 6 times, which gives 6a for the amount 
 earned. 
 
 1. What will ten yards of cloth cost at c dollars per yard? 
 
 Ans. lOc dollars. 
 
 2. What will d hats cost at 9 dollars per hat 1 
 
 Ans. 9d dollars. 
 
 3. What will b cravats cost at 40 cents each ? 
 
 Ans. 40b cents. 
 
 4. What will b pair of gloves cost at a cents a pair ? 
 
 QUEST. 33. What is the object of multiplication in algebra ? It a 
 man earns a dollars in one day, how much will ho. earn in 4 days'? In 
 5 days 1 In 6 days * 
 
MULTIPLICATION. 29 
 
 Here it is plain that the cost will be found by repeating b 
 as many times as there are units in a : Hence, the cost is 
 ab cents. Ans. ab cents. 
 
 NOTE. If we suppose a=6, c=4, and d=3 t wha-t would 
 be the numerical values of the above answers ? 
 
 5. If a man's income is 3a dollars a week, how much 
 will it be in 4b weeks. Here we must repeat 3a dollars as 
 many times as there are units in 4b weeks ; hence, the pro- 
 duct is equal to 
 
 If we suppose a=4= and b = 3 the product will be equal 
 to 144. 
 
 34. REMARK. It is plain that the product I2ab will not 
 be altered by changing the arrangement of the factors ; that 
 is, I2ab is the same as a6x!2, or as fozx!2, or as 
 axl2xb (See Arithmetic, 22). 
 
 35. Let us now multiply 3a 2 b 2 by 2a 2 b, which may be 
 placed under the form 
 
 3a 2 b 2 X 2a 2 b=3 X2aaaabbb ; 
 
 in which a is a factor four times, and b a factor three times : 
 hence (Art. 13). 
 
 in which, we multiply the co-efficients together and add the 
 exponents of the like letters. 
 
 QUEST. 34. Will a product be altered by changing the arrangement 
 if the factors 1 Is 3ai the same as 3ba 1 Is it the same as a X 3i t 
 As 5x3a 1 35. In multiplying monomials what do you do with the co- 
 efficients 1 What do you do with the exponents of the common letters ? 
 If a letter is found in one factor and not in the other, what do you do 1 
 
 4 
 
80 ELEMENTARY ALGEBRA. 
 
 Hence, for the multiplication of monomials, we have the 
 following 
 
 RULE. 
 
 I. Multiply the coefficients together for a new coefficient. 
 
 II. Write after this coefficient all the letters which enter 
 into the multiplicand and multiplier^ affecting each with 
 an exponent equal to the sum of its exponents in both fac- 
 
 tors. 
 
 
 
 EXAMPLES. 
 
 2. ZltfVcdx 8ab<?= 168oW</. 
 
 3. 4abcX7df = VSabcdf. 
 
 
 (4) 
 
 (5) 
 
 (6; 
 
 Multiply 
 
 3a 2 b 
 
 I2a 2 x 
 
 6a:y z 
 
 by 
 
 2a 2 b 
 
 I2x 2 y 
 
 ay 2 z 
 
 
 6a 4 b* 
 
 I44a 2 x 3 y 
 
 6axy 3 z z 
 
 (7) 
 
 (8) 
 
 (9) 
 
 
 
 LtX 
 
 87o; 2 y 
 
 2a?x 2 y 
 
 10. Multiply 5a 3 bW by 6c 5 x 6 . Ans. 
 
 11. Multiply 10 4 6 5 c 8 by 7acd. Ans. 7Qa 5 b 5 c*d. 
 
 12. Multiply 9a 3 bxy by 9a 3 bxy. Ans. 
 
 13. Multiply 36a 8 6 7 c 6 d 5 by 2Qab 2 c 3 d*. Ans. 
 
 14. Multiply 27axyz by 9a 2 b 2 c 2 d 2 xyz. 
 
 Ans. 
 
 15. Multiply 13 3 & 2 c by Sabxy. Ans \Q4a*b 3 cxy 
 
MULTIPLICATION. 31 
 
 16. Multiply 2Qcfb 5 cd by 12aVfy. dns. 
 
 17. Multiply l4aWy by 2Qa 5 c 2 x 2 y. 
 
 tins. 
 
 18. Multiply Scfby by 7a*bxy*. Ans. 56a 7 & 4 a*/ 9 . 
 
 19. Multiply 75axyz by 5a 5 bcdx 2 y\ 
 
 tins. 275a?bcdx s y s z. 
 
 20. Multiply Slayer 2 by a*bc*y?y. ./2/is. 459a 4 6cVy. 
 
 21. Multiply 2a s &y by ISabx. * .tfrcs. 
 
 22. Multiply 64aVx 4 yz by 
 
 23. Multiply 9a 2 #W 3 by 12aW. .flns. 
 
 24. Multiply 216a6W by 3a 3 6 2 c 5 . ^ns. 648aWd 8 . 
 
 25. Multiply 70a 8 b 7 c*d 2 fx by 1 
 
 36. We will now consider the most general case of two 
 polynomials. 
 
 Let a represent the sum of all the additive terms of the 
 multiplicand, and b the sum of the subtractive terms. Let c 
 denote the sum of the additive terms of the multiplier, and d 
 the sum of the subtractive terms. The multiplicand will 
 then be represented by a 6, and the multiplier by c d; and 
 it is required to take a b as many times as there are units in 
 c d. 
 
 Let us first take a b as many times a b 
 as there are units in c. c d 
 
 We begin by writing ac, which is aclc 
 too great by b taken c times; for it is _ ad-\-hd 
 
 only the difference between a and b acbcad+bd. 
 which is to be taken c times. Hence, 
 ac be is the product of a b by c. 
 
 But it was proposed to take a b only as many times as 
 there are units in the difference between c and-c/: hence the 
 
32 ELEMENTARY ALGEBRA. 
 
 last product ac be is too large by a b taken d times ; which 
 without regarding the sign of d, is ad bd. Changing the 
 signs of this last product, and subtracting it from that of a b 
 J)y c (Art. 3O), and we have 
 
 (a b) x (c d) = ac be ad+ bd. 
 ~ 37. Hence, we have the following rule for the signs. 
 
 When two terms of the multiplicand and multiplier are 
 affected with the same sign, the corresponding product is 
 affected with the sign +; and when they are affected with 
 contrary signs, the product is affected with the sign . 
 
 Therefore we say in algebraic language, that -f multiplied 
 by H- , or multiplied by , gives + ; multiplied by 
 + , or -f multiplied by , gives . 
 
 Hence, for the multiplication of polynomials we have the 
 following 
 
 RULE. 
 
 Multiply all the terms of the multiplicand by each term of 
 the multiplier, observing that like signs give plus in the pro- 
 duct, and unlike signs minus. Then reduce the polynomial 
 result to its simplest form. 
 
 EXAMPLES IN WHICH ALL THE TERMS ARE PLUS. 
 
 1 Multiply ...... 
 
 by ....... 20 + 56 
 
 The product, after reducing, . +l5a z b+2Qab 2 -\-5b 3 
 
 becomes ..... Go 3 -f 23a 2 b + 22ab 2 + 56 3 . 
 
 QUEST. 37. What does -f multiplied by -f give ? -f multiplied 
 by ? multiplied by -f ? multiplied by ? Give the rule for 
 the multiplication of polynomials. 
 
MULTIPLICATION. 33 
 
 2. Multiply z?+2ax+ a 2 by x+a. 
 
 Jlns. ar } +3a# 2 +3a 2 ;r-f-a s . 
 
 3. Multiply x 3 +y 3 by x+y. - *fl.ns. x 4 -\-xy 3 +x s y-i-y*. 
 
 4. Multiply 3ab*+6tf<* by 3a6 2 +3o 2 c 2 . 
 
 5. Multiply tftf + fd by a +b. 
 
 Ans. 
 
 6. Multiply 3ax*+9ab s +cd 5 by 6aV 
 
 7. Multiply 64aV+27a 2 #+9a& ly 8a 3 cJ 
 
 rfns. Sltofcd 
 
 8. Multiply aHSa^+or 2 by a + x. 
 
 Jlns. 
 
 9. Multiply (f + Scfx+Satf+x 3 by 
 
 10. Multiply a^+y 2 by x+y. tins. x s +xy 2 +x 2 y+y 3 . 
 
 11. Multiply x*-\-xy 6 -\-7ax by 
 
 12. Multiply a 3 +3a 2 6 + 36 2 + 6 3 by a +b. 
 
 Jlns. a*+ 
 
 13. Multiply x 3 -}-x 2 y+xif+y 3 by 
 
 14. Multiply a; 3 +2a; 2 + a; + 3 by 3a*+l. 
 Ans. 
 
 GENERAL EXAMPLES. 
 
 1. Multiply 20# Sab 
 
 by 3x b. 
 
 The product Gax 2 9abx 
 
 becomes after 2&r-f 3a5 2 
 
 reducing Gax 2 llabx+3ab 2 . 
 
34 ELEMENTARY. ALGEBRA. 
 
 2. Multiply a 4 2b 3 by a b. 
 
 Ans. a 5 2ab 3 
 
 3. Multiply x 2 3x 7 by x2. 
 
 Ans. x 3 5x 2 a?-J-14 
 
 4. Multiply 3a 2 5ab + 2b 2 by a 2 7ab. 
 
 Ans. 3a*26a 3 b+37a 2 b 2 Uab* 
 6. Multiply & 2 +& 4 +& 6 by 2 1. Ans. b 8 b 2 
 
 6. Multiply x*2x 3 y + 4x 2 y 2 8xy 3 +16y* by #+2y. 
 
 Arcs. rr 5 +32y 5 . 
 
 7. Multiply 4a: 2 2y by 2y. Ans, 8x*y 4y 2 . 
 
 8. Multiply 2x+4y by 2x 4y. Ans. 4x 2 l6y z . 
 
 9. Multiply x 3 -\-x 2 y-\-xy 2 -}-y 3 by # y. 
 
 -An^. a? 4 y 4 . 
 
 10. Multiply ff 2 + #y+y 2 by x 2 xy+y 2 . 
 
 Ans. # 4 -f-# 2 y 2 -|-y 4 . 
 
 11. Multiply 2a z 3ax+4x* by 5a 2 6^-2 2 . 
 
 Ans. 10a 4 27a 3 x+34a z x 2 I8ax 3 8x*. 
 
 12. Multiply 3x 2 2a?y45 by a: 2 +2a:y 3. 
 
 An^. 3* 4 +4a; 3 y 4a; 2 4a: 2 y 2 +16a:y 15. 
 
 13. Multiply 3tf 3 4-2a: 2 y 2 -f 3y 2 by 2x 3 3a; 2 y 2 
 
 ' C 
 
 14. Multiply Sax Gab c by 
 
 Ans. 1 6a 2 # 2 4a 2 bx 6a z b z -\- 6acx 7abc c 2 . 
 
 15. Multiply 3a 2 55 2 + 3c 2 by a 2 b 2 . 
 
 Ans. 3a 4 
 
 16. 3a 2 55d!+ c/ 
 
 8cf. 
 
 Pro. red. ^-15a 4 +37a 2 ^ 29a?cf 
 
MULTIPLICATION. 35 
 
 38. To finish with what has reference to algebraic mul- 
 tiplication, we will make known a few results of frequent 
 use in Algebra. 
 
 Let it be required to form the square or second power of 
 the binomial (a-f-Z>). We have, from known principles, 
 
 That is, the square of the sum of two quantities is equal to 
 the square of 'the first , plus twice the product of the first by the 
 second, plus the square of the second. 
 
 1. Form the square of 2a+3b. We have from the rule 
 
 (2a -f 36) 2 = 4a 2 + I2ab -f 95 2 . 
 
 2. (5ab+3ac) 2 = 25aW+ 30a*bc+ 9a 2 c 2 . 
 
 3. (5a 2 +8a 2 6) 2 =25a* + 80a 4 b 
 
 4. (6aa;-f9 2 a; 2 ) 2 = 36a 2 a; 2 +108a 3 a: 3 
 
 39. To form the square of a difference a &, we have 
 
 That is, the square of the difference between two quantities is 
 equal to the square of the first, minus twice the product of the 
 first by the second, plus the square of the second. 
 1. Form the square of 2ab. We have 
 
 2. Form the square of 4ac be. We have 
 
 (4ac 6c) 2 = 16a 2 c 2 8abc z +b z c z . 
 
 3. Form the square of 7a z b 2 I2ab*. We have 
 
 QUEST. 38. What is the square of the sum of two quantities equal to ? 
 39. What is the square of the difference of two quantities equal to ? 
 
36 ELEMENTARY ALGEBRA. 
 
 4O. Let it be required to multiply a-\-b by a b. We 
 have 
 
 Hence, the sum of two quantities, multiplied by their differ* 
 ence, is equal to the difference of their square* 
 
 1. Multiply 2c+5 by 2c b. We have 
 
 (2c-f&)x(2c-&)z=4c 2 - 2 . 
 
 2. Multiply 9<zc-f-35c by Qac 3bc. We have 
 
 3. Multiply 8a 3 +7ab 2 by 8a 3 7ab 2 . We have 
 (8a 3 +7a& 2 )(8a 3 7a6 2 ) = 64a 6 49a 2 Z> 4 . 
 
 41. It is sometimes convenient to find the factors of a 
 polynomial, or to resolve a polynomial into its factors. 
 Thus, if we have the polynomial 
 
 ac-\-ab -\-adj 
 
 we see that a is a common factor to each of the terms : 
 hence, it may be placed under the form 
 
 a(c+b+d). 
 
 1. Find the factors of the polynomial a z b 2 +a z d-{-a 2 f. 
 
 Ans. 
 
 2. Find the factors of 3a 2 b+6a 2 b*+b 2 d. 
 
 Ans. b(Z 
 
 3. Find the factors of 3<z 2 &+9a 2 c+18a 2 o:y. 
 
 Ans. 
 
 QUEST. 40. What is the sum of two quantities multiplied by theii 
 difference equal to! 
 
DIVISION. 37 
 
 4. Find the factors of 8a*cx 18acx 2 +2ac 5 y 30a 6 c 9 a?. 
 
 Ans. 
 
 5. Find the factors of a 2 -\ 
 
 Ans. (a+b)x(a+b). 
 
 6. Find the factors of a 2 b*. Ans. (a+b)x(ab). 
 
 7. Find the factors of a 2 2ab+b z . 
 
 Ans. (ab)x(ab). 
 
 DIVISION. 
 
 42. Algebraic division has the same object as arithmeti- 
 cal, viz : having given a product, and one of its factors, to 
 find the other factor. 
 
 We will first consider the case of two monomials. 
 
 The division of 72a 5 by 8a 3 is indicated thus : 
 
 72a 5 
 W 
 
 It is required to find a third monomial, which, multiplied 
 by the second, will produce the first. It is plain that the 
 tiiird monomial is 9a 2 ; for by the rules of multiplication 
 
 72 a 5 
 
 Hence, we have =9a 2 , 
 
 8a J 
 
 a result which is obtained by dividing the coefficient of the 
 dividend by the coefficient of the divisor, and subtracting the 
 exponents of the like letter. 
 
 QUEST. 42. What .is the object of division in Algebra] Give the 
 rule for dividing monomials! 
 
38 
 
 ELEMENTARY ALGEBRA. 
 
 Also, 
 
 Tab 
 for, 7ab X 5a 2 bc = 
 
 Hence, lor the division of monomials we have the following 
 
 I. Divide the coefficient of the dividend by the coefficient 
 of the divisor, for a new coefficient. 
 
 II. Write after this coefficient, all the letters of the divi- 
 dend, and affect each with an exponent equal to the ex- 
 cess of its exponent in the dividend over that in the 
 divisor. 
 
 From these rules we find 
 48oWd 
 
 1. Divide Wx 2 by Sx. 
 
 2. Divide 15a*y by Say. 
 
 3. Divide *84ab 3 x by I2b 2 . 
 
 4. Divide 36a 4 6 5 c 2 by 9a 3 Z> 2 c. 
 
 5. Divide 88a 3 b 2 c by 8a 2 b. 
 
 6. Divide 99a*b*x 5 by Ila 3 5 2 a; 4 . 
 
 7. Divide 108a? 6 y 5 ^ 3 by 54x 5 z. 
 
 8. Divide 64x 7 y 5 ^ 6 by I6x 6 y*z 5 . 
 
 9. Divide 96a 7 5 6 c 5 by 12a 2 5c. 
 
 10. Divide 54a 7 c 5 ^ 6 by 27acd. 
 
 11. Divide 38a 4 6 6 d 4 by 2a 3 b 5 d. 
 
 ns. 
 
 dns. 
 Ans. 7abx. 
 
 Ans. 4ab 3 c. 
 
 Ans. llabc. 
 
 Ans. 9ab 2 x 
 Ans. 2xy 5 z 2 . 
 
 Ans. xyz. 
 Ans. 8a 5 b 5 c*. 
 Ans. 2a 6 c*d 5 . 
 Ans. I9abd*. 
 
DIVISION. 39 
 
 12. Divide 42a 2 6 2 c 2 by 7abc. Ans. 6abc. 
 
 13. Divide 64a 5 6 4 c 8 by 32a*bc. ^n.?. 2a 3 c 7 . 
 
 14. Divide 128a 5 z 6 y 7 by IQaxy*. Ans. 8a 4 tf 5 y 3 . 
 
 15. Divide 132W 5 / 6 by 2d 4 /. Ans. 66 bdf 5 . 
 
 16. Divide 256a 4 6W 7 by I6a 3 bc 6 . Ans. 16ab 8 c 2 <F 
 
 17. Divide 200a 8 m 2 n 2 by dOtfmn. Ans. 4amn. 
 
 18. Divide 300^ 3 y% 2 by 60^y%. Ans. 5x 2 y 2 z. 
 
 19. Divide 27a 5 2 c 2 by 9a6c. An*. 3a 4 5c. 
 
 20. Divide 64a 3 y 6 ^ 8 by 32ay 5 z 1 . Ans. 2a z yz. 
 
 21. Divide 88a 5 6 6 c 8 by Ila 3 & 4 c 6 . An5. 8 fl 2 6^. 
 
 43. It follows from the preceding rule, that the division 
 of monomials will be impossible, 
 
 1st. When the coefficients are not divisible by each other. 
 
 2nd. When the exponent of the same letter is greater 
 in the divisor than in the dividend. 
 
 3rd. When the divisor contains one or more letters which 
 are not found in the dividend. 
 
 When either of these three cases occurs, the quotient re- 
 mains under the form of a monomial fraction ; that is, a 
 monomial expression, necessarily containing the algebraic 
 sign of division, but which may frequently be reduced. 
 
 Take, for example, 12a 4 & 2 cc?, to be divided by 8a 2 bc 2 . 
 which is placed under the form 
 
 \2aWcd 
 
 QUEST. 43. What is the first case named in which the division of 
 jflonomials will not be exact 1 What is the second 1 What is the third 1 
 If either of these cases occur, can the exact division be made 1 ? Under 
 what form will the quotient then remain 1 Mav this fraction be often 
 reduced to a simpler form 1 
 
2c 
 5a 
 
 40 ELEMENTARY ALGEBRA. 
 
 this may be reduced by dividing the numerator and denomi- 
 nator by the common factors 4, a 2 , b, and c, which gives 
 
 3a*bd 
 
 Also, 
 
 44. Hence, for the reduction of a monomial fraction to its 
 simplest form, we have the following 
 
 RULE. 
 
 Suppress every factor, whether numerical or literal, that 
 is common to both terms of the fraction, and the result will 
 be the reduced fraction sought. 
 
 From this new rule, we find, 
 
 (1) (2) 
 
 370 
 
 = T: : and 
 
 36a*b 3 c z de~~ 3bce 
 
 (3) (4) 
 
 also .'"^ = \ : and ^^ = 
 2ao bao* oo 
 
 7bc 2 
 5. Divide 49a 2 6 2 c 6 by 14a 3 ic 4 . Ans. 
 
 6. Divide Gamra by 3abc. . Ans. 
 
 7. Divide 18a 2 5 2 mn 2 by 12a*Mcrf. An*. 
 
 2a 
 2mn 
 
 QUEST. 44. Give the rule for the reduction of a monomial fraction. 
 
DIVISION. 41 
 
 8. Divide 28<7 5 # 5 c 7 <Z 8 by l6ab 9 ccFm. Ans. 
 
 6 
 9 Divide 72a 3 c 2 b 2 by I2a 5 c*b 3 d. Ans. 
 
 10. Divide I00a 8 b 5 xmn by 25a 3 b 4 d. Ans. -^ 
 
 32a 3 b*c s df 
 
 11. Divide 96a 5 b*c*df by 75a 2 cxy. Ans. J - . 
 
 1 7n 2 x 2 v 3 
 
 12. Divide 85m 2 w 3 /x 2 y 3 by 15aw 4 n/. Ans. f- . 
 
 127 
 
 13. Divide I27d 3 x 2 y 2 by 16c/ 4 #V- Ans - T^ 
 
 45. If we have an expression of the form 
 
 a a 2 a 3 a 4 a 5 
 
 , or , or , or , or , 
 a a 2 a 3 a 4 a* 
 
 and apply the rule for the exponents, we shall have 
 
 But since any quantity divided by itself is equal to 1, it 
 follows that 
 
 =aO = i fl =a a-a =tf o == i & Cj 
 
 a a 2 
 
 >r finally, if we designate the general exponent by m, we 
 have 
 
 =a m - m =a=l ; 
 " 
 
 that is, any /?0MJer o/" which the exponent is is equal to 1; and 
 hence, a factor of the form a, being equal to 1, may be omitted. 
 
 QUEST. 45. What is a equal to ? What is 6 equal to ? What is 
 the power of any number equal to, wfyeu tle exponent of the power is ? 
 
42 ELEMENTARY ALGEBRA. 
 
 2. Divide 6a?b 2 c*d by 2a z b z d. 
 
 2a z b z d 
 
 3. Divide So^W by 4a 2 5 8 c*<P. An*. 2a 2 
 
 4. Divide 16a 6 W by 8aW. Ans. 2<2 8 . 
 
 5. Divide 32m 3 n 3 oc' 2 y z by 4m 3 n 3 xy. Ans. 8,ry. 
 
 6. Divide 96a 4 5 d 8 c 9 by 24aWd 5 c*. Ans. 4bd 3 
 
 SIGNS IN DIVISION. 
 
 4G. The object of division, is to find a third quantity 
 called the quotient, which, multiplied by the divisor, shall 
 produce the dividend. 
 
 Since, in multiplication, the product of two terms having 
 the same sign is affected with the sign -f, and the product 
 of two terms having contrary signs is affected with the 
 sign , we may conclude, 
 
 1st. That when the term of the dividend has the sign -f, 
 and that of the divisor the sign of -f-j the term of the quo- 
 tient must have the sign ~f- 
 
 2nd. When the term of the dividend has the sign -\- , and 
 that of the divisor the sign , the term of the quotient 
 must have the sign , because it is only the sign , 
 which, multiplied with the sign , can produce the sign -f- 
 of the dividend. 
 
 Q0EST. 46. What will the quotient, multiplied by the divisor, be 
 equal to ? If the multiplicand and multiplier have like signs, what will 
 be the sign of- the product? If they have contrary signs, what will be 
 the sign of the product? When the term of the dividend and the term 
 of the divisor have the same sign, what will be the sign of the quotient ? 
 WVion thev have different signs, what will be the sign of the quotient ? 
 
DIVISION. 43 
 
 3rd. When the term of the dividend has the sign , and 
 ihat of the divisor the sign -}-, the quotient must have the 
 sign . Again we say for brevity, that, 
 
 + divided by -}-, and divided by , give + ; 
 divided by -f-, and -f- divided by , give . 
 
 EXAMPLES. 
 
 1. Divide 4ax by 2a. Ans. 2a?. 
 Here it is plain that the answer must be 2x ; for, 
 
 2a X 2x= + 4ax, the divisor 
 
 2. Divide 360 3 z 2 by 12a 2 *. Ans. 3ax. 
 
 3. Divide 58a 3 b 5 c 2 d 2 by 29a 2 4 c. Ans. 2abcd 2 . 
 
 4. Divide 84a 4 W 3 by 42aWd. Ans. 2aWd z . 
 
 5. Divide 64c*d*x 3 by 16c*dx. Ans. 4d*x 2 . 
 
 6. Divide 88 4 *y by 24& 3 c<fo 5 . Ans. +f- 
 
 7. Divide 77a*y 3 z 4 by ; 
 
 8. Divide 84a*b 2 c 2 d by 42a 4 b 2 c 2 d. 
 
 9. Divide 60a 7 2<W by : 
 
 10. Divide 88a 6 6 7 c 6 by 
 
 11. Divide 16# 2 by 8ar. 
 
 12. Divide 15a 2 #y 3 by Say. 
 
 13. Divide 84ab 3 x by I2b 2 . 
 
 14. Divide 96a 4 2 c 3 by I2a 3 bc. 
 
 15. Divide 144a 9 6 8 c 7 d 5 by 36a 4 i 6 c 6 tZ. 
 
 16. Divide 256d 3 bc 2 x 3 by 16a 2 ca: 2 . Ans. \6abcx. 
 
 17. Divide 300a 5 6 4 c 3 * 2 by 30aWc 2 x. Ans. lOabcx. 
 18 Divide 500a 8 /A- by 100a 7 i 8 c 4 . Ans. 5a?>c 2 . 
 
44 ELEMENTARY ALGEBRA. 
 
 19. Divide 64a 5 6 8 c 7 by 8a 4 6 7 c 6 . Ans. 8abc. 
 
 20. Divide -f-96 5 W by 24a*b 2 d. Ans. 4</W. 
 
 21. Divide 72a 5 b 3 d* by 8a*b 2 d. Ans. 9abd 3 . 
 
 Division of Polynomials. 
 
 FIRST EXAMPLE. 
 
 47. Divide a 2 2ax+x 2 by ax. 
 
 It is found most convenient, Dividend. Divisor. 
 
 in division in algebra, to place a 2 2ax-}-x 2 \a x 
 
 
 the divisor on the right of the a 2 ax 
 
 dividend, and the quotient di- ax+x 2 Quotient. 
 
 rectly u ader the divisor. ax-\-x 2 
 
 We first divide the term a 2 of the dividend by the term a 
 of the divisor : the partial quotient is o, which we place 
 under the divisor. We then multiply the divisor by a, and 
 subtract the product a 2 ax from the dividend, and to the 
 remainder bring down x 2 . We then divide the first term of 
 the remainder, ax by a, the quotient is x. We then 
 multiply the divisor by a;, and, subtracting as before, we 
 find nothing remains. Hence, a x is the exact quotient. 
 
 In this example, we have written the terms of the dividend 
 and divisor in such a manner that the exponents of the same 
 letter shall go on diminishing from left to right. This is 
 what is called arranging the dividend and divisor with 
 reference to a certain letter. By this preparation, the first 
 term on the left of the dividend, and the first on the left of 
 the divisor^are always the two which must be divided bv 
 each other in order to obtain a term of the quotient. 
 
 QUEST. 47. What do you understand by arranging a polynomial with 
 reference to a particular letter ? 
 
DIVISION. 45 
 
 48. Hence, for the division of polynomials we have the 
 following 
 
 RULE. 
 
 I. Arrange the dividend and divisor with reference to a cer- 
 tain letter, and then divide the first term on the left of the 
 dividend by the first term on the left of the divisor, the result 
 is the, first term of the quotient ; multiply the divisor by this 
 term, and subtract the product from the dividend. 
 
 II. Then divide the first term of the remainder by the first 
 term of the divisor, which gives the second term of the quotient ; 
 multiply the divisor by the second term, and subtract the pro* 
 duct from the result of the first operation. Continue the same 
 process until you obtain for a remainder ; in which case the 
 division it said to be exact. 
 
 SECOND EXAMPLE. 
 
 Let it be required to divide 
 51a 2 2 -}-100 4 48a 3 b 15b*+4ab 3 by 4ab 
 We here arrange with reference to a. 
 
 Dividend. Divisor. 
 
 > 15M 
 
 ^ 8a 3 b 
 
 2a?+8ab 
 Quotient. 
 
 QUEST. 48. Give the general rule for the division of polynomials 1 
 If the first term of the arranged dividend is not divisible by the first term 
 of *.he arranged divisor, is the exact division possible 1 If the first term 
 of any partial dividend is not divisible bv the first term of the divisor, is 
 the e*act division possible ? 
 
 5* 
 
46 ELEMENTARY ALGEBRA. 
 
 REMARK. When the first term of the arranged dividend 
 is not exactly divisible by that of the* arranged divisor, the 
 complete division is impossible ; that is to say, there is not 
 a polynomial which, multiplied by the divisor, will produce 
 the dividend. And in general, we shall find that a division 
 is impossible, when the first term of one of the partial 
 dividends is not divisible by the first term of the divisor. 
 
 GENERAL EXAMPLES. 
 
 1. Divide I8x z by 9x. Ans. 2x. 
 
 2. Divide Wx 2 y 2 by 5x 2 y. Ans. 2y. 
 
 3. Divide 9ax 2 y 2 by 9x 2 y. Ans. ay. 
 
 4. Divide Sx 2 by 2x. Ans. + 4x. 
 
 5. Divide I0ab+I5ac by 5a. Ans. 2 + 3c. 
 
 6. Divide 30aa? 54x by 6x. Ans. 5a 9. 
 
 7. Divide I0x 2 y I5y 2 5y by 5y. Ans. 2x 2 3y l. 
 
 8. Divide \2a-{-3ax I8ax 2 by 3a. Ans. 4 + x 6x 2 . 
 
 9. Divide 6ax 2 + 9a 2 x+a 2 x 2 by ax. Ans. 6x+9a + ax. 
 
 10. Divide a 2 -f 2 ax -\-x 2 by a-\-x. Ans. a+x. 
 
 11. Divide a 3 3a 2 y+3ay 2 y 3 by a y. 
 
 Ans. a 2 2ay-{-y 2 . 
 
 12. Divide 24a 2 b I2a 3 cb 2 Gab by Gab. 
 
 Ans. 4a+2a 2 c6-f 1. 
 
 13. Divide 6* 4 96 by 3* 6. Ans. 2ar 3 +4a: 2 +8x+16 
 
 14. Divide . . . a 5 Sa^+lOaV }0a 2 x 3 +5ax*x* 
 by a 2 2ax+x 2 . Ans. a 3 3a 2 x+3ax 2 x 3 . 
 
 15. Divide 48x 3 76ax 2 -64a 2 x+l05a 3 by 2x-3a. 
 
 Ans. 
 
DIVISION. 47 
 
 16. Divide y 6 3y*x 2 +3y 2 x* * 6 by y 3 3y 2 x+3yx z x*. 
 
 Ans. y 3 +3y 
 
 17. Divide 64aW25a 2 b B by 8/* 2 / 3 +5<z6 4 . 
 
 18. Divide 6a 3 +23a 2 6+22a& 2 +55 3 by 
 
 19. Divide 6aff 6 +6a# 2 y 6 -f-42a 2 a? 2 by 
 
 20. Divide . . 15a*+37a?bd 29a 2 c/ 
 
 Sc 2 / 2 by 3a 2 5W+c/. An^. 5a 2 +4W 8c/. 
 
 21. Divide o^ + a&y+y* by a; 2 xy+y 2 . 
 
 Ans. x 2 -\-xy-\-y 2 . 
 
 22. Divide a? 4 y 4 by a: y. Arcs. a; 3 -f-rr 2 y-ha?y 2 -t-y 3 . 
 
 23. Divide 3a 4 - 8a 2 6 2 -f 3a 2 c 2 +5i 4 --3i 2 c 2 by a 2 b 2 . 
 
 Ans. 3a 2 5b*+3c 2 . 
 
 24. Divide . . 6x 6 -5x 5 y 2 6^ 4 y 4 +6a; 3 y 2 4-15a; 3 y 3 9# 2 y 4 
 r 10x 2 y 5 -f 15y 6 by Sa^-f 2a? 2 y 2 +3y 2 . 
 
 Ans. 2x 3 3x 2 y 2 +5y 3 . 
 
 25. Divide . c 2 +ldA 2 7abc4a 2 bx 6a 2 b 2 +6acx 
 by Sax Sab c. Ans. 2ax-\-ab+c 
 
 26. Divide .... 3x 4 -f-4a; 3 y 4* 2 4a: 2 y 2 -j-16a:y-15 
 by 2a:y+a; 2 3. Ans. 3x 2 
 
 27. Divide a; 5 +32y 5 by x+2y. 
 
 Ans. x* 
 
 28. Divide 30 4 26a 3 b I4ab 3 +37a?b 2 by 2b 2 5ab * 
 4 3a 2 . Ans. a 2 7ab. 
 
48 ELEMENTARY ALGEBRA. 
 
 CHAPTER II. 
 
 Algebraic Fractions. 
 
 49. Algebraic fractions should be considered in the same 
 point of view as arithmetical fractions, such as J, \ ; that 
 is, we must conceive that the unit has been divided into as 
 many equal parts as there are units in the denominator, and 
 that one of these parts is taken as many times as there are 
 units in the numerator. Hence, addition, subtraction, mul- 
 tiplication, and division, are performed according to the rules 
 established for arithmetical fractions. 
 
 It will not, therefore, be necessary to demonstrate those 
 rules, and in their application we must follow the procedures 
 indicated for the operations on entire algebraic quantities. 
 
 50. Every quantity which is not expressed under a 
 fractional form is called an entire algebraic quantity. 
 
 51. An algebraic expression, composed partly of an 
 entire quantity and partly of a fraction, is called a mixed 
 quantity. 
 
 QUEST. 49 How are algebraic fractions to be considered 1 What 
 does the denominator show ? What does the numerator show ? How 
 then are the operations in fractions to be performed 1 50. What is ars 
 ntire quantity? 51. What is a mixed quantity? 
 
ALGEBRAIC FRACTIONS. 49 
 
 CASE I. 
 
 To reduce a fraction to its simplest terms. 
 
 52. A fraction is said to be in its lowest terms, when there 
 is no common factor in the numerator and denominator. The 
 rule for reducing a monomial fractior t to its lowest terms has 
 already been given (Art. 44). 
 
 With respect to polynomial fractions, the following are 
 cases which are easily reduced. 
 
 1. Take, for example, the expression 
 
 This fraction can take the form 
 
 (a+b) (a-b) 
 (a-b)* 
 
 (Art. 39 and 4O). Suppressing the factor a b, which 
 is common to the two terms, we obtain 
 
 a+b 
 a-b' 
 
 2. Again, take the expression 
 
 This expression can be decomposed thus : 
 
 OT ' 
 
 Sa\ab) ' 
 
 5a (a-by 
 8a z (a-b) ' 
 
 QUEST. 52. How do you reduced, fraction to its simplest terms' 
 
50 
 
 ELEMENTARY ALGEBRA. 
 
 Suppressing the common factors a(a b), the result is 
 
 5(a-b) 
 
 8a ' 
 
 Hence, to reduce any fraction to its simplest terms, we sup- 
 press or cancel every factor common to the numerator and 
 denominator. 
 
 NOTE. Find the common factors of the numerator and 
 denominator as explained in (Art. 41). 
 
 EXAMPLES. 
 
 
 1. Keduce - - , to its simplest terms. 
 12a 4 -{-6a d c 2 
 
 2. Reduce 
 
 3. Reduce 
 
 4. Reduce 
 
 4a 
 to its simplest terms. 
 
 An,. 
 
 
 to its simplest terms. 
 
 to its simplest terms. 
 
 o. Keauce 
 7. Reduce 
 
 to its simplest terms. Ans. 8. 
 Ans. 
 
ALGEBRAIC FRACTIONS. 51 
 
 CASE II. 
 
 53. To reduce a mixed quantity to the form of a fraction 
 
 RULE. 
 
 Multiply the entire part by the denominator of the fraction ; 
 then connect this product with the terms of the numerator by 
 the rules for addition, and under the result place the given 
 denominator. 
 
 EXAMPLES. 
 
 1. Reduce 6 to the form of a fraction 
 
 A Q 
 
 6x7=42: 42 + 1=43: hence, 6^=y. 
 
 / fl 2 X 2\ 
 
 2. Reduce x - to the form of a fraction. 
 
 XXX 
 
 3. Reduce- x ^~ to the form of a fraction. 
 
 2a 
 
 ax-x* 
 
 Ans. . 
 
 2<z 
 
 4. Reduce 5 -\ to the form of a fraction. 
 
 Ans. . 
 
 5. Reduce 1 to the form of a fraction. 
 
 Ans 
 
 a 
 
 QUEST. 63. How do you reduce a mixed quantity to the form of a 
 fraction 1 
 
2 ELEMENTARY ALGEBRA. 
 
 6. Reduce 1-f 2x ^- to the form of a fraction. 
 
 10* 2 +4ar-}-3 
 Ans. . 
 
 7. Reduce 2a-\-b to the form of a fraction. 
 
 16a+8b 3c 4 
 Ans. _ 8 
 
 8. Reduce 6ax+ b to the form of a fraction. 
 
 40 
 
 18a z x+5ab 
 
 Ans. 
 
 4a 
 
 9. Reduce 8+3ab ' x to the form of a frac- 
 
 tion, Ans. , 
 
 I2abx 4 
 
 10 Reduce 9-\ to the form of a fraction. 
 
 a b 2 
 
 CASE III. 
 
 54. To reduce a fraction to an entire or mixed quantity. 
 
 RULE. 
 
 Divide the numerator by the denominator for the entire part, 
 and place the remainder, if any, over the denominator for the 
 fractional part. 
 
 QUEST. 54. How do you reduce a fraction to an entire or mixed 
 quantity 1 
 
ALGEBRAIC FRACTIONS. 53 
 
 EXAMPLES. 
 
 , 8966 
 
 1. Keduce to an entire number. 
 
 o 
 
 8)8966( 
 
 1120.. .6rem. 
 
 Hence, 1120f= Ans. 
 
 2. Reduce to a mixed quantity. 
 
 Ans. a . 
 
 fig, _ j, 
 
 3. Reduce - to an entire or mixed quantity. 
 
 Ans. a 
 
 ab2cP 
 
 4. Reduce - - - to a mixed quantity. 
 
 2a 2 
 
 Ans. a -- = . 
 o 
 
 _ 
 
 5. Reduce - to an entire quantity. Ans. 
 
 3 _ 3 
 
 6. Reduce - ? to an entire quantity. 
 
 x y 
 
 Ans. 
 
 10* 2 
 7 Reduce - - to a mixed quantity. 
 
