«'i,Wh «i' 1,1' ,, 1 Si I mi*':,y?^.:.:s ft' w jfif a '' ' '' 4'i fal ! r ^ll-li. ,"' 4 PRESENTED TO TH3S CORNELL UNIVERSITY, 1870, The Hon. William Kelly Of Rhinebeck, I University Library A treatise on algebra. o.in,an? ^924 031 275 278 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031275278 TREATISE ALGEBRA, By BrSESTINI, S.J. Author of Analytical Geometry, and Elementary Algebra. FBOFESSOB IN fiEOKGETOWN COLLEfiE. BALTIl^ORE: PUBLISHED BY JOHN MURPHY & 00. No. 178 MARKET STEEBT. PITTSBUEG....GEOEaE QUIGLBY. Sold by Booksellers generally. 1855. Kntered according to Act of Congress, in the year 1855, by JOHN MUKPHY & Co. in the Clerk's Office of the District Coxirt of the United States for the Eastern District of Maryland. BTEBEOTTPSD BY L. JOHNSOX ft CO. PHILADELPHIA. PEEFACE. This treatise may be considered as a sequel to the small Ele- mentary Algebra, whose second edition,! revised and enlarged, has just preceded the present publication. However, it is not so connected with the Elementary Algebra that it might not be taken alone, for it does not depend on the former in any of its parts, and is complete, as far as is allowed by the nature of a book destined for the use of those who desire to be initiated in the study of Algebra. The reader, even before perusing the present introduction, has probably noticed the difference of type, intended to separate those subjects which are more accessible to pupils at large from those which suppose in the student either quicker parts or already some advancement in the study of Algebra. That is to say, the most elementary principles adapted even for those who, for the first time, open a book of Algebra, are printed in larger type : the other parts, which enter a little more into the secrets of the science, are printed in smaller characters. We beg the reader, however, to observe, first, that the under- standing even of the most elementary principles of Algebra and Geometry supposes always a certain degree of aptitude. Of this, one who for any time has had experience of the tedious labour of teaching, will render, without hesitation, abundant testimony. Another observation to be made is, that the separation adopted in the present treatise, with distinction of type, does not trace a limit to be scrupulously followed, so that the teacher or the stu- dent be compelled to go over all that is printed in large characters before commencing the rest. But it is left to the discretion of the teacher to enter, more or less, into the subject where and when he will judge fit to do so. The teacher is fully aware that he must unquestionably labour, and must not be satisfied merely 3 4 PREFACE. with what he is to teach, hut he should know much more. He should be master of the subject, and be competent to adapt it to the capacity of his pupils. The Treatise is divided into two parts, the first of whick con- tains algebraical operations, with several questions and doctrines connected with them, so that each section may prove complete in its own subject, and the inconvenience of returning elsewhere to speak of matter left unfinished before, may be avoided. The same method is followed in the second part, of which we will im- mediately say a few words. With this method, every thing is put in its own place, so that any one who would go over the whole uninterruptedly might have the advantage of order, and of seeing, at a single glance, all that each subject embraces. Nay, the same advantages may be enjoyed by those also who will be able to overcome the first difficulties at a second or third reading. This method, we believe, has also the advantage of contracting the bulk of the volume, which the same subjects, disorderly scat- tered, would render much larger. The second part contains the most indispensable theories of equations, proportions, and progressions, logarithms, and some few principles on the series. The doctrine of equations has been treated more copiously than the others, not so much on account of its importance, as because it is well adapted to give an idea of algebraic analysis, and thus prepare the mind of the student who would afterwards apply himself to higher studies. Geoboetowh Colleob, July, 1865. CONTENTS. PAGE Ihtbobitctory Article 13 Mathematics: its object, and various branches 13 Algebra: its object and generality — its symbols... 13 Numerical relations of quantities 13 Relation of oppositions and signs 14 Opposite quantities — how mutually influenced , 14 Coefficient and similar terms 15 Signs of addition and subtraction 15 Equations and inequalities 15 Monomials and polynomials 16 Members of equations 16 Constant and variable quai^tities. — Functions 16 Modification of quantities — their mutual relations 17 Division of the treatise 18 FIRST PART. ALQEBRAIC OPERATIONS. CHAPTER I. — DEFINITIONS AKD OPEEATIOHS OK MOSOMIALS. Art. 1. Addition and Subtraction. — Addition 19 Algebraic and arithmetical addition compared 19 Definition of algebraic addition 20 Different cases 20 Rules 1 and 2 20 Doctrine of signs — product of signs 21 Examples and problems 22 Letters used for unknown quantities 24 Numerical values and general character of algebraical questions 24 Subtraction. — Difference— definition — result of signs 25 Rule, and general examples 26 1* 5 6 CONTENTS. PAOE Criterion of magnitude 26 Twofold acceptation of numbers 27 The term zero taken as a mere term of comparison 27 Examples and problems 27 Art. 2. Multiplication and Division. — Multiplication 28 In what multiplication consists — nomenclature. 28 Various manners of representing the product 29 Definition of numerical miiiltipUcation 29 Case of the multiplier, whole number, and positive 29 Consequence concerning the sign of product 29 Case of the multiplier negative 30 Case of the multiplier fractional 30 Case of the irrational factors 30 Arithmetical rule applicable to quantilies 30 Mutual influenco of factors 31 Bule of signs 31 Various forms of the numerical value of quantities 31 General formulas of numerical multiplication 32 How the arithmetical rules of multiplication follow from the definition 32 Mechanical artifice, showing the aliquot part of a fraction 33 The multiplicand may become the multiplier, and vice versa, without affecting the product 33 Pactors, whole numbers, and other cases 34 General inference i 34 The product of any number of terms is the same, whatever be the or- der of the factors 34 Sign to be given to the product of several factors — three cases 36 Rule 37 Product of repeated factors — exponent 37 Examples and problems 38 Division. — Its definition 40 Algebraical expression of division 40 Rule — numerical values 41 Compound monomials 42 Their product and quotient 43 Reduction of results to a simpler form 44 Reduction to the same denominator 44 Examples and problems 45 Art. 3. Formation of Poicera and Extraction of Roots 47 Poto era.— "What a power is — root, exponent, degree, power 47 Formation of powers, embracing all cases of numerical values, its de- scription and definition »••. 47 CONTENTS. 7 PAGE Apparent powers 48 Powers of unity , 48 Product of powers— case of the exponent's whole numbers 49 " " case of the exponent's fractional numbers 49 General inference and rule 50 Product of powers when the same exponent is applied to different roots 50 Inference and rule , 50 Powers of powers ; 50 Inference and rule 52 Simplification of exponential fractions — different cases — examples.... 52 Extraction of roots is the inverse operation of raising to a power 55 Signs and nomenclature 56 The root of a quantity may be represented by a fractional exponent given to the same quantity bQ Fractional index of the root.... 56 A radical having a fractional index can bo transformed into one hav- ing a whole number for its index .' 5lr Product of roots — different cases 57 The index of the root and that of the quantity under the radical sign may be multiplied or divided by the same quantity or number.... 58 Different eases embraced by a radical expression 58 Imaginary radicals or roots... 59 Operations on imaginary quantities ■ 69 Irrational radicals and quantities 61 Greatest common measure of numbers *. 61 Prime numbers 63 How a prime number can accurately divide the product of two other numbers 64 A prime number dividing a productj divides at least one of the factors 65 Powers of fractional expressions 66 Irrational roots 67 A Series of rational or assignable numbers may be conceived ap- proaching constantly to any irrational root. 68 Miscellaneous examples 69 CHAPTER II. — OPERATIONS ON POLYNOMIALS. Art. 1. Addition and Subtraction 73 Keduction of similar terms — rule and examples 73 Subtraction. — Rule — examples 75 Art. 2. Multiplication and Division 76 Eule and examples..... , 76 Kule of signs demonstrated 77 8 CONTENTS. Remark on common factors 78 Dxamples of multiplication 7S Singular property of numbers 80 Dvviaion. — Polynomials arranged 81 Operation explained 81 Kule and examples 81 Bemark concerning the arrangement of polTnomials 84 The dividend not exactly divisible by its divisor 85 Number of the terms of the indefinite quotient .' 87 The quotient of the same form as the imaginary expressions divided by one another 89 Art. 3. Formation of Powers and Extraction of Mooter 90 Newtonian formula. 91 General term 92 Square and cube of any binomial 92 Binomial theorem demonstrated 93 Singular property of the coefficients of the binomial 9* Permutations 94 Corollary 97 Combinations 98 On some property of numbers — theorem 99 Evolution of the binomial when the exponent is negative 100 Extraction of Hoots. — The root of a polynomial another polynomial. — Example 102 The process to find the square root of any polynomial is always the same 103 General role for the extracUon of square roots of polynomials. — Ex- amples 105 Extraction of square roots of numbers 106 Practical rules 107 Polynomial expression of the square of a number 110 Extraction bf cubical roots of polynomials 112 Extraction of cubical roots of numbers 114 SECOND PART. ALGEBEAIO THEORIES. CHAPTER I. — EQUATIONS. Division of the chapter. Ug Practical rules 117 Known and unknown quantities 118 Determinate and indeterminate equations 119 CONTENTS. 9 PAGE Art. 1. Equations of the Jvrst degree 119 General formula of determinate equations of the first degree 120 Resolution of aqy determinate equation of the first degree. — Kule.. 120 Examples and problems 121 Indeterminate equations of the first degree 122 Equations containing several unknown quantities 123 Different methods of resolution. — First, by comparison 124 Elimination by substitution ,,..., 125 Elimination by addition and subtraction 127 The same methods applicable to all cases 128 Incompatible equations 128 Eules and examples 129 Problems — general rule 131 Art. 2. Equations of the second degree 135 General formula of the determinate equations of the second degree.. 135 General rule — remark 136 Resolution of the general equation ? 137 First case. — Rule and examples 137 Second case 138 Real and imaginary roots 139 Examples and problems 141 Rule 142 Art. 3. On some properties of determined equations of any degree, 144 Preliminary theorems 144 General formula of a determined equation of any degree 147 Equations resolvable with at least one real root 147 Equations resolvable with at least two real roots 147 When equations cannot be resolved with real roots, they are resolva- ble with one or more imaginary roots 148 Any determined equation of any degree admits of as many roots as there are units in the degree of the equation 149 Connection between the roots and the coefficients of any determined .equation 151 Corollaries. — Case of one or more roots equal to zero 153 Signs of the roots changed 153 Roots multiplied and denominators eliminated 153 The sums of various powers of the roots can be known, although the roots themselves are unknown * 155 Corollary and criterion 158 Art. 4. Resolution of determinate equations of the third and fourth degree, having real coe^dents 161 General formulas of the equations of the third degree 161 10 CONTENTS. PASE Koots of the general equation of the third degree 161 Besolution of the same general equation 163 The first condition always fulfilled, hut not the second,. 164 Examples > 167 General remark 16S Equationa of the fourth degree 169 Quality of the roots: how found out 171 Example 172 CHAPTER II. — BATIOS, PEOPORTIONS, AND PBOGBESSIOHS. Division of the chapter, and definitions 173 Art. 1. Arithmetical EatioB, ProportionSt and Progressions 173 BatioB. — Definitions and property 173 Proportions, — Simple arithmetical proportions 174 Continual and compound proportions 175 Progreasiona. — Terms of an unlimited progression 175 General formula 176 Sum — Examples 177 How and when the terms of a progression may be found — Example 178 Art. 2. Geometrical BatioSf Proportions^ and Progressions 178 Patios. — Definitions and property 178 Variable ratios — direct and reciprocal terms 179 Continual geometrical ratios — Theorem 180 Direct and reciprocal compound ratios — Theorem 181 Proportions. — Simple geometrical proportions 182 General properties 183 Continual and compound proportions 183 . Other properties of geometrical proportions 184 Numerical proportions, the first ratio of which is irreducible 186 Progressions. — Terms of any geometrical progression 186 General formula 187 Sum of n terms and sum of an indefinite number of terms 187 Examples 188 How and when the terms of a certain progression may be found...'... 189 Example 189 CHAPTER in. — LOSAniTHMS. Exponential quantities 190 Logarithms 190 Base and sign 191 The logarithm of unity and the logarithm of the base are the same in all systems 191 Positive and negative logarithms 192 CONTENTS. 11 PAGE Theorems 192 In any system when the logarithms form an arithmetical progres- sion, the corresponding numhers form a geometrical one 193 Useful theorems , 193 Common or ordinary tables of logarithms 195 Constant ratio of logarithms in every system 197 How the logarithms of one system may he inferred from the loga- rithms of other systems 197 Explanatory remarks — Eules 198 Characteristic 201 Application of logarithms 203 CHAPTER IV. — SERIES. What algebraic series are, and their Tarious orders 206 Various questions concerning the series 207 General term of any series 208 Sum of any number of terms 209 Dxamples 210 Problems , 213 TREATISE ON ALGEBRA. INTKODUOTOKY AKTICLE. Mathematics: their § 1- MATHEMATICS treat of quantities, name- ''^^^*- ly, of all that whioli can be numerically esti- mated or measured. Their Tnrious §2- Hence, Mathematics, in their general branches, acceptation, embrace as many branches as there are species of quantities taken into consideration, and these various branches are also distinguished by appropriate denomi- nations, as Geometry, Hydraulics, Optics, &c. § 3. Algebra considers quantities in an ab- stract manner ; that ia, it considers in quantities those properties and relations which are common to all the various species ; and we may add : That which Logic is to mental philosophy and mental sciences of every description, in some measure Algebra is to the mathematical sciences. Generality of Ai- § 4. Algebraic questions are consequently gebraic questions. . n i i i Aigeijraie symbols, quite general, as well as the symbols used to represent the quantities. These symbols are commonly the letters of the Latin and Grreek alphabet. Algebraic qnes- § 5. Algebraic questions and operations are, tions connected with ,.^ ., , t • ,^ ■ t arithmetical ques- besides, stnctly connected With numerical or tions. lielation of . , , . , , . .j.. , magnitude. arithmetical questions, iseeause, whenever one quantity is compared with another of the same kind, for 2 13 14 TREATISE ON ALGEBRA. instance, weiglit with weight, space with space, &o., the rela- tion is no other than numerical. This relation is a relation of magnitude. Relation of oppo- §6- Another relation, we may say of oppo "'i™- sition, depends on the different manner of the existence of quantities. This opposition is designated by the denominations of positive and negative quantities. So, for example, two forces acting in the same straight line, but in a direction opposite to each other, if compared, are respectively positive and negative. § 7. When a quantity, for instance, a, is destined '''^^' to represent a positive one, the sign -(- (^plus) is fre- quently placed before it. When the quantity is negative, the sign — (mimiii) is always prefixed to the symbol. § 8. When, therefore, in the same question Use of the sigDS. , , ^ , . . . 7 -x • we meet with the quantities -\- a — o, it is always understood that a is positive with reference to b, and h is negative with reference to a. And, vice versa, if two quantities are given opposite to each other, and before the first we put no sign or the positive sign, the negative sign is then to be constantly put before the second. How opposite § 9. When quantities of different signs, sup- quantities are mu- n . 1 1 i' . tuaiiy influenced, pose two, are collectively taken, their value is then equivalent to a third quantity, which is the difference of the absolute value of tbem, and whose gign is either positive or negative according as the greater absolute value of the two quantities is that of the positive, or that of the negative. For example ; two forces, -\- B and — h, if applied to the same material point and along the same straight line, their effect is the result of their simultaneous and collective action. But, if we suppose B to impel the point twice as much as h, or, which is the same, the absolute value of B to be twice as great as that of 6, since the forces act in opposition, the effect of h will be counteracted by that of B; and one-half of B (which INTRODUCTORY iUlTIOLE. 15 is equal to b) will produce alone the effect in- the positive direction of B. That is, the collective action of the two forces is equal to that of their difference, and this difference acts in the direction of the greater force. § 10. When a symbol, for example, 6, is adopted Coefficient. . . -, • i to represent a certain quantity, and it happens that in the^ same investigation another quantity occurs whose magnitude is twice, three times, &c. the magnitude of the former b ; instead of making use of another symbol, or repeating the same, we write only once the symbol b, placing before it a figure to show how many times the quantity is taken. This number is called coefficient, which means making together with the symbol, the whole of a quantity. If, for instance, the quantity B is three times as great as h, or five times as great as c, instead of writing B and C, we would write 36 and 5c. When two or more terms differ only in the coefficient, they are termed similar. For ex- ample, 5J, 26, or 3c, 7c, 12c, are similar terms. § 11. Let us remark here also, than when a quantity is to be added to another quantity, 6 for instance to a, or several quantities are to be added to another, this addition is commonly expressed by interposing between the quantities or terms the positive sign -\-, which, for this reason, is termed' also a sign of addition. Suppose, for example, that the quantities 6, c, d are to be added to a, this will be indicated by writing, a-\-b-\-c-\-d. Sign of sattme- § 12. When, On the contrary, a quantity is to *'""• be subtracted from another, the quantity from which we subtract is first written, then the other, and the negative sign is placed between them. If, for instance, 6 is to be subtracted from a, we will write a — 6. Equations and § l^- Comparing together quantities of the inenualities. same kind, for example, weights with weights, 16 TREATISE ON ALGEBRA. surfaces with surfaces, &c., we will fiad them either equal or not. Suppose now, for the sake of simplicity, only two such quantities, which we will call a, h. If they are equal, then we write a^l, and the sign {=) of equality is read equal to; the terms so compared, considered as forming a single expres- sion, are called an equation. If the same terms represent two unequal quantities, then a is either greater or less than h ; in the former case the inequality is expressed by writing a^-h, in the second, a<^h; that is, we place between the terms the angle, or sign of inequality, with the vertex towards the less of the unequal terms. Monomials and § 1*- -^iy algebraical expression whose sym- poiynomiais. ^jgjg g^^g ^^^ separated by positive or negative signs, or the signs of equality or inequality, is called term or monomial. For example, the symbol h, together with the coefficient 5, constitute the monomial 56. When two terms are separated by a positive or negative sign, the expression is then called binomial; if three such terms are separated in the same manner, they form then a trinomial, &c. ; and in general these expressions are oaWei pol^noTniah. Members of equa- § 1^. When two or more tcrms are separated ''™^- by the sign of equality or inequality, these terms constitute the members of the equation or inequality. For example, in the equation, a-\-b -\-c ^=m — n, the tri- nomial a-}-b -\- c forms the first member, and the binomial m — n the second member of the equation. Likewise, in the inequality p -\- q >/ — d, the first binomial is the first mem- ber, and the second binomial forms the second member of the inequality. Constant and § 16. Any algebraical expression whose value variable quantities. i /• • 7 ^ . > Functions. depends on the value or a variable quantity, is called function of that variable. For instance, the monomial 6a; depends on the value given to x. So, likewise, in the INTRODUCTORY ARTICLE. 17 equation, y^a -\- x, supposing a an invariable or constant quantity, and x variable, y necessarily depends on the value of X ; hence, y is a function of x. When such quantities as y are functions of another, or other quantities, this is expressed by the index /, and, instead of writing, for example, i/ or a-\- X, we simply write, /(x), or y=/(x). The index / must be varied when different functions occur in the same question, and we then make use of F or p, or some other letter. When, therefore, a quantity a, or several quantities a-\-b, &c., are submitted to any operation, the result is a function of those quantities, because it depends on the same quantities; so that, if instead of ra ot: a-\-h, we should submit to the same operations other quantities, for instance, A and A + B, the result would necessarily be different. But if two or more quantities are equal among themselves, and are submitted to the same operations or equally modified, the result must necessarily be the same. Hence, if a is equal to 6 and c, &o., and a is submitted to such an operation as to give for result, /(a), if we submit h and c to the same operation, with the equation, a = 6 =; c ^ . . . . we must have also, f(a)^f(h)=f(c) = .... That is to say, the members of an, equation equally modified form, another equation. This deduction cannot for the present be developed nor illustrated by examples, but its frequent application will soon supply copious illustrations. Modification of § 17. Quantities are essentially capable of quautities; their . ...... . -i • mutual relations, increase and diminution ; and considenng any quantity in an abstract manner, we cannot conceive any other modification of it, except that which is performed by addition or by subtraction, or by equivalent operations. Again, 2* 18 TREATISE ON ALGEBRA. quantities m'ay be compared together, either by a simple or complex comparison. This is all that concerns quantities, Division of the generally considered; hence, Algebra may be '^'^ '^°' conveniently divided into two parts : the first of which has for its object operations on quantities ; the second, to investigate and discover the properties, connections, and dependences of quantities, according to their various com- parisons and combinations. FIRST PART. ALaEBEAIC OPEEATIOB'S. CHAPTER I. DEFINITIONS AND OPERATIONS ON MONOMIALS. AKTICLE I. Addition and Subtraction. Aigeiraicana § 1^- ADDITION. — Numerical or arithmetical diUo^''*'°com- addition consists in finding out a number containing pared: jj^ itself as many units and fractions of units as there are in all the numbers to be added together. Eor example, to find out the number 12, which contains in itself as many units as there are in the numbers 2, 4, 6, is to- make the sum or addition of these numbers, and this sum is expressed (11, 18) by 2 -]- 4 -[- 6 = 12. But, with regard to algebraical quanti- ties ; for instance, a,h,c,...., although the sum is repre- sented as in numbers, namely, a-\-h-{- c-\- .. . , yet, on account of the more general signification of the algebraic symbols, the operation is not equally simple as for numbers. The quantities represented by algebraic symbols have, indeed, a numerical value ; nay, this value is the one taken into account in addition, as well as in other algebraic operations. But quantities may have either a positive or a negative value ; so that a, for example, may be negative with regard to 6 and c. Then the numerical and relative value of a is to be expressed, for instance, by — 3, while the others are expressed, suppose by -|- 4 and -(- 5 : in this sum or collection the negative part is 19 20 TREATISE ON ALGEBRA. destroyed by the positive, and since 5 -)- 4 ^ -(-9 and 9 — 3 = -(- 6, a quantity m, whose numerical value is -)- 6, will represent the sum of the given quantities a, b, c, and we will have a-^h -{- c = m. From these remarks we deduce the following definition, and two practical rules : DeflnitioQ of The Sum of a number of quantities is a mo- algebraic addi- .77.7 tion. normal, whose numerical value ts the excess of the numerical value of the quantities affected by one sign, over the numerical value of the quantities affected by the opposite sign, and the sign of which is the sign of the same excess. Terms all af- Some consequences easily derived from this feotod iy the . .„ , , same sign. definition Will make it more clear. First, if all the quantities to be added have either a positive or a negative sign or value, the sum has, likewise, a positive or negative value, and the numerical value of this sum contains as many units and fractions as there are in all the numerical values of the quantities, added exactly as for numbers. Secondly, if the Equal nume- numerical value of the quantities to be added rical value of « i i • i i quantities af- amount to the Same for those which have a posi- site signs. tive, and for those which have a negative value, the sum is then equal to zero. A third consequence needs not to be pointed out, since it obviously appears from the definition itself Rule 1. — When the quantities to be added are merely represented by symbols, we consider them as Jiaving a positive sign, and their sum is expressed by writing in succession the same quantities, and placing the positive sign between them. For example, the sum of the quantities a, b, c, d, . . . . is expressed by a-\-b-\- c-\- d-\- . . . . Rule 2. — When the sign is placed before the quantities to OPERATIONS ON MONOMIALS. 21 be added, then the sum is represented hy writing, likewise, the synibols in succession, each with its own sign ; hut the sign of the Jirst is not written unless it he negative. For example, the sum of the quantities, a, h, c, — g — h — k, is expressed by a-\-h -\-c — g — h — k. It is evident that a sum will be always equal to a certain unvariable monomial expression, whatever be the order in which the terms are written. So, for instance, calling m the equivalent monomial expression, we may write a -\-h -\- c — g — h — k = 'm, or a — g -\-b — h-\-c — 7<;=;m, &c. The proposed examples are the most general. In more particular cases there occur simplifications or reductions of terms, which we will soon see in other examples. Doctrine of § ^^' ^^ liave already remarked, that the sign placed sisns. before a symbol is not always the same sign as that of the numerical value of the quantity represented by it ; and although, generally, the quantity a or -f- a is considered as having a positive numerical value, and the quanity — 6, a negative numerical value, it happens, however, in mathematical investigations, that the numeri- cal value of a positive quantity is sometimes found negative, and vice versa. Hence, some questions arise concerning the final sign to be given to a quantity, which deserve to be noticed here. And, to give to our researches a quite general character, let us first remark, that one or more signs by which an algebraical symbol can be occasionally affected necessarily affect the numerical value itself, and vice versa ; secondly, an algebraical symbol is frequently a symbol of another, or other symbols, sometimes affected by the same, sometimes by the opposite signs. For instance, we may have (r = -l-A or a ^ — A ; hence, 4-a = + (-fA), or +« = + (— A) — a = — (+ A), or —a = — (—A). Produot of The expressions of these two sets are manifestly signs. opposite, but 4-(4--A-) is equivalent to -j-A; hence, (-f-A) must be equivalent to — A; again, -|- ( — A) is equivalent 22 TREATISE ON ALGEBRA. to — A ; hence, — ( — A) must be equivalent to + A. Tlierefoi-e, we derive tliis general inference : A double sign placed before an algebraical symbol, is equivalent to the positive sign, when the two are either both positive or both negative; it is equivalent to the negative sign, tvhen one of the two is positive and the other negative. But, suppose A to be a symbol of another algebraical symbol ; for example, -f- * or — "-^ '^'^ '"^i^l have + +A = ++(+.), or + + A = ++(-=.), and, continning the same process, we see that any number of signs may affect an algebraical symbol, but the same signs may be easily reduced to a single one. When, for example, b is affected by a num- ber of signs, as follows : + + -i; ■ make first, -| J = c, we will have + +-* = + "; make again, e = d, we will have -\ 1 b = +d = d; but -\ b = — b; hence, c = — b, and c = -\~ c; hence, c ^ d, and consequently, d ^ — b; but d = + — 1 6; hence, -| ■ 1 6 = — 6; and in general, when the original number of signs contains an even number of negative ones, the final sign is always positive ; and when the original number of signs contains an odd number of negative, the final sign is always negative. In fact, no sign is changed from posi- tive into negative, and vice versa, except by the influence of a preceding negative sign ; hence, the first negative sign determines a negative sign for the symbol, the second changes it into positive, the third into negative, &e. The analogy between the mutual influence of signs when applied to the same quantity, and the influence of terms affected by different signs when multiplied together, has given to the final sign in question the name of product of signs, although this result is altogether different from that of multiplication. Examples and §20- Let - 2i) + m) +36) -3m) +/) problems. _|_ 2^) fee terms to be added. In this example, OPERATIONS ON MONOMIALS. 23 the similar terms, — 26, -[- 36, and -\- m, — 3m, -|- 2»i may be reduced to a less number, because (9) 36 — 26 is evidently equal to -\-b or 6, and m -|- 2m = 3m ; hence, m -f- 2m — 3m = 0; therefore, the sum of all the terms is given by h-\-/, that is, — 26 + m + 36 — 3m-|-/+2m = 6+/. Let the terms given for another addition be 16ot) -|- 12c) — ir) + s) — 12m) — 13c) — 3s) + r) + c) — m) + 7r) — 13s). Select first the similar terms, and dispose them as follows : + 16m -fl2c —ir + s — 12m —13c + r — 3s ■ — m -(- c -\-7r — 13s Sum + 3m 0' +4»- —15s. And since the collection of the separtial sums giyes the total sum, we will have 16m + 12c — 4y + s — 12m — 13c — 3s + ?• + c — m + 7r — 13s := 3m -|- 4>- — 15s. From these examples it is plain, that the addition of simple monomials consists in a bare reduction of similar terms, and this reduction is performed by taking the sum of their co- efficients when they are affected with the same sign, or by taking the difference of the same coefficients when affected with opposite signs. Let us now propose some problems to be resolved with simple addition of monomials. Twelve divisions of soldiers, containing each 2)1 soldiers, are in a castle when the enemy com- mences the attack ; 2 of these divisions take to flight during the assault; 4J divisions perish in the conflict. The assailants gain the battle, and their general with 8 divisions, each con- taining r men, enters into the fort when it is still occupied by the defenders. We ask what is the amount x of combatants in the fortress after the entrance of the victorious general ? This problem, besides giving another example of addition, 24 TREATISE ON ALGEBRA. gives also occasion to exemplify still more the relative signifi- cation of positive and negative terms. The quantity here inquired is a number of men ; considering, therefore, as posi- tive all that tend to add to this number, we must consider as negative all that tend to diminish the same number, being evidently quantities opposite to one another. Hence, the terms given by the problem, will be as follows : Twelve divisions of men, each containing 2n soldiers, give the term, -|- 12 . 2n, or -f 24ra. Two divisions of men leaving the fort, give the term — 2 ..2», or — 4n. Four and a half divisions lost in the battle, give the terms, — 4 . 2n, — 11 ; that is, — 8«, — n. The general entering into the eastle, gives the term, -(- 1. And the eight divisions containing each r soldiers, give the term, -|- 8?-. Hence, we have for the required amount 24ra — 4» — 8w — B + 1 + Br; which gives x = lln -|- 1 -|- S'"- Letters used We may here remark, that x, as well as ^ and quantities. z, and some of the other last letters of the alpha- bet, are usually employed to represent the quantities to be found, or generally unknown quantities. Numerical ap- But to render the case more determined, sup- plications of the ^rt 1 nrt mi J? problem. posc K = 50 and r = 80, we will have irom x = lln+l-j-8r, x = 1191. IS, instead of ra = 50 and r ^ 80, we take re = 80 and r = 60, then we have x = 1361. General cha- -^"'^ ^<> ^^ oould resolve any number of cases braiMi°'ques^ ^J Substituting other values for r and n. And tions. from this the learner may appreciate the general character of algebraical questions. Four hunters agree to meet together at the verge of a river after hunting. The first of them shoots OPERATIONS ON MONOMIALS. 25 11 birds J the second, twice the same number, and 2r birds besides ; the third shoots as many birds as the first and second killed together j the fourth does not shoot any, but seeing the good success of his companions, takes the birds brought by the first, and one-half of those brought by the second, and throws them into the river. How many birds remained after this 1 and how many birds were brought by the hunters ? Ans. to the first question : x = in-\- 3r. Ans. to the second question : 2/ = Qn -\- 4?-. Suppose n = 10, r = 9, then a; =67, y =96. Suppose m = 9, ?• ^ 10, then X = 66, y = 94, &c. §21. Subtraction. — To subtract a quantity h from another quantity a, means, to find the difference between the two quantities; and this difference, which we will call d, is such, that if added to the quantity h, the sum will be a. Hence, we may briefly give the following definition : To subtract b from a, means to _find out ai quantity d, which added to b gives a. Namely, «Z -)- & = »• Now let us add the term — h to both members of this equa- tion, we will have d -\-h — 6=a — &, that is d^a — 6. Suppose, again, h equal first to -|- g, and secondly to — g; we will have, in the two cases, -6 = -(+^) But in the second case, the value of 6 is oppo- e signs. ^.^^ ^^ ^^^ ^^ ^^ ^wsx^ b in the first case, con- 26 TREATISE ON AIGEBRA. sequently, since — Q-\- g) = — (g), or since in the first case — 6 ^ — g, in the second we will have — b=^-\- g ; that is, — ( — gr)^-|-^, which exactly corresponds to the doctrine of signs (19) ; hence, with b = -\- g, we have d^a — 6 = o — g; and with b = — g, we have d ^a — h =^a -\- g. In the first case we ohtain the difierence d, by adding to a a quantity opposite to -\- g; in the second case the required diflference is obtained by adding to a a quantity opposite to — g; hence, follows this general rule : Rule for sub- ^^* quantity h is subtracted from a, by adding traction. fo a. a quantity opposite to b. General ex- Thus, for example, — h is subtracted from h, ampies. |jy gimply writing and in is subtracted from n by writing n — m. These are the most general examples of algebraical sub- traction of monomials. We will soon propose other examples and problems, in which the difference can be expressed by a single term. , Cr"te ■ of ? ^^- ^^ ™^ ^®''® obsorrc, that when the diiference magnitude. n — m is positive, then n is said to be greater than m ; when the same difference is negative, then n is said to be less than m. The difference, therefore, between two quantities, is the criterion of their relative magnitude ; and since by substituting for n any positive number or numerical value, and for m any negative number or Any positive numerical value, the difference is certainly always posi- er"tha™ au'ne^ *'™' ^° ^* foUows that any positive number or quantity gative. relatively considered is greater than all negative ones. Again, substituting for n and m negative numerical values, but the absolute value of n less than the absolute value of m, the difference is, likewise, always positive ; therefore, the greater of two negative quantities or numbers, relatively considered fo a certain term, is that OPERATIONS ON MONOMIALS. 27 ™ ^,j which has u, less mimerioal value. We may illustrate Twofold accsp- ■' tatinu of num- the Same doctrine as follows : observing first, that num- bers are either considered as terms of comparison, or as symbols by which one or more existing object^ may be designated. When we comsider nambers under this last point of view, the only cipher zero, which excludes all numerical signification, is and signifies nothing. When we consider numbers as terms of comparison, zero is a term to which we may refer all the others as to any number. So it is evident that to say three units above zero, or two units below five, conveys the same conception of the number three. Nay, the term The term zero ^^^''^ is the central term between the ascending series of teim" of com" positive, and the descending series of negative num- parison. bers. This being admitted, we observe, moreover, that with regard to positive numbers, all agree that the greater among them is that which is farther from zero, the term below all positive num- bers ; but zero is in an equal manner above all negative numbers, and the more above, the more they increase in absolute value ; referring, therefore, to the same term, zero, negative numbers, we infer that among negative numbers relatively considered those are less that have a greater numerical value. Examples and § 23. From 6i .subtract 46 ; proMems. .^g .