lE. 
 
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 Each volume in the series is so constructed that it may be used 
 with equal ease by the youngest and least disciplined who should 
 be pursuing its theme, and by those who in more mature years 
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 SHAW'S NEW SERIES 
 
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 ENGLISH AND AMERICAN LITEEATUEE. 
 
 Shaw's New History of English and Amemcan Lit- 
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 using as a basis Shaw's Manual, edited by Dr. William Smith. 
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 4. The history of great Authors is marked by the use of larger-sized 
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 5. It also contains diaeraras, showing the easiest way to classify and 
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 II. 
 Shaw's Specimens of American LiteratiirCf find 
 Literary Header, Greatly Enlarged. By Prof. Benj. 
 N. Martin, D.D., L.H.D., Professor in the University of the City 
 of New York. 1 vol. 12mo. 
 
 This book contains specimens from all the chief American writers. Espe- 
 cially those authors who have given tone and character to American 
 literature are so represented that scholars may obtain a just idea of their 
 style. 
 
 As a LITJ£RAltY READER for use in our Higher Seminaries, it is 
 believed that no superior book can be found. 
 
 III. 
 
 Shaw's Choice Specimens of English Literature, 
 
 A Companion Volume to the New History of Literature. Selected 
 from the chief English writers, and arranged chronologically by 
 Thos. B. Shaw and Wm. Smith, LL.D. Arranged and en'arged 
 for American students by Benj. N. Martin, D.D., L.H.D., Prof, 
 of Philosophy and Logic in the University of the City of New 
 York. 1 vol. large 12mo. 
 
 We shall still continue to publish 
 Shaw's Complete Manual of English and American 
 Literature, By Thos. B. Shaw, M.A., Wm. Smith, LL.D., 
 author of Smith's Bible and Classical Dictionaries, and Prof. 
 Henry T. Tuckerman. With copious notes and illustrations. 
 1 vol. large 12mo, 540 pp. 
 
A 
 
 u:niyersity 
 
 ALGEBRA 
 
 COMPRISING 
 
 I. -A COMPENDIOUS, YET COMPLETE AND THOROUGH COURSE IN ELEMENTARY 
 ALGEBRA, AND 
 
 II.— AN ADVANCED COURSE IN ALGEBRA, suppiciently extendbd to meet tub 
 
 WAMT3 or OUB UKirEBSITlES, COLLEGES, AND SCHOOLS Of SCIENCE. 
 
 EDWARD pLNEY, 
 
 PROVE880R OP MATHBMATICS IN THK UNITERSITY OP MICHIQAN, AND AUTHOR OV 
 A SERIES OF MATHEMATICS. 
 
 NEW YORK: 
 
 SHELDON & COMPANY, 
 
 No. 8 MURRAY STREET. 
 1885. 
 
oj-e _?. 
 
 PROF. OLNEY'S MATHEMATICAL COURSE. 
 
 INTRODUCTION TO ALGEBRA 
 
 COMPLETE ALGEBRA 
 
 KEY TO COMPLETE ALGEBRA --.--- 
 
 UNIVERSITY ALGEBRA 
 
 KEY" TO UNIVERSITY ALGEBRA 
 
 A VOLUME OF TEST EXAMPLES IN ALGEBRA 
 
 ELEMENTS OF GEOMETRY AND TRIGONOMETRY 
 
 ELEMENTS OF GEOMETRY AND TRIGONOMETRY, University 
 Edition 
 
 ELEMENTS OF GEOMETRY, separate 
 
 ELEMENTS OF TRIGONOMETRY, separate 
 
 GENERAL GEOMETRY AND CALCULUS 
 
 BELLOWS' TRIGONOMETRY 
 
 * ^ » m * 
 
 PROF. OLNEY^S SERIES OF ARITHMETICS. 
 
 PRIMARY ARITHMETIC 
 
 ELEMENTS OF ARITHMETIC -------- 
 
 PRACTICAL ARITHxMETIC 
 
 SCIENCE OF ARITHMETIC - ^ 
 
 Entered according to Act of Congress, in the year 1873, 
 
 By SHELDON & COMPANY, 
 
 In the Office of the Librarian of Congress, at Washington. 
 
 UN. A. 
 
The Author's Complete School Algebra was written to meet the wants 
 of our Common and High Schools and Academies, and to afford adequate prepara- 
 tion for entering our best Colleges, Schools of Science, and Universities. 
 