 
 Ans. 2x H . 
 
 8. Reduce - to a mixed quantity. 
 
 Ans. 4x 2 8-\ ^-. 
 
54 ELEMENTARY ALGEBRA. 
 
 t CASE IV. 
 
 55. To reduce fractions having different denominators 
 to equivalent fractions having a common denominator. 
 
 RULE. 
 
 Multiply each numerator into all the denominators except us 
 own, for the new numerators, and all the denominators togethe* 
 for a common denominator. 
 
 EXAMPLES. 
 
 1. Reduce ^, , and J, to a common denominator. 
 
 1x3x5 = 15 the new numerator of the 1st. 
 7x2x5=70 2nd. 
 
 4x3x2=24 3rd. 
 
 and 2x3x5=30 the common denominator. 
 
 Therefore, ^, J$, and , are the equivalent fractions. 
 
 NOTE. It is plain that this reduction does not alter the 
 values of the several fractions, since the numerator and 
 denominator of each are multiplied by the same number. 
 
 2. Reduce ~ and to equivalent fractions having 
 a common denominator. 
 
 axc=ac the new numerators. 
 
 bxb=b* 
 and bxc=bc the common denominator. 
 
 QUEST. 55. How do you reduce fractions to a common denominator 1 
 
ALGEBRAIC FRACTIONS. 55 
 
 Hence, T and are the equivalent fractions. 
 be be 
 
 3. Reduce and to fractions having a com- 
 
 ae ab+b z 
 
 mon denominator. Ans. and = . 
 
 be be 
 
 4. Reduce -, , and rf, to fractions having a 
 
 tliCL oC 
 
 9cx 4ab . 6acd 
 
 common denominator. Ans. -- , , and . 
 
 6ac 6ac 6ac 
 
 5. Reduce , , and a-\ , to fractions having 
 
 43 a 
 
 a common denominator. 
 
 9a 8ax I2a 2 -\-24x 
 
 Ans. , - , and 
 
 I2a ' I2a ' I2a 
 
 1 a 2 a 2 -\-x 2 
 
 6. Reduce , - , and , to fractions 
 
 having a common denominator. 
 
 3a+3x 2a 3 + 2a 2 x 6a 2 +6a; 2 
 
 otz ~r~ o^r \ya ~\~ \)oc ocz ~r~ \}oc 
 
 a 6ax a 2 x 2 
 
 7. Reduce p, , and = to a common 
 
 30 5c a 
 
 denominator, 
 
 5acd ISabdx \5a 2 bc15bcx* 
 
 AnS ' -^~^- 15M ' 
 
 8. Reduce , , and r-, to a common de- 
 
 5a c a-\-o 
 
 nominator. 
 
 5a 3 -5a& 2 
 
56 ELEMENTARY ALGEBRA. 
 
 CASE V. 
 
 56. To add fractional quantities together 
 
 RULE. 
 
 Reduce the fractions, if necessary, to a common denomina- 
 tor ; then add the numerators together, and place their sum 
 over the common denominator. 
 
 EXAMPLES. 
 
 1. Add f, f, and f together. 
 By reducing to a common denominator, we have 
 
 6x3x5 = 90 1st numerator. 
 
 4 X 2 x 5= 40 2nd numerator. 
 
 2x3x2 = 12 3rd numerator. 
 
 2x3x5 = 30 the denominator. 
 
 Hence, the fractions become 
 
 90 40 12 142 
 
 30 30 30 30 
 which, by reducing to the lowest terms become 4|-|. 
 2. Find the sum of , , and -~^ 
 
 Here axdxf=adf\ 
 
 c X b X f=cbf > the new numerators 
 
 exbxd=ebd ) 
 And b x d xf= bdf the common denominator. 
 
 adf cbf ebd adf+cbf+ebd , 
 
 = - ~~ 
 
 QUEST. 56. How do you add fractions. 
 
ALGEBRAIC FRACTIONS. 67 
 
 3 To a- add 
 
 h c, 
 
 , . , 2abx3cx 2 
 Ans. a + b-\ -- 1 - . 
 be 
 
 4. Add , and - together. Ans. x-}-. 
 
 Z o 4 \.<i 
 
 x2 4.u I9x 14 
 
 5. Add - and together. Ans. - 
 
 o 7 *1 
 
 x 2 2* 3 , lOz-17 
 
 6. Add a?H -- to 3x-{ -- - . Ans. 4x-{ -- . 
 
 34 I* 
 
 7. It is required to add 4#, - , and - together. 
 
 2 ax 
 
 , , . ,. . , 
 
 8. It is required to add , , and - together. 
 
 49a?-fl~2 
 
 Ans . 
 
 9. It is required to add 4x, , and 2+ together. 
 
 44^+90 
 Ans . 
 
 10. It is required to add 3x-\ -- and x together! 
 
 o 9 
 
 Ans. 3x-\ 
 
 4 O 
 
 11. Required the sum of ac - and 1 -- . 
 
 oa a 
 
 8a 2 cd6bd+8ad8ac 
 Ans. - 
 
 Sad 
 
ELEMENTARY ALGEBRA. 
 CASE VI. 
 
 57, To subtract one fractional quantity from another 
 
 RULE. 
 
 I. Reduce the fractions to a common denominator. 
 
 II. Subtract the numerator of the fraction to be subtracted 
 from the numerator of the other fraction, and place the dif- 
 ference over the common denominator. 
 
 EXAMPLES. 
 
 3 2 
 
 1 . What is the difference between and . 
 
 3 2 24 14 10 5 
 
 __ M _ LI __ ___ _ n ___ ^^. ____ . A j\ 
 
 7 8 ~56 56~~56~28* ' 
 
 ff> . ft 2^2 4- /y* 
 
 2. Find the difference of the fractions and - . 
 
 2b 3c 
 
 (x a 
 
 ^ 7 the numerators 
 
 (2a 
 
 And, 25x3c=z6Z>c the common denominator. 
 
 3cx3ac 4ab8bx 3cx3ac4ab + 8bx 
 Hence, - --- -= - = -- -= -- . Ans 
 6bc 6bc 6bc 
 
 ,,,.,,. f I2x , 3o? 39a; 
 
 3. Required the difference of - - and . Ans. . 
 
 4. Required the difference of 5y and . Ans. ^-, 
 
 8 8 
 
 5. Required the difference of and . Ans. -- . 
 
 79 63 
 
 QUEST. 57. How do you subtract fractions ? 
 
ALGEBRAIC FRACTIONS. 69 
 
 6. Required the difference between - and =-. 
 
 U a 
 
 dx-\-adbc 
 
 Ans - - ~ 
 
 7. Required the difference of and - . 
 
 DO o 
 
 24x f 8a Wbx 355 
 - -406- - 
 
 J8. Required the difference of 3a?+ and x -- . 
 
 cx+ bx ab 
 Ans. 2aH -- 7 -- . 
 
 CASE VII. 
 
 58. To multiply fractional quantities together. 
 
 RULE. 
 
 If the quantities to be multiplied are mixed, reduce them to 
 fractional forms ; then multiply the numerators together for 
 a numerator and the denominators together for a denominator 
 
 EXAMPLES. 
 
 1. Multiply i- of y by 8J. 
 
 Operation. 
 
 We first reduce the com- 1.33 
 
 pound fraction to the sim- 6 7 ~~~42' 
 
 pie one ^, and then the 25 
 
 mixed number to the equiva- 8 J = ~3~" 
 lent fraction 2 ^ ; after 
 
 whicii, we multiply the Hence, x =-^ = ~. 
 
 numerators and denomira- 42 3 126 42 
 
 tors together. ^^^ 2 
 
60 ELEMENTARY ALGEBRA. 
 
 2. Multiply +*i by -L. First, a4 ^=?!d 
 
 a d a a 
 
 TT a 2 -}-& c a 2 c+bcx 
 
 Hence, . ' X-^= -, . Ans. 
 
 a d ad 
 
 . 3x 3a 9aa? 
 
 3. Kequired the product of and -r-. Ans. -. 
 
 5 b 5b 
 
 2 V ^Y^ 
 
 4. Required the product of and 
 
 o 2/<i 
 
 3x* 
 
 Ans. . 
 5a 
 
 . 2a? Sab Sac 
 
 5. Find the continued product of , and ~. 
 
 a c 2b 
 
 Ans. 9ax. 
 
 6. It is required to find the product of b-\ and . 
 
 a x 
 
 
 
 ab+bx 
 Ans. . 
 
 x 
 
 7. Required the product of ; and s . 
 
 be b-\-c 
 
 ** 
 Ans. 
 
 
 x I j x _ j 
 
 8. Required the product of x-\ -- , and , . 
 
 ax 2 ax+x* 1 
 
 9. Required the product of a-\ -- by 
 
 a-x p+aT 
 
 n. n-T* 
 
 A. 
 
 x 2 x 3 
 
 QUEST 6S. How do yoa multiply fractions together? 
 
ALGEBRAIC FRACTIONS. 61 
 
 CASE VIII. 
 
 59. To divide one fractional quantity by another. 
 
 RULE. , 
 
 Reduce the mixed quantities, if there are any, to fractional 
 forms ; then invert the terms of the divisor, and multiply the 
 fractions together as in the last case, 
 
 EXAMPLES. 
 
 10 5 
 
 1. Divide. . . . by . 
 
 If the divisor were 5, the Operation. 
 
 quotient would be y 1 ^. But, 5 _ 1 
 
 since the divisor is of 5, 8 ~ 8 
 
 the true quotient must be 8 10 10 
 
 times 1 1 ^, for the eighth of 24 ~ ~~12(J 
 
 a number will be contained 10 80 2 
 
 ^ __ \^ O _ __ _ - ___ 
 
 '\n the dividend 8 times more 120 "120 3 ' 
 
 than the number itself. In 
 
 this operation we have actually multiplied the numerator 
 
 of the dividend by 8 and the denominator by 5 ; that is, we 
 
 have inverted the terms of the divisor and multiplied the fractions 
 
 together. 
 
 2. Divide . . a-- by . 
 
 2c g 
 
 b 2acb 
 
 b f 2ac b 
 Hence, -+=- 
 
 QUEST. 59. How do you divide one fraction by another 1 
 
62 ELEMENTARY ALGEBRA. 
 
 7x 12 91a? 
 
 3. Let - be divided by . Ans -^--. 
 
 5 13 60 
 
 4. Let - be divided by 5a?. Ans. -. 
 
 5. Let TT- be divided by . Ans. 
 
 6. Let be divided by -. Ans. -, 
 
 o?l 2 a; 1 
 
 5x 2a 5bx 
 
 7. Let be divided by -r-. Ans. - 
 
 -=-. 
 60 
 
 x b 3cx x b 
 
 8. Let - - be divided by j-. Ans. 
 
 8cd 4d 
 
 9. Let 2 - be divided by . 
 
 x 2 2bx-\- W- xb 
 
 Ans. x-\ . 
 
 10. Divide 6<z 2 + by c 2 ^^. 
 5 2 
 
 11. Divide 18c 2 a: + -^- by a 2 . 
 
 12. Divide 20* 2 -j^- by a? 2 -^. 
 
 Ant J 
 
EQUATIONS OF THE FIRST DEGREE. 63 
 
 CHAPTER III. 
 
 Of Equations of the First Degree. 
 
 GO. An Equation is the algebraic expression of two equal 
 quantities with the sign of equality placed between them. 
 Thus, x=a+b is an equation, in which x is equal to the 
 sum of a and b. 
 
 6 1 . By the definition, every equation is composed of two 
 parts, separated from each other by the sign =. The part 
 on the left of the sign, is called the first member ; and the 
 part on the right, is called the second member. Each mem- 
 ber may be 'composed of one or more terms. Thus, in the 
 equation x=a-\-b, x is the first member, and a-\-b the 
 second. 
 
 62. Every equation may be regarded as the enunciation, 
 in algebraic language, of a particular question. Thus, the 
 equation a: +# = 30, is the algebraic enunciation of the 
 following question : 
 
 QUEST. 60. What is an equation 1 ! 61. Of how many parts is every 
 equation composed 1 How are the parts separated from each other 1 
 What is the part on the left called 1 What is the part on the right called ? 
 May each member be composed of one or more terms 1 In the equation 
 x=a-\-b, which is the first member? Which the second? How many 
 terms in the first member? How many in the second 1 
 
64 ELEMENTARY ALGEBRA 
 
 To find a number which being added to itself, shall give, a 
 sum equal to 30. 
 
 Were it required to solve this question, we should first 
 express it in algebraic language, which would give the 
 equation 
 
 x+x=30. 
 
 By adding x to itself, we have 
 2* =30. 
 
 And by dividing by 2, we obtain 
 cc=\.5. 
 
 Hence we see that the solution of a question by algebra 
 consists of twa distinct parts : viz. the STATEMENT and the 
 SOLUTION of an equation. 
 
 I. The STATEMENT consists in expressing algebraically 
 the relation between the knoivn and unknown quantities. 
 
 II. The SOLUTION of the equation consists in finding the 
 value of the unknown quantity in terms of those which are 
 known. 
 
 The given or known parts of a question, are represented 
 either by numbers or by the first letters of the alphabet, 
 a, b, c, &c. The unknown or required parts are repre- 
 sented by the final letters, x, y, z, &c. 
 
 EXAMPLE. 
 
 Find a number which, being added to twice itself, the 
 sum shall be equal to 24. 
 
 QUEST. 62. How may you regard every equation ? What question 
 does the equation x+x=3Q state"! Of how rr.any parts does the solu- 
 tion of a question by algebra consist 1 Name them. What is the 2nd 
 part called 1 By what are the known parts of a question represented ? 
 By what are the unknown parts represented ? 
 
EQUATIONS OF THE FIRST DEGREE. 05 
 
 Statement. 
 Let x represent the number. We shall then have 
 
 This is the statement. 
 
 Solution. 
 
 Having .... x+2x=24, 
 we add ..... a;+2a?, 
 which gives . . . 3^=24 , 
 
 and dividing by 3, . x8. 
 
 63. An equation is said to be verified when the answer 
 found, being substituted for the unknown quantity, proves 
 the two members of the equation to be equal to each other. 
 
 Thus, in the last equation we found #=8. If we substi- 
 tute this value for x in the equation 
 
 we shall have 8 + 2x8 = 8 + 13 = 24. 
 
 which proves that 8 is the true answer. 
 
 6 4. An equation involving only the first power of the 
 unknown quantity, is called an equation of the first degree. 
 
 Thus, 6a;+3a? 5 = 13, 
 
 and ax-}- bx + c = d, 
 
 are equations of the first degree. 
 
 By considering the nature of an equation, we perceive 
 that it must possess the three following properties : 
 
 QUEST. 63. When is an equation said to be verified 1 64. When 
 an equation involves only the first power of the unknown quantity, wnat 
 is it called ? What are the three properties of every equation ? 
 
 7 
 
66 ELEMENTARY ALGEBRA. 
 
 1 st. The two members are composed of quantities of the 
 same kind : that is, dollars = dollars, pounds = pounds, &c. 
 2nd. The two members are equal to each other. 
 3rd. The two members must have the same sign. 
 
 65. An axiom is a self-evident truth. We may here 
 state the following. 
 
 1. If equal quantities be added to loth members of an equa- 
 tion, the equality of the members will not be destroyed. 
 
 2. If equal quantities be subtracted from both members of 
 an equation, the equality will not be destroyed. 
 
 3. If both members of an equation be multiplied by the same 
 number, the equality will not be destroyed. 
 
 4. If both members of an equation be divided by the same 
 number, the equality will not be destroyed. 
 
 Transformation of Equations. 
 
 66. The transformation of an equation consists in chang 
 ing its form without affecting the equality of its members. 
 
 The following transformations are of continual use in the 
 resolution of equations. 
 
 First Transformation. 
 
 67. When some of the terms of an equation are frac- 
 tional, to reduce the equation to one in which the terms shall 
 be entire. 
 
 1. Take the equation 
 
 3 
 
 QUEST. 65. What is an axiom ? Name the four axioms. 66. Whaf 
 is the transformation of an equation ? 67. What is the first transforma- 
 tion ? What is the least common multiple of several numbers 1 How 
 do von find the least common multiple 7 
 
 
EQUATIONS OF THE FIRST DEGREE. 67 
 
 First, reduce all the fractions to the same denominator 
 by the known rule ; the equation then becomes 
 
 48s 54a? , 12a;__ 11 
 "72 72 72"- 
 
 and sinc we can multiply both members by the same num- 
 ber without destroying the equality, we will multiply them 
 by 72, which is the same as suppressing the denominator 
 72, in the fractional terms, and multiplying the entire term 
 by 72 ; the equation then becomes 
 
 48a? 54a?+12a?=792, 
 or dividing by 6 Sx 9a?-f- 2#=132. 
 
 But this last equation can be obtained in a shorter way, by 
 finding the least common multiple of the denominators. 
 
 The least common multiple of several numbers is the 
 least number which they will separately divide without a 
 remainder. When the numbers are small, it may at once 
 be determined by inspection. The manner of finding the 
 least common multiple is fully shown in Arithmetic 87 
 
 Take for example, the last equation 
 
 2x 3 x 
 --- #-!_ -=11. 
 346 
 
 We see that 12 is the least common multiple of the deno- 
 minators, and if we multiply all the terms of the equation 
 by 12, and divide by the denominators, we obtain 
 
 8a; 9. 
 the same equation as before found. 
 
08 ELEMENTARY ALGEBRA. 
 
 68. Hence, to make the denominators disappear from an 
 equation, we have the following 
 
 
 RULE. 
 
 I. Find the least common multiple of all the denomi' 
 nators. 
 
 II. Multiply every term of the equation by the common 
 multiple reducing at the same time the fractional to entire 
 terms. 
 
 EXAMPLES. 
 
 1. Clear the equation of - f 4 = 3 of its denomi- 
 
 o i. 
 
 nators. Ans. 7x+5x 140^105. 
 
 2. Clear the equation -^-+-r ^;= 8 f i ts denomi- 
 
 o y 4ti 
 
 nators. Ans. 9x -\-6x- 2x = 432 
 
 3. Clear the equation -- f --- jrH r^ 20 ofitsde- 
 
 -w O 3 1 <t 
 
 nominators. Ans. 
 
 4. Clear the equation f =4 of its denomi- 
 
 O 7 Z 
 
 nators. Ans. 14x+lQx 35ar=280. 
 
 5. Clear the equation --- - ^--=zl5 of its denomi- 
 
 4 5 o 
 
 nators Ans. I5x 
 
 QUEST. 68. Give the rule for clearing an equation of its denoini 
 nators. 
 
EQUATIONS OF THE FIRST DEGREE. 69 
 
 6. Clear the equation -^ TT+'S'+'Tr 12 of its de ~ 
 
 4 o o y 
 
 nominators. Ans. 18o? 12*4- 9oj-f-8a:=864 
 
 7. Clear the equation +f=g. 
 
 Ans. adbc+bdf=bdg. 
 
 8. In the equation 
 
 ax 2c 2 x 4bc*x 5a 3 , 2c 2 
 
 7 K4a = 5 rr ! 3 * 
 
 b ab a 3 b 2 a 
 
 the least common multiple of the denominators is a 3 b* ; 
 hence clearing the fractions, we obtain 
 
 = 4b 3 c 2 x 
 
 Second Transformation. 
 
 69. When the two members of an equation are entire 
 polynomials, to transpose certain terms from one member to 
 the other. 
 
 1. Take for example the equation 
 5x 6 = 8 + 2*. 
 
 If, in the first place we subtract 2x from both members, 
 the equality will not be destroyed, and we have 
 
 5x 6 2x^8. 
 
 Whence we see that the term 2ar, which was additive 
 in the second member becomes subtractive in the first. 
 
 QUEST. 69. What is the second transformation? What do you 
 understand by transposing a term 1 Give the rule for transposing from 
 
 one member to the other. 
 
 7* 
 
70 ELEMENTARY ALGEBRA. 
 
 In the second place, if we add 6 to both members, 
 equality will still exist, and we have 
 
 5x 6 2^+6 = 8 + 6. 
 
 Or, since 6 and -{-6 destroy each other, we have 
 5x 2x= 
 
 Hence the term which was subtractive in the first mem- 
 ber, passes into the second member with the sign of 
 addition. 
 
 2. Again, take the equation 
 
 ax-\-l = d ex. 
 
 If we add ex to both members and subtract b from 
 them, the equation becomes 
 
 ax-\-b-\-cx b=zd cx-\-cx b. 
 or reducing ax-}-cx=:d b. 
 
 When a term is taken from one member of an equation 
 and placed in the other, it is said to be transposed. 
 
 Therefore, for the transposition of the terms, we have the 
 following 
 
 RULE. 
 
 Any term of an equation may be transposed from one mem- 
 ber to the other by changing its sign. 
 
 7O. We will now apply the preceding principles to the 
 resolution of equations. 
 
 1. Take the equation 
 
EQUATIONS OF THE FIRST DEGREE. 71 
 
 By transposing the terms 3 and 2#, it becomes 
 
 4x 2x=5 + 3. 
 Or, reducing 2x = 8. 
 
 
 
 Dividing by 2 x ==4. 
 
 Verification. 
 
 If now, 4 be substituted in the place of x in the given 
 equation 
 
 it becomes 4x43=2x 
 
 or, 13 = 13. 
 
 Hence, the value of x is verified by substituting it for the 
 unknown quantity in the given equation. 
 
 2. For a second example, take the equation 
 
 5x 4x 7 I3x 
 
 12 3 " ~ 8 ~ 6 ' 
 
 By making the denominators disappear, we have 
 
 10* 32* 312=21 52*, 
 or, by transposing 
 
 Wx 32a?+52#=:21 + 312 
 by reducing 30a? == 333 
 
 333 111 
 ~-30-~-~-lO-~- U > 1 ' 
 
 a result which may be verified by substituting it for x in the 
 given equation. 
 
 3. For a third example let us take the equation 
 
 (3a x)(c 
 
ELEMENTARY ALGEBRA. 
 
 It is first necessary to perform the multiplications indica- 
 ted, in order to reduce the two members to two polynomials, 
 and thus be able to disengage the unknown quantity x, from 
 the known quantities. Having done that, the equation 
 becomes, 
 
 3a 2 ax 
 or, by transposing 
 
 by reducing ax 3bx = 7ab 3a*. 
 
 Or, (Art. 41). (a 3b)x = 7ab 3a z . 
 
 Dividing both members by a 3b we find 
 
 _7ab 3a? 
 x ~ a 36 ' 
 
 Hence, in order to resolve an equation of the first degree, 
 we have the following general 
 
 RULE. 
 
 I. If there are any denominators, cause them to disappear, 
 and perform, in both members, all the algebraic operations 
 indicated. 
 
 II. Then transpose all the terms affected with the unknown 
 quantity into the first member, and all the known terms into 
 the second member. 
 
 III. Reduce to a single term all the terms involving x : 
 this term will be composed of two factors, one of which will 
 be x, and the other all the multipliers of x, connected with 
 their respective signs. 
 
 IV. Divide both members of the equation by the multi* 
 plier of the unknown quantity. 
 
 QUEST. 70. What is the first step in resolving an equation of the 
 first degree ? What the second ? What the third ? What the fourth ? 
 
EQUATIONS OF THE FIRST DEGREE. 73 
 EXAMPLES. 
 
 1. Given 3x 2+24=31 to find a:. Ans. x=3. 
 
 2. Given a?+18 = 3a: 5 to find x. Ans. a?=lly. 
 
 3. Given 6 2ar+10=20 3x 2 to find x. 
 
 Ans. x=2. 
 
 6. Given 2x x-\-L=5x 2 to find x. Ans. #=-=-. 
 
 2 7 
 
 6. Given 3ax-\ 3 = for a to find x. 
 
 6-3a 
 Ans. x = - 
 
 7. Given z+-=20 z to find 
 23 2 
 
 8. Given 4.=4- to find 
 
 3 4 
 
 9. Given - --|-ar=~-- 3 to find x. 
 4 
 
 10. Given ^ ^ 4=/ to find a:. 
 
 6a2l>' 
 
 cdf \-4cd 
 
 Ans. x= J -. 
 
 3ad2bc 
 
74 ELEMENTARY ALGEttRA. 
 
 n . Given !2^--2*rL=4-* to find *. 
 
 56 + 95 7c 
 : 16a ' 
 
 12. Given -^ -~5 l"7r=-5- to find x. 
 O 3 2 3 
 
 An*. a; =10. 
 
 13. Given - 4~+~ ^T=f to find * 
 
 a be d J 
 
 __ abcdf 
 
 ~ bcd-acd+abd-abc* 
 
 NOTE. What is the numerical value of x, when a=l, 
 5=2, c=3, d=4, 5=5, and/=6. 
 
 14. Given ^-^-^i=-12ff to find *. 
 
 7 y *> 
 
 . =14. 
 
 15. Given a-+^+l to find . 
 
 16. Given ar+i+i ^-=2^-43 to find *. 
 
 4 5 o 
 
 An*, a: =60 
 
 17. Given 2a? -- ^~ = - to find x. 
 
 5 
 
 Ans. x=3. 
 
 18 Given 3x-\ ~ =x+a to find x. 
 
 o 
 
 ax b a bx bx a -. , 
 
 19. Given ^=~2 3 ** *' 
 
 35 
 Ans. x=- 
 
 3a -! 
 
EQUATIONS OF THE FIRST DEGREE. 75 
 
 20. Find the value of x in the equation 
 
 (a+b) (xb) 4abb 2 a z bx 
 
 - - -- 3a = - - -- 2x-\ -- - - . 
 a b a-\-o b 
 
 Of Questions producing Equations of the First Degree 
 involving but a single unknown quantity. 
 
 71. It has already been observed (Art. 62), that the 
 solution of a question by algebra consists of two distinct 
 parts : 
 
 1st. To make the STATEMENT : that is, to express the con- 
 ditions of the question algebraically ; 
 
 2d. To solve the equation : that is, to disengage the known 
 from tne unknown quantities. 
 
 We have already explained the manner of finding the 
 value of the unknown quantity, after the question has been 
 stated ; and it only remains to point out the best methods 
 of putting a question in the language of algebra. 
 
 This part of the algebraic solution of a question cannot, 
 like the second, be subjected to any well defined rule. 
 Sometimes the enuroiation of the question furnishes the 
 equation immediately ; and sometimes it is necessary to dis- 
 cover, from the enunciation, new conditions from which an 
 equation may be formed. 
 
 QUEST. 71. Into how many parts is the resolution of a question in 
 a.gebra divided 1 What is the first step ? Whai the second 1 Which 
 part has already been explained ? Which part is now to be considered ? 
 Can this part be subjected to exact rules ? Give the general rule for 
 stating a question 
 
76 ELEMENTARY ALGEBRA. 
 
 In almost all cases, however, we are able to make the state- 
 ment ; that is, to discover the equation, by applying the fo/- 
 lowing 
 
 RULE. 
 
 Represent the unknown quantity by one of the final letters 
 of the alphabet ; and then indicate by means of the algebraic 
 signs, the same operations on the known and unknown quan- 
 tities, as would verify the value of the unknown quantity, 
 were such value known. 
 
 QUESTIONS. 
 
 1. To find a number to which if 5 be added, the sum will 
 be equal to 9. 
 
 Denote the number by x. 
 Then by the conditions 
 
 
 This is the statement of the question. 
 To find the value of x, we transpose 5 to the second 
 member, which gives 
 
 a; = 9-5 = 4. 
 
 Verification. 
 4+5 = 9. 
 
 2. Find a number such, that the sum of one half, one 
 third, and one fourth of it, augmented by 45, shall be equal 
 to 448. 
 
 Let the required number be denoted by x. 
 
 Then, one half of it will be denoted by -. 
 
 one third by -. 
 
 o 
 
 one fourth by -. 
 
EQUATIONS OF THE FIRST DEGREE 77 
 
 And by the conditions. 
 
 This is the statement of the question. 
 To find the value of x, subtract 45 from both members : 
 this gives 
 
 By clearing the terms of their denominators, we obtain 
 
 6a:+4a;+3a:=4836, 
 or 13a:=4836. 
 
 Hence, x= =372. 
 
 Verification. 
 
 372 372 372 
 
 +-+--1+45 = 186 + 124+93 + 45=448. 
 
 4i O 4 
 
 3. What number is that whose third part exceeds its 
 fourth by 16. 
 
 Let the required number be represented by x. Then, 
 
 -#= the third part. 
 3 
 
 1 
 -j-a?= the fourth part. 
 
 And by the question 
 
 This is the statement. To find the value of a?, we clear 
 the terms of the denominators, which gives 
 
 4a? 3a;=192. 
 and a: =192. 
 
 8 
 
78 ELEMENTARY ALGEBRA. 
 
 Verification. 
 
 
 4. Divide $1000 between A, B and C, so that A shall 
 have $72 more than B, and C $100 more than A. 
 
 Let x = B's share of the $1000. 
 
 Then x+ 72= A's share, 
 
 and a:+172 = C's share, 
 
 their sum is 3a?+244=$1000. 
 This is the statement. 
 By transposing 244 we have 
 3x= 1000 244=756 
 
 py f /> 
 
 and x =-=252= B's share. 
 
 9 
 
 Hence, a+ 72 =252+ 72 = $324= A's share. 
 And *-t- 172 =252-{-172 = $424= C's share. 
 
 Verification. 
 252 + 324+424 = 1000. 
 
 5. Out of a cask of wine which had leaked away a third 
 part, 21 gallons were afterwards drawn, and the cask being 
 then guaged, appeared to be half full: how much did it 
 hold? 
 
 Suppose the cask to have held x gallons. 
 
 Then, - what leaked away. 
 
 
 
 x 
 
 And ~+ 21= what had leaked and been drawn 
 
 3 
 
 Hence, + 21= -x by the question. 
 o 2 
 
 This is the statement. 
 
EQUATIONS OF THE FIRST DEGREE. 79 
 
 To find #, we have 
 
 2.c+126=:3tr, 
 
 and x =126, 
 
 or x = 126, 
 
 by changing the signs of both members, which does not 
 destroy their equality. 
 
 Verification. 
 
 6. A fish was caught whose tail weighed 9lb., his head 
 weighed as much as his tail and half his body, and his body 
 weighed as much as his head and tail together ; what was 
 the weight of the fish ? 
 
 Let 2ar= the weight of the body. 
 
 Then, 9-{-x= weight of the head ; 
 
 and since the body weighed as much as both head and tail, 
 
 which is the statement. Then, 
 
 2x x=lS and ar=18. 
 
 Verification. 
 
 2x = 36/6. = weight of the body. 
 
 9-f ar=27/6.= weight of the head. 
 
 9lb.= weight of the tail. 
 
 Hence, 72/6. = weight of the fish. 
 
80 ELEMENTARY ALGEBRA. 
 
 7. The sum of two numbers is 67 and their difference 
 19 : what are the two numbers ? 
 
 Let x= the least number. 
 
 Then, #-f-19 = the greater, 
 
 and by the conditions of the question 
 
 2*+ 19 = 67. 
 This is the statement. 
 
 To find a?, we first transpose 19, which gires. 
 2a?=67 19=48; 
 
 4R 
 
 hence, *=- = 24, and x+t9=43. 
 
 Verification. 
 434-24=67, and 4324=19. 
 
 Another Solution. 
 
 Let a? represent the greater number : 
 then, or 19 will represent the least, 
 
 and, 2a? 19=67, whence 2;r=67-f-I9; 
 
 fifi 
 
 therefore, a:= =43 
 
 2 
 
 and consequently x 19=43 19=24. 
 
 General Solution of this Problem. 
 
 The sum of two numbers is , their difference is b. 
 What are the two numbers ? 
 
 
EQUATIONS OF THE FIRST DEGREE. 81 
 
 Let x be the least number, 
 
 x-\-b will represent the greater. 
 Hence, 2x + h=a, whence 1xa &; 
 
 a b a b 
 therefore, a= =_--, 
 
 a b a b 
 
 and consequently, x-\-b= --- +6:= -}--. 
 2 2 2 
 
 As the form of these two results is independent of any 
 value attributed to the letters a and b, it follows that, 
 
 Knowing the sum and difference of two numbers, we obtain 
 the greater by adding the half difference to the half sum, and 
 the less, by subtracting the half difference from half the sum. 
 
 Thus, if the given sum were 237, and the difference 99, 
 
 237 99 237 + 99 336 
 
 the greater is -T+T' or - - r^- = 168; 
 
 237 99 138 
 
 and the least --- , or =69. 
 
 22 2 
 
 Verification. 
 
 168+69=237 and 16869=99. 
 
 8. A person engaged a workman for 48 days. For 
 each day that he laboured he received 24 cents, and for 
 each day that he was idle, he paid 12 cents for his board. 
 At the end of the 48 days, the account was settled, when 
 the labourer received 504 cents. Required the number of 
 working days, and the number of days he was idle. 
 8* 
 
82 ELEMENTARY ALGEBRA. 
 
 If these two numbers were known, by multiplying them 
 respectively by 24 and 12, then subtracting the last product 
 from the first, the result would be 504. Let us indicate 
 these operations by means of algebraic signs.. 
 
 Let x = the number of working days. 
 
 48 x = the number of idle days. 
 Then, 24 x oc = the amount earned, 
 and 12(48 x)= the amount paid for his board. 
 Then, 24# 12(48 #) = 504 
 
 what he received, which is the statement. Then to find % 
 we first multiply by 12, which gives 
 
 24a? 576 + I2a?=504. 
 
 
 or, 36^=5044-576 = 1080, . 
 
 a?= =30 the working days. 
 
 oO 
 
 whence, 4830 = 18 the idle days. 
 
 Verification. 
 
 Thirty day's labour, at 24 cents 
 a day, amounts to ...... 30x24=720 cents 
 
 And 18 day's board, at 12 cents 
 a day, amounts to ...... 18 X 12=216 cents. 
 
 The difference is the amount received 504 cents. 
 
 General Solution. 
 
 This question may be made general, by denoting the 
 whole number of working and idle days by n. 
 
 The amount received for each day he worked by a. 
 The amount paid for his board, for each idle day, by b 
 
EQUATIONS OF THE FIRST DEGREE. 83 
 
 And the balance due the laborer, or the result of the 
 account, by c, 
 
 As before, let the number of working days be repre- 
 sented by a:. 
 
 The number of idle days will be expressed by n x. 
 . Hence, what he earns will be expressed by ax. 
 
 And the sum to be deducted, on account of his board, by 
 
 The equation of the problem therefore is 
 ax b(n x)=c, 
 
 which is the statement. To find x we first multiply by 6, 
 which gives 
 
 ax bn-\-bx=c 
 or, (a-\-b)x=c -{-bn 
 
 c -\-bn . . 
 
 whence, x= TTT~~~ working days. 
 
 c -4-bn an-4-bn c^-bn 
 
 and consequently, n x=n -7= , 
 
 < a -4-0 a+b 
 
 anc ... . 
 
 or n x= = idle days. 
 
 a-{-b 
 
 Let us now suppose n = 48, a = 24, 5 = 12, and c=r504 
 These numbers will give for x the same value as before 
 found. 
 
 9. A person dying leaves half of his property to his wite, 
 one-sixth to each of two daughters, one-twelfth to a servant 
 and the remaining $600 to the poor : what was the amount 
 of his property? 
 
84 ELEMENTARY ALGEBRA. 
 
 Represent the amount of the property by x. 
 
 Then, - = what he left to his wife, 
 Z 
 
 - -= what he left to one daughter, 
 and = - what he left to both daughters, 
 
 D o 
 
 = what he left to his servant. 
 
 $600 to the poor. 
 Then, by the conditions of the question 
 
 Y+T+TI +600= * 
 
 the amount of the property, which gives x =$7200. 
 
 10. A and B play together at cards. A sits down with 
 $84 and B with $48. Each loses and wins in turn, when 
 it appears that A has five times as much as B. How much 
 did A win ? 
 
 Let x represent what A won. 
 Then A rose with $84+ a? dollars, 
 
 and B rose with $48 a: dollars. 
 
 But by the conditions of the question, we have 
 
 84+;r=5(48 a:), 
 
 hence, 84-j-o?=240 5#; 
 
 and, 6* =156, 
 
 consequently, oc = $26 what A won. 
 
 Verification. 
 
 84+26 = 110; 48-26=22; 
 110 = 5(22) = 110 
 
 
EQUATIONS OF THE FIRST DEGREE. 85 
 
 11. A can do a piece of work alone in 10 days, B in 13 
 days : in what time can they do it if they work together ? 
 Denote the time by cc, arid the work to be done by 1. 
 
 Then in 1 day A could do of the work, and B could 
 do ; and in x days A could do of the work, and 
 
 B, : hence, by the conditions of the question 
 lo 
 
 KT 13 ' 
 which gives 1 3x + 1 Ox = 1 30 : 
 
 hence, 23x= 130, x= = 5| days. 
 
 o 
 
 12. A fox, pursued by a greyhound, has a start of 60 
 leaps. He makes 9 leaps while the greyhound makes but 
 6 ; but, three leaps of the greyhound are equivalent to 7 
 of the fox. How many leaps must the greyhound make to 
 overtake the fox ? 
 
 From the enunciation, it is evident that the distance to 
 be passed over by the greyhound is composed of the 60 
 leaps which the fox is in advance, plus the distance that the 
 fox passes over from the moment when the greyhound starts 
 in pursuit of him. Hence, if we can find the expression for 
 these two distances, it will be easy to form the equation of 
 the problem. 
 
 Let x= the -number of leaps made by the greyhound 
 before he overtakes the fox. 
 
 Now, since the fox makes 9 leaps while the greyhound 
 
 9 3 
 
 makes but 6," the fox will make or leaps while 
 
 6 2 
 
86 ELEMENTARY ALGEBRA. 
 
 the greyhound makes 1 ; and, therefore, while the greyhound 
 
 3 
 makes x leaps, the fox will make # leaps. 
 
 2 
 
 Hence, the distance which the greyhound must pass over 
 
 3 
 
 will be expressed by 60 -\ -- x leaps of the fox. 
 
 2 
 
 It might be supposed, that in order to obtain the equation, 
 
 3 
 it would be sufficient to place x equal to 60+ z; but 
 
 in doing so, a manifest error would be committed ; for the 
 leaps of the greyhound are greater than those of the fox,'and 
 we should then equate numbers of different denominations ; 
 that is, numbers referring to different units. Hence it is 
 necessary to express the leaps of the fox by means of those 
 of the greyhound, or reciprocally. Now, according to the 
 enunciation, 3 leaps of the greyhound are equivalent to 7 
 leaps of the fox, then 1 leap of the greyhound is equiva- 
 
 lent to leaps of the fox, and consequently x leaps of 
 o 
 
 7x 
 
 the greyhound are equivalent to of the fox. 
 
 o 
 
 Hence, we have the equation 
 
 
 making the denominators disappear 
 
 whence . 5# 360 and x=72. 
 