^^jjj jjg^^g ^jjg difference 56 — 46 = 6. l?rom 46 subtract 56 ; we will obtain 46 — 56 = — h. From 56 subtract — 46 ; we will have 56 -(- 46 = 96. From — 56 subtract — 46 ; we will have — 56 -f- 46 = — 6. From — 46 subtract — 56; we will have — 46 -|- 56 ^ 6. Ten men pull, with a rope, a heavy stone in a straight line from A towards B, and with a fores lOn. Seven more men pull the same stone in an opposite direction, namely, from B towards A, with a force 7re. What is the difference x of the action of these two forces ? 28 TREATISE ON ALGEBRA. Ans. It is plain that, considering the action which moves the weight towards B as positive, the opposite action must be negative. Hence, the two terms in question are -|- lOn — Tn. Now the second is to be subtracted from the first, therefore, X = 10» + 111 = 17ii. Numericai^iH Suppose )i = 10 pounds, we will have plications. ^ ^ ]^y() _ ^_ Suppose n = 15 pounds, we will have X = 255 . . .p. Four workmen cut each n pieces of timber, and three boys cut each r pieces. What is the differ- ence between the two numbers ? Ans. x=^in — 3)-. Numericaiap. Suppose n = bO,r = 30, then plications. ™ __ 11 A Suppose ji ^ 90, r ^ 40, then X = 240. Let us observe, that when we merely intend to take the difference between two numbers affected by the same sign, we only attend to the numerical value. ARTICLE n. Multiplication and Division. inwbatmu.- § ^4. MULTIPLICATION. — To multiply a mo- 8ists?''nomen- Domial a by another monomial 6, means to find a ciature. quantity p, whose numerical value is equal to the product of the numerical values of a and b. The monomial a is termed multiplicand, h multiplier, and hoth, /actors. The OPERATIONS ON MONOMIALS. 29 Various man- quantity p is termed product, and this product Sing'thepS is represented by the factors in any of the foUow- daot. jj^g manners : a .h, ay^h, ah, and each one of tbese expressions is read a multiplied by h, or simply ah. Deflnition of § 25. The definition and description of nnmerical mul- tiplkation. tiplioation is frequently given as follows : Multiplica- tion is the addition of the multiplicand repeated as many times as there are units in the multiplier. This definition (when we merely consider the absolute value of the product) is correct so far as the multiplier is a whole number ; but when it becomes a fractional one, that is, when the multiplier is a fraction of unity or even contains some units, but a fraction of unity besides, the given definition can- not then be rigorously applied. A definition which comprehends the object in its full extension, supposing, namely, the multiplicand A and the multiplier B to be any two numbers, is the following : To multiply A by B, is to derive from A through addition a number in the same man- ner as, through the addition of the same element, the number B is derived from positive unity. That is, the operation to which positive unity must be submitted in order to give through addition the number B, is the same operation to which A must be submitted to obtain the pro- duct of the numbers A and B. Now, B represents a rational number, (either whole or fractional,) or an irrational one. Let us examine each of these cases, and we will have a complete explanation of the last definition. Case of tha Suppose, first, B a whole and positive number. The Whole'number simple addition of unity repeated as many times as there and positive. are units in B, is the operation to be made about unity to derive from it B through addition. The multiplication, there- fore, of A by B, consists in this case in making the addition of A taken as many times as there are units in B, which accords exactly with the first definition. From this we derive a consequence concern- ing the sign which affects the product ; a consequence applicable also to the cases to be considered hereafter. Consequence Since positive unity, taken as it is, forms by repeated signoftte pK^ addition the positive multiplier B, so A, taken as it is dnot. given, and repeatedly added to itself, gives the product 3® 30 TEEATISE ON ALGEBRA. of A by B. Hence it follows, tliat -when A also is positive, the pro- duct is positive. But when the multiplier A is given negative, and B is still positive, then the product, being a sum of negative terms, is necessarily negative. Suppose now, B a whole negative number, then B cannot *be immediately obtained from positive unity, but we must The multiplier ^^^^ change its sign. But according to the definition, negative. jj, obtain the product of A by B, we ought to operate about A as about positive unity to obtain B. So in the case of B negative, the sign of the multiplicand A is to be changed ; then observ- ing how many units are in B, add A to itself, as in the preceding case, but with the sign changed, which, consequently, is the sign of the product. Therefore, when B is negative, and A also negative, the product is positive ; when B is negative but A positive, the product then is negative. Hence, the known rule, like signs give a positive product ; unlike signs, negative. Case of the When B is a fractional number, having, for example, n tional. ■ for its numerator, and m for denominator, in this case, to obtain B from unity, we must take, first, one m" part of unity and add it n times to itself, because in this way only, through the addi- tion of the same element, we can derive B from unity. Operating now upon A in like manner, we will have first - , which represents the «!" part of A ; taking then n times this element, which is expressed by A A placing the coefficint n before -, we will have the product n-, corres- n ponding to the factors A, - . •^ m Case of the In one of the following paragraphs we will dwell on tors. irrational numbers. For the present it is enough to •observe that they cannot be expressed like rational numbers, although -we may conceive a series of rationals, continually and indefinitely approaching to any irrational. Hence, whenever an irrational number is to be used for any purpose, we must necessarily make use of a rational near it. Therefore, in the case of irrational numbers, the multiplieation will be performed with rational numbers, and, conse- Kjuently, the foregoing remarks are applicable to this case also. Arithmetical § 26. Considering the numerical value and -rules applicahle , , , . . , . , 'to quantities. Sign 01 quantities, it IS plain that the same arithmetical rules are to be followed with regard to quantities OPERATIONS ON MONOMIALS. 31 for that which concerns the form and sign of the product. The rule of signs may be derived indeed from the definition. But since all agree in admitting that -(- a, multiplied by -(- h, gives a positive product, we may infer the same rule as fol- lows : Miituai influ- ^hc factors are mutually influenced in effecting ence of factors, jjjg product, and this influence is twofold : the one numerical, or of magnitude ; the other of sign. Suppose, now, the numerical value of the factors -{-a, -\-l> to remain un- varied, and change the sign of either of them ; this change must necessarily affect the product -\-p, and this cannot be done except by the change of the sign of the same product, and so admitting we must admit, also, the two following equations : + a X — S = —p ■> — OS X +'b = —p- Take again either of these two equations, for instance, the last, and change the sign of h; this will again produce an equivalent change in the product, and we will have — aX— 6 = +i5- Treating of the multiplication of polynomials, we will come to the same consequence by another process; meanwhile we may infer the general rule. The sign of the product is positive when both factors are affected with the same sign : it is nega- tive when the factors are affected with opposite signs. In the practical application of this rule we usually say, plus by plus, or minus by minus, give plus ; plus by minus, or minus by plus, give minus. Various forms §27. Thus far we have considered the factors mi*''Ta?ue™of ^^ ^^^^^ ^°^^ general acceptation, and only two. auantities. ]g^(, j.]jg numerical value, which is the one taken into account, specifies in some measure the quantities, because S2 TREATISE ON ALGEBRA. this value is either whole or fractional ; hence, four cases can take place with regard to the factors a and 6. We may first suppose the numerical values of both of them to be whole numbers ; secondly, both fractional ; thirdly, the numerical value of the multiplicand a whole number, and that of the multiplier fractioaal, and we may finally suppose the multi- plier a whole number, and the multiplicand fractional. The student being familiar with the numerical operations, it is not necessary here to dwell upon them : it will be profitable, however, to place before his eyes the general formulas of those which concern our present question, leaving, if necessary, to the teacher the care of making numerical substitutions. General foiv Suppose m, n, k, h to represent whole num- mulas of nu- '^^ J plication. bers, and —,,, fractional ODCS. With them we may m /c represent the above-mentioned cases ; and calling p the pro- duct of m by n, we will have m .n=p h n .h , h n h . n ■ and I k m.k km k. m h m .h . h h .m ; , n .h 1 -, n h . n - . ft = and A . — z=: . t. m mm From these formulas we will soon derive a general and use- ful consequence. Howthearith- ? ^^' ^^^ ^^ observe, meanwhile, that the arithmetical metical rules of rules expressed by the (f) cannot be arbitrary, and if multiplication . , , ., , , , . . . follow from the right, they must necessarily follow from the defimtion of 6 ni on. multiplication. Examining (25) the case of a fractional multiplier, we have touched this subject, which we will develop here. And, first, suppose - to be the multiplicand, and — the multiplier, which indicates that the n" part of unity has been taken m times. Hence, to obtain the product of - by — , according to the definition tc n OPERATIONS ON MONOMIALS. 33 A (25), the «"" part of - is first to be determined, and the same is then 7i A to he taken m times. Now, the re" part of r is .; — . Suppose, in /c k .n Mechanical arti- fact, a straight line divided into A equal parts, these fice, showing the , i aliquot part of a parts may repi-esent the h of the fraction -; suppose, besides, each one of these parts to be subdivided into n equal parts, (which is the same as dividing unity into n.k parts,) the h part of our line will then become h . ii, but each one of these is equal to the (i.n)" part of unity; therefore, the h.n parts of the h. n line wUl represent the fraction — — , but the same line represents also the fraction -; therefore, -- = r-^. Compare now together, the fractions = — and ^-^ ; the first is n times less than the second, but the k.n k.n second is equal to - ; therefore, ; — is n times less than 7, or which is K k.n k the same, - — is the n" part of -r. To complete now the multiplica- tion of r by — , the «"■ part of the first fraction is to be taken m times, k n which is evidently obtained by multiplying by m the numerator A of the ^ „ Am Am , , , ., ,, fraction ■= — . Hence, - . — = -j—, exactly as the rule prescribes ; the product of one fraction multiplied by another is equal to another frac- tion whose numerator is the product of the numerators of the factors, and the denominator is the product of the denominators of the same factors. Let us now come to the cases of the multiplicand whole number, and of the multiplier fractional, and vice verm. In the first of these . , , A m. A , . , two oases, reasoning as above, we will nave m . - = — j— ; and m the k k second, it is plain that — . A = -^. That is, the product of a whole m m number by a fraction, and vice versa, is obtained by multiplying the numerator of the fraction by the other factor. Themuitipii- § 29. The product J} of the first formula (f) come muiti- is given bv m units, repeated ra times. But to plier, and vice .. . . . , i 1 1 mrsa, without add m units n times, is tne same as to aao. n units affecting the -i 1. i • i? product. m times. We can see it by making use 01 a 34" TREATISE ON ALGEBRA, mechanical artifice. Range on a horizontal line a vow of in Case of the dots, and from each dot draw a vertical line; fectors : whole . , „ ,, i* i i ^ nombers. range again on each one oi these verticals n dots, commencing with that already marked on the horizontal line. In this manner we have m dots repeated n times, and conse- quently the whole number of dots is the product of m by n. But since, on each vertical line there are n dots, and these lines are m in number, we have also n dots repeated m times; that is, the product of re by m given by the same num- ber of dots ; hence, we may always write m .11 ^ n.m. other cases. Therefore we may write also n.h^h.n, and m .k = 11 .m, and consequently — '—r = t- — i hut from the , . . j.^ , n:h n h , h . n h n second of ( / ), we have ^ = — . 7, and -^ = - . — ; ^ m . K m le k.m, km, , . n h h n therefore, — . r = r . — . m k k m Reasoning ia the same manner, we deduce from the remain- ing formulas (/) that h h m .- = - . mr. k k General infe- Whatever be, therefore, the numerical value of braicai actors, the algebraical terms a and 6, we generally infer that a . b ^ b . a. The product § ^^- '^^^ Same inference may be applied to ft te°rm?Ts"the ^^J number of algebraical terms a, b, e, d, e . . . S'the^OTawTn Observe first, that if any number of quantities, tSs*are° S^ having all a whole numerical value, bring the same posed. product whatever be the order in which they are taken ; any number of algebraical quantities will bring the same product also when their numerical values are fractional or partly fractional, and partly whole numbers, whatever be the order in which they are taken, since the operation is always OPERATIONS ON MONOMIALS. S5 performed upon the whole numbers of numerators and denomi- nators. It is enough, therefore, to demonstrate here, that whatsoever he the order in which whole numbers, or quantities having whole numerical values, are taken, their product will be always the same. Let three such quantities be a, h, c. To multiply a by 6, is to take a as many times as there are units in the numerical value of b; that is, a -{- a -{- a -{- a -\- . Again, to multiply this product by c is to add the whole series of terms a-\- a-\- a -^ a repeated as many times as there are units in c. Now, h terms repeated c times give a number of them equal to the product J X c. To multiply, therefore, the product a . 6 by c, or a by the product b . c, gives the same result ; hence, generally, u, .by(_c ^ ay^b . c; and since a .b = b . a, 6 . c ^ c . &, so we will have also, b.ay_c^ay!,c.b^a.c'X,b, and, in like manner, a,c'Xb = c.ay(,b=c'X,a.b; and so on. So that we may evidently infer that three factors multiplied in any order whatsoever, give constantly the sam* product. Add now another factor, and make Pour actors. , , ^ a.b.c.a = i'. The first three may be changed at pleasure, and the factor d will always multiply the same quantity ; but calling p the partial product of the first two factors, the same product P can be represented also hj p .c.d, orhj p. d.c; that is, resolving again J) into its faciora, a.b.d.e = ^. But again, whatever be the order in which a, h, d are taken, tteir product will remain unvaried; the factor d, therefore, which was the last, can become the third, the second, and the first, while the other three factors may be arranged from the beginning in any manner whatsoever ; but this evidently em- 36 TREATISE ON ALGEBRA. braces all the possible cases of combinations of the four fac- tors ; therefore, the product P made by four factors will be always the same, whatever might be the order in which the factors are taken. We may reason iu the same manner when the factors become five, because the first four may be changed at pleasure ; considering then the first three as a single term, the place of the fourth may be changed with that of the fifth, which, together with the three preceeding, will always give the -|- . . . same product, whatever be the manner in which it is combined with the others; the same consequence, therefore, can be inferred with regard to five as with regard to four factors, and the same with regard to six, with regard to seven, and generally with regard to any number n of factors. Sign to be given § 31. It remains now for us to see what is the to the product of . , . - i i. i i several ikctois. Sign to be given to the product, when several terms are multiplied. The factors are either all positive or all negative, or partly positive and partly negative ; in the first case the product is evidently positive; in the second it is positive likewise, if the number of terms is even, because the first factor with the second make a positive product, which the third changes into another negative, and this, with the fourth factor, makes again another product positive ; and so on. If the negative factors are three, their ' product is negative; if four, positive; if five, negative; and hence, generally, when all the factors are negative, their pro^ duct is positive, when their number is even ; their product is negative, when their nunnber is odd. The same is to be said when only a portion of the factors is negatiTC ; that is, when the number of these factors is even, the total product is posi- tive ; when the number of negative factors is odd, the total product is negative. In fact, the first negative factor after some positive factors makes the whole product negative, and if other positive factors occur, the successive products will remain still negative ; but when another negative factor occurs, then OPERATIONS ON MONOMIALS. 37 the product becomes positive, and such will remain until the third negative factor comes; and therefore, the following rule will determine the sign to he given to the total product in all cases : The sign is negative, -whenever the number of negative factors is odd } otherwise, it is always positive. Prod-uctofthe § ^2. When different factors are given, it may Kime factors. Qccur that some of the factors are repeated : in this case, instead of writing, for example, a. a. a, we write a'; we write, namely, a only once, and above it the number of times the same quantity is taken as a factor, and this number is called exponent. Of such exponential quantities, xponen . ^^^ ^^ their reduction, we will speak more fully in its proper plaee in the next article ; however, we cannot omit here adding a few remarks concerning this subject, inasmuch as it is connected with simple multiplication. And first, if two or more exponential quantities, for example, 6* and i" are to be multiplied together, their product will be represented either by 6^.5% or by 6'; since the signification of these ex- pressions is the same, that is, the sum 9 of the partial expo- nents 5, 4, signifies that h is taken as a factor nine times in, both cases. Therefore, in cases similar to this, it is enough to write once the quantity, and give to it for- exponent the sum of the partial exponents. Yice versa, since the number 9 is equal to the sum of 5 and #, or 6 and 3, &c. We may, for the same reason, write P = 6' . 6* ^ 6^ js ^ . . . . ; and this also can be evidently applied to all similar cases. Observe also, that, since the order in which the factors are taken does not change the product, (30,) the products a.a.h.h and a.h.a.h are equal to one another. Now, a.a.h.h:^ a^V, and ah.ah = (aby-, therefore, a^y = (aby, and for the same reason, if any number of factors having the same ex- ponent are to be multiplied together, we may write once the 4 38 TREATISE ON ALGEBRA. product of the simple quantities, and apply to this product the common exponent; for instance, the product a^.i^.c' is equivalent to (a.b. c)=. Examples. Exampiesand § 33, (jiyen factors. Product. problems. " (1.) 3a, m.n, — Ig, — r -j- ^agrmn. (2.) 16m, — 12n, — i6, |c, — ^c?/) 3^, \k — mnbcgdfhli. (3.) 4a6, —2hd>,—mad, d?.. -\- 9,(a.hcd)^m,.d^. (4.) —lahc, — 3a&cc?=, \ahcd^, — labcdef*.. . . — ^^(ahcde/y. (5.) ahcde, — abed, + ahc, — ah, -{-a + a%Vd^e. (6.) a%, —hc^, + cd\—g'/K . + (ahy.(cdy.(g/y. (I.) ga?he, — b^d', + {a^^c', — iad^ + J^ad + (abcd)« . (abc)' . (ah) . a. (8.) 4a', — ba%, — 8a&», 2J^ . + 320(a.6)''. (9.) a^6, — aZ-", + c, — rf, -|- cd, — ahcd, -\- dc. . . — Qibcdy. (10.) 4a5n, — 166c, + ^m?, — mn-\- ^hrf, — bed . . — m*.}? .(?.a.n.r .f.d. A general, in order to exercise his soldiers, ranges them on a field before the eastle, and divides the whole army in two sections fronting each other, the one under the walls of the castle, the other opposite to it. During the exercise the general rides up and down between the opposite ranks, and when the exercise commences he rides, having the castle at his right hand, and he goes n times up in this man- ner, and returns n times to his former station. Each time the general rides from his first position to the second, g ranks of v men pass from the left to the right hand of the general, and r ranks, each containing p men, pass from the right tQ the left hand. When the general returns to his former station, each I passing on the castle side. OPERATIONS ON MONOMIALS. 39 time / ranks of s men pass from his left to his right hand, and q ranks of t men pass from his right to his left. We ask, first, how many ranks, and how many men go towards the castle : again, how many ranks and men march over to the other divisions from the castle during the n times ' that the general goes from his first station to the opposite side of the field ? Ans. Ranks. ..m.(/ Men n. Eanks...ra.r > . j. ^ o I soma; from the castle. Men n.r.p ) ^ ^ 2d. How many ranks and men go to and from the castle the Ji times the general returns to his former station ? Ans. Eanks...7i.o' ) ■ ^ ^, ^^ - ^ }■ going to the castle. Men n.q.ti ^ ^ Ranks. ..n/" ")•»., ^i „ y going from the castle. Men n.fs J ^ *= 3d. How many ranks and men go towards the castle, and how many go from the castle during the whole military exercise ? Ans. Ranks... wa 4- no 1 . j. ^t_ ,, !■ going to the castle. Men...,ngv + nqti ^ ^ Ranks...n,r 4-«/ ) . „ ,, ,, „ Y going from the castle. M.en....nrjp-{-nfs J 5 6 4th. What is the difference between the number of the ranks and men passing to the castle side, and that of the ranks and men passing to the opposite side ? Ans. Ranks. . . (ng -(- nq) — (nr -\- nf). Men... (ngv -|- nqt) — (nrp -f- nfs). Or, if (ng — nq) and (ngv -f- nqi) are the less numbers, Ranks. . .(nr -j- nf) — (ng -\- nq). Men (nrj) -|- n/s) — (ngv -\- nqt). Observe that, if we consider the passing from the left to the right hand of the general as positive, we 40 TREATISE ON ALGEBRA. must consider as negative the passing from the right to the lefc. But referring the movement of the ranks to the castle, then when the going towards it is considered as positive, the going from the castle must be considered as negative. Again, considering as positive the going of the general from his for- mer station to the opposite end of the field, we ought to con- sider as negative his I'eturning to the same station. Taking now the movements of the army first with reference to the castle, and then with regard to the general, and following the rules of signs, we will find in both cases the same resolution of the problem. A steamboat travels at the rate of n miles per hour. How many miles does the steamboat run over in m days, travelling 16 hours a day ? Ans. x= 16re.OT. Suppose n = 15, m ^= 12, then X = 2880. Suppose n = 10, m = 20, X = 3200, &c. § 34. Division. — To divide a quantity a by another quantity 6, means to find out such a quantity q, which, if multiplied by 6, ought to give a for product ; a is called dividend; b, divisor, and q, quotient. From the given definition it follows, that when the dividend is given, this is considered as the product of two factors, one of which is the divisor, and the object of the division is to find out the other factor. Algebraical The Operation of division is designated as follows : expressions of ^'^'"'- p or a : b, and each of these expressions is read a divided by a. The rule of signs for division must necessarily be the same as that for multiplication. Suppose, in fact, first a and b both positive, since the quotient q multi- OPERATIONS ON MONOMIALS. 41 plied by a positive quantity must give a positive product, q _ also, in this case, cannot be but positive, that is. Suppose both dividend and divisor affected by a negative sign ; in this case also the quotient must be positive. Because, — b multiplied by q ought to give a negative product, which cannot be obtained unless q is positive. Therefore, — a — 6 = +^- Suppose the dividend positive, and the divisor negative, in this case the quotient must be negative, because h, a negative quantity multiplied by q, ought to produce a positive, which cannot be obtained except with q negative ; hence, + a The last case is when 6 is positive and a negative, and in this case also, the quotient is negative ; because 6, a positive quantity multiplied by q, ought to give the negative a, which necessarily supposes q negative; hence, — a Tb = -'- We infer, therefore, the general The sign of the quotient is positive when both dividend and divisor are affected hy the same sign ; the sign of the quotient is negative when the dividend and divisor are affected hy different signs. Some, for brevity's sake, express this rule common to multi- plication and division, as follows : — Like signs produce plus, . . and unlike signs, minus. Various nu- § ^^- Observe here again that the numerical mencaivaiuefl. y^lues assignable* to the algebraical terms em- ployed in division may be either whole numbers or fractional, and consequently, the quotient -, numerically considered, em- 4* 42 TREATISE ON ALGEBRA. braces four cases corresponding to those already considered (27'/) for multiplication, and which we will represent here with corresponding formulas, and according to arithmetical rules : m m n' n m h n.k ' k m.h' m h mk 'k~ h' n ■.h- '^ , m' m,.h These rules for division are founded in the definition. In fact, if we multiply the quotient or the second member of each one of the preceding equations by the corresponding divisor, the product will result equal to the dividend. m m.ji n — . ra = ^ m .- = m, n n n n.k h n.k.h n.k.h n kh n m.h' k m.h.k m.k.h m' kh m' m.k h m.k.h kh h -1= h.k ^™-/..A = ™' n , n.h n h n m.h' m.h m' h m' Numerical § ^^- '^^^ numerical values of each eloment of a polml "rnono^ compouud monomial sometimes are given separately. minis. jjj (.jjjg gj^gg ^.{^g ultimate reduction for multipli- cation and division involves some complication. Let us here examine the case of monomials, having the fractional form, and suppose t and - to be such monomials. Representing by m, n, p, r, s, t, u, v whole numbers, let the numerical value of a be represented by — , and that of b by -, that of c by «puS'tion°"''" ~. and that of d by -. And to commence with OPERATIONS ON MONOMIALS. 43 the case of multiplication, let 7 be the multiplicand, and --, the multiplier. Now, m a n m.p m.r b p n ' r n.p' r s c t s . w s.v d u t ' V t.u V Therefore, a c m.r s.v mrsv b ' d n.p ' t.u nptu' mnv msrv ms.pu m s p u nptu ntpu nt' rv n t' r v consequently, a c ac b-d~M' ac . - = a.c: o.a=T-^; bd Tie product whatever be the numerical value of each ele- representea^'as ment a, b, c, d of the fractional expressions, or SerMvliiSI algebraical factors. Let us now come to the case of division, and let ■=- be the dividend, and -5 the divisor, we have, as above, a m.r c s.v Case of division. 7- = r , — := ~ — ; o n.p d t.u a.c m.r s.v mrtu therefore, - . - — : — = . : 6 d n.p t.u npsv , mrtu murt m,u ps m u p s , , ad but = = — r— =:— .-^;-. -=a.rt:6.c= ^-: npsv nvps nv rt n v r t be me quotient a c ad is obtained and hence — : ~ = * expressed as for ' b d ' bc ' i»rs. as for simple numerical division, whatever be the numerical values of each element of the monomial^ t and -,. 6 a 44 TREATISE ON ALGEBRA. Eeduction of § 37. We must here add some remarks con- results to a . . n i • i . stapler form. cemiDg the reduction oi the quotient, and in general the reduction of the result of any other operation to a simpler form. And first, any quantity or number multiplied by unity, gives a product equal to itself j for instance, as X 1 := a. Secondly, any quantity divided by itself, gives a quo- tient equal to unity. Because, the quotient, for example, of a : a must be such, that multiplied by a it gives a, which is none else except unity ; hence, it follows, that c a. - = a; c , , . a.c a c and, consequently, since 7 — ::=-.-, and a b a.c a c c c b.c h' c' ... , , a.c a a b we will have also, ^ — =-,-:- = «:&. b.c b c c Rule. That is, a quantity having the fractional form -j o remains unchanged, multiplying or dividing the numerator and denominator by any other quantity. Eeduction to Hcncc, any number of such algebraical quan- the same deuo- a C G minator. tjties 7, -^t J, ■ ■ ■ ■ may be easily reduced to the same denominator like numbers ; multiplying, namely, the numerator and denominator of each by the product of all the other denominators. So the above quantities, without being changed, can be expressed as follows : a.d.f c.b.f e.b.d hAj' dTTf jt:!' having all the same denominator. ApjUoabb to We may here observe also, that to obtain the addition and ' subtraction. sum or difference of algebraical quantities having a fractional form and the same denominator, it is enough to OPEEATIONS ON MONOMIALS, 45 tske the sum or difference of tlie numerators and divide it by the common denominator. Hence, if the quantities =-, -5 are to be added or subtracted from one another, we may first re- duce them to the same denominator, taking -j-^ for j, and -=^- for -, and in the case of addition we will have a.b a a c ad cb ad-\- cb _ I'^d'^M'^bd"" W ' in the case of subtraction, a c ad cb ad — cb ^ l~d^ bd~bd'^ bd~' exactly as for numbers. pSZ^"'°* §38. aiven. Answers. CI.) ab:m, — . (2.) abcm : m, abc. (3.) a''bc : ab, ac. ab c ' . ^ bed m^n?. (5.) mn : — , -r-j-. ^ -^ mn bed „ , m?h^ m?b^ ac (6.) (7-) (4.) aVc^ : M, ac ' a^c^ ' mb Zmpq 4a'b ia'b^ ' Zmi'p' \Qa%* (8.) mn-.-rr^, bed. ^ bed (9.) -fg-.TB, rs ,,., „, mb^ Zma (10.) m%--^, -^. ia , 79*m%d (11.) _^s^«:-^^, +-4^- 46 TREATISE ON ALGEBRA (12.) (IS.) (14.) (15.) 4bdr _ ijnq^ ibH'^r^ 3iV _ mdh'' Qb^cqr* A certain number of balls is taken n times from an urn. At the end, the amount of the balls extracted is found to be m. How many balls where taken each time ? Ans. X = — . n Suppose m = 50, n = 10, then a; = 5. Suppose m^=56, n = S, then a; = 7, &c. Three messengers A, B, C leave at once the same Problem 2. , . , , t . /..it. town. A arrives at a distance oi n miles, B at a distance of r miles, and C at a distance of s miles, at the end of the same number g of days, travelling each one of them an equal number of miles every day. How many miles did A travel each day ? how many B ? how many C ? Ans. A travelled a; = - miles. 9 B " 9 C " x" =i « Suppose g = 15 and n = 450, m = 630, s = 420, then X = 30, x' ^ 42, a/' = 28. OPERATIONS ON MONOMIALS. 47 Accents or When syiubols are used to represent similar dashes : their ... i j? i • i_ meaning. or anslogous quantities, instead of changing the symbol, we make use sometimes of the same symbol with one or more accents above it, as in the preceding example, and such symbols are read x prime, x second, &c. ARTICLE III. Formation of Powers and Extraction of Eqpts, What a cer- § 39. PowERS. — Let m be a whole number, tain power of a i i • • m • i quantity is. and a any algebraic quantity. Tp raise a to tne power m, or to form the jb"" power of a, means to multiply a Root, expo- by itself m times. In this operation a is termed nent, degree, . ^ , power. root, m the exponent or degree, or index; the povxr, (which may be called jp) is expressed by o", and this expression is read a to the m* power, or simply a to m. The operation is the same as for numbers. Eornmtion of i ^^' Numerical relations are in all oases applicable powers, enibra- to quantities, since, as we haTe frequently observed, of numerical whenever algebraical quantities are submitted to any scrintlon^' and Operation or comparison, their numerical value is taken definition. ii^o account. Let I, n, p represent three numbers, and suppose ^ to be obtained from I through multiplication, as n is derived from unity through addition. Then p is the ra** power of I, and the expression of th« power being Z" , we have P- z= p. To raise, therefore, the root I to the power n, is to find out the number p. Now n, a rational number, and if irrational, represented by a rational one, is either a whole Case of the in- number or a fraction. Suppose, first, «» to be a whole ^ex w e nam- ^^jjjjjgj,^ j^. jg ig^^^ derived from unity through addition by taking 1 + 1 + 1 + 1 + B times ; therefore, the power p or Z" is given by the product where I is taken n times, and this is the case considered in the pre- ceding number. 48 TREATTSE ON ALGEBRA. Case of the in- Suppose, now, n to be a fraction having r for nume- dex fractional, rator, and s for denominator. To derive from unity T throTigh addition - , it is necessary first to divide unity into a parts, and take r times one of these parts, namely, r To obtain, therefore, p or Z* it is first necessary to determine a quantity a, which, if multiplied s times by itself would give I in the same manner as — added » times to itself ^ves 1. Then, since to 7-1 T.. obtain - Wfe take - r times, so the power Z» will be equal to ^X^X^XaX where a. is taken r times. And, consequently, r I' = i'. Hence, we derive the following definition: — The power Blofinition. , . .,,,, , of any given quantity a w the product of fabtors equal ia the same quantity a, or the product of factors equal to such an element, which multiplied by itself a certain number of times produces the given Apparent quantity a. From this definition it follows, that when the powers. index n of the power V is either equal to I, or equal to 1 - , the power t3ien is merely apparent or nominal. Because, in the first case the root I is not moiltiplied by itself, but simply taken as it is ; and so, likewise, the element », in the second case is not multiplied by itself. Therefore, according to the given dcfimtiwi, neither of the expressions L /i = i, Z" = «, CulDB and is a power. However, as we term P the third power or nal powers. cube of I, and P the second power or squa/re of I; so hy analogy we term P the first power of I, and U the ( - j power of I, which last is so far in reality from being a power of I, that on the contrary i is a power of x, as we have seen, and we will better' see hsreafter. Powersof ? ^^- ^® signification of unity is either collective or ""''y- simple. In the first case 1 is, like any other root, capable of being raised to any power n, and consequently susceptible of cor- responding modifications. In the second case 1" is again an apparent power, since simple unity is incapable of being raised to any power OPERATIONS ON MOKOMIALS. 49 wkatsoever, or in other words, simple unity cannot be affected by any Casaofsimple index, either wliole number or a fraction. Suppose in ""ity- fact, the index n a whole number, wo will have then 1» = 1 X 1 X 1 X but 1, multiplied by simple unity is equal to 1, and, consequently, 1" = 1. Suppose n a, fraction, we must first find the flement a., which multiplied a certain number of times by itself gives 1, but this element cannot be any except simple unity ; therefore, a. in this case is equal to 1. But in the present supposition, the power 1" is obtained by repeatedly multiplying a. by itself; therefore, in this case also, 1" = 1. Simple unity, therefore, remains unchanged when raised to any power. Collective •'^^^ when unity has u, collective meaning, then 1" or unity. more simply 1^, 1^, . . . have by no means the same value and signification as 1. However, even in this case we write 1^ = 1, 1^ = 1, 1» = 1, not being able to express these units of various orders with other symbols. But, whenever in mathematical investiga- tions such units of various degrees occur, we take notice of their different meaning, or the exponent is left to indicate the order. Product of § 42. Let now m t represent two wtole num- Cexpo^entf bers. We have seen (32) already that the pro- apSie(j"to''the ^^'^^ oi m -\- 1 factors, equal to a, can be expressed same root. ^,y gn_gt^ ^s wsU as by «"+'; nay, -whatever be the number of such exponents m, t, r, s, . . ., we will always evidently have a™, a', a^ a*. .. = »" + ' + ' + '+•■'. * Fractional ex- I^ ^^ exponents become fractional, (and supposing ponents. them reduced to the same denominator,) we will have, likewise, ■ H L L 2^.L^!i4.... a'.a'.a'... = a' ' ' Because, for each factor of the first number of this equation, we must first (40) find out a number which multiplied s times by itself gives a. Let this number be v. We will have — t. ///s:=; V™, a^ ^ V*, "^ = v% . . . and, consequently, m - - 5 50 TEEATISE ON ALGEBRA. T> X T S,5.,!L,. .. "+'+'•+■ ■ iJut also, «"•"»"'"«+ , __,"•+<+'■+...; mil — _L-j.-j.. . . therefore, i , , ."'"«+«''■ ' a . a . a . . . =a General infer- Hence, TTB generally infer the folio-wing proposition ence and rule, andrule : The pAduct of any number of powers of the same root, is this very root raised to a power equal to the sum of all the partial exponents, whether whole numbers or fractions. rroduot of § 43. Let us resume again the wliole number powers : ease of l e ±t l 1 the same ex- in as common exponent oi the roots a, b, c, ... Tamber '"'a"^ The powers a", Jr, C, . . . multiplied together, pifed tovarious ^ jjj iQ^nifestly give a'\ 6™. c" . . . = (a . 6 . c . . )», since in both cases the same number of factors equal to a, equal to h, &o., are multiplied. But if the exponent becomes fractional, for instance. Fractional ex- ponent. _ . g,jjj^ consequently, the preceding powers are changed into a', b', c', . . . . Let, as before, v represent the numerical value, which multiplied s times by itself gives a, and let y, i, . . . \>e the numerical values or numbers, which multiplied, each « times by itself, give i, c, . . . , we will have m m m a' = v", 4' = }•", c' = iP and consequently, m 7>i m a' :b' .c' == v". }-" . r =(v.y.S. .)". Now, since s factors equal to v . > . J . . . . give a . b. c... the last num- ber of this equation represents the power (a. b . c . . .)'. Therefore, fn m TO m a' .b' .c' = {a.b.c.y. Generalinfet- Hence, we generally infer, that ence and role. j^g produet of any number of powers of the same de- gree, either viliole or fractional, is equal to the product of the roots raised to the same power. Powers of pow- § ^^- ^S* '^"^ *^ P°^8'^ ""* ^^ ^^^^^^ ^^ ^^^ e"'nentswhrfe'"°°* °^ another power of the degree h. Sup- numbers. posing h a whole number, (a")* signifies that the OPEKATIONS ON MONOMIALS. 