 The present volume is designed for use in these advanced courses of training. 
 Thus, while it is thought that the former affords as extended a course in Algebra 
 as is expedient for the preparatory schools, it is believed that this will be found 
 to contain all that these higher schools require. 
 
 It was deemed necessary to make the work a complete treatise, including the 
 Elements, for purposes of reference, and for reviews, and also in consideration 
 of the fact that our higher institutions have various standards of requirement 
 for admission. In fact, there are few students of Higher Algebra who do not 
 find it necessary to have the Elements at hand for occasional consultation. 
 
 This Elementary portion is embraced in the first 150 pages, and contains all 
 the definitions, principles, rules, and demonstrations of the Complete School 
 Algebra, with an abundant collection of Neio Examines ; but from it all ele- 
 mentary illustrations, explanations, solutions, and suggestions, are omitted. 
 The whole is so arranged as to secure readiness of reference and convenience 
 of review by somewhat mature students. 
 
 The subjects treated in Part III., which constitutes the Advanced Course 
 proper, will be best seen by turning to the Table of Contents. In this place 
 the author wishes merely to call attention to a few of the distinguishing fea- 
 tures of this Part. 
 
 1, The conception of Function and Variable is introduced at once, and is 
 made familiar by such use of it as mathematicians are constantly making. No 
 one needs to be told that this conception lies at the foundation of all higher 
 algebraic discussion ; yet, strangely enough, the very terms are scarcely to be 
 found in our common text-books, and the practical use of the conception is 
 totally wanting. 80054S 
 
iV PREFACE. 
 
 2. The first chapter in the Advanced Course is given to an olcmpntary and 
 practical exposition of the Infinitesimal Analysis. Tlie autlior knows from liis 
 own experience, and from tliat of many others, that tliis subject presents no 
 peculiar diflBculties to ordinary minds ; and everybody knows tliat it is cmly by 
 this analysis that the development of functions, as in the Binomial Formula, 
 Logarithmic Series, etc., the general relation of function and variable, tlie 
 evolution of many of the principles requisite in solving the Higher Equations, 
 and many other subjects, are ever treated by mathematicians, except wlien thoy 
 attempt to make Algebras. No mathematician thinks of using the clumsy 
 and antiquated processes by which we have been accustomed to teach our i>upil3 
 in algebra to demonstrate the Binomial Formula, produce the Logarithmic Series, 
 deduce the law of derived polynomials, examine the relative rate of change of a 
 function and its variable, etc., except when he is teaching the tyro. Why not, 
 then, dismiss forever these processes, and let tin; pupil enter at once upon those 
 elegant and productive methods of thinking which he will ever after use ? 
 
 3. By the introduction of a short chapter on Loci of Equations, which any 
 one can read even without a knowledge of Elementary Geometry, and which 
 in itself is always interesting to the pupil, and of fundamental use in the sub- 
 sequent course, all the more abstruse principles of tlie Tfieory of EqwUions are 
 illustrated, and the student is thus enabled to see the truth, as well as to demon- 
 strate it abstractly. How great an advantage this is, no experienced teacher 
 
 to be told. 
 
 4. In the treatment of the Higher E<iuations, while some things have been 
 discarded which everylxxly knows to be worthless, but which have in some 
 way found a place in our text-books, a far more full and clear discussion of 
 practical principles and methods is given, than is found in any of the trea- 
 tises in common use. 
 
 5. The important but difficult subject of the Discussion of Equations lias 
 been reserved till late in the course, for several reasons. Thus, when the pupil 
 reaches this topic, he has become familiar with most of the principles to he 
 applied, and has Ijecome sufficiently imbued with the spirit of the algebraic 
 analy.sis to be enabled to grasp it. To discuss an equation independently and 
 well, is a high mathematical accomplishment, and should not be expected of tlie 
 tyro. It is nothing else than to think in mathematical formula?, and hence is 
 one of the later products of mathematical study. It is hoped that the position 
 assigned to this subject in the course, and the manner of treating it, will insures 
 better results tlian we have hitherto been able to obtain. 
 