 Therefore the greyhound will make 72 leaps to overtake 
 the fox, and during this time the fox will make 
 
 72 X or 108 
 
EQUATIONS OF THE FIRST DEGREE. 87 
 
 Verification. 
 The 72 leaps of the greyhound are equivalent to 
 
 leaps of the fox. And 
 
 60+108 = 168, 
 
 the leaps which the fox made from the beginning. 
 
 13. A father leaves his property, amounting to $2520, to 
 four sons, A, B, C, and D. C is to have $360, B as much 
 as C and D together, and A twice as much as B less 
 $1000 : how much does A, B, and D receive ? 
 
 Ans. A $760, B $880, D $520. 
 
 14. An estate of $7500 is to be divided between a widow, 
 two sons, and three daughters, so that each son shall receive 
 twice as much as each daughter, and the widow herself 
 $500 more than all the children : what was her share, and 
 what the share of each child ? 
 
 / Widow's share $4000. 
 Ans. J Each son's $1000. 
 
 t Each daughter's $500. 
 
 15. A company of 180 persons consists of men, women, 
 and children. The men are 8 more in number than the 
 women, and the children 20 more than the men and women 
 together : how many of each sort in the company ? 
 
 Ans. 44 men, 36 women, 100 children. 
 
 16. A father divides $2000 among five sons, so that each 
 elder should receive $40 more than his next younger bro- 
 ther: what is the share of the youngest? Ans. $320. 
 
 17. A purse of $2850 is to be divided among three per- 
 sons, A, B, and C. A's share is to be to B's as 6 to 1 1. 
 
88 ELEMENTARY ALGEBRA. 
 
 and C is to have $300 more than A and B together : what 
 is each one's share ? 
 
 Ans. A's $450, B's $825, C's $1575 
 
 18. Two pedestrians start from the same point; the first 
 steps twice as far as the second, but the second makes 5 
 steps while the first makes but one. At the end of a certain 
 time they are 300 feet apart. Now, allowing each of the 
 longer paces to be 3 feet, how far will each have travelled? 
 
 Ans. 1st, 200 /e^; 2nd, 500. 
 
 19. Two carpenters, 24 journeymen, and 8 apprentices, 
 received at the end of a certain time $144. The carpen- 
 ters received $1 per day, each journeyman half a dollar, 
 and each apprentice 25 cents : how many days were they 
 employed ? A ns. 9 a 1 ays. 
 
 20. A capitalist receives a yearly income of $2940 : four- 
 fifths of his money bears an interest of 4 per cent, and the 
 remainder at 5 per cent : how much has he at interest ? 
 
 Ans. 70000. 
 
 21. A cistern containing 60 gallons of water has three 
 unequal cocks for discharging it ; the largest will empty it 
 in one hour, the second in two hours, and the third in three : 
 in what time will the cistern be emptied if they all run 
 together ? AJIS. 32^min. 
 
 22. In a certain orchard are apple trees, J peach trees, 
 J plum trees, 120 cherry trees, and 80 pear trees : how 
 many trees in the orchard ? Ans. 2400. 
 
 23. A farmer being asked how many sheep he had, 
 answered that he had them in five fields ; in the 1 st he 
 had i, in the 2nd J, in the 3rd J, and in the 4th j^, and in 
 the 5th 450 : how many had he ? Ans. 1200. 
 
 24. My horse and saddle together are worth $132, and 
 the horse is worth ten times as much as the saddle : what 
 is the value of the horse ? Ans. 120. 
 
EQUATIONS OF THE FIRST DEGREE. 8<) 
 
 25. The rent of 'm estate is this year 8 per cent greater 
 than it was last. This year it is $1890 : what was it last 
 year? Ans. $1750. 
 
 26. What number is that from which, if 5 be subtracted, 
 jj of the remainder will be 40 ? Ans. 65. 
 
 27. A post is 1 in the mud, J in the water, and 10 feet 
 above the water : what is the whole length of the post ? 
 
 Ans. 24 feet. 
 
 28. After paying \ and \ of my money, I had 66 guineas 
 left in my purse : how many guineas were in it at first ? 
 
 Ans. 120. 
 
 29. A person was desirous of giving 3 pence apiece to 
 some beggars, but found he had not money enough in his 
 pocket by 8 pence : he therefore gave them each 2 pence 
 and had 3 pence remaining : required the number of beggars. 
 
 Ans. 11. 
 
 30. A person in play lost J of his money, and then won 
 3 shillings ; after which he lost J of what he then had ; 
 and this done, found that he had but 12 shillings remaining : 
 what had he at first ? Ans. 20.?. 
 
 31. Two persons, A and B, lay out equal sums of money 
 in trade; A gains $126, and B loses $87, and A's money 
 is now double of B's : what did each lay out ? 
 
 Ans. $300. 
 
 32. A person goes to a tavern with a certain sum of mo- 
 ney in his pocket, where he spends 2 shillings ; he then 
 borrows as much money as he had left, and going to another 
 tavern, he there spends 2 shillings also ; then borrowing 
 again as much money as was left, he went to a third tavern, 
 where likewise he spent two shillings and borrowed as 
 mirch as he had left ; and again spending 2 shillings at a 
 fourth tavern, he then had nothing remaining. What had 
 he ai first? Ans. 3*. 9rf. 
 
 9 
 
90 ELEMENTARY ALGEBRA. 
 
 Of Equations of the First Degree involving two or 
 more unknown quantities. 
 
 72. Although several of the questions hitherto resolved 
 contained in their enunciation more than one unknown quan- 
 tity, we have resolved them all by employing but one sym- 
 bol. The reason of this is, that we have been able, from 
 the conditions of the enunciation, to express easily the other 
 unknown quantities by means of this symbol ; but we are 
 unable to do this in all problems containing more than one 
 unknown quantity. 
 
 To ascertain how problems of this kind are resolved, let 
 us take some of those which have been resolved by means 
 of one unknown quantity. 
 
 1 . Given the sum of two numbers equal to 36 and their 
 difference equal to 12, to find the numbers. 
 
 Let x= the greater, and y= the less number. 
 
 Then, by the 1st condition x+y 36, 
 
 and by the 2nd, x y = 12. 
 
 By adding (Art. 65, Ax. 1), .... 2#=48. 
 
 By subtracting (Art. 65, Ax. 2), ... 2y=24. 
 
 Each of these equations contains but one unknown quantity 
 
 46 
 
 From the first we obtain #==24. 
 
 24 
 And from the second y=I2 
 
 Verification. 
 
 ar+y=36 gives 24+12 = 36 
 x V =12 24 12 = 12 
 
2 
 a-b 
 
 EQUATIONS OF THE FIRST DEGREE. 91 
 
 General Solution. 
 Let x=. the greater, and y the less number. 
 
 Then by the conditions x-{-y=.a, 
 
 and xy=b. 
 
 By adding, (Art. 65, Ax. 1), . . . 2x=a+b. 
 
 By subtracting, (Art. 65, Ax. 2), . . 2y=ab. 
 
 Each of these equations contains but one unknown quantity 
 From the first we obtain x= 
 
 And from the second y= 
 
 '* 
 
 Verification. 
 
 a+b , a b 2a a+b ab 2b 
 
 +-5- = T =; and =y=*. 
 
 For a second example, let us also take a problem that has 
 oeen already solved. 
 
 2. A person engaged a workman for 48 days. For each 
 day that he labored he was to receive 24 cents, and for each 
 day that he was idle he was to pay 12 cents for his board. 
 At the end of the 48 days the account was settled, when 
 the laborer received 504 cents. Required the number of 
 working days, and the number of days he was idle. 
 
 Let x=. the number of working days, 
 
 y= the number of idle days. 
 Then, 24a?=r what he earned, 
 and 12y= what he paid for his board. 
 
 Then, by the conditions of the question, we have 
 
 x+y =48, 
 and 24# 12y = 504. 
 
 This is the statement of the question. 
 
92 ELEMENTARY ALGEBRA. 
 
 It has already been shown (Art. 65, Ax. 3), that the two 
 members of an equation can be multiplied by the same num- 
 ber, without destroying the equality. Let, then, the first 
 equation be multiplied by 24, the coefficient of x in the 
 second : we shall then have 
 
 24a 12y 504. 
 And by subtracting, 36y= 648, 
 
 648 
 and y=~W= 18 ' 
 
 Substituting this value of y in the equation 
 
 24a> 12y = 504, we have 24# 216 = 504, 
 which gives 
 
 24*=504-f-216:=720, and x=^- = 30. 
 
 Verification. 
 
 x+ y= 48 gives 30 + 18= 48, 
 
 24* 12y=504 gives 24 x 30-12 x 18 = 504. 
 
 Elimination. 
 
 7 3. The method which has just been explained of com- 
 bining two equations, involving two unknown quantities, and 
 deducing therefrom a single equation involving but one, is 
 called elimination. 
 
 QUEST. 73. What is elimination 1 How many methods of ehm ; na- 
 tion are there 1 Give the rule for elimination by addition and subtrac- 
 tion t What is the first step] What the second] What the third' 
 
EQUATIONS OF THE FIRST DEGREE. 93 
 
 There are three principal methods of elimination : 
 
 1st. By addition and subtraction 
 
 2d. By substitution. 
 
 3d. By comparison. 
 We will consider these methods separately. 
 
 Elimination by Addition and Subtraction. 
 
 1. Take the two equations 
 
 3x-2y=7 
 
 If we add these two equations, member to member, we 
 obtain 
 
 which gives, by dividing by 11 
 
 and substituting this value in either of the given equations, 
 we find 
 
 2. Again, take the equations 
 
 8a:-h2y-48 
 3x+2y=23. 
 
 If we subtract the 2nd equation from the first, we obtain 
 
 5x=25, 
 which gives, by dividing by 5 
 
 x=5: 
 and by substituting this value, we find 
 
 9* 
 
94 ELEMENTARY ALGEBRA. 
 
 3 Take the two equations 
 
 If, in these equations, one of the unknown quantities was 
 affected with the same coefficient, we might, by a simple 
 subtraction, form a new equation which^would contain but 
 one unknown quantity. 
 
 Now, if both members of the first equation be multiplied 
 by 9, the coefficient of y in the second, and the two mem- 
 bers of the second by 7, the coefficient of y in the first, we 
 will obtain 
 
 45z+63y=387, 
 77o?+63y=483. 
 
 Subtracting, then, the first of these equations from the 
 second, there results 
 
 32#=96, whence x=3. 
 
 Again, if we multiply both membeis of the first equation 
 by 1 1 , the coefficient of x in the second, and both members 
 of the second by 5, the coefficient of x in the first, we will 
 form the two equations 
 
 55;r+77y = 473, 
 
 Subtracting, then, the second of these two equations from 
 the first, there results 
 
 32y=128, whence y=4. 
 Therefore x =3 and y 4, are the values of x and y. 
 
 Verification* 
 
 5;r+7y=43 gives 5 x 3 + 7x4 = 15+28=43 ; 
 ll*+95=69 11x3 + 9x4= 
 
EQUATIONS OF THE FIRST DEGREE. 95 
 
 The method of elimination just explained, is called the 
 method by addition and subtraction.- 
 
 To eliminate by this method we have the following 
 
 RULE. 
 
 . I. See which of the unknown quantities you will eliminate. 
 
 II. Make the coefficient of this unknown quantity the same 
 in both equations, either by multiplication or division. 
 
 III. If the signs of the like terms are the same in both 
 equations, subtract one equation from the other ; but if the 
 signs are unlike, add them. 
 
 EXAMPLES. 
 
 4. Find the values of x and y in the equations 
 
 3* y=3, 
 y+2x=7. 
 
 Ans. x=2, y=3. 
 
 5. Find the values of x and y in the equations 
 
 4x 7y= 22, 
 5a? |-2y=37. 
 
 Ans. x=5 t y=6 
 
 6. Find the values of x and y in the equations 
 
 8x6y= 3. 
 
 Ans. x=4%, y=5|. 
 7. Find the values of a? and y in the equations 
 
 8x9y=l. 
 6x3y = 4x. 
 
 Ans. x=, y~. 
 
96 ELEMENTARY ALGEBRA. 
 
 8. Find the values of .r and y in the equations 
 
 14* 15y 12, 
 7x+ 8y = 37. 
 
 Ans. x =3, y=%> 
 
 9. Find the values of x and y in the equations 
 
 10. Find the values of x and y in the equations 
 
 1+1,=4, 
 
 a y 2. 
 
 Ans. x=14, y-16. 
 
 11. Says A to B, you give me $40 of your money, and 
 I shall then have 5 times as much as you will have left. 
 Now they both had $120 : how much had each ? 
 
 Ans. Each had $60. 
 
 12. A Father says to his son, "twenty years ago, my 
 age was four times yours ; now, it is just double ;" what wer 
 their ages ? 4 5 Father's 60 years 
 
 \ Son's 30 years. 
 
 13. A Father divides his property between his two sons. 
 At the end of the first year the elder had spent one quarter 
 of his, and the younger had made $1000, and their property 
 was then equal. After this the elder spends $500 and the 
 younger makes $2000, when it appears the younger has 
 just double the elder : what had each from the father ? 
 
 . ( Elder $4000 
 t Younger $2000 
 
EQUATIONS OF THE FIRST DEGREE. 97 
 
 14. If John give Charles 15 apples, they will have the 
 same number; but if Charles give 15 to John, John will 
 have 15 times as many wanting 10 as Charles will have 
 left. How many had each ? . ( John 50. 
 
 (Charles 20. 
 
 15. Two clerks, A and B, have salaries which are to 
 gether equal to $900. A spends ^ per year of what he 
 receives, and B adds as much to his as A spends. At the 
 end of the year they have equal sums : what was the salary 
 of each? 
 
 'Elimination by Substitution. 
 7 4. Let us again take the equations 
 
 5a?+7y=43, 
 Ila:-h9y=69. 
 
 Find the value of x in the first equation, which gives 
 
 Substitute this value of x in the second equation, and we 
 have 
 
 43 7y 
 11 X r-^-+9y=69, 
 
 D 
 
 or, 
 or, 
 Hence, 
 
 and, 
 
98 ELEMENTARY ALGEBRA. 
 
 This method is called the method by substitution : we 
 have for it the following 
 
 RULE. 
 
 Find the value of one of the unknown quantities in either 
 of the equations, and substitute this value for the same unknown 
 quantity in the other equation : there will thus arise a new 
 equation with but one unknown quantity. 
 
 REMARK. This method of elimination is used to great 
 advantage when the coefficient of either of the unknown 
 quantities is unity. 
 
 it ' 
 
 EXAMPLES. 
 / 
 
 1 . Find, by the last method, the values of x and y in the 
 equations 
 
 3a? y=l and 3y 2a?=4 
 
 Ans. =1, y=2. 
 
 2. Find the values of x and y in the equations 
 
 5y 4a?= 22 and 3y-f4a?=38. 
 
 Ans. x8, y=2. 
 
 3. Find the values of x and y in the equations 
 
 a?+8y=18 and y 3x= 29. 
 
 Ans. x=lQ, y=l. 
 
 4. Find the values of x and y in the equations 
 
 5*-y=13 and 8x+^-y=29. 
 
 y 
 Ans. # 3J, y=4j. 
 
 QUEST. 74. Give the rule for elimination by substitution' When is 
 it desirable to use this method 11 
 
EQUATIONS OF THE FIRST DEGREE. 99 
 
 6. Find the values of x and y from the equations 
 
 IQx ^-=69 and lOy ^-=49. 
 o 7 
 
 Ans. #=7, y=5. 
 6. Find the values of x and y from the equations 
 
 -= I and - 
 
 Ans. a?=8, y=10. 
 7. Find the values of x and y in the equations 
 
 8 Find the values of x and y in the equations 
 i+ + .= ana -. 
 
 9. Find the values of x and y from the equations 
 
 10. Find the values of a and y from the equations 
 
 --i' and 5 ^-= 29 - 
 
 ?=6, y=7. 
 
 1 1 . Two misers A and B sit down to count over their 
 money. They both have $20000, and B has three times as 
 much as A : how much has each ? 
 
 A . . $5000. 
 
 B . $15000 
 
100 ELEMENTAKV ALGEBRA. 
 
 12. A person has two purses. If he puts $7 into the 
 first purse, it is worth three times as much as the second : 
 but if he puts $7 into the second it becomes worth five 
 times as much as the first : what is the value of each purse ? 
 
 Ans. 1st, $2 : 2nd, $3. 
 
 13. Two numbers have the following properties : if the 
 first be multiplied by 6 the product will be equal to the 
 second multiplied by 5 ; and one subtracted from the first 
 leaves the same remainder as 2 subtracted from the second : 
 what are the numbers ? Ans. 5 and 6. 
 
 14. Find two numbers with the following properties : 
 the first increased by 2 to be 3 times greater than the 
 second : and the second increased by 4 to be half the first : 
 what are the numbers ? Ans. 24 and 8. 
 
 15. A father says to his son, " twelve years ago I was 
 twice as old as you are now : four times your age, at that 
 time, plus twelve years, will express my age twelve years 
 hence :" what were their ages ? ^ ( Father 72 yeyrs. 
 
 nS ' \ 
 
 Son 30 
 
 Elimination by Comparison. 
 75. Take the same equations 
 
 llx+9y=69. 
 Finding the value of x in the first equation, we have 
 
 _43-7y . 
 ~5~ 
 
 and finding the value of x in the second, we obtain 
 
 x= TT~' 
 
EQUATIONS OF THE FIRST DEGREE. 101 
 
 Let these two values of x be placed equal to each other, 
 and we have 
 
 43 7y _ 69 -9y 
 5 ~~~ll ' 
 
 Or, 
 
 473 77y = 34 
 
 ) 45y; 
 
 Or, 
 
 32y= 128. 
 
 Hence. 
 
 y=4. 
 
 
 And, 
 
 69 36 
 
 = 3. 
 
 This method of elimination is called the method by 
 comparison, for which we have the following 
 
 RULE. 
 
 I. Find the value of the same unknown quantity in each 
 
 equation. 
 
 II. Place these values equal to each other; and a new 
 equation will arise with but one unknown quantity. 
 
 EXAMPLES. 
 
 1 . Find, by the last rule, the values of x and y in the 
 equations 
 
 JL_|_6=42 and y^-=U\. 
 o 22 
 
 Ans. x=ll, y=15 
 
 QUEST. 75. Give the rule for elimination by comparison ? What is 
 th< first stop 7 What the second 1 
 
 10 
 
102 ELEMENTARY ALGEBRA. 
 
 2. Find the values of x and y in the equations 
 
 l_f+5=6 and f +4=^+6. 
 
 Ans. a: 28, y=20. 
 
 3. Find the values of x and y in the equations 
 
 v x 22 
 
 - +=1 and 3 -*= 6 ' 
 
 4. Find the values of x and y in the equations 
 
 y-3 = l*+5 and fltlUy-S}. 
 
 Ans. #=2, y=9. 
 
 5. Find the values of x and y in the equations 
 
 . ar=16, y = 7. 
 6. Find the values of a; and y from the equations 
 
 y + x yx 2y 
 
 ^ f~ -=x+, and a:+y=16. 
 
 . a;=10, y=6 
 7. Find the values of x and y in the equations 
 
 8. Find the values of x and y in the equations 
 
 2y+3*=y+43, y-^^-y- 
 Ans. x=W, 
 
EQUATIONS OF THE FIRST DEGREE. 103 
 
 9. Find the values of x and y in the equations 
 4y x ^y =x -^l8 l and 27 y=ff-fy+4. 
 
 Ans. a?=9, y=7 
 1.0. Find the vralues of x and y in the equations 
 
 ) . 
 
 Ans. ff=10, y=20. 
 
 76. Having explained the principal methods of elimina- 
 tion, we shall add a few examples which may be solved by 
 either ; and often indeed, it may be advantageous to use 
 them all even in the same question. 
 
 GENERAL EXAMPLES. 
 
 1. Given 2o?+3y=16, and 3x 2y=ll to find the 
 values of x and y. Ans. x=5, y=2. 
 
 2x , 3y 9 , 3x 2y 61 
 
 2. Given -+JL =- and -^+i= l ^ to find the 
 
 values of x and y. Ans. x= , y= . 
 
 3. Given ~-{-7y=99, and -|--f 7a?=51, to find the 
 values of x and y. Ans. x =7, y=14. 
 
 4. Given 
 
 -12=+8, and 
 
 to find the values of x and y. Ans, x = 6Q, y 40. 
 
104 ELEMENTARY ALGEBRA. 
 
 QUESTIONS. 
 
 1. What fraction is that, to the numerator of which, if ] 
 be added, its value will be , but if one be added to its 
 
 o 
 
 denominator, its value will be ? 
 
 4 
 
 Let the fraction be represented by . 
 
 Then, by the question 
 
 a+1 I . x I 
 
 = and = . 
 
 y 3 y+1 4 
 
 Whence 3x+3 = y and 4a?=y+l. 
 
 Therefore, by subtracting, 
 
 x3 = l or oc=4. 
 Hence, 12 + 3=y; 
 
 therefore, y 15. 
 
 2. A market-woman bought a certain number of eggs ai 
 2 for a penny, and as many others, at 3 for a penny ; and 
 having sold them again altogether, at the rate of 5 for 2d, 
 found that she had lost 4d : how many eggs had she ? 
 
 Let 2o?= the whole number of eggs. 
 
 Then x= the number of eggs of each sort. 
 
 Then will "9-*= the cost f tne ^ rst sort > 
 
 and x = the cost of the second sort. 
 
 o 
 
 But 5 : 2x : : 2 : the amount for which the eggs 
 o 
 
 wore sold. 
 
EQUATIONS OF THE FIRST DEGREE. 105 
 
 Hence, by the question 
 
 11 4x 
 _*+_*__ = 4 . 
 
 Therefore, 15x+Wx24x=120 
 
 or a::=120 ; 
 
 the number of eggs of each sort. 
 
 3. A person possessed a capital of 30,000 dollars, foi 
 which he drew a certain interest ; but he owed the sum of 
 20,000 dollars, for which he paid a certain interest. The 
 interest that he received exceeded that which he paid by 
 800 dollars. Another person possessed 35,000 dollars, for 
 which he received interest at the second of the above rates ; 
 but he owed 24,000 dollars, for which he paid interest at 
 the first of the above rates. The interest that he received 
 exceeded that which he paid by 310 dollars. Required the 
 two rates of interest. 
 
 Let oc and y denote the two rates of interest ; that is, the 
 interest of $100 for the giv#n time. 
 
 To obtain the interest of $30,000 at the first rate, denoted 
 by a?, we form the proportion 
 
 100 : x : : 30,000 : : 3 f X or 300*. 
 100 
 
 And for the interest $20,000, the rate being y, 
 
 100 : y : : 20,000 : : gS. or 200y. 
 
 But from the enunciation, the difference between these 
 i wo interests is equal to 800 dollars. 
 
 We have, then, for the first equation of the problem 
 
 300* 200y 800 
 10* 
 
106 ELEMENTARY ALGEBRA. 
 
 By writing algebraically the second condition of the pro- 
 blem, we obtain the other equation, 
 
 Both members of the first equation being divisible by 1 00, 
 and those of the second by 10, we may put the following, 
 in place of them : 
 
 3x 2 = 8 35 
 
 To eliminate x, multiply the first equation by 8, and then 
 add it to the second ; there results 
 
 19y=95, whence y=5. 
 
 Substituting for y, in the first equation, its value, this 
 equation becomes 
 
 3x 10 8, whence x=6. 
 Therefore, the first rate is 6 per cent, and the second 5. 
 
 Verification. 
 
 $30,000, placed at 6 per cent, gives 300x6 $1800. 
 $20,000, 5 200x5 = $1000. 
 
 And we have 1800 1 000 800. 
 
 The second condition can be verified in the same manner. 
 
 4. What two numbers are those, whose difference is 7, 
 and sum 33 1 Ans. 13 and 20. 
 
 5. To divide the number 75 into two such parts, that 
 three times the greater may exceed seven times the less 
 by 15. Ans. 54 and 21. 
 
 6. In a mixture of wine and cider, J of the whole plus 
 25 gallons was wine, and J part minus 5 gallons was cider : 
 how many gallons were there of each ? 
 
 Ans. 85 of wine, and 35 of cider 
 
EQUATIONS OF THE FIRST DEGREE. 107 
 
 7. A bill of 120 waa.paid in guineas and moidores, and 
 the number of pieces of both sorts that were used was just 
 100. If the guinea be estimated at 21*, and the moidore 
 at 27s, how many were there of each ? Ans. 50 of each. 
 
 8. Two travellers set out at the same time from London 
 and York, whose distance apart is 150 miles. One of them 
 goes 8 miles a day, and the other 7 : in what time will they 
 meet? Ans. In 10 days. 
 
 9. At a certain election, 375 persons voted for two can- 
 didates, and the candidate chosen had a majority of 91 : 
 how many voted for each ? 
 
 Ans. 233 for one, and 142 for the other. 
 
 10. A person has two horses, and a saddle worth 50. 
 Now, if the saddle be put on the back of the first horse, it 
 will make his value double that of the second ; but if it be 
 put on the back of the second, it will make his value triple 
 that of the first. What is the value of each horse ? . 
 
 Ans. One .30, and the other 40. 
 
 1 1 . The hour and minute hands of a clock are exactly 
 together at 12 o'clock : when are they next together ? 
 
 Ans. Ihr. 5 T 5 T mm. 
 
 12. A man and his wife usually drank out a cask of beer 
 in 12 days ; but when the man was from home, it lasted 
 the woman 30 days : how many days would the man alone 
 be in "drinking it? Ans. 20 days. 
 
 13. If 32 pounds of sea-water contain 1 pound of salt, 
 how much fresh water must be added to these 32 pounds, 
 in order that the quantity of salt contained in 32 pounds of 
 the new mixture shall be reduced to 2 ounces, or l of a 
 pound? Ans. 224lb. 
 
 14. A person who possessed 100,000 dollars, placed the 
 greater part of it out at 5 per cent interest, and the other 
 
1 08 ELEMENTARY ALGEBRA. 
 
 at 4 per cent. The interest which he received for tr whole 
 amounted to 4640 dollars. Required the two parts. 
 
 Ans. 64,000 arid 36,000. 
 
 15. At the close of an election, the successful candidate 
 had a majority of 1500 votes. Had a fourth of the votes 
 of the unsuccessful candidate been also given to him, he 
 would have received three times as many as his competitor 
 wanting three thousand five hundred : how many votes did 
 each receive? ( 1st, 6500 
 
 < 2d, 5000. 
 
 16. A gentlemen bought a gold and a silver watch, and 
 a chain worth $25. When he put the chain on the gold 
 watch, it was worth three and a half times more than the 
 silver watch ; but when he put the chain on the silver watch, 
 it was worth one half the gold watch and 15 dollars over: 
 what was the value of each watch ? 
 
 Gold watch $80. 
 
 Ans. 
 
 Silver $30. 
 
 17. There is a certain number expressed by two figures, 
 which figures are called digits. The sum of the digits is 
 11, and if 13 be added to the first digit the sum will be three 
 times the second : what is the number ? Ans. 56. 
 
 18. From a company of ladies and gentlemen 15 ladies 
 retire ; there are then left two gentlemen to each lady. 
 After which, 45 gentlemen depart, when there are left 5 
 ladies to each gentleman : how many were there of each at 
 first? A (50 gentlemen. 
 
 ( 40 ladies. 
 
 19. A person wishes to dispose of his horse by lottery 
 If he sells the tickets at $2 each, he will lose $30 on his 
 horse ; but if he sells them at $3 each, he will receive $30 
 more than his horse cost him. What is the value of the 
 horse and number of tickets? A __ $ Horse . . . $150. 
 
 its 60 
 
 ( Horse . . . 
 Ans. < 
 
 (No. of ticket 
 
EQUATIONS OF THE FIRST DEGREE. 109 
 
 20. A person purchases a lot of wheat at $1 , and a lot of 
 rye at 75 cents per bushel, the whole costing him $117,50. 
 He then sells of his wheat and of his rye at the same 
 rate, and realizes $27,50. How much did he buy of each ? 
 
 ( 80bu. of wheat 
 Ans. < 
 
 50bu. of rye. 
 
 Equations involving three or more unknown quantities. 
 
 77. Let us now consider the case of three equations 
 involving three unknown quantities. 
 Take the equations 
 
 3z= 19, 
 
 To eliminate z by means of the first two equations, mul- 
 tiply the first by 3 and the second by 4 ; then, since the 
 coefficients of z have contrary signs, add the two results 
 >,ogether. This gives a new equation : 
 
 43ar 2y=121. 
 
 Multiplying the second equation by 2, a factor of the co- 
 efficient of z in the third equation, and adding them together, 
 we have 
 
 16a;-f9y = 84. 
 
 The question is then reduced to finding the values of x 
 and y, which will satisfy these new equations. 
 
 Now, if the first be multiplied by 9, the second by 2, and 
 the results be added together, we find 
 
 whence x=3 
 
110 ELEMENTARY ALGEBRA. 
 
 We might, by means of the two equations involving a 
 and y, determine y in the same way we have determined x ; 
 but the value of y may be determined more simply, by ob- 
 serving that the last of these two equations becomes, by 
 substituting for x its value found above, 
 
 48-f-9y=84, whence y= - =4. 
 
 y 
 
 In the same manner the first of the three proposed equa- 
 tions becomes, by substituting the values of x and y, 
 
 24 
 15 24-{-4z = l5, whence z= = 6. 
 
 4 
 
 Hence, to solve equations containing three or more un- 
 known quantities, we have the following 
 
 RULE. 
 
 I. To eliminate one of the unknown quantities, combine any 
 one of the equations with each of the others ; there will thus be 
 obtained a series of new equations containing one less unknown 
 quantity. 
 
 II. Eliminate another unknown quantity by combining one 
 of these new equations with the others. 
 
 III. Continue this series of operations until a single equa- 
 tion containing but one unknown quantity is obtained, from 
 which the value of this unknown quantity is easily found. 
 Then, by going back through the series of equations which have 
 been obtained, the vdtues of the other unknown quantities may 
 be successively determined. 
 
 QUEST. 77. Give the general rule for solving equations involving 
 three or more unknown quantities'! What is the first step] What the 
 second 1 What the third ? 
 
EQUAT1OJNS OF THE FIRST DEGREE. Ill 
 
 78. REMARK. It often happens that each of the pro- 
 posed equations does not contain all the unknown quantities. 
 In this case, with a little address, the elimination is very 
 quickly performed. 
 
 Take the four equations involving four unknown quantities: 
 
 (1.) 2x 3y+2z= 13. (3.) 
 
 (2.) 4u 2x= 30. (4.) 5y+3=32. 
 
 By inspecting these equations, we see that the elimina- 
 tion of z in the two equations, (1) and (3), will give an 
 equation involving x and y ; and if we eliminate u in 
 the equations (2) and (4), we shall obtain a second equation, 
 involving x and y. These two last unknown quantities 
 may therefore be easily determined. In the first place, the 
 elimination of z in (1) and (3) gives 
 
 7y2x=l ; 
 That of u in (2) and (4), gives 
 
 Multiplying the first of these equations by 3, and adding, 
 
 41y=41 ; 
 
 Whence y= I. 
 
 Substituting this value in 7y 2x=l, we find 
 
 x=3. 
 Substituting for x its value in equation (2), it becomes 
 
 4w 6 30 : 
 
 Whence u = 9. 
 
 And substituting for y its value in equation (3), there 
 results z =5. 
 
1. Given 
 
 3. Given 
 
 ELEMENTARY ALGEBRA. 
 
 EXAMPLES. 
 
 x-\ yH z = \5 }> to find a?, y and z. 
 
 3 45 
 
 Ans. a?=12, y=r20, ^ = 
 
 4. Divide the number 90 into four such parts that tht 
 first increased by 2, the second diminished by 2, the third 
 multiplied by 2, and the fourth divided by 2, shall be equa\ 
 to each other. 
 
 This question may be easily solved by introducing a new 
 unknown quantity. 
 
 Let a?, y, z, and u, be the required parts, and desig 
 nate by m the several equal quantities which arise from 
 the conditions. We shall then have 
 
 =m. 
 
EQUATIONS OF THE FINEST DEGREE. 113 
 
 Krom which we find 
 
 x=.m 2, y=rm-f-2, z , ac2m. 
 And by adding the equations, 
 
 x + y+z + u .m-\-m-\ 4-2m=4jm. 
 
 And since, by the conditions of the question, the first 
 member is equal to 90, we have 
 
 41/72 = 90, or fm = 90; 
 hence m=20. 
 
 Having the value of m, we easily find the other values : 
 viz. 
 
 * =18, y=22, *=10, = 40. 
 
 5. There are three ingots composed of different metals 
 mixed together. A pound of the first contains 7 ounces of 
 silver, 3 ounces of copper, and 6 of pewter. A pound of 
 the second contains 12 ounces of silver, 3 ounces of cop- 
 per, and 1 of pewter, A pound of the third contains 4 
 ovnces of silver, 7 ounces of copper, and 5 of pewter. It 
 is required to find how much it will take of each of the 
 three ingots to form a fourth, which shall contain in a 
 pound, 8 ounces of silver, 3^ of copper, and 4^- of pewter. 
 
 Let ar, y, and z represent the number of ounces which it 
 is necessary to take from the three ingots respectively, in 
 order to form a pound of the required ingot. Since there 
 are 7 ounces of silver in a pound, or 16 ounces, of the 
 first ingot, it follows that one ounce of it contains r 7 ^ of an 
 ounce of silver, and consequently in a number of ounce* 
 
 denoted by a?, there is - ounces of silver. In the sam 
 lo 
 
 11 
 
I 14 ELEMENTARY ALGEBRA. 
 
 manner we would find that 2- and , express the num 
 
 16 16 
 
 her of ounces of silver taken from the second and third, to 
 form the fourth ; but from the enunciation, one pound of this 
 fourth ingot contains 8 ounces of silver. We have, then, 
 for the first equation, 
 
 7x I2y 4* 
 16 + 16 + 16- 
 
 or, making the denominators disappear, 
 
 As respects the copper, we should find 
 3tf + 3y +7* = 60, 
 and with reference to the pewter 
 
 As the coefficients of y in these three equations, are 
 the most simple, it is most convenient to eliminate this un- 
 known quantity first. 
 
 Multiplying the second equation by 4, and subtracting 
 the first, we have 
 
 5x + 24z=:ll2. 
 
 Multiplying the third equation by 3, and subtracting the 
 second from the product, 
 
 Multiplying this last equation by 3, and subtracting the 
 preceding one from the product, we obtain 
 
 40*:= 320, 
 whence x=S. 
 
EQUATIONS OF THE FIRST DEGREE. 
 
 Substitute this value for x in the equation 
 
 it becomes 120 + 8* = 144, 
 
 whence z=3. 
 
 Lastly, the two values a; =8, z=3, being substituted in 
 the equation 
 
 give 48-hy-fl5=68, 
 
 whence y=5- 
 
 Therefore, in order to form a pound of the fourth ingot, 
 we must take 8 ounces of the first, 5 ounces of the second, 
 and 3 of the third. 
 
 Verification. 
 
 If there be 7 ounces of silver in 1 6 ounces of the first 
 ingot, in 8 ounces of it, there should be a number of ounces 
 of silver expressed by 
 
 7x8 
 16 ' 
 In like manner, 
 
 12x5 4x3 
 
 and 
 
 16 16 
 
 will express the quantity of silver contained in 5 ounces of 
 the second ingot, and 3 ounces of the third. 
 Now, we have 
 
 7X8 12x5 4x3_ 128 _ 
 16 ~~I6~~ 16 " = 16 ~ ' 
 
 therefore, a pound of the fourth ingot contains 8 ounces of 
 silver, as required by the enunciation. The same condi- 
 tions may be verified relative to the copper and pewter 
 
1 1 6 ELEMENTARY ALGEBRA. 
 
 6 A's age is double B's, and B's is triple of C's, and the 
 sum of all their ages is 140. What is the age of each ? 
 
 Ans. A's~84, B's = 42, and C's=14. 
 
 7. A person bought a chaise, horse, and harness, for 
 60 ; the horse came to twice the price of the harness, 
 and the chaise to twice the price of the horse and harness 
 What did he give for each ? 
 
 .13 6s. 8d. for the horse. 
 Ans. < . 6 13s. 4d. for the harness 
 ' 40 for the chaise. 
 
 8. To divide the number 36 into three such parts that 
 of the first, i of the second, and J of the third, may be 
 all equal to each other. Ans. 8, 12, and 16. 
 
 9. If A and B together can do a piece of work in 8 days, 
 A and C together in 9 days, and B and C in ten days ; how 
 many days would it take each to perform the same work 
 alone? " Ans. A 14f|, B 17|f, C 23^ T . 
 
 10. Three persons, A, B, and C, begin to play together, 
 having among them all $600. At the end of the first game 
 A has won one-half of B's money, which, added to his own, 
 makes double the amount B had at first. In the second 
 game, A loses and B wins just as much as had at the 
 beginning, when A leaves off with exactly what he had at 
 first. How much had each at the beginning ? 
 
 Ans. A $300, B $200, C $100. 
 
 11. Three persons, A, B, and C, together possess $3640. 
 If B gives A $400 of his money, then A will have $320 
 more than B ; but if B takes $140 of C's money, then B 
 and C will have equal sums. How much has each ? 
 
 Ans. A $800, B $1280, C $1560. 
 
 12. Three persons have a bill to pay, which neither 
 alone is able to discharge. A says to B, " Give me the 
 4th of your money, and then I can pay the bill." B says 
 to C, " Give me the 8th of yours, and I can pay it. Bui 
 
EQUATIONS OF THE FIRST DEGREE. 1 1 7 
 
 C says to A, " You must give me the half of yours before 
 I can pay it, as I have but $8." What was the amount of 
 their bill, and how much money had A and B ? 
 
 . 1 Amount of the bill, $13. 
 ' \ A had $10, and B $12. 
 
 13. A person possessed a certain capital, which he placed 
 out at a certain interest. Another person, who possessed 
 10000 dollars more* than the first, and who put out his capi- 
 tal 1 per cent, more advantageously, had an income greater 
 by 800 dollars. A third person, who possessed 15000 dol- 
 lars more than the first, putting out his capital 2 per cent, 
 more advantageously, had an income greater by 1 500 dollars. 
 Required the capitals of the three persons, and the rates of 
 interest. 
 