51 product of m factors equal to a is multiplied h times by itself. But this comes to the same as to take the product of m.Ji factors equal to a, which product is expressed by the power a"*-*; hence, (a")" ^ a" ■ ". Fractional ex- '^^^ Same multiplication of exponents wUl take place pononis. when tliey become fractional numbers. Suppose, in fact, m to be changed into — , and h into — , or the power a« raised to h — the power -. And making again, as in preceding numbers «« = v"*. T we will have , m. ft ft UV' =(»"•)'.... (o). Let now /3 be another number which multiplied r times by itself gives p ; in this supposition, we have and, consequently, »"'= /S'-™ = ,&'"■'■ = {^Y . From this we infer that ,2"* is a, number which multiplied r times by itself, gives v". Resume now, again, the second member of the equation (o). To ft obtain the power expressed by the monomial (v"')'', it is enough to raise to the power h, the number which multiplied r times by itself, gives v", but this number is ,8", therefore, ft (>'»)'■= (yg™)»= ^".ft; m but 1'"= a"; hence, C"^)'=^"'-" (o,) Now, since 0'' is equal to v, we wUl have, also but V is that nrunber, wMcK multiplied s times by itself, gives a ; therefore, 0^-' = a; that is, ^ is such a number, which multiplied r. s times by itself, m. ft gives a. Hence, the power a* ' '' is obtained by raising ^ to the power m . h, that is, ro^ m . ft ^m.ft^^..r ^^. r_ Substituting, finally, this value in the second number of (0,) we have Qf m h mh a = a I 52 TREATISE ON ALGEBRA. General infei^ ^^ ^°^' ^^^ "'"^^ °^ exponents, whole numbers. Hence, ence and rule, generally, The power of a power of any root is equal to the same root, having for exponent the product of the given separate exponents. SimpiiScation § "^5. We Lave seen already (37) how a quo- of exponential j^^jj); qj fjaotioD may be reduced to a simpler tients. form \<\ the elimination of common factors. Let us see here how in the same fraction we may change, in all cases, cither the numerator or denominator into unity. Let us commence with the most simple case First case. Kx- ponents whole of thswhole numbers m and n, and the first numbers, and „, the numerator" sxGa,\AT than the sccond. The fraction — may be a" easily transformed in this case to the form of a whole quan- tity, because, calling d, the difference m — n, we have «■*=;«"»-»', 'but (42), «"■== «<"*-'". a" 3 hence, o^^a^.a", — = = a* = a*"-"' . a" a" a" Second case. .Q^^l-ll^r^QO nnnr flio n-iTron fvn/if-.inn fn \\a Tr(i Hi|hw *de^™ Suppose now, the given fraction to be in the deuomz- nator. wjll have then a" a" a" or a'a" a" a"'-" Similar modifications will take place with fractional ■ Third case. 7^ n Fractional ex- exponents. Suppose, in fact, — and - to be the expo- ponents, and S S the higher in ra n /7 /-»• 5^ - - the numerator, j^^j^jg^ then- — - = - , and aC " V. a' = a' ; there- 5 S S " m ^ H fore, a* = a*, a", and 3 6 8 ^ /"* "\ a a' . a " (.IT) « 5 a' a' n Fourth ease. a* Higher degree If the given fraction is — , then we have in the denomi- — nator. a* OPERATIONS ON MONOMIALS. 53 m d n d m n a' a' . a' a' a' '. rifth case. Following the analogy of tlie preceding eases, Equal expo- ^w nentB. the algebraic fraction — , will give either a"*"", 1 a™ 1 or ; but — =; 1 ; therefore, a"-" = „ „ ^= 1, or a° — 1. Ne»-atiTe en- Extending still farther the analogy, and sup- poiients. . . , . O" a™ ^ , posing m the quotients -^, — m>TO, as above; as from the latter, we infer -^ = a""" = a'', so we represent a" the former — bv a""" = a~''. And likewise the former ra" 1 1 fraction gives — = = —7, so bv analogy, we write -;^ ^ -^^. Therefore the quotient — is represented at 1 a" 1 once by a^, and by -^^, and the quotient -^ by -^, and by a~'', or the expressions are considered as synonyms. It is not necessary to extend here the same ohservations to the case of fractional exponents, to which they conld be as easily applied, as it is evident. From the said convention it follows, first, that Inferences. o^.a-" = 1. Secondly, since the difference d between any two numbers m, n, has the same absolute value, but different signs, accord- 54 TREATISE ON ALGEBRA. ing as m is subtracted from n, or vice versa, it follows, tliat in all cases we may write : a" 1 — = a*""" = . a" a"-" § 46. Write in the upper or lower line all the terms of the given fractions. (!•) ^3' Answers: a^hc'-'d-^,or ^^j^,^,. a^-y-'d--', or V"-J f^d"' (3.) d^e^' (-t-) a'^—'.d" ffy' ' ff,^ d-^e-" a'"c''d-^e-% or ar^l^j^d" 1 1 a"- '.cZ^£r-ys or;^ Write the following expressions all with positive exponents, and reduce the exponential quantities according to the pre- ceding rules : Given. Answers. a .0 ^ a-.J-.c'. a-) a-^b-' 1 a'"b" ' a'"+Kb"+'' ^"^ a-^-"' d" (9.) (10.) (11.) a'^b-'d? {a.gfd^ (12.) (13.) (14.) OPEEATIONS ON MONOMIALS. 55 Given, Answers. jj(»— m)_ f^—ln + 2m) ^ g— (n— m)^ — (ni — m)' c" . d'^ ' o-*6.c^''.f-'.^- ('■-'"> (a.&)^(Z'" &—.y. (/'■)-' ' p'+' Ojii ^—(n—m) ^m 7,— (n— 171) (15.) _ ^_,_;^_„^_,,+.„ (a.6)-./- There is also another modification of algebraic expressions of this kind. Observe that the power of any fractional mo- nomial, for instance, (- ) is equal to the power m of the numerator, divided by the power m of the denominator ; since z- . J. ... is equal to the product of a taken m times, divided by the product of b taken also m times. The expression then (-) may be transformed into 5—, or vice versa. So, for example, we may write /2\" KS:!^ _ «" ^ ('\' &, The extrae- § 47. EXTRACTION OF KOOTS. — To extract the tiou of roots 18 . . „ , , the inverse ope- root 01 a quantity ct IS to find out another quan- ration of rais- , i - t • i ing to power, tity ?', whioh raised to a certain power, gives a. Hence, the given quantity a is considered as the power of another quantity r, raised to the exponent indicated by the root ; and since the extraction of the root consists in finding this r, th§ operation, therefore, is the inverse of raising to power. 56 TREATISE ON ALGEBRA. convenUoDai Thg manner in wliich this operation is indicated Figns and no- *■ menciature. jg \,y the expression 1^' . . which is read m"" root of a, and with generical terms radical expression, or simply radical. So, likewise, ^/~ is called radical siijn, m index or degree of the root, which is the ex- ponent to be given to the unknown quantity r or root, to snuare and obtain o. If m is equal to 2, the root is called cubical roots, square, and in this case the index is omitted, so that the expression -j/a without any number in the radical sign, signifies square root of a. When m is equal to 3, the root is termed cubical, -[/m, is also represented by a", and in f/a" by a",* fractional powers on which we may operate, as on whole powers."!" The root of a 2 48. Suppose m n to be whole uumliers, we say that quantify may m be always re- „ , — . „ . i iT ■¥> i i presented by a [/ a" tho n"' root of av la equal to a - Because, let n''ent'°gWen''''to » be the numerical value which multipUed « times by the same quan- itself, rives' a, we have tity. ' ° m a" = a» ; and, consequently, (*">)» = Vo"/ = a". Therefore, a" is that numerical value, which raised to the power n, m gives a" ; hence, a" is the «"■ root of a", but a.'" = a", therefore, m 1^ I'/a"' = a» = (a")" ; and supposing m = 1, _ i ^/a = ffi»; that is, the re" root of a, is a raised to the power -, and vice versa, (40.) Case of the Observe, also, that fractional index. *L * See the following number, t?? 42, 4.3, 44, 45. OPERATIONS ON MONOMIALS. 57 A radical hav- consequently, ing a fracLiouiil ,j_ index can he ^ i transformed iu- — ^ I 2* to onehayinga 'JV iit" ^^^ "V [a."')'" ^= et"^ * whole numljei' ^ V / » for iadux. vi VI bat oL*^ = a and ct"* = a" ; again, a" ^ [^ 'i'^ so from tlie last equation, we infer (n\' """- a" = ya- — 1 root of a is equal to the ( — J power of the same «, and equal to the n*"^ root of a^. I V7. [A', f/^. . . = ^SIEE^/^ Products of 2 49. Hence, (42, 43,) roots. ' . ' V ' >y = ^ m + e+r+. . Pirst case. Because, Different roots i. « ^ 5i L *L of the same ,„/— (/— r/— „s „« „• quantity. |/a.j/a.(/a.... = a.« .a ... =+'-+- + •• Wt + f + ^-l = a ' , And, since fractions may be always reduced to the same numerator, no less than to the same denominator, similar reductions can be per- formed in all oases ; and, supposing in the preceding formulas s = 2, ■we will have g , In the case 6t equal roots of diflFerent quantities, we Equal roots of will have different quan- l/a.'i/''b.yc...='i/a.b.c... Because, = |^(a. b . e . . .)'" g _s ^ m Trl m l/a. yT. 1/7. .. = a' .b' .c'... m = {a.b.c...y; 58 TREATISE ON ALGEBRA, and, supposing tb = 1. {/a . \/b . {/c. . . = \/a .b.c... Root of a root. Let now the root of a root be given, for instance, ^/t/^, wo- will find I h/ — mft/ — (|/a = -j/a; B 8_ because, (44) "/i _ "j^ /is™ /». "V i/o = \a'' = (,»'■;• = a". Supposing the denominators m and h equal to unity, then T- - The index of Observe here also, that, since the root, and . that of the ^ Vh m m^ quantity under |/a™ = a« and a" = a"** = ""ya*"* , the radical sign, may be nmiti- SO we nave also pUed or divided by the same iV^a"* — ^ "iVa*^ ' * ■ quantity. y y ' n , 5* / ™\* ft „ again, ^" _ (^/ j« _ -^f^ir; hence, n Four cases em- §50. Let US now analyze the formula, braced by a radi- cal expression. i/a = fit. Four cases may occur about this formula. First, n (which is sup- posed to be a whole number) is either An even number = 2i, or an odd number ^ 2i-{-l: again, with n = 2i, we can suppose a posi- tive or a negative, and the same supposition can be made when n = 2i-\- 1. Let us commence to examine the ease of a First case. . ,— positive, and n = 21 Since the formula |/a = a, sup- poses a." = fl, we will have a.^= a. But -j- Aj as well as — a, raised at the power 2?, give the same positive product ; hence, (+«)« = (_«)« = «. OPERATIONS ON MONOMIALS. 59 Therefore, in the first case, we have \/ -\-a = ±a; that is, the [2if' root of a is either positive or negative. Let again a bo positive, but the index n == 28+ 1, or let becoiid cnse. ° ^ ^+\/jf- » = «, we will have (aj^'+i = + a, but a in this case must be necessarily positive, because, 2i-\-\, factors of the same qviantity, cannot give a positive product unless the same quantity is positive; hence, (-)-a)^+' = a, Let now a he negative, and the index » still equal to Third rasa. 2i-|-l. Since a negative produet of the same quantity multiplied 2i+ 1 times, can be obtained only by a negative quantity, BO we will have, in this case and ='+,/=^=-«. The last case is that of a negative when n^li and Fourth case. nt / expressed by j/ — a. But neither among positive nor negative numbers or quantities, any one is to be found representing this root,, because we have seen in the first case, that with a either positive or negative, we have always a^ = 4- " ; hence the radical expression ^/_a. Imaginary radi- ^^ termed imaginary root, or radical or imaginary cals or roots. quantity or expression. ^ Although such expressions, considered in one point of view, are paradoxical, yet they may he also considered as symbols of terms heterogeneous to real quantities, that is, ^ — a is the symbol of a term not included in the category of real quantities, which term being incapable of taking a, real form, cannot be represented but by an ambiguous or paradoxical form. These mysterious symbols, so frequently met with in mathematical investigations, are used and profitably managed like real quantities. Operations on 2 51. Let us then see here some of the elementary imaginary (quan- tities, operations about such quantities, and first, let us take Multiplication tte product of ^ — a by ^^^. Since ■ — a and — 6 and division. are equivalent to a X — l, o X — Ij and ^ — a, y^-— o are equivalent to — o*, — h~, so we will have 60 TREATISE ON ALGEBRA. v/~« = (a X — 1)* = "■■ - 1* = n/^- x/=^. ^=1 = (J X - 1)* = 6*- — 1* = n/5' v/=^- Therefore, ^/^^ v'^^^ = \/^ \/^i\^—^T > but ^a . v^= v^'^' (\/— '!)' = — 1 ; 'ie'i'=o y/ — a . y/ — b = — ,yah. In like maimer, we •wUl have — Oy/ — 1 . JyA^l = a.b; because — a . 6 = — ab and (\/— 1)' = — 1- Again, the ratio a^ — 1 : b^ — 1, gives a^ — 1 a = i' Powers, Since (^^Zl).= _l, we will have, also (v/— 1)* =(-!)' = +1, and so on. And generally, {x/~^f = (v/'FT)^)* = (-1)' = ±1, in which the sign is positive when i is an even number. In an equal manner, we will have where, likewise, the positive sign is to be taken with i an even number. Because, (.^^^1 )(*"+« = (^=^1)^. y/—l ; hence, A corollary follows from this doctrine, applicable to the case of real quantities. We have seen (50) that the n"' real root of a is either double or only one or none ; but if together with the real, we reckon , also the imaginary roots of the same quantity, their number is quite different. So, for example, (+2)', (-2)*, (+2^-1)', (-2v'■=^)^ OPERATIONS ON MONOMIALS. 61 give all + 16 ; hence, the fourth root of 16 admits four different values, two real, and two imaginary — ^namely, ^16 = ±2, Irrational radi- g 52. It now remains for us to add some remarks Utis."" ''"™' concerning irrational radicals. These remarks are con- nected with other questions somewhat foreign to the subject of the preceding numbers. For this reason they have been left for the last of the present article. We will commence by illustrating the method of finding the greatest common measure of two numbers. Greatest com- Suppose M and N to be two whole numbers, men measure. ^^ numerical values of two quantities, and M > N. Let the whole number g' be the quotient of M divided by N and K' the remainder; or, in other terms, let M6 be equal to g' times N pins K', which is less than N. Dividing now N by R', let q" be the quotient and E." the remainder ; dividing in the next place, R' by R'", let q'" be the quotient, and R'" the remainder, &c., in this manner, we have the equations M = 2' N + R' N = 2" R' + R" R' = 5"'R" + R'" \- (0.,) R"=2"R"' + R" &c. Now, if R', for example, would be found equal to zero, namely, if R" divides exactly R'" without any remainder, we say first, that the same R'' is an exact divisor of M and N ; and secondly, this divisor is the greatest common divisor or measure of M and N. The first assertion is easily demon- strated, observing that R" cannot exactly divide R'", without dividing also R", because, R" = j^'R'" -j- R'' ; but by suppo- sition, R'" = 2'R" ; therefore, R" = j". g'. R" + R" ; that is to say, R" is equal to o'". g' + 1 times R", and for brevity's R" sake, calling Q the sum, j". g' + 1 : R" = QR", or =^ = Q. e 62 TREATISE ON ALGEBRA. But, again, W^ cannot be an exact divisor of E." without dividing also R'. This remainder is equal to g"'R" -\- R'", but by supposition, R'" = g'R'", and consequently, R" ^ QR'" ; hence, R' = q'" QR'' + j'R'" ; that is to say, R' is equal to 2'". Q-f"2' times R% and for brevity's sake, calling Q' the R' sum q"'Q + g', R' = Q'R", or =55; = Q'. Following the same process, we find that N and .M are, likewise, multiples of the ^ same number R'', which, consequently, is an exact divisor of them. But the same remainder is the greatest common divisor of the same number; because, no number — K for instance — greater than the last said remainder, can divide both numbers M and N. To demonstrate this, observe, first, that subtracting from both members of the preceding equations, the same number, namely, q'N, from the members of the first, g"R'; from those of the second, and so on, we easily deduce (16,) the following equations : R' =M— 5'N R" = N — 2"R' R"'=R'— 2"'R" I. (0,.) -2" &c. R"=R" — o"R"' Making now the supposition that K divides exactly M and N, we must admit that it divides also R'. Because M, for example, will contain exactly K two or three, or r times, and N will exactly contain the same K, for instance, s times ; then M = rK, N = sK, and M — q'N^rK — q'sK, but from the last equations, M — g'N = R' ; hence, R' = rK — g'sK ; that is to say, R' contains (r — q's) times K, or R' is an exact multiple of R'. Reasoning in the same manner, since N and R' are multiples of K, it follows from the second equation, that K must be an exact divisor of R" also, and for the same reason an exact divisor of R'" and R'^; but K is greater than OPERATIONS ON MONOMIALS. 63 R'', and consequently, R'' cannot be divided by K ; hence, the supposition of K a common measure of M and N, and greater than the last remainder cannot be admitted ; and ws generally infer, that when two numbers M and N are given, , the liist remainder found in the above-described process ia their greatest common measure. Examples. Let M ^ 189, N =". 147, we will have Namely, M, and divide N by M, and again, M by the first remainder, and this by the second, and so on ; call ff, g", cf", .... the quotients, and /, p", p'" , .... the remainder, the last of which must be as in the preceding case, equal to 1, on account of N being a prime number. With these elements we can easily form equations similar OPERATIONS ON MONOMIALS. 65 to the preceding, and those corresponding to the equations (Oj), will be Vp' = NP — (/'MP ^ P^" = MP_(;yP I p/'==/p— yyp [<-°^--' &c. J Now N, which certainly divides exactly NP, divides by supposition, accurately, MP ; therefore, the second member of the first (Og) is exactly divisible by N ; hence, also, its equiva- lent Vp'. Rgasoning as in the preceding case, we will find all the following products, Vp", P/)'", exactly divisible by N ; but the last p is equal to 1, and consequently the last of those products exactly divisible by N is P. This factor, therefore, is exactly divisible by N, when the other factor M, either greater or less than N, is not divisible by the same prime numbers. It is plain that in the same manner in which we have demonstrated that P is exactly divisible by N, when the other factor M is not divisible by the same N; so we could demonstrate that when P is not divisible by N, then 51 is certainly divisible by it; and we can generally infer, there- fore, that when the product PM is accurately divisible by N, a prime number, and one of the factors is not divisible by it, the other is necessarily divisible by the same number N. Wo may observe, that even in the case in which N is not a prime number absolutely, but prime only to M; and N divides exactly the product MP, P is exactly divisible by N. Which thing is proved in the same manner as the preceding theorem. Aprimenum- CoTollary. — We can now easily infer this product, divides coroUary : When a prime number N divides the factorr^ ° cxactly the product Q = K . M . K . H . . , the same must necessarily divide at least one of the factors. Call P' the partial product M.K.H. . ., and P" the partial product K.H. . . ; P"' the product H. . . ., and so on, we will have the given 66 TREATISE ON ALGEBRA. Q =E.F P' = MP" P" = KF" P"'= HP^'&c. w Now R, and the first factor of each one of the partial pro- ducts, is one of the factors of our product Q = R . M . K . H . . . ., the number of which is filled by the last of those partial products which contains the two last factors of Q. Suppose now that none of the first factors R, M, K, .... is exactly divisible by N ; it follows first, that P' is divisible by N, and consequently the second product MP";" but M by supposition is not divisible by N, therefore the factor P" must be a multiple of N ; and consequently the product KP'", and the factor P'" of this" product, and so on, till the last factor of the last product, which is one of the factors of Q. If none, therefore, of the factors R . M . K .... of the given product Q, until the last, is divisible by N, the last must certainly be divisible by it. Vice versa, if none of the factors of Q is divisible by the prime number N, neither Q can be divisible by it. It follows, besides, that if M or K are not divisible by a prime number N, neither M" and K" are divisible by it ; because n representing a whole number, M" and K" are the products of n factors, none of which is exactly divisible, by N. Powers of frac- § ^^- Let US now take the fractional expression tional expres- Q, sions. - reduced to its simplest terms : that is, to such simple elements a and 6, as to admit no common divisor except unity. We say that the numerical values of the powers ^r-, T-, ■ ■ ■ T- arc necessarily fractional numbers, irreducible Irreducible to *° ^ simpler form : that is to say, admitting no a simple form, common measure except unity. Suppose, in fact, that a" and 6" are exactly divisible by a number greater OPERATIONS ON MONOMIALS. 67 than 1 ; in this case a also and 6 must have a common measure above unity, ■which is against the present supposition. Before we demonstrate this, let us observe first, that whole numbers are eitker prime numbers or products of prime numbers ; con- sequently we cannot suppose a number divisible by another, which is not prime, without supposing that same number divisible also by some prime number. Because, let the whole number N be divisible by the other whole number N P = m . ?i : that is, let the quotient y^ ^ Q be a whole num- ber; since from this equation we deduce the other Q.P = N, N N or Q.m.ra = N, we will have also, — ^Q.m — = Q.m; TO n that is, N is exactly divisible by the factors of P. a" Let us now resume our fraction — , the supposed common measure is either a prime number or not; in both oases we must admit that some prime number is common divisor to both a" and &"; but we have already seen that a" or i" cannot be divisible by any prime number, unless a and 6 are divisible by the 'same number. Therefore, when the fractional ex- pression 7 is reduced to its simplest terms, any power — of the same fraction is another fractional expression, whose terms cannot be reduced to a simpler form, and consequently it is essentially fractional. § 55. We are now able to see how, among Irrational roots. , t i ... , the radical quantities, there are some whose numerical vaiues can never be exactly assigned, and are, con- sequently, to, be reckoned among the irrational expressions. Powers of the Observe, that taking the squares, the cubes, natural num- , i p . i i /. , , . i . bei-s. the fourth powers, and so on, of the natural series 1, 2, 3, 4, 5 of numbers, we have 68 TREATISE ON ALGEBRA. Squares 1, 4, 9, 16, .... Cubes 1, 8,27, 64, .... Fourth powers. 1, 16, 71, 256, , &o. That is, the square root of 1 is 1, and the square r(^t of 4 is 2 ; hence, the square roots of the numbers 2, 3 are between 1 and 2, or they are numbers of an essentially fractional form, however reduced to their lowest terms ; but we have seen that a fractional expression raised to any power gives constantly a fractional result; no number, therefore, between 1 and 2, if squared, can give for power either the whole numbers 2 or 3 ; therefore the numbers 2 and 3 have neither their square roots among whole numbers nor among numbers of fractional form; these roots, therefore, cannot be exactly expressed by any number either whole or fractional, although we may represent them by numbers more and more approaching to their value; the same roots, therefore, are irrational quantities. The same is to be said of the square roots of the numbers between 4 and 9, between 9 and 16, &c. ; the same of the cubical roots of the numbers between 1 and 8, between 8 and 27, &o. ; the same of the fourth roots of the numbers between 1 and 16, between 16 and 71, and so on. And generally, we infer, that %1'hra ifhoJc numbers have not their roots amony other whole numbers, neither can they have them among fractional ones. A series of 2 56. Let us see now how an indefinite series of rational or as- signable num- rational numbers can be conceived constantly approacn- conceifr?d^ a^ i°g *<> any irrational root . proachiriK con- Divide unity into any number of small equal parts, stantly to any j j -a f > ifiatidnal root, "wliicli we will call a. It is plain, first, that a is a frac- tion greater or smaller, according to the less or greq,ter number of parts into which unity is divided ; secondly, not only a, but 2a, 3m, io!, . . . . are all fractions until we take the whole number of them ; thirdly, the diiFerence between two successive terms of the series m, 2ai, ?a>, . . . . is smaller in proportion to the greater number of parts into which unity has been d'mdcJ ; fourthly, representing by n any number the diiFerence between n-\- u and n -\- 2ai, is the same as that OPERATIONS ON MONOMIALS. 69 between the simple terms a and Sa. Finally, if n represents a whole number, n-\- m, n-\- 2ai, n -f- 3ai, and so on, are all fractional numbers comprised between n and n-\-\, until we add to n all tlio parts a into which unity has been divided. Each one, therefore, of the numbers n ■\- x, n ■\- 1m, n -\- Zx, reduced to its lowest terms, must have the a fractional form t ; none of them, therefore, raised to any power, gives an exact whole number. Now the square of 2, the first term of the series 2, 2 + », 2 + 2», 2 + 3i», 2 + 1 (?,) is 4, and the square of the last 2-|- 1, is 9. The squares, therefore, of the included terms must be included between 4 and 9, and the square of the second term 2 -|- <» is greater than 4, but nearer to it than the square of the next 2 -|- 2a) ; the square of 2-|- Sai is still greater, and so on. But increasing indefinitely the number of the particles ai, the differ- ence of the squares of the successive terms diminishes also indefinitely ; and in the same manner as by increasing the number of these particles, the square of the second term 1-\- a approaches more and more to 4, io the squares of some of the following terms will more and more approach to 5, to 6, to 7, to 8, and the square of the term before the last will approach more and more to 9. But, in the same manner as this square, and that of the second term 2 + m, never reach 9 and 4, however great may be the number of the particles a, so none of the squares of the intermediate terms will ever reach the whole numbers 5, 6, 7, 8. Therefore, the radical expressions v/s; s/b; ^t, ^\ and the same we say of all similar roots, are numbers which cannot be exactly expressed, and, consequently, neither measured ; for this reason they are called incommenmrable or irrationcd, and surd. Beside these irrational numbers, other irrational quantities occur in mathematical questions, and all are reckoned among transcendental expressions.. Miscellaneous § ^7. Grive the exponential form to the foUow- exampies. jng roots : (See 47, 48.) t/' Exponential AnS. •j/a = a^, yaF= a^, |/o"= a", form given to l m i roote. — — — - — 70 TREATISE ON ALGEBRA. 1 a" / 1 -1' V«^="- J/a' = a", C'a "' a" a" Give the radical form to the following powers : Radical form giyon to powers. a' , — , a", S^a-", a . 6» 62 Observe, that a" = a " , &o. - — a" „ //as" 6" f Radicals re- Reduoe the following radicals to the same STeV-ee*' degree. (See §49.) Ans. Ans. Ans. ,/ /(Ui, ... , '^^, i^T^ H/'^, ^^, j/^; Maitipiication Examples of Multiplication. (See §§ 42, 43, ofradicals. 44^ 45^ 49) Given factors : (10 ^/^7^. V^^, Vb-'c-^, OPERATIONS ON MONOMIALS. Given factors : (■^■) \J^' •\Sy=^' -V/v-"' Xb-" • (4.) V^,^l. (5.) f/^=, i/«> V^- 71 j-mu' Xa'". ¥'' Answers : = a.h.f.r. (5.) ^i^ X lA X '^/^ = o«. KedurtioTi of Reduction of the roots of roots to a simpler SoV' "'form (§§44,47.) Given roots : (1-) 4i (2-) V^S'"- 72 TREATISE ON ALGEBRA. (30 ^l^ -^- (4.) Vs/t-"- . (5.) j,n^^" s. /"^^~' Answers. (10 ^ra=/a- (2.) ^I/W^IK (3.) r^'^s-==rv^ (40 ^/^/¥=i^•^|■ (5.) ;,;^^,.*. « iTr- Xa-.i-a-«* a/t- ''WVV(i)"(l)"(7)'(5)'=-<(s^l=s/i Greatest com- j^gt «s add some examples concerning the mon measure of ^ « numbers. greatest common measure of numbers. Given numbers : 544 416. abs. 32. Given « 916 2201. Ans. 31. Given « 1261 1079. Ans. 13. Given " 1267 916. Ans. 1. OPERATIONS ON POLYNOMIALS. 73 CHAPTBE II. OPERATIONS ON POLYNOMIALS. AKTICLE I. Addition and Subtraction. § 58. Addition. — The addition of polynomials is sub- stantially the same as that of monomials, since to add one polynomial to another is nothing more than to add a number of monomials to another number of monomials. And to add together several polynomials signifies to add together as many monomials as are those contained in the given polynomials. All, therefore, that has been said (18) with regard to the addition of monomials is applicable to the case of polynomials. Hence, the addition of the polynomials a»-f 4ai + 3c» -~d, — Sfls" -\- ac — Sab -\- I, 7a' + ab— I +d, is obtained by writing in succession all their terms, each with its, proper sign. Kesuction of Howcver, before making this operation, and in Eimuar terms. Qjjgj jq obtain 3, simpler result, observe whether similar terms, (10,) or equal, are to be found in the polynomials. Because, all such terms are expressed by a single term similar to them, and having for coefficient the sum or the difference of the partial coefficients. For example, the first of the given polynomials contains the terms -|- o,"} + 4aS ; the second, the similar terms — 3a^ — Sab, and the third polynomial, the terms -|- 7a' -f- ab. Now, these six terms can be represented by only two equivalent to them, since the sum of the similar terms o" — Sa" -j- 7a" = Sa", and that of the terms iab — Sab -\- ab = 2ab. Again, the first polynomial contains the term — d, and the third the term + d, which are mutually elimi- 74 TREATISE ON ALGEBRA. Examples. nated, as well as the terms -\- I and — I, the first in the second, and the last in the third polynomial. Hence, the sum of the given polynomiads fe Sa" 4- 2ab + 3c=' + ac. And generally, to obtain the sum of given poly- nomdalSf write, first, any of them as given, then a second, so that the similar terms shall fall under the corre- sponding terms of the first, and so all the other polynomials. Reduce then the similar terms, and annex those terms which are alone. §59. Add together the polynomials f 3ci'% 4- 1¥c — 9c=3 — 132=6, (1.) j — la% + 3c»3 + 2^6 — la, Arranging these polynomials according to the preceding rule, we will have 3a»6 + We — 9c»g — \^H> — la'b +3cSg-|- q^— la — 86% + 6c^2+ 2fb + 3hi Sum — 4a«Z>— l^ —10fb + 2la. Add together the polynomials ( ia'd -f 3cob — 9m'n, 4m»» + aV + bc^b -\- Id^d, Gm^n — 5(flb -j- imn' — 8ab% Imn" -\- 6(?b — 5ot% — 6a'6,. 7, o' + 6= + Sa% 2o» — 46' — 5ai«, 6a'b + 1006=, [^ _ 6a' — la^'b + 4a6» + 26'. OPERATIONS ON POLYNOMIAliS. 75 Answers : (2.) Sum, lla'd + 18(^6 — 12m»n — 2Sab^ + Umn' — 10c? + mn. (3.) Sum, a' + 6= + a»6 + ah\ § 60. Subtraction. — To subtract a polynomial B from another polynomial A, means to find the difference between the two polynomials; that is, another polynomial D, which, if added to B, gives A for sum. Applying now to these ex- pressions the reasoning made (21) 'with regard to simple monomials, we may easily infer, that The polynomial B is subtracted from A hy adding to this a polynomial opposite to B. We do not need to prove that the polynomial opposite to B contains the same terms of B, but with opposite signs. Take B = Qm^h — 5a^5» + I. Examples. „ « . From A = Zm'h + 4m'c — 6a=6^. The polynomial opposite to B is — Qm^c -\- ba^y — I. Hence, A — B = 3m=Z> -)- 4ot'c — Qa^V — 6ot'c + 5a»6» — I. And D = 3m''6 — ImH — 0=4^ — I. Adding, in fact, this D to B, we will obtain A. .J ^ f Take a + 2i + c + 5c^ ^ '' t from 4c5 + 36 — 2c + 8(?. (2-) (3-) (4.) f Take — hal + IV — 19a« + 2m 1 from 12a6 + 3&» — ITa" + 3m. ( Take 10m=6 + lOm^ — 10m=6'' 1 from 10m' + \m?h — hrrfib"^. < Take — a^ + lah — 6» I from a" + 2ah + 6». ,g V f Take — fqg — r^s + 12m» *- '^ 1 from ilr^s + 126^—15i>qg. 76 TREATISE ON ALGEBRA. Answers : (1.) I> = Sa-\-b — 3c + 3d. (2.) D = 17a6 — 45» + 2a» + m. (3.) I) = bm^l^ — 6m^b. (4.) D = 2o» + 26a. (5.) D^12r^s + '[2s'> — 12fqg + 12mn. ARTICLE 11. Multiplication and Division. § 62. Multiplication. — The multiplication of a poly- nomial A by another polynomial B, consists like that of mo- nomials (24) in finding the product P, an expression either monomial or polynomial, whose numerical value is equal to the product of th<3 numerical values of the factors A and B. But multiplying A by each of the terms of B, and summing up all these partial products, the sum must be the product of A by B. Therefore, to obtain the product of a polynomial A by another polynomial B, Take the sum of the product of all the terms of A hy each of the terms of B, or vice versa. Taking A for multiplicand, and B for multiplier, or vice versa, in both cases we will have the same terms to be added. The product of polynomials is usually indicated by enclosing them within parentheses, as follows : (A) (B). ^ § 63. Let A = a — 6, B ;= c — ) ( Multiply A = a'J=4- %a?l^ + 3ai* + 6' ^ \ by B = o6» — 4a''6 + 2a». ( Multiply A ^ 3a»,/S'+ Zm^'^aF— ?!^L^ (3.) - m «V^ ( by B = i/6-^^^-S. , J -, f Multiply together ^ '^ 1 (a— J), (6 — c), (c — d), (c? — e). ,g . ( Multiply A = a" + 2a6 4- 6" *• '^ 1. by B=a — &. f Multiply A ^ a 4- 6 1 by B = o — 6 'c'a* (6.) OPEEATIONS ON POLYNOMIALS. 79 Answers : (2.) P = 2a%3 _(- 1a%^— 5a*S* — Ta^V — a'lfi + aV. (3.) P=3a=6-65r(^^+|^. (4.) P =ahcd — ¥cd—ad'd-\-ld'd—ahcP-\-l^J^-\-ac(P — hcd^ — ahce -\- V^ce -\- ac^e — hcH -\- abde — Vde — acd& + We. Or else, P == a lbd(e — e) + de(b — c)] + 6'[c(e — d) _f- d(d — e)] + c" [a(e — d) — b(d — e)] + dialed — 6fZJ + 6(c6— c(Z)]. (5.) P = a' + a»6 — a6^ — b\ (6.) P = a» — ft''. The following examples deserve to be noticed on account of their frequent and useful applications. ,j , r Multiply A = 1 + 2+0"+ 33+.. .+z" ^ ■■' 1 byB = 1—2. ( Multiply A = a + b^^^ byB = a—by^^^l. ] Multiply A = «+ 6v^=^r A + /c^=ri. f Multiply A = ffi — 4v/:=T 1 by B = A — A:v/=ri. Observe, that the exponent n of the last term of A, in the first of these examples, is a ■whole number, containing one unity less than the number of terms of the same polynomial. Answers : (1.) P = l — 2"+'. (2.) P = a» + i'. (3.) F = ah — bk+ {ak + bh)^'=\. (4.) V = ah — blc — {ale + bh)^/^^. We may remark, with regard to the second of these products, that in changing the factors a into h, and h into Ic, we would have had P = A' + /c'. Singular pro- i 64. We can demonstrate now a singular property of perty of num- o i- r j liers. numbers, which is contained in the follovring theorem : If two numbers M and N are such, that each one of them may be (2-) i j byB: 80 TREATISE ON ALGEBRA. resolved into two square numbers, that is, M = (8"+ i°, N = K'-\- k', the product M . N of the same number may likewise be resolved into two square numbers. From the last example (2) and its equivalent we have. (a'+ i=)(A'+ P) = [(a+ 6^/=rT) (a — i)>/=T)] [(* + *\/^^ (A_AV— 1)] (A-V=^)]- Again, from the last examples (3) and (4), we have (a+ b,^/^::^)(_h -\- iv^^) = {ah—bk) -{- {ak+ bh).y=l (a — S^:ri)(A _ i^:ri) = (oA — bk) — {ak+ bh)y/^-l. Therefore, (a" + 4') (7i' + A*) = [(aA— JA) + (ak + 5/j)^^l] [(oA — bk) — {ak + 6AX7— 1]. But we have seen in the preceding number that the product of the sum by the difference of any two quantities, is equal to the difference of the squares of the same quantities ; hence, the product, or second member of the last equation, is {ah — bk)' — ( (fflA 4- 44)^=1)2 ; that is, (a2+i2)(A»+A') = (oA — ii)*— ((ai + ^^=1)" But ( {ak+ bh)^=T.)' = (ak + bhf . ^'=lf = — {ak+ bhf ; hence, (««+ i«)(/i' + 7c') = [ah — bk)' + {ak + bhf. That is to say, the product of two numbers M = (a* + i"), N = (ff'+ P), is equal to the sum of two square numbers. For example, take M = 40 and N = 58, we will have M = (6*+ 2"), N = (7' 4- 3',) and consequently, M . N, or 40 . 58 = (6 . 7 — 2 . 8f + (6 . 3 +'2 . 7)* = 36' + 32= = 1296 + 1024 = 2320. § 65. Division. — To divide a polynomial A by another polynomial B, is to find a polynomial, or even monomial ex- pression Q, the quotient, which, multiplied by the divisor B, gives for product the dividend A. Polynomials ^° obtain the quotient, it is expedient to arranged. arrange both dividend and divisor, according to OrERATIONS ON POLYNOMIALS. 81 the powers of the same letter; that is, writing, in the first place, the term in which the power of the letter is the highest, then the term where the power of the same letter is the highest of the remaining, and so on. This is the arrangement usually made, although it would be equally profitable to arrange the terms in the inverted order, commencing, namely, with the lowest power and increasing in order to the highest. Thus, for example, the polynomials A = a" + 2a6 + b\ B = 2a3 + 2a% + a¥ + 5% are both arranged according to the decreasing powers of o, and according to the increasing powers of h. When the polynomials are thus arranged, the operation of division is easily performed. But to see better on what prin- operatioD ^'P^® ttis operation rests, let us multiply together Bxpiained. ^fjg ^^q preceding polynomials arranged A and B, marking each partial product, we will have p' 2a^-\-4ta*b-\-2.a^b'>, p" 2a'i+4as5^+2ai>6', p'" a^V-\-2a''h^-{-ab% p" a<'b^-{-2ab*+b^ P'+P"+P"'+P"'= 2a'-\-Qa'b+7a^l^+5a!'b'+Sab^+¥ = P. Let us now observe, first, that the product which results, is arranged according to the powers of the letters of the factors, and it is not difficult to see that it cannot be otherwise. Secondly, the first and the last term of the same product are merely produced by the multiplication of the first terms of the factors, and by the multiplication of the last terms. Hence, dividing the first term of the product by the first term of one of the factors, the quotient must be the first term of the other factor. Hence also, generally, when both dividend and divisor are arranged according to the powers of the same letter, the 82 TREATISE ON ALGEBRA. first term of the quotient is obtained by dividing the first term of the dividend by the first of the divisor. Now, after having obtained the first term of the quotient., we can have also the partial product p' of the divisor by the quotient. For example, in the case before us dividing P by the polynomial A, we will obtain the first term of the quo- tient, that is, 2a», by dividing the first term of P by the first of A. Now, multiplying A by 2a^, we obtain ^', which sub- tracted from P, gives for remainder a polynomial P ^jj" +i''" + i'*' j *'i8,t is to say, the product of the same A by the remaining terms of the quotient. Repeating the operation with the divisor A and the dividend P', we will obtain the Bext term of the quotient, which in our case is 1aH>, and, in consequence, the second partial product -p" , and so on. Hence, the rule. To divide the polynomial M hy another poly- nomial N, arrange, first, both according to the powers of the same letter. Then divide the first term of the dividend by the first of the divisor, and marh the quotient. Multiplg then by this term, the divisor N, and subtract the product from M, and taking the remainder for dividend, repeat the same operation until the end. Divisor. Dividend. Quotient. Examples. ^ ^ ^^^ ^^ _^ j^^^ — hz — hz (h — Z 1st product h^-\- hk 1st remainder — hz- — kz 2d product. — hz — 7cz 2d remainder The product may be taken with changed signs, and so the remainders may be obtained by simple addition. When the dividend contains many terms, it is not necessary to write each time all those which belong to the remainder, but it is enough to write as many terms as there are in the divisor, as ia the following example : OPERATIONS ON POLYNOMIALS. 