 6. In the selection of Subjects to be Presented, constant regard has been had 
 
PREFACE. V 
 
 to tlie demands of the subsequent nKithcniatical coarse. This has led to the 
 omission of a number of theorems and methods, which, though well enough 
 in themselves as mere matter of theory, find no practical application in a sub- 
 sequent course, however extended ; and has, at the same timy, led to the 
 introduction of not a few things which the advanced student always finds occa- 
 gion to use, but for which he searches his Algebra in vain, if he has at hand 
 nothing but our common American text-books. 
 
 7. In Method of Treatment the following principles have been kept constantly 
 in mind : 1. That the view presented be in line with the mathematical thinking 
 of to-day. 2. That everything be rigidly demonstrated and amply and clearly 
 illustrated. 3. When long experience has shown that the majority of good 
 students have difficulty in comprehending a subject, special pains should be 
 taken to elucidate it. 4. No principle is thoroughly learned by a pupil until he 
 can apply it ; and nothing so fixes principles in the mind as the use of them. 
 Hence an unusually large number of examples has been introduced. 5. It is 
 often necessary to multiply examples in order to meet the requirements of the 
 class-room. 
 
 8. Answers. — The answers to examples are not generally annexed to them in 
 the text. There are, however, two editions of the volume, one with the answers 
 at the end, and the other without any answers, except an occasional one in the 
 body of the book. 
 
 9. Finally, the Order of Topics is such that a student requiring a less extended 
 course than the entire volume presents, can stop at any point, and feel assured 
 that what he has studied is of more elementary importance than what follows. 
 Thus students who do not desire to study the Higher Equations can conclude 
 their course with the first chapter of Part III.; and a course which includes the 
 first three chapters of this part will be found as extended as most of our 
 Academies, and perhaps many of our Colleges, will find expedient. 
 
 &uch works as those of Seiiret, Cirodde, Comberousse, Wood, Hymers, 
 Hind, Todhunter, Young, and most of our American treatises, have been at 
 hand during the preparation of the entire volume. To WrirrwoRTn's charming 
 little treatise on Choice and Chance, the author is indebted for a number of 
 examples in the last section. 
 
 The quick eye and cultivated taste of my friend, Mr. W. W. Beman, AM., 
 Instructor of Mathematics in the University, have done me excellent service in 
 reading the proof-sheets, and have, I trust, given the work a dtrgree of typo- 
 graphical accuracy not usually found in first issues of such treatises. 
 
Tl PREFACE. 
 
 Witli these words of explanation as to wlmt I have attempted to do, I commit 
 tlie volume to the hands of my f el low-laborers in the work of teaching, assured 
 from the generous and ai>preciative reception which they have given my previous 
 efforts, that this will not fail of a candid consideration. 
 
 EDWARD OLXEY. 
 
 Univeiisity op MmiKJAN, 
 Ann Arbot'y July, 1873. 
 
^ 
 
 ONTIenTS')) 
 
 INTRODUCTION. 
 
 SECTION I. 
 General Definitions, and the Algebraic Notation. 
 
 PAGB 
 
 Branches of Pure Mathematics: 
 
 Pure Mutheiuatics. — Definitiou (1) ; branches enumerated (3, 3) 1 
 
 Quantity.— Definition (4) 1 
 
 Number. — Definition (5) ; Discontinuous and Continuous (6, 7, 8) 1, 2 
 
 Definition of the several branches. — Arithmetic (9) ; Algebra (10) ; 
 Calculus (11); Geometry (12) 2 
 
 Lo^hco-Mathematical Terms : 
 
 Proi)osition. — Definition (13) 
 
 Varieties of Propositions. — Enumerated (14); Axiom (15); Theo- 
 rem (16); Lemma (18); Corollary (19) ; Postulate (20) ; Problem (21). 2, % 
 
 Definition of.— Demonstration (17) ; Rule (22) ; Solution (23) ; Scho- 
 lium (24) 2,3 
 
 PART L— LITERAL ARITHMETIC. 
 
 CTIAPTEU I. 
 FUNDAMENTAL RULES, 
 
 SECTION I. 
 
 Notation. 
 
 System of Notation.— Definition (25) 4 
 
 Symbols of Quantity.— Arabic (2G) ; Literal (27) ; advantages of latter (28). 4, 5 
 
 l^awH.— Of Decimal Notation (29) ; Of Literal Notation (30, 31) 5 
 
 Symlx)l 20 , and its meaning (32) «*> 
 
 Symbols of Operation (33) 6 
 
 Definitions.—Exponent (e34); A Positive Integer (35); a Positive Frac 
 
 tion (3G) ; a Negative Number (38) ; Radical Sign (37). . . fl 
 
via CONTENTS. 
 