 , < Sums at interest, $30000, 40000, 4500U. 
 ( Rates of interest, 4 5 6 pr. ct. 
 
 14. A widow receives an estate of $15000 from her de- 
 ceased husband, with directions to divide it among two sons 
 and three daughters, so that each son may receive twice as 
 much as each daughter, and she herself to receive $1000 
 more than all the children together. What was her share, 
 and what the share of each child ? 
 
 ^ The widow's share, $8000. 
 
 Ans. ? Each son's, 2000. 
 
 ' Each daughter's, 1000. 
 
 15. A certain sum of money is to be divided between 
 three persons, A, B, and C. A is to receive $3000 less 
 than half of it, B $1000 less than one third part, and C to 
 receive $800 more than the fourth part of the whole. What 
 is the sum to be divided, and what does each receive ? 
 
 Sum, $38400. 
 
 A receives 16200. 
 B 11800. 
 C 10400. 
 
11 fc ELEMENTARY ALGEHRA. 
 
 CHAPTER IV. 
 
 Of Powers. 
 
 79. If a quantity be multiplied several times by itself 
 the product is called the power of the quantity. Thus, 
 
 a = a is the root, or first power of a. 
 
 axa = a 2 is the square, or second power of a. 
 
 aXaXa=a 3 is the cube, or third power of a. 
 
 aXaXaXa=a* is the fourth power of a. 
 
 aXaXaXaX a=a 5 is the fifth power of a. 
 
 In every power there, are three things to be considered : 
 
 1st. The quantity which is multiplied by itself, and which 
 is called the root or the first power. 
 
 2nd. The small figure which is placed at the right, and 
 a little above the letter. This figure is called the exponent 
 of the power, and shows how many times the letter enters 
 as a factor. 
 
 3rd. The power itself, which is the final product, or 
 result of the multiplications. 
 
 QUEST. 79. If a quantity be continually multiplied by itself, what is 
 the product called 1 How many things are to be considered in every 
 power 1 What are they 1 
 
OF POWERS. 119 
 
 For example, if we suppose <z = 3, we h?ve 
 
 o 3 the root, or 1st power of 3. 
 
 u z 3 2 =3x3= 9 the second power of 3. 
 
 a* = 3 3 = 3 x3x3= 27 the third power of 3. 
 
 a* = 3* 3 x3 x3x3=: 81 the fourth power of 3. 
 
 = 3 5 =:3x 3x3x3x3=243 the fifth power of 3. 
 
 In these expressions, 3 is the root, 1, 2, 3, 4 and 5 are 
 the exponents, and 3, 9, 27, 81 and 243 are the powers. 
 
 To raise monomials to any power. 
 
 8O. Let it be required to raise the monomial 2a?b 2 to 
 the fourth power. We have 
 
 which merely expresses that the fourth power is equal to 
 the product which arises from writing the quantity four 
 times as a factor. By the rules for multiplication, this pro 
 duct becomes 
 
 = 2 4 a 3 + 3 + 3 + 3 6 2 + 2 + 2 + 2 = 2 4 a 12 6 8 ; 
 
 from which we see, 
 
 1st. That the coefficient 2 must be raised to the 4th 
 power ; and, 
 
 2nd. That the exponent of each letter must be multiplied 
 by 4, the exponent of the power. 
 
 As the same reasoning would apply to every example, 
 we have, for the raising of monomials to any power, the 
 following 
 
120 ELEMENTARY ALGEBRA. 
 
 RULE 
 
 I. Raise the coefficient to the required power. 
 
 II. Multiply the exponent of each letter by the exponent oj 
 the power. 
 
 EXAMPLES. 
 
 1. What is the square of 3a 2 y 3 ? Ans. 9a 4 / 
 
 2. What is the cube of 6a 5 y 2 a? ? Ans. 216a 15 ya: 3 
 
 3. What is the fourth power of 2a 3 y 3 i 5 ? 
 
 4. What is the square of 2 A 5 y 3 ? 4ns. </ 4 i 10 y 6 . 
 
 5. What is the seventh power of a 2 bcd? ? 
 
 4ns. a 1 * We'd? 1 . 
 
 6. What is the sixth power of aWc z dl Ans. a 12 18 c 12 </ 6 . 
 
 7. What is the square and cube of 2a 2 b 2 1 
 
 Square. Cube. 
 
 By observing the way in which the powers are formed, 
 we may conclude, 
 
 1st. When the root is positive, all the powers will be positive. 
 
 2nd. When the root is negative, all the even powers will be 
 positive and all the odd powers negative. 
 
 QUEST. 8O. What is a monomial 1 Give the rule for raising a 
 monomial to any power. When the root is positive, how will the powers 
 be 1 When the root is negative, how will the powers he ' 
 
OF POWERS. 121 
 
 8. What is the square of 2<z 4 5 5 ? Ans. 4a*b w . 
 
 9. What is the cube of 5a 5 y 2 c ? Ans. 125a l *y e c 3 . 
 
 10. What is the eighth power of a 3 xy 2 1 
 
 Ans. +a 24 a? 8 y 16 
 
 11. What is the seventh power of a z yx 2 ? 
 
 Ans. aWyV*: 
 
 12. What is the sixth power of 2ab*y 5 ? 
 
 Ans. 64a*b 36 y 30 . 
 
 13. What is the ninth power of cdx 2 y 3 1 
 
 Ans. 
 
 14. What is the sixth power of 3ab 2 d ? 
 
 Ans. 
 
 15. What is the square of WaWc 3 ? Ans. 100a 4 6 4 c 6 . 
 
 16. What is the cube of 9a<W 3 / 2 ? 
 
 Ans. 
 
 17. What is the fourth pow.er of 4a 5 b 3 c*d 5 ? 
 
 18. What is the cube of 4a 2 b 2 c 3 d ? 
 
 Ans. - 
 
 19. What is the fifth power of 2a 3 b*xy ? 
 
 Ans. 
 
 20. What is the square of 20a; 4 y 4 c 5 ? Ans. 400x 8 y 8 c }0 . 
 
 21. What is the fourth power of 3a z b 2 c 3 ? 
 
 Ans. 
 
 22. What is the fifth power of c^x 2 ^ ? 
 
 Ans. 
 
 23. What is the sixth power of ac 2 df1 
 
 Ans. 
 
 24. What is the fourth power of 2a 2 c 2 d 3 ? 
 
 Ans. 
 
1 22 
 
 ELEMENTARY ALGEBRA. 
 
 To raise Polynomials to any power. 
 
 8 1 . The power of a polynomial, like that of a mono- 
 mial, is obtained by multiplying the quantity continually by 
 itself. Thus, to find the fifth power of the binomial a-\ b, 
 we have 
 
 a 4- b 
 
 a + b 
 
 a 2 + ab 
 
 4- 
 
 a z +2ab 
 a 4- b 
 
 1st power. 
 
 2nd power. 
 
 ab* 
 
 4- 
 
 4- b 3 
 
 a 3 4-3a 2 &4- 
 a + b 
 
 -f 
 
 . 3rd power. 
 
 -f & 4 4th powey 
 
 a + 
 
 6a 3 6 2 -j- 
 
 4- 
 
 a 3 6 2 + 1 Oa <2 6 3 + 5a 
 
 Ans. 
 
 REMARK. 82. It will be observed that the number of 
 multiplications is always 1 less than the units in the expo- 
 
 QUEST. 81. How is the power of a polynomial obtained ? 
 
OF POWERS. 
 
 123 
 
 nent of the power. Thus, if the exponent is 1, no multipli- 
 cation is necessary. If it is 2, we multiply once ; if it is 
 3, twice ; if 4, three times, &c. The powers of polyno- 
 mials may be expressed by means of the exponent. Thus, 
 lo express that a+b is to ba raised to the 5th power, we write 
 
 which expresses the fifth power of a-\-b. 
 2. Find the 5th power of the binomial a b. 
 
 a b ........ 1st power. 
 
 a - b 
 a 2 ab 
 
 - ab +6* 
 a?2ab -{-b 2 . .... 2nd power. 
 
 a - b 
 
 a 3 2a z b + ab z 
 
 a 3 3a 2 b+ 3ab 2 b 3 ... 3rd power. 
 
 a - b 
 
 ab 3 
 
 3ab 3 
 
 6a 2 & 2 4ab 3 + b* 4th power. 
 
 a - b 
 
 4a 2 b 3 + ab* 
 
 a 5 
 
 b 5 Ans. 
 
 QUEST. 82. How does the number of- multiplications compare with 
 the exponent of the power 1 ? If the exponent is 4, how many multipli- 
 cations 1 
 
124 
 
 ELEMENTARY ALGEBRA. 
 
 3. What is the square of 5a 
 
 5a 2c + d 
 
 5a 2c + d 
 
 25a 2 10ac+ 5ad 
 
 10<zc+ 4c 2 2cd 
 
 25 2 
 
 4cd+cP Ans. 
 
 4. Find the 4th power of the binomial 3a 2b. 
 
 3a 25 . . . . . . . . 1st power. 
 
 9a 2 6ab 
 
 9a 2 
 
 3a 2b 
 
 2nd power. 
 
 27 a 3 36a*b+12ab* 
 
 27a 3 
 3a 2b 
 
 . . 3rd power. 
 
 Sla 4 
 
 An*. 
 
 5. What is the square of the binomial a-\- 1 ? 
 
 Ans. a 
 
 6. What is the square of the binomial a 1 ? 
 
 Ans. a*2a+l. 
 
 7. What is the cube of 9a 36? 
 
 Ans. 729a 3 729a 2 5+2436 2 275 3 
 
 8. What is the third power of a 11 
 
 Ans. a 3 3a 2 -f3a 1. 
 
OF POWERS. 125 
 
 9 What is the 4th power of xyl 
 Ans. x* 
 
 10. What is the cube of the trinomial x-\-y-\-z1 
 
 Ans. x 3 + 3o% + 3x 2 z -f 3xf + 3xz 2 + 3y*z -f 3y* 2 + Gays 
 
 11. What is the cube of the trinomial 2a 2 4aZ>-f 3b 2 ? 
 . 8a 6 
 
 To raz^e a Fraction to any Power. 
 
 83. The power of a fraction is obtained by multiplying 
 the fraction by itself ; that is, by multiplying the numerator 
 by the numerator, and the denominator by the denominator 
 
 Thus, the cube of -=-, which is written 
 o 
 
 a a a a 
 
 is found by cubing the numerator and denominator sepa- 
 rately. 
 
 2. What is the square of the fraction -r - ? 
 
 b-\-c 
 
 We have 
 
 /a c\ 2 _ (acf _ atZac+c 2 
 ~ -~ 
 
 3. What is the cube of -- ? Ans. 
 
 . - 
 
 3 be 27b 3 c 3 
 
 QUEST. 83 How do you find the power of a fraction 1 
 12 
 
ELEMENTARY ALGEBRA. 
 
 4. What is the fourth power of 
 
 *<"t> 
 
 5. What is the cube of 
 
 ^' 
 
 2 ax 
 
 > What is the fourth power of 
 
 7. \Vhat is the fifth power of 
 
 8. What is the square of 
 
 9. What is the cube of 2 
 
 Binomial Theorem. 
 
 16aV 
 x v 
 
 84. The method which has been explained of raising a 
 polynomial to any power, is somewhat tedious, and hence 
 other methods, less difficult, have been anxiously sought 
 for. The most simple which has yet been discovered, is 
 the one invented by Sir Isaac Newton, called the Binomial 
 Theorem. 
 
 Q UEST ._84. What is the object of the Binomial Theorem 1 Who 
 discovered this theorem 1 
 
BINOMIAL THEOREM. 127 
 
 85. In raising a quantity to any power, it is plain that 
 there are four things to be considered : 
 
 1st. The number of terms of the power. 
 2nd. The signs of the terms. 
 3rd. The exponents of the letters. 
 4th. The coefficients of the terms. 
 
 Of the Terms. 
 
 86. If we take the two examples of Article 81, which 
 we there wrought out in full ; we have 
 
 By examining the several multiplications, in Art. 8 1 , we 
 shall observe that the second power of a binomial contains 
 three terms, the third power four, the fourth power five, the 
 fifth power six, &c ; and hence we may conclude That 
 the number of terms in any power of a binomial, is one greater 
 than the exponent $f the power. 
 
 Of the Signs of the Terms. 
 
 87. It is evident that when both terms of the given bi- 
 nomial are plus, all the terms of the power will be plus. 
 
 2nd. If the second term of the binomial is negative, then 
 all the odd terms, counted from the left, will be positive, and 
 all the even terms negative. 
 
 QUEST. 85. In raising a quantity to any power, how many things 
 are to be considered ? What are they 1 86. How many terms are 
 there in any power of a binomial ? If the exponent is 3, how many 
 terms'! If it is 4, how many terms I If 5] &c. 87. If both terms 
 of the binomial are positive, how are the terms of the power? If the 
 second term is negative, how are the signs of the terms 1 
 
128 ELEMENTARY ALGEBRA. 
 
 Of the Exponents. 
 
 88. The letter which occupies the first place in a bino- 
 mial, is called the leading letter. Thus, a is the leading 
 letter in the binomials a-\-b, ab. 
 
 1st. It is evident that the exponent of the leading lettei 
 in the first term will be the same as the exponent of the 
 power ; and that this exponent will diminish by unity in 
 each term to the right, until we reach the last term, which 
 does not contain the leading letter. 
 
 2nd. The exponent of the second letter is 1 in the second 
 term, and increases by unity in each term to the right 
 until we reach the last term, in which the exponent is the 
 same as that of the given power. 
 
 3rd. The sum of the exponents of the two letters, in any 
 term, is equal to the exponent of the given power. This 
 last remark will enable us to verify any result obtained by 
 the binomial theorem. 
 
 Let us now apply these principles in the two following 
 examples, in which the coefficients are omitted : 
 
 (a b) 6 . . . a 6 5 &-f a 4 // a?b*+a 2 b 4 a 
 
 As the pupil should be practised in writing the terms, 
 with their proper signs, without the coefficients, we will add 
 a few more examples. 
 
 QUEST. 88. Which is the leading letter of the binomial 1 What is 
 the exponent of this letter in the first term 1 How does it change in the 
 terms towards the right 1 What is the exponent of the second letter in 
 the second term? How does it change in the terms towards the right 1 
 What is it in the last term 7 What is the sum of the exponents in any 
 term equal to 7 
 
BINOMIAL THEOREM. 
 
 2. (a*) 4 . . 4 -a 
 
 3. 
 
 4. (a 
 
 O/" /*e Coefficients. 
 
 89. The coefficient of the first term is unity. The co- 
 efficient of the second term is the same as the exponent of 
 the given power. The coefficient of the third term is found 
 by multiplying the coefficient of the second term by the 
 exponent of the leading letter, and dividing the product by 
 2. And finally If the coefficient of any term^be multiplied 
 by the exponent of the leading letter, and the product divided 
 by the number which marks the place of that term from the 
 left, the quotient will be the coefficient of the next term. 
 
 Thus, to find the coefficients in the example 
 
 (a I) 1 . . . a' a*b + a 5 b* a*y > \ a 3 b*-a?b 5 -\-al 6 - V 
 
 we first place the exponent 7 as a coefficient of the second 
 term. Then, to find the coefficient of the third term, we 
 multiply 7 by 6, the exponent of a, and divide by 2. The 
 quotient 21 is the coefficient of the third term. To find the 
 coefficient of the fourth, we multiply 21 by 5, and divide 
 the product by 3 : this gives 35. To find the coefficient of 
 the fifth term, we multiply 35 by 4, and divide the product 
 by 4 : this gives 35. The coefficient of the sixth term, 
 found in the same way, is 21 ; that of the seventh, 7 ; and 
 that of the eighth, 1. Collecting these coefficients, we 
 have 
 
 - 7a G b + 2 1 a 5 6 2 - 
 
 12* 
 
I 30 ELEMENTARY ALGEBRA. 
 
 REMARK. We see, in examining this last result, that the 
 coefficients of the extreme terms are each unity, and that 
 the coefficients of terms equally distant from the extreme 
 terms are equal. It will, therefore, be sufficient to find the 
 coefficients of the first half of the terms, from which the 
 others may be immediately written. 
 
 j EXAMPLES. 
 
 1. Find the fourth power of a+b. 
 
 Ans. 
 
 2. Find the fourth power of a b. 
 
 Ans. a 4 
 
 3. Find the fifth power of a+b. 
 
 Ans. 5 + 5a 4 &+1 
 
 4. Find the fifth power of a b. 
 
 Ans. a 5 5a 4 &+10a 3 #>-10a 2 & 3 +5flM b 5 . 
 
 5. Find the sixth power of a-\-b. 
 
 Ans. 6 +6a 5 5+15a 4 i 2 +20a 3 ^ 3 +15a 2 ^+6a5 5 -f-^. 
 
 6. Find the sixth power of a b. 
 
 Ans. a 6 6a 5 6+15a 4 6 2 20 3 6 3 4-15a 2 4 6ab 5 +b* 
 
 7. Let it be required to raise the binomial 3a 2 c2bd to 
 the fourth power. 
 
 It frequently occurs that the terms of the binomial are 
 affected with coefficients and exponents, as in the above 
 
 Q UEST 89. What is the coefficient of the first term? What is the 
 coefficient of the second 1 How do you find the coefficient of the third 
 term 1 How do you find the coefficient of any term 1 What are the 
 coefficients of the first and last terms 1 How are the coefficients o/ 
 terms equally distant from the extremes 1 
 
BINOMIAL THEOREM. 131 
 
 example. In the first place, we represent each term of the 
 binomial by a single letter. Thus, we place 
 
 3a 2 c=x, and 2bd=i/, 
 we then have 
 
 But, x 2 = 9a*c 2 , x 3 =27a 6 c 3 , aj4 = 
 
 and y 2 =4bW,y 3 = 8b 3 d?, y* 
 
 Substituting for x and y their values, we have 
 
 (3a 3 c 2W)* = (3a 2 c)*+4(3a 2 c) 3 ( 2bd) + 6 (3a 2 c) 2 ( 2bd) z 
 + 4(3a 2 c) (-2bd) 3 + (-2bd)\ 
 
 and by performing the operations indicated, 
 
 2l6a 6 c 3 bd + 2l6a*cWd? 
 
 8. What is the square of 3a 6b ? 
 
 Ans. 9 2 
 
 9. What is the cube of 3x 6y T 
 
 Ans. 27x 3 l62x 2 y+324xy* 
 
 10. What is the square of xy ? 
 
 Ans. x 2 
 
 11. What is the eighth power of m-\-n ? 
 Ans. 
 
 12. What is the fourth power of a 3b ? 
 
 Ans. a* l2a 3 b+54a z b 2 
 
 13. What is the fifth power of c 2d 1 
 
 Ans. c 5 1 Oc 4 d + 40c 3 d 2 
 
 14. What is the cube of 5a 3d ? 
 
 Ans. I25a 3 225a?d+l35ad 2 27d 3 
 
1 32 ELEMENTARY ALGEBRA. 
 
 REMARK. The powers of any polynomial may easily he 
 found by the Binomial Theorem. 
 
 15. For example, raise a -\-b-\-c to the third power 
 
 First, put .... b-\-c d 
 Then, (a+ + e) 3 = (a + </) 3 = 
 Or. by substituting for the value of d, 
 
 This expression is composed of the cubes of the three 
 terms, plus three times the square of each term by the first 
 powers of the two others, plus six times the product of alt 
 three terms. It is easily proved that this law is true for any 
 polynomial. 
 
 To apply the preceding formula to the development of 
 the cube of a trinomial, in which the terms are affected 
 with coefficients and exponents, designate each term by a 
 single letter, then replace the letters introduced, by their values, 
 and perform the operations indicated. 
 
 From this rule, we find that 
 
 (2a 2 4ab + 3b*) 3 = 8a 6 -48a 5 b+132aW 208a 3 b 3 
 + 1 98a 2 M - 1 QSab* + 27 b 6 . 
 
 The fourth, fifth, &c, powers of any polynomial can be 
 found in a similar manner. 
 
 16. What is the cube of a 2b+c ? 
 Ans. 3~8Z/ 3 + c 3 
 6bc 2 I2abc. 
 
EXTRACTION OP THE SQUARE ROOT. 
 
 CHAPTER V. 
 
 Extraction of the Square Roof of Numbers. Forma" 
 tion of the Square and Extraction of the Square 
 Root of Algebraic Quantities. Calculus of Radicals 
 of the Second Degree. 
 
 90. The square or second power of a number, is the 
 product which arises from multiplying that number by itself 
 once : for example, 49 is the square of 7, and 144 is the 
 square of 12. 
 
 9 1 . The square root of a number is that number which, 
 being multiplied by itself once, will produce the given num- 
 ber. Tnus, 7 is the square root of 49, and 12 the square 
 root of 144: for, 7x7 = 49, and 12x12 = 144. 
 
 92. The square of a number, either entire or fractional, 
 is easily found, being always obtained by multiplying this 
 number by itself once. The extraction of the square root 
 of a number is, however, attended vyith some difficulty, and 
 requires particular explanation. 
 
 QUEST. 90. What is the square, or second power of a number? 
 01 . What is the square root of a number ^ 
 
134 . ELEMENTARY ALGEBRA. 
 
 The first ten numbers are. 
 
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10; 
 and their squares, 
 
 1, 4, 9, 16. 25, 36, 49, 64, 81, 100; 
 
 and reciprocally, the numbers of the first line are the square 
 roots of the corresponding numbers of the second. We 
 may also remark that, the square of a number expressed by a 
 single figure, will contain no figure of a higher denomination 
 than tens. 
 
 The numbers of the last line, 1, 4, 9, 16, &c, and all 
 other numbers which can be produced by the multiplication 
 of a number by itself, are called perfect squares. 
 
 It is obvious that there are but nine perfect squares among 
 all the numbers which can be expressed by one or two 
 figures : the square roots of all other numbers expressed 
 by one or two figures, will be found between two whole 
 numbers differing from each other by unity. Thus 55, 
 which is comprised between 49 and 64, has for its square 
 root a number between 7 and 8. Also 91, which is com- 
 prised between 81 and 100, has for its square root a number 
 between 9 and 10. 
 
 93. Every number may be regarded as made up of a 
 certain number of tens and a certain number of units. 
 Thus 64 is made up of 6 tens and 4 units, arid may be ex- 
 pressed under the form 60 + 4. 
 
 QUEST. 92. What will be the highest denomination of the square 
 of a number expressed by a single figure 1 What are perfect squares 1 
 How many are there between 1 and 100 1 What are they ? 
 
EXTRACTION OF THE SQUARE ROOT. 135 
 
 Now, if we represent the tens by a and the units by b, 
 we shall have 
 
 a+b = 64, 
 and 
 
 or 
 
 Which proves that the square of a number composed of 
 tens and units, contains the square of the tens plus twice the 
 product of the tens by the units, plus the square of the units. 
 
 94. If, now, we make the units 1,2, 3, 4, &c, tens, or 
 units of the second order, by annexing to each figure a ci- 
 pher, we shall have 
 
 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 
 and for their squares, 
 100, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100, 10000. 
 
 From which we see that the square of one ten is 100, the 
 square of two tens 400 ; and generally, that the square of 
 tens will contain no figures of a less denomination than hun- 
 dreds, nor of a higher name than thousands. 
 
 Ex. 1. To extract the square root of 6084. 
 
 Since this number is composed of more than 
 two places of figures, its root will contain 60 84 
 
 more than one. But since it is less than 1 0000, 
 which is the square of 100, the root will contain but two 
 figures : that is, units and tens. 
 
 Now, the square of the tens must be found in the two 
 
 QUEST. 93. How may every number be regarded as made up f What 
 is the square of a number composed of tens and units equal to 1 ! 
 94. What is the square of one ten equal to ? Of 2 tens 1 Of 3 
 tens ' &c. 
 
136 
 
 ELEMENTARY ALGEBRA. 
 
 left-hand figures, which we will separate from the other two 
 by putting a point over the place of units, and a second over 
 the place of hundreds. These parts, of two figures each, are 
 called periods. The part 60 is comprised between the two 
 squaies 49 and 64, of which the roots are 7 and 8 : hence, 
 7 is the figure of the tens sought ; and the required root is 
 composed of 7 tens and a certain number of units. 
 
 The figure 7 being found, we 
 write it on the right of the given 
 number, from which we separate 
 it by a vertical line : then we 
 subtract its square, 49, from 60, 
 which leaves a remainder of 1 1 , 
 to which we bring down the two 
 
 next figures 84. The result of this operation, 1184, con- 
 tains twice the product of the tens by the units, plus the square 
 of the units. 
 
 But since tens multiplied by units cannot give a product of 
 a less name than tens, it follows that the last figure, 4, can 
 form no part of the double product of the tens by the units : 
 this double product is therefore found in the part 118, which 
 we separate from the units' place, 4. 
 
 Now if we double the tens, which gives 14, and then di- 
 vide 118 by 14, the quotient 8 is the figure of the units, or 
 a figure greater than the units. This quotient figure can 
 never be too small, since the part 118 will be at least equal 
 to twice the product of the tens by the units : but it may be 
 too large ; for the 118, besides the double product of the 
 tens by the units, may likewise contain tens arising from 
 the square of the units. To ascertain if the quotient 8 ex- 
 presses the units, we write the 8 on the right of the 14, 
 which gives 148, and then we multiply 148 by 8. Thus, 
 we evidently form, 1st, the square of the units; and, 
 2nd, the double product of the tens by the units. Thi* 
 
EXTRACTION OF THE SQUARE ROC f. 137 
 
 multiplication being effected, gives for a product 1184, a 
 number equal to the result of the first operation. Having 
 subtracted the product, we find the remainder equal to : 
 hence 78 is the root required. 
 
 Indeed, in the operations, we have merely subtracted 
 from the given number 6084, 1st, the square of 7 tens, or 
 70 ; 2nd, twice the product of 70 by 8 ; and, 3d, the square 
 of 8 : that is, the three parts which enter into the composi- 
 tion of the square 70 -(-8, or 78 ; and since the result of 
 the subtraction is "0, it follows that 78 is the square root of 
 6084. 
 
 1)5. REMARK. The operations in the last example have 
 been performed on but two periods, but it is plain that the 
 same reasoning is equally applicable to larger numbers, for 
 by changing the order of the units, we do not change the 
 relation in which they stand to each other. 
 
 Thus, in the number 60 84 95, the two periods 60 84 
 have the same relation to each other as in the numbtr 
 60 84 ; and hence the methods used in the last example 
 are equally applicable to larger numbers. 
 
 96. Hence, for the extraction of the square root of 
 numbers, we have the following 
 
 RULE. 
 
 I. Separate the given number into periods of two figures 
 tach, beginnijig at the right hand : the period on the left will 
 often contain but one figure. 
 
 II. Find the greatest square in the first period on the lej>\ 
 and place its root on the right, after the manner of a quotient 
 
 QUEST. -95. Will the reasoning in .he example apply to more than 
 two period* 7 
 
 13 
 
138 ELEMENTARY ALGEBRA. 
 
 in division. Subtract the square of the root from the firsi 
 period, and to the remainder bring down the second period for 
 a dividend. 
 
 III. Double the root already found, and place it on the lift 
 for a divisor. Seek how many times the divisor is contained 
 in the dividend, exclusive of the right-hand figure, and place 
 the figure in the root and also at the right of the divisor. 
 
 IV. Multiply the divisor, thus augmented, by the last figure 
 of the root, and subtract the product from tfte dividend, and to 
 the remainder bring down the next period for a new dividend 
 But if any of the products should be greater than the dividend, 
 diminish the last figure of the root. 
 
 V. Double the whole root already found, for a new divisor, 
 and continue the operation as before, until all the periods are 
 *-t ought down. 
 
 97. 1st REMARK. If, after all the periods are brought 
 down, there is no remainder, the proposed number is a per- 
 fect square. But if there is a remainder, you have only 
 found the root of the greatest perfect square contained in 
 the given number, or the entire part of the root sought. 
 
 For example, if it were required to extract the square 
 root of 665, we should find 25 for the entire part of the 
 root, and a remainder of 40, which shows that 665 is not 
 a perfect square. But is the square of 25 the greatest per- 
 fect square contained in 665 ? that is, is 25 the entire part 
 of the root ? To prove this, we will first show that, the 
 difference between the squares of two consecutive numbers, is 
 equal to twice the less number augmented by unity. 
 
 QUEST. 96. Give the rule for extracting the square root of numbers 
 What is the first step ? What the second ? What the third 1 Wha 
 the fourth ! What the fifth ? 
 
EXTRACTION OF THE SQUARE ROOT. 139 
 
 Let . . a= the less number, 
 
 and . . a-j-1 = the greater. 
 
 Then . (a+l) 2 =a?+ 2a + l, 
 
 and . . (a) 2 =a 2 . 
 
 Their difference is =^ 2a-\-l as enunciated. 
 
 Hence, the entire part of the root cannot be augmented, 
 unless the remainder exceeds twice the root found, plus 
 unity. 
 
 But 25x2+l=51>40 the remainder: therefore, 25 is 
 the entire part of the root. 
 
 98. 2nd REMARK. The number of figures in the root 
 will always be equal to the number of periods into which 
 the given number is separated. 
 
 EXAMPLES. 
 
 1. To find the square root of 7225. Ans. 85. 
 
 2. To find the square root of 17689. Ans. 133. 
 
 3. To find the square root of 994009. Ans. 997. 
 
 4. To find the square root of 85673536. Ans. 9256. 
 
 5. To find the square root of 67798756. Ans. 8234. 
 
 6. To find the square root of 978121. Ans. 989. 
 
 7. To find the square root of 956484. Ans. 978. 
 
 8. What is the square root of 3^372961 ? Ans. 6031. 
 
 9. What is the square root of 22071204 1 Ans. 4698. 
 
 10. What is the square root of 106929? Ans. 327. 
 
 11. What is the square root of 12088868379025 ? 
 
 Ans. 3476905. 
 
 QUEST. 98. How many figures will you always find in the root] 
 
140 
 
 ELEMENTARY ALGEBRA. 
 
 99. 3rd REMARK. If the given number has not an exact 
 root, there will be a remainder after all the periods are 
 brought down, in which case ciphers may be annexed, 
 forming new periods, each of which will give one decimal 
 place in the root. 
 
 1. What is the square root of 36729 ? 
 
 In this example there are 
 two periods of decimals, 
 which give two places of 
 decimals in the root. 
 
 36729 
 1 
 
 191,64 + 
 
 2 91267 
 |261 
 
 38 1 
 
 3826 
 
 629 
 381 
 
 24800 
 22956 
 
 3832 4 
 
 184400 
 153296 
 
 31104 Rem. 
 
 2. What is the square root of 2268741 ? 
 
 3. What is the square root of 7596796 ? 
 
 4. What is the square root of 96 ? 
 
 5. What is the square root of 153 ? 
 
 6. What is the square root of 101 ? 
 
 Ans. 1506,23 + . 
 
 \ 
 
 Ans. 2756,22 + . 
 
 Ans. 9,79795 + . 
 Ans. 12,36931 + . 
 Ans. 10,04987 + . 
 
 QUEST. 99. How will you find the decimal part of the root 
 
EXTRACTION OF THE SQUARE ROOT. 141 
 
 7. What is the square root of 285970396644 ? 
 
 Ans. 534762. 
 
 8. What is the square root of 41605800625 ? 
 
 Ans. 203975. 
 
 9. What is the square root of 48303584206084 ? 
 
 Ans. 6950078. 
 
 Extraction of the square root of Fractions. 
 
 1 OO. Since the square or second power of a fraction is 
 obtained by squaring the numerator and denominator sepa- 
 rately, it follows that the square root of a fraction will be 
 equal to the square root of the numerator divided by the 
 square root of the denominator. 
 
 For example, the square root of - is equal to -=- : for 
 
 1. What is the square root of ? Ans. . 
 
 9 3 
 
 2. What is the square root of ? Ans. ~. 
 
 10 4 
 
 3. What is the square root of ? Ans. . 
 
 4. What is the square root of ? Ans. . 
 
 1 a * 
 
 5. What is the square root of - ? Ans. . 
 
 o4 2 
 
 QUEST. 10O. If the numerator and denominator of a fraction are 
 perfect squares, how will you extract the square root * 
 13* 
 
142 ELEMENTARY ALGEBRA. 
 
 ' 6. What is the square root of ^7:7^ ? Arts. ~ 
 
 ' 
 
 582169 763 
 
 7. What is the square root of - ? Ans. - 
 
 956484 978 
 
 1O1. If neither the numerator nor the denominator is a 
 perfect square, the root of the fraction cannot be exactly 
 found. We can, however, easily find the approximate root. 
 For this purpose, 
 
 Multiply both terms of the fraction by the denominator, 
 which makes the denominator a perfect square without altering 
 the value of the fraction. Then, extract t,he square root of 
 the numerator, and divide this root by the root of the denomi- 
 tor ; this quotient will be the approximate root. 
 
 3 
 
 Thus, if it be required to extract the square root of , 
 
 
 
 we multiply, both terms by 5, which gives . 
 v 25 
 
 We then have 
 
 V r 15 = 3,8729-f- : 
 hence, 3,8729-f -r- 5 = ,7745+ = Ans. 
 
 7 
 
 2. What is the square root of ? Ans. 1,32287+. 
 
 14 
 
 3. What is the square root of ? Ans. 1,24721+. 
 
 y 
 
 4. What is the square root of 11 ? 
 
 Ib 
 
 Ans. 3,41869 + 
 
 QUEST. 101. If the numerator and denominator of a fraction are not 
 perfect squares, how do you extract the square root ! 
 
EXTRACTION OF THE SQUARE ROOT. 143 
 
 5. What is the square root of ?i| 1 Ans. 2,7131 3 -f-. 
 
 6. What is the square root of 8 ? Ans. 2,88203 + . 
 
 49 
 
 7. W^at is the square root of ? Ans. 0,64549 f. 
 
 12 
 
 8. What is the square root of 10 1 
 
 Ans. 3,20936 + . 
 
 1O2. Finally, instead of the last method, we may, if we 
 please, 
 
 Change the vulgar fraction into a decimal, and continue the 
 division until the number of decimal places is double the number 
 of places required in the root. Then, extract the root of the 
 decimal by the last rule. 
 
 Ex. 1. Extract the square root of to within ,001. 
 
 This number, reduced to decimals, is 0.785714 to within 
 0,000001 ; but the root of 0,785714 to the nearest unit, is 
 
 ,886 ; hence 0,886 is the root of to within ,001. 
 
 14 
 
 2. Find the V/2 to within 0,0001. 
 
 Ans. 1,6931 + 
 
 3. What is the square root of ? Ans. 0,24253+ , 
 
 4. What is the square root of - % Ans. 0,93541+. 
 
 o 
 
 5. What is the square root of ? Ans. 1,29099+. 
 
 o 
 
 QUEST. 102. By what other method may the root be found 1 
 
144 ELEMENTARY ALGEBRA. 
 
 Extraction of the Square Root of Monomials. 
 
 1O3. In order to discover the process for extracting the 
 square root, we must see how the square of the monomia. 
 is formed. 
 
 By the rule for the multiplication of monomials (Art. 35), 
 we have 
 
 that is, in order to square a monomial, it is necessary to 
 square its coefficient, and double each of the exponents of the 
 different letters. Hence, to find the root of the square of a 
 monomial, we have the following 
 
 RULE. 
 
 I. Extract the square root of the coefficient. 
 
 II. Divide the exponent of each letter by 2. 
 
 Thus, -y^G4a 6 Z> 4 =. 8a 3 b 2 for 
 
 2. Find the square root of 625a 2 6 8 c 6 . Ans. 25ab*c*. 
 
 3. Find the square root of 576a 4 b G c 8 . Ans. 24a 2 b 3 c*. 
 
 4. Find the square root of 196x 6 y 2 z 4 . Ans. I4x 3 yz z . 
 
 5. Find the square root of 44la 8 b e c lo d 16 . 
 
 Ans. 
 
 6. Find the square root of 784a l2 b u c l6 d 2 . 
 
 Ans. 
 
 7. Find the square root of 81a 8 & 4 c 6 . 
 
 Ans. 
 
 Q DEST . 103. How do you extract the square root of a monomial 
 
EXTRACTION OF THE SQUARE ROOT. 145 
 
 1O4. From the preceding rule it follows, that when a 
 monomial is a perfect square, its numerical coefficient is a 
 perfect square, and all its exponents even numbers. Thus, 
 25o 4 2 is a perfect square, but 98a6 4 is not a perfect square, 
 because 98 is not a perfect square, and a is affected with 
 an uneven exponent. 
 
 In the latter case, the quantity is introduced into the cal- 
 culus by affecting it with the sign -y/ , and it is written 
 thus : 
 
 Quantities of this kind are called radical quantities, or irra- 
 tional quantities, or simply radicals of the second degree. 
 They are also, sometimes called Surds. 
 
 These expressions may often be simplified, upon the prin- 
 ciple that, the square root of the product of two or more factors 
 is equal to the product of the square roots of these factors ; or, 
 in algebraic language, 
 
 *\/abcd . . . =-y/a . y' b . -y/c . -y/ d . . . 
 
 This being the case, the above expression, 
 be put under the form 
 
 Now V49M may be reduced to 7b z ; hence, 
 
 In like manner, 
 
146 ELEMENTARY ALGEBRA. 
 
 The quantity which stands without the radical sign is 
 called the coefficient of the radical. Thus, in the expressions 
 
 the quantities 7 2 , 3abc, 12a 2 c 5 , are called coefficients of 
 the radicals. 
 
 Hence, to simplify a radical expression of the second 
 degree, we have the following 
 
 RULE. 
 
 I. Separate the expression into two parts, of which one shall 
 contain all the factors that are perfect squares, and the other 
 the remaining ones. 
 
 II. Take the roots of the perfect squares and place them 
 before the radical sign, under which leave those factors which 
 are not perfect squares. 
 
 1O5. REMARK. To determine if a given number has 
 any factor which is a perfect square, we examine and see 
 if it is divisible by either of the perfect squares 
 
 4, 9, 16, 25, 36, 49, 64, 81, &c ; 
 
 and if it is not, we conclude that it does not contain a fac- 
 tor which is a perfect square. 
 