8S + + o 1= 4- o 1-4 + 'e + + e »^ + •n j & r<3 a m us "O^ ■4 ua e iS e 8 e 1-1 o O o (N (N T-H 1-4 + 1 + 1 I.O rO !? rO (D '^i "e e e "% 1:~ (M >a in + [ X^ r« e s ^-\ \ & 1 •r-t Q .E; rO e e t~ in + 1 & S> <", « e cs T— 1 o T-( 1— 1 + 1 * * ►o ^0 e % IQ in 1 l^ *» fO pO + 1 ts o 53 ts — hk-\-7c'. (3.) Q = a» — Ea6 + 26^. (4.) Q = x'—xi/ + i/^. Bemark con- The arrangement of polynomials according to S™g^men't° "^f 'be powers of the same letter, is not an indis- poiynomiais. pensable requisite to obtain the quotient. This can be also obtained without such an arrangement. Nay, this must necessarily be done when the polynomials cannot be arranged. Such is, however, the nature of the process, that the same quotient will be obtained with different arrange- ments, although the disposition, the form of the terms, and their signs may appear different in the quotient for different arrangements. For example, dividing the same polynomial A, differently arranged, as follows : A = a' -(- 2ab -f- bf = 2ab-\-a'>-{-V by B = a + 6. With the first arrrangement, we have Q = a 4- 6. OPERATIONS ON POLYNOMIALS. 85 Witli the second, Q = 2h-j-a — b So likewise, let the dividend be A = 2aM+mnl-\-ql + 2abr-{-mnr -\-qr + 2abs -\- m,ns--\- qs. And differently arranged, A = mnr-\- qr-[- 2ahr -\- mns -\-qs-\-2als-{- mnl -\-qi-\- 2abl. Divide it by B = I -\- r -\- s. With the first arrangement, we will find Q = 2ab + "»" -K?' With the second, mnr qr 2abr mns qs 2abs *^ = — +T + ~r + "r + T + ~r -)- mm -|- J + 2ab , mnr mns qr qs 2abr 2abs 1 I 1 ~1 I T' which is manifestly equal to 2ab -|- mn -|- q, as for the first arrangement. In order, however, to diminish useless labor, it is always expedient to arrange as much as possible the given poly- nomials. So, for example, the following dividend and divisor, A = abed -(- cdm -)- mn -\- abmn -\- m^n -\- 3a% -\- Sabm B ^ m -|- ab, may be arranged in this manner : A = Sa^b -(- abed -j- abmn -(- Sabm -\- cdm -\- m'n -\- mn H ^ ab -\- m. And we will find Q = Sab 4- cd -\- mn -j = — ; . ab-\- m The dividend § ^^- 1° '^'^ ^^^ example, after having found TWbif by'^tht t'le first three terms of the quotient, the remain- dmsor. jg^ jg ^jg ,jjyj(je(j jg jjjra, which cannot be divided by B ; we add, therefore, as the last term of the quotient, a 86 TREATISE ON AtGEBRA. fractional expression having the remainder for the numerator, and the divisor for the denominator. This case takes place whenever the dividend is not the exact product of the divisor and the quotient. Let us give another example. Divide A = /.-^ 4- ikh^ -f 3ffi«A + ik" by B = A + 7f, we will find Q = A= + 2M -|- Ic^ -\- -j—— . The correctness of the process to find out the quotient in all cases, can be demonstrated also in the following manner : CaU t', t", t'^', . . . the first, the second, the third term, &o. of the quotient obtained by dividing A by B, and call r', r", r"', . . . the remainders corresponding to each term of tlie quotient, we will have r' =A — t'Ti r" = A — CB — ;"B = A — ((' + t")^ r'" = A — ('B — i"B — 1"''& = A — (i/ + t" + t'")^, &o. iVnd j-(") = A — B((' + t" + t'" + .... + <(»)). Suppose now, that after having obtained the »"' term of the quotient, we stop the operation. The,polynomial ;' + «"+<"'+. ... + (("), represents the quotient, and 7-("' the last remainder. Now, from the preceding equation, we infer A = B((' + i" + i'" + . . . + (<"') + »•!»'. But the quotient of A divided by B, must be such a quantity, which, if multiplied by B ought to give A, or, which is the same, '&{t' -f- t^' + i"' + • . . + *'"') + »•'"', but this product is evidently obtained by multiplying {f + t>'' + t'" +. . . + (<">) + '^ by B. Therefore, ! = (/ + k^ k' ft' + ft + A" M" — h k* k^ k? Ifi A2 + fts A + A ■ A3 ' ! *! ll? A"+2 A3 h-\-k and generally, '''^ft4r4~A A2+A3 ■•• ""p"^ A + A" A"" From the simple inspection of the order in which the signs of the preceding equations follow one another, and from the uniformity of the 88 TRKATISE ON ALGEBRA. process to obtain any number of terms for the quotient, it is plain that the upper of the two signs placed before the last terms (a) is to be used when n is an uneven number, and the other when » is eyen. With regard to the formula [a), three different cases can happen. That is, we may have A = A, or A <' A, or A ]> A : k in the first case, 7 = 1: n k in the second, t- ^ 1 ; k in the third, ;- <[ 1. In the last of these cases, by increasing indefinitely the number of k^ the terms, and consequently, « in the formula (a), the factor t^ of the last term will more and more approach to zero, and consequently likewise the whole term A+ft ■ A" ^'"-'' But if, by increasing the number of the terms in (a), the last of them constantly approaches to zero, we may say that the polynomial A ~ A2 + A3 ~ ■ • A" ' which contains all the terms of (a) with the exception of the last, is such that by increasing the number of its terms it constantly ap- proaches to the determined value of the first number of (a), namely, to A + A • ■ • ■ W- We call this fractional expression (s), to designate that it is equal to the sum of aU the terms (a ; ^ , _ ^^^, ^^, _ ^^, ^^, . . of the series indefinitely protracted ; that is, (" I h + k — h h?'^ h? The term \r) here omitted is called residual term, or residuum, after the n"" term. In the other two oases of - = 1 or ^ 1, this residuum cannot be k J^ omitted, because when r = 1, (r) is constantly equal to ± j— — - , A A -f- k OPERATIONS ON POLYNOMIALS. 89 and when t >• 1, the value of (r) constantly increaaes by adding units to n. From the formula (a), dividing first iboth members by !c^, and making then A = 1, we deduce the two following useful equations : ^ '_1 l_i. *^_ -i-^m 1 !^ h+7c ~ A"~ A2+ A3 ■ ■ ■ A» ^ A + /c ■ A» («'") 1 jfcn ';4.i«_... ± A"-' .1 + A~ ^ -+-* + *• The remarks made with regard to the general formula (a) are evidently applicable to the last two, and their residuums can be jfc omitted whenever 7 <[ 1- With regard to the second of {a'"), the k fraction t cannot be <^ 1, unless k itself is <[ 1 ; that is, unless k be a fraction ; because, since in that formula A = 1, the expression r is nothing else but k. Supposing, therefore, A to be a fraction, we wUlhave — y-j = 1 — A + A» — 43+ an indefinite series. The (juotient i 68. When the imaginary expression a + bs/ — 1 ofthe same form ' s> J if "TV ary 'exprSsions '^ divided by another imaginary expression c + e^'^-l, divided by one the quotient or fractiou snotlier. a+ V— 1 remains unaltered when its numerator and denominator are multiplied by the same expression ; for example, by c — e^ — 1. Effecting this multiplication, the numerator (63) becomes ac-\-he-\- [cb — ae)\/ — 1 ; and the denominator, c* -f e", and, consequently, a + b^ — 1 ac + ie cb — ue :p Calling A the real quantity ^J" and B the coefficient " "^ likewise real, of ^ — 1, the same quotient will be represented by 8* 90 TREATISE ON ALGEBRA. That is to say, the quotient, no less than the product of two imaginary expressions of the form a -|- by/ — 1, is another imaginary expression of the same form. ARTICLE III. Formation of Powers and Extraction of Roots. § 69. Powers. — There is no diiFerence between the forma- tion of powers of monomials and that of polynomials, since the same operation, which in the former case is to be made about a monomial root, is to be made about a polynomial in the second. All, therefore, that concerns this operation with regard to monomials (Chap. I. 39) is to be applied to the case of polynomials. Hence, when the positive ex- ponent m is a whole number, the operation to be performed to obtain the power of a given polynomial, or root A, is to multiply that same root by itself, as many times as there are units in m. Let us take, for example, the most simple case. That is, let the given rout be the binomial 1 + z, and let us take in succession for exponent ?n =2, m =3, &c., we will have Examples. C (1 +")' = (1+^) (1+^) = l+2z+Z» (a-) J (1+2? = (l+^Kl+^) = 1+3^+3^'+ ^ ^ I (!+«)* = (l+2)Xl+«) = l+4«+62''-f4a'4-«*, Observe now, that l+3.+3.=+^ =. l+3.+?2=l),,+ 3(3-l)(3-2)^ 4(4-1X4-2X4-3) + 2374 "^^ OPERATIONS ON POLYNOMIALS. 91 Therefore, (i+z)s = i+2z+ ^^^r-^V (1-|-Z)4=J+4Z ■ ' ,2 , 1 3(3- -1)., 3(3- -1)(3- -2).3 ■1 2 , 1 4(4- -1)., , I 4(4- 2.3 -1)(4- -2). '' 2 " 1 2.3 I 4(4-l)(4-2)(4-3) + 2.3.4 ''^- Prom the uniformity observable in these evolutions of the binomial 1 -|- z, we could infer by analogy a general law, ex- tending alike to all positive and whole powers, so that the m"" power of (1 -|- z) would be given by the formula ., , , , m(m — V) , m(m — T)(m — 2) . (a') . . . (l+z)" = l+mz+ ^ ^ ^ z'+ -^ ^ ^z» I w(m— l)(m— 2)(m— 3) + ■ 2X4 '^+---' containing (m -\- 1) terms in the evolution, because, were they m -|- 2, the last of them would have among its factors (m — m), which renders the whole term equal to zero. The same factor (m — m) would be found also in all the follow- ing terms. Therefore, the said evolution of (1 -\- z)" cannot contain more than m -)- 1 terms, and consequently, three terms when m^ 2, and four when m=3, &c., as we have seen in the preceding examples. It is moreover plain, that in the same evolution of (1 -(- z)", the highest and last exponent of z is equal to m. foS^a."'"" I^et us now make z = -, and, substituting this value in (a'), we will have / y\"'_ , y TO(m— 1) y" , m(OT— l)(m— 2) ^ , V'^x) ~~ '^ x'^ 2 x"'^ 2.3 a?""""- but (1-|--) ^ ' m p multiplying then both members of 92 TREATISE ON ALGEBRA. the equation by x", we will find (e). . ■(a:+y)"=a°'4-mz"-V + ™^'"o~ ^^'''V m(m-l)(m-2) _ , + 2.3 3'+--- The form and the order in which the terms of the binomial (x -\- y)" evolved, succeed one another, was first discovered by Sir Isaac Newton ; hence, this formula (e) bears the name of its discoverer. By mere induction, however, the correctness of the evolu- tion is not demonstrated. A rigorous demonstration of it may be seen in the following number. Let us now observe, first, that the formula , ,. mim, — Y)(m, — 2).... Cm — (p — 1) General term. (e' ) . . . — ^^ ^\ „ „ . ^^ ^^ -x"^ - Vi represents the {jp -j- 1)" term of (e). Substituting, in fact, 1, 2, 3, 4, &c. instead of jp, we will find the second, the third, the fourth term, &c. of the evolution. Hence (e*) is called the general term, of the series. „ , m,{m — 1) „ , , m{m — 1)(to — 2) „ , . mx'^-^y, —^-5 i-x^-^y\ — ^ jr^ ix'^-^y' u Jj .0 Secondly, substituting 2 and 3, instead of m, in the formula (e), we have (x+yy = x<' + 2xi/ +^* (je-\-yy = a? + Zoi?y + Zx^ -\- y\ Just as we would obtain from the first and second (a), substi- Square and ... X . if rr cuiie of any tuting in them - instead 01 z. Hence, the square binonual. V of a binomial is given hy the square of the first term, the double product of the first by the second term, and the square of the second term. The cube of a binomial contains the cube of the first term; the triple product of the square of the first by the simple OPERATIONS ON POLYKOMIALS. 93 second term, the triple product of the simple first term hy the square of the second, and finally, the cube of the second term. Bmomialthe- g 70. Let us now resume the formula (a'). We say strSed. " ° ' that wben m is a whole and positive number, the evolu- tion of (1 + z)" is exactly represented by the second member of {a'). Observe, first, that (a') X (1+2) <"' il+,n.+ '!<^ {.'+... )(! + .) = 1 + ^. + ^J^^.^+ "'("'-l)(j»-2) ^+ . . . And, adding together the similar terms, K) + (l + z) = l+(m+l)2+ (^z+lVz'+ (^^+l) ^i+(^+1).+ (!:L+1)!!;.^+ ("+^M;-^W ... That is to say, (a') -{- (l-{-z) is equal to a polynomial which contains m-f-2 terms, one term more than those of («'), and this is easily proved in the same manner as we have demonstrated that the number of terms in (a') do not exceed (m + 1). Now, substituting m + 1 instead of m in (a'), we have i + ^,,, + ^.+ (J!L+l>^+ (^ + 'M^-% +.. that is, the product of (a') by (1 + z). Therefore, whatever be the whole and positive number m, the product of (a') by (1 + z) is obtained by changing in («') m into m-\-l. We may now proceed to demonstrate the theorem as follows : If the polynomial (a') is the exact evolution, for example, of the third power of 1 + z when m = 3, it will be also the evolution of the fourth power of the same binomial, making m = 4 ; but if (a') is the exact evolution of the fourth power of 1 -}- z when m = 4, it must be also the evolution of the fifth power of 1 + z, making m = 6, and so on. Therefore, whatever be the number of units in m above 3, provided vrith m ^ 3, the polynomial («') gives the evolution of 94 TREATISE ON ALGEBRA, (1 + 2)^ it will g'.ve the evolution also of (1 -\- z)", so that this power in our supposition is exactly represented by the polynomial {<"). But, indeed, (a') is the evolution of (1 + z)', (1 + z)3, when we make m = 2, m z=3, which may be easily verified, compai-ing {a') with the formulas (a). Therefore, when m is any whole and positive number, the evolution of (1 + z)" is given by the polynomial (a'). And since we have seen, that making in (a') z = -, the same (a') is changed into the Newtonian formula (e) ; the evolution, therefore, of the binomial (x4- y)™! according to the known law, is also rigorously demonstrated, whatever may be the whole and positive value of m. Singular pro- § 71. The coefScients of the formula (c) have the eflkfenta'of (S singular property of representing the numbers of oom- furmula (e). binations of m different quantities. Hence, before we speak of the extraction of roots, we will dwell here upon this subject, as well as upon the use of the preceding formulas, in order to find out some numerical properties. Let ii,b,c,d, represent m different symbols. The first a taken in succession with each one of the following b, c, d, ... will give us (m — 1) binaries ab, ac, ad, ... . But, joining in equal manner the second symbol 6 with all the others, we will have again (m — 1) binaries ; and let the same be said of the third, fourth, and so on. In this manner we will have a joined to i, and again b joined to a, a joined to c, and also c to a, &o. ; for this reason this arrangement of symbols is termed permutation. But for each symbol the number of permutations of the letters taken two and two is m — 1, and the symbols are m in number ; therefore, the whole number of permutations of m letters taken two and two is m[m — 1). Again, the binary ab joined in succession with each one of the remain- ing symbols, will give us m — 2 ternaries ; and the same we must say of all the other binaries. Now the number of permutations of m symbols taken two and two is m(m — 1); therefore, the number of permutations of m symbols taken three and three, is 7»(m — 1)(ot — 2). It is now easy to see, that the number of permutations of m symbols taken four and four, is OT(m— l)(m— 2)(m — 3), and generally, the number of permutations of m symbols taken p and ^, is m{m — !)('» — 2) . . . . (m — [p — 1)), OrEKATIONS ON POLYNOJilALS. 95 But if to the preceding binaries, ternaries, &c., we would add all those which come out from the repetition of the same symbols — for example, aa, bb, .... aaa, aah, . . . bbc, ... to tlie number ?«(«» — 1) of the terms taken two and two — we must then add m more binaries ; and since m(m — 1) -\- m = m", the number of permutations of m terms taken two and two, with the repetition of the same symbol, is We would find in a similar manner, that m? gives the number of permu- tations with the repetition of the same symbols of the m terms taken three and three. But let us investigate the subject, assuming it in a more general point of view. Permutations 2 ''2.- Suppose m letters to be taken^ — 1, and^' — 1, with repetitious. [^ ^\i possible manners, without excluding the repeti- tion of the sanie terms. To obtain the same m symbols taken p and p, it is enough to add in succession the m terms to each of the collections of the same terms taken p — 1 and p — l, and make the addition in. the last place. To demonstrate this proposition, let / be one of the m symbols. Among the terms taken (^ — 1) and (p — 1), there must be some collections in which / does not enter at all, others in which / enters only once, others in which it enters twice, &c. Tlie same must be with regard to the symbols taken p and p ; but all those collections of terms taken p and p, and excluding f, must certainly terminate with .5ny symbol except f. To obtain, therefore, all the collections of terms taken p and p, with the exclusion of f, it is eKough to add in succession to each one of those collections, taken^ — 1 and^ — 1, and which exclude f, all the m symbols except f; brat adding to the s.ime collections also /, we will obtain all those taken p and p, containing f once, and in the last plaee. After the collections of symbols taken p — 1 and^ — 1, and exclud- ing f, come those which contain /only once, some of which must have / for the first, some for the second, some for the third term, and so on. Now, adding to the end of each one of them the m terms, one after another with the exclusion of /, we will evidently obtain all the possible collections of m terms taken pand p, and in which / enters only once, either in the first, second, or third place, and so on, except those which contain f in the last place ; but we have seen how they are obtained in the first addition, and now if to the same terms take^ — 1 96 TREATISE ON ALGEBRA. and p — 1, and containing / only once, we add / once more for the last term, we will liare those collections of terms taken p and p, iu which / enters twice, with one of them, however, constantly at the end. If now to each collection of terms taken p — 1 and p — 1, and containing / twice, we add in succession ail the m terms with the ex- ception of /, these, together with the last mentioned, will give us all the collections of terms taken p and p in which / enters twice in all possible ways. It is now plain, that adding, likewise, to all the re- maining collections of terms taken p — 1 and p — 1 all the m letters in succession, we will obtain all those taken p and p, when / enters three, four, five times, &c., in all possible ways. But whatever is demonstrated with regard to the symbol / is evidently applicable to all the others. Hence, adding in succession the m terms to each collection of the same terms taken p — 1 and p — 1 with all kinds of permutations and repetitions, we will obtain all the same permutations with the repetitions of the terms taken p and p ; and the number of the collections of the terms taken p and p is evidently m times as great as the number of collections of the terms taken p — 1 and p — 1. Let us now call N the number of collections of the m terms taken p — 1 and^^l. The number of collections of the same terms taken p and p will be given by N . ot. But supposing p =S, and, consequently, p — 1 = 2; as we have seen in the preceding number N in this case is = m% Therefore, the number of permutations with repetitions of m terms taken three and three is vi^ . m = m^. But if the number of collections of m terms taken three and three is N =: ttfi, it follows, likewise, that the number of collections of the same terms taken four and four must be m'. m = m*, and so on ; and consequently, we generally infer that the number of permutations, with repetitions of vi tenns taken p and p^ is N = ml. How the same ? 73. We are now able to infer again the general BcmMRlf'irmulas formula of simple permutations. Suppose the permu- of pBrmuratinus fitions of m symbols taken p — \ and p — 1. Each without repeti- •' i- r tioBs symbol enters only once in these collections, and the symbol /. for example, in some of them will be the first, in others the second, in others the third, and so on, till the last. To all these / carniut be added to obtain the permutations of symbols taken p and p ; but adding other symbols we will obtain the terms taken p and p con- OPERATIONS ON POLYNOMIALS. 97 taining / in the first, second, and third place, and so on till the last, exclusiTely. Hence, to all the other permutations of terms taken p — 1 and p — 1, and excluding /, this symbol cannot be added except at the end. The same thing ought to be said of any other symbol : if the symbol is already in the permutation containing p — 1 terms, it is not to be added ; if it is not in it, it must be added only at the end. Now each permutation contains by supposition p — 1 symbols. The number, therefore, of symbols to be successively added at the end of each one of them is m — (p — 1), and so we will obtain all the permutations of terms taken p and p. So that, calling v the number of permutations of m terms taken p — 1 and p — 1, for the number of simple permutations of the same m terms taken p and p, we will have v[m — (^— 1)]. Let us take for example p = Z, and, consequently, p — 1=2. In this case we have seen (71), that v = m(m — 1) ; therefore, the num- ber of simple permutations of m symbols taken three and three is m(m — 1)(™ — 2). And if the number v of permutations of m symbols taken three and three is m[m, — l)(m — 2), that of the same symbols taken four and four is m(m — l)(m — 2)(m — 3), and so on. And generally, the number of permutations of m terms taken p and p, is that already found (71), with another process — m(m — 1)(™ — 2) .... (m — [p — 1)). Taking p =z m, we will have the number of permuta- tions which may be obtained by the collection of all the m symbols. But observe, that in this case the last factor (m — (p — 1)) of the general formula becomes (m — m + 1) = 1, and, consequently, the factor before the last is 2, and the preceding one 3, &c. Hence, the number of simple permutations which can be formed with all the m symbols is m(m — 1)(™ — 2) . . . . 3.2.1. This same formula gives us, besides, the number of permutations, which can be obtained from the same two, the same three, and the same p letters. So that, if we wish to know, for example, how many permutations are made by the first five of the m symbols taken five and five, it is enough to substitute 5 instead of m in the preceding formula. Calling now, v,, Vj v, , the number of permutations 9 98 TREATISE ON ALGEBRA. formed with any two and the same symbols, vrtth any three .... with any p, we will haTC V, = 2 . 1 =2 .r, = 3.2.1 =2.3 ,,=p(p-l)....2.1 = 2.3....(p-l)p. It is now easy for us to determine the simple combinations of m given symbols. I 74. In the simple combinations we exclude all the m ma ions. jQjjggtig^g of terms in which at least one symbol is not different from the symbols of another collection. For example, the symbols a and b can be combined with c, d, . . . but after having taken abc, aid, we exclude all the permutations which can be formed witlt the same tei-ms a, b, c, or a, b, d. Call now m, the number of simple combinations of terms taken three and three. From, that which we have just observed, and from the preceding formula »^ it follows first, tliat the number of permuta- tions of any three and the same symbols is 2 . 3, and, consequently, "a+(2.3) is the number of permutations of all the m terms taken three and three. But this same number is expressed also by ro(m — !)('» — 2) ; therefore, n,+ (2.3) = m{m — l){m — 2), m(m — l)(»n — 2) and, consequently, n, = — ^ =-^^ -. In equal manner, calling tip the number of combinations made with m symbols taken/) and p, and multipyling /ip by y, we vriU obtain the number of corresponding permutations Wj, -|- (2 . 3 , . . p), but the same number is also given by m(m — 1).... (m — (p — 1)); therefore, nj,4-(2.3 p = m{m — l)... (m — (p — l)), and, consequently. m(m — !)(»> — 2) . . . (m — (p — 1)) ""= 2.3.4. ...i) ■ Now, this is the general formula of the coefficients of (e), and making in it^ := 2, =3, = 4, =:, &c., we obtain the coefficients of the third, of the fourth term, and so on, of the same evolution. But at the san^e time, making^ = 2, =3, =, &c., we have the numbers of com- binations of m terms taken two and two, three and three, and so on ; therefore, the coefficient of the third term of the evolution of the binomial (x -\- yy gives the number of combinations of m symbols OPERATIONS ON POLYNOMIALS. 99 taken two and two, the coefficient of the fourth term gives the number of combinations of m terms taken three and three, and so on. On Bome pro- J 75. Let US now make use of the formula (e) to find perty of nam- ^ e -u bers. a certain property of numbers. The formula (e) may be changed indefinitely by giving different values to y, to x, and to m. Let us take instead of the exponent m, the number k prime in itself and make x = h and y = 1 ; the evolu- tion (e) will become Pm these periods, the square of ► will be evidently expressed by (/') .' =p„10^-^ + p„_^l(P"-*+ .... +p,W +p,W'+p. But ,'= [A.10»-^ + B.10— =^+ ..\ + M.IO + N]'; and (77) the square of any polynomial is expressed by the formula (/) from which, in our case, we have .= = A'. l(P»-8-l-2AB. 10="— «+B'. lCf^-*+(2t,G . W-'+C. 10="-=) -f- {2t,T) . 10"-*-f V>'. W-') +.... + (2C-iN+ N^), where fj = A.lO^-'-f B.IO"-'' f, = A. 10»-> -f B . 10"-=+ C . 10™-», &c. ,- and, consequently, 2^. C. 10"— » = 2AC. 10="-'+ 2BC. 10«»'-> 2<3D.10"— '= 2AD.10="-» + 2BD.102»-'' + 2CD. 10»"— ', &c. Substituting now these values in the preceding formula, we will have (/") >'= AM0="^=4-2AB.10="^+B'. 10''"" -I-2AC .10="-*+2BC . 10="^+ CS^. 10=»-» 4-2AD. 10="^4-2BI>10="^ +, &c. Both formulas (f) and (/''') give the same value of >'' with a differ- ence, however, which is to be remarked here. Suppose, for example, A ^ 3 and B = 4. The first and second terms of the square of » given by (/"), taken separately from the rest, will give us A2. 10="'-=+2AB. 10=»-»= 9 . 10^"^ + 24 . 10=»-» = 9.10="^=-f (2.10 4.4)10="-» = 9.10*»-=-f 2.10="-=-f- 4. 10="^ = 11 . 10="-= + 4 . 10="— OPERATIONS ON POLYNOMIALS. Ill It appears from this reduction, that some units of the second term 2AB . 10''™-' join the units of n. higher order, and enter into the first term. The same thing occurs with regard to the following terms, and the difference between the formulas (/'), (/") is that in the first the periods p„, Pm-i, ■ • ■ • contain all the units which may possibly be reduced to their respective order ; in the second, some of the units of a higher order form part of the suooessiTe terms. But when any square number is given, its periods are as in (/'), in which the terms of the root do not so distinctly appear as in [f"). We may still safely say, that A^ . 10'" -'■', or the square of the first term of the root, is altogether included in the period p„, and the double product 2AB . lO^"*-* of the first and second term of the I'oot does not go beyond the first figure of the following period p,^x, and the square B". 10'™-'of the second term of the root does not go beyond the same period ^m_i, and so on. Let us observe, also, that comparing, for instance, the number 2AB. 10="— ' + B2. lO'""-', ^■, which is the same, (2A . 10 + B)B . lO'"—*, with (2A.10 + B)B, the only difference to be found between the two numbers is that in the first, the product (2A. 10 -\- B)B is followed hy 2m — 4 ciphers ; in the second, the same product is followed by no cipher. In the case, therefore, in which the ciphers indicated by 10^-' would not be taken into account, the number (2A. 10 + B)B may be used instead of (2A.10 4-B)B.10=»-'. We may now proceed to see the reason of the operation to be per- formed and expressed by the rule, in order to extract the square root of any given number. We must commence by taking for the first figure of the root the square root of the last period, or rather the square root of the highest square number contained in the period. Then, after having subtracted the same square number from the said period, we wiLl obtain the second figure, by dividing by the double of the number already obtained for the root the remainder as far as the first figure, inclusively of the follow- ing period ^m_i ; since the second figure of the root, multiplied by the double of the first, is within these limits. In this manner we have the part of the root which is obtained and expressed in separate terms by A . 10 -|- B, although, rigorously speaking, the same terms should be expressed by A. lO"-'-)- B. 10"; since A is the m*, and B the 112 TREATISE ON ALGEBRA. (m — l)"" figure after units iu the root; but in (/") we have the terms 2AB. 10="— »4- BM(P"^, or their equiyalent (2A . 10 + Ji)'R.W-*, which are contained within the period ^„_i of (/■'), and haTing no consideration for the following ciphers, as we do with regard to the figures in the root ; this is the product of the double of the first cipher A, obtained for the root, plus the second cipher B, multiplied by B, which beiag subtracted from the remainder, as far as the whole period jo„_„ wiU give for the next remainder 2AC . 10=»-» + 2BC . 10="—'+, &c., or2{A. 10 + B)C . 10="— ' +, &c. Hence, taking this remainder as far as the first figure of the period p^-i, we have in it the product of the double of the root already obtained, multiplied by the third figure to be yet found ; to find, therefore, this third cipher, we divide the remain- der as far as the said figure of the period p„-s by the double of the root obtained. The number then thus far obtained for the root is ex- pressed by A. lO'-f- B . 10-|- c. But taking the whole period ^„_2 in the remainder, we have in it 2AC. 10="-* + 2BC . 10=" -« -f c'. 10="-», or (2(A . 10''+ B . 10) + C)C . 10="-", and having no consideration for the following ciphers expressed by 10="-*, the said remainder con- tains the product of the double of the two first figures of the root, plus the third figure, all multiplied by the same third figure ; and this product being subtracted from the remainder will give us another re- mainder, which taken as far as the first figure of the following period Pm-3, contains the product 2(A . 10= + B . 10 + C)D . 10=—', which, having no consideration for the following ciphers, is the product of the double of the root obtained by the figure to be next found. Dividing, therefore, the last remainder as far as the said limit by the double of the root obtained, we will have for quotient, the fouirth figure, &c. It is not necessary to go On farther to see on what principles the rule given to extract square roots of numbers rests, and to see also, that the process of the operation is identically the same as that of the extraction of roots of polynomials, although somewhat more compli- cated, on account of the units of the squares of each figure of the root, separated in different orders and periods. Extraction of §80. In the Same manner as the rule to ex- poiynomiais. tract the Square root from polynomials, is inferred from the formation of the square power, the rule to extract the cubical root of a given polynomial is deduced from the formation of the corresponding power. OPERATIONS ON POLYNOMIALS. 113 Cubical power ^ow the cubical powcr of a binomial contains an/ofanypoiy- (^9) t^e cubo of the first term, the triple product nommi. ^f ^j^g gq,ja,j.e of the first by the simple seeond term, the triple of the product of the square of the second by the simple first term, and finally, the cube of the second term. But a polynomial can be at pleasure divided into two parts, considering each part as a single term. In this manner the evolution of the cubical power of a binomial becomes applica- ble to any polynomial, and the rule to extract the cubical root inferred from the formation of the cubical power of a binomial is likewise generally applicable to the extraction of a cubical root of any polynomial. This will be better seen with an example. Let the given polynomial be P = 08 -(- 3a=5 + Ba*b^ 4- a^h" -\- Ba%^c -\- 3aic^ Examples. _^ ^^ + Stt^c + Gft^Sc + 3a^c=; which, arranged according to the powers of a, gives us „ f aS+3a=J+3a*6^+a5J3+3a''6'=c+8c}6c»+cV ) (O { 3a%-{-Sa*I)^-\-a^i'>+3a%'c+Sahc^+» + a»J», we will find f^, or R = 2a» + a6. Bxtractionof 3 81. The same process is applicable to the extrao- «ubical roots of ,. «..,,« , * -, numbers. tion of cubical roots of numbers. And recalung to mind that which we have already remarked (79) with regard to the extraction of square roots of numbers, it will be easy to see the identity of the operation with an example. Bat let us first observe, that the cubical power N of a number t of m, figures ABC .... cannot contain more than 3m figures, nor less than 3m — ■ 2 ; therefore, dividing the given power N into periods of figures taken three and three, the last of these periods will either con- tain three figures like the others, or only two, or even one ; secondly, the number of periods of N wiU be equal to that of the figures of its OPERATIONS ON POLYNOMIALS. 115 cubical root p. By obserrations similar to those previously made (79), wo find, besides, that the cubical power of the first figure A of v is entirely in the last period of N, and the triple product of A" by B does not go beyond the first figure of the following period, and the cubical power of (lOA + B) is entirely within the same period, &e But let ug see an example : Let the given power or number N be 34012224, or separating it into periods, let N = 34,012,224, V or l^N must then contain three figures, and we can represent it by V = ABC. The operation to find out these ciphers proceeds as follows : N , A = 3 34,012,224(324 A' = 27 —27 3A' = 3 . 9 = 27, B = 2 (r,) 7 0/12 3(10'A)''B + 3(10A)B" + B^ = 5768 — 5 7 68 3(10A + B)* = 3(32)' = 2187, (rj 1 2 442/24 — 1 2 442 24 C = 4 (rj oT S(10'A + 10B)» . c + 3(10°A+ 10B)c» + ' + ', n ^ that is, mx -\- - — 3 — »--l-< = (5'). Comparing (I) and (b') with (a), we see that Practical rules. t the terms - and g of the first member of (a) are n in the second member of (6) with changed signs, and the terms r and t of the second member of (a) are in the first member of (b') likewise with changed signs. We infer, there- fore, this practical and general rule The terms of an equation can he transposed ' from one to another member, without destroying the equality, provided their signs be changed. Dividing or multiplying both members of (a) by the same quantity, the products or quotients will form another equation, and dividing first both members by m, we will have mx h q r t m nm m m m' that is x -\ — = (c). nm, m m m Multiply now both members of (a) by n, we will have , nh ^ mnx A nq ^ nr — nt; n that is, mnx -\-h — nq = nr — nt . . . (-/, (3.) la- + g = l-h, we will have m — q — h Examples. (1.) X» = (2.) y = . (3.) a? = ■ n (2.) y=,mp — mf= m(p — /). m(l — g — h) n Known »nd § 83. Equations commonly contain known or unknownquan- . .,. - - .,.^. titles. given quantities, and unknown quantities oi quantities to be found: the known quantities are generally expressed (20) by the first letters a, b, c, ... of the alphabet, the unknown quantities by the last .... v, x, y, z. Hence, in the equation ax^ -\- b — zz=c, we would consider a, b, and c as given quantities, x and z as quantities to be determined. EesoiiiUon of '^^^ determination of these unknown quantities is equations. called the resolution of the equation; thus, for in- EaUATIONS. 119 20 stance, to find out -=- or the value of x, which makes the first o member of the equation 3:^ — 4 = 16, equal to the second, is to resolve the same equation. Determinate Now, when an equation contains only one un- and indetermi- , , , ,^ , ^ natceiinations. known quantity, it IS called a determinate equa- tion; when it contains more than one unknown quantity, it is called indeterminate. The reason of such an appellation is, that an equation which contains only one unknown quan- tity, has either only one or a determinate number of resolu- tions; and an equation which contains more than one unknown Roots of eqna- quantity, has no determinate number of resolo- *'°''°' tions. The value of the unknown quantities are termed also roots of the equation. ARTICLE I. Equations of the First Degr^. Begi-eeofthe § §4. The degree of an equation is given by ejuations. ^jjg highest exponent of the unknown quantity or quantities. Thus, for example, the equation x^ — ax = h, in which the highest exponent of the unknown quantity x is 2, is an equation of the second degree, and the equation ys — x' -)- ayx ^m -\- q, in which the highest exponent of the unknown quantities x and y is 3, is an equation of the third degree. Equations, therefore, of the first degree are all those in which the exponents of the unknown quantities do not sur- pass unity ; such, for instance, are the equations 120 ' TEEATISE ON ALGEBRA. ax -)- 6 = J, at/ -\-bx = q — x, &c. ; and generally, ax" — 6y -f abx"-^ — j^»-2-|-/= r, is an equation of the n*^ degree. General for- § ^^- -^"7 determinate equation of the first ""^equatten^f degree may always be reduced to the simple form the flret degree. X = A (l), because it cannot contain other terms except known quantities, and those in which the only unknown quantity x is either alone or afieoted by a coefficient j as, for instance, in the equation ax -\- b — c -\- dx ^ ^x — ^ -\-/- Now we may first transpose all the known terms to the second, and the remaining to the first member, and have the eqjuation ax -\-dx — -X = c — 5 — ^-\-/) and again, (a-\-d — jx = c — b — i-{-/- The known terms [a-^d — „ ) ''^'^ easily be reduced to a single term C, and likewise the terms — 6-|-c — l-\-f, to the single term K, so that the same equation can simply be written as follows : Cx = K. The unknown x being cleared of its coefficient, we will have _K a?— Q, an equation of the same form as (i). Resointion of -^0^ (0 ^^ ^ resolved equation. Therefore, to SaUonfThe resolve any determinate equation of the first first degree. degree, Transpose all the known terms to the second member, and all the others to the first ; reduce each member to a single term, and clear the v.nlenown quantity/ of its coefficient. EQUATIONS. 121 Resolve the equations Examples. -i ' 4 (2.) (3.) 6. :13y— 26 1 7 Answers : (3.) (5.) (4.) (5.) 7y-7:= 2^-4+1 = ^-4.- lx + V2=lx ay -{-1 — = my — /. X = 4. 7 (2.) . = ¥■ (4.) X = 48. y c-h-f a — m § 86. Equations can be profitably used in the resolution of problems, since the conditions of any problem can be expressed by one or more equations. Let us see some examples : What number is that which first multiplied by 2, and then divided by 7, gives 18 for the difierence between the product and the quotient ? Ans. The product is 2x, the quotient is =, the difierence X is 2x — z ; therefore, the equation is First. 