 PAOS 
 
 Symbols of Relation.— Sign : (39); Sign .. (40); Signs =, : : (41); 
 
 Sign a (42) ; Signs < > (43) 6, 7 
 
 Synilx)la of Aggregation , (),[], | 1 , | , (44, 45) 7 
 
 Symbols of (Continuation (46) 7 
 
 Synib<jls of Deduction (47) 7 
 
 Positive and Negative. — How applied (48) ; Double use of signs + and 
 — (4U) ; Essential sign of a quantity (50) ; Illustrations ; abstract 
 quantities no sign (51) ; less than zero (52) ; increase of a nega- 
 tive quantity (53) 7-9 
 
 Names of Different Forms of Expre.s.sion. — Polynomial (54) ; Monomial, 
 
 Binomial, Trinomial, etc. (55) ; Coefficient (56) ; Similar Terms (57) . 9. 10 
 
 SECTION II. 
 
 Addition. 
 
 Definitions.— Addition (58) ; Sum or Amount (59) 10 
 
 Prop. 1.— To add similar terms (60) ; Cor. 1. Sign of the sum (61) ; Sch. 
 Addition sometimes seems like Subtraction ; Cor. 3. Algtibraic 
 sum of a i)ositive and negative quantity (62) ; Cor. 3. Addition 
 
 does not always imply increase (63) 10, 11 
 
 Prop. 2. — To add dissimilar terms (64) ; Sch. The sign — before a term ; 
 
 Cor. Adding a negative quantity (65) 11, 12 
 
 Prob. — To add polynomials (66) 12 
 
 Prop. 3.— Terms but partially similar (67) 12 
 
 Prop. 4. — Compound terms (68) ; Sch. Examples 12, 14 
 
 SECTION III. 
 
 S U R T 11 A C T I O N. 
 
 Definfttons.— Subtraction (69) ; Difference (70) 14 
 
 Prou. — To perform Subtraction (71) ; Cor. 1. Removing a i)arcntlicsis 
 preceded by the sign — (72) ; Cor. 2. Introducing a paren- 
 thesis (73) ; Examples ; Theory of Subtraction 14-16 
 
 SECTION IV. 
 MuiiTII'MrATJON. 
 
 Definition.— (74) ; Corft. 1 and 2. Character of factors and product (75, 76) 16 
 
 Prop. 1.— Order of factors immaterial (77) 16 
 
 Prop. 2.— Sign of Product (78) ; Com. 1, 2, and 3. Sign of product of 
 
 several factors (79-81) 16, 17 
 
 Prop. 3.— Product of quantities afFecte*! with exponents (82) ; Examples ; 
 
 Sell. Fractional exponent 17, 18 
 
 Prob.— To multiply monomials (83) 18, 19 
 
 Prod.— To multiply polynomials (8-1) 19 
 
CONTENTS. IX 
 
 PAQE 
 
 Three Important Theorems. — Square of the sum (85) ; Square of the dif- 
 ference (8(5) ; Product of sum and difference (87) ; Examples. ... 19, 20 
 Multiplication by Detached Coetiicients (88) ; Examples 21 
 
 SECTION V. 
 
 Division. 
 
 Definitions. — Division (89) ; Problem, how stated (90) ; Cars. 1-5. De- 
 ductions from definition (91-95) ; Cancellation (96) 22 
 
 Lemm.v 1.— Sign of Quotient (97) 22 
 
 Lemma 2.— a"' -»- a" (98) ; Cor. 1. Exponent (99) ; Cor. 2. How negative 
 exponents arise (100) ; Cor. 3. To transfer a factor from dividend 
 
 to divisor (101) 22, 23 
 
 Puor.. 1. — Division of monomials (102) 23 
 
 Piion. 2. — To divide a polynomial by a monomial (103) ; Arrangement of 
 
 terms (104) 23 
 
 Pnon. .J.— To divide a polynomial by a polynomial (105) ; ScJi. Similar to 
 
 ' ' Long Division ; " Examples 24, 25 
 
 Division by Detached Coefficients (100) ; Examples 25, 26 
 
 Synthetic Division (107) ; Examples 26, 27 
 
 CHAPTER II. 
 FACTORING. 
 