 QUEST. 104. When is a monomial a perfect square 7 When it is 
 not a perfect square, how is it introduced into the calculus 1 What are 
 quantities of this kind called 7 May they be simplified ! Upon whal 
 principle 1 What is a coefficient of a radical 1 Give the rule for reducing 
 radicals. 105. How do you determine whether a given number has a 
 factor which is a perfect square 1 
 
EXTRACTION OF THE SQUARE ROOT. 147 
 
 EXAMPLES. 
 
 1. Reduce -\/75d 3 bc to its simplest form. 
 
 Ans. 
 
 2. Reduce yTSsFo^ to its simplest form. 
 
 Ans. 
 
 3. Reduce -\/32a 9 b 8 c to its simplest form 
 
 Ans. 
 
 4. Reduce -\/256a 2 b*c 8 to its simplest form. 
 
 Ans. 
 
 5. Reduce Vl024a 9 6 7 e* to its simplest form. 
 
 Ans. 
 
 6. Reduce -v/729a 7 6 5 c 6 J to its simplest form. 
 
 7. Reduce 
 
 'ft | 
 
 ,/ I 
 
 : s "|/!ri"|lj 
 4l<!l 
 
 to its simplest form, 
 ^n^. 
 
 i ^ ts simplest form. 
 
 tii 
 
 its simplest form 
 
 An.y. ISO 4 */ 3 * 
 
 its simplest form 
 
 Ans. 
 s simplest form. 
 Ans. 
 
148 
 
 ELEMENTARY ALGEBRA. 
 
 1O6. Since like signs in both the factors give a plus 
 sign in the product, the square of a, as well as that of 
 -\-a, will be a 2 ; hence the root of a 2 is either -{-a or a 
 Also, the square root of 25a?b 4 is either -\-5ab 2 or -dab 2 . 
 Whence we may conclude, that if a monomial is positive 
 its square root may be affected either with the sign -f or 
 ; thus, V9 4 =dc3a 2 ; for, + 3a 2 or 3a 2 , squared, 
 gives 9a 4 . The double sign with which the root is 
 affected is read plus or minus. 
 
 If the proposed monomial were negative, it would be im- 
 possible to extract its root, since it has just been shown that 
 the square of every quantity, whether positive or negative, 
 is essentially positive. Therefore, 
 
 are algebraic symbols which indicate operations that cannot 
 be performed. They are called imaginary quantities, or 
 rather imaginary expressions, and are frequently met with 
 in the resolution of equations of the second degree. These 
 symbols can, however, by extending the rules, be simplified 
 in the same manner as those irrational expressions which 
 indicate operations that cannot be performed. Thus, V 9 
 may be reduced by (Art. 1O4). Thus, 
 
 also, 
 
 QUEST. 106. What sign is placed before the square root of a mono- 
 mial 1 Why may you place the sign plus or minus'? What is an ima- 
 ginary quantity 1 Why is it called imaginary 1 
 
RADICALS OF THE SECOND DEGREE. 149 
 
 Of the Calculus of Radicals of the Second Degree 
 
 107. A radical quantity is the indicated root of an 
 imperfect power. 
 
 The extraction of the square root gives rise to such ex- 
 pressions as yX 3 -\/b, 7 -\/2, which are called irra- 
 tional quantities, or radicals of the second degree. We will 
 now establish rules for performing the four fundamental 
 operations on these expressions. 
 
 108. Two radicals of the second degree are similar^ 
 when the quantities under the- radical sign are the same in 
 both. Thus, 3 \/b and 5e yT~~ are similar radicals; and 
 BO also are 9 ^/2 and 7 -v/57 
 
 Addition. 
 
 1O9. Radicals of the second degree may be added 
 together by the following 
 
 RULE. 
 
 I. If the radicals are similar add their coefficients, and to 
 the sum annex the common radical. 
 
 II. If the radicals are not similar, connect them together 
 with their proper signs. 
 
 Thus, 
 
 QUEST. 107. What is a radical quantity 1 What are such quantities 
 called? 1-08. When are radicals of the second degree similar 1 
 109. How do you add similar radicals of the second degree * How do 
 vou add radicals which are not similar 1 
 
 14 
 
150 ELEMENTARY ALGEBRA. 
 
 In like manner. 
 
 Two radicals, which do not appear to be similar at first 
 sight, may become so by simplification (Art. 1O4). 
 For example, 
 
 and 
 
 When the radicals are not similar, the addition or sub- 
 traction can only be indicated. Thus, in order to add 
 3 -\/b to 5 -y/a, we write 
 
 EXAMPLES. 
 
 1. What is the sum of ^/27a 2 and -y/48a 2 ? 
 
 Ans. 
 
 2. What is the sum of -/50a 4 2 and </72a*b 2 ? 
 
 3. \Vhat is the sum of -- and 
 
 5 V 15 
 
 4. What is the sum of Vl25 and l/500a 2 ? 
 
RADICALS OF THE SECOND DEGREE. 151 
 
 / 50 /I 00 
 
 5. VV hat is the sum of \ I and \ /- - ? 
 V 147 V 294 
 
 6. What is the sum of y98a?x and i/36x 2 36a 2 
 
 " What is the sum of -y/98a 2 ar and 
 
 ^4^. 
 S Required the sum of -v/72" and -y/128. 
 
 . 14 
 9. Required the sum of -v/27 and -/1 47. 
 
 Ans. 10 
 
 27 
 
 /2* 
 10. Required the sum of \ / -^nd 
 
 Ans. g 
 11. Required the sum of 2 y/lffi and 3 
 
 12. Required the sum of y^43" and 10-/3637 
 
 jlns. 119/3 
 
 13. What is the sum of ^/32Qa 2 b 2 and 
 
 14 What is the sum of -TSo^F and 
 
152 ELEMENTARY ALGEBRA. 
 
 Subtraction. 
 
 1 1 O. To subtract one radical expression from another, 
 we have the following 
 
 RULE. 
 
 I. If the radicals are similar, subtract their coefficients, 
 and to the difference annex the common radical. 
 
 II. If the radicals are not similar, their difference can only 
 be indicated by the minus sign. 
 
 EXAMPLES. 
 
 1. What is the difference between 3ai/~b and a 
 
 Here' ^a^/^ba-^T)=2a^/~JT Ans. 
 
 2. From 9 a y27F subtract 6a V 27b z . 
 First, 9ay7P"=:27&'/~37 and 6a ^/l27b* 
 and 27abi~3'18ab<~3 = 9abi~3 Ans. 
 
 3. What is the difference of \f~75 and -\/~48 1 
 
 Ans. 
 
 4. What is the difference of ^24aW and V 54& 4 I 
 
 Ans. 2ab 
 
 QUEST. 110. How do you subtract similar radicals 1 How do you 
 subtract radicals which are not similar ? 
 
RADICALS OF THE SECOND DEGREE. 153 
 
 5. Required the difference of it-r- and \=< 
 
 40 
 
 6. What is the difference of ^f\^S~aW and -/32 9 ? 
 
 7. What is the difference of ^/48a 3 b 3 and 
 
 ^4/i.s. 4ab 
 
 8. What is the difference of V242a 5 6 5 and 
 
 /3~ /3" 
 
 9. What is the difference of V/-T- and \/ ? 
 
 10. What is the difference of V 320 2 and 
 
 1 1 What is the difference between 
 and 
 
 12. What is the difference between 
 and 
 
 13. What is the difference between 
 and 
 
 14* 
 
1 54 ELEMENTARY ALGEBRA. 
 
 Multiplication. 
 
 111. For the multiplication of radicals, we have the 
 following 
 
 RULE. 
 
 I. Multiply the quantities under the radical signs together, 
 and place the common radical over the product. 
 
 II. If the radicals have coefficients, we multiply them to- 
 gether, and place the product before the common radical. 
 
 Thus, yV X V^~= Va5 ; 
 
 This is the principle of Art. 1O4, taken in the inverse 
 order. 
 
 EXAMPLES. 
 
 1. What is the product of 3 ^5aB and 4 
 
 Ans. 
 
 2. What is the product of 2a ^/fic and 
 
 Ans. 6a?be 
 
 3. What is the product of 2a ^a 2 +b* and 3a 
 
 Ans. 6a z (a z +b 2 ). 
 
 QUEST. 111. How do you multiply quantities which are under radi- 
 cal signs 1 When the radicals have coefficients, now do you multiple 
 them 1 
 
RADICALS OF THE SECOND DEGREE. 155 
 
 4. What is the product of 3 -\^~ and 2 -y/ST 
 
 Ans. 24. 
 
 5. What is the product of f -y/l^ and T 2 o V% c * b 
 
 Ans. 
 
 6. What is the product of 2x4 ^fb and 2x 
 
 Ans. 4x z b. 
 
 7. What is the product of 
 
 and 
 
 8. What is the product of 3a-/270 3 by -/^ ? 
 
 Division. 
 
 112. To divide one radical by another, we have the 
 following 
 
 RULE. 
 
 I. Divide one of the quantities under the radical sign by the 
 other, and place the common radical jver the quotient. 
 
 II. If the radicals have coefficients, divide the coefficient of 
 the dividend by the coefficient of the divisor, and place the 
 quotient before the common radical. 
 
 QUEST. 112. How do you divide quantities which are under the 
 radical sign 1 When the radicals have coefficients, how do you divide 
 them? 
 
156 ELEMENTARY ALGEBRA. 
 
 Thus, =\/ ; for the squares of these two 
 
 expressions are equal to the same quantity ; hence 
 the expressions themselves must be equal. 
 
 EXAMPLES. 
 
 1. Divide 5yT by 2 & V^- Ans - ^T 
 
 2. Divide Wac^Wc by 4ci/2F. Ans. 
 
 3. Divide 6a-v/96F by 3^8P. Ans. 4abi/3. 
 
 4. Divide 4a 2 -/50F by 2a 2 -y/57 Ans. 
 
 5. Divide 26a?b^/8laW by 1 
 
 6. Divide 84ff 3 ^ 4 ^27 ac by 
 
 7. Divide -y/F 2 by ^2- 
 
 8. Divide Ga^y^a 3 by 12-/5a. Ans. 
 
 9. Divide Ga-v/ToF" by 3-/5T A 
 
 10. Divide 48b^^/l5' by 2i 2 VS ^^- 3606 2 . 
 
 11. Divide Sa^c 3 -/^ by 20^2837 
 
 2ab*c 3 d. 
 12. Divide 96a 4 c 3 -v/98F by 
 
RADICALS OF THE SECOND DEGREE. 157 
 
 13. Divide 27a 5 5 6 i/2Ttf by ^/^a~. 
 
 Ans. 
 
 14. Divide ISaW-^&T* by &ab^/tf~ 
 
 Ans. 
 
 To Extract the Square Root of a Polynomial. 
 
 113. Before explaining the rule for the extraction of 
 the square root of a polynomial, let us first examine the 
 squares of several polynomials : we have 
 
 The law by which these squares are formed can be enun- 
 ciated thus : 
 
 The, square of any polynomial contains the square of the 
 first term, plus twice the product of the first term, by the second, 
 plus the square of the second ; plus twice the first two terms 
 multiplied by the third, plus the square of the third ; plus twice 
 the first three terms multiplied by the fourth, plus the square 
 of the fourth; and so on. 
 
 QUEST. 113. What is the square of a binomial eqaalto? What 
 is the square of a trinomial equal to ? What is the square of any 
 polynomial equal to ? 
 
1 58 ELEMENTARY ALGEBRA. 
 
 114. Hence, to extract the square root of a polynomial 
 we have the following 
 
 RULE. 
 
 I. Arrange the polynom ' zl with reference to one of its letters 
 and extract the square root of the first term : this will give the 
 first term of the root. 
 
 II. Divide the second term of the polynomial by double the 
 first term of the root, and lite quotient will be the second term 
 of the root. 
 
 III. Then form the square of the two terms of the root 
 found, and subtract it fi jm the first polynomial, and then 
 
 divide the first term of the remainder by clouble the first term 
 of the root, and the quotient will be the third term. 
 
 IV. Form the double products of the first and second terms, 
 by the third, plus the square of the third ; then subtract all 
 these products from the last remainder, and divide the first 
 term of the result by double the first term of the root, and the 
 quotient will be the fourth term. Then proceed in the sanw 
 manner to find the other te*ms. 
 
 EXAMPLES. 
 
 X Extract the square loot of the polynomial 
 
 First arrange it with reference to the letter a. 
 
 25a 4 
 25a 4 
 
 10a 2 
 40a 2 & 2 -^4tfZ> 3 -|-16& 4 1st Rem. 
 
 2d Rem. 
 
RADICALS OF THE SECOND DEGREE. 159 
 
 After having arranged the polynomial with reference to a, 
 extract the square root of 25a 4 , this gives 5a 2 , which is 
 placed at the right of the polynomial ; then divide the 
 second term, 30 3 6, by the double of 5a 2 , or 10a 2 ;'the 
 quotient is 3ab, and is placed at the right of 5a 2 . Hence, 
 the first two terms of the root are 5a 2 Sab. Squaring this 
 binomial, it becomes 25a 4 30 3 i-|-9a 2 & 2 , which, subtracted 
 from the proposed polynomial, gives a remainder, of which 
 the first term is 40a 2 6 2 . Dividing this first term by 10<z 2 , 
 (the double of 5a 2 ), the quotient is -f-4& 2 ; this is the third 
 term of the root, and is written on the right of the first two 
 terms. By forming the double product of 5a 2 3ab by 4& 2 , 
 and at the same time squaring 4& 2 , we find the polynomial 
 4Qa 2 b 2 24ab 3 +l6b*, which, subtracted from the first re- 
 mainder, gives 0. Therefore 5a 2 3ab + 4b 2 is the required 
 root. 
 
 2. Find the square root of a*-{-4a 3 x-\-6a 2 x 2 -r4ax 3 -\-x 4 . 
 
 Ans. at + Zax+x 2 . 
 
 3. Find the square root of a*~-4a 3 x-\-6a?x 2 4ax 3 -[-x*. 
 
 Ans. a 2 2ax-{-x 2 . 
 
 4. Find the square root of 
 
 4# 6 + 12a; 5 +5* 4 2x 3 +7x 2 2a?+ 1 . 
 Ans. 
 
 5. Find the square root of 
 
 9a 4 12 3 6-f28a 2 6 2 
 
 Ans. 3a 2 
 
 QUEST. 114. Give the rule for extracting the square root of a poly- 
 nomial 1 What is the first step 1 What the second 1 What the third ' 
 What the fourth 1 
 
1 60 ELEMENTARY ALGEBRA. 
 
 G. What is the square root of 
 
 a; 4 4ax 3 + 4a 2 x 2 4x 2 -f Sax -f 4. 
 
 Ans. x 2 2ax2. 
 7. What is the square root of 
 
 Ans. 3a?+y 2. 
 8. What is the square root of y 4 2y 2 x 2 + 2x 2 2y 2 +l 
 
 9. What is the square root, of 9a 4 6 4 30<z 3 3 +25a 2 2 ? 
 
 Arcs. 3a 2 & 2 5a 
 
 10. Find the square root of 
 
 25a*b 2 40 3 6 2 c -f 76a 2 5 2 c 2 48ab z c 3 + 3 6i 
 36 2 ^c 3 + 9 4 c 2 . 
 Ans. 5a 2 b 3a 2 c 
 
 115. We will conclude this subject with the following 
 remarks. 
 
 1st. A binomial can never be a perfect square, since we 
 know that the square of the most simple polynomial, viz : 
 a binomial, contains three distinct parts, which cannot ex- 
 perience any reduction amongst themselves. Thus, the 
 expression a 2 -\-b 2 is not a perfect square ; it wants the term 
 2ab in order that it should be the square of ab. 
 
 2nd. In order that a trinomial, when arranged, may be a 
 perfect square, its two extreme terms must be squares, and 
 the middle term must be the double product of the square 
 roots of the two others. Therefore, to obtain the square 
 root of a trinomial when it is a perfect square ; Extract the 
 roots of the two extreme terms, and give these roots the same 
 or contrary signs, nrcnrding as the middle term is positive or 
 
RADICALS OF THE SECOND DEGREE. 
 
 161 
 
 negative. To verify , see if the double product of the two 
 roots gives the middle t o *-m of the trinomial. Thus, 
 
 Qa 6 4Q a 4: 2 4-G4 2 & 4 is a perfect square, 
 
 since -y/906 = 3a 3 , and <v/64a 2 & 4 = 8ab z , 
 
 and also 2 x 3a 3 X 8ab 2 = 48 4 5 2 = the middle term. 
 
 But 4a z -\-l4ab+9b 2 is not a perfect square : for although 
 4a 2 and -I- 9b 2 are the squares of 20 and 3, yet 2 X 2a X 3i 
 is not equal to I4ab. 
 
 3rd. In the series of operations required in a general ex- 
 ample, when the first term of one of the remainders is not 
 exactly divisible by twice the first term of the root, we may 
 conclude that the proposed polynomial is not a perfect 
 square. This is an evident consequence of the course of 
 reasoning, by which we have arrived at the general rule for 
 extracting the square root. 
 
 4th. When the polynomial is not a perfect square, it may 
 be simplified (See Art. 1O4.) 
 
 Take, for example, the expression \/a 3 b + 4a 2 b 2 -\-4ab 3 . 
 
 The quantity under the radical is not a perfect square ; 
 but it can be put under the form ab(a 2 -\-4ab-{-4:b 2 ). Now, 
 the factor between the parenthesis is evidently the square 
 of a+26, whence we may conclude that, 
 
 2. Reduce V2a 2 6 4ab 2 -f 2b 3 to its simple form 
 
 Ans. (a 
 
 QUEST. 115. Can a binomial ever be a pertect power! Why not 1 
 When is a trinomial a perfect square 1 When, in extracting the square 
 root we find that the first term of the remainder is not divisible by twice the 
 root, in the polynomial a perfect, power or not 7 
 
 15 
 
162 KLKiMENTARY ALGEBRA. 
 
 CHAPTER VI. 
 Equations of the Second Degree. 
 
 116* An Equation of the second degree is c.ie in which 
 the greatest exponent of the unknown quantity is equal to 2. 
 
 If the equation contains two unknown quantities, it is of 
 the second degree when the greatest sum of the exponents 
 with which the unknown quantity is affected, in any term, is 
 equal to 2. Thus, x 
 
 a?=a, ax 2 -\-bx=c, and xy + x=d*, 
 are equations of the second degree. 
 
 117. Equations of the second degree are divided into two 
 classes : 
 
 1st. Equations which involve only the square of the un- 
 known quantity and known terms. These are called, Incom- 
 plete Equations. 
 
 2d. Equations which involve the first and second powers 
 of the unknown quantity, aad known terms. These are 
 called, Complete Equations. 
 
 QUEST. 1 16. What is an equation of the second degree ? 1 1 7. Into 
 how many classes are equations of the second degree divided? What 
 i an incomplete equation ? What is a complete equation ? 
 
EQUATIONS OF THE SECOND DEGREE. 163 
 
 Thus, a^+S* 2 -5=7 
 
 and 5s 2 3a* 4=a 
 
 are incomplete equations : and 
 
 3x*-5ar 3.r 2 + a=6 
 Sx 2 Sx 2 - X c=d 
 
 are complete equations. 
 
 O/* Incomplete Equations. 
 
 118. If we take an incomplete equation of the form 
 
 14x 2 -8x 2 =40-2x il 
 we have, by collecting the coefficients of x 2 , 
 
 8x 2 =40, or ^=5. 
 Again, if we have the equation 
 
 ax 2 + fcx 2 + </=/, 
 we shall have, 
 
 ==W and ar 2 ==m 
 
 by substituting m for the known terms which compose the 
 second member. Hence, 
 
 Every incomplete equation can be reduced to an equation 
 involving two terms, of the form 
 ar 2 =m, 
 
 and from this circumstance the incomplete equations are often 
 called equations involving two terms. 
 
 From which we have, by extracting the square root of both 
 members, x 
 
 QUEST. 118, To what form may every incomplete equation be 
 reduced ? What are incomplete equations often called ? 
 
164 ELEMENTARY ALGEBRA. 
 
 1. What number is that which being multiplied by itself 
 the product will be 144. 
 
 Let x= the number: then 
 
 x Xx= a?=144. 
 
 It is plain that the value of x will be found by extracting 
 the square root of both members of the equation : that is 
 
 2. A person being asked how much money he had, said 
 if the number of dollars be squared and 6 be added, the sum 
 will be 42 : How much had he ? 
 
 Let x= the number of dollars. 
 Then by the conditions 
 
 * 2 +6=42: 
 
 hence, a*=42-6=36 
 
 and #=6. 
 
 Ans. $6 
 
 3. A grocer being asked how much sugar he had sold to a 
 person, answered, if the square of the number of pounds be 
 multiplied by 7, the product will be 1575. How many pounds 
 had he sold ? 
 
 Denote the number of pounds by x. 
 Then by the conditions of the question 
 
 7^=1575: 
 
 ience, s*=225 
 
 and x =15. 
 
 Jlns. 15 
 
EQUATIONS OP THE SECOND DEGREE. 165 
 
 4. A person being asked his age said, if from the square 
 of my age you take 192, the remainder will be the square 
 of half my age : what was his age ? 
 
 Denote his age by x. 
 
 Then, by the conditions of the question 
 
 and by clearing the fractions 
 
 hence, 4x 2 x z =768, 
 
 and 3a? 2 =768 
 
 * 2 =256 
 x = 16. 
 
 Ans. 16. 
 
 5. What number is that whose eighth part multiplied by 
 its fifth part and the product divided by 4, shall give a quotient 
 equal to 40 ? 
 
 Let x= the number, 
 
 By the conditions of the question 
 
 hence, _g. =40 
 
 by clearing fractions, 
 
 * a =6400 
 x= 80. 
 
 Ans. 80. 
 15* 
 
166 ELEMENTARY ALGEBRA. 
 
 119. Hence, to find the value of x we have the fol- 
 lowing 
 
 RULE. 
 
 1. Find the value of x 2 ; and then extract the square root 
 of both members of the equation. 
 
 4. What is the value of x in the equation 
 
 3x*-\-8 = 5x 2 10. 
 
 By transposition 3x 2 5x 2 = 1 8, 
 by reducing 2x 2 = 18, 
 
 by dividing by 2 and changing the signs 
 
 a: 2 = 9, 
 by extracting the square root x = 3. 
 
 We should, however, remark that the square root of 9, 
 is either +3, or 3. For, 
 
 -f3x+3 = 9 and 3x 3 = 9. 
 Hence, when we have the equation 
 
 ^=9, 
 we have x =4-3 and x =3. 
 
 1 2O. A root of an equation is any expression which being 
 substituted for the unknown quantity, will satisfy the equa- 
 tion, that is, render the two members equal to each other. 
 Thus, in the equation 
 
 rr 2 = 9 
 
 there are two roots, -f3 and 3 ; for either of these 
 numbers being substituted for x will satisfy the equation. 
 
EQUATIONS OF THE SECOND DEGREE. 167 
 
 7. Again, if we take the equation 
 
 x*=m, 
 we shall have 
 
 x=+/m and ar= /nT 
 
 For, 
 and 
 
 Hence we may conclude, 
 
 1st. That every incomplete equation of the second degree 
 has two roots. 
 
 2d. That these roots are numerically equal, but have con- 
 trary signs. 
 
 8. What are the roots of the equation 
 
 . a?=+4 and x= 4. 
 9. What are the roots of the equation 
 
 dns. #=-f9 and x 9. 
 
 10. What are the roots of the equation 
 
 4^+13-2^=45. 
 
 Ans. x=+4 and x= 4 
 
 QUEST. 1 19. How do you resolve an incomplete equation 1 120. What 
 is the root of an equation? What are the roots of the equation x 2 =9? 
 Of the equation z 2 m? How many roots has every incomplete equa- 
 tion ? How do those roots compare with each other ? 
 
1 68 ELEMENTARY ALGEBRA. 
 
 8. What are the roots of the equation 
 
 Ans. *=4-2, ar=Lj 2 
 9. What are the roots of the equation. 
 
 Ans. #=-f-5, x= 5 
 
 1 0. Find a number such that one-third of it multiplied 
 by one-fourth shall be equal to 108 ? 
 
 Ans. 36. 
 
 1 1 . What number is that whose sixth part multiplied by 
 its fifth part and product divided by ten, shall give a quo- 
 tient equal to 3 ? 
 % Ans. 30. 
 
 12. What number is that whose square, plus 18, shall be 
 equal to half its square plus 30J. 
 
 Ans. 5. 
 
 13. What numbers are those which. are to each other as 
 1 to 2 and the difference of whose squares is equal to 75. 
 
 Let a?= the less number. 
 
 Then 2x= the greater. 
 
 Then by the conditions of the question 
 
 hence, 3a? 2 =75 ; 
 
 and by dividing by 3, * 2 =25 and at =5, 
 
 and 2op=10. 
 
 Ans. 5 and 10. 
 
EQUATIONS OF THE SECOND DEGREE. 169 
 
 14. What two numbers are those which are to each other 
 as 5 to 6, and the difference of whose squares is 44. 
 
 Let x= the greatest number. 
 
 ^ 
 
 Then -#= the least. 
 6 
 
 By the conditions of the question 
 
 *'-H^=44. 
 36 
 
 by clearing fractions, 
 
 36JT 2 25ff 2 =1584; 
 
 hence, 1 10*= 1584, 
 
 and a*=144, 
 
 hence, x =12, 
 
 and - = 10 - 
 
 o 
 
 Ans. 10 and 12. 
 
 15. What two numbers are those which are to each 
 other as 3 to 4, and the difference of whose squares is 28 ? 
 
 Ans. 6 and 8. 
 
 16. What two numbers are those which are to each other 
 as 5 to 11, and the sum of whose square is 584 ? 
 
 Ans. 10 and 22. 
 
 17. A says to B, my son's age is one quarter of yours, 
 and the difference between the squares of the numbers re- 
 presenting their ages is 240 : what were their ages ? 
 
 Younger 4. 
 
170 ELEMENTARY ALGEBRA. 
 
 When there are two unknown quantities. 
 
 121. When there are two or more unknown quantities, 
 eliminate one of them by the rule of A rticle 7 7 : there will 
 thus arise a new equation with but a single unknown quantity, 
 the value of which may be found by the rule already given. 
 
 1. There is a room of such dimensions, that the differ- 
 rence of the sides multiplied by the less is equal to 36, and 
 the product of the sides is equal to 360 : what are the 
 sides ? 
 
 Let a?= the less side ; 
 y=. the greater. 
 
 Then, by the 1 st condition, 
 
 and by the 2nd, <ry=360. 
 
 From the first equation, we have 
 
 and by subtraction, ar 2 =:324. 
 
 Hence, x=^324~=18; 
 
 _ J60 _ 
 
 Ans.x\Q t y '20. 
 
 QUEST.- -121. How do you resolve the equation when there are two 
 or more unknown quantities 1 
 
EQUATIONS OF THE SECOND DEGREE. 171 
 
 2. A merchant sells two pieces of muslin, which together 
 measure 12 yards. He received for each piece just so 
 many dollars per yard as the piece contained yards. Now, 
 he gets four times as much for one piece as for the other 
 how many yards in each piece * 
 
 Let a: the number in the larger piece ; 
 y the number in the shorter piece. 
 
 Then, by the conditions of the question, 
 
 xXx=x 2 = what he got for the larger piece ; 
 yXy=y z = what he got for the shorter. 
 And a: 2 =i4y 2 , by the 2nd condition. 
 
 a; =2y, by extracting the square root 
 Substituting this value of x in the first equation, we have 
 
 y+2y=12; 
 and consequently, y 4, 
 
 and *= 8. 
 
 AHS. S and 4. 
 
 3. What two numbers are those whose product is 30, and 
 quotient 3 1 Ans. 10 and 3. 
 
 4. The product of two numbers is a, and their quotient 
 : what are the numbers ? 
 
 Ans. i/ ab and \l - 
 
 a 
 ~b 
 
 5. The sum of the squares of two numbers is 117, an 
 u*e difference of their squares 45 : what are the -numbers ? 
 
 Ans. 9 and 6 
 
172 ELEMENTARY ALGEBRA. 
 
 6. The sum of the squares of two numbers is , and the 
 difference of their squares is b : what are the numbers ? 
 
 7. What two numbers are those which are to each other 
 as 3 to 4, and the sum of whose squares is 225 ? 
 
 Ans. 9 and 12. 
 
 8. What two numbers are those which are to each other 
 as m to , and the sum of whose squares is equal to a 2 1 
 
 ma na 
 
 Ans. 
 
 9. What two numbers are those which are to each other 
 as 1 to 2, and the difference of whose squares is 75 ? 
 
 Ans. 5 and 10. 
 
 10. What two numbers are those which are to each other 
 as m to n, and the difference of whose squares is equal 
 to #>? 
 
 mb nb 
 
 11. A certain sum of money is placed at interest for six 
 months, at 8 per cent, per annum. Now, if the amount be 
 multiplied by the number expressing the interest, the pro- 
 duct will be 562500 : what is the amount at interes* ? 
 
 Ans. $3750. 
 
 12. A person distributes a sum of money between a num- 
 ber of women and boys. The number of women is to the 
 number of boys as 3 to 4. Now, the boys receive one- 
 half as many dollars as there are persons, and the women 
 twice as many dollars as there are boys, and together they 
 receive 138 dollars : how many women were there, and 
 how many boys ? 
 
 (36 women 
 f 48 boys. 
 
EQUATIONS OF THE SECOND DEGREE. 173 
 
 Of Complete Equations. 
 
 122. We have already seen (Art. 117), that a complete 
 equation of the second degree, contains the square of the 
 unknown quantity, the first power of the unknown quantity, 
 and known terms. 
 
 1. If we have the complete equation 
 we have, by transposing and reducing, 
 
 and by dividing by 3, 
 
 ar 2 3ar=8, 
 an equation containing but three terms. 
 
 2. If we have the equation 
 
 a z x 2 + 3abx + x 2 = ex -f d, 
 by ollecting the coefficients of x 2 and x, we have 
 
 (a 2 +l)x 2 +(3abc)x=d; 
 and dividing by the coefficient of x 2 , we have 
 3ab c d 
 
 QUEST. 122. How many terms does a complete equation of the 
 second degree contain 1 ? Of what is the first term composed 1 The 
 second 1 The third ? 
 
 16 
 
174 ELEMENTARY ALGEBRA. 
 
 If we represent the coefficient of x by 2/>, and the known 
 term by </, we have 
 
 an equation containing but three terms. 
 
 Hence, we see that every complete equation of the second de- 
 gree can be reduced to an equation containing but three terms 
 
 123. We wish now to show that there are four forms 
 under which this equation will be expressed, each depend- 
 ing on the signs of 2p and q. 
 
 1 st. Let us for the sake of illustration, make 
 
 2/>=-f-4, and ? +5: 
 we shall then have # 2 4-4a?:r_-5. 
 2nd. Let us now suppose 
 
 2p= 4, and ^ = +5: 
 we shall then have x 2 4x=5. 
 3rd. If we make 
 
 2p= + 4, and q^= 5, 
 we have x 2 +4x= 5. 
 
 4th. If we make 
 
 2p= 4, and q= 5, 
 we have x 2 4:X= 5. 
 
 QUEST. 123. Under how many forms may every equation of the 
 second degree be expressed 1 On what will these forms depend ? What 
 are the signs of the coefficient of x and the known term, in the first 
 form 1 What in the second ? What in the third 1 What in the fourth ' 
 Repea* the four forms. 
 
EQUATIONS OF THE SECOND DEGREE. 175 
 
 We therefore conclude that every complete equation of 
 the second degree may be reduced to one of these forms : 
 
 #? -f- 2pw = + #, 1st form. 
 
 a: 2 2px=+q, 2nd form. 
 
 x z -{-2px= q, 3rd form. 
 
 x 2 2px=q, 4th form. 
 
 124. REMARK. If, in reducing an equation to either of 
 these forms, the second power of the unknown quantity 
 should have a negative sign, it must be rendered positive 
 by changing the sign of every term of the equation. 
 
 125. We are next to show the manner in which the 
 value of the unknown quantity may be found. We have 
 seen (Art. 38), that 
 
 and comparing this square with the first and third forms, we 
 see that the first member in each contains two terms of the 
 square of a binomial, viz : the square of the first term plus 
 
1 76 ELEMENTARY ALGEBRA. 
 
 called completing the square. Then, by extracting the 
 square root of both members of the equation, we have 
 
 and x-\-p= -Y/ 
 
 which gives, by transposing p, 
 
 x=p < 
 
 126. If we compare the second and fourth forms with 
 the square 
 
 we also see that half the coefficient of x being squared and 
 added to both members, will make the first members perfect 
 squares. Having made the additions, we have 
 
 Then, by extracting the square root of both members we 
 have 
 
 and x p= V q+p* ; 
 
 and by transposing p, we find 
 
 and x=p ^/ 
 
 QUEST. 120. In the second form, how do you make the first mem- 
 ber a perfect square 1 
 
EQUATIONS OF THE SECOND DEGREE. 177 
 
 1 27. Hence, for the resolution of every equation of the 
 econd degree, we have the foll9\ving 
 
 RULE. 
 
 I. Reduce the equation to one of the known forma. 
 
 II. Take half the coefficient of the second term, square it, 
 and add the result to both members of the equation. 
 
 III. Then extract the square root of both members of the 
 equation ; after which, transpose the known term to the second 
 member. 
 
 REMARK. The square root of the first member is always 
 equal to the square root of the first term, plus or minus half 
 the coefficient of x. 
 
 EXAMPLES IN THE FIRST FORM. 
 
 1. What are the values of x in the equation 
 
 If we first divide by the coefficient 2, we obtain 
 
 x*+4x=32. 
 Then, completing the square, 
 
 tf 2 
 Extracting the root, 
 
 = + 6 or 6. 
 Hence, x= 2 + 6=+4; 
 
 or, a?= 2 6 = 8. 
 
 QUEST. 127. Give the general rule for resolving an equation or the 
 second vlegree. What is the first step ? What the second 1 What the 
 third 7 What is the square root of the first member always equal to ' 
 16* 
 
1 73 ELEMENTARY ALGEBRA. 
 
 That is, in this form the smaller root is positive, and the 
 larger negative. 
 
 Verification. 
 
 If we take the positive value, viz: #=+4, 
 he equation a? 2 +4#=32 
 
 gives 4 2 +4x4 = 32: 
 
 and if we take the negative value of x, viz : #= 8, 
 the equation x 2 -\-4x=32 
 
 gives (8) 2 +4( 8)^64-32-32. 
 
 From which we see that either of the values of x, viz : 
 a: =+4 or 0?= 8, will satisfy the equation. 
 
 2. What are the values of x in the equation 
 By transposing the terms, we have 
 
 and by reducing, 
 
 4z 2 +24a:=108; 
 
 and dividing by the coefficient of a; 2 , 
 
 Now, by completing the square, 
 
 z 2 -f-6.r +9-36; 
 extracting the square root, 
 
 v-{-? = '\/36 = + 6 or 6: 
 hence, x= + 6 3=1 + 3; 
 
 or, x= 6 3 = 9. 
 
EQUATIONS OF THE SECOND DEGREE. 179 
 
 Verification. 
 If we take the plus root, the equation 
 
 x z +6x=27 
 
 gives (3)*-r-6(3)=27; 
 
 and for the negative root, 
 
 gives ( 9) 2 +6(~9) = 81 54=27. 
 
 4. What are the values of x in the equation 
 
 a? 2 KXr-f 15=^ -- 34z+155. 
 o 
 
 By clearing the fractions, we hare 
 
 by transposing and reducing, we obtain 
 4x* f 120ar = 700; 
 then, dividing by the coefficient of a? 2 , we have 
 
 and by completing the square, 
 
 # 2 +30o:+225: 
 and by extracting the square root, 
 
 x+15 = -v/400^4-20 or 20 
 Hence, x=-\-5 or 35. 
 
 Verification. 
 For the plus value of a?, the equation 
 
 gives (5) 2 4 30 x 5=25+ 150 = 175 
 
1 80 ELEMENTARY ALGEBRA. 
 
 A.nd for the negative value of #, we have 
 
 (_35)2_j_30(_35) = 1225 1050 = 175. 
 
 5. What are the values of a? in the equation 
 
 Clearing the fractions, we have 
 
 10^6^+9=96 &r 12a 2 +273 ; 
 transposing and reducing, 
 
 22x 2 +2a?=360; 
 dividing both members by 22, 
 
 2 360 
 
 2 
 Add { ) to both members, and the equation becomes 
 
 2 ,/iy 360 /I 
 22^(22) = -22- + 
 
 whence, by extracting the square root, 
 
 1 7360 / 1 \ 2 
 
 ^22 = 
 
 therefore, 
 
 _ J_ . /36<) . / 1 \ 2 
 
 A i 360 
 
 and ^ == __ 
 
EQUATIONS OF THE SECOND DEGREE. 181 
 
 It remains to perform the numerical operations. In the 
 
 urst place, \-( \ must be reduced to a single num- 
 
 22 \22/ 
 
 her, having (22 ) 2 for its denominator. 
 