2a;- X "r 13, which, resolved, gives . . . x = 7. That is, 7 is the number required. There is a certain number of apples to be dis- tributed among another number of boys. If we give to each boy three apples, there are nine wanting ; but if we give only two to each, then there are two remaining. How many are the apples, and how many the boys ? n 122 TREATISE ON ALGEBRA. It is plain that after having found the number x of the apples, the number of boys is also found ; because, dividing X by two, we have the number of boys, plus one; that is, the number of boys is ^ i 1~ ' the unknown quantity of the problem is then only x. But addiag 9 to x, and dividing a; -(- 9 by 3, we have again, according to the condition of the problem, the number of boys; that is, x -)- 9 x ^ ~3~ ^2~ An equation which, resolved, gives a; == 24 . . . number of apples ; and consequently, ^r — 1 = 11, number of boys What is the number, which multiplied by 3 and divided by 7, gives a product and quotient whose difference is 20 ? Ans. a; = 7. Find such a number that the sum of one-third, one-sixth, and one-twelfth of it shall be equal to 21. Ans. X =^ 36. A soldier receives every day twelve cents; but when he is engaged in the service, the first time in the month he receives twice as much; the second time three times as much ; and the remaining days of service four times as much. At the end of a month of 30 days he receives five dollars and four cents. How many times was he engaged in the service ? Ans. a; = 5. Indeterminate § 87. The preceding examples and problems equations of the -i p \ t first degree. show that the value of the unknown quantity in determinate equations is determinate; namely, only one. But this is not the case with indeterminate equations. Let us take, for example, ax — 5 -J- cy ^ 2 — X — y, EQUATIONS. 123 in which x and y are both unknown quantities ; hence, the equation may be resolved either with regard to x or with re- gard to y. Let us resolve it with regard to x ; we will have _l+q c+1 Now X depends on the value of y, and giving, for example, to y a numerical value equal to 1, the corresponding value of x is _h±q c+l and giving to y a numerical value equal to 6, the correspond- ing a; is b -\- q c-j-1 a — 1 a — 1 Unless, therefore, the value of y be determined by some con- dition, the value of x also remains undetermined, depending on any arbitrary value given to y. But if two equations are given, containing each the same two unknown quantities x and y, then the value of both can be determined; nay, more generally, when a number of different equations is given equal to the number of the un- known quantities contained in them, all the values of the un- known quantities can be determined. Eauationscon- § ^8. The determination of these unknown unknownTi^n- quantities can be obtained in different manners, *'''^^- as we may see in the following example : Let each of the equations — az -\- a'y + a"x -\- a'" = \ bz + h'y + h"x + V" = [ (o), cz -f c'y -f c"x + c'" = 3 contain the same unknown quantities x, y, z; and observe, that any equation of the first degree containing three unknown quantities, can be reduced to the form of the equations (o); so that the different methods applicable to obtain the values of the unknown quantities contained in (o), are applicable to all 124 TREATISE ON ALGEBRA. First method. a a a Elimination by Z = y — X — comparison. a a a V b" V" 2 = -pj- r"- h c' c" c"' z — y — — X c c c similar or equivalent cases, and generally to any number of equations containing an equal number of unknown quantities. Different me- Resolve, first, each equation with regard to the tliods of resolu- . j» i 'n tions. same unknown quantity; tor example, z, we will have (o'). In this manner, since the first member of each equation (o') is the same, and, consequently, the second members are all equal to each other, we have b'" 'b > a"' ■ C c" c"' u X = V ^ ■ a a a a c c And, consequently, by transposing the terms to the second member, we have a a" a" ■ —X — a a u b" -p- r" -d c" -y- (- b'-. , /a" b'\ , a'" b" ■b>+{a-r>+a — r c"n »=e-;>+e-:->+i And making b\ b" e-^)=^'(^-^)=''(^-^)=^■. dy + rf'x + d" = I 3j, _(- S-x + 5" = I (-^ ^• Now resolving each equation (o") with regard to y, we have d' d" EQUATIONS. 125 5' d" and consequently, d' d" _ d' S" from which (-_-)x+(^--)==0, and making \d j)-"' Kd s )-" ' Dx + D' = I (o'"), which resolved, gives a determinate value of x; namely, D' Now this value suhstituted in one of the preceding (o"), gives us an equation containing only the unknown quantity y, whose value, therefore, can be obtained; and this, together with X substituted in any of the given equations (o), gives us an equation containing the unknown quantity z alone. This method of elimination is called elimination by comparison. Let us pass to the second method. Elimination by '^^^ second method coDsists in resolving one of substitution, jj^g gjyen equations (o), for instance, with regard to z, and then substituting the found value of z in the other equations. Thus, we obtain two equations containing only the unknown quantities x and y, to which the same method of elimination can be applied, in order to obtain an equation with only one unknown quantity. But let us see the process of the operation. Resolve the first equation (o) with regard to z, we will have a a = y X - a a }(!>)■ This value of Zj substituted in the second and third (o), gives us 11* 126 TREATISE ON ALGEBRA. \ W (Jb Qj / c(_4- -X- — ) + c'y +c"x+ c"'= 0; V a a a ^ from which we deduce the two equivalents 0'-?>+('"-^>+('"-^)=«. ~ ('■-?>+('"-v>+(«"'-^)=«. and making we will have dy -\- d! x -\- ), gives us an equation with the unknown quantity y alone, and EQUATIONS. 127 substituting x and y in (^), we obtain the value of the third unknown quantity z. Elimination The elimination by addition and subtraction in by addition and subtraction. some casss IS preferable to the two preceding. This method of elimination consists in giving the same co- efficient to the same unknown quantity in different equations, and then subtracting one equation from another if the terms affected with the same coefficient have the same sign, or adding the equations if the terms affected with the same co- efficient have different signs. Let us resume the equations (o) ; and first to reduce the unknown quantity z to the same co- efficient in the first and second equation, multiply all the terms of the first (o) by the coefficient of z of the second, and all the terms of the second (o) by the coefficient of z of the first, we will obtain the following equations : haz + la'y + hd'x + 6a'" = 0, haz + aVy -(- aV'x + aV" = 0, which, subtracted from one another, give y(ha'—ah') + x(ha"— ah") + (ba'"— ab'") = 0. In equal manner, reducing to the same coefficient the first term of the second and third (o), we will have cbz + cb'y + cb"x + ch'" = 0, cbz + b(/y -J- 6c"x -f be'" = 0, and, consequently, y(ch'— he') + x(cb"— hd') + (cb'"~ be'") = 0, making now , (ha'— ah') = d, (ha"— ah") = d', (ba'"— ah'") = d", {cV— bd) ^ S, {ch"— he") = 3', (ch'"— be'") = S". . The obtained equations will become dy-]-d'x+d"= I dy -\- 3'x -\- S" = 1^9^)- Eeduce now the first term of both (q') to the same coefficient, 128 TREATISE ON ALGEBRA. we will have ddy -\- d'dx + d"S = 0, dSy -\- dS'x + dS" = 0, which being subtracted from one another, give x(d'd — dd') -f (d"d — dS") = 0, and making d'd — dd'='D, d"S — 7/-\-Bv^ 32. r a: + 3/ = 36, (7.) ] x + z=m, ( _y 4- z = 53. Answers : (5.) a; = 3, y = 4, z = 8. (6.) ;b = 3, 5- = 1, z = 5, « = 9. (7.) x = 16, y = 20, z = 33. § 91. Problems frequently contain more than one unknown quantity. In this case the conditions of the problem must commonly contain as many equations as there are unknown quantities to be determined. The skill required in the resolution of the problem consists in knowing how to give the algebraical form to the equations proble- matically expressed. Practice and natural aptitude, rather than any rule, facili- tate the resolution of problems. We may, however, observe that the difficulty in the resolution of problems General rule. . if-.Tii i- ^ t IS greatly dimmisned by this general rule : Separate first the unknown quantity, and then modifiy and combine them according to the conditions of the problem. An application of this rule may be seen in the following example : 132 TREATISE ON ALGEBRA. The three ciphers of a number are such that their sum is 14 ; the sum of the first and last divided by the second, gives 6; and subtracting 594 from the given num- ber, the difference contains the same three unknown ciphers, disposed in an inverted order. What is the number ? The three figures of the number are the unknown quantities of the problem, which we separate from the known quantities contained in the problem, calling them x, y, z. Now the first condition is, that the sum of the figures is equal to 14. Hence, the first equation x + y + z = U (1). Another condition expressed in the problem is, that the sum of the first and third figures, divided by the second, gives 6 for quotient ; hence, the second equation — J^ = 6, or x + z = 6i/ (2). The last conrlition is, that subtracting 594 from the unknown numbers, the remainder is the same unknown number iakeu in an inverted order. The equation contained in this condi- tion is not so obvious as the preceding ; to deduce it, observe that the number 594 may be decomposed, as follows : 594 = 500+90 + 4 = 100.5 + 10.9 + 4; hence, the number also, whose first cipher is x, the second y, and the last z, is likewise resolvable into three ; namely, 100.x + 10 .y + z; and therefore, the inverted number is 100.« + 10.y + rc. Hence, the equation contained in the last condition is lOOx+lO.y + z — 594 = 100.2 + 10.y + a-, or 99a; — 99« = 594 which is easily reduced to x — 2 = 6 (3). EQUATIONS. 133 Thus we have obtained as many equations as there are un- known quantities. To have them resolved, subtract first equa- tion (2) from equation (1); we will have y= 14 — 6y; that is, y = 2, which value of y, substituted in equation (2), gives + z = 12. Subtract now from this, and then add to the same equation (3), we will have 1% = 6, 2a; = 18 ; that is, z = 3, x ^ 9 ; the required number, therefore, is N = 923. We have, in fact, K4-y + 2 = 9 + 2 + 3 = 14 3; + ^ _ 9+3 _« y - 2 -° 923 — 594 = 329. What two numbers are those whose difierence is 9, and sum three times as much ? Ans. x=18, y=9. What three numbers x, y, z are those whose sum IS d4 ; the sum oi the last, and twice the first is 30 ; and the sum of the first and twice the second is 26 ? Ans. a; = 6, y = 10, zz= 18. The weight of four globes A, B, C, D is 340 Problem 4 , . , „ . -r^ . pounds, and the weight of A -f D is equal to that of B -f C; C is ten pounds less than B; and the weight of D, plus one-third that of B, make the weight of A. What is the weight of each globe ? Ans. Calling x, y, v, z the respective weights of A, B, C, D, we will find x = 100, y = 90, d = 80, z = 70. 12 134 TREATISE ON ALGEBEA. Id a mixture of wine and water, one-tenth of the whole, plus 10 gallons, is water, and one-half of the whole, plus 30 gallons, is wine. How many gallons are there of each ? Ans. x = 80, y = 20. Divide the number 144 into four such parts, that, if the first be divided by 5, and the second multi- plied by 5, the third diminished by 5, and the last increased by 5, the quotient, the product, the difference, and the sum are all equal. Ans. Calling x, y, v, z the first, second, third, and fourth parts, a; = 100, y = 4, v = 2b, z = 15. Three persons A, B, C have each a certain sum of money: one-third of the money of A and C, minus 6 dollars, is the sum of B; one-half the money of C, minus the money of A, and minus 9 dollars, give, again, 2 the sum of dollars of B ; the sum of C, multiplied by -, gives y twice the sum of A. What is the sum of each? Ans. Calling x, y, z the sums of A, B, C, we have x = 18, y = 54, a =162. Five wheels A, B, C, D, E are so combined, that while A performs x revolutions, B performs y, C performs v, D, w, and B, z. Now ten times the revo- lutions of A, plus three times those of B, and four times those of E, give the same number as 9 times the revolutions of D, plus the product of the number of revolutions of C 33 multiplied by ^n; twice the revolutions of A, plus twice those of C, give the same number as the revolutions of D, added to one-fourth of tho.se of E ; the revolutions of D and E, plus ten times those of B, are equal to seven times the revolutions of C ; the revolutions of A, plus five times those of B, give the revolutions of C ; and the revolutions of E, minus three EQUATIONS. 135 times those of C, give 20 revolutions. How many revolu- tions does each wheel perform in the same time ? Ans. a; = 10, y = % « = 20, k; = 40, a = 80 Problem 9. What fraction is that whose value is g, if we add 1 . . . 1 to its numerator, and 7, if we add one to its denominator ? ' 4 X 4 Ans. - = — ^. y 15 There are two horses and two saddles : the best saddle costs 40 dollars, and the other only 6; placing the best saddle on the first horse, and the other on the second, the first horse costs 6 dollars more than the other^ and chang- ing the saddles, the second horse costs three times more than the first. What is the price of each horse ? Ans. X ^ 25, y = 53. ARTICLE II. Equations of the Second Degree. General foi- § 92. Any equation in which the highest ex- mula of the do- .... terminate equa^ ponent of the uuKnown quantity or quantities is tions of the *^ . „^ . \. i in second degree. 2, IS (84) an equation or the second degree. Hence, the general formula of the equations of the second de- gree containing only one unknown quantity, is cc»+Aa; = B (h); because all the terms to be possibly found in this class of equations are either known quantities or terms containing the simple unknown quantity x, and terms containins the square of X ; as, for instance, in the equation ma? — nx -\-px^ — q = x-\- ri? — fx -f- g, in which q and g represent known quailtities, like the coefficients 136 TREATISE ON ALGEBRA. m, n, p, r,f; the terms nx, x,/x contain the simple unknown quantity x; and the remaining turms the square of jc. It is now easy to reduce this equation to the form of the preceding (/t), because, through a simple transposition, we can first write mx^-\- ip3? — rx"^ — nx — x -\-fx =^ q-\- g ; that is, {m-\-p — r)jcP -f (/ — n — l)a: := 2-\- g, from which x" + x = — ^'" — m-\-p — r m.-\-p — r an equation of the same form as (/t). Hence, the resolution of any equation of the second degree is the same as the reso- lution of the general equation (A); and the first operation to be miide when an equation of the second degree, containing only one unknown quantity, is given to be resolved, is to re- duce the given equation to the form of (7t) ; hence, also, the first and general rule : Transpose all the known terms to the second member, and all the others to the first ; reduce to a single term all those that contain the square of the unknown, quantity, and likewise all those which contain the first power of the same quantity ; then clear the square of the unknown quantity of its coefficient. It is to be observed that equations of the second degree do not sometimes apparently contain terms with the square of the unknown quantity ; as, for instance, in the equation , n ax — 6 = - ; X but the same equation is reducible to the following : ax^ — bx ^ n. In cases similar to this, before proceding to resolve the equa- tion, a similar transformation is to be made. The given equations being thus prepared, we may pass to see how the general equation (h) is resolved. EQUATIONS. 137 Keeoiutionofthe §93. Two oases Can take place with regard to general ec^uation. Two cases. the first member of Qi), according as the co- efficient A is either equal or not to zero. In the first case the equation is simplified, and becomes x»— B Qi'), which is easily resolved, because, from Qi'), we have -j/^ = ± t/B; or X = it i/K That is, the value of the unknown quantity x is the positive as well as the negative square root of the known quantity B, or the second member of Qi'). We have, in fact, [ + T/By= + B, [-l/Fl»= + B; both values, therefore, -(- j/B, — |/B, fulfil the equation (A'). When, therefore, the given equation, reduced to the general form, assumes that of Qi'), the double value of the unknown quantity is immediately found, as follows : Rule and ex- Tahe the positive and negative square root of ampiea. ^^^ second member. The equations, for example, (1.) 3x»— 3 = 6— 2a;a, (2.) 4x= — 36 = 2a;» — 4, (3.) 2aa;» — £c+6==ax» — x + SJ, reduced to the general formula, give (1.) a;''=9, 2.) x»=16, (30 .^=f, and resolved, (1.) X=±y/W=z±Z. (2.) x = ± i/IB = ± 4. (3.) .==±V? 12* 138 TREATISE ON ALGEBRA. But when A is not equal to zero, the equation Second case. i, , i-i-iii.- (/ij cannot generally be resolved without being modified, as we will presently see. Observe that (69) (x + i/^ = a;» + 2x2/ + &% and the second term of this evolution is the double product of the two terms of the binomial; so that the last term of the same evolu- tion can be easily inferred from the first and the second, by dividing, namely, this by the double root of the first, and squaring the quotient. For example, a" -\- 2ma, representing the incomplete evolution of the square of a binomial, it will become complete by adding m' to it, because -r — gives m for quotient ; and consequently, m/' for the last term of the unfinished evolution. In like manner, b' -f- ch, representing c' another unfinished evolution, by adding to it the term -r, it will become complete. That is, to render b' -^ cb a perfect square, add the square of half the coefficient of b. Now, the first member of the general formula (h), or a;'-j- Ax, has the form of an incomplete square, which is easily A» finished by adding -^j the square of half the coefficient of x. But in order to preserve the equality, if we finish the square in the first member, we must add the same term to the second member also. This addition being made, we will have A" A" or, since x^ + Ax + -^=^{x + -^)°; Hty-B+-. from which x + i=±jB + ^; EQUATIONS. 139 and consequently, the values of x, which resolve the equation (Ji) given by the known quantities A and B. We say the values, because, when we take the upper sign, A , L , A» when we take the other, 1 A'' =^v B + 4-' Eeaiandimagi- These values of X, or roots of the equation (K) nary roots. .^^jjj ^^ either both real or both imaginary ; and again, when real, both positive or negative, or one positive and one negative. A' Let us here examine all these cases. When the binomial B + -!-> under the radical sign, is a positiTe quantity, the radical, and conse- quently the value of x in both equations, is real. But when the same binomial is negatiye, then (50) the radical is imaginary, and the values of X contain the real term — -r-, plus or minus the imaginary root; and, therefore, both values of x are imaginary, because neither a positive nor a negative real term or quantity can ever represent the difference or the sum of two expressions, the one real, and the other imaginary. A* Suppose, first, the binomial B -|- -j- to be a positive quantity ; in this A? supposition B is positive, or if negative, less than — . When B is positive, one of the values of x is positive and the other negative ; when B is negative, the values of x are either both positive or both negative. We have, in fact, in the first case. ^F?>^|?(-I)^ 140 TREATISE ON ALGEBRA, and, therefore, we may write ■■y[^+^ = ^{^ + 'y' hence, from (A'")' '^ ^^ coefficient A in (A) is positive, ^ = _|-|-rf = -A-d; and if the coefficient A in (A) is negative, a: = +|+|+«[ = A + (?, x = + j—^-d= -d. „, . ., , Hence, When in the equation (h), that it, First cnterton. ' _ 7?+ Ax = B; B is positive, one of the roots is likewise positive, and the other negative^ whatever be the sign of the coefficient A. A2 In the second case, when B is negative and less than — , we have I 3* A thatis, \/^+T ~ 2 ""''' therefore, if the coefficient of i in (A) is positive, we will have from x=.-- + --d=-d. =-^^F?• both valnes of x heing negative. In the same supposition of B negative, if the coefficient A of sc also is negative, we will have from (A"') *+v^. 2 x = \-\+d = + d, both values of x being positive. Therefore, EQUATIONS. 141 'g , .. _. . When in the eqitafion (h), B is negative and less than — , and A is positive^ tlic roots are both real and negative. With the same B and with A negative, tht roots are both real and positive. A* Let us now suppose B negative, and equal to — . In this case, 4 .+T-»^ and consequently, both values of x from (A'") are equal to each other, and have the same sign. That is, A2 Thi-d 'te ' When in (h), B is negative and equal to — , both valuer of the roots are equal to - positive, being the coefficient of x A 2 negative, or equal to -^ negative when the coefficient ofxis positive. A2 The last case is that of B negative, and greater than — , in which A^ 4 case the binomial B4- -r is necessarily a negative quantity, — d; hence, 4 „ , A2 an imaginary expression. Therefore, When in (h) the value of B is negative, and greater Fourth criterion. ., A^ , «., _,. i ^i - than -J-, the roots of the equation are both imaginary. The preceding criterions applied to the equations — (1.) z^+1x = \2, (5.) a:2^12a; = — 36, (2.) a;2— 15a: = 18, (6.) a:^ — 8s;=— 16, (3.) a;2+163: = — 40, (7.) a;2+14i = — 50, (4.) «2— 20a;=— 90, (8.) a:^ — 6a; = — 18, reduced already to the general form (A), show that the roots of the equations (1) and (2) are real, and affected with different signs ; the roots of (3) are also real, and both negative; the roots of (4) are real, and both positive ; those of (5) are also both negative, and, besides, equal to each other ; those of (6) are both positive, and equal to each , other. The roots of the equations (7) and (8) are imaginary. Examples and § ^4. Let US proceed to resolve some equations, problems. Q^^^^ equations : (1.) 2x» + a; = a;» + 54 — 2x. (2.) a;» — a; = 56. 142 TREATISE ON ALGEBRA. Given equations, (3.) 2x» — 4.C— 9 = a;=+2x — 17. (4.) 16a;'' — 4a; + 36 = 14a;'' — 32a; — 60. (5.) X* — ab -\- ax^ hx. (6.) x^ -\- mn — nx = mx. (7.) a:« -I- ex — c» = — 2cx — 3c». (8.) a-» + 26 = 4x + 13. (9.) x''—2ax + a'+b<' = 0. The first of the proposed examples is easily reduced to the general form (^), as follows : a? + 3a; == 54, and adding the term (-) in order to have a complete square in the first member, we have x» + 3x + |=54 + | or (^a:+-) = 54+^5 and consequently, , 3 /22r 15 hence, the double value of x : The process is the same for the other examples, and is con- t^ned in the annexed practical rule : Reduce the given equation to the general form, finish the square of the first member, extract the root of both members, and leave the unJcnown quantity/ alone in the first member. In this manner, the remaining examples, being resolved, will give : EQUATIONS. 143 x=1 {X ^ — 6 c X ^= — a x = -S. (5-) {.= +6. (6-){, = „ (7.){,= _2.. ■ 3;/— T. t x = a — V^=^ When the conditions of a problem, whose resolu- tion is reducible to that of an equation of the second degree, are such as to excludi.', for instance, the negative sign for the unknown quantity, and the equation resolved gives the values of the unknown quantity affected with opposite signs, the positive alone resolves the problem. The square number of my dollars added to 180, gives 27 times the number of my dollars. How many dollars have I ? Ana. a; ^ 12, or x =: 15. I have as many dogs as he has cats. All my dogs, plus four of his cats, multiplied by the whole number of dogs and oats, give 12 times the number of dogs, plus 160. What is the number of my dogs ? Ans. The equation resolved gives x = — 8, a;==10; the first value is to be excluded. Hence, x = 10. Find a number whose product by 5, minus six units, mutiplied by the same number added to 1, gives for product seven times its negative square. Ans. x = -\-g, x=z — ^. The product of a certain number by 7, minus 75, Problem*. . , „- , , . . . „ -,„^ is equal to 95, plus the quotient arising from 125 divided by the same number. What is the number ? 5 Ans. X = -)- 25, X = — =. 144 TREATISE ON ALGEBRA. An army commencing battle, contains an equal number of men in each rank, and it contains as many ranks as there are men in one rank. During the battle, the first three ranks and 350 men beside are killed. ~ The army after the battle contains 2000 soldiers. Find the original number. Ans. X := 2500, each rank containing 50 men. With regard to equations of the second degree, containing more than one unknown quantity, the same methods of elimi- nation given in the preceding number (88) can be applied ARTICLE III. On some Properties of Determined Equations of any Degree. PreEminary 2 ^^- Several discussions of the present article rest on theorems. some general theorems, useful for other investigations, no less than to find out the properties of equations of any degree. We commence, therefore, this article by demonstrating the same theorems. And, first, Let the coefRcieuta a, b, c, . . . h . of the variable and real quantity 2, and the last term k of the polynomial 02" -(- Jz"-' -|- cz"-' -f- . . . h.z-\-k . .. . (p), be all real and invariable quantities. We say, that if by changing the value of «, (p) assumes a positive and then a negative value, „, , There must be some value z„ of z, which substituted in (v). Theorem 1. ^ m ./ r ^r/* makes this polynomial equal to zero. To demonstrate this proposition, let us suppose the value of z to be changed in such a manner that the difference between any two such values, taken in succession, be capable of an indefinite diminution. In this manner the polynomial (p) also will be changed by degrees capable of indefinite attenuation. But by supposition, the polynomial (p) may be changed from positive into negative; so that, making, for instance, 2 = 2j; the polynomial assumes the positive value {-\-p„); EQUATIONS. 145 and making z = z„, the polynomial becomes negative, tint is, ( — ^„). The difference, therefore, between the two Taluea of the polynomial (P), ia [Ph) + {Pn), which may become smaller and smaller either by «« approaching to «„, or z„ to Zj, or both of them to each other. Suppose that lea-ring «„ unchanged, 2j approaches constantly to «„, the difference (^,.) + {Pn) will indefinitely approach to zero, and by degrees capable of indefinite attenuation ; that is to say, the said difference is capable of assuming all the values contained between (^j) + (pj and zero ; now ( j)„) is one of these values ; therefore, among the values which the difference (y^) -)- (^„) will take by Zj approaching incessantly to «„, is also (p„) ; and since the difference [p^) -\- [p^) cannot become (^„) unless (^j) becomes zero, therefore, the value of (^/,) constantly changed with Zj vrill once become zero. Call «„ the value which makes [p,^ ^ 0, we will have az„»+ 6s„"-'+ c«»— '+ 1- fe„+ A = 0. „ When the decrease of the variable z in fp) is carried to a Theorem 2. '' ^^^ certain limits the polynomial retains from that limit con- stantly the same sign, equal to the sign of Us last term. The polynomial [p) without its last term is ffiz" 4- 62"-' + c^~'' +... + &, which evidently approaches to zero by constantly diminishing the value of 0. Now it cannot uninterruptedly approach to zero without becoming smaller than any fixed value different . from zero ; hence; by diminishing constantly in [p), the value of z, all its terms, with the exception of the last h, will finally become a smaller quantity than the same k. And, consequently, from this limit, whatever might be the sign of the rest, the sign of the, whole polynomial {p) will be that of k, if h is positive; (p) also, from that limits will be constantly positive ; if A; is negative, (p) from the same limit will be also negative-. When % in (p) is increased to u, certain limit, the poly- nomial from that limit will constantly retain the sign of its first term. The polynomial [p] is manifestly equivalent to the follo-vring product : Now, by constantly increasing the value of z, each term within the parenthesis, except the first, approaches constantly to zero, and, con>- 13 146 TREATISE ON ALGEBRA. sequently, also the sum of all of them. Hence, the same sum, yrhen z is increased to a, certain value, will be equal to and then become smaller than the fixed quantity a. If now, for the sake of brevity, we call S the sum of the diminishing terms, we will have (p) = 2»(a+ S). In which, when z is increased to the said limit, and much beyond that limit, S is smaller than a ; hence, from this limit, the sign of a + S must be the same as that of a, whatever be the sign of S ; hence, also, the sign of the product i"(a -|- S), that is, of the polynomial (p), is the same as the sign of ai', which is the first term of the same polynomial. When two polynomials, such as "0+ a.2+ 0^'+ • • +a»s" ■) . , «»+ «i2 + Cj^^ + • . • + c,.2» ; ^^)' remain equal to each other, substituting in them n-|- 1 different values ofz, the two polynomials are identical. Let «„, u„ Zj, . . . . «„ represent the « -f- 1 different values, which, substituted in succession in the polynomials, make them equal to each other ; that is, o„ + a,2„ + (V„2_|_ ...-}- a„z„" = c„+ c,z. + c^„2 + . . . . -}- (;„z». Hence, also, calling p„ p^, p„ . . . . the first members of these equa- tions, we wiU have Theorem 4. and likewise. a„ + fflA 4- "Vo^ + • • • + "'^0" = J'o ffl„+ «A+ ) > > termtafid e^uit terms K of the equation ttc,aof anyde- ^^^ ^_^ Ax«-1+ B^-^+ . . . + Ha; + K = 0, of the n" degree be real quantities. If taking for x two real values, the one makes [e] positive, and the other negative, the same [c] is resolvable with at least one real root ; that is, there is at least one real value of x, which makes the iirst member of [e] equal to zero. To prove this, it is enough to apply to [e] the demonstration of the first theorem of the preceding number. Bguations re- I'^* ^^ ''■°'^ see how one of two values of z makes [e] solvable with at positive, and the other negative. When the degree n of root. the equation is an uneven number, the sign of the first term x" is the same as the sign of x; but when x has a sufSciently great value according to the third theorem of the preceding number, the sign of the whole polynomial [e] is the same as that of the first term ; hence, positive if x is positive, negative if x is negative. When, therefore, the degree of the equation is uneven, the first member of [«], by the substitution of one certain value of x, can be made positive, and by another negative ; hence, the equation in this case is certainly resolvable with at least one real root. Equations re- When the degree n of the equation is an even number, solTaWewithat amj the last term k is negative, the equation is then re- ISSiSti two TBEli roots. solvable with at least two real roots. Because, taking a very small value for x, the sign of the formula [e], according to the second among the preceding theorems, is the same as that of its last term K ; that is, negative. And taking a sufBciently great value of x, then the sign of the polynomial is the same as that of the first term. But the sign of the first term is positive, whether the value of x be positive or negative ; therefore, a value of x between one very small and another large and positive, wiU make the first member of [e] equal to zero ; and again, another value of x between the same very small value and another large and negative, will make the first member of [e] 148 TREATISE ON ALGEBRA. equal to zero. When, therefore, the degree of the equation [e] is an even number, and its last term K a negative quantity, the equation [e] = can be resolved with at least two real roots. When eqoa- X 97. Tpfe have seen that when the degree of the equa- tions cAunot be . . resolved with tion [e] :^ IS an uneven number, the equation is are "resol'vaWe al'ways resolvable with at least one real root ; and when with one or jjje demee n of the same equation is an even number, more imagi- nary roots. the resolution may be obtained with two different real roots, provided the last term K of [«] be a negative quantity ; but if K should be positive, and n an even number, we would then be unable to demonstrate the possibility of resolution of [e] = with real roots. Because, although [e] involves a function of x, and always of the same real form, and consequently in the equation [e] = the variable x is necessarily a reciprocal function of A, B, C, . . . K, that is, 2;=/(A, B, ...H, K), of a determined and unvariable form, it may occur that the change of the sign of k changes the real value of x into an imaginary one. For example, the real value of the expression in which A and B are supposed to be positive, becomes imaginary when B, being greater than A, has its sign changed. Bat whatever be the value of x, either real or imaginary, it is certain that by substituting in [e] = instead of x the function /(A, B, . . H, K) the equation is fulfilled ; and, therefore, when the degree of the equation [e] = is an even number, and the last term K is positive, the equation is resolvable, at least with an imaginary root. Last supposi- '^^ ^''■^^ supposition which can be made with regard ti™- to the last term K of the general equation [«] = is, that the same term be equal to zero. In this supposition our equation becomes equivalent to the following one : s:[Aa:"-i + Bx"-2 4- + H] = 0, or (making Aa?*-' + Bif -'-{-....-{- K = [e']) equivalent to a;[e'] = 0. It is now plain that any value of x which makes («'] = 0, makes also x[^e''\ = 0. That is, any root which fulfils the equation [«'] = 0, fulfils also the equation [e] = 0. But [«'] is a polynomial, having EQUATIONS. 149 tlie same form as [e] ; and we hare seen above, that the equation [e] = admits always at least of one root, either real or imaginary. Therefore, the same equation is likewise resolvable when K = 0. Any deter- 3 98. We may now pass to see that the number of mined equation . .... of any degree roots 01 any determined equation is always equal to the SinyVots as degree of the same equation. there are units yfe have Seen that the equation [e] ^ admits in all in the degree ^ ■- -* of the equation, cases of at least one root ; call A this root, and from [e] = 0, we will have A»+ AA»-'+ BA»-2+ . . . . + HA+ K = 0; and consequently, K= — /i"— AA"-i — BA»-2— .. ..—HA. Substituting now this value of K in [e], we will have [e] = x"+ Ai"-'+ B3f -''-{- + Ha;— A»— AA"-> — BA''-2 — ... — HA, or, [e] = (a:"— A») + A(a;"-i— A»-i) + B(a;»-2— A»-2)+...4-H(k — A), in which x may have any value. Now we have seen (63), among the last examples of multiplication, that (l-(-2-f-s2+ . . . +3»)(l — 2) = 1— j;«+i, from which, by changing the signs of both members, and taking n — 1 instead of n, we infer »" — 1 = (l+z-|-32+ . . . + 2»-i)(z — 1) ; or, substituting - instead of s;, ©■-'-(>+l+(i)'+.... + (in(I-:), ;«nd from this or a:" — A»= (a: — A)(A''-' + A"-%;4- A"-V+ .. . + a:»-'). Inverting the order of the terms of the last polynomials, and then substituting in succession n — 1, n — 2, « — 3, . . . . instead of n, we win have x^—'h!^ = (a:— A)(a:"-i+ Ax»-=+ AV-'+.. . + A"-') a:»-i_ A»-i — (a; — A)(a:»-2 4- Aa^--^- A'a?— ^4- . . . + A"-") jn-z_ A"-2 = (a;— A)(a?'-» + Aa:»-'+ A'a;"-^ + . . . + A"-'), &o. 13-s 150 TREATISE ON ALGEBRA. Making now, a substitution of the values of these binomials in the expression of [c] last obtained, we will have [e] = (x — k)(ii^~^+h3f'-' + ....)+A{x — h){3f-'+ha^-'+...) + B(x- A)(x«-» ....) + ... + (x — h)n; or else, [e] = (x — h)[tr-'+ {h+ A)3f-'+ {K' + Ah + 3)31'-'+ .. + (A»-' + AA»-^+ BA—» + H)].; or more simply, [c] = (x — A)[a?'-i + B,s»-2+ C,a:»-s+ . . . + H,a; + K.] : making, namely, A + A = B„ A" + AA + B = C„ &o. Eepresent now the last polynomi&l by [f,], that is, make af-i+ B,7i—'+C,x''-'+... + H.I+ K. = [e.]; then we will have [e] =: (a; — A) [e,], whatever be the value of x. Therefore, A being a root of the equation [e] = 0, the polynomial can be decomposed in two factors, one of which is a; — h, the second another polynomial [e,] of the same form as [e], but of a degree one unity lower than that of [e]. Now, resuming again [e] = 0, or its equivalent {x-h)M=0, it is plain, that not only by making x equal to A, we will have the equation fulfilled ; but also, any value of x which renders [e,] equal to zero, fulfils likewise the equation ; that is to say, any root of the equation, [e,] = 0, is a root also of [«] = 0. Now, [e,] = admits certainly of at least one either real or imaginary root, which we may call i ; and applying to [e,] z= that which we have said with regard to [e] ^ 0, we will have [cj = (x — i)[iif'-^+ Cji"-' + .... + H^j;+ K,], and making 3if-' + C^-'+ . . . + K, = [«J [e.'] = (a; — i)[6j; and since [e] = (x — A)[e,], also M = (a;_A)(a;-i)M. But the polynomial [ftj], like the other two [e] aijd [«i], can be de- composed into two factors, the first of which having the form (a; — I), and the second the same form as the preceding polynomials [c], [e,], [ea], with this diflference, that the highest degree of x in [e] is b ; in [e,], n — 1; in [e^], n — 2, in the following polynomial [«,] is n — 3. EQUATIONS. 151 Coutinuing the same process, we will finally obtain the polynomial [e] decomposed iato n factors of the first degree, and of the same form. That is, the polynomial [e] will become equal to a product of n factors in the following manner : a?* 4- As" -' + Bs--^ + ^- Ha; + K = (a: — A) (» — i) {x — I) ("-')■ From the supposition, therefore, that the polynomial [«] is equal to zero, it follows that it may be decomposed as above. Now from the last formula it is evident that substituting for x any of the n values h, i, I, . . . . t, the equation [e] = is fulfilled. Hence, the roots of the equation of the n"" degree are n in number. Connection le- § 99. The product of /i equal factors of the form tween the roots , , , . ,„„ ,_„, and the coeffl- ["^ — "j IS [}>'>, l^) cientsof anyde- , ,, , , n{n — 1),, , , , termined equa- (a; — A)" = a?* — nto"-' + -i— - — '-h'x^-' — . . . =t A", tion. 2 In the last equation of the preceding number, we have a product of n biuomials; in which, however, the second term is different in each of them ; but since the first term x is the same in all, the product of those n binomials with regard to x, must be equivalent to that of n factors, all equal to the same binomial x — h. That is, x wiU com- mence with the highest degree n, and orderly diminish it tiU the lowest possible degree. The difference of the two products must be in the coefficients of x, which, when all the binomials are equal to x — h, are nh, ' — iA". . . . But the coefficients of the various powers of x, as well as the last term of the product, are formed iu the same manner in both cases. That is, when the term subtracted from x is the same A in aU the binomials, it is repeatedly used as a factor as many times and iu the same manner as the different terms A, i, I. ... in the other case. It is, besides, to be observed, that when the terms subtracted from X are all unequal, each of them must have an equal share in the formation of the coefficients and last term of the product. We may now institute an analytical comparison between the co- efficients — nh-\ — i— = — —h', . . . together with the last term A", when the factors are all equal, and those which are produced when the terms subtracted from x are different in all the binomials. The first of these coefficients — nh shows that — A is used n times as a factor of ar*-'. But when the subtractive terms are all different, no term can be found in the product multiplied by — A more than 152 TREATISE ON ALGEBRA. once, but all the same subtraotiTe terms concur in like manner to form the coefficients ; therefore, the coefficient equivalent to — nh must be in this case — (A + j -j- Z + . . . . + 0; that is, the negative sum of all the roots of [ej = 0. The next coefficient, or coefficient of the third n(n — 1) term in the supposition of aU the binomials being equal, is -| — '—^ — ^ A" which is "'■" ~~ ' times the square of h. Now in the case of tho terms taken from x, being all unequal, A can never multiply itself, and A" must be necessarily changed into the product of two differen* terms, for instance, hi ; but again, aU the terms taken from x, con- cur in an equal maimer to the formation of the coefficient; and as in the coefficient — — - — -If, the square of A is taken -^ — - — - times, the products of the terms, taken two and two, ought to be as many in number; and in fact, the number of combinations of « symbols, taken two and two, is (74) ^ — '-. Hence, the coefficient of the third term is [hi-\-hl-\- -^ht-^-il-lf . . .it-\- . ...); that is, the positive sum of the products of all the roots of [e] = 0, taken two and two. It is now easy to see, in the same manner, that the co- efficient of the fourth term is the negative sum of the products of the same roots, taken three and three ; the next, the positive sum of the products of all the roots, taken four and, four, and so on. And the last term is the product of all the roots ; a positive product if n is an even number, and a negative product if » is an uneven number. Our equation, therefore, a:" + Ax"-i + Ba:"-'i+ + K = (a; — A)(x — 1) {x — t) is equivalent to «" + Aaf-'-f Bar— =+ + K = a?