 SECTION I. 
 Fundamental Propositions. 
 
 Definitions.— Factor (108) ; Common Divisor (109) ; Common Multi- 
 ple (110) ; Composite Number (111) ; Prime (112, 113) 28 
 
 Prop. 1.- To resolve a monomial (114) 29 
 
 Prop. 2. — To remove a monomial factor from a polynomial (115) 29 
 
 Prop. 3. — To factor a trinomial scpiare (1 10) 29 
 
 Prop. 4.— To factor the difference of two squares (117) 29 
 
 Prop. 5.— Given one factor to find the olher (118) 20 
 
 Prop. G. — By what the sum and difference of like j)Ovvers are divisible(119) ; 
 
 Cor. Prop. applied to fractional and negative exponents (120). .29, 30 
 
 Prop. 7. — To resolve a trinomial (121) 
 
 Prop. 8. — To resolve a polynomial by separating it into parts (122) ; 
 
 Examples 31-33 
 
 SECTION II. 
 Greatest or Highest Common Divisor. 
 
 Definition (123) 33 
 
 Lemma 1. — The H. C. D. the product of all the common factors (124) ; 
 
 Exami)le8 33, 34 
 
X CONTENTS. 
 
 PAGB 
 
 Lemma 2.— A divisor of a polynomial of the form Ax^ -\ Bx'"-^ + Ca^-^ + 
 
 + Ex+ F (136) 34 
 
 Lemma 3. — A divisor of a number divides any multiple (127) 35 
 
 Lemma 4. — A C. D. divides the sum and also the difference (128) 35 
 
 General Rule for H. C. D. demonstrated and applied (129, 130) 35-38 
 
 SECTION III. 
 
 Lowest or Least Common Multiple. 
 
 Definition (131) 39 
 
 Pkoh. — To find the L. C. M. hy resolving numbers into factors (132) ; 
 
 Examples 39, 40 
 
 Finding the L. C. M. facilitated by the method of H. C. D 40 
 
 CHAPTER III. 
 
 FJi ACTIONS. 
 
 Definitions and Fundamental Principles : 
 
 Fraction (133) ; The true conception of a literal fraction (134) ; 
 Corff. 1 and 2. Chanires in terms of a fraction (135,136) ; Integral 
 Form, Mixed Kuiulxr. Proi>er Fraction, Im])roper, Simple, Com- 
 pound, Complex, Lowist Terms, Lowest Common Denomina- 
 tor (137-146) 41, 42 
 
 Reduction.— Definition (147) ; Kinds of (148) 42, 43 
 
 Prop,. 1.— To lowest terms (149) ; Seh, 1. The use of the H. C. D.; 
 
 Sc/t. 2. The converse process 43 
 
 Prop 2. — From improper to mixed form (150); Cor. Use of nega- 
 tive exponents (151) 43 
 
 Prop. 3. — From integral or mixed to fractional form (152) 43 
 
 Prop. 4. — To common denominator (153) ; Cor. To L. C. D. (154). . 44 
 
 Prop. 5. — Complex to simple (155) 44 
 
 Addition : 
 
 Prop. — To add fractions (150) ; Co7'. To add mixed nunilwrs (157). 44, 45 
 
 SURTRACTION : 
 
 Prop.— To subtract fractions (158) ; Cor. To subtract mixed num- 
 bers (159) 45 
 
 Multiplication : 
 
 Prop. 1. — A fraction by an integer (100) 45 
 
 Prop. 2. — To multiply by a fraction (161); Cor. To multiply 
 
 mixed numbers (162) 45, 46 
 
 Division : 
 
 Prop. 1.— To divide by an integer (163) 46 
 
 Prop. 2. — To divide by a fraction (164) ; Reason for inverting the 
 divisor ; Sell. To reverse the operation of multiplication ; Cor. 
 Reciprocal, what (165) 46, 47 
 
CONTENTS. XI 
 
 PAGK 
 
 Signs of a Fraction : 
 
 Three things to consider (166) 47 
 
 Essential character (167) ; Examples 47-58 
 
 CHAPTER IV. 
 POWERS AND ROOTS. 
 
 SECTION I. 
 