 360 / 1 \ 2 360x22-f-l_7921 
 ' W> ~22~ + V22/ = (22) 2 ""(22)2 J 
 
 extracting the square root of 792 1 , we find it to be 89 ; 
 therefore, 
 
 /360 /J_\ 2 _ 8 
 V~22~ \22/ "^22* 
 
 Consequently, the plus value of x is 
 
 * = ~: 
 
 and the negative value is 
 
 __J_ 89_88 
 ~~22 22~22~ ' 
 
 __1 __ 9__ 45 
 ~~~~" 
 
 that is, one of the two values of x which will satisfy the 
 proposed equation is a positive whole number, and the other 
 a negative fraction. 
 
 6. What are the values of x in the equation 
 
 7. What are the values of x in the equation 
 
 = 5 -?- + 197. 
 4 8 
 
 Ans 
 
 i: 8 _, 
 
182 ELEMENTARY ALGEBRA. 
 
 8. What are the values of x in the equation 
 
 Ans. 
 9. What are the values of x in the- equation 
 
 10. What are the values of a; in the equation 
 
 a? jr__a?2_jr 13 
 ~~ ~ + ' 
 
 EXAMPLES IN THE SECOND FORM. 
 
 1. What are the values of x in the equation 
 
 An*. 
 
 x=-2$ 
 
 By transposing, 
 
 x 2 8x=19 10 = 9, 
 then by completing the square 
 
 x 2 8x+16 = 9-j-16=25, 
 and by extracting the root 
 
 x 4= rfc-v/25 = +5 or 5. 
 Hence, 
 
 a?=4-|-5 = 9 or o?=4 5 = 1. 
 
 That is, in this form, the largest root is positive and the 
 smaller negative. 
 
EQUATIONS OF TUB SECOND DEGREE. 183 
 
 Verification. 
 
 If we take the positive value of a;, the equation 
 
 *> -8a;=9 gives (9) 2 8x9 = 81 72 = 9; 
 and if we take the negative value, the equation 
 
 tf 2 --8a: = 9 gives (-1)2_8(-1)-1 + 8 = 9 ; 
 
 from which we see that both values alike satisfy the equa- 
 tion. 
 
 2. What are the values of x in the equation 
 
 x 2 x x 2 
 
 _ + __ 1 5 = _ + *-14 i . 
 
 By clearing the fractions, we have 
 
 177 
 
 and by transposing and reducing 
 
 and dividing by the co-efficient of a; 2 , we obtain 
 T 2 8 r _i 
 
 ~Y 
 
 Then, by completing the square, we have 
 
 md by extracting the square root, 
 
 Hence, 
 
 25 5 5 
 
 =+ or -- 
 
 4,5 451 
 
 +=+ 3 or *=~ = - 
 
184 ELEMENTARY ALGEBRA. 
 
 Verification. 
 For the positive value of a?, the equation 
 
 --!=' 
 
 o 
 
 gives 3 2 x 3=9 8 = 1: 
 
 3 
 
 and for the negative value, the equation 
 
 8 1 1,8 
 
 T X - T=T+T =1. 
 
 3. What are the values of x in the equation 
 
 Clearing the fractions, and dividing by the coefficient of 
 a: 2 , we have 
 
 Completing the square, we have 
 
 then, by extracting the square root, we have 
 
 hence, 
 
 7 7 
 
 - == + or -" ; 
 
 1 , 7 9 lt 17 5 
 
 T 4 T = T =li, or ,=---= -Y. 
 
EQUATIONS OF THE SECOND DEGREE. 185 
 
 Verification 
 If we take the positive value of #, the equation 
 
 gives (!}) xl}=2}-l=lj: 
 
 and for the negative value, the equation 
 
 * 2 5 25 10 45 
 
 4. What are the values of x in the equation 
 =l8ab l8b z T 
 
 By transposing, changing the signs, and dividing by 2, it 
 becomes 
 
 x 2 ax=2a 2 
 
 whence, completing the square, 
 
 4 4 
 extracting the square root, 
 
 * = T 
 
 
 
 Now, the square root of 9ab+9b 2 , is evidently 
 
 ~ 36. Therefore, 
 
 ?= 2a 36, 
 
 ' = 1 -3*), <* J , or 
 
 f #:=: a-f-30. 
 
 17 
 
186 ELEMENTARY ALGEBRA. 
 
 What will be the numerical values of x, if we suppose 
 a = 6 and b=l ? 
 
 5. What are the values of x in the equation 
 
 .Lt_4_38+2a-- 1*2 = 45- 3*2+4* ? 
 
 o O 
 
 C x= 7,12 > to within 
 .Aws. < > 
 
 lx= 5,73* 0,01. 
 
 6. What are the values of x in the equation 
 
 7. What are the values of x in the equation 
 
 T 
 
 Ans. \ x = S ' 
 
 ( ar= 4. 
 
 8. What are the values of x in the equation 
 
 x z 
 2 
 
 f x Q 
 
 Ans. < 
 
 9. What are the values of x in the equation 
 
 ( a* 2a-L-b. 
 An* ] 
 
 ( x=b. 
 
 10. What are the values of x in the equation 
 
 ~n 2 m z 
 Ans. 
 
EQUATIONS OF THE SECOND DEGREE. 187 
 
 EXAMPLES IN THE THIRD FORM. 
 
 1. What are the values of x in the equation 
 
 First, by completing the square, we have 
 x*+4x+4 = 3 + 4=1 ; 
 and by extracting the square root, 
 
 x+2= -v/T= + l or 1 : 
 nence, x= 2-j-l = 1 ; or x= 2 1 = 3 
 That is, in this form both the roots are negative. 
 
 Verification. 
 If we take the first negative value, the equation 
 
 X* + 4X= 3 
 
 gives (-l)+4(-l) = l-4 = -3; 
 
 and by taking the second value, the equation 
 
 x*+4x= 3 
 
 gives (_3)2_|_4(_3)-9_i2 = 3 : 
 
 hence, both values of x satisfy the given equation. 
 2 What are the values of * in the equation 
 
 By transposing and reducing, we have 
 -* 2 -ll;r=28; 
 
 then since the coefficient of the second power of x is nega- 
 tive, we change the signs of all the terms which gives 
 
 r= 28, 
 
1 88 ELEMENTARY ALGEBRA. 
 
 then by completing the square 
 
 a? 2 +lla?+30,25=2,25, 
 hence, 
 
 ar-f 5,5= -y/2,25 = -|-l,5 or 1,5. 
 onsequently, 
 
 x= 4 or x= 7. 
 
 3. What are the values of x in the equation 
 
 < x= 2 
 Ans. 1 
 
 I x=z 5. 
 
 4. What are the values of x in the equation 
 
 2# 2 +8ar= 2j x. 
 
 3 
 
 Ans. { 
 
 U=-J. 
 
 5. What are the values of x in the equation 
 
 5 
 
 6. What are the values of x in the equation 
 
 2_4_JL 
 
 ~T* ~" 2 
 
 7. What are the values of x in the equation 
 
 a: 2 -*- 7a?-{-20:= a? 2 Ha? 60. 
 
 9 9 
 
 Ans. < ~~ 
 
 ?= - 8. 
 
EQUATIONS OF THE SECOND DEGREE. 189 
 
 8. What are the values of a? in the equation 
 
 , *= 8 
 Ans. 
 
 9. What are the values of x in the equation 
 
 Ans. I * " 
 
 I a:= T %. 
 
 lO. What are the values of a; in the equation 
 
 1 1 . What are the values of x in the equation 
 : 90= 93. 
 
 = 3 
 Ans. 
 
 EXAMPLES IN THE FOURTH FORM. 
 
 1 . What are the values of x in the equation 
 
 x 2 8x= 7. 
 By completing the square we have 
 
 x*8x+l6=i 7+16=9; 
 then by extracting the square root 
 
 x 4 = -y/9^: + 3 or 3; 
 
 hence, 
 
 =+7 or a: =+1. 
 
 That is, in this form, both the roots are positive. 
 17* 
 
190 ELEMENTARY ALGEBRA. 
 
 Verification. 
 If \i e take the largest root, the equation 
 
 x*8x=7 gives 7 2 8 x 7 = 49-56= -7; 
 and for the smaller, the equation 
 
 a 2 8a?= 7 gives I 2 8 x 1 = 1 8 = 7 : 
 hence, both of the roots will satisfy the equation. 
 2. What are the values of x in the equation 
 
 I-'-* 2 + 3# 10 = IJz 2 
 
 By clearing the fractions, we have 
 
 . ~ 3 X 2+G X 20=3* 2 36a?+40; 
 then by collecting the like terms 
 
 6# 2 +42a:=60 ; 
 
 then by dividing by the coefficient of a: 2 , and at the same 
 time changing the signs of all the terms, we have 
 
 x 2 7x= 10. 
 By completing the square, we have 
 
 x z lx+ 12,25 =2,25, 
 and by extracting the square root of both members, 
 
 a? 3,5=db V2,25 = + l,5 or 1,5. 
 hence, 
 
 *=3,5-{- 1,5 = 5, or x=3,5 1,5=2. 
 
EQUATIONS OF THE SECOND DEGREE. 191 
 
 Verification. 
 
 If we take the larger root, the equation 
 
 x*7x= 10 gives 5 2 7x5=25 35 = - 10 ; 
 . and if we take the smaller root, the equation 
 
 X* 7x= 10 gives 2 2 7x2=4 14.= 10. 
 3. What are the values of x in the equation 
 
 3x+2x z +l = l7x 2x 2 3, \ 
 By transposing and collecting the terms, we have 
 
 4a; 2 20|*= 4; 
 then dividing by the coefficient of x 2 we have 
 
 *2-5j*=-l. 
 
 By completing the square, we obtain 
 
 HS-l.+5-Sf. 
 
 and by extracting the root 
 
 /T44~ 12 12 
 
 * 2 _ 2i = x /_ = +_ or -_; 
 
 hence, 
 
 *= 2 1+T= 5 ' or ' = 8 *-T=f- 
 
 Verification. 
 
 If we take the larger root, the equation 
 
 a; 2 5|-a?= 1 gives 5 2 5| x5=25 26 = 1 
 suid if we take the smaller root, the equation 
 
1 92 ELEMENTARY ALGEBRA. 
 
 4. What are the values of x in the equation 
 
 Ans. 
 
 5. What are the values of x in the equation 
 
 1 
 7 
 
 Ans. 
 
 6. What are the values of x in the equation 
 
 81 11, 
 
 7. What are the values of x in the equation 
 
 Ans. 
 
 8. What are the values of x in the equation 
 
 17 x z 2r 2 
 
 --+100=- 
 5 5 
 
 An,. 
 
 x= 
 
 9. What are the values of x in the equation 
 
 7a* 
 
 ~~ 3 
 
 Ans - 
 
 10. What are the values of x in the equation 
 
 JLi 
 
 10 
 
EQUATIONS OF THE SECOND DEGREE. 193 
 
 Properties of the Roots. 
 
 128. We have thus far, only explained the methods of 
 finding the roots of an equation of the second degree. We 
 
 . are now going to show some of the properties of these roots 
 
 The first form. 
 
 129. The first form 
 
 gives 1st root x= p f 
 
 2nd root x= p 
 
 and their sum = 2p. 
 
 Since, in this form q is supposed positive, the quantity 
 q+p 2 under the radical sign will be greater than j> 2 , and 
 hence its root will be greater than p. Consequently the 
 first root, which is equal to the difference between p and 
 the radical, will be positive and less than \/q-{-p 2 . In the second 
 root, p and the radical have the same sign ; hence, the 
 second root will be equal to their sum and negative. If we 
 multiply the two roots together, we have 
 
 p -f- 
 -p - 
 
 +P 2 -pV<l+P 2 
 
 Product equal to ..... q. 
 
 QUKST. 129. In the first form, have the roots the same or contrary 
 signs'! What is the sign of the first root] What of the second I 
 Which is the greater 1 What is .heir sum equal to 7 What is their 
 product equal to 1 
 
194 ELEMENTARY ALGEBRA. 
 
 Hence we conclude, 
 
 1st. That in the first form one of the roots is always posi 
 ttve, and the other negative. 
 
 2nd. That the positive root is numerically less than tht 
 negative. 
 
 3rd. That the sum of the two roots is equal to the coefficient 
 of x in the second term, taken with a contrary sign. 
 
 4th. That the product of the two roots is equal to the known 
 term in the second member, taken with a contrary sign. 
 
 EXAMPLES. 
 
 1. In the equation 
 
 we find the roots to be 4 and 5. Their sum is 1, 
 and their product 20. 
 
 2. In the equation 
 
 we find the roots to be 1 and 3. Their sum is equal to 
 2, and their product to 3. 
 
 3. The roots of the equation 
 
 x*+x 90, 
 
 are _L_9 and 10. Their sum is 1, and their product 
 90. 
 
 4. The roots of the equation 
 
 are 6 and 10. Their sum is -4, and their product is 
 -60. 
 
EQUATIONS OF THE SECOND DEGREE. 195 
 
 ! -et. these principles be applied to each of the examples 
 
 Udder " EXAMPLES IN THE FIRST FORM." 
 
 Second Form. 
 13O. The second form is, 
 
 and by resolving the equation we find 
 1st root, ix= -\-p-\- 
 
 2nd root, 
 and their sum 
 
 In this form, the first root is positive and the second 
 negative. Tf we multiply the two roots together, we have 
 
 Hence we conclude, 
 
 1st. That in the second form one of the roots is positive 
 and the other negative. 
 
 2nd. That the positive root is numerically greater than th& 
 negative. 
 
 3rd. That the sum of the roots is equal to the coefficient of 
 X in the second term, taken with a contrary sign. 
 
 4th. That the product of the roots is equal to the known 
 t?rm in the second member, taken with a contrary sign. 
 
 QUEST. 130. What is the sign of the first root in the second form 1 
 \\Trat is the sign of the second 1 Which is the greater 1 What is their 
 sum equal to 1 What is their product equal to 1 
 
196 ELEMENTARY ALGEBRA. 
 
 EXAMPLES. 
 
 1. The roots of the equation 
 
 x z x=I2, 
 
 are +4 and 3. Their sum is -f-l> and their product 
 12. 
 
 2. The roots of the equation 
 
 are +10 and . Their sum is 9j^, and their product 
 is 1. 
 
 3. The 'roots of the equation 
 
 are +8 and 2. Their sum is +6, and their product 
 is 16. 
 
 4. The roots of the equation 
 
 are -{-16 and 5. Their sum is -|- 1 1 , and their product 
 is 80. 
 
 Let these principles be applied to each of the examples 
 under " EXAMPLES IN THE SECOND FORM." 
 
 Third Form. 
 131. The third form is, 
 
 and by resolving the equation we find, 
 1st root, #:= 
 
 2nd root, x= p i/ 
 
 Their sum is = 2p 
 
EQUATIONS OF THE SECOND DEGREE. 197 
 
 In tins form, the quantity under the radical being less 
 than p 2 j its root will be less than- p : hence both the roots 
 will be negative, and the first will be numerically the least. 
 
 If we multiply the roots together, we have 
 
 Hence we conclude, 
 
 1 st. That in the third form both the roots are negative. 
 2nd. That the first root is numerically less than the second. 
 
 3rd. That the sum of the two roots is equal to the coefficient 
 of x in the second term, taken with a contrary sign. 
 
 4th. That the product of the roots is equal to the known 
 term in the second member ', taken with a contrary sign. 
 
 EXAMPLES. 
 1. The roots of the equation 
 
 #=: 20, 
 
 are 4 and 5. Their sum is 9, and their product 
 
 + 20. 
 
 2. The roots of the equation 
 
 #2+13*:= 42, 
 
 are 6 and 7. Their sum is 13, and their product 
 
 +42. 
 
 QUEST. 131. In the third form, what are the signs of the roots ? 
 Which root is the least 1 What is the sum of the roots equal to 1 
 What is their product equal to 1 
 
 18 
 
1 98 ELEMENTARY ALGEBRA. 
 
 3. The roots of the equation 
 
 are and 2. Their sum is 2|, and their product 
 
 4. The roots of the equation 
 
 = 6, 
 
 are 2 and 3. Their sum is 5, and their product 
 is +6 
 
 Let these principles be applied to each of the examples 
 under " EXAMPLES IN THE THIRD FORM." 
 
 Fourth Form. 
 
 132. The fourth form is, 
 
 x 2 2px= q ; 
 
 and by resolving the equation we find, 
 1st root, x=p-{--\/ 
 
 2nd root, 
 
 Their sum is 
 
 In this form, as well as in the third, the quantity under 
 the radical being less than p 2 , its root will be less than p : 
 hence both the roots will be positive, and the first will be 
 the greatest. 
 
 If we multiply the two roots together, we have 
 
 -1-7- 
 
t 
 EQUATIONS OF THE SECOND DEGREE. 199 
 
 Hence we conclude, 
 
 1 st. That in the fourth form both the roots are positive. 
 
 2nd. That the first root is greater than the second. 
 
 3rd. That the sum of the roots is equal to the coefficient of 
 x in the second term, taken with a contrary sign. 
 
 4th. That the product of the roots is equal to the known 
 term in the second member, taken with a contrary sign. 
 
 EXAMPLES. 
 
 1 . The roots of the equation 
 
 * 2 7x= 12, 
 
 are +4 and +3. Their sum is +7 and their pro- 
 duct + 12. 
 
 2. The roots of the equation 
 
 * 2 14*= -24, 
 
 are +12 and +2. Their sum is +14 and their pro- 
 duct + 24. 
 
 3. The roots of the equation 
 
 are +18 and +2. Their sum is +20 and their pro- 
 duct + 36. 
 
 4. The roots of the equation 
 
 x 2 \7x=42, 
 
 are +14 and +3. Their sum is +17 and thoir pro 
 duct +42. 
 
 QUEST. 132. In the fourth form, what are the signs of the roots 1 
 Which root is the greatest 1 What is the sum of the roots equal to ? 
 What is their product e<]ual to 7 
 
200 ELEMENTARY ALGEBRA. 
 
 133. In the third and fourth forms the values of x some- 
 times become imaginary, and in such cases it is necessary 
 to know how the results are to be interpreted. 
 
 If we have q>p 2 , that is, if the known term is greater 
 than half the coefficient ofx squared, it is plain that ^/ q+p 2 
 will be imaginary, since the quantity under the radical 
 will be negative. Under this supposition the values of x 
 in the third and fourth forms will be imaginary. 
 
 We will now show that, when in the third and fourth 
 forms, we have q^>j) 2 , the conditions of the question will be 
 incompatible with each other. 
 
 134. Before showing this it will be necessary to estab- 
 lish a proposition on which it depends : viz. 
 
 If a given number be decomposed into two parts and those 
 parts multiplied together, the product will be the greatest pos- 
 sible when the parts are equal. 
 
 Let 2p be ihe number to be decomposed, and d the differ- 
 ence of the parts. Then 
 
 p +-o- = the greater part (page 80, Ex. 7.) 
 
 and p = the less part. 
 
 2 
 
 d 2 
 and p 2 = P, their product (Art. 4O.) 
 
 Now it is plain that P will increase as d diminishes, and 
 that it will be the greatest possible when d=0 : that is, 
 
 p xp =p 2 is the greatest product. 
 
 QUEST. 133. In which fr .as do the values of x become imaginary 7 
 When will the values of x be imaginary ? Why will the values of x be 
 then imaginary 1 
 
EQUATIONS OF THE SECOND DEGREE. 201 
 
 Now, since in the equation 
 
 2p is the sum of the roots, and q their product, it follows 
 that q can never be greater than p 2 . The conditions of the 
 equation, therefore, fix a limit to the value of q, and if we 
 make q^p 2 , we express by the equation a condition which 
 cannot be fulfilled, and, this contradiction is made apparent 
 by the values of x becoming imaginary. Hence we may 
 conclude that, 
 
 When the values of the unknown quantity are imaginary, 
 the conditions of the question are incompatible with each other. 
 
 EXAMPLES. 
 
 1. Find two numbers whose sum. shall be 12 and pro- 
 duct 46. 
 
 Let x and y be the numbers. 
 
 By the 1st condition, x-\- y=l2 ; 
 
 and by the 2d, xy = 46. 
 
 The first equation gives 
 
 x = l2y. 
 Substituting this value for x in the second, we have 
 
 12y-y 2 = 46; 
 and changing the signs of the terms, we have 
 
 *l2 = 46. 
 
 QUKST. 134. What is the proposition demonstrated in Article 134] 
 If the conditions of the question are incompatible, how will the values 
 of the unknown quantit) be 1 
 
 18* 
 
202 ELEMENTARY ALGEBRA. 
 
 Then by completing the square 
 
 y z l2y+36= 46 + 36 = 10 
 which gives y = 6 + -\/ 10, 
 
 and y 6 -y/ 10; 
 
 both of which values are imaginary, as indeed they should 
 be, since the conditions are incompatible. 
 
 2. The sum of two numbers is 8, and their product 20 
 what are the numbers ? 
 
 Denote the numbers by x and y. 
 By the first condition, 
 
 and by the second, a?y=20. 
 
 The first equation gives 
 
 cc = 8 y 
 Substituting this value of x in the second, we have 
 
 8y y 2 =20 ; 
 changing the signs, and completing the square, we have 
 
 and by extracting the root, 
 
 y=4-{--v/^4 and y=4 -y/ 4. 
 These values of y maybe put under the forms (Art. 1O6) 
 
 3. What are the values of x in the equation 
 -10. 
 
 Ans. 
 
EQUATIONS OF THE SECOND DEGREE. 203 
 
 Examples with more than one unknown quantity. 
 
 tofind * and y ' 
 
 By transposing y in the first equation, we have 
 
 a?=14 y ; 
 and by squaring both members, 
 
 * 2 =196 28y+y 2 . 
 Substituting this value for a? 2 in the 2nd equation, we have 
 
 196 28y+y 2 +y 2 =100 ; 
 from which we have 
 
 y 2 14y= 48 ; 
 and by completing the square, 
 
 y 2 14y+49 = l ; 
 and by extracting the square root, 
 
 y 7=yT=+l or 1: 
 hence, y 7+1=8, or y=7 1=6. 
 
 If we take the larger value, we find a; =6 ; and if we 
 
 take the smaller, we find a? =8. 
 
 Verification. 
 
 For the largest value, y = 8, the equation 
 x+y = 14 gives 6 + 8 = 14; 
 and # 2 4-y 2 = 100 gives 36+64 = 100. 
 
 For the value y = 6, the equation 
 
 #+yr=14 gives 8 + 6 = 14; 
 and a; 2 +y 2 = 100 gives 64 + 36 = 100. 
 
 Hence, both sets of values will satisfy the given equation 
 
204 ELEMENTAEY ALGEBRA. 
 
 / y> */ 3 I 
 
 2. Given < \ to find x and y. 
 
 ( x 2 y 2 = 45 ) 
 
 Transposing y in the first equation, we have 
 
 and then squaring both members, 
 
 Substituting this value for x 2 in the second equation, we 
 have 
 
 whence we have 
 
 6y=36 and y=6. 
 Substituting this value of y in the first equation, we have 
 
 #-6 = 3, 
 and consequently x =3-}- 6 =9. 
 
 Verification. 
 
 xy = 3 gives 9 6 = 3; 
 and a: 2 y 2 = 45 gives 8136=45. 
 
 3 - Given \ Jt5+2y* = 40 i * find X a " d ' 
 Subtracting the first equation from the second, we have 
 
 2y 2 =18, 
 
 which gives y 2 =9 
 
 and y=-f-3 01 3. 
 
 Substituting the plus value in the first equation, we have 
 2 
 
EQUATIONS OF THE SECOND DEGREE. 205 
 
 from which we find 
 
 x=+2 and #= 11. 
 
 If we take the negative value, y= 3, we have from the 
 first equation, 
 
 a; 2 9#=22 ; 
 from which we find 
 
 and a= 2. 
 
 For the values y=+3 and x =+2, the equation 
 
 gives 2 2 +3x2x3=4 + 18=22: 
 
 and for the second value, ar= 11, the same equation 
 
 gives ( ll) 2 +3x -11x3 = 12199=22. 
 
 If now .we take the second value of y, that is, y= 3 
 and the corresponding values of ar, viz, #=-{-11, and 
 x= 2 ; for #= -f- 1 1 , the equation 
 
 gives H 2 +3 x 11 x -3 = 121 -99=22 ; 
 
 and for x= 2, the same equation 
 
 gives (-2) 2 +3 X 2 X 3=4+18=22. 
 
 4. Given < x +y ~\-z = 7 (2) > to find x t y, and z. 
 ( x *+ v zi z 2 = 2i ( 3 ) j 
 
206 ELEMENTARY ALGEBRA. 
 
 Transposing y in the second equation, we have 
 
 *+*=7-y (4); 
 then squaring the members, we have 
 
 x*+2xz+z 2 =49 14y+y 2 . 
 
 If now we substitute for 2xz its value taken from the 
 first equation, we have 
 
 a: 2 +2y 2 +;s 2 =49 Hy+y 2 ; 
 and cancelling y 2 in each member, there results 
 
 But, from the third equation we see that each member of 
 the last equation is equal to 21 : hence 
 
 49 14y=21, 
 and 14y=49 21=28. 
 
 28 
 hence, y= =2. 
 
 Placing this value for y in equation (1) gives 
 
 xz=4 j 
 and placing it in equation (4) gives 
 
 x+z=5, and x = 5z. 
 
 Substituting this value of x in the previous equation, we 
 obtain 
 
 5zz*=4 or z*5z= 4; 
 and by completing the square, we have 
 s 2 5^+6,25=2,5, 
 
 and z2,5 = \/2 = + 1,5 or 1,5; 
 
 nence, z= 2,5+1,5=4 or *= -1-2,5 1,5 = 1. 
 
EQUATIONS OF THE SECOND DEGREE 207 
 
 If we take the value 
 
 z=4, we find x=l : 
 if we take the less value 
 
 *=1, we find x=4. 
 
 3. Given x + ^fxy-\-y 19 
 
 19 ) 
 
 \ to 
 =l33 S 
 
 o 
 and x*+ xy+y z 
 
 Dividing the second equation by the first, we have 
 
 -/ay+ y= 7 
 
 but x+</xy+ y=l9 
 
 nence, by addition, 2x +2y=26 
 
 or a 4- y=13 
 
 and substituting in 1st equa. y^y-f-13 = 19 
 
 or y r icy"= 6 
 
 and by squaring a:y=36 
 
 From 2d equation, x 2 +a:y +y 2 =133 
 
 and from the last 3xy =108 
 
 Subtracting a; 2 2ary-f y*= 25 
 
 hence, ar y=db 5 
 
 but a?4-y= 13 
 
 hence o?=9 or 4 ; and y=4 or 9. 
 
 6. Given the sum of two numbers equal to a, and the 
 sum of their cubes equal to c, to find the numbers 
 
 By the conditions 
 
 ix+y=a 
 
208 
 
 ELEMENTARY ALGEBRA. 
 
 Putting x=s-\-z, and y=s z, we have 
 
 a=2s, or s= ; 
 2 
 
 and 
 
 3s z z-\-3sz 2 z 3 . 
 
 hence, by addition, x 3 + y 3 = 2s 3 -f 6sz 2 = c, 
 whence z z =" ~" and z=. 
 
 or 
 
 ; and y= 
 
 c: 
 6~* 
 
 c \ 
 6^~' 
 
 or by putting for s its value, 
 
 and 
 
 NOTE. What are the numbers when a=5 and c=35. 
 What are the numbers when a=9 and c=243. 
 
 QUESTIONS. 
 
 1. Find a number such, that twice its square, added to 
 three times the number, shall give 65. 
 
 Let x denote the unknown number. Then the equation 
 of the problem will be 
 
 whence 
 
 65 9 
 
 3 23 
 
EQUAUON& OF THE SECOND DEGREE. 209 
 
 Therefore, 
 
 3 , 23 3 23 13 
 
 .= - T + T =5, and *=- T - T= _ T . 
 
 Both these values satisfy the question in its algebraic 
 sense. For, 
 
 / 13\ 2 13 169 39 130 
 
 and 2 (__) +3 x --=_- y =_=65. 
 
 REMARK. If we wish to restrict the enunciation to its 
 arithmetical sense, we will first observe, that when x is 
 replaced by x, in the equation 2x 2 .-}-3x=65, the sign of 
 the second term 3x only, is changed, because ( x) 2 =x*. 
 
 q oq 
 
 Therefore, instead of obtaining x= T^~T> we s ^ ou ^ 
 
 3 23 13 
 
 find x= , or x= , and x= 5, values which only 
 
 differ from the preceding by their signs. Hence, we may 
 
 13 
 say that the negative solution -- , considered indepen 
 
 dently of its sign, satisfies this new enunciation, viz : To 
 find a number such, that twice its square, diminished by three 
 times the number, shall give 65. In fact, we have 
 
 13 169 39 
 
 REMARK. The root which results from giving the plus 
 sign to the radical, generally resolves the question both 
 in its arithmetical and algebraic sense, while the second 
 root resolves it in its algebraic sense only. 
 
 19 
 
210 ELEMENTARY A.LGEBRA. 
 
 Thus, in the example, it was required to find a number, 
 of which twice the square added to three times the number 
 shall give 65. Now, in the arithmetical sense, added means 
 increased ; but in the algebraic sense it implies diminution, 
 when the quantity added is negative. In this sense, the 
 second root satisfies the enunciation. 
 
 2. A certain person purchased a number of yards of cloth 
 for 240 cents. If he had received 3 yards less of the same 
 cloth for the same sum, it would have cost him 4 cents more 
 per yard. How many yards did he purchase ? 
 
 Let x= the number of yards purchased. 
 
 240 
 Then will express the price per yard. 
 
 If, for 240 cents, he had received 3 yards less, that is 
 x3 yards, the price per yard, under this hypothesis, would 
 
 240 
 have been represented by . But, by the enunciation, 
 
 OC O 
 
 this last cost would exceed the first by 4 cents. Therefore, 
 we have the equation 
 
 240 240 _ 
 
 ^=3 ~ : :4; 
 
 whence, by reducing a; 2 3a?=180, 
 
 therefore a?=15 and sc = 12. 
 
 The value x =15 satisfies the enunciation; for, 15 yards 
 
 240 
 of 240 cents gives , or 16 cents for the price of 
 
 I O 
 
 one yard, and 12 yards for 240 cents, gires 20 cents for the 
 prirr of one yard, which exceeds 16 by 4. 
 
EQUATIONS OF THE SECOND DEGREE. 21 1 
 
 As to the second solution, we can form a new enuncia- 
 tion, with which it will agree. For, going back to the 
 equation, and changing x into #, it becomes 
 
 240 240 240 240 
 
 - x -3 -x x x+3 
 
 an equation which may be considered the algebraic transla- 
 tion of this problem, viz : A certain person purchased a num- 
 ber of yards of cloth for 240 cents : if he had paid the same 
 sum for 3 yards more, it would have cost him 4 cents less per 
 yard. How many yards did he purchase ? 
 
 Ans. 07=12, and x= 15. 
 
 3. A man bought a horse, which he sold after some time 
 for 24 dollars. At this sale, he loses as much per cent, 
 upon the price of his purchase as the horse cost him. 
 What did he pay for the horse ? 
 
 Let x denote the number of dollars that he paid for the 
 horse, # 24 will express the loss he sustained. But as 
 
 he lost x per cent, by the sale, he must have lost 
 
 i CO 
 
 upon each dollar, and upon x dollars he loses a sum de- 
 noted by ; we have then the equ :on 
 
 x 2 
 
 =o?24, whence x 2 10(L =2400. 
 
 100 
 
 and ar=50-v/2500- >"V>_ 
 
 Therefore, a; =60 and a =40. 
 
 Both of these values satisfy the question. 
 
 For, in the first place, suppose the man gave $60 for the 
 horse and sold him for 24, he loses 36. Again, from the 
 enunciation, he should lose 60 per cent, of 60, that is, 
 
212 ELEMENTARY ALGEBRA. 
 
 of 60, or TTTTT-J which reduces to 36 ; there- 
 
 100 100 
 
 fore, 60 satisfies the enunciation. 
 
 Had he paid $40, he would have lost $16 by the sale ; 
 
 40 
 for, he should lose 40 per cent, of 40, or 40 x -- , which 
 
 reduces to 16 ; therefore, 40 verifies the enunciation. 
 
 4. A man being asked his age, said the square root oi 
 my own age is half the age of my son, and the sum of 
 our ages is 80 years : what was the age of each ? 
 
 Let x= the age of the father. 
 
 y= that of the son. 
 Then by the first condition 
 
 .. 
 
 and by the second condition 
 
 z+y = 80. 
 If we take the first equation 
 
 ^f 
 
 and square both members, we have 
 
 ,= 
 
 If we transpose y in the second, we have 
 
 x=80y: 
 from which we find 
 
 y= 
 
 by taking the plus root, which answers to the question in 
 its arithmetical sense. Substituting this value, we find 
 oc=64. j J Father's age 64 
 
 nS ' ( Son's 16 
 
EQUATIONS OF THE SECOND DEGREE. 213 
 
 5. Find two numbers, such that the sum of their pro- 
 ducts by the respective numbers a and &, may be equal to 
 2s, and that their product may be equal to p. 
 
 Let x and y be the required numbers, we have the equa- 
 tions 
 
 ax -\- by = 2s. 
 and xy=p. 
 
 From the first y= 7 
 
 whence, by substituting in the second, and reducing, 
 aoc 2 2sx= bp. 
 
 Therefore, 
 and consequently, 
 
 _ S 1 -g ,- 
 
 b b 
 
 This problem is susceptible of two direct solutions, be- 
 cause s is evidently > -\/s 2 abp ; but in order that they 
 may be real, it is necessary that s 2 > or =abp. 
 
 Let a=b=l. ; the values of x and y reduce to 
 
 Whence we see, that the two values of x are equal to 
 those of y,_taken in an inverse order ; which shows, that if 
 $4- -\/s 2 p represents the value of x, s -\/s 2 p will re- 
 present the corresponding value of y, and reciprocally. 
 
 This circumstance is accounted for, by observing that in 
 this particular case the equations reduce to 
 
 19* 
 
214 ELEMENTARY ALGEBRA. 
 
 and then the question is reduced to, finding two numbers of 
 which the sum is 2s, and their product p, or in other M r ords 
 to divide a number 2s, into two such parts, that their produc. 
 may be equal to a given number p. 
 
 Let us now suppose 
 
 2s=l4 and p = 48: 
 what will then be the values of x and y ? 
 
 ( x= 
 Ans. < 
 
 ( = 
 
 x = 8 or 6 
 y=6 or 8 
 
 6. A grazier bought as many sheep as cost him 60, and 
 after reserving fifteen out of the number, he sold tho re- 
 mainder for j54, and gained 2s. a head on those he sold : 
 how many did he buy ? Ans. 75. 
 
 7. A merchant bought cloth for which he paid j33 15.sv, 
 which he sold again at 2 8s. per piece, and gained by the 
 bargain as much as one piece cost him : how many pieces 
 did he buy? Ans. 15. 
 
 8. What number is that, which, being divided by the pro- 
 duct of its digits, the quotient is 3 ; and if 18 be added to 
 it, the digits will be inverted ? Ans. 24, 
 
 9. To find a number, such that if you subtract it from 1 0, 
 and multiply the remainder by the number itself, the product 
 shall be 21. Ans. 7 or 3. 
 
 10. Two persons, A and B, departed from different places 
 at the same time, and travelled towards each other. On 
 meeting, it appeared that A had travelled 18 miles more 
 than B ; and that A could have gone B's journey in 1 5| 
 days, but. B would have been 28 days in performing A's 
 journey How far did each travel ? 
 
 ( A 72 miles 
 Ans. ^ 
 
 B 54 miles 
 
EQUATIONS OF THE SECONI* DEGREE. 215 
 
 11. There are two numbers whose difference is 15, cind 
 half the'ir product is equal to the cube of the lesser num- 
 
 'ber. What are those numbers 1 Ans. 3 and 18. 
 
 12. What two numbers are those whose sum, multiplied 
 by the greater, is equal to 77 ; and whose difference, multi 
 plied by the lesser, is equal to 12 ? 
 
 Ans. 4 and 7, or f yTand y -/~2. 
 
 13. To divide 100 into two such parts, that the sum 01 
 their square roots may be 14. Ans. 64 and 36. 
 
 14. It is required to divide the number 24 into two such 
 parts, that their product may be equal to 35 times their dif- 
 ference. Ans. 10 and 14. 
 
 15. The sum of two numbers is 8, and the sum of their 
 cubes 152. What are the numbers 1 Ans. 3 and 5. 
 
 16. Two merchants each sold the same kind of stuff; 
 the second sold 3 yards more of it than the first, and to- 
 gether they receive 35 dollars. The first said to the second, 
 "1 would have received 24 'dollars for your stuff;" the 
 other replied, " And I should have received 12 J dollars for 
 yours." How many yards did each of them sell ? 
 
 ( 1st merchant x =15 x=5. 
 
 Ans. ? or 
 
 I 2nd ' y=18 y=8. 
 
 17. A widow possessed 13,000 dollars, which she divided 
 into two parts, and placed them at interest, in such a man- 
 ner, that the incomes from them were equal. If she had 
 put out the first portion at the same rate as the second, she 
 would have drawn for this part 360 dollars interest ; and if 
 she had placed the second out at the same rate as the first, 
 she would have drawn for it 490 dollars interest. What 
 were the two rates of interest ? " 
 
 Ans. 7 and per cen 
 
216 ELEMENTARY ALGEBRA 
 
 CHAPTER VII. 
 Of Proportions and Progressions. 
 
 135. Two quantities of the same kind may be compared 
 together in two ways : 
 
 1st. By considering how muck one is greater or less than 
 the other, which is shown by their difference ; and, 
 
 2nd. By considering how many times one is greater or 
 less than the other, which is shown by their quotient. 
 
 Thus, in comparing the numbers 3 and 12 together with 
 respect to their difference, we find that 1 2 exceeds 3 by 9 ; 
 and in comparing them together with respect to their quo- 
 tient, we find that 12 contains 3 four times, or that 12 is 4 
 times as great as 3. 
 
 The first of these methods of comparison is called Arith- 
 metical Proportion, and the second Geometrical Proportion. 
 
 Hence, Arithmetical Proportion considers the relation of 
 quantities with respect to their difference, and Geometrical 
 Proportion the relation of quantities with respect to their 
 quotient. 
 
 QUKST. 135. In how many ways may two quantities be compared 
 together 1 What does the first method consider ] What the second ! 
 What is the first of these methods called 1 What is the second called * 
 How then do you defmo the two proportions'? 
 
ARITHMETICAL PROPORTION. 217 
 
 Of Arithmetical Proportion and Progression. 
 
 136. If we have four numbers, 2, 4, 8, and 10, of 
 
 which the difference between the first and second is equal 
 to the difference between the third and fourth, these num 
 bers are said to be in arithmetical proportion. The first 
 term 2 is called an antecedent, and the second term 4, with 
 which it is compared, a consequent. The number 8 is also 
 called an antecedent, and the number 10, with which it is 
 compared, a consequent. 
 
 When the difference between the first and second is equal 
 to the difference between the third and fourth, the four num- 
 bers are said to be in proportion. Thus, the numbers 
 
 2, 4, 8, 10, 
 are in arithmetical proportion. 
 
 137. When the difference between the first antecedent 
 and consequent is the same as between any t\vo adjacent 
 terms of the proportion, the proportion is called an arith- 
 metical progression. Hence, a progression by differences, or 
 an arithmetical progression, is a series in which the succes- 
 sive terms continually increase or decrease by a constant 
 number, which is called the common difference of the 
 progression. 
 
 Thus, in the two series 
 
 1, 4, 7, 10, 13, 16, 19, 22, 25, ... 
 60, 56, 52, 48, 44, 40, 36, 32, 28, ... 
 