>— (A-f-«-f- + <)x»-' + (Aj + AZ + . . + A« -1- 47 + . . . )ii'-' + . . . =t (A . j. Z i), in whatsoever manner x be taken. But (95, Th. 4, cor.) when two such polynomials are found equal to each other with any value of x, the corresponding coefficients of the same x are respectively equal to each other. So we will have A = -(A-l-i-(-Z-f ...-1-0 B = (Ai+ U+ + A* + a + + it+ ) C = — {Ml + Aii? + Jf. hit -\- hlg -[-..+ hit + ) (- (r). &o. K= =t A. j.Z (: EQtfATIONS. 153 That is, the coefficient of the second term of [e] = is the negative sum of all the roots of the same equation ; the coefficient of the fol- lowing term is the positive sum of the products of the roots taken two and two, &c. _, „ , 1 100. From this mutual connection between the roots Corollaries. ^ and iha coefficients of the equation [c] := 0, we infer some corollaries : „ If one of the roots should be equal to zero, the last term K of [e] must be also equal to zero : and if two of Case of one "^ -* ^ or more roots the roots are equal to zero, the coefficient H also of the term before the last is equal to zero, for it contains the products of all the roots taken (n — 1) and (ra — 1). In each one, therefore, of these products, there must be at least one of the roots equal to zero, and the whole coefficient is consequently equal to zero. Let the same be said of the coefficient preceding the two last terms, when three roots are equal to zero, and so on. CoroUarv 2 ^^ changing the signs of all the roots, the sign of the second term of Pel will be also changed ; that of the Signs of the "- -* o ' roots changed, third will remain unvaried ; the sign of the fourth will be changed ; that of the fifth will remain as it is, &c. The reason of this is, that by changing the sign of all the factors, the signs of the products will be changed only when the number of factors is an un- even number. Corollary 3 Multiplying each one of the roots h, i, I, . . . . t of the equation [e] = by the same quantity a, the coeffi- cients A, B, C, . . . H, and the last term K of [c], wiH then become aA, a^, a?G, .... a"-'H, a"K. Hence, vice vend,, if the terms of the equation a:" + Ax"-' -f Ba:"- ^ -f- . . . -{- Ha; + K = 0, be orderly multiplied by the terms of the series 1, a, ti?, a"-!, o». Hoots multl- ^^^ i^i ^^ fii'st by the first, the second by the second, *'''"*■ &o., the resulting polynomial ^ -)- aAa;"-i -f- a^x"-' + -f a»-'Ha: -f a^K, made equal to zero, will be resolved with the same roots of [e] ^ 0, each one of them being multiplied by a. Denominators Hence, also, if the coefficients of a given equation eliminated. contain denominators, they may be all cleared of them 154 TREATISE ON ALGEBRA. without giving any coefficient to the first term. Suppose, in fact, that the different denominators are b, c, d, . . . Multiplying the terms of the given equation by the terms of the series 1, (4c . . . ), (ic ...)',.. . the coefficients vriU be evidently aU cleared of their denominators, while the first term of the equation remains unchanged. The roots, however, of the equation thus modified to be reduced to those of the given equation, must be divided by the product (i. t. . .) Oo oUarv 4 ■''^* fourth coroUary deserves to be particularly noticed, on account of its use in the resolution of the equations of the third and fourth degree. In the equation [e] = 0, that is, af + Aa"-i + Bx?'-^ -\ + Ha; + K = 0, X stands to represent any of the n roots h, i, . . . of the equation, which roots depend on the coefficients A, B, ... in the manner above seen. Let us now suppose another equation of the same degree and form as [e] = 0, whose roots are h -\- a, i -{- a, &c. ; that is, the same roots as [c] = 0, but each increased by the quantity a. We may represent this new equation as follows : x'" + AV-^ + 'BV-'+ ... + HV + K', x' standing to represent any of the roots A +' a, i -|- a, . . . . and as the coefficient A of [e] = is the negative sum of all the roots A, i, I, ... of that equation, A' = -((A+ a) + (»+«) + • ••• + («+«)), Or, A' = — {h+i+l+ + t)—na,' But — (A + i +?+... + «) = A, therefore. A' := A — na. In other words, changing the roots of [e] = from ic into x' = x -\- a, the coefficient A' of the second term of the new equation must be changed from A into A' = A — bos. Let us now suppose a to be taken equal to — ; in this case A' =: A — A = 0. That is, when the roots of the equation [c] ^ are changed from x into x^ = x-\- —, the new equation must be without the second term. And this equation being resolved, it will be enough to subtract from the different values of x^ or roots the constant quantity — , to obtain the roots of the former equation. EQUATIONS. 155 The sums of J 101. Call S, the sum of all the simple roots of [e] of the roots can = 0, S, the sum of aE the squares of the same roots, though"'"' t£ S, the sum of all the cubes, &c. We will haye known. S, = K'+i'+P+ + (', S, = A'4- «' + «'+ + i', &0- Now, although the roots h, i, . . . t may all remain tminown, yet the sums S„ Sj . . . may be made known by the coefScients A, B ... of the equation. With regard to the first, it is well known that the negatire sum of all the roots is equal to the coefficient A of the second term. Hence, S. = — A. But to demonstrate the proposition with regard to all the sums, ob- serve, first, that (98) But [e] =a:»4-A2"-^ + B2"-2+...4-Ha; + K, (X - A)[e.] = ^+(B,- h)x--^ + (C. - AB,)x"-= + . . . + (K. — AHJs — AK,. Now, since the two first members of these equations are equal to each other for any value of x, so also are the second members. Hence, according to the fifth theorem (95), A = B. — A, B = C, — AB. H = K, — AH, K = — AK,. From which we infer B, = A + A, C, = B+AB. = B + AA + A2, B. = C 4- AC. = C + AB 4- A^A + h\ &c., K. = H -I- AG + . . . + A"-2A + A»-i. Tel But from [e] = [x — A)[e,] we have '^ = [ej and (98) [«J = !!:"-i + B,a;''-= + . . . + H,a;-f-K.. Hence, ^-M_ = [e,] = x"-i + (A + A)af'-» + (B -f- AA + A2>?»-= + (C + AB 4- A'A + A3);c»-''+ ....4-H + AG+.... + A"-2A4- A"-'. And since what we say of A may be equally said of any other root, we will have in equal manner, 156 TREATISE ON ALGEBRA. -^ = ar— 1 + (A + i)x"-' + (B + iA + t2)a:»-»+ . . . . + H +iQ + + !"-=A + 8»-S and so on. Now tlie roots of [ej == are the same as those of [«] = 0, ■with the exception of h; hence, the roots of _ , = are the same as the roots of [e] = 0, with the exception of h. Hence, also, on ac- count of the well-known dependence of the roots on the coef&cients of the equation, Tel Reasoning in the same manner with regard to the equations '- ■' X — i = 0, . . . . -!^^!— ^ 0, we will haTe ' x — t ' K".)- 'lw. The coefficient B in [e] is equal to the sum of the products of the roots, taken two and two. If we suppose one of the roots wanting, for instance h, the products of the remaining roots, taken two and two, win be given by B — A(i + Z + -f Q. Now the coefficient of the tiiird term of -^ = is equal to the sum of the products of the roots of [«] := 0, with the exception of A, taken two and two. Hence, B -f AA-f A» = B — h(i-\- 1 + ... t}}(c,). In like manner from the equations — t-i^ = . . . . ^ -* := 0, we have X — 1 X — i B + iA + i' = B — i{h + l+... + t)^ B+- ^ A, lab = K, (a 4- h^'—Vf' = A + BV=1, (a _ V^^^)" = A — KV^^. from these, (a + l^-=rxf = (a + iv^=l)(A + EV=1), Now (63) [a ± V^^)(* ± Kv^^^^) = («A — iK) ± (aK+ hh) And making oA — 6K = Z, aK -j- JA = m ; (a ± 6^=ri)(A ± K^=^) = I ± m^/=l, Hence, (a + i^/=^)= = Z + m^'^^^l, (a — 6^/7^1)3 = I — OTv'^— '1- In like manner, we have (a ^ V^^)" = « + Ov'^l, (a _ b^=lY = n — Ov'^l, and generally (a 4- V^^^)" = A + Bv'^^, (a _ 5^=^)™ = A — B^:^. [cl^^O^mnn™ ? ^"3- I'^ ^^^ supposition that one of the roots of [e] teo"oo^ugl°f = Ohas the imaginary form tt+V~l, substituting without admit ^ r^-i _ 0, that is, in x»4- Ak»-i + + K = 0, ting also of the L J ' > i i i ' other. that value instead of x, the. equation will take the form for taking separately each term of the equation, we will have Ax"-' = A(m + v^~^-VY—- = u" + v" v/^n, Bx«-i= =«'// + «'/V—^. &<=• K = K. And, consequently, calling U the sum of the terms u', u" .. . .K and V the sum of the coefficients v', v" . . . . «(»', we wiU have 160 TREATISE ON ALGEBRA. Now U 4- V^^^ cannot be equal to zero, unless separately U and V are each equal to zero ; because U, a real term, can never be elimi- nated by V^^n;, an imaginary one ; hence, [e] = 0, -which, in our rapposition, is U + Vv'— ~1 = 0. necessarily supposes U ^ and V = 0. It is now «asy to see that when x ^ u-\- B^^^^ is a root of [e] = 0, X ^ u — v,/^^^ is a root of the same equation likewise. Because, substitating this value of x in each term of [e], we have, X" z= (u — »y/^T.)" = m' — «' n/— T-i Ai»-i = A(« — t)v^^^l)»-' = «" — t)"v'— 1. Bar»-' = = !4'" — t)"V— T^. &o. E = K. And therefore, x» 4- Aa^-'+ + K = U — Vv''^. But when a; ^ « + "v'— 1 is a root of [e] = 0, U and V are sepa- rately each equal to zero; hence, U — V^ — 1, as well as U 4- V y/ — 1, is equal to zero. But U — V^ — 1 is that which [e] becomes when « — »y/ — 1 is substituted for x ; hence, x = u — v^ — 1 is a root of [«] = 0. Therefore, when one of the conjugate radical ex- pressions is a root of the equation, the other also is necessarily a root of the same equation. From this connection it follows, first, that the number Corollaries. of the roots of the imaginary form (o ± b^ — 1) must necessarily be even. Secondly, since whatever be the roots of [«] = 0, we have always [«] = (x — h)(x — i) (x — s)(x — t). Supposing that the first two, or four, or eight and so on, are imagi- nary, we will have for example, A = a -f- b-^/^^l, i= a — 6y/-^^ ; hence, (x—h)(x—i) = {x — a—b^—l)(x — a+b^^^) == (a; — o)'+ 6^, and, consequently, W = l(x-af + i^](x-l) (x-t). That is to say, whatever be the nature of the roots of [e] ^ 0, the polynomial [e] is always capable of being decomposed into real fac- tors, either of the first or of the second degree. EQUATIONS. 161 ARTICLE rV. Resolution of Determined Equations of the Third and Fourth Degrees, having Real Coefficients. Qeneralformu- 1 104. A GENERAL formula expressing any equation ttons oftheth?rd °^ ^'^^ *^'^'^ degree, may be as follows : degree. a;3 + Aa;2 -f. Ba; + C = (r). Now we have seen (100) that equations of any degree can be cleared of the second term, and (r) can heoome a;3^Ha;4-K = {r'). Which being resolved, we may obtain the roots of r, by taking - from each of the roots of (r'), for [r') is deduced from (r) by substi- tuting x' ox X ■\- — instead of x. Q A A Now, from x' = x-\- -^, we have also x = x' — -j, which, if sub- stituted in if), win give us the equation, ^ x" + Hi' + K = 0, A" A' in which x' is the same as the x of {r) and H=B ^, K:=.j- A^ AB — HS 5~ 4" C. But when the coefficient H and the term h are thus determined, it is immaterial to call the variable either x or x', since the roots must be such as to correspond to H and K; we may therefore use ()•') as well as the last equation. Observe also, that the formula {r') is as general as if) ; and since the resolution of (r') gives the resolution of if) also, all that we may say with regard to the resolution of (r') can be applied to the resolution of any equation of the third degree. Eoots of the \ 105. Since the degree of the equation is an uneven tion^oftlwthlra number, the equation (r') = contains certainly (96) degree. Qjjg ^gj^j j-qq); ^j jgg^gj . jjjg ot^^er two wiU be (103) either both real or both imaginary. Calling, therefore, A the real root, and the other two i and I; the equation (r') = will be (98) equivalent to {x--h)\{x-i){x-l)-\ = ^, in which {x — i) {x — Z) =: a;" — {i-\-l)x-^ il. U* 162 TREATISE ON ALGEBRA. Again, (r') does not contain the second term, which supposes (99 . r) equal to zero, the sum of the roots of (r') = ; that is, h-\- i-\-l= 0; and consequently, h = — (t'+Z). To find out the quality of the roots i and I, make — (i-\- I) = 2a and U = =0; and consequently, K= 6ae — 2a? i ^^ >' Resolving now the equation {x — i){x — l) = a;'' + 2oa!-f (a'— 3c) = 0, we have (93) i = — a -)- n/^ I z= — — x/3c, which are either real or imaginary, according as c is positive or nega- tive. Now from H and K that are given, and from the equations (/), we may find out whether c is positive or negative. The equations (/) may be changed as follows : H K - = — o2— c, -=Zac — a', from which H3_ _„e_3a4,_g<,.,._^, and consequently, K2 B? -^+^ = -9«^<;+6aV-c', = — c(9a4 — &a?c + c^), = — c(Za^ — cf. W H'. Now (3a2 — c)2 is essentially positive. When, therefore, x ' 97 '^ positive, the factor c of the second member must be negative, and tvhen the same binomial is negative, the factor c must be positive. are all real when the cube I — ) of one-third of the coefficient of x. UQXJATIONS. 163 But when c is positive, all the roots of (r^) = are real. Henoe, the roots of the equation, 3?+nx+K = (1)" plus the square of one half of the last term, give a negative sum ; if the sum is positive, then two of the roots of (r') ^ are imaginary. Let us apply the criterion to the following examples : (1.) a:3 — 3a;+52 = 0, (2.) x» — 19s+30 = 0, from the first in which H = — 3, K = 52, we have 4 +27 = 676-1 = +675. The sum is positive ; therefore!', two of the roots of (1) are imaginary ; and, in fact, the roots of this equation are K = — 4, X = 2 + 3v/"=^l, X = 2 — 3v/=^l. From the second in which H ^ — 19, K = 30, we have K' W 6859 784 The sum is negative ; therefore, the roots of (2) are real, and in fact, the roots of this equation are »» X = 2, X = S, X = — 5. Kesolntion of 3 106. It remains now for us to see in what manner the same gene- . ... ra} equation. these roots, either real or imagmary, may he found and determined. And here observe, that to have any quantity exactly determined, it Two conditions '^ not enough to have it explicitly given by a function required. ^f other quantities which are known ; but it is neces- sary, besides, that the function itself be reducible to a determined and explicit value. Thus, for example, in the equation x = ^1S5, we have the unknown quantity r. explicitly given by a function of a known quantity. But this function can never be exactly determined, for ^20 is an irrational number. And more generally the unknown quantity explicitly given by any function of known quantities follows the nature of the function ; and in cases in which the value of the func- tion could not be determined, either exactly or in any way, the unknown quantity also would remain undetermined or altogether unknown. 164 TREATISE ON ALGEBRA. The first con- Now with rtegard to the resolution of our general fuiSled," "but equation, we can always obtain the values of the roots not the second, explicitly given by a function of H and K, which are known quantities ; but the function itself is not reducible to a definite term, except in some oases. Let us first see how the first condition is always verified. Take with the general equation, the other of the second degree, having for the coefficient of z the last term of (»"'), and for the last term the cube of one-tliird of the coefficient of 2 in (r'). Now the equation (r") can be resolved, and caUiiig2„ z, its roots, the roots also of (r') will be given by the different values of the binomial In fact, the equation (r') = is fulfilled when the binomial a,^ + z,^ is substituted instead of x. To see this, make z,' u, z^ == v, or Z* + Zj* = « + i). and the substitution of this binomial being made in (r,) we will have (u + v)'+n(u + v) + K. Now from the equation (r") we have- (99) K = — z, — z^ = — u? — v% A^y Z,«, =: mV ; that is, H = — Suv, and substituting these values of H and K in the last formula, it will become (u + v)^ — Suv{u -\-v) — («»-[- tjS), which, if (« + «)) is a root of (r') = 0, must be equal to zero. Now evolving the first and second terms of this trinomial, we have «3 + Su-v + Suv'-\- »3 — Su'v — 3m»2 — u' — 1)^=0. Hence, the binomial a + « or z/ + zj substituted in (r') fulfils the equation, and zj-f z,* is a root of the equation. But zj + z,5 admits of different values, some of which must be i:QtJA'noN&. 165 excluded. That is, all those values, and only those, which make — (w* -\-v') = K and — Suv = H, will make also u-{-v = Oa. root of (r'). From the equations u = 2,^, v = z^, we have also but -^Fhas the foUowing, different values: ^T=l, because each of them, raised to the third power, gives 1. Therefore, u and v admit each of three different values; that is, the three values of u, are .,',.,»[=i±|^],..*[=i4^] and the three values of v, Now among these values those only may be used from which we obtain — (m" -|- v^) = K, — Siw = H. The term K will be always obtained in the same manner, whatever be the values chosen for u and V ; since, in all cases u'-{-v' ^ ^1 4" ^^ ^^^ — (^1 + ^a) = E ; but with regard to H, not all the values of u and v can give it, but those only whose product is 2,%a* Now this product is obtained in the three following manners only : Multiplying the first value of u by the first of V ; the second value of u by the third of v, and the third value of u by the second of v. The roots, therefore, K„ x,, x, of (?•') = 0, win be represented as follows : - 1 + 3 V=l ^ I - 1 - 3^—1 i 9 2i -t- ■ 9 % • 166 TREATISE ON ALGEBRA. These are the expressions of the three roots of the general equation (r') of the third degree, in which the coefficients of z,* and 2, are either equal to unity or of an imaginary form. With regard to ?, and z,, which are the roots of {t") = 0, we have their values (93), as follows : 1?" '^ — — 3 — ^U/ +T' imaginary or real, accordingly as the binomial under the radical sign is either negatire or positiye. But from the criterion given in the last number, when this same binomial is negative, the roots of (r') = are all real, and when the binomial is positive, two of the roots of (r') = are imaginary. That is, when the roots of ir") are imaginary, all the roots of (r') are real ; and when the roots of (r") are real, two of the roots of {r') are imaginary. Again, whenever the /H\' K° binomial I "S" ) + j- is iiot equal to zero, and all the roots of {r') = are real, they are exclusively given by terms and factors of an imagi- nary form. From all this, it follows that the roots of the equation {r') may be always given by explicit functions of the known terms H and K, and the first of the two conditions is, consequently, fulfilled in all cases. But we wiE see, by some examples, that the functions themselves are not always reducible to explicit and definite values, which is the second condition to be fulfilled to have the equation {r') completely resolved. In the case of the binomial ( — 1 -f- -j- = 0, the roots of (r") are real and equal to each other ; namely, _ K Zj ^ Zg ^ 2 • The roots also of {r') are all real, and two of them equal to each other; that is, \ and consequently, — a:, ==a x ■\- x,. idTJAnoNS. 167 _ , ? 107. Gfiven eqnatious : (1.) a:'— 6a:2 + 3a;+20 = a ^ (2.) x'+3x — U = 0. (3.) ;,.>— 12a;+16 = 0. The first of these equations is to be cleared of the second term, ■which is easily done by sulistituting (104) x'-\- g, or x'-f- 2, instead of X. In tMs manner, we will have (x'+ 2)*— 6(2;'+ 2)'+ 3(:c'+ 2) + 20 = 0, or a:"— 9x'+ 10 = 0, containing the roots of the given equation (1), but diminished each by the number 2 ; for from 2; =: 2:'+ 2, it follows that x' = x — 2. Hence, after having found the roots of the kist equation, it is enough to add to each of them the number 2, to have the roots of the given equation (1). Now, to resolve the last equation, let us compare it with the general equation (r'), and we will have H = — 9, K = 10 ; and therefore, K , H „ 2=5. i-=-3^ hence, (^-^ + _ = _2. The binomial being negative, the roots of the equation are all real. And with regard to these roots, we have first, from the preceding number, «»=— 5 — v — 2)^=:^, which comes to the same as to take y= — 6y = — 5, 3y' — 2 =1. 168 TREATISE ON ALGEBRA. Now from this last we have i/' = 1, and consequently, But since the positiye valne of y alone substituted iny' — 6y makes it equal to — 5^ therefore + 1 is the only admissible value for y ; hence, — 5 ± v'^^ = (1 ± v/=^)' ; and consequently, , and 3,* = 1 + ^=^, zi = 1 — v/=2 ; hence, also a:.= 2, x^= — 1—^W, j;,= — 1 + ^/6; and consequently, adding 2 to each of these, we will have, for the roots of the given equation (1), 4, i-v/e; i+v/^. General re- ^^^ "^ remark here that since the last term of tlie ™"'''- equation is the product of the roots of the same equation (99 . r), we may succeed in finding the roots among the factors of the last term, by trying if any of them fulfils the equation. Thus, among the factors of the last term 20 of the preceding example (1), there is the number 4 which fulfils the equation ; to find the other two, divide the equation by a; — 4, and we wUl have x° — 2x — 5 = 0, which, resolved, gives X = — 1 ± v'6- The observation just made is general ;. that is, applicable to equa- tions of any degree. „ , „ The equation (2) does not contain the second term ; and Example 2. i \ j consequently we may immediately compare it with (r-'), from which comparison, we have H = 3, K = — 14, and T'^27 ~ The binomial is positive ; therefore two of the roots of the equation are imaginary. With regard to «, smd z^ we will have and mating 7 ± 5y/T= (y ± ^2)s = (3^+6y)±(3y4-2V2; EQUATIONS. - 169 or, which is the same, S/> + 6y = 7, 3^2+2 = 5, we have from the last ys ^ 1 ; that is, y = ± 1, bat since + 1 only fulfils the other equation, and consequently, 7 ± 5.^ = (1 ± y/T)', and 2.* = 1 + v/J, z,* = 1 — v^. Hence, a;, = 2, a:^ = — 1 + v'S^v'— !> »^3 = — 1 — i/^y/ — !> for the roots of the equation (2). The last equation (3), compared with (r'), gives H = —12, K = 16; Example 3. hence. therefore (106), z, = z, = — 8, and a:i = — 4, ai, = a;, = 2, for the roots of the equation (3). EOXTATIONS OF THE FOUETH DEGEEE. Resolution of § 308. The preceding method to resolve equations of thrfourth'da^ *^ ^^"^ degree is applicable, with some modifications, gree. also to those of the fourth degree. The folloTring, 2:4+G3?+H^+K = (j), is the general formula of the equations of the fourth degree, cleared of the second term. To resolve it, take the equation, ■•+f'S-!)-S-« w of the third degree, together with (g), and let the roots of (j') be called «„ ^a, i,. The roots of (?) wiU be given by the addition, either positive or negative, of z,*, z, , zf, and the difference between the same ex- pressions, variously taken. To prove it, observe first, that (99 . r) 1 r Z*^" 1 K^ 15 170 TREATISE ON ALGEBRA. From wliich, taking : 2j* ^ V, we have G ' 2 U»-}- t,'-}- !«»= — 4/ ~I i^, (?'0; and from these G = — 2m» — 2»" — 2w', K = «* + ti« + t»* — 2u'b2 — 2j<2m)2 — 2))'m)', H = Sieuw, or H :^ — 8mkw ; but let us take the signs of the factors «, «, w in such a manner as to have H = — Smdw. Substituting now in (j) the values of G, H, K, given by the last equations, we have a:< — 2(«» + «' 4- K>2)x' _ ?Mvw% + Ml + 1,4 ^ a>< _ 2m»ii^— 2a»w2 — 'iv'w^ = 0. Now making in this equation (which does not differ from (y), except in form) x^ u-\-v-\-w, the first member becomes zero, and the equation is resolved ; x, therefore, equal \au-\-v-\-v>,'\'i the root of the equation (j). But H is either positive or negative : in the first case we may have 11= — Smjjw, -taking u, v, w m four different manners, as follows : tt = + v'JT, » = + \/zii, VJ = —%/z„ « = + y/e[, V = — v^ii w = + ^J„ u = — ^7^, J) = -f v'^ w = + v^3, u = — y/Ti, V := — v'ii, w = — y/z,. Hence, when H is positive, the roots of the equation (y) are X, = y/zl + ^T, — y/J„ Xa = v/«. — x/z, + v'Zj, X, = — v'5r+ v'^+vAi- «• = — \/2i — •vA — sA".- When H is negative, we may likewise have H = — 8uvw in four different manners, taking the values of u, v, w, as follows : tt = + v'T,, » = + y/ll w = + ^T^, EQUATIONS. 171 I _ _ u = — ^z„ V = — ^0j, w = 4- \/^s. « = — v'zT, f ^ + v'^' ^ = — \/^ And the roots of the equation (q) are in this case, *i = v'2r+ \/^+ \/^> ^a = \A; — ^2^ — ^zl, a^ = — ■v/i; — v'^ + v'«r, a;. = — v^i^ + V^^ — v'z^. The addition, therefore, either positive or negative, of z,*, s^', zf, anU the difference of the same expressions, give the roots of the equa- tion (y). _ _ _ Quality of the ■"■* ''^ "°^ evident that when ^z,, ^^Zj, y/z, are roots; how found real expressions, their sum, either positive or negative, and their difference in whatever manner it is taken, will always give a real result, and consequently, real roots. On the contrary, when one or more of the radicals are imaginary, the same results from addition and from taking their difference will be likewise imaginary, unless the imaginary terms be mutually destroyed. Hence, to know when the roots of (j), or at least some of them, are real and when imaginary, it is enough to know whether the radicals \/2i> x/^a. \/z3 are real or imaginary. Observe, now, that from the third of the equations {q"), we have z, . z, . ^ = + ^ g-j ; that is to say, the roots of (j') give a positive product ; but the posi- tive product of three factors cannot be obtained, unless one of them is positive, and the other two both real and positive, or both real and negative, or imaginar^ When z, is positive, and the two remaining roots of {q') also real and positive, the radicals y/z^, ^z^, y/T, are all real, and likewise the four roots of [q). If z, is positive, and the two remaining z„ Zj, real and negative, then the radical ^/Zi is real, but the two y/^T, y/ii are both imaginary ; and consequently, all the roots of [q), or at least two of them, are imaginary; .for when z^=:z^, -|- \/za — x/S = ^ ' ^^^ therefore, in two of the preceding values of the roots of [q] the imaginary terms must disappear. If z, is real, and the other two roots of (q') are imaginary, first, z, must be posi- tive ; because, supposing A -|- k^ — 1 to be the form of Za, the form of Za (102) ought to be A — h^^^ ; hence, z^.z, (= v? -\- v') is a 172 TREATISE ON ALGEBRA. positive product ; and, consequently, «„ z^ 2, cannot be positive un- less z, is positive. Secondly, in this case, two of the roots of (j) will be real and two imaginary ; because (h ± ky/ — 1)* is equivalent to an imaginary expression of the same form (102) : for instance, a ± 6 ^ — 1 ; therefore, in those values of the roots of (j) in which y^^, y/z^ are taken with the same sign, the imaginary term 4^/— T^ dig- appears ; hence, two of the roots of (g) are real and two imaginary. ^ 109. To resolve now the equation Example. t^ — \2x^ — 16 . 3^1 — 16 = 0, compare it with the general equation (j). We will have G = — 12, H = — 16 . 3*, K = — 16. Hence [q') will be z3 — 6z»+13z — 12 = 0, whose roots are .=3,z.=ii4^,.=i^4^. NOW, (li%^y = J(7=t.2.7V-l-l). = J(6±2.7V=T)> _ 3±7^yEJ ~ 2 Hence, Now H the last term of the given equation is negative; therefore, the formulas giving the roots are (108) — X^ = s/zT — y/z~— ^T„ *a = — -vA — \/^+ -v/^. RATIOS, PEOPORTIONS, AND PEOGEESSIONS. 173 Hence the roots of the giyen equation : x^ = v'^— v^r a;. = — v'F+ x/=n. CHAPTEK II. RATIOS, PROPORTIONS, AND PROaRESSIONS. DiviBionofthe 8 110. Katios are the elements out of which chapter: defini- i . i tions. proportions are made, which are either simple, or compound, or continual. The terms of a continual proportion form a progression. Now ratios are of two different kinds — namely, arithmeti- cal and geometrical. Hence the corresponding proportions and progressions are likewise of two different kinds, distin- guished from each other by the same appellations, viz. : arith- metical and geometrical. The present chapter, therefore, may be conveniently divided into two articles ; in the first of which we will treat of arithmetical, arid in the second, of geometri- cal ratios, proportions, and progressions. ARTICLE I. Arithmetical Ratios, Proportions, and Progressions. Deflnitions § HI" Katios. — The difference a — b between end property, ^^g quantities is called also arithmetical ratio, and the first of two terms is called antecedent, the latter, con- sequent. Now a — b, which we may express also by d, is 15» 174 TREATISE ON ALGEBRA. such, that if we add to both terms or take from them the same quantity, tTie ratio or difference d is not changed. Hence, generally, The terms of any arithmetical ratio may he increased or diminished by any quantity q without changing the ratio it- self. Simpieaiith- g 112. PROPORTIONS. — Two or more ratios metical propoi> " Uons. equal to one another form a proportion j for in- stance, a — b =^ a' — 6' is a simple arithmetical proportion, which is either written with the sign of equality between the ratios, or more commonly as follows : a — b .-. a' — 6' ; and we read it a is to & as a' is to 6' ; that is, the sign — stands for is to, and .•. for as. The terms a and b', the first, namely, and the last, are called extremes, and the two remaining mean terms; and since from a — b = a! — 6', we have a + S' = a' + 6, so also in arithmetical proportions, The sum of the extremes is equal to the sum of the mean terms. And since from any equation, like a -{- h' z= a' ~{- 1, we ■deduce a — b = a' — V , so, vice versd. Whenever the sum of two terms is equal to the sum of two ether terms, the four terms are arithmetically proportional. When the mean terms of the proportion are equal to each other, the proportion becomes a — b = h — b', from which 26 = a -(" ^'j ^°d . = -±1. The term 6 is called mean arithmetical proportional be- RATIOS, PROPORTIONS, AND PROGRESSIONS. 175 tween a and 6', but b is given by — ^ — ; hence, to find out the mean arithmetical proportional between two given terms m and n, it is enough to divide their sum by 2. Continual and The proportions having the mean terms equal portions. to One another, are called also continual propor- tions. Let now different arithmetical proportions be given, as fol- lows : a—b=af—b',c — d = cf — d',e — /= e' — /' It is easy to see that we will have also („ + e + e + ....)-(6 + rf+/+....) = (a'+c'+e'+...) -(&' + d'+/'+...), which is a compound proportion of those given. The ratios also, for the same reason, are called compound ratios. Terms of an § 113. PROGRESSIONS. — A progression, as we unlimited pro- ,,,_. iii n gression. have Said already, is represented by the terms or a continual proportion. Let now a continual proportion, containing an unlimited number of ratios, be given as follows : a — b = b — b' = b'—b"= b"— b"'= &c. ; in this case a, b, V , b", V", &c., are the terms of an unlimited arithmetical progression. But the general formula of any such progression may be differently expressed. For since the difference is the same for every one of the ratios a — b,b — 6' .... , the binomials also b — a, V — b. . . . must all give the same difference. Calling 8 this last difference, we will have b — a:=d, b'—b=d, b"—b' = S But from these equations we have b =a + S, V=b + d = a + 28, 6" = 6' -f 5 = a + 3d, &o. 176 TREATISE ON ALGEBRA. Hence the terms of any arithmetical progression, from the first to the «."', may be generally expressed as follows : General for- mula. a, a + S, a + 2d, a + Sd, "| -^. a+ (n — 2.)5,a + (n—l)d J^^ With such a form given to the terms of the geometrical pro- gression, it is easy to obtain the sum of any number of its terms, commencing with the first j for instance, the sum of all the n terms as above. Observe, in fact, that the sum of the first and second terms, is 2o-f (71 — 1)5; but the same is the sum of the terms a -\- d and a -\- (n — 2)5, that is of the second term, and of the term before the lastj and the same is that of the third term, and of the next term before the last, and so on. Suppose now, that the number n of the terms is an even number, we will evidently have ^ sums, and each one of them equal to 2a -\- (n — 1)3 j and therefore, the sum of the sum of the whole progression will be h[2« + (» — I)*]- But let n be an uneven number, then in the progression there will be a central term, having — ^ — terms before, and ^ 1 — - — terms after it. These terms, added respectively to one n 1 another, as above, will give — ^ — sums, each equal to 2a -j- (n — 1)5 ; and consequently, the sum of the n terms of the progression, with the exception of the central one, is n 1 — ^ — pa + (n — 1)5]. But the central term, added to it- self, must give the sum 2a + (n — 1)5, as the equidistant terms do when added to one another ; therefore, the central RATIOS, PROPORTIONS, AND PROGRESSIONS. 177 term is equal to ^pa + (re — 1)5], Hence, the sum of all the n terms of the progression, is !!L=d[2a + („ _ 1)3] + ^[2« + („ _ 1)5] ; that is, ^[2a + (re — 1)5], expressed, namely, in the same manner as when n is even. Hence, generally, whatever be the number n of the terms of the progression, their sum is given by the formula, sum. S = 1\2a + (re — 1)3]. u That is, to know what the sum of n terms of the arithmetical progression is, it is enough to know the first term a, and the difference S between two successive terms : for 2a -\- (n — 1)5, multiplied by one-half the given number re, gives for product the required sum. Let us see ap example. Suppose a clock striking the hours and the quarters in this manner : The hours alone, and the quarters also alone ; first one, then two, and lastly three. Hence, 7 will be the number of the strokes, from the first hour, or hour one to two, and 8 will be the number of strokes from two to three, and then 9, and so on. How many strokes are contained in 12 hours ? The num- ber re of the terms is 12, the first term a is 7, and the dif- ference d between two successive terms is 1 ; therefore, the sum of the number of strokes in twelve hours is S =6(14 +11) = 150. But suppose that the hour is repeated each time when the clock strikes the quarters, and that it strikes four quarters before each hour. From the hour one to two, including the four quarters before the hour, we will have 14 strokes ; from 178 TREATISE ON ALGEBRA. two to three, 18 ; and then 22, and so on. If, therefore, we ask the sum of the strokes in 12 hours, it will be given by S = 6[2.14 + 11.4] =432. How and when When the first and last terms of an arithmetical pro^Samay progTession ate given, and their number is also be found. given, we may find all the intervening terms. For let a be the first given term, and u the last, and let n be the given number.of terms. The form and value of the last term u is from the preceding (<), u =■ a -{- (n — 1)5, in which equation u and o and n are known, and conse- quently d is easily found : and since from the same (i) a -\- d, a-\-28 .... are the intervening terms between the first and the last, they also are all equally determined. Let, for instance, the given values be as follows : Example. „ ., . - a = 2, w ^ 14, «. := 5 ; from v,^ a-\- (n — V)S, we will have 14 = 2 + 4.5; and consequently, 5 = 3; hence, for the intervening terms between a and u, we have 5, 8, 11. ARTICLE 11. Geometrical Ratios, Proportions, and Progressions. p"op*rty.°°'°°* § 114. Eatios.— The quotient ^ of two quan- tities a and 6 is called also their geometrical ratio, and a the antecedent, and h the consequent of the ratio. Whenever a ratio is mentioned without adding the quality of arithmetical or geometrical, it is always understood to be a geometrical ratio. RATIOS, PROPORTIONS, AND PEOQRESSIONS. 179 Multiplying now botli terms of - by q, we will have a.q a g a and dividing the same terms by q, we have a.b a q a q' q h' q h' that is, The terms a and b of the ratio y can he multiplied or divided by the same quantity q without changing the ratio. Variable ratios § 115. The terms of a ratio may be either — direct and re- • -t -, ii -, .ii ciprocai terms, constant (» variable, and when they are variable they may vary with a certain dependence on one another, or not. If they vary independently of one another, the ratio itself is variable. But with changeable terms, - depending on one another, the ratio may be constant. Suppose, for example, 1 ^ m ' and in this equation m to be constant. It is evident first, that for any change of x, a corresponding change must be made in y. Call >/ the value to be given to y when x is changed into of, and y" the value to be given to y when x is changed into od', and so on. Now, generally, whatever be the values of X and y, from the given equation we always have X - ^ m. y Although, therefore, the terms x and y are variable, their ratio x' x" m is constant, and —;, -jj, .... will be all equal to m. Now, Whenever two variable quantities are so connected together as to give constantly the same ratio, tJiey are said to vary together directly. T, ,. 11 Isut taking - = —x ° y m 180 TREATISE ON AIGEBRA. in the same supposition of m being constant, and x variable together with y, whatever be x and y, we will always have 1 x:- ^ m; that is, x.y = m. The terms x and y, therefore, vary in this case, in such a manner as to give constantly the same product. It is then plain, that one of them cannot increase without a corre- sponding diminution in the other, and vice versd. Hence, generally. When two variable quantities are depending on each other iu such a manner as to give constantly/ the same product, tJiey are said to vary invorsely or reciprocally. It is to be observed here also, that since 1 x.y -^^ X : - ■= m, y 1 ■when the variables x and y vary inversely, x and - vary directly. Continual geo- § H". We have seen (56) that irrational numbers are metrical ratios, those Emits to which an indefinite series of rational numbers of fractional form may constantly approach. So, for instance, the square roots of 2 and 3 are such numbers contained between I and 2, which cannot be exactly determined, but to which an indefinite series of rational numbers contained between the same limits may con- stantly approach. Now all the numbers, both rational and irrational, contained within the limits 1 and 2 form a continual series ; and if we conceive the number 1 to be successively changed into every one of the terms of this series, proceeding orderly from the first to the last, the number 1 would be said to increase eontirmally, or to increase by degrees smaller than any assignable quantity. Upon this, let x and y be two quantities depending on each otner in such a manner that when x becomes 2a; or Zx, &c., y also becomes XX y y 2y, Zy, &c. ; and when x becomes =, 5, &c., y also becomes „, „, With regard to these variables x and y, we say that UATIOS, PROPORTIONS, AND PROGRESSIONS. 