 Involution. 
 General Definitions : 
 
 Power, Degree (168) ; Root (169) ; Exponent or Index ; How to read 
 an Exponent (170) ; Radical Number, Rational, Irrational, 
 Surd (171) ; Radical Sign (172) ; Imaginary Quantity (173) ; 
 Real (174) ; Similar Radicals (175) ; Rationalize (176) ; To affect 
 with an exponent (177) ; Involution (178) ; Evolution (17i)) ; Cal- 
 culus of Radicals (180) 54, 55 
 
 Involution : 
 
 Pkob. 1. — To raise to any power (181) ; Cor. Signs of powers (182). 55, 56 
 
 Prob. 2.— To affect with any exponent (183) 5(» 
 
 Pros. 3.— The Binomial Formula (184) ; Cor. 1. Wlien the series 
 terminates (185) ; Cor. 2, Number of terms (186) ; Cor. 3. Equal 
 Coefficients (187) ; Cor. 4. Sum of exponents in any tenn (188) ; 
 Coi'. 5. Statement of the Rule (189) ; Cor. 6. Signs of the terms 
 in the expansion of (a—b)'^ (190) ; Examples 57-60 
 
 SECTION II. 
 Evolution. 
 
 Prob. 1.— To extract root of perfect power (101) ; Sch. Signs of root (192) ; 
 
 Cor. 1. Roots of monomials (193) ; Cor. 2. Root of Product (194) ; 
 
 Cor. 3. Root of a Quotient (195) ; Examples 60,61 
 
 Prob. 2, — To extnu-t lootn whose indices are composed of factors 2 
 
 and 3 (196) 62 
 
 Prob. 3.— To extract the mih root of a number (197) ; Examples 62-67 
 
 SECTION in. 
 
 Calculus of Radicals. 
 Reductions : 
 
 Pnow. 1 .—To remove a factor (198) ; Cor. To giniplify a fraction (190) 67 
 
 Pkob. 2.— When the index is a composite number (200/ 68 
 
 Prob. 3.— To any required index (201) ; Cor. To put the coefficient 
 
 under the radical sign (202) 68 
 
XH CONTENTS. 
 
 PAQK 
 
 Prob. 4.— To a common index (303) G8 
 
 Prob. 5. — To rationalize a monomial denominator (204) 69 
 
 Prob. 6. — To rationalize a radical binomial denominator (205).. ... 00 
 
 Prop. 1. — To rationalize any binomial radical (206) 69 
 
 Prop. 2.— To rationalize Va-\- Vb + Vc (207) ; Examples 70-72 
 
 SECTION IV. 
 
 Combinations of Radicals. 
 
 Addition and Subtraction : 
 
 Prob. 1.— To add or subtract (208) 73 
 
 Multiplication : 
 
 Prop. 1.— Product of like roots (209) 73 
 
 Prop. 2.— Similar Radicals (210) 73 
 
 Prob. 2.— To multiply Radicals (211) 73 
 
 Division : 
 
 Prop.— Quotient of like roots (212) 73 
 
 Prob. 3.— To divide Radicals (213) 73 
 
 Involltion : 
 
 Prob. 4. — To raise to any power (214) ; Cor. Index of power and 
 root alike (215) 73 
 
 Evolution : 
 
 Prob. 5. — To extract any root of a Monomial Radical (216) 74 
 
 Prob. 6. — To extract the square root of a ± n Vb, or 
 mVa± 7it^(217); Examples 74-76 
 
 SECTION V. 
 
 Imaginary Quantities. 
 
 Definition (218) ; not unreal (219) ; a curious property of (220) 76 
 
 Prop.— Reduced to form m^^^^X (221) ; fkh, Tlie form mV^^l (222).. 77 
 Prob. — To add and subtract imaginary monomials of second degree (223); 
 
 Examples 77, 78 
 
 Prop.— Polynomial reduced to form a±&^^l(224); Sch. Conjugate 
 
 Imaginaries, Modulus (225) ; Examples 79 
 
 Multiplication and Involution : 
 
 Prob.— To determine character of product (226) ; Examples 70, 80 
 
 Division op Imaginaries : 
 
 Prob. — To divide one imaginary by another (227) ; Examples 80, 81 
 
CONTENTS. XI 11 
 
 PART IL— ELEMENTARY COURSE IN ALGEBRA. 
 
 CHAPTER I. 
 SIMPLE EQUATIONS. 
 
 SECTION I. 
 