 QUEST. 136. When arc four numbers in arithmetical proportion? 
 What is the first called 1 What is the second called 1 What is the 
 third called 1 What is the fourth called ? 
 
218 ELEMENTARY ALGEBRA. 
 
 the first is called an increasing progression, of which trie 
 common difference is 3, and the second a decreasing pro- 
 gression, of which the common difference is 4. 
 
 In general, let 0, &, c, e?, e,f, . . . designate the terms 
 of a progression by differences ; it has been agreed to write 
 them thus : 
 
 a.b.c.d.e.f.g.h.i.k... 
 
 This series is read, a is to b, as b is to c, as c is to d, as d 
 is to e, &c. This is a series of continued equi-differences, 
 in which each term is at the same time a consequent and 
 antecedent, with the exception of the first term, which is 
 only an antecedent, and the last, which is only a consequent. 
 
 138. Let r represent the common difference of the 
 progression 
 
 a.b.c.d.e.f.g.h, &c, 
 
 which we will consider increasing. 
 
 From the definition of the progression, it evidently follows 
 that 
 
 b=a+r, c=b + r=a+2r, d=c+r=a+3r ; 
 
 and, in general, any term of the series is equal to the first 
 term plus as many times the common difference as there are 
 preceding terms. 
 
 Thus, let / be any term, and n the number which marks 
 the place of it : the expression for this general term is 
 
 l=a+(n l)r. 
 
 QUEST. 137. What is an arithmetical progression 1 What is the 
 number called by which the terms are increased or diminished 1 What 
 13 an increasing progression 1 What is a decreasing progression I 
 Which term is only an antecedent 1 Which only a consequent 7 
 
ARITHMETICAL PROGRESSION. 219 
 
 Hence, for finding the last term, we have the following 
 
 RULE. 
 
 I. Multiply the common difference by one less than the 
 number of terms. 
 
 II. To the product add the fast term: the sum will be the 
 last term. 
 
 EXAMPLES. 
 
 The formula l=a-\-(n l)r serves to find any term 
 whatever, without our being obliged to determine ail those 
 which precede it. 
 
 1. If we make n=l, we have l=<!,, that is, the series 
 will have but one term. 
 
 2. If we make n=2, we have l=a-*-r ; that is, the series 
 will have two terms, and the second term is equal to the 
 first plus the common difference. 
 
 3. If a=3 and r=2, what is the 3rd term ? Ans. 7. 
 
 4. If a=5 and r=4, what is the 6th term? Ans. 25. 
 
 5. If a=* and r=5, what is the 9th term? Ans. 47. 
 
 6. If a=8 and r=5, what is the 10th term ? 
 
 Ans. 53. 
 1 If a = 20 and r=4, what is the 12th term ? 
 
 Ans. 64. 
 8. If a=40 and r=20, what is the 50th term ? 
 
 Ans. 1020. 
 
 QUEST. 138. Give the rule for finding the last term of a series when 
 the progression is increasing. 
 
220 ELEMENTARY ALGEBRA. 
 
 9 If 0=45 and r= 30, what is the 40th term ? 
 
 Ans. 1215 
 
 10. If a=30 and r=20, what is the 60th term? 
 
 Ans. 1210 
 
 11. If a = 50 and r=10, what is the 100th term? 
 
 Ans. 1040 
 
 12. To find the 50th term of the progression 
 
 1 . 4 . 7 . 10 . 13 . 16 . 19 . . ., 
 we have 2=1+49x3=148. 
 
 13. To find the 60th term of the progression 
 
 1 . 5 . 9 . 13 . 17 . 21 . 25 . . ., 
 we have 1= 1+59x4 =237. 
 
 139. If the progression were a decreasing one, we 
 bhould have 
 
 l=a (n I)r. 
 
 Hence, to find the last term of a decreasing progression, 
 we have the following 
 
 RULE. 
 
 I. Multiply the common difference by one less than the num- 
 ber of terms. 
 
 II. Subtract the product from the first term . the remaindei 
 will be the last term 
 
 Q UEST . 1"39. Give the rule for finding the last term of a series, 
 when the progression is decreasing. 
 
ARITHMETICAL PROGRESSION. 221 
 
 EXAMPLES. 
 
 1. The first term of a decreasing progression is 60, the 
 number of terms 20, and the common difference 3 : what 
 is tne last term ? 
 
 l-a-(n-l)r gives 1=60 (20 1)3 = 60 57 = 3. 
 
 2. The first term is 90, the common difference 4, and 
 the number of terms 1 5 : what is the last term 1 Ans. 34. 
 
 3. The first term is 100, the number of terms 40, and the 
 common difference 2 : what is the last term 1 Ans. 22. 
 
 4. The first term is 80, the number of terms 10, and the 
 common difference 4 : what is the last term 1 Ans. 44. 
 
 5. The first term is 600, the numl^r of terms 100, and 
 the common difference 5 : what is the last term ? 
 
 Ans. 105. 
 
 6. The first term is 800, the number of terms 200, and 
 the common difference 2 : what is the last term ? 
 
 Ans. 402. 
 
 1 4O. A progression by differences being given, it is 
 proposed to prove that, the sum of any two terms, taken at 
 equal distances from the two extremes, is equal to the sum of 
 the two extremes. 
 
 That is, if we have the progression 
 
 2 . 4 . 6 . 8 . 10 . 12, 
 
 we wish to prove that 
 
 4+10 or 6 + 8 
 
 is equal to the sum of the two extremes 2 and 12. 
 
 20 
 
222 ELEMENTARY ALGEBRA. 
 
 Let a.i>.c.d.c.j....i.k.l be the pro- 
 posed progression, and n the number of terms. 
 
 We will first observe that, if x denotes a term which has 
 p terms before it, and y a term which has p terms after it 
 we have, from what has been said, 
 
 x=a+pxr, 
 
 and y=lpxr\ 
 
 whence, by addition, x-}-y=a-}-l. 
 
 Which demonstrates the proposition. 
 
 Referring this proof to the previous example, if we sup- 
 pose, in the first place, x to denote the second term 4, then 
 y will denote the term 10, next to the last. If x denotes 
 the 3rd term 6, thei* y will denote 8, the third term from 
 the last. 
 
 Having proved the first part of the proposition, write the 
 progression below itself, but in an inverse order, viz : 
 
 a.b.c.d.e.f...i.k.l. 
 I . k . i ......... c . b . a. 
 
 Calling S the sum of the terms of the first progression, 
 2S will be the sum of the terms in both progressions, and 
 we shall have 
 
 Now, since all the parts a-\~l, b-\-k, c-\-i . . . are equal 
 to each other, and their number equal to n, 
 
 or = 
 
ARITHMETICAL PROGRESSION. 223 
 
 Hence, for finding the sum of an arithmetical series, we 
 have the following 
 
 RULE. 
 
 I. Add the two extremes together, and take half their s'<im. 
 
 II. Multiply the half-sum by the number of terms ; the 
 product will be the sum of the series. 
 
 EXAMPLES. 
 
 I. The extremes are 2 and 16, and the number of terms 
 8 : what is the sum of the series ? 
 
 S=( )X, gives Sr=- 
 
 3. The extremes are 3 and 27, and the number of terms 
 12 : what is the sum of the series ? Ans. 180. 
 
 3. The extremes are 4 and 20, and the number of terms 
 10 : what is the sum of the series ? Ans. 120. 
 
 4. The extremes are 100 and 200, and the number of 
 terms 80 : what is the sum of the series ? Ans. 12000. 
 
 5. The extremes are 500 and 60, and the number of terms 
 20 : what is the sum of the series ? Ans. 5600. 
 
 6. The extremes are 800 and 1200, and the number of 
 terms 50 : what is the sum of the series ? Ans. 50000. 
 
 QUEST. 140. In every progression, what is the sum of the twc ex- 
 tremes equal to 1 What is the rule for finding the sum of an arithmeti- 
 cal scries ? 
 
224 ELEMENTARY ALGBBRA. 
 
 141. In arithmetical proportion there are five number 
 to be considered : 
 
 1st. The first term, a. 
 2nd. The common difference, r. 
 3rd. The number of terms, n. 
 4th. The last term, I. 
 
 5th. The sum, S. 
 
 \ 
 
 The formulas 
 
 J =a +n-l)r and S= 
 
 contain five quantities, a, r, n, I, and S, arid consequently 
 give rise to the following general problem, viz : Any three 
 of these Jive quantities being given, to determine the other 
 two. 
 
 We already know the value of *S in terms of a, n, and 7 
 From the formula 
 
 J=a+(n-l)r, 
 we find a = l (n l)r. 
 
 That is : The first term of an increasing arithmetical pro- 
 gression is equal to the last term, minus the product of the 
 common difference by the number of terms less one. 
 From the same formula, we also find 
 
 I a 
 
 
 n l 
 
 That is : In any arithmetical progression, the common differ- 
 ence is equal to the difference between the two extremes divided 
 by the number of terms less one. 
 
 QUEST. 141. How many numbers are considered in arithmetical 
 proportion 1 What are they 1 In every arithmetical progression, what 
 is the common difference equal to 1 
 
ARITHMETICAL PROGRESSION. 225 
 
 The last term is 16, the first term 4, and the number of 
 terms 5 : what is the common difference ? 
 
 The formula r= 
 
 n 1 
 16-4 
 
 r= 
 
 2. The last term is 22, the first term 4, and the number 
 of terms 10 : what is the common difference ? Ans. 2. 
 
 1 42. The last principle affords a solution to the follow- 
 ing question : 
 
 To find a number m of arithmetical means between two 
 given numbers a and b. 
 
 To resolve this question, it is first necessary to find the 
 common difference. Now, we may regard a as the first 
 term of an arithmetical progression, b as the last term, and 
 the required means as intermediate terms. The number of 
 terms of this progression will be expressed by w-j-2. 
 
 Now, by substituting in the above formula, b for I, and 
 Tn-f-2 for n, it becomes 
 
 m + 2 1 
 
 that is, the common difference of the required progression is 
 obtained by dividing the difference between the given num- 
 bers a and &, by one more than the required number of 
 means. 
 
 QUEST. 142. How do you find any number of arithmetical means 
 oetween two g'iven numbers 1 
 
 20* 
 
226 ELEMENTARY ALGEBRA. 
 
 Having obtained the common difference, form the second 
 term of the progression, or the first arithmetical mean, by 
 
 adding r, or -, to the first term a. The second mean 
 
 m-\- 1 
 
 ; s obtained by augmenting the first by r, &c. 
 
 1. Find three arithmetical means between the extremes 
 2 and 18. 
 
 The formula r= 
 
 18-2 
 gives r= - =4 ; 
 
 hence, the progression is 
 
 2 . 6 . 10 . 14 . 18. 
 
 2. Find twelve arithmetical means between 12 and 7V. 
 
 b-a 
 The formula 
 
 gives r=^-^-=5. 
 
 Hence the progression is 
 
 12 . 17 . 22 . 27 .... 77 
 
 143. REMARK. If the same number of arithmetical 
 means are inserted between all the terms, taken two and 
 two, these terms, and the arithmetical means united, will 
 form but one and the same progression. 
 
 For, let a.b.c.d.e.f... be the proposed 
 progression, and m the number of means to be inserted 
 between a and b, b and c, c and d . . . 
 
ARITHMETICAL PROGRESSION. 227 
 
 From what lias just been said, the common difference of 
 each partial progression will be expressed by 
 
 which are equal to each other, since a, 5, c ... are in 
 progression : therefore, the common difference is the same 
 in each of the partial progressions ; and since the last term 
 of the first, forms the first term of the second, &c, we may 
 conclude that all of these partial progressions form a single 
 progression. 
 
 EXAMPLES. 
 
 1. Find the sum of the first fifty terms of the progression 
 2 . 9 . 16 . 23 ... 
 
 For the 50th term we have 
 
 Hence, S = 2 + 345 X = 347x25 = 8675. 
 
 2. Find the 100th term of the series 2 . 9 . 16 . 23 ... 
 
 Ans. 695 
 
 3. Find the sum of 100 terms of the series 1 . 3_ 5 . 
 7.9 ... Ans. ] 0000 
 
 4. The greatest term is 70, the common difference 3, and 
 the number of terms 21 : what is the least term and the 
 sum of the series ? 
 
 Ans. Least term 10 ; sum of serifs 840 
 
228 ELEMENTARY ALGEBRA. 
 
 5 The first term is 4, the common difference 8, and the 
 number of terms 8 : what is the last term, and the sum of 
 the series ? 
 
 Last term 60. 
 
 Sum = 256. 
 
 6. The first term is 2, the last term 20, and the number 
 of terms 10 : what is the common difference ? 
 
 Ans. 2. 
 
 7. Insert four means between the two numbers 4 and 1 9 : 
 what is the series ? 
 
 Ans. 4 . 7 . 10 : 13 . 16 . 19. 
 
 sion is 10, the common difference , and the number of 
 
 o 
 
 8. The first term of a decreasing arithmetical progres- 
 sn is 10, the common difference , and 
 
 o 
 
 terms 21 : required the sum of the series. 
 
 Ans. 140. 
 
 9. In a progression by differences, having given the 
 common difference 6, the last term 185, and the sum of the 
 terms 2945 : find the first term, and the number of terms. 
 
 Ans. First term =5 ; number of terms 31. 
 
 10. Find nine arithmetical means between each antece- 
 dent and consequent of the progression 2.5.8.11.14 ... 
 
 Ans. Common dif., or n=0,3. 
 
 11. Find the number of men contained in a triangular bat- 
 talion, the first rank containing one man, the second 2, the 
 third 3, and so on to the n*, which contains n. In other 
 words, find the expression for the sum of the natural num- 
 bers 1, 2, 3 . . ., from 1 to n inclusively. 
 
 A*.. S= 
 
GEOMETRICAL PROPORTION. 229 
 
 12. Find the sum of the n first terms of the progression 
 of uneven numbers 1, 3, 5, 7, 9 ... Ans. S n z . 
 
 13. One hundred stones being placed on the ground in a 
 straight line, at the distance of 2 yards from each other, 
 .how far will a person travel who shall bring them one by 
 One to a basket, placed at 2 yards from the first stone ? 
 
 Ans. 11 miles, 840 yards 
 
 Geometrical Proportion and Progression. 
 
 144. Ratio is the quotient arising from dividing one 
 quantity by another quantity of the same kind. Thus, if 
 the numbers 3 and 6 have the same unit, the ratio of 3 to 6 
 will be expressed by 
 
 And in general, if A and B represent quantities of the same 
 kind, the ratio of A to B will be expressed by 
 
 B_ 
 A' 
 
 145. If there be four numbers 
 
 2, 4, 8, 16, 
 
 having such values that the second divided by the first is 
 equal to the fourth divided by the third, the numbers are 
 
 QUEST. 144. What is ratio 1 What is the ratio of 3 to 6? Of 4 
 to 12? 
 
230 ELEMENTARY ALGEBRA. 
 
 .said to be in proportion. And in general, if there be foui 
 quantities, A, B, C, and D, having such values that 
 
 B D 
 
 then A is said to have the same ratio to B that C has to D 
 or, the ratio of A to B is equal to the ratio of C to D, 
 When four quantities have this relation to each other, they 
 are said to be in proportion. Hence, proportion is an equality. 
 of ratios. 
 
 To express that the ratio of A to B is equal to the ratio 
 of C to Z), we write the quantities thus : 
 
 and read, A is to B as C to D. 
 
 The quantities which are compared together are called 
 the terms of the proportion. The first and last terms are 
 called the two extremes, and the second and third terms, the 
 two means. Thus, A and D are the extremes, and B and 
 C the means. 
 
 1 46. Of four proportional quantities, the first and third 
 are called the antecedents, and the second and fourth the 
 consequents ; and the last is said to be a fourth proportional 
 to the other three taken in order. Thus, in the last pro- 
 portion A and C are the antecedents, and B and D the 
 consequents. 
 
 QUEST. 145. What is proportion 1 How do you express that foui 
 numbers are in proportion 1 What are the numbers called 1 What arc 
 the first and fourth called? What the second and third! 146. In four 
 proportional quantities, what are the first and third called ? What the 
 second and fourth 1 
 
GEOMETRICAL PROPORTION. 231 
 
 1 47 . Three quantities are in proportion when the first 
 has the same ratio to the second that the second has to the 
 third ; and then the middle term is said to be a mean pro- 
 portional between the other two. For example, 
 
 3 : 6 : : 6 : 12; 
 and 6 is a mean proportional between 3 and 12. 
 
 148. Quantities are said to be in proportion by inver- 
 sion, or inversely, when the consequents are made the ante- 
 cedents and the antecedents the consequents. 
 
 Thus, if we have the proportion 
 
 3 : 6 : : 8 : 16, 
 the inverse proportion would be 
 
 6 : 3 : : 16 : 8. 
 
 1 49. Quantities are said to be in proportion by alterna- 
 tion, or alternately, when antecedent is compared with ante- 
 cedent and consequent with consequent. 
 
 Thus, if we have the proportion 
 
 3 : 6 : : 8 : 16, 
 the alternate proportion would be 
 
 3 : 8 : : 6 : 16. 
 
 QUEST. 147. When are three quantities proportionall What is the 
 middle one called? 148. When are quantities said to be in proportion 
 by inversion, or inversely 1 149. When are quantities in proportion by 
 alternation * 
 
232 ELEMENTARY ALGEBRA. 
 
 150. Quantities are said to be in proportion by compo- 
 sition, when the sum of the antecedent and consequent is 
 compared either with antecedent or consequent. 
 
 Thus, if we have the proportion 
 
 2 : 4 : : 8 . 16, 
 die proportion by composition would be 
 
 2 + 4 : 4 : : 8 + 16 : 16; 
 that is, 6:4:: 24 : 16. 
 
 151. Quantities are said to be in proportion by division, 
 when the difference of the antecedent and consequent is 
 compared either with antecedent or consequent. 
 
 Thus, if we have the proportion 
 
 3 : 9 : : 12 : 36, 
 the proportion by division will be 
 
 9 3 : 9 : : 36-12 : 36; 
 that is, 6 : 9 : : 24 : 36. 
 
 152. Equi-multiples of two or more quantities are the 
 products which arise from multiplying the quantities by the 
 same number. 
 
 Thus, if we have any two numbers, as 6 and 5, and mul- 
 tiply them both by any number, as 9, the equi-multiples will 
 be 54 and 45 ; for 
 
 6x9 = 54, and 5x9 = 45. 
 
 Q UE8T . 150. When are quantities in proportion by composition? 
 151. When are quantities in proportion by division"! 152. What 
 are equi-multiples of two or more quantities 1 
 
GEOMETRICAL PROPORTION. 233 
 
 Also, m X A and mxB are equi-multiples of A and B, the 
 common multiplier being m. 
 
 153. Two quantities, A and B, are said to be recipro- 
 cally proportional, or inversely proportional, when one in- 
 creases in the same ratio as the other diminishes. When 
 this relation exists, either of them is equal to a constant 
 quantity divided by the other. 
 
 Thus, if we had any two numbers, as 2 and 4, so related 
 to each other that if we divided one by any number we must 
 multiply the other by the same number, one wduld increase 
 just as fast as the other would diminish, and their product 
 would be constant. 
 
 154. If we have the proportion 
 
 A : B : : C : D, 
 
 7? D 
 we have ~A = ~C' ( Art 14 ^) ; 
 
 and by clearing the equation of fractions, we have 
 BC=AD. 
 
 That is, Of four proportional quantities, the product of the 
 two extremes is equal to the product of the two means. 
 
 This general principle is apparent in the proportion be- 
 tween the numbers 
 
 2 : 10 : : 12 : 60, 
 which gives 2x60 = 10x12 = 120. 
 
 QUEST. 153. When are two quantities said to be reciprocally pro- 
 portional 1 154. If four quantities are proportional, what is the product 
 of the two means equal to ? 
 
2,'M ELEMENTARY ALGEBRA. 
 
 1 55. If four quantities, A, B, C, D, are so related**- 
 each other that 
 
 AxD=BxC t 
 
 we shall also have -r=~77> 
 
 A C 
 
 and hence, A : B : : C : D. 
 
 That is : If the product of two quantities is equal to the. pro- 
 duct of two other quantities, two of them may be made the 
 extremes, and the other two the means of a proportion, 
 Thus, if we have 
 
 2x8=4x4, 
 we also have 
 
 2 : 4 : : 4 : 8. 
 
 156. If we have three proportional quantities 
 A : B : : B : C, 
 
 B C 
 we have = -^- ; 
 
 hence, B 2 =AC. 
 
 That is : The square of the middle term is equal to the product 
 of the two extremes. 
 
 Thus, if we have the proportion 
 
 3 : 6 : : 6 : 12, 
 
 we shall also have / 
 
 QUEST. 155. If the product of two quantities is equal to the product 
 of two other quantities, may the four be placed in a proportion 1 How 1 
 156. If three quantities are proportional, what is the product of thf 
 extremes equal to 1 
 
GEOMETRICAL PROPORTION. 235 
 
 157. If we have 
 
 B D 
 
 A : B : : C : D, and consequently - =TT , 
 
 A \j 
 
 c 
 
 multiply both members of the last equation by -=-, we 
 then obtain, 
 
 and hence, A : C : : B : D. 
 
 That is : If four quantities are proportional, they will be in 
 proportion by alternation. 
 
 Let us take, as an example, 
 
 10 : 15 : : 20 : 30. 
 We shall have, by alternating the terms, 
 
 10 : 20 : : 15 : 30. 
 158. If we have 
 
 A : B : : C : D and A : B : : E : F, 
 we shall also have 
 
 B _D B _F 
 
 ~A~~C ~A~~E' 
 
 D F 
 
 hence, -7^=-^ and C : D : : E : F. 
 
 L> TJ 
 
 That is : If there are two sets of proportions having an 
 
 QUEST. 157. If four quantities are proportional, will they be in pro- 
 portion by alternation 1 
 
236 ELEMENTARY ALGEBRA. 
 
 antecedent and consequent in the one equal to an antecedent 
 and consequent of the other ', the remaining terms will be pro 
 portional. 
 
 If we have the two proportions 
 
 2 : 6 : : 8 : 24 and 2 : 6 : : 10 : 30, 
 we shall also have 
 
 8 : 24 : : 10 : 30. 
 159. If we have 
 
 TO T\ 
 
 A : B : : C : D, and consequently =- , 
 
 we have, by dividing 1 by each member of the equation 
 
 A C 
 
 =-pr, and consequently B : A : : D : C. 
 
 Jj D 
 
 That is : Four proportional quantities will be in proportion^ 
 when taken inversely. 
 
 To give an example in numbers, take the proportion 
 
 7 : 14 : : 8 : 16; 
 then, the inverse proportion will be 
 
 14 : 7 : : 16 : 8, 
 in which the ratio is one-half. 
 1 GO. The proportion 
 
 A : B :: C : D gives AxD=BxC. 
 
 QUEST. 158. If you have, two sets of proportions having an ante- 
 cedent and consequent in each, equal ; what will follow 1 159. If four 
 quantities are in proportion, will they be in proportion when takei in- 
 versely 1 
 
GEOMETRICAL PROPORTION. 237 
 
 To each member of the last equation add BxD. We 
 shall then have 
 
 and by separating the factors, we obtain 
 
 A+B : B : : C+D : D. 
 
 If, instead of adding, we subtract B x D from both mem- 
 bers, we have 
 
 which gives 
 
 A-B : B : : C-D : D. 
 
 That is : If four quantities are proportional, they will be tn 
 proportion by composition or division. 
 
 Thus, if we have the proportion 
 
 9 : 27 : : 16 : 48, 
 we shall have, by composition, 
 
 9 + 27 : 27 : : 16 + 48 : 48: 
 that is, 36 : 27 : : 64 : 48, 
 
 in which the ratio is three-fourths. 
 The proportion gives us, by division, 
 
 27 9 : 27 : : 48 16 : 48; 
 that is, 18 : 27 : : 32 : 48, 
 
 in which the ratio is one and one -half. 
 
 QUEST. 160. If four quantities are in proportion, will they he in rn>- 
 portion by composition 1 Will they be in proportion by division 1 What 
 > the fWFerence between composition and division 1 
 21* 
 
238 ELEMENTARY ALGEBRA. 
 
 161. If we have 
 
 B_D 
 ~A~~C* 
 
 and multiply the numerator and denominator of the fiist 
 member by any number m, we obtain 
 
 v and mA . mB .. c . Dm 
 
 mA C 
 
 That is : Equal multiples of two quantities have the same 
 ratio as the quantities themselves. 
 
 For example, if we have the proportion 
 
 5 : 10 : : 12 : 24, 
 
 and multiply the first antecedent and consequent by 6, we 
 have 
 
 30 : 60 : : 12 : 24, 
 
 in which the ratio is still 2. 
 162. The proportions 
 
 A : B : : C : D and A : B : : E : F, 
 give AxD=BxC and AxF=BxE-, 
 adding and subtracting these equations, we obtain 
 A(DF) = B(CE), or A : B :: CE : DF 
 
 That is : If C and D, the antecedent and consequent, be aug- 
 mented or diminished by quantities E and F, which have the 
 same ratio as C to D, the resulting quantities will also have 
 the same ratio. 
 
 QUEST. 161. Have equal multiples of two quantities the same ratii 
 as the quantities'? 162. Suppose the antecedent and consequent bi 
 augmented or diminished by quantities having the same ratio .' 
 
GEOMETRICAL PROPORTION. 239 
 
 Let us take, as an example, the proportion 
 9 : 18 : : 20 : 40, 
 
 b which the ratio is 2. 
 
 If we augment the antecedent and consequent by 15 and 
 3U vrhich have the same ratio, we shall have 
 
 9 + 15 : 18 + 30 : : 20 : 40; 
 that is, 24 : 48 : : 20 : 40, 
 
 in which the ratio is still 2. 
 
 If we diminish the second antecedent and consequent by 
 the same numbers, we have 
 
 9 : 18 : : 20 15 : 4030; 
 that is, 9 : 18 : : 5 : 10, 
 
 in wLich the ratio is still 2. 
 
 163. If we have several proportions 
 
 A : B :: C : D, which gives Axi* = BxC, 
 A : B : : E : F, AxF-BxE, 
 A : B : : G : H, AxH=B*G. 
 
 &c, &c, 
 we shJfl have, by addition, 
 
 A(D+F+H)=B(C+E+G); 
 and by separating the factors, 
 
 A : B : : C+E+G : D+F+H. 
 
 That is : In any number of proportions having the same 
 ratio, any antecedent will be to its consequent, as the sum uj 
 the antecedents to the sum of the consequents. 
 
y-' ELEMENTARY ALGEBRA. 
 
 Let us take, for example, 
 
 2 : 4 : : 6 : 12 and 1 : 2 : : 3 : 6, &c 
 Then, 2:4:: 6+3 : 12 + 6; 
 
 that is, 2 : 4 : : 9 : 18, 
 
 iu which the ratio is still 2. 
 
 164. If we have four proportional quantities 
 
 A : B : : C : D, we have ^=^ ; 
 and raising both members to any power, as , we have 
 
 and consequently 
 
 A* : B n : : C* : D\ 
 
 That is : If four quantities arc proportional, any like powers 
 or roots will be proportional. 
 
 If we have, for example, 
 
 2 : 4 : : 3 : 6, * 
 we shall have 2 2 : 4 2 : : 3 2 : 6 2 ; 
 that is, 4 : 16 : : 9 : 36, 
 
 in which the terms are proportional, the ratio being 4. 
 
 165. Let there be two sets of proportions, 
 
 7? 7} 
 A : B : : C : D, which gives =, 
 
 E : F : : G : //, ~ = ~' 
 
 QUEST. 163. In any number of proportions having the same rati 
 how will any one antecpdent be to its consequent 1 164. In four pr<. 
 portional quantities, how are like powers or roots'? 
 
GEOMETRICAL PROGRESSION. 241 
 
 Multiply them together, member by member, we have 
 BF DH 
 
 AE~ CG 
 
 which gives AE : BF : : CG : DH. 
 
 That is : In two sets of proportional quantities, the products 
 of the corresponding terms will be proportional. 
 
 Thus, if we have the two proportions 
 
 8 : 16 : : 10 : 20 
 and 3 : 4 : : 6 : 8, 
 
 we shall have 24 : 64 : : 60 : 160. 
 
 Geometrical Progression. 
 
 166. We have thus far only required that the ratio of 
 the first term to the second should be the same as that of 
 the third to the fourth. 
 
 If we impose the farther condition, that the ratio of the 
 second term to the third shall also be the same as that of the 
 first to the second, or of the third to the fourth, we shall have 
 a series of numbers, each one of which, divided by the 
 preceding one, will give the same ratio. Hence, if any 
 term be multiplied by this quotient, the product will be the 
 succeeding term. A series of numbers so formed is called 
 a geometrical progression. Hence, 
 
 A Geometrical Progression, or progression by quotients, is 
 a series of terms, each of which is equal to the product of 
 
 QUEST. 165. In two sets of proportions, how are the products of the 
 corresponding terms 7 
 
24x5 ELEMENTARY ALGEBRA. 
 
 that which precedes it by a constant number, which number 
 is called the ratio of the progression. Thus, 
 
 1 : 3 : 9 : 27 : 81 : 243, &c, 
 
 is a geometrical progression, which is written by merely 
 placing two dots between each two of the terms. Also, 
 
 64 : 32 : 16 : 8 : 4 : 2 : 1 
 
 is a geometrical progression, in which the ratio is one-half. 
 
 In the first progression each term is contained three times 
 in the one that follows, and hence the ratio is 3. In the 
 second, each term is contained one-half times in the one 
 which follows, and hence the ratio is one-half. 
 
 The first is called an increasing progression, and the 
 second a decreasing progression. 
 
 Let a, b, c, d, e t f, . . . be numbers in a progression by 
 quotients ; they are written thus : 
 
 a : I : c : d : e : f : g . . . 
 
 and it is enunciated in the same manner as a progression by 
 differences. It is necessary, however, to make the distinc- 
 tion, that one is a series of equal differences, and the other 
 a series of equal quotients or ratios. It should be remarked 
 that each term is at the same time an antecedent and a con- 
 sequent, except the first, which is only an antecedent, and 
 the last, which is only a consequent. 
 
 QUEST. 166. What is a geometrical progression? What is the ratio 
 of the progression'? If any term of, a progression be multiplied by the 
 ratio, what will the product be 1 If any term be divided by the ratio, 
 what will the quotient be 1 How is a progression by quotients written ? 
 Which of the terms is only an antecedent ? Which only a consequent T 
 How may each of the others be considered 1 
 
GEOMETRICAL PROGRESSION. 243 
 
 IG7. Let q denote the ratio of the progression 
 a : b : c :<?...; 
 
 <t being >1 when the progression is increasing, and ^< 1 
 when it is decreasing. Then, since 
 
 b__ _e__ d__ _e__ 
 a b c d 
 
 we have 
 
 that is, the second term is equal to aq, the third to aq 2 , the 
 fourth to aq 3 , the fifth to a^ 4 , &c ; and in general, any term 
 , that is, one which has n 1 terms before it, is expressed 
 by aq n ~ l . 
 
 Let / be this term ; we then have the formula 
 
 by means of which we can obtain any term without being 
 obliged to find all the terms which precede it. Hence, to 
 find the last term of a progression, we have the following 
 
 RULE. 
 
 I. Raise the ratio to a power whose exponent is one less than 
 the number of terms. 
 
 II. Multiply the power thus found by the first term : the 
 product will be the required term. 
 
 QUEST. 167. By what letter do we denote the ratio of the progres- 
 sion 1 In an increasing progression is q greater or less than 1 * In a 
 decreasing progression is q greater or less than 11 If a is the firt term 
 and q the ratio, what is the second term equal to 1 What the i ird \ 
 What the fourth 1 What is the last term equal to 1 Give the rtdo foi 
 finding the last term 
 
244 ELEMENTARY ALGEBRA. 
 
 EXAMPLES. 
 
 1. Find the 5th term of the progression 
 
 2 : 4 : 8 : 16 . . 
 
 in which the first term is 2 and the common ratio 2. 
 5th term=2x2*=2x 16=32 Ans. 
 
 2. Find the 8th term of the progression 
 
 2 : 6 : 18 : 54 ..." 
 
 8th term=2 x 3 7 =2 x2187=4374 Ans. 
 
 3. Find the 6th term of the, progression 
 
 2 : 8 : 32 : 128 ... 
 6th term=2 x 4 5 =2 x 1024=2048 Ans. 
 
 4. Find the 7th term of the progression 
 
 3 : 9 : 27 : 81 ... 
 
 7th term=3x3 6 =3x 729=2187 Ans. 
 
 5. Find the 6th term of the progression 
 
 4 : 12 : 36 : 108 ... 
 6th term=4x3 5 = 4x243 = 972 Ans. 
 
 6. A person agreed to pay his servant 1 cent for the first 
 day, two for the second, and four for the third, doubling 
 every day for ten days : how much did he receive on the 
 tenth day? Ans. $5,12. 
 
GEOMETRICAL PROGRESSION. 245 
 
 7. What is the 8th term of the progression 
 
 9 : 36 : 144 : 576 ... 
 8thterm=:9x4 7 =:9xl6384=:147456 Ans. 
 8 Find the 12th term of the progression 
 
 64 : 16 : 4 : 1 : . . . 
 4 
 
 12th termz=64( ) = = = Ans. 
 
 168. We will now proceed to determine the sum of n 
 terms of the progression 
 
 a : b : c : d : e : f : . . . : i ' k : I', 
 
 I denoting the nth term. 
 
 We have the equations (Art. 167), 
 
 b aq, c = bq, d = cq, e=dq, . . . k=iq, l=kq\ 
 
 and by adding them all together, member to member, we 
 deduce 
 
 Sum of 1st members. Sum of %nd members. 
 
 +c+d+e+ . . . +k+l=(a+b+c+d+ . . . +'+%; 
 
 in which we see that the first member wants the first term 
 , and the polynomial within the parenthesis in the second 
 member wants the last term /. Hence, if we call the sum 
 of the terms S, we have 
 
 whence 
 
 S-a=(S l)q=Sq Iq, or S? S = fy a; 
 Iq a 
 
246 ELEMENTARY ALGEBRA. 
 
 Therefore, to obtain the sum of the terms of a geometrical 
 progression, we have the following 
 
 RULE. 
 
 I. Multiply the last term by the ratio. 
 
 II. Subtract the frst term from the product. 
 
 III. Divide the remainder by the ratio diminished by unity, 
 and the quotient will be the sum of the series. 
 
 1 . Find the sum of eight terms of the progression 
 2 : 6 : 18 : 54 : 162 ... 2x3 7 ^4374. 
 
 q-l 2 
 
 2 Find the sum of the progression 
 
 2 : 4 : 8 : 16 : 32. 
 
 
 q-\ 1 
 
 3. Find the sum of ten terms of the progression 
 
 2 : 6 : 18 : 54 : 162 ... 2x3 9 = 39366. 
 
 Ans. 59048. 
 
 4. What debt may be discharged in a year, or twelve 
 months, by paying $1 the first month, $2 the second month, 
 
 QUEST. 168. Give the rule for finding the sum of the series. What 
 the first step 1 What the second 1 What the third 7 
 
GEOMETRICAL PROGRESSION. 
 
 $4 the third month, and so on, each succeeding payment 
 being double the last ; and what will be the last payment ? 
 
 ( Debt, . . $4095. 
 
 ! Last payment, $2048. 
 
 5. A gentleman married his daughter on New Year's day, 
 and gave her husband Is. towards her portion, and was to 
 double it on the first day of every month during the year : 
 what was her portion? Ans. 204: 15s. 
 
 6. A man bought 10 bushels of wheat on the condition 
 that he should pay 1 cent for the 1st bushel, 3 for the second, 
 9 for the third, and so on to the last : what did he pay for 
 the last bushel and for the ten bushels ? 
 
 * ( Last bushel, $196,83. 
 '* \ Total cost, $295,24. 
 
 7. A man plants 4 bushels of barley, which, at the first 
 harvest, produced 32 bushels ; these he also plants, which, 
 in like manner, produce 8 fold ; he again plants all his crop, 
 and again gets 8 fold, and so on for 16 years : what is his 
 last crop, and what the sum of the series ? 
 
 Last, 140737488355328^. 
 Sum, 16084284*3834660. 
 
 169 Whep the progression is decreasing, we have 
 and l-^a ; the above formula 
 
 Iq a 
 
 ?-i ' 
 
 for the sum is then written under the form 
 g ^ a Iq 
 
 in order that the two terms of the fraction may be positive. 
 
 QUEST. 163. What is the formula for the sum of the series of a 
 decreasing progression "? 
 
248 ELEMENTARY ALGEBRA. 
 
 1. Find the sum of the terms of the progression 
 32 : 16 : 8 : 4 : 2. 
 
 31 
 
 . 
 
 lq 
 
 ~2 ~2 
 
 2. Find the sum of the first twelve terms of the progression 
 
 l-l l ' 
 
 eifM 11 or ] 
 
 4 
 
 1 1 
 65536 4 
 
 \4/ ' 65536 
 
 1 
 65536 65535 
 