181 ProposltioQ If X ohanaes constantly, ij also 'will cliange, continually and its demon- , , . stration. keeping pace with x. Let, in fact, m and m' represent any two wliole numbers : we will have y changed into m'y, when x is changed into m'x\ and if in m'x, we change x into — , in m'y the variable y must become -. Therefore, when X is changed into — x, y\s necessarily changed into — y. Now, ni 10 ^ ^ m and m' are any two whole numbers : hence, — stands to represent m any rational fraction ; hence, also — may be any of the terms of an m indefinite series, approaching constantly to some irrational number ft, and consequently, if x is changed into ftx, y also wiU become /xy. „ ,, Generally, representing by v and u', any two numbers, either rational or iirational, when x is changed into «■'= vx, y will become y' ^ vy, and when x is changed iato x" = v'x, y will become y" = v'y. Hence, x^ y' x^' ~ y^'' That is, when x and y change together and equally, the ratio between any two values is always equal to the ratio between the corresponding values of y. Directandre- \ 117. Let now y, a, a, . . . represent any number pound ratio^ "^ variables, all independent of one another, and let Theorem. x be another variable, depending directly on each one of them, so that, for any value given to the independent variables y, z .... we always have the ratios -,-,.... unchanged. Call now P the product y . z . m ...... of the independent variables ,- this product depends on each one of the variaMes y,z,u... directly, and in the same manner as x. 'Whatevei- be, therefore, the values given to the independent variables, the ratio p will remain constantly unchanged. That is. If X varies directly as each one of the independent variables y, z, u, . . . it varies also directly as their product. "We may arrive at the same conclusion in another manner. Since x varies directly as any of the variables y, *!,... independent of one another, if in the ratio ? = E, y .z .u 16 182 TREATISE ON ALGEBRA. we give any value to y, the ratio will constantly retain the same valne as B. In fact, the same ratio can be decomposed as follows : y z . u. . .. Now giving at pleasiire to y any value, ^, » . . . . suffer no change, z 1 and X varies directly as y ; therefore both factors - and remain y z.u. . . unchanged for any value whatever of y ; that is, the ratio y.z.u.., will be constantly equal to the same E, whatever be the value wo give to y. But the same reasoning is applicable to z, to u, &c. ; therefore, whatever be the values given to the independent variables y, z,u . . . X in the ratio , its value will remain constantly unchanged; that is, X varies directly as the product y .z .u. . . . Corollary ■"^^^ ^^ * depends directly on a, t), . . . and inversely on y, ■i, ....X varies directly (115) as each of the following: 1 1 y 2 Therefore, it varies directly as their product a.v . . . ~ . -... u. V . . . = ; hence, y-z . . .' If X varies directly M u, v, . . . and inversely as y, z, . . . Theorem. u . V . . . . it mil vary directly as' the quotient — '- — -^-^. Simple geo § 118. PROPORTIONS. — Two Or more ratios metrical propor- i_ ,1. j. i.- t • tions. eqnal to each other, lorm a proportion; tor in- stance, a a' 6 "" b" and this is the general expression of any simple geometrical proportion. The manner, however, of writing these propor- tions is as follows : a ih :: a' :V, and we read it a is to & as a' is to Z/ ; that is, the two dots [:] stand for is to, and the four dots [: :] stand for as. In geo- metrical proportions also, the terms a and U are called ex- tremes, and the other two mean terms. RATIOS, PEOPORTIONS, AND PEOGRESSIONS. 183 General pro- ^'om the proportion equivalently represented peities. jjy jjjg equation a a we have b'l' a' b ^ b'-b' and consequently a.V = a! .b. That is, in geometrical proportions, The, product of the extremes is equal to that of the mean terms. But from a . &' = a' . 6 we have likewise, ^ = t; ■ Hence, When four terms a, b, a', b' are such that the product of the first by the last is equal to the product of the other two terms, the four terms are geometrically proportional. Continual and SupposB HOW that the mean terms are equal ODmpound pro- i i i -n i portions. to each other, then we will have a:b::b:b'; and consequently, h^ = aV , that is, b = \/aJ7. b is the mean geometrical proportional between a and V. If, therefore, a and V are given, to find out their mean geometri- cal proportional, take the square root of their product. Such proportions also, having both mean terms equal, are called continual proportions. Let now several proportions be given, as follows a a' c d e e' b^l"d'^d"f^f' We have from them ace d d d b-~d''f' ^Vd'-f'---- And consequently (a.c.e...) -.(b.d.f...) :: (of . c' . d . . .) : (b'.d' .f . . .) 184 TREATISE ON ALOEEBA. That is, If the first terms of several or any number of geo- metrical proportions he multiplied together, and likewise the second, third, and fourth terms, the products are proportional. The ratios and the proportion itself, made out of these pro- ducts, are called compound ratios and compound proportion. otii6rpropei> S 119. Prom a : h : : a' : V , OX from ties of geome- trical proper- a a tions. — = — . h b" , a b we have —; = — •, a' b that is, a : a' :: b : b' . Hence the terms of any geometrical proportion are such, that The antecedent of the first ratio is to the antecedent of the second ratio as the consequent of the first is to the consequent of the second ratio. Again, from' the given proportion or equation r = ry, we b V have - = -; ; a a that is, b : a •.•.b' : a'. Hence, Tlie consequent of the first ratio of any given propor- tion is to its antecedent as the consequent of the second ratio is to its own antecedent. From the same proportion or equation j-^jf,-we deduce the two following : -4-1--4-1--1---1- b^ b'^ ' b b' ' and consequently, a + b _a'-^b' a — b a' — b' b ~ V ' b ~ V that is, a -\- b -.b :: a' -\-b' :¥, a — b:b ■.-..a' — b' lb'. RATIOS, PROPORTIONS, AND PROGRESSIONS. 185 In equal manner, since j = — may be changed into - = — , we have h -\- a : a ::}/ -\- a' : a!, h — a : a •.xV — a' : a'. That is, the terms of any giyen geometrical proportions are such, that 2%e sum or the difference of the terms of the first ratio is to the first or to the second term of the same ratio, as the sum or the difference of the terms of the second ratio is to the first or to the second term of the same ratio. We may observe that in the last proportion b — a : a :: I' — a' : a', the differences or terms b — a, V — a', may be changed into a — b and a' — V , the terms remaining still in geometrical proportion ; for this ' inversion affects only the sign of the ratios, which being equally changed in both of them, the equality of the ratios still exists, and conse- quently the proportion also. The same observation may be made with regard to the proportion a — & : 5 : : a' — b' '.V . So that the last inference is altogether general. Prom the proportions or equations a + b _ a' -\- V a—b _ a'— b' ~6~ ~ - b' ' IT' ~ b' ■' just inferred from the given proportion or equation t == 77, we have a +b b a—b b + b' b"a'~b' h' hence, also. a + 6 a — b a' +b' a'— b" and a + b a' -\-b' a — b a'—b" consequently. a + b ■.a'-\-b'::a — b:a'- a + b •.a — b::a'-\-b':a'- v. That is. The sum of the terms of the first ratio of any propor- 16* 186 TREATISE ON ALGEBRA. tion is to the sum of the terms of the second ratio, as the dif- ference of the terms of the first ratio is to the difference of the terms of the second. And The sum of the terms of the first ratio is to their difference, as tJie sum of the terms of the second ratio is to their difference. These, and the preceding inferences, are of great practical use. Numerical pro- ? 120. Let the terms of the proportion portions whose o : d : : o' : 6' first ratio is i]> «- . w . . « . reducible. be whole numbers, and let the terms a, b of the first ratio he prime numbers to each other. Of the numbers o', b', the first will be equal to na, the second to nb, » being a whole number. In fact, calling n the quotient — , or makmg — = n, a' may always be expressed by na. But na cannot express a', unless nb expresses i' ; for by supposition, a' a b'^ b' and consequently, if a' = na, b' cannot be but equal to nb. We say now, that k is a whole number ; for from the same equation, |;=?wehaye ^^, ^, a ab^ Now 7 by supposition is an irreducible fraction ; therefore, -7- cannot be equal to the whole number a', unless (53) the number ¥ is exactly V divisible by b ; that is, unless the quotient t- is a whole number ; but b' b' ^ nb, and from this equation, we have j- = n; the number n, therefore, is a whole number. Terms of any §121. PROGRESSIONS. — The terms of any geometrical pro- 1 • 1 • .1 it j igresdon. geometrical progression are the same as those of any continual proportion. And a continual proportion is generally represented by a_ 6 _ y _ h~ v~ y orelseby a:h::l:V ■.-.V -.V :: . . . . The terms, therefore, a, h, V, h", .... are the terms of any unlimited geometrical progression. RATIOS, PEOPOBTIONS, AND PROGRESSIONS. 187 Let US now call h the eommon value of the ratios t, =-., ... 6 we will have a ,6 ,6' . „ 1 = '"' l' = '"> 3" = ^' ^°- > and consequently, cs , h a . (z -A ->?;»-^-^^----' hence, the terms of any geometrical progression, from the first to the ra*, may be generally represented as follows : "' 7c' p •••• fc^i ^'■-'' inSa°°'^°^ *"^" or else, (making j = z), the general formula of tte terms of any geometrical progression containing n terms, is a, az, az") at?, .... aa'^^ (t'). Now the sum of these n, terms is easily obtained from the known product (63) (1 + »+ 2"+ . . . . + z^'-)(\—z) = 1 — z", \ ^n which gives 1 + a + z^ + ... + 2"^' = -— — , from which , «_, 1 — z" a az^ a -\- az -\- a^ 4- ... + az*^' = a= = = = . ' ■ 1 — z 1 — z 1 — z Somofrnterms Now, the first number of this equation is the tadeMte °Li sum S of the terms (/); therefore, ter of terms. a az'' 1^ — z 1 — z Suppose now that the numerical value of a is a number less T than unity, it may be represented by a fraction — , in which m is greater than r. In this supposition, the last term of Qs) Vill be /r\''m — r m.a /rs" \m/ ' m m — r\m/ 188 TREATISE ON ALGEBRA. Now, '- — is a constant coeflEicient, as well as the fraction — , m — r m but the exponent n has different values according as the terms summed up are more or less in number. Now the more we increase n, the more the power (—1 approaches to zero, and with it the whole term '■ — ( — ) . Hence, takine an in- m — r\m/ definite number of terms — that is, supposing the number of the terms summed up to be without limit — the last term of ( 1 we take o~' instead of a', then we will have a~^ = — ^ -. Now -is less than unity; hence, the a^ a^ a j > > number or power corresponding to a~' is smaller than unity; the same is to be said of any other power a"""' in which m is taken between and 1, or greater than 1. That is, when the base is greater than 1, and the logarithm is negative, the cor- responding number ia less than unity. Prom this and the preceding inference we deduce the two following : When the base is greater than unity, and the number is likewise greater than unity, the logarithm of this number is positive. When the base is greater than unitjf, and the number less than unity, the logarithm of the number is negative. But let the base a be smaller than unity; then giving to a any exponent eontained between and 1, the resulting power will be contained between a''^=\ and a' = a. Now all the numbers between 1 and a are smaller than unity ; a, therefore, raised to any exponent between and 1, gives a fraction for power. But if a is raised to any exponent greater than unity, the power also will be a fraction. That is, when the base is less than unity, and the logarithm is positive, the correspond- ing number also is less than unity. lOGAElTHMS. 193 Grive now to o <^ 1 a negative exponent, we will have generally a~" = -^; but a", as we have just seen, is always less than unity ; therefore, — ^ is always greater than unity ; hence, when the base is less than unity, and the logarithm is- negative, the corresponding number is greater than unity. From this and the preceding deduction, we infer also that When the base is less than unity, and the number also is less than unity, the logarithm of the number is positive. When the base is less than unity, and the number greater than unity, the logarithm is negative. when^thTft § ^^^' '^^^^ °°^' successively for x, the rithms form equation a'^z, x, x -\- 5, x 4- 2d, an arithmetical ^ . J J i 7 i 7 progreasion.the -which are the terms of any arithmetical pro- numbers form a gression, we will have geometrical one. ° ' a" = z. Now (42) a'+*=a'.ffl*, a«+^'' = a«. a^'' = a%a'y, 0"+^' = a^^a'y, and making a" = C, we will have Hence, the powers ;i, s/, 2", a'" .... are repTCseated by the terms a', aX, a%^, a"^, which are the terms of a geometrical progression. But the same powers are the numbers corresponding to the logarithms X, x-\- S, x-\-15 ... in the system having for base any number a. Therefore, In any system of logarithins, when the logarithms form an arithmetical, the corresponding numbers form a geometrical, progression. TJsafui theo- § ^'~^- ^^^ ^®* ^^ come to those theorems which "'"^' show how advantageously logarithms may be used. 17 194 TREATISE ON ALGFEBRA. Let X and y be the logarithms of the numbera z and v in the system having a for base, we will have a" ^ z, a? = v; and consequently, a^ .a!' = 0'+'' = z .v. Now, from these equations, we have, also X = l.z, y = l.v, aj -j- y = l.(z.v), and therefore, l.(z.v) = l.z-\-I.v; that is to say, The logarithm of the product is equal to the sum of the logarithms of the factors. Again, from the same equations, we have '»' ^ ^„ " — = -, or »»-» = - ; a" V V and consequently, x — y = I.-, or, 1- = l.z- — l.v : that is, The logarithm of the quotient is egual to the Theorem 2. , .' .", ■' ^ . , ^, logarithm oj the numerator, minus the logarithm of the denominator. Kaise to the exponent c both members of the equation a":^ z, we will have (a")" = z' or a" = z", and l.z° := xc ; but from a'^ z, we have x = l.z ; hence, l.z° := cl.z ; that is, The logarithm of the power of any numler is equal to the logarithm of the number multiplied by the exponent. But if we take the root of the degree c of both members of the equations a" = z, we will have X 1 LOGARITHMS. 195 ' X 1 and consequently, Lz' = - =z -x; now x = l.z} Hence, i -1 l.z' or I. i/a ■= -l.z ; •^ c that is, The logarithm of the root of any number is equal to the logarithm of the number divided by the degree or index of the root. Erom these theorems we infer, that when the logarithms of the numbers are determined in any system, numerical calcula- tions become much easier; for multiplications and divisions are performed with simple additions and subtractions — powers and roots are obtained with multiplications and divisions. Common or The logarithms of numbers have been care- ordinary tables _,,- .- -, T ofiogarinims. fully determined, and the common and most use- ful system of logarithms is that whose base is a ^ 10 ; hence, the general formula a" = z, in this system becomes 10^= = z, and taking in it successively 0, 1, 2 .... instead of x, we will have 1, 10, 100 .... for the corresponding number z. To find out the logarithms of the intervening numbers be- tween 1 and 10, between 10 and 100, &o., it will be enough to take the numbers between these limits in a geometrical progression, and an equal number of terms between and 1, between 1 and 2, &c., in an arithmetical progression ; the terms of the latter progression will be respectively logarithms (124) of the corresponding terms of the geometrical pro- gression. Now the terms of any progression, either arithmetical or geometrical, are the same (113, 121) as those of a continual proportion ; and when the extreme terms « and b' of a con- tinual proportion are given, we obtain the mean arithmetical term by taking (112) "T ■ , and the mean geometrical by 196 TREATISE ON ALGEBRA. taking (118) -j/ai'. Hence, in our case, the mean geo- metrical term between 1 and 10 is |/10 ( = B, 16 . . . )> ^^^ + 1 1 2 ~2' the mean arithmetical term between and 1 is and the numbers, 1, 3, 16 ... . 10, 0, I, 1, are terms of two progressions, the first geometrical and the second arithmetical ; and consequently, since and 1 are re- spectively logarithms of 1 and 10, so -, or 0.5, is the loga- rithm of 3, 16 ... Now, again, taking the mean geo- metrical proportional between 1 and 3, 16 ... and between 3,16 .... and 10 ; taking, also, the mean arithmetical pro- portional between and 0.5, and between 0.5 and 1, we will have two more numbers contained between 1 and 10 and their logarithms ; continuing in this manner, we may have as many numbers as we like between 1 and 10 and their loga- rithms. The same may be -said of the numbers contained within the limits 10 and 100, 100 and 1000, &c., and of their logarithms. This method shows well how logarithms of any quantity of numbers may be found ; in practice, how- ever, methods more expeditious are preferred. ' It is yet to be remarked that even the method just explained is not neces- sarily to be applied to all the numbers ; but it is enough to find the logarithms of prime numbers, for these being de- termined, we have the logarithms also of all the numbers which can be resolved into factors, and the logarithms of frac- tions also. Take the number 15, for example, which may be decomposed into the two prime numbers and factors 3 . 5, we will have (125,. Th. 1) l.\b = ?.3 + ?.5 ; take the fraction 7 7 g, we will have (125, Th. 2) l.^ = 1.1 — Z.9. But it is not necessary for us to dwell any longer on this subject, for LOGARITHMS. 197 copious tables of logarithms are made with most exquisite accuracy and with all desirable improvements. Constant ra- § ^^B? Let a, a', a" represent the bases of three Sihn^taew different systems of logarithms, and l, I.', I." the Bystem. signs of the corresponding logarithms. From the equation a'" = r, we have, x = l.'r. But if we take the logarithm of each member of the same equation in the system having a for base, and then in 'the sys- tem having a" for base, we will have, Kjx") = l.r, l.'Xa") = l."r, or, (125, Th. 3,) xl.d = l.r, xl."a' = l."r. Substituting now in these equations the preceding value of x, we have, l.'r .la' ^ l.r, l.'r . I." a' = l"r. Hence, — -l.'r, — = l.'r. And consequently, UT^' ' that is to say. The logarithms of any two numbers r, a', divided by each other, give constantly the same ratio in all systems. Howtheiogar Suppose the logarithms I. of the system having sys'iSa maybe '^ ^"^ ^^^e, to be knowu or determined, and let a' ttaXliritS 156 the base of another system of logarithms I.' of others. which are to be determined; let also n be any number. The logarithm of n is known in the system having a for base, and unknown in the system whose base is a' ; that is, in the equation a'" = n, the exponent x(== l.'n) is unknown. But from the same equation, taking the logarithms in the system whose base is a, we have xl.a' = l.n; ir* 198 TREATISE ON ALGEBRA. and consequently, x = j—, ; but the logarithms I. of n and a! are known ; hence their ratio also, or X, is known. Knowing, therefore, the logarithms of numbers in any system, we may from them infer the logarithm ' of any number n taken in any other system ; and consequently, when tables of logarithms are made for one system, we may derive from them other tables for any other system of loga- rithms..; Explanatory % 127. Let US resume the two progressions, remarks. -,„..«„ -„..« Rules. 1, 10, 100, 1000 0, 1, 2, 3.... the first representing the numbers, and the second the corresponding logarithms, in the system having a = 10 for base. In the same system the logarithms of all fractions must be (123) negative, and the following terms may be added to the preceding pro- gressions, 1 11 1000' 100' 10' ... -3, -2,-1; so that the number 1 in the geometrical, and in the arithmetical, progressions, are the central or middle terms of two progressions, indefinite in both ways. From the same progressions we see that in the same manner in which the logarithms of the numbers between 1 and 10 are greater than 0, and smaller than 1, the logarithms of the numbers between 10 and 100 are greater than 1, and smaller than 2, and so on. In like manner, the logarithm of the fractions between — and 1 are contained between • — 1 and zero, and the logarithms of the fractions between -r-z and — are contained between — 2 and — 1, &c. Calling V any number between 1 and 10, since all the numbers be- tween 10 and 100 are ten times greater than the corresponding numbers between 1 and 10, the number contained between It) and 100 and corresponding to v, will be lOj or av, and so likevrise the next corre- fiponding number between 100 and 1000 wiU be lOOv or ah, and so on In like manner, since the numbers between — and 1 are ten times LOGAMTHMS. 199 less tiian tlie corresponding numbers between 1 and 10, the fraction between — and 1, corresponding to v, is — or -, and the next fraction corresponding to the same v, and contained between j^ and ^, is m ""^ %' '■''^ ^° ""• So that, we may generally represent by O'l any number contaiaed If between the decades 10, 100, 1000 .... and by — , any number con- tained between the decadal fractions 1, ttt, r-^rj;, . . . giving, namely, to n any of the numerical values 1, 2, 3 And to represent all the numbers, we have a", a"? 1 V for those above umty, and — , — for the fractions. Whatever, therefore, may be said concerning these numbers and their logarithms is evidently applicable to all numbers and logarithms in our system. From the same formulas we infer general rules, useful both for the understanding and the use of the tables. But first observe that the immediate object of logarithmioal tables is twofold. To point out, namely, the logarithm corresponding to a given number, or, vice vers&, to point out the number corresponding to a given logarithm. It is scarcely necessary to say any thing concerning the numbers a", — ( = ffl~") of a mere decadal form, it being evident that a" is equal to iinity followed by as many zeros as there are units in n, and — is equal to 1 divided by unity, followed by as many zeros as there are units in n. And, vice versS,, when any whole number of a mere decadal form is given, its logarithm n will be a whole number containing as many units as there are zeros in the given number. Hence, When the given whole number N is of a mere decadal form, it has for logarithm a number con- taining exactly/ as many units as there are zeros in N ; and when the given logarithTui n is an exact whole number, the corresponding number is unitt/ followed by n zeros. 200 TREATISE ON ALGEBRA. It is plain that for such logarithms and numbers wo need not have recourse to the tables, and so also for the fractional numbers ^j^ of simply decadal form, for which, and for their logarithms, we have the following rule : Kuie 2. When tlie given fractional number :j^ is of simply decadal form, it has — n for its logarithm, containing exactly as many units as their are zeros in N j and when the given logarithm — n contains an exact number of units, the corresponding number is 1 divided by unity, followed by n zeros. We may observe, that such fractions of simply decadal form may be expressed also by 0,1, 0,01, 0,001 . . . .; and using the decimal form instead of that of ordinary fractions, the second rule will be modified as follows : When a decimal fraction ends with 1, preceded by n ciphers, all equal to zero, the logarithm of the fraction is — n. And when, vice versd, — n is given, the corresponding number or decimal fraction ends with 1, preceded by n ciphers, all equal to zero. a" Let us now come to the numbers a"ii, — , and to their logarithms, in which v, we must recollect, is any number greater than 1, and less than 10. But a"' * also, in which o, J represents any decimal fraction, is a number, and any number contained between 1 and 10 ; therefore, we may generally write ► = a"' ', and in this equation, the exponent o, S cannot be changed except when f is changed. Now with v ^ a"' *, we have also aPy = a" . a"' * = a'"^°< * = a"' ^, and whatever be n, the decimal fraction i will always be the same when f remains the same ; but from a"' ' =: ffi'V, we have n,S=l.{a'.i) {\). And a". V is a number contained between o^ and 0"+^ either a simple whole number or a whole number with a. fraction l\ added to it. LOGARITHMS. 201 In both cases tlie integral part (let tis call it N') of a" . j must have the same number of ciphers that are in a" ; that is, in a» . » = N', A M. the number of ciphers of N' is the same as the number of those of a", namely, n-\-l. The figures, besides, of the number N', A ^"^6 tlis same as the figures of t, the first n -|- 1 of which form the integral part N', and the other, if there are any, the decimal A- New from (\) and (a'), we have w, i = I. (N', A)- And from the preceding remarks, it follows first, that the figures of the integral part N' of N', A are one more than the units contained In the integral part n of the logarithm, and vice versa. Secondly, the figures of N', A are inTariably the same when the fractional part J of the logarilim remains the same, and vice versS,, for the figures of N', A S'^^s the same as those of s, and i does not change except with n. Hence, it is enough to know what is the v corresponding to the fraction o, / to have immediately the numbei's corresponding to all the following logarithms, 1, J, 2, J, 3, J ... . m, J, and vice vers&, when the number is given, and consequently, the ciphers of > also are given, it is enough to know what is the fraction 0, J corresponding to v, to find out also the logarithm of the given number. Now this is precisely that which is given by. the tables. That is, the first column, marked N, contains the numbers or figures of a, and the other columns the decimal part o, S of the logarithms. Hence, To find the logarithm when the nirmher is given, we have the following rule : When the number N', A is given, write n contain- ing one unit less than the number of figures in N', and this n is the integral part of the logarithm. Then, tak- ing N', A o,^ OM uninterrupted number, add to n the fraction 0, d, corresponding to the same number, and given by the tables. The whole number n. or integral part of Characteristic. ,, , .,, .. iii,i 7 .. the logarithm n, a, is called the cfiaractenstvc. Hence, 202 TREATISE ON ALGEBRA. The characteristic of the logarilhm, of any given nnmher contains one unit less than the number of figures forming the integral part of the given number. When the logarithm n, S is given, find in the tables the number corresponding to S; cutoff the first n -j- 1 figures of this number from the following : the first part will be the integral, and the rest the decimal part of the number having n, S for its logarithm,. y With regard to the fraction — ire may remark first, that since o" is y a number of simply decadal form, like 10, 100, &o., the quotient — reduced to the form of a decimal fraction will contain the same figures that are in y, preceded by one or more zeros ; that is, as many in num- ber as there are units in n. ^ y Observe, secondly, that — = oj-". »; hence. Now, a—". a°'^ = a— "+'''. Instead, however, of writing expli- citly the difference — n-\- o,f; the same exponent is represented by the simple expression n, 3, with the negative sign above the character- istic, to signify that it does not affect the decimal part i added to it. We will have then, — = 0".'': '< a" ' the two numbers y and i depending on each other, as above, whatever (f n should be. But — , reduced to the form of a decimal fraction, may a" ' be represented by o, d ; hence, 0*^' = o, d; that is, _ n,i = l (o, d). Now, from the tables we may have the ciphers of v corresponding to }, or, vice versa, we may have S corresponding to the ciphers of y that are in D ; and since, as we have observed, the figures of r commence in o, D after n zeros, hence, - lOGABITHMS- 203 Eor negative logarithms and their coiTesponding members or fractions, we have the following rules. First, when the fraction o, D is given, and its logarithm is to be found : iSise how many zeros precede the first figure of units in o, D, and write the number n of these zeros as the characteristic of tJie logarithm j talcing then from, B the number which commences with the first figure of units, find from the tables the corresponding d, and add it to the charac- teristic. And tofind the number corresponding to a given logarithm, we have the rule — When the logarithm n, S is given, write first as Eule 6. .■ . . . , many zeros as there are units m n, separating witk a comma the first from the others; then add to these ciphers the number corresponding to S, as given by the tables. We may observe, that the logarithms of fractional numbers are differently expressed by different writers. We have ex- pressed them by n, d. But when w = 1, or 2, &c., others express these logarithms by 9, 5 ; 8, 5, &c. But this manner of writing such logarithms is somewhat ambiguous, and we may say partial. For this reason we have preferred to make itse of the above-mentioned expression. Application of §128. The practical application of the preced- logarithms. jjjg ^^jgg jjjygj ^,g jgfj entirely to the direction of the teacher, and to the diligence of the pupil ; since any at- tempt to apply them without having logarithmical tables at band would prove altogether useless. To give, however, some idea of the useful application of logarithms, let us observe that exponential equations can be resolved by means of logarithms. That is, those equations in which the unknown quantity is the exponent of some other quantity ; as, for instance, in 204 TREATISE ON AIGEBEA. C_ J_ h ~ 2<-«' in which equation x is the unknown quantity. Applying the logarithms, we will have 'i = '[i^] -r hence, (125, Th. 2, 3,) l.e — l.h = l.l — l.q^'-^^ = 1.1 — (x — 1) t.q; and since l.l ^= 0, l.c — l.h = (a: — 1) l.q ; , from which _ l.c — l.h x—1 — , l.q and l.c — l.h , , X = — h 1- l.q Let another exponential equation be as follows : Vioo/ Taking the logarithms, we will have that is, X p.iai — Z.IOO] = 1. Now, from the tables Z.IOO = 2, and ?.101 =2,0043214; hence, ?.101 — I.IOO = 0,0043214 ; therefore, _ 1 _ 1,0000000 ^ ~ 0,0048214 ~ 0,0043214 ' and finally the approximate value of a; = 231. But suppose that instead of ^-rrrr, the given fraction is /= Yiyi > then, from the equation we have x p.lOO — ?.101] = 1. LOGAUITHMS. 205 Now Z.IOO — ?.101 = 2 — 2,0043214 - = — 1 + 3—2,0043214) = — 1 + 0,9956786 V .= 1,9956786 J hence, for the value of x, _ 1 ^ ~ 0,0048214 = — 231 nearly. 1 T f + /loiy /loix- Infect, (^_) = (^_) aoi /lOOx-^ aiotJ ■ =-jTj;) ^10, we have a; = 231; the exponent, therefore, to be given to the fraction ymi '^^^'^ made equal to 10, is the same number 231, but taken with a negative sign. For other examples the student may combine at pleasure several numbers, and make them equal to unknown quantities, and then resolve the equations by means of the logarithms. For instance, let a, b, c, d, e represent given numbers; we may form with them the following equations t a.b — -= ex. c.d (1.) (2.) a.l.c=d.e'. (3.) aKc.d=-, (40 c.e a.h.d X ' (5.) af.c^^x'; and so on ; and applying the logarithms to them, we will find (1.) l.x = I. a + l.b — l.G — l.d — I.e. 18 206 TREATISE ON ALGEBRA. _l-'^+ l-^ + ?-e — l-d (3.) l.x = ll.a + l.c + Id + ;.e. (4.) l.x = la + l.h + Z.c + Z.cZ + I.e. (5.) ?.^_^il±^=. With the exception of the second of these examples, in all the others we have not the value of x, but the value of the loga- rithm of x; now the corresponding number of any logarithm is given by the tables; hence, with the l.x, we may have x also. It is plain, moreover, that any example like the preceding is the general formula of as many as the pupil will like, by substituting numbers for the symbols a, J,, c, &c., and chang- ing them at pleasure. CHAPTER IV. SERIES. viTiat aio'e- § •^^^- Aeithmetical progressions are the ar*''and tt^i? ™ost simple of all algebraical series, and are various orders, galled scries of the first order. The series of the second order are those whose second differences are equal, and the series of the third order are those whose third differences are equal, &c. That is, let t', f, r, r, f ... represent the numbers or terms of an algebraical series. If the differences f—i^ i"—r, r—f'.... between the successive numbers are all equal, the same num- SERIES. 207 ber or terms belong to an algebraical series of the first order. But suppose the differences between the successive given terms to be unequal, so as to have i'-~ 1? = t', r— f = t", i!'"— iC" = t"' If the second differences, or the differences between the terms t', t", t"'. . . . are equal, that is, if we have r"— r' = t"'— t" = T*'— t"' = . . . . , the given terms i, i' , f . . . then belong to an algebraic series of the second order. That is, the second differences also are unequal, and we have r"—^=ef, r"'_T"=:fl", r'^— T"'=e"', &c.; but the third differences, or differences between the terms ff, 0", ff" are equal : the given terms i, f, i" belong in this case to a series of the third order. It is. now easy to see when a series will be of the fourth, of the fifth, and gene- rally of the m"' order. It is likewise easy to infer from the foregoing remarks that the second, the third differences, and so on, of any algebraical series of the first order are alT equal to zero, and the third, and following differences of any algebraical series of the second order are also equal to zero, and generally, the (wi — l)*" and following differences of any algebraical series of the m!^ order are equal to zero. Various que.'i- The most common investigations concerning tjons concern- , , . - . i , ,i i ing the series, these algebraical series are about the general term, and the sum of any number n of their first terms ; but the principal object in view is that of reducing other functions to the form of a series. This doctrine is copiously treated by modern writers, and with exquisite analysis in differential calculus. It is not oar intention to enter here into long discussions on the subject, and it -will be enough for us to give an idea of it, treating briefly the first and second questions above mentioned. 208 TREATISE ON ALGEBRA. General term § 130. Let us commence -vrith the general term. The of any series. polynomial, (p) = A+ A.n + A,«'+ . . . .+A^n'», in which the coefficients A, A„ A, . . . A„ are constant quantities, re- presents the general term or the «"■ term of any algehraio series of the m"" order. To have it demonstrated, it is enough to show that the m'" differences of the series corresponding to this term are all equal. Now to say that (p) represents the n* term of a series is the same as to say that it represents the first, the second term, and so on, when ji is made equal to 1, to 2, &c., and the term immediately preceding the »"> will he obtained from (p), by changing in it n into n — 1. Now, call (^i) the term preceding the n*\ it will be (p,) = A+ A,{n-1) + K{n — ir +..'. + A^in — l)", and {p) — (pi) will be the first difference of any two successive terms of the series.. But this diff'erence — we may call it (p^y — after reduc- tion will take the following form : ■B + -B,n + B,n'+...+ B„_i n"-' ; hence, (p,y, or (p) — (p,) = B + B, n -f B,n'+ . . . + B„_i ti"-': in which difference, if we make n equal to 2, or 3, or 4, and so on, we will have the difference between tlie second and the third terms, be- tween the third and the second, between the fourth and the third, and so on; we will have, also, the terms of another- series, because (piY has the same form as (p). Repeating, therefore, on (^i)' the same operation which we have made on (p) — that is, changing n into n — 1, to have the term imme- diately preceding (j?,)', — we may call it (p^) — the difference will be (Pa)', or (p,y—(p^) = C + 0. n-f C,«»-|- . . . . + C„_, «»-', representing any one of the second differences of the series having (p) for general term, as it represents any one of the first differences be- tween two successive terms of the series having for general term (p,)'. It is now easy to see that the third differences of the series, corre- sponding to the general term (p), are given by (p,y, or (p^y— iPs) = D + D.71+ D,n'-f . . . . -f-D„^ n"-^, &o.; and the (m — l)* differences by (i'm-i )'. or (Pm-i y — (Pv^i ) = Q 4- Q. n ; # SERIES. 209 and, finally, the m"" differences by ' that is, any of the m"" differences of the series, whose general term is(^), is given by Q ; that is, inTariably by the same quantity ; for whatever be n, Qi depends always equally on the oonstEint coefficients A, A,, A,, .... of (p) only ; hence it is constant, like them. The m"" differ- ences, therefore, of the series corresponding to the term'(^), are all equal ; hence the same (p) is the general term of any series of the order m. Nay, not only (^), but any expression reducible to the form of {p), represents likewise the general term of algebraic series of any order. Now, the following formula, i^ = «.+ ». [n"— (n — 1)2] + a, [nS — (ra_ 1)3] -\. . is reducible to the form of [p) ; hence (o) also represents the general term of any algebraic series. Snm of any ^ 131. Prom the Same (o) we have the first, the second, number of . , . . terms. the third term, and so on, of the series, by making m succession «^ 1, « = 2, ra = 3, &c., and these terms will be repre- sehted by the first member of (o), as follows : fij '3> tg, . . . r,j_i, t„ . Now, we say that the sum «, + 'a +•••• + *» of these n terms, is given by s„ = a, B-f a^ Ji'-f a, n3+ + a^x «"+' } (o,).- To demonstrate it, it is enough to show that (o,) is equal to the first term «, of the series when n is made equal to 1, and equal to ij-j- t^ when n is made equal to 2 ; equal to <, 4" 'a -|- 's when n is made equal to 3, &o. Now that (o,) is equal to <, when n is made equal to 1, is evident by observing that (o) is equal to (o,), when in both of them we make n = 1. Before showing that (o,) is equal to <, ■\- 1^ when n is made equal to 2, &o., observe that if in (o,) we change n into n — 1, we will have s^x = ffi, (ra — 1) + Oj (n — )2+ a, (m — l)s+ . . . + a^^ (ra— 1)-^' ; hence, s„— s,^, = a,+ u. In' (n — 1)'] + a, [nS _ („ _ 1)3] + . . . but this last member is the general term (0) of the series ; hence 18» 210 TREATISE ON ALGEBRA. % »» — «n-i ^ t„ (o«); and consequently if in (o,) we make n = 2, we will have Sg — St = ^a ! but Si^ t,; hence Sj = <, 4" '«• If in (o,) we make n = 3, then from the same (Oj) we have but Sj = "" +•■• = '■-•-*!