 Equations with one Unknown Quantity. 
 Definitions : pagb 
 Equation (1) ; Algebra (3) ; Members (3) ; Numerical Equation (4) ; 
 Literal Equation (5) ; Degree of an Equation (6) ; Simple Equa- 
 tion (7) ; Quadratic (8) ; Cubic (9) ; Higher Equations (10) 82, 83 
 
 Transformations : 
 
 What (11, 12) ; Axioms (13) 83 
 
 Prob, — To clear of Fractions (14) ; Transposition (15) 83 
 
 Prob. — To transpose (16) 84 
 
 Solution of Simple Equations ; 
 
 What (17) ; When an equation is satisfied (18) ; Verification (19). . . 84 
 Prob, 1. — To solve a simple equation (20) ; 8ch. 1, Kinds of changes 
 which can be made (21); Car. 1. Clianging signs of both mem- 
 bers (22) ; 8ch. 2. Not always expedient to make the transforma- 
 tions in the same order (23) ; Sch. 3, Equations which become 
 simple by reduction (24) 85 
 
 Simple Equations containing Radicals : 
 
 Prob. 2.— To free an equation of Radicals (25, 26) 85 
 
 Summary of Practical Suggestions (27, 28) ; Examples , 86-88 
 
 Applications to the Solution of Exampi.es (29) ; Statement, Solu- 
 tion (30) ; Knowledge required in making statement (31) ; Direc- 
 tions to guide in making statement (32) ; Not always best to use 
 X (33) ; Examples 89-92 
 
 SECTION IT, 
 
 Independent, Simultaneous, Simple Equations with two Unknown 
 
 Quantities, 
 Definitions : 
 
 Independent Equations (34) ; Simultaneous Equations (35) ; Elimi- 
 nation (36) ; Methods (37) 93 
 
 Elimination : 
 
 Prob. 1.— By Comparison (38) 93 
 
 Prob. 2.— By Substitution (39) 94 
 
XIV CONTENTS. 
 
 PAGE 
 
 Prob. 3.— By Addition or Subtraction (40) 9 i 
 
 PROB. 4. — By Undetermined Multipliers (41) 95 
 
 Prob. 6. — By Division (43) ; Examples and Applications 96-99 
 
 SECTION III. 
 
 Independent, Simultaneous, Simple Equations with more than two 
 Unknown Quantities. 
 
 Prob.— To solve (43) ; Examples and Applications 1(X)-103 
 
 CHAPTER II. 
 RATIO, PROPORTION, AND PROGRESSION. 
 
 SECTION I. 
 Ratio. 
 
 Definitions — Ratio (44) ; Sign (45) ; Cor. Effect of Multiplying or 
 Dividing the Terms (40) ; Direct and Reciprocal Ratio (47) ; 
 Greater and I^ss Inequality (48) ; Compound Ratio (49) ; Du- 
 plicate, Subduplicate, etc. (50) 104, 105 
 
 Examples 105, 106 
 
 SECTION II. 
 
 Proportion. ^ 
 
 Oepinitions.— Proportion (51) ; Extremes and Means (52) ; Mean Pro- 
 portional (53) ; Third Proportional (54) ; Inversion (55) ; Alter- 
 nation (56) ; Composition (57) ; Division (58) ; Inversely Pro- 
 portional (59) ; Continued Proportion (60) 106, 107 
 
 Prop. 1. — Product of extremes equals product of means (61) ; Cor. 1. 
 
 Square of mean proportional (62); Cor. 2. Value of any term (63) 107 
 
 Prop. 2.— To convert an equation into a projMjrtion (64) ; Cor. Taken 
 
 by alternation and inversion (65) 107, 108 
 
 Prop. 3. — What transformations can be made without destroying the 
 
 proportion (66, 67) 108 
 
 Prop. 4. — Products or Quotients of corresponding terms of two pro* 
 
 portions (68) ; Cor, Like powers or roots (69) 108 
 
 Prop. 5. — Two proportions with equal ratio in each (70) 108 
 
 Prop. 6. — Taken by composition and division (71) ; Cor. Series of equal 
 ratios as a continued proportion (72) ; Sch. Method of testing 
 any transformation (73) ; Examples and Applications 109-113 
 
CONTENTS. XV 
 
 SECTION III. 
 Progressions. 
 