 : 85 J . 
 
 4 
 
 REMARK. 17O. We perceive that the principal diffi- 
 culty consists in obtaining the numerical value of the last 
 term, a tedious operation, even when the number of terms 
 is not very great. 
 
 3. Find the sum of 6 terms of the progression 
 
 512 : 128 : 32 ... 
 
 Ans. 682^ 
 
 4. Find the sum of seven terms of the progression 
 
 2187 : 729 : 243 ... 
 
 Ans. 3279. 
 
 5. Find the sum of six terms of the progression 
 
 972 ; 324 : 108 
 
 Ans. 1456. 
 
 6. Find tho sum of 8 terms of the progression 
 
 147456 : 36864 : 9216 . . . 
 
 Ans. 196605 
 
GEOMETRICAL PROGRESSION. 249 
 
 Of Progressions having an infinite number of terms 
 
 171. Let there be the decreasing progression 
 
 a : b : c : d : e : f : . . . 
 
 containing an indefinite number of terms. In the formula 
 
 a-lq 
 
 -- 
 
 substitute for / its value aq n ~ l (Art. 167), and we have 
 
 which represents the sum of n terms of the progression. 
 This may be put under the form 
 
 Now, since the progression is decreasing, q is a proper 
 fraction ; and q n is also a fraction, which diminishes as n 
 increases. Therefore, the greater the number of terms we 
 take, the more will x q n diminish, and consequent- 
 ly the more will the partial sum of these terms approximate 
 
 to an equality with the first part of S, that is, to . 
 
 \-q 
 
 Finally, when n is taken greater than any given number, or 
 n = infinity, then X^" will be less than any given 
 
 number, or will become equal to ; and the expression 
 
 will represent the true value of the sum of all the terms of 
 
 the series. Whence we may conclude, that the expression 
 
 22* 
 
250 ELEMENTARY ALGEBRA. 
 
 for the sum of the terms of a decreasing progression, in uihich 
 the number of terms is infinite, is 
 
 That is, equal to the first term divided by 1 minus the ratio. 
 
 This is, properly speaking, the limit to which the partial 
 sums approach, by taking a greater number of terms in the 
 progression. The difference between these sums and 
 
 - - can become as small as we please, and will only 
 become nothing when the number of terms taken is infinite 
 
 EXAMPLES. 
 1. Find the sum of 
 
 I: T : T :: T toinfinit y- 
 
 We have for the expression of the sum of the terms 
 
 The error committed by taking this expression for 
 value of the sum of the n first terms, is expressed by 
 
 3/ 
 
 First take n=5 ; it becomes 
 
 3 / 1 x 5 1 1 
 
 2 \ 3 / 2 . 3* 162 
 
 QUEST. 165. When the progression is decreasing and the number o< 
 terms infinite, what is the value of the sum of the series 1 
 
GEOMETRICAL PROGRESSION. 251 
 
 When n = 6, we find 
 
 2\3/ 162 3 486 
 
 q 
 
 Whence we see that the error committed, when -~ is 
 
 2 
 
 taken for the sum of a certain number of terms, is less in 
 proportion as this number is greater. 
 2. Again take the progression 
 
 .1 J_ I. JL .1 & 
 2 : 4 : 8 : 16 : 32 : 
 
 We have 
 
 3. What is the sum of the progression 
 
 re- W' !' TdSoc 
 
 '- ' ' ' te ">' 
 
 10 
 
 172. In the several questions of geometrical progres 
 sion there are five numbers to be considered : 
 
 1st. The first term, ........ a. 
 
 2nd. The ratio, .......... q. 
 
 3rd. The number of terms, ...... n. 
 
 4th. The last term, ........ I 
 
 5th. The sum of the terms, ... S. 
 
 QUEST. 166. How many numbers are considered in geometrical pro- 
 gression? What are they' 
 
252 ELEMENTARY ALGEBRA. 
 
 173. We shall terminate this subject by the question, 
 
 To find a mean proportional between any two numbers, 
 as m and n. 
 
 Denote the required mean by x. We shall then have 
 (Art. 156), 
 
 x z = 
 
 and hence 
 
 That is, Multiply the two numbers together, and extract the 
 square root of the product. 
 
 1. What is the geometrical mean between the numbers 
 2 and 8 ? 
 
 Mean= ^fSx2 yT6 4 Ans. 
 
 2. What is the mean between 4 and 16? Ans. 8 
 
 3. What is the mean between 3 and 27 ? Ans. 9. 
 
 4. What is the mean between 2 and 72? Ans. 12. 
 
 5. What is the mean between 4 and 64 ? Ans. 16. 
 
 QUEST. 167. How c.0 you find a mean proportional between -iwo 
 numbers ' 
 
OF LOGARITHMS, 253 
 
 CHAPTER VIII. 
 
 Of Logarithms. 
 
 1 74. The nature and properties of the logarithms in com 
 tnon use, will be readily understood, by considering atten- 
 tively the different powers of the number 10. They are, 
 
 10 = 1 
 
 10^ = 10 
 
 10 2 = 100 
 
 10 3 = 1000 
 
 10 4 = 10000 
 
 10 5 = 100000 
 &c. &c. 
 
 It is plain that the indices or exponents 0, 1, 2, 3, 4, 5, &c. 
 form an arithmetical series of which the common difference 
 b 1 ; and that the numbers 1, 10, 100, 1000, 10000, 100000, 
 &.c. form a geometrical series of which the common ratio is 
 10. The number 10, is called the base of the system of log- 
 arithms ; and the indices 0, 1, 2, 3, 4, 7, &,c., are the loga- 
 
 QrEsi. 174. What relation exists between the exponents 1, 2, 3, 
 &<.? How .are the corresponding numbers 10, 100, 10001 What is 
 the common difference of the exponents'! What is the common ratio of 
 the corresponding numbers 1 What is the base of the common system 
 of logarithms] What are the indices? Of what number is the index 
 1 the logarithm * The index 2 ? The index 3 ? 
 
254 ELEMENTARY ALGEBRA. 
 
 rithms of the numbers which are produced by raising 10 to 
 the powers denoted by those indices. 
 
 1 75; If we denote the base of the system by a, and the 
 logarithm of any number by wi, then the number itself will 
 e the mth power of a : that is, if we represent the corres- 
 onding number by M, 
 
 a=M 
 
 Thus, if we make ?w=0, M will be equal to 1 ; if m=>l, 
 M will be equal to 10, &c. Hence, 
 
 The logarithm of a number is the exponent of the power 
 to which it is necessary to raise the base of the system in order 
 to produce the number. 
 
 1 7G. Letting, as before, a denote the base of the system 
 of logarithms, m any exponent, and M the corresponding 
 number : we shall then have, 
 
 a m =M 
 in which m is the logarithm of M. 
 
 If we take a second exponent w, and let JV denote the cor- 
 responding number, we shall have, 
 a"=JV 
 in which n is the logarithm of JV. 
 
 If now, we multiply the first of these equations by the 
 second, member by member, we have 
 
 but since a is the base of the system, m-\-n is the logarithm 
 hence, 
 
 QUEST. IfS. If we denote the base of a system by a, and the expo- 
 nent by m, what will represent the corresponding number ? What is the 
 logarithm of a number 1 176. To what is the sum of the logarithms of 
 any two numbers equal 1 To what then, will the addition of logarithms 
 correspond ? 
 
OF LOUA.RITHM8. . 255 
 
 The sum of the logarithms of any two numbers is equal to 
 the logarithm of their product. 
 
 Therefore, the addition of logarithms corresponds to the 
 multiplication of their numbers. 
 
 177. If we divide the equations by each other, member 
 by member, we have, 
 
 -W-J?; 
 
 a" JV' 
 
 but since a is the base of the system, m n is the logarithm 
 of ; hence, 
 
 If one number be divided by another, the logarithm of the 
 quotient will be equal to the logarithm of the dividend dimi- 
 nished by that of the divisor. 
 
 Therefore, the subtraction of logarithms corresponds to the 
 division of their numbers. 
 
 178. Let us examine further the equations 
 
 10 =1 
 
 10 1 =10 
 
 10 3 =100 
 
 10 3 =1000 
 
 &c. &c. 
 
 It is plain that the logarithm of 1 is 0, and that the loga- 
 rithms of all the numbers between 1 and 10, are greater than 
 and less than 1. They are generally expressed by decimal 
 fractions : thus, 
 
 log 2=0.301030 
 
 Q UEST . \*i*t If one number be divided by another, what will the 
 logarithm of the quotient be equal tol To what then will the subtrac- 
 tion of logarithms correspond ] 1?. What is the logarithm of 1 ? 
 Between what limits are the logarithms of all numbers between 1 and 10 ] 
 How are they generally expressed? 
 
256 ELEMENTARY ALGEBRA. 
 
 The logarithms of all the numbers greater than 10 and less 
 than 100, are greater than 1 and less than 2, and are generally 
 expressed by 1 and a decimal fraction : thus, 
 log 50=1.698970. 
 
 The part of the logarithm which stands on the left of the 
 decimal point, is called the characteristic of the logarithm. 
 The characteristic is always one less than the places of integer 
 figures in the number whose logarithm is taken. 
 
 Thus, in the first case, for numbers between 1 and 10, 
 there is but one place of figures, and the characteristic is 0. 
 For numbers between 10 and 100, there are two places of 
 figures, and the characteristic is 1 ; and similarly for other 
 numbers. 
 
 Table of Logarithms. 
 
 1 70. A table of logarithms is a table in which are written 
 the logarithms of all numbers between 1 and some other 
 given number. A table showing the logarithms of the num- 
 bers between 1 and 100 is annexed. The numbers are written 
 in the column designated by the letter N, and the logarithms 
 in the columns designated by Log. 
 
 QUEST. How is it with the logarithms of numbers between 10 and 
 100"? What is that part of the logarithm called which stands at the left 
 of the characteristic? What is the value of the characteristic? IT'O- 
 What is a table of logarithms 1 Explain the manner of finding the loga- 
 rithms of numbers between 1 and 100? 
 
OP LOGARITHMS- 
 
 TABLE. 
 
 257 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 1 
 2 
 
 3 
 4 
 5 
 
 0.000000 
 0.301030 
 0.477121 
 0.602060 
 0.698970 
 
 26 
 27 
 
 28 
 29 
 30 
 
 1.414973 
 1.431364 
 1.447158 
 1.462398 
 1.477121 
 
 51 
 52 
 53 
 54 
 55 
 
 1.707570 
 1.716003 
 1.724276 
 1.732394 
 1.740363 
 
 76 
 
 77 
 78 
 79 
 80 
 
 1.880814 
 1.886491 
 1.892095 
 1.897627 
 1.903090 
 
 6 
 
 7 
 8 
 9 
 10 
 
 0.778151 
 0.845098 
 0.903090 
 0.954243 
 
 1.000000 
 
 31 
 32 
 33 
 34 
 35 
 
 1.491362 
 1.505150 
 1.518514 
 1.531479 
 1.544068 
 
 56 
 57 
 
 58 
 59 
 60 
 
 1.748188 
 1.755875 
 1.763428 
 1.770852 
 1.778151 
 
 81 
 
 82 
 83 
 84 
 
 85 
 
 1.908485 
 1.913814 
 1.919078 
 1.924279 
 1.929419 
 
 11 
 12 
 13 
 14 
 15 
 
 1.041393 
 1.079181 
 1.113943 
 1.146128 
 1.176091 
 
 36 
 37 
 38 
 39 
 40 
 
 1.556303 
 1.568202 
 1.579784 
 1.591065 
 1.602060 
 
 61 
 62 
 63 
 64 
 65 
 
 1.785330 
 1.792392 
 1.799341 
 1.806180 
 1.812913 
 
 86 
 87 
 88 
 89 
 90 
 
 1.934498 
 1.939519 
 1.944483 
 1.949390 
 1.954243 
 
 16 
 17 
 
 18 
 19 
 
 20 
 
 1.204120 
 1.230449 
 1.255273 
 1.278754 
 1,301030 
 
 41 
 42 
 43 
 44 
 45 
 
 1.612784 
 1.623249 
 1.633468 
 1.643453 
 1.653213 
 
 66 
 67 
 68 
 69 
 
 70 
 
 1.819544 
 1.826075 
 1.832509 
 1.838849 
 1.845098 
 
 91 
 92 
 93 
 94 
 95 
 
 1.959041 
 1.963788 
 1.968483 
 1.973128 
 1.977724 
 
 21 
 22 
 23 
 24 
 25 
 
 1.322219 
 1.342423 
 1.361728 
 1.380211 
 1.397940 
 
 46 
 47 
 48 
 49 
 50 
 
 1.662758 
 1.672098 
 1.681241 
 1.690196 
 1.698970 
 
 71 
 72 
 73 
 74 
 75 
 
 1.851258 
 1.857333 
 1.863323 
 1.869232 
 1.875061 
 
 96 
 97 
 98 
 99 
 100 
 
 1.982271 
 1.986772 
 1.991226 
 1.995635 
 2.000000 
 
 EXAMPLES. 
 
 1. Let it be required to multiply 8 by 9, by means of loga 
 rithms. We have seen, Art. 176, that the sum of the loga- 
 rithms is equal to the logarithm of the product. Therefore, 
 find the logarithm of 8 from the table, which is 0.903090 
 and then the logarithm of 9, which is 0.954243; and their 
 sum, which is 1.857333, will be the logarithm of the product* 
 
 23 
 
2.58 ELEMENTARY AI GEBRA 
 
 In searching along in the table, we find that 72 stands oppo 
 site this logarithm : hence, 72 is the product of 8 by 9. 
 2. What is the product of 7 by 12 ? 
 
 Logarithm of 7 is, . . . 0.845098 
 
 Logarithm of 12 is, . . . . 1.079181 
 
 Logarithm of their product, . . 1.924279 
 and the number corresponding is 84. 
 3. What is the product of 9 by 11 ? 
 
 Logarithm of 9 is, . . . 0.954243 
 
 Logarithm of 11 is, . . . . 1.041393 
 
 Logarithm of their product, . . 1.995636 
 and the corresponding number is 99. 
 
 4. Let it be required to divide 84 by 3. We have seen in 
 Article 177, that the subtraction of Logarithms corresponds 
 to the division of their numbers. Hence, if we find the lo- 
 garithm of 84, and then subtract from it the logarithm of 3, 
 the remainder will be the logarithm of the quotient. 
 The logarithm of 84 is, . . . 1.924279 
 The logarithm of 3 is, . . . 0.477121 
 
 Their difference is, ... 1.447158 
 
 and the number corresponding i 28. 
 5. What is the product of 6 by 7 ? 
 
 Logarithm of 6 is, . . . . 0.778151 
 Logarithm of 7 is, . . . . 0.845098 
 
 Their sum is, 1.623249 
 
 and the corresponding number of the table, 42. 
 
SUPPLEMENT. 
 
 EXAMPLES IN ADDITION AND SUBTRACTION 
 
 1. What is the sum of 
 
 ax 
 
 2. What is the sum of 
 
 3. What is the SUIT, of 
 
 10a 4 -f. 3a*+ Go 4 d 4 -M 
 
 4. What is the sum of 
 
 5. What is the sum of 
 
 a n & m 9a"'-}~5a n i- + 6a ra -(- 
 
 6. What is the sum of 
 
 7. What is the sum of 
 5a*b + 3a 2 b*c 7ab Qa 
 
 8. What is the sum of 
 3 
 
 9. What is the sura of 
 9W 
 
 10. From Qa" 1 ^ 13+2^^ 46 m co^ 
 take 3b m cx* 
 
ELEMENTARY ALGEBRA SUPPLEMENT. 
 
 11. From 5a 4 7a 3 b 2 3cd*+7d 
 take l5</b 2 + 3a 4 3a 2 Vci/ 2 
 
 12. From 9a m 6 2 +6a& m d 5 +18a 4 6 n 
 take 7a 4 b"+d 5 8ub'+9ab 2 
 
 13. From 126 5 <f 16a 8 6 8 5a m & n +6ac 
 take 6a m 6 n 6ac b + 16a 8 6 6 + 126 5 <Z 6 
 
 14. From 8a?b 3 c s - 
 
 take Sa^- 8aW 12a m ^ 
 
 15. From 12a m ^ n 9oar 5 4a6-|-6a 2 Z> 2 a 
 take 3a 6a 2 6 2 + 12a m 6 n 9a^-f 5* 
 
 EXAMPLES IN MULTIPLICATION. 
 
 1. What is the product of 
 
 2. What is the product of 
 
 2a*x7a 9 x 3a* 
 
 3. What is the product of 
 
 4. What is the product of 
 
 _-aP-i x SaP- 2 X/X Sa^+'c 
 
 5. What is the product of 
 
 5a 3 6 4 xl0a 2 6 5 cx 3a 7 
 
 6. What is the product of 
 
 7. What is the product of 
 
 a m b p cq X a n 6 r c q X a n '6 X 
 
 8. What is the product of 
 
 (a 2 
 
 9. What is the product of 
 
 (2a 3 // 
 
EXAMPLES IN MULTIPLICATION. 261 
 
 10. What is the product of 
 
 11. What is the product of 
 
 (aWcb 5 d 3 f+ 3c m ) X 
 
 12. What is, the product of 
 
 13. What is the product of 
 
 (3F 5M+2P) x(# 7kl) 
 
 14. What is the product of 
 
 15. What is the product of 
 
 (4a 2 16aa?+ Sz 2 ) x (So 3 2 2 z) 
 
 16. What is the product of 
 
 17. What is the product of 
 
 18. What is the product of 
 
 (2aV 3 jy ) x (2<*V -h 35y ) 
 
 19. What is the product of 
 
 +6at 2 2b 3 ) X (3a 4 4a 3 J-f 
 
 20. What is the product of 
 
 EXAMPLES IN DIVISION. 
 
 1. Divide a m by a n . 
 
 2. Divide a m by a 2 ". 
 
 3. Divide 8a 16 by 2O 4 . 
 
 4. Divide ca 18 by rfa 4 . 
 
 23* 
 
262 ELEMENTARY ALGEBRA - SUPPLEMENT. 
 
 5. Divide 6(a+Z>) 9 by 
 
 6. Divide (a+x) 2 x(a+y) 3 by (a+x)x(a+y) 2 . 
 
 7. Divide 6a 3 2 15a 2 /+27a 4 fo, by 3a 2 . 
 
 8. Divide be 3 c s x by I x. 
 
 9. Divide a 3 +a 2 & atf -W by a . 
 
 10. Divide 3a 6 +16a 4 533a 3 i 2 -hl4^ 3 by a 2 -f 7a. 
 
 11. Divide a 7 6a 6 3 +14a s i 6 12a 4 Z> 9 by a 3 2a 2 3 . 
 
 12. Divide a 4 2a 2 6 2 + 4 by a 2 J 2 . 
 
 13. Divide a*b*+15a n tf 48a 14 Z 5 20a n i 7 
 by lOa 9 * 2 a 6 *. 
 
 14. Divide a 8 16* 8 by a 2 2 2 . 
 
 15. Divide 2a 4 13a 3 J+31a 2 ^ 38ai 3 +24J 4 
 by 20 2 3ai+4^ 
 
 16. Divide 4c 4 9&V+6 3 c fl* by 2c 2 3Jc f P. 
 
 17. Divide l+aV by 1 + an. 
 
 18. Divide a e +2aV+z 6 by a 2 02 + a*. 
 
 19. Divide i 6z 2 +27^ by i-f-22 + 32 2 . 
 
 20. Divide a 6 16aV+64a? 6 by 
 
 21. Divide a 3 d 3 3a 2 cd 3 +3ac'^ 3 
 by a 2^_ 2acd*+c 2 d?+ac z d. 
 
 EXAMPLES IN REDUCTION OF FRACTIONS. 
 
 1. Reduce to its simplest terms the fraction 
 
 ISacf Qbdcf 2ad 
 3adf 
 
 2. Reduce to its simplest terms the fraction 
 
 2a 
 
EXAMPLES IN REDUCTION OF FRACTIONS. 263 
 
 3. Reduce to its simplest tetms the fraction 
 12ac 
 
 4. Reduce to its simplest terms the fraction 
 ab ac 
 bc 
 
 5. Reduce a 6-f- to the form of a fraction. 
 x a 
 
 6. Reduce x L. to the form of a fraction. 
 
 ax c 
 
 7. Reduce a-f ft-f- to the form of a fraction. 
 
 8. Reduce x ab - -- to the form of a fraction. 
 r, x 
 
 9. Reduce a l_n to the form of a fraction. 
 
 10. Reduce 6/ 2 ar-f 9a/ -+ ax to the form of a fraction. 
 
 11. Reduce 5acx y _ -- to the form of a fraction. 
 
 fat 
 
 12. Reduce to an entire or mixed quantity the fraction 
 
 fee 
 
 2a 2 
 
 13. Reduce to an entire or mixed quantity the fraction 
 
*2(54 ELEMENTARY ALGEBRA - SUPPLEMENT. 
 
 14. Reduce to an entire or mixed quantity the fraction 
 
 a 3 - 2a*x+ab+ax*-bx 
 a x 
 
 15. Reduce the following fractions to a common denomi- 
 nator : viz. 
 
 a x b c &ax e 
 
 . , _ , ________ . 
 
 a+x f a x 
 
 16. Reduce the following fractions to a common denomi- 
 nator: viz. 
 
 _JL_, ?H?and-__ 
 3ax b 3Z> ax 
 
 17. Reduce the following fractions to a common denomi- 
 nator : viz. 
 
 7a c c y 
 
 18. Reduce the following fractions to a common denomi- 
 nator : viz. 
 
 *"+* and 8ac -/ 
 Sac -f 4a# 
 
 19. Reduce the following fractions to a common denomi- 
 nator: viz. 
 
 a x a+x a 
 
 20. Reduce the following fractions to a common denomi- 
 nator: viz. 
 
 and 1? 
 
 x a c x-\-a 
 
EXAMPLES IN ADDITION AND SUBTRACTION. 265 
 
 ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION 
 OF FRACTIONS. 
 
 1. What is the sum of 2, ___ and c 
 x a a x 
 
 2. What is the sum of *""*, ^^ and y 
 b a c 
 
 3. What is the sum of ??, =2?, - 
 
 b d 3c / 
 
 4. What is the sum of ax f, ac g, day 
 
 c+a 
 
 5. What is the sum of 3a-f , 2a . 
 
 x a a+b 
 
 6. From 8a+ - take 
 
 b a i 
 
 7. From ^~^ take 3 
 
 b c 
 
 8. From take 
 
 ay 
 
 9. From ay- "*< take 3ay+ 
 oa? 
 
 10. From take 7aJ - 
 
 ga? 8a a? 
 
 11. Multiply 7a+ by 
 6 
 
 12. Multiply _L by 3ar/-~ 
 a; a 
 

 266 ELEMENTARY ALGEBRA SUPPLEMENT. 
 
 13 Multiply 9a ~ a? by 2a+ 
 
 14. Multiply 3ax+ 5a y-~ x by 5+ -fL 
 
 y & 
 
 15. Multiply 6a+ ^""" 8a by 6a ~"* 
 
 b a lr 
 
 16. Divide Sac 2adc /+ by 2a 
 
 17. Divide ?-h^ Sac 4-7 by ?f 
 
 a 2c d 
 
 18. Divide 3a 2 - M - ?lf - _^+ ?^ - ^ 
 
 24282 
 
 by 3a 5+ ? 
 
 19. Divide - --- 
 
 12 9 8 
 
 20. Divide - + - + +26fJ 
 "334 
 
 by ~ 
 
 EXAMPLES IN EQUATIONS OP THE FIRST DEGREE. 
 
 1. Given { }. to find a: and y. 
 
 Z?'4-.ti =41 
 
EXAMPLES IN EQUATIONS OF THE FIRST DEGREE. 267 
 
 Given <| to find x and y. 
 
 I to 
 
 3 Given ax+JL- -f j (# /) = g to find a?, 
 "c a 
 
 4. Given """" * +<"- -^ u ~" * J, to find ar. 
 4 5 c 
 
 5. Given J I to find x and 
 
 6. Given J ^ i to find x and y. 
 
 / /- r* t> ~7/ * 
 
 3 
 
 7. Given < . I to find ar and y. 
 
 8. Given Zf_ -f x . ^, to find x. 
 
 9. Given 3a ~ 6a? ~ 2a ~ 3x + x - d /, to find . 
 
 10. Given 
 
 + ^ Z ^ } to find x and y. 
 
 11. Given ll + =L + 4(^3) = 68, to find x. 
 
 J <6 
 
ELEMENTARY ALGEBRA - SUPPLEMENT. 
 
 12. Given ~3~ to find ?, y, and 2. 
 
 ( or + 2y +3 Z = 14 } 
 13. Given J * y + 2 = 2 \ to find ar, y, and z. 
 
 x y + g +g _ 4 
 14. Given ^4 5 I to find *, y and 2. 
 
 17. Given J /^ = ^ j to find x and y. 
 
 18. Given j+ - | to find * and 
 
 C _a_ _i 1 
 
 19. Given < J+ y 3a+:r > to find x and y. 
 
 ( a.r+2 ly = d ) 
 
 C lex = cy 2J J 
 
 20. Given j ^ + a(c 3 -y) = 2^ ^ [ to find a; and y. 
 
 21. Given % / 
 
 2bf2 } to find x and y. 
 
 I y x= sL 
 
 V f* r 
 
 J j 
 
EXAMPLES IN EQUATION?. OF THE FIRST DEGREE. 269 
 
 'C x + y + z = 29H 
 22. Given < x + y z = 18j > to find a:, y and z. 
 
 i i 1Ox m 
 
 (a? y + z = 13| ) 
 
 23. Given 
 
 (3* + 
 iven < 7x + 
 (2y + 
 
 % = 161 ) 
 
 2z = 209 > to find a;, y and z. 
 
 2 = 89 
 
 24. Given < 
 
 1,1 
 
 - + - 
 
 x z 
 
 to find a?, y and 
 
 25. Given ^ dx + ey = f ^ to find a?, y and 
 
 gy + ^ = 
 
 EXAMPLES IN EQUATIONS OF THE SECOND DEGREE. 
 
 1. Given ar 2 5fz=18 to find x. 
 
 2. Given Sz 2 2#=65 to find a?. 
 
 3. Given 622^=15^+6384 to find x. 
 
 4. Given ll$x3%3?= 41J, to find x. 
 
 5. Given 9^901*+ 195=0, to find x. 
 
 6. Given 20748 1616a?+21^=0, to find x. 
 
 7. Given 9^90^+195=0, to find x. 
 
 18078* 
 
 8. Given .+ + 4728=0, to find x. 
 
 65 
 
 24 
 
270 ELEMENTARY ALGEBRA-SUPPLEMENT. 
 
 9. Given x 2 Sx= 14 to find x. 
 
 10. Given 3z 2 -f x=7 to find x. 
 
 11. Given 11822^=20 to find ar. 
 
 12. Given 6:r 30=3^ to find x. 
 
 13. Given 8^7^4-34=0 
 
 14. Given 4i^ 9#=5# 2 255f Sx to find x. 
 
 15. Given 80*+ + glgir 
 
 16. Given -.. l to find 
 
 17. Given -- to find ,. 
 
 10^81 5a? 8 5 
 
 18. Given A0 " ra> ^^T^ w _ to find x. 
 
 6(3 x) 19 7x 4(3 #) 
 
 19. Given # 2 7a?-f3J=0 
 
 20. Given 4^ 9a?=5a^ 255| 8a? to find x. 
 
 21. Given ^-= - to find x. 
 
 3X 5 
 
 22. Given -12- +??=13 to find a?. 
 
 x 5 x' 
 
 23. Given 6=?2 to find *. 
 
 x+2 3x 
 
 24. Given iL = J^- 5 to find . 
 
271 
 
 PROMISCUOUS QUESTIONS. 
 
 EQUATIONS OF THE FIRST DEGREE. 
 
 1 . A person expends 30 cents for apples and pears, giving 
 one cent for four apples, and one cent for five pears : he then 
 sold, at the prices he gave, half his apples and one-third his 
 pears, for 13 cents. How many did he buy of each ? 
 
 2. A tailor cut 19 yards from each of three equal pieces 
 of cloth, and 17 yards from another of the same length, and 
 found that the four remnants were altogether equal to 142 
 yards. How many yards in each piece ? 
 
 3. A fortress is garrisoned by 2600 men, consisting of 
 infantry, artillery, and cavalry. Now, there are nine times 
 as many infantry, and three times as many artillery soldiers, 
 as there are cavalry. How many are there of each corps ? 
 
 4. All the journey ings of an individual amounted to 2970 
 miles. Of these he travelled 3J times more by water than 
 on horseback, and 2J times more on foot than by water. 
 How many miles did he travel in each way ? 
 
 5. A sum of money was divided between two persons, A 
 and B. A's share was to exceed B's in the proportion of 5 to 
 3, and to exceed five-ninths of the entire sum by 50. What 
 was the share of each ? 
 
 6. There are 52 pieces of money in each of two bags, out 
 of which A and B help themselves. A takes twice as much 
 
272 ELEMENTARY ALGKBRA SUPPLEMENT. 
 
 as B left, and B takes seven times as much as A left. How 
 
 much did each take ? 
 
 x t 
 
 7. Two persons, A and B, agree to purchase a house to 
 gether, worth $1200. Says A to B, give me two-thirds of 
 your money and I can purchase it alone; but, says B to A, 
 if you give me three-fourths of your money I shall be able to 
 purchase it alone. How much had each ? 
 
 8. A father directs that $1170 shall be divided among his 
 three sons, in proportion to their ages. The oldest is twice 
 as old as the youngest, and the second is one-third older than 
 the youngest. How much was each to receive ? 
 
 9. Three regiments are to furnish 594 men, and each to 
 furnish in proportion to its strength. Now, the strength of 
 the first is to the second as 3 to 5 ; and that of the second to 
 the third as 8 to 7 ? How many must each furnish ? 
 
 10. A grocer finds that if he mixes sherry and brandy in 
 the proportion of 2 to 1, the mixture will be worth 78s. per 
 dozen ; but if he mixes them in the proportion of 7 to 2, he 
 can get 79s. a dozen. What is the pri'ce of each liquor per 
 dozen ? 
 
 11. A person bought 7 books, the prices of which were in 
 arithmetical progression, (in shillings.) The price of the one 
 next above the cheapest, was 8 shillings, and the price of the 
 dearest, 23 shillings. What was the price of each book ? 
 
 12. A number consists of three digits, which are in arith- 
 metical proportion. If the number be divided by the sum of 
 the digits, the quotient will be 26 -, but, if 198 be added to it, 
 the digits will be inverted. 
 
 13. A person has three horses, and a saddle which is worth 
 $220. If the saddle be put on the back of the first horse, it 
 
EQUATIONS OF THE EIRST DEGREE. 273 
 
 will make his value equal to that of the second and third ; if 
 it be put on the back of the second, it will make his value 
 double that of the first and third ; if it be put on the back of 
 the third, it will make his value triple that of the first and 
 second. What is the value of each horse ? 
 
 14. The crew of a ship consisted of her complement of 
 sailors, and a number of soldiers. There are 22 sailors to 
 every three guns, and 10 over ; also, the whole number of 
 hands is five times the number of soldiers and guns together. 
 But after an engagement, in which the slain were one-fourth 
 of the survivors, there wanted 5 men to make 13 men to , 
 every two guns. Required, the number of guns, soldiers, 
 and sailors. 
 
 15. Three persons have $96, which they wish to divide 
 equally between them. In order to do this. A, who has the 
 most, gives to B and C as much as they have already : then 
 B divides with A and C in the same manner, that is, by giving 
 to each as much as he had after A had divided with them: C 
 then makes a division with A and B, when it is found that 
 they all have equal sums. How much had each at first ? 
 
 16. To divide the number a into three such parts, that the 
 first shall he to the second as m to n, and the second to the 
 third as p to q. 
 
 17. Five heirs, A, B, C, D, and E, are to divide an in- 
 heritance of $5600. B is to receive twice as much as A, and 
 $,200 more; C three times as much as A, less $400; D the 
 half of what B and C receive together, and 150 more ; and E 
 the fourth part of what the four others get, plus $475. How 
 much did each receive? 
 
 18. A person has four casks, the second of which being 
 
 24* 
 
274 
 
 ELEMENTARY ALGEBRA SUPPLEMENT. 
 
 filled from the first, leaves the first four-sevenths full. The 
 third being filled from the second, leaves it one-fomrth full, 
 and when the third is emptied into the fourth, it is found to 
 fill only nine-sixteenths of it. But the first will fill the third 
 arid fourth, and leave 15 quarts remaining. How many 
 quarts does each hold ? 
 
 19. A courier who had started from a place 10 days, was 
 pursued by a second courier. The first travels 4 miles a 
 day, the other 9. How many days before the second will 
 overtake the first? 
 
 20. If the first courier had left n days before the other, 
 and made a miles a day, and the second courier had travelled 
 b miles, how many days before the second would have over- 
 taken the first ? 
 
 21. A courier goes 31| miles every five hours, and is 
 followed by another after he had been gone eight hours. 
 The second travels 22J miles every three hours. How 
 many hours before he will overtake the first ? 
 
 22. Two places are eighty miles apart, and a person leaves 
 one of them and travels towards the other, at the rate of 3 J 
 miles per hour. Eight hours after, a person departs from the 
 second place, and travels at the rate of 5^ miles per hour. 
 How long before they will meet each other ? 
 
 23. Three masons, A, B, and C, are to build a wall. A 
 and B together can do it in 12 days ; B and C in 20 days ; 
 and A and C in 15 days. In what time can each do it alone, 
 and in what time can they all do it if they work together? 
 
 24. A laborer can do a certain work expressed by a, in a 
 time expressed by b ; a second laborer, the work c in a time 
 '/ ; a third, the work e in a time f. It is required to find the 
 
EQUATIONS^ OF THE FIRST DEGREE. 275 
 
 time it would take the three laborers, working together, to 
 perform the work g. 
 
 25. Required to find three numbers with the following 
 conditions. If 6 be added to the 1st and 2d, the sums are to 
 one another as 2 to 3. If 5 be added to the 1st and 3d, the 
 sums are as 7 to 11; but, if 36 be subtracted from the 2d 
 and 3d, the remainders will be as 6 to 7. 
 
 26. The sum of $500 was put out at interest, in two 
 separate sums, the smaller sum at two per cent, more than 
 the other. The interest of the larger sum was afterwards 
 increased, and that of the smaller diminished, by one per 
 cent. By this, the interest of the whole was augmented one- 
 fourth. But if the interest of the greater sum had been so 
 increased, without any diminution of the less, the interest of 
 the whole would have been increased one-third. What were 
 the sums, and what the rate per cent. ? 
 
 27. The ingredients of a loaf of bread weighing 15lbs. 
 are rice, flour, and water. The weight of the rice, augmented 
 by 5lbs., is two-thirds the weight of the flour; and the 
 weight of the water is one-fifth the weight of the flour and 
 rice together. Required, the weight of each. 
 
 28. Several detachments of artillery divided a certain num- 
 ber of cannon balls. The first took 72 and of the re- 
 mainder; the next 144 and - of the remainder; the third 
 216 and | of the remainder; the fourth 288 and i of what 
 was left ; and so on, until nothing remained ; when it was 
 found that the balls were equally divided. Required, the 
 number of balls ind the number of detachments 
 
 29. A banker has two kinds of money; it takes a pieces 
 
276 ELEMENTARY ALGEBRA SUPPLEMENT. 
 
 of the first to make a crown, and b of the second to make 
 the same sum. He is offered a crown for c pieces. How 
 many of each kind must he give ? 
 
 30. Find what each of three persons, A, B, and C is worth, 
 knowing, 1st, that what A is worth, added to Z times what 
 B and C are worth, is equal to p ; 2d, that what B is worth, 
 added to m times what A and C are worth, is equal to q; 3d, 
 that what C is worth, added to n times what A and B are 
 worth, is equal to r. 
 
 31. Find the values of the estates of six persons, A, B, C, 
 
 D, E, and F, from the following conditions. 1st. The sum 
 of the estates of A and B is equal to a ; that of C and D to 
 b ; and that of E and F to c. 2d. The estate of A is worth 
 m times that of C ; the estate of D is worth n times that of 
 
 E, and the estate of F is worth p times that of B. 
 
277 
 
 PROMISCUOUS QUESTIONS. 
 
 INVOLVING EQUATIONS OP THE SECOND DEGREE. 
 
 1. FIND three numbers, such, that the difference between 
 the third and second shall exceed the difference between the 
 second and first by 6 : that the sum of the numbers shall be 
 33, and the sum of their squares 467. 
 
 2. It is required to find three numbers in geometrical pro- 
 gression, such that their sum shall be 14, and the sum of 
 their squares 84. 
 
 3. What two numbers are those, whose sum multiplied by 
 the greater, gives 144, and whose difference multiplied by the 
 less, gives 14 ? 
 
 4. What two numbers are those, which are to each other 
 as m to ?i, and the sum of whose squares is b ? 
 
 5. What two numbers are those, which are to each other 
 as m to w, and the difference of whose squares is b ? 
 
 6. A certain capital is out at 4 per cent interest. If we 
 multiply the number of dollars in the capital by the number 
 of dollars in the interest, for five months, we obtain $117041|' 
 What is the capital? 
 
 7. A person has three kinds of goods, which together cost 
 $230/ T . One pound of each article costs as many times ^ 
 of a dollar as there are pounds of that article. Now, he has 
 
278 ELEMENTARY ALGEBRA SUPPLEMENT. 
 
 one-third more of the second kind than of the first, and 3j 
 times more of the third than of the t jcond. flow many 
 pounds had he of each ? 
 
 8. Required to find three numbers, such, that the producl 
 of the first and second shall be equal to a ; the product of the 
 first and third equal to b ; and the sum of the squares of the 
 second and third equal to c. 
 
 9. It is required to find three numbers, whose sum shall be 
 38, the sum of their squares 634, and the difference between 
 the second and first greater by 7 than the difference between 
 the third and second. 
 
 10. Find three numbers in geometrical progression, whose 
 sum shall be 52, and the sum of the extremes to the mean, 
 as 10 to 3. 
 
 11. The sum of three numbers in geometrical progression 
 is 13, and the product of the mean by the sum of the ex- 
 tremes is 30. What are the numbers ? 
 
 12. It is required to find three numbers, such, that the 
 product of the first and second, added to the sum of their 
 squares, shall be 37 ; and the product of the first and third, 
 added to the sum of their squares, shall be 49 ; and the pro- 
 duct of the second and third, added to the sum of their 
 squares, shall be 61. 
 
 14. Find two numbers, such, that their difference, added 
 to the difference of their squares, shall be -equal to 150, and 
 whose sum, added to the sum of their squares, shall be equal 
 to 330. 
 
 15. It is required to find a number consisting of three 
 digits, such, that the sum of the squares of the digits shall be 
 
EQUATIONS OF THE SECOND DEGREE. 279 
 
 04 ; the square of the middle digit to exceed twice the 
 product of the other two by 4 ; and if 594 be subtracted from 
 the number, the three digits become inverted. 
 
 16. The sum of two numbers and the sum of their squares 
 being given, to find the numbers. 
 
 17. The sum, and the sum of the cubes, of two numbers 
 being given, to find the numbers. 
 
 18. To find three numbers irv arithmetical progression such, 
 that their sum shall be equal to 18, and the product of the 
 two extremes added to 25 shall be equal to the square of the 
 mean. 
 
 19. Having given the sum, and the sum of the fourth pow- 
 ers of two numbers ; to find the numbers. 
 
 20. To find three numbers in arithmetical progression, 
 such, that the sum of their squares shall be equal to 1232, 
 and the square of the mean greater than the product of the 
 two extremes, by 16. 
 
 21. To find two" numbers whose sum is 80, and such, that 
 if they be divided alternately by each other, the sum of the 
 quotients shall be 3i. 
 
 22. To find two numbers whose difference shall be 10, 
 and if 600 be divided by each of them, the difference of the 
 quotients shall also be 10. 
 
02512. 
 
 .'850