+■••• + '»; that is, (o,) is the expression of the sum of n terms of any aeries of the m"" order. Let us now pass to see how, by means of the formulas (o) and (o,), we may find the general term and the sum of some given series. Examples. § ^^^- I'®* ^' ^> 1^' 1^' ^1 be the first terms of a given First. series, in which the second differences are all equal to 1 ; hence, in the general term of this series, m must be equal to 2 ; that is, the formula (o) will be, in this case, t„=a, + a,[B'— (« — 1)"] + a, [nS _ (» _ l)s]. To find out the coeflcieuts a„ a„ a„ make in succession m = 1, n = 2, n=:3; and since, with these substitutions, the general term ought to represent the first three terms of the given series, we will have the equations, a, + Bj -|- o, =3, Ui ~{- Bct^-\- 7^3 = 6, a.+ 5a,-i-19ffl, = 10, from which a^ = x, a,=:l, a, = — . Hence, the general term of the given series is <„=^ + 2»-l+i(3«2-3«+lj, (« + !)(« + 2) — 2 and the sum of n terms. SKRIES. 211 Let also, 1, 5, 14, 30, 55, 91 . . . be the first terms of another series whose third differences are all equal to 2. Making, therefore, in (o) m = 3, we will hare for the general term of this given series «„ = a.+ a,[ra2 — (« — 1)2] + a3[„3 _ („ _ 1)3] ^ a-,ln*—(n — l)% from which, making in suooessioti n ^1, n = 2, « =: 3, m = 4, we will have the equations, Oi+a-, +«> + «. = 1, a, 4" Sflj -j- 7a, + ISa, = 5, "i + Sffj + 19a, + 65a. = 14, ■'i 4- 7", 4- 37a, 4- 175a, = 30 ; and from these the following values of the coefficients : 15 11 '''=6' ''»=12' "'=3' "'=12 = and substituting these values in the general term, we have 1 , 1 . ■ 1 , <»= e" + 2™ +3™ ' and for the sum of the first n terms of the same given series, we will fiid 1 , 5 , , 1 , , 1 *»=6"+i2"+3"'+i2"'- Other ex- Some of the coefficients a^, a, . . of (o), and the first amples. t^iiji ^^ ^Iso, may be equal to zero, or may be such that some terms of (o) evolved be mutually eliminated. In this supposi- tion the general term may apparently have a different form from that of (p). So, for example, we may have in = "'. «» = «'■ Suppose now that such general terms are given, we may obtain the sums also ; for substituting successively the natural numbers 1, 2, 3 . . . instead of n, from t„ = «', we have the series 1, 4, 9, 16, 25 ... , and from <„ = n% 1, 8, 27, 64, 125 , the first of which has the second differences, and the last the third differences constant; hence, with the same process followed in the preceding examples, we will find for the first ^ 111 212 TREATISE ON ALGEBRA. * and for the second, n 1 1 1 and, therefore, the sum of n terms of the series having t„ = ti' for its general term, is 1 , 1 . , 1 , «»= gn + 2" +3™''; and the snm of n terms of the series having t„ ^ n? for its general term, is 1 2 . ^ 3 r ^ ^ »» = 4" + 2" + 4«''- PROBLEMS. 213 PEOBLEMS. 1st. Two merchants, A and B, possess a capital of 38700 pounds, but the capital of A is twice that of B ; how much does each one of them possess ? Ans. A . . . . 25800, B . . . . 12900. 2d. Philip makes a present to Ms children, M and N, of 2500 dollars, but M gets as many times 20 dollars as N gets 5. What is the share of each? Ans. M.... 2000, N.... 500. 3d. Divide the number 237 in two such parts, that the first be greater than the second by one quarter of the second. Ans. 1st. 131 + I, 2d. 105 + l- o o 4th. Two friends wish to buy a horse, but the first cannot pay but one-fifth of the price, and the second one-seventh only ; to have the horse they should add £20. What is the price of the horse ? Ans. a: = 30+^ 5th. A merchant after his speculations finds that he has gained 15 per cent, on his capital, and the amoimt of his actual fortune is £15571. What was the original capital ? Ans. a: = 18540. 6th. A man sells a certain amount of goods in three successive days. The first day he loses one-sixth of the value of the articles he is going to sell during the three days ; the second day he loses one- tenth of the same value ; but the last day he gains one-third of the price. At the end he finds that he has gained no more than three dollars. What is the price of the articles sold in the three days ? Ans. X = 45. 7th. Twice the number of years of my age, diminished by the fourth of the same number, gives twelve years more than those of my age. What is my age ? Ans. x = 16. 8th. A father sends to his five children 1000 dollars, with the con- dition that the eldest should have 20 dollars more than the second, and the second 20 dollars more than the third, and so likewise the rest How many shall the first of the children have ? Ans. X = 240. 214 TREATISE ON ALGEBRA. 9tli. I had 42 shillings, and I paid a part of them. If you divide the remainder by the number of those which I have paid, y KathoUscher Katechismus 19- Biblische Geachichte des AUen und Ntue/n l^tameavUs 2& 2d Murphy & Co.'s Standard School Books. KERNE Y'S ARITHMETICS. The Columbian Arithmetic^ designed for the use of Academies and Schools. By M. J. KerneTj A,M. Sixth improved edition.. 38 This 7ork possesses merits of a sspeiior nature in tbat department of science to which it belongs. Itim'bod^ot practical irutruction; one in -which the scieuce of flgures is thoroughly explained, and clearly elucidated. The examples for practice are generally such as the pupil will meet in tlie various business transactions of lire. The arrangement of the woi'lt is entirely progi'csaire, all questions being solved by rules previously explained. Introduction to the Columbian Arithmetic^ designed for the use of Academies and Schools. By M. J. KERXKr, A. M. Sixth edition 13 This little worlc is designed as an introdnction to the former, and 1b intended for children abont to commence the study of Arithmetic. The first principles of the science are familiarly explained Id the form of question and answer, and the pupils are conducted in the study as far as the end of compound numbers. It is replete with practical examples, adapted to the capacity of that class of learners for which it is designed, and it also contains all the Tables. In calling public attention to the foregoing works, the publishers take great pleasure in stating tbat they have already passed through several large editions, which is the most conclusive evidence of the high estimation in vthich they arc held by the instructors of youth, as far as they are known. The present editions have been carefully revised and corrected by the author, and no pnius will be spared to render them, at all times, deserving of the high reputation they have already acquired. The publishers are fully aware of the prejudices existing among many teachers iu favor of old works on Arithmetic; hence the great dilllculty of introducing new ones, however meritorious- from sucb they earnestly solicit a careful examination of these works — and respectfully refer them to the following testimonials of their merits from practical tciichers. * Alexandria, Va., November ISth, 1819. An examination of the " Columbian Arithmetic," by M. J, Kerney, has convinced us of its sterling utility, and we shall accordingly make arrangements for its immediate introduction into our school, (numbering, at present, eighty pupils.) We believe it to be second to none extant, and would, therefore, recommend it to uU teachers of Academies, &c. L. WHITTLESEY & SUN. 1 Iiave examined the " Columbian Arithmetic," and " Introduction" to it, by M. J. Kemey, find consider them excellent books ; they are judiciously arranged, and practical in their application. The rules are plain, sufficiently concise, and well adapted to the comprehension of young pcruons. The explanation of the theory of Proportion is simple, perspicuous, and accurate, we inteu J to in- troduce those books into our school. JOHN SLATTERY, Principal Washington Seminary. Number of pupils about 300, I have examined with much care the "Columbian Arithmetic," by M. J. Kerney. It apveavs to me to be a work of considerable merit, and is hetter calculated for schools of the United States, and for counting-bouses, than any other book on the subject tbat I have yet seen. The general ar- rangement is systematic, and according to the afflnitioa of different rules. Under the impres.'iion that it is an improvement upon every other work of the kind now before the public, I will immedi- ately adopt it in my school, numbering at present about ninety pupils, Waxhingum, Nov. 13, IW9. S. B. BITTBNHOUaE, Principal Washington luBtltute. As an evidence of the high opinion I entertain of the " Columbian Arithmetic," I have superseded the MSQotDaviea', by its immediate introduction. F. KNIGHTON, Alexandria XjiatiOiie. The " Introduction to the Columbian Arithmetic" is so admirably adapted to its purpose, that we have introduced it in the place of others in this seminary. Number of pupils, over one hundred. McLeod's Seminary, WoBhiJigton. J. O. WILSON, L. H. CHURCHILL, Associate Principals. I have examined the Arithmetics by U. J. Kerney, and unhesitatingly give them the preference over all the various works of the kind wliich I have met with. llrs. J. UcLEOB, Select School. I believe that the Columbian series of Arithmetics, by M. J. Kemey, better calculated to assist the pupil in that branch of science, than any other. A. W. HALL, Alexandria, Va. I am much pleased with Eemey's Arithmetics, and think them much better adapted to the use of pupils generally, than any others I have examined. RICHARD L. CaRNE, Jk. St. JoJm'a Academy, Alexandria, Va., November 14th, 1849. 1 am much pleased with the Arithmetics by M. J, Kemey, and shall introduce them in my school, believing them to be better calculated to facilitate the progress of the pupil in that branch of science, than any other book on that sutyect. Georgetown, November 13th, 1849, J. B. COMPTON. I have examined Kerney's " Columbian Arithmetic," and I consider it such a one as has been much wanted in schools. I decidedly prefer it to any heretofore used, and shall introduce it into my school. M. R. SHYNE, Navy Yard Academy, Washington. I have examined the " Introduction to the Columbian Arithmetic," by M.J. Kerney, and have adopted it in preference to others. I shaU also use the " Columbian Arithmetic" in my more ad- vanced classes. Boarding and Day School of the Misses HAWLEY, WaaJtington, D. 0. I have examined, as far as my leisure would permit, the " Columbian Arithmetic," and am muob pleaded with many features of the work. I have introduced it into my school. HENBY B. WOODBUBT, Principal Washington Select School. 26 Mtirpliy & Co.'s Standard School Books. PRBDET'S UNIVERSAL HISTORIES, &C. The disfcinguished and wide-spread reputation of the author as aa historiaa and Professor of History in St. Mary's College for tlie laat twenty years ; — the universal favor with which these worka have been received, and their immediate introduction into many of the principal literary institu- tions in-tbe United States, precludes the necessity of giving many of the numerous complimentary and Qattering tcstimoidals that have been so freely extended to them, both in this country, and in England, where they are extensively used. QZr" Prof- I'i'edet's Histories havo been adopted as Test-Books in the Irish TTuivcrsIt^. Ancient Histrrry : from the dispersion of the Sons of Noe, to the Battle of Ac- tiura, and the change of the Eoman Republic into an Empire. By Petee JFbedet, D. D., Professor of History in St. Mary's College, Baltimore. Fourth edition, care- fully revised and enlarged i2mo 88 Modem History : from the coming of Christ, and the change of the Roman Re- public into an Empire, to the year of our liOrd, 1854. By Petee Fbbdet, D.D., Professor of 'History in St. Mary's College, Baltimore. Tenth enlarged and improved edition '. 12mo. 88 Mw and Improved Editions, carefuUy revised and con-ected hy the Avihor. These two volumes for a Complete Course op History, or a continuous chain of Historical Events, from the Ceeation of the Would to the Year 1854. The publishers are happy to announce that they have just issued new, enlarged, and improved editions of the above works, in uniform style. Each volume contains upwards of five hundred pages, and may justly be considered the cheapest, most authentis, and reliable histories published. The London Catholic Standard says : " These two excellent manuals of history have a wide and increasing circulation in America, and are everywhere held in the highest esteem. The compiler, Dr. Frodet, has acUiei-ed a task of no ordinary difficulty, in oompressiog so much recondite matter in- to so small a space ; in leariag untold nothing that was of note of the immense and varied annala of the n'Orld. No college, school, or library ought to be without these excellent works." The DuUin Telegraph says : " Fredefs Histories have been adopted, as a class-book, by the Irish Catholic University; and we entertain no doubt, that they will soon supersede, even in other estab- lishments, those miserable compilations which wilful perverters of truth have long palmed upon the public— both Catholic and Protestant — as histories and abridgments of histories." The JDuhZin Tahlet says : "These two volumes are plain, copious, and useful summaries of history, and the number of editions through which they have passed attest their popularity." The Catholic Inst/ructor says : " We hope these Histories will soon find their way into every literary institution among us, in order that the young may learn the past from pure and uncovrupted sources." The Catholic Sentinel says : " These beautiful treatises are quite deserving of the patronage which they obtain. They are most commendable for their Christian and unbiassed spirit. And we are not Orstonished that Dr. Fredet has his name taken up by the Irish University, proud that America has made therein such an inroad upon the abridged histories heretofore existing." '!Che Metropolitan says: "The style is veritably charming l^its simplicity, and by the quiet love of his subject which the re^-erend author constantly displays, ft is the language of a talented and suc- cessful teacher, who relates to his class the great events of time, succinctly but graphically, without bombast, yet in a lively and picturesque manner. It is thus that history should be written for youth." I^ingard's England Abrid£;ed, for the Use of Schools. An Abridgment of the History of England. By John Lingaed, D.D. With a con- tinuation from 16SS to the reign of Queen Victoria, by James Burke, Esq., A.B. With Marginal Notes, adapted to the use of schools in the United States, by M. J. Kernet, a. M 12mo, half arabesque 1 00 An abridgment of Dr. Lingard's great work, adapted to the use of schools, has been long and anxiously looked for in this country. The publishers take great pleasure in inviting the earnest at- tention of the conductors of schools, and others interested in the cause of education, to this edition. Although Lingard's England has been nearly half a century before the public, not one fact stated by him has been proved to be erroneous, while the critics of all creeds have joined in expressing their approbation of his great work. In style without a superior, in truthfulness without an equal, Lingard stands before the historic student as the model of what an historian should be. Having thus spoken of the atyl& of Lingard, it is right to add that the student will find that the ipsiesima verba of the great Catholic historian of England have been religiously preserved in the Abridgment. Of the continuation we shall merely say that it has been written by an author who has been long and tSa- vorably knoWn in literature. The publishers therefore feel confident that Mr. Burke wUl be found to have written in strict accordance with the spirit which dictated the great work of the historian whose pages he has followed. The sketch of the British Constitution, the abstract of the geography of England in Saxon times, the list of eminent natives, and the marginal notes, will add much to the interest of the work, and will be found useful by way of reference. McSherry's History of Maryland, with Quebtiohb, &c 75 This work is used in the Public Schools of Baltimore, and is strongly recommended by the Com- missioners. 27 Murphy & Co.'s Standard School Books. IRHNG'S SERES OF SCHOOL CATECHISMS, IN TWELVE PARTS. Revised by M. J. Kerney, A. M. TnE long-established reputation of Irting'S Catechisus, and the Terj extensive clrouliv- tion which they bare had, not only in England, but alsointhiBOOuntrj, is the best proof of their ntilitj. The plan of his works is the very best that could be adopted. The oatechetioal form of instruction is now admitted by tho most experienced teachers, to be the best adapted to the nature and capacity of youth ; — a system by which ohildreu will aoquire a knowledge of a science in less time than by any other. Murphy & Co., haTing become the publishers of this standard and highly popular series of Gate cbisnis, wish to inform the public that they have issued entirely new editions, with all new dlaco- veriea and modem improvements in each branch, audcr the careful supervision of M. J. KEBN'PIY, Esq., who has prepared for the series a CATECHISM OF THE HISTORY OP THE XJNITED STATES— an entirely new work. iVie folloifsing constitute the Series : Astronomy: containing the Motions, Magnitude, Periods, Distances, and other Phenomena of the Heavenly Bodies, founded on the laws of Gravitation. With on- graved Illustrations 13 This little volume possesses the pecnliar merit of reducing to tho comprehension of children the principles of a difficult, but, at the same time, a most interesting science. It explains the solar system, the courses and the revolutlocs of the planets, eclipses, the theory of tides, and many other Interesting astrouomical principles. Botany: containing a Description of the most familiar and interesting Plants, arranged according to the Linnccan System, with an Atpendix on the formation of an Herbarium. "With engraved Illustrations 13 This popular little work is intended for children who are about to enter on the study of the In- teresting science of Botany. The plan of the work is admirably adapted to that class of learners for which it is designed. It presents to the mind of the pupil, in nn easy and atlraotlve style, the various beauties of the science, and the many advantages to be derived from its study. Practical Chemistry: being a Familiar Introduction to that interesting Science —with an Appendix, containing many safe, easy, and pleasing Experiments. "With engraved Hlnstrations 13 This little treatise is admirably adapted for those who are about to enter on the study of Chemis- try, being a familiar introduction to that science. The grand and leading principlcH of cliemical knowledge are explained on a plan that will be found to be both interesting and instructive. Though originally designed for the young. It will be found to contain lessons that may be rood wlti pleasure and profit by the more advanced in years. Mythology: beiog a Compendious History of the Heathen Gods, Goddesses, and Heroes; designed chiefly as an Introduction to the Study of the Ancient Classics. With engraved Illustrations 13 To the English scholar this work will prove highly Interesting; but to the classical student it «rill be found a most desirable compendium. It embraces all that is interesting or important in the fiubject of which it treats ; while, at the same time, the brevity and clearness of Its style render it preferable to other works of the same kind which are of much greater dimensions. Classical Biography: containing an Account of the Lives of the most Celehrated Characters among the Ancient Greeks and Romans. With engraved Illustrations . . , 13 To the classical student, in particular, the above named work will be found to possess peculiar merits. It presents, in a small compass, the most interesting events in the lives of those whose names have cast a lustre over the historic pages of Greece and Borne. History of the. United States : with a Chronological Table of American History, from its discovery in 1492, to the year 1854 ' . 13 This valuable little work comprises within a small compass all the most Important and interest- dng events in the history of the United States, from the discovery of America to the present time, The arrangement and style are admirably adapted Co the capacity of children about to commence the study of history. It is sufficiently compreheDsive for that class of learners for which it Is de- signed. From its instructive pages the child will learn to revere the names and Imitate the actions •of those illustrious men of America who have gone before us in the path of usefulness and of fame. Gredan Histnry : from the Earliest Times to the Period when Greece became a Roman Province. With engraved Illustrations 13 Mojnan Histwy ; containing a concise Account of the most Striking Events, from the Foundation of the City to the Fall of the Western Empire. With engraved Illustrations '3 These two works contain all the most important and Interesting events in the bistoiy of ■Greece and Rome. As introductory works, to be placed in the hands of children, they will be found «o possess peculiar merits. The arrangement and style are happily adapted to that class of learnen for which they are designed. ■Catechism of Sacred History : Abridged for the use of Schools, translated from the Freuoh, by a Friend of Youth : designed to accompany Irviug's Series of Catechisms 13 2*8 Murphy & Co/s Standard School Sooks. Bi&tory of England : containing the most Striking Events from the Earliest Pe- riod to the Present Time 13 This work compriseg, in a few pages, the most important events in the history of England, from a period prior to the invasion of the Romans to the present time. The present edition has been oju-efully revised and corrected ; every thing resecting on the American character has been erased, and every thing of a sectarian nature has been removed, Jewish Antiquities: containing an Account of the Classes, Institutions, Rites, Ceremonies, Manners, Customs, &q., of the Ancient Jews, With engraved Illustrations 13 Ch'ecian Antiquities : being an Account of the Religion, Government, Judicial Proceedings, Military and Naval Affairs, Dress, Food, Baths, Exercises, Marriages, Funerals, Coins, Weights, Measures, &,a., of the Greeks — to which is prefixed a De- scription of the Cities of Athens and Sparta. With engraved Illustrations 13 KoTTian Antiquities; or, An Account of the Religion, Civil Government, Military and Naval Affairs, Games, Names, Coins, Weights and Measures, Dress, Food, Exer- cises, Baths, Domestic Employments, Marriages, Funerals, and other Customs and Ceremonies of the Roman People ; with a Description of the Public Buildings of the city of Rome. With engraved Illustrations 13- The above works are highly interesting In themselves, and may be read with pleasure and profit by every member of the community. Bat for the classical student they possess particular attrao- tions. For his benefit they were chiefly intended, and years of experience prove that they are pe- onliarly adapted to the end for which they were designed; A familiarity with the laws, manners, and customs of the ancient nations will often render clear and ezpUdt the most obsctu'e passages, so frequently met with in the authors of antiquity. CIiASSICAIi BOOKS, &c. In calling attention to the following Works, it is deemed it sufBoient to state, that the pre- sent editione have been issued under the careful supervision of the eminent Professors of St. Mary's College, Baltimore, and may justly be considered the hest and cheapest editions published. Epitome Histori(B SacrcB Auctore, L'homond, edito Nova JProsodicB, signes vo- cumque interpretatione adornata 30 As an elementary work, Historise Sacraj is beyond exceptions. The easy arrangements of its style in the beginning, and the gradual introduction of the Latin construction, relieve the pupil of much embiirrassment and labor, and tend in a material degree to facilitate his advancement. This posECBses advantages over any previous edition. The vocabulary has been carefully revised, and the work has received such improvements as greatly enhance its merits. Ph(edri Axigusti Liberii Fabularum iEsopium, Libri Quinque 30 A new edition, carefully revised and greatly improved. This little work has long been held in high estimation in our colleges and schools. The man; moral and interesting lessons it contains render it a text-book peculiarly adapted to the young ; and, indeed, no work could be better designed to initiate the pupil into the study of Latin poetry. De Viris lUustHbus Urbis Eomm,, A Romulo ad Augustum, Auctore L'homond, in Universitate, Farisiensi Frofessore Emerito 38 This work possesses the rare quality of being admirably adapted to the capacity of those com- mencing the study of the Latin language, without deviating from the purity of the Latin style. The materials of which it is compiled are most interesting and instructive in their nature, thus affording the pupil the double advantage of acquiring a knowledge of the Latin tongue, and, at the same time, of storing his mind with historical facts. This edition haa been lately revised, and put into a neat, convenient foi'm. These improvements, it Is believed, will add to its merits, and wiU tend to advance the pupil in his study. Fables Choisies de la Fontaine, Nouvelle Edition 63 Few works have elicited more general admiration, or have been more generally used in schools, than the Fables of La Fontaine. Forthe pupil engaged in the study of the French language they possess peculiar advantages. Many beautiful and moral lessons are inculcated in a style at once easy and attractive, while, at the same time, a taste for poetical composition is cultivated. This edition has been carefully revised, and contains much desirable improvement.' Rvddiman^s Bvdiments of the Latin Sbngne ; or a Plain and Easy Introduction to Latin Grammar : wherein the principles of the language are methodically digested, both in the English and. Latin. With useful Notes and Observations. Thirtiem Edi- tion, Corrected and Improved. By Wm, Mann, M. A 12mo, half arab. 38 The cheapest and best Latin grammar published. EleTti&ntos de Sicologio, Elements of Pyschology 75 PiwLiTo^s Dialogues. Select Ori^nal Dialogues, or Spanish and English Conver- sations : followed by a collection of pieces in prose and verse — adapted to the use of Spanish classes in schools and academies. By J. A. Pizahro, Professor of the Spanish Language in St. Mary's College, Baltuuore. Third edition, im- proved and enlarged by the author 12mo 75 This new edition of a very popular work, by one of the most distinguished instructors in the •«untry, is greatly improved, and particularly adapted to the present style of teaching andself-im- b^ovement. The prior editions have become established as standard in some of the best institution! •M the United States, and the present doubles its advantages. 3» 29 DIurpliy & Go/s Standard Sckool Books. SESTINI'S AIiGEBRA. Elementary Algebra. By B. Sestini, S. J., author of Analytical Geometry, Professor of Natural Philosophy and Astronomy in Georgeto^m College _ 12mo 60 The m&in object of this treatise is to render the science of Algebra Intelligible to pupils Mhose minds are yet nnaccustomcd to such studies. The beginner will here be faiiiistaed with auch prooh &B are salted to his capacity; examples will afford new light to what might be otherwiac obscure ; with regard to the operations founded on higher principles, he will, for the present, conten^ himself with merely practical roles, exemplified lathe same manner. With a mind thus gradually led ou to strict mathematical discussion, he may then resume his course with profit, by the aid of a treatise now in preparation, which is intended as a sequel to this, and, by more exact aud thorough luvestl* gatioQ, complete h^ Btndy of Algebra. BRIEF EXTBACTS PEOM NOTICES OF THE PRESS. " This work recommends itself to favor by the admirable order of its parts, and the conciseness and eleamesB with which its principles are expounded. One needs but open the book to perceive that the author has brongbt to the execution of his task a ripened judgment and well-tried experience. He Ib not a compiler — his work has the rare merit of originality, and every student of Algebra wUl thauk him for having given in a few pages what has usually occupied a large volume, and for hav- ing rendered intelligible what has often proved an enigma to many," MelropolUan. " This book might very properly be called " Algebra without a master." One very Important Ira- provement that the author has made upon all our text-books, aud which deserves to be mentioocd, is this, that he keeps monomials and polynomials disttnat, and explains and applies to them separately the various rules as laid down In his Algebra. The work only wants to be known, in order to ha universally approved." WeBtem Tablet. " We feel much pleasure in recommending it as containing nearly all necessary t6 be known on the subject of which it treats. It is emlneatly adapted for the use of young persons who wish to ac- quire a knowledge of the difficult science of Algebra." ffcdi/ax Catholic. " To persons commencing the study of Algebra, we cheerfully recommend Mr. Scstinl's work, as one well calculated to smooth the difBculties which beginners have to overcome in their first attempts to master that science." PUtahurg Catholic. "As a rudimentalbookitwillbefoundcmincntlyusefulln schools andcolleges." Dct. TiTidicator. , ' ' The author is well known as a man of great ability, and his work cannot fail to be of good scr vice in schools." Bujfaio Sentinel. SESTINI'S ANALYTICAL GEOMETRY. A Treatise of Analytical Geometry, proposed by B. Sestini, S. J., author of Elementary Algebra, Professor of Natural Philosophy and Astro- nomy in Georgetovm College 8vo, paper 1 25 "This treatise of Prof. Sestini discusses the various topics under consideration by a purely analytical method, and is well adapted to the modern plan embraced by learned professors, who do not content themselves with a anperficial view of the subject, but dive into its deepest recesses with no other Instrument of research than analysis, and naked analysis. The new treatise Is an acquisi- tion for the lovers of the exact sciences, taught in the most exact manner ; it introduces some new methods of the Baron Cauchy, a savant well known, and highly esteemed in Paris for his scientillB acquirements. As the new treatise is intended for the use of Georgetown College, we are inclioed from this circumstaace to form a very favorable opinion of the i>roflciency of the students in the mostabtruse branches of mathematics, and it is a subject on which we congratulate the teachers and their scholars," Me^f^tMta/n. jg^ J. muBFHT k Co. have the pleasure to announce, that in addition to tHeir own list of School Books, their arrangements with the principal publishers are each as to receive ALL New Wohks on Education as soon as issued— and to keep a large stock constantly on hand, which enables them to supply orders with the least possible delay. SCHOOL AND CLASSICAL BOOKS, PAPER, STATIONERY, &c. A large stock, comprising every variety, constantly on hand. FRENCH SCHOOL BOOKS.— The latest and best editions of French School Books, con- stantly on hand — or imported to order at short notice. jg®" Orders are respectfully solicited — to which they pledge themselves to give Hie sama oareful and prompt attention as if selected in person. Particular attention given to the packing and shipment of orders to distant points. J, MURPHY & CO., Publishers, 178 Marltet Street, BaltxTnore. 30 X 8 S 5. JUST PUBLISHED. A SCHOOL EDITION OF LINGARD'S ENGLAND, In a neat volume of nearly 700 pages ISrao.j only f 1. Abridgment of the History of England. By John Lingard, D. D. With a Continuation from 1688 to the Reign of Queen Victoria. Bir James Burke, Esq. To which are added. Marginal Notes and Questions, adapted to the, use of Schools ; by M. J. Kerney, A. M. The student will find that the ipsissima verba of the great Catholic Historian of England to have been religiously preserved in the Abridgment. Of tbe Contmuation we shall merely say, that it has been written by an author who has been long and favorably known in literature. OPINIONS OP THE ENGLISH AND IRISH PRESS. The Duhlin Review says : — " Mr. Burke's Abridgment is comijletely successful. We do not hesitate to pronounce the work as a whole, one of the most valuable ad- ditions to our scanty school literature which we have met with for many years." The London edition of Brovmson's Review says : — " Mr. Burke appears to have entered on his task with an enthusiasm equal to the abihty which he has displayed m executing it. He has formed a Manual of British History, not merely the best for the object aimed at— the instruction of youth — but a volume of safe reference to those of riper yeais." The Dublin Freeman's Journal says : — "Mr. Burke has performed his laborious task well, compressing into a comparatively small space, the substance of such a large work iu the author^s own [language, adding a clear and rapid abstract of tbe national history down to the present year." The Drogheda ^flrgus says:— "Mr. Burke's attempt to condense Dr. Lingard'a work, has proved eminently successful. " The London Lamp says :— " We thank Mr. Burke for the admirable manner in which he has accomplished his very difficult task of abridging the writing^ of Dr. Lingard, and we congratulate him on his valuable Continuation." The Dvilin Euening Post says: — "This Abridgment has been prepared for the use of schools, and while serving this purpose to perfection, it will do more, and persons who have long passed their school days may readily profit by the condensed information which it contains." The Hull Advertiser says :— " Having compared the Abridgment with the large work, we can testify that Mr. Burke has performed the task imdertaken by him with great skill, and the constant exercise of a sound judgment." The London Critic says :— « The author has carefully and Tsuccessfully produced a volume that must be very acceptable to those for whose use it was designed." The Cork Southern fieporier says : — "A really -meritorioos and excellent work." The London Rambler says :— " Mr. Burke has done his work well, and the result is very satisfactory." The Dublin Tablet says :— " A Catholic History of England suitable for schools has long been a desideratum. The present volume supplies that want which has been so much and so constantly lamented." The Midland Counties Gazette says : — " In the discharge of so heavy a task, re- quiring great assiduity and labor, combined with judgment and literary taste, Mr. Burke has fully acquitted himself to the satisfaction of the Catholic public." Payson and Dunton's Penmanshipj in 6 parts. M. & Co. having been appointed Agents for the sale of these Books, take great pleasure in recommend- ing them to Teachers, as the most complete and perfect System of Penmanship published. Just Published, in & neat iSimo. Volume, price 50 eta. Rudiments of the Greek Language^ arranged for the Students of Loyola College, Baltimore,— upon the basis of Wetenthall. Extract from the Preface.—" It is not intended by Uiis publication to present a new Greek Grammar to the classical student; atler the elaborate volumes of Mattliiae, Bultman, Kuhner, Gail, Bumouf and other Echolars of Germany and France, it would be altogether vain to expect any new discovery in tliat language. The most that we can do is to avail ourselves of their labors in order to smooth the difficulties, which are usually met in its study. Tlie greatest of these we have learned from a long experience in teaching, is the large size of the grammars, which are put in the student's hands, when be commences. Excellent as these may be for the pro- fessor or more advanced scholar, they only tend to deter the beginner from approach- ing it. We trust that this will be obviated by the present compendium, in which we have endeavoured to comprise v^tbln as short a compass as possible, all that is of absolute necessity to the learner. If it induce him to apply with more alacrity to study a language, second to none in the literary beauties and treasures which It contains, our intentions will have been amply fulfilled." *3 Treatiseon Algebra, by E. Sebtini, S. J., author of " Elementary Algebra," "Analytical Geometry," &c. (Nearly ready.) In the preface to the Elementary Algebra, published in 1854, the author an- nounced his intention of preparing the present work, as a sequel, and thus present to the student a complete ^Treatise on Algebra. This work is divided into two parts, the first embracing Algebraical Operations, and many other inieresting ques- tions connected with thera. The Second contains tlie most important Theorems. Spanish Spelliiig and Reading Books for Boys and Girls. Silabario Castellano, para el Uso de los Ninos, It^ Silabario Castellano, para el Uso de las Ninas, 18| New and Improved editions, nearly ready. FTl-^lSr GJH. ©GHOOL BOOIKS- The latest and best editions of French School Books kept constantly on hand— or imported to order at short notice— A liberal discount to Teachers. Noel et Chapsal. Grammaire et Exercises Francais, 1 vol $1 00 " « Corrig^des *' " 63 L'homond. Elements de Grammaire Francaise 13 Dictionnaire I'Academie Francaise, 9 vols. '4to 15 00 Noel. Gradus ad Parnassum . .' 2 95 Cortambert. Lecons de Geographic, 8vo 3 00 Gaultier. Lecons deG£ographie...$0 50 Meissas. Petit Atlas G^ographie Gaultier. Atlas " ... 3 00 Modeme, 8vo 150 Drioux. Atlas UniverselledeG^ogra- Geruzez. Lecons de Mythologie, phie 300 8vo 1 00 Drioux. Atlas Elementaire, do 1 00 Boileau, Oeuvres Poetiques 50 Meissas. PetltAtlas Geographic An- Sevigne,JVIme., Lettres Choisies. 87 cienne, 8vo 1 50 La Fontaine, Fables Choisies. ... 63 7 The following STANDARD SCHOOIi BOOKS, published by Messrs. DuxUgan & Bro., New York, are kept constantly on sale and supplied at Publishers' Prices, Wholesale and Retail: Cannon's Practical Speller— Catholic School Book. Lessons for Young Learners, Nos. 1 & 3 — Universal Reading Book. Christian Brothers' Books, Nos. 1, 3 & 3, cheap edition. Christian Brothers' Books, Nos. 1, 2, 3 and 4, improved edition. Walker's Dictionary^Challoner's Bible History— Graces' Outlines of History. History of Modern Europe — Selects e Classicis Latinitates Auctoribus 1 & 2. Viri Romoe — Phedrus— Cornelius Nepos — Cicero de Senectute. History of the United States, to correspond with Graces' Outlines, nearly ready. ESSutSB, 'M'^affilbiahlSU, 1 ^sfl JffrS^:t|rl¥pi^^^'i .:gS|l