 PAGE 
 
 Definitions. — Progression, Arithmetical, Geometrical, Ascending, De- 
 scending, Common Difference, Ratio (74) ; Signs and Illustra- 
 tions (75) ; Arithmetical and Geometric Mean (76, 77) ; Five 
 things considered (78) .♦. 113, 114 
 
 Arithmetical Progression. — Prop. 1. To find the last term (79) ; 
 Prop. 2. To find the sum (80) ; Cor. 1. These formulas sutE- 
 cient (81) ; Cor. 2. To insert means (82) ; Formulae in Arith- 
 metical Progression (83) ; Examples 114-117 
 
 Geometrical Progression. — Prop. 1. To find the last term (84) ; 
 Prop. 2.— To find the sum (85) ; Cor. 1. These formulas suflS- 
 cient (86) ; Cor. 2. Another formula for sum (87) ; Cor. 3. To 
 insert means (88) ; Cor. 4. Sum of an infinite series (89) ; Geo- 
 metrical Formulae (90) ; Examples 117-122 
 
 SECTION IV. 
 
 Variation. 
 
 Definitions. — Variation, directly, inversely, jointly, directly as one 
 and inversely as another (91-93) ; Sign (94) ; Prop. Variation 
 expressed as Proportion (95) ; Exercises 123-125 
 
 SECTION V. 
 
 Harmonic Proportion and Progression. 
 Definitions. — Harmonic Proportion (96) ; Harmonic Mean (97) ; 
 Prop. Quantities in Harmonic Proportion, their Reciprocals in 
 Arithmetical (98) ; Harmonic Progression (99) ; Derivation of 
 the term Harmonic (100) ; Exercises 125, 126 
 
 CHAPTER III. 
 QUADRATIC EQUATIONS. 
 
 SECTION I. 
 
 Pure Quadratics. 
 
 Definitions.— Quadratic (101); Kinds (102); Pure (103); Affected (104); 
 
 Root (105) 127 
 
 Resolution of a Pure Quadratic Equation (106) ; Cor. 1. A Pure 
 
 Quadratic has two roots (107) ; Cor. 2. Imaginary roots (108). . 127, 128 
 Examples and Applications 128-130 
 
XVI CONTENT& 
 
 SECTION II. 
 Affected Quadratics. 
 
 PAGE 
 
 Definition (109) 130 
 
 Resolution. — Commou Method (110) ; Sch. 1. Completing the Square ; 
 Cor. 1. Two roots, character of (111) ; Cor. 2. To write the 
 roots of x'^ + px = q, witliout completing the square (112) ; 
 Cor. 3. Special methods (113, 114) ; Examples 130-134 
 
 SECTION III 
 
 Equations of other Degrees which may be Solved as Quadratics. 
 
 Prop. 1. — Any pure equation (115) 134 
 
 Prop. 2. — Any equation containing one unknown quantity with only 
 
 two different exponents, one of which is twice the other (116). 135 
 
 Prop's 3-5.— Special Solutions (117-122) ; Examples 135-139 
 
 SECTION IV. 
 
 Simultaneous Equations of the Second Degree between two 
 Unknown Quantities. 
 
 Prop. 1. — One equation of the second degree and one of the first (123). 140 
 
 Prop. 2. — Two equations of the second degree usually involve one of 
 
 the fourth, after eliminating (124) 140 
 
 Prop. 3.— Homogenous quadratics (125, 126) 140, 141 
 
 Prop. 4. — When the unknown quantities are similarly involved (127). . 141 
 
 Examples, Special Solutions, Applications 142-147 
 
 CHAPTER IV. 
 
 INEQUALITIES. 
 
 Dephtition (128) ; Fundamental Principle (129) ; Members (130) ; Same 
 transformations as equations (131) ; Same and opposite 
 
 sense (132) 148 
 
 Prop. — Sense of an inequality not changed (133) 148, 149 
 
 Prop. — Sense of an inequality changed (134) ; Exercises 149, 150 
 
CONTENTS. XVll 
 
 PART IIL— ADVANCED COURSE IN ALGEBRA. 
 
 CHAPTER I. 
 INFINITESIMAL ANALYSIS, 
 
 SECTION I. 
 Differentiation. 
 
 PAG« 
 
 Definitions.— Constant and Variable Quantities (135-137) ; Sch. Dis- 
 tinction between constant and variable, known and un- 
 known (138) ; Function (139) ; How represented (140) ; Inde- 
 pendent and Dependent Variable (141); Infinitesimal (142); 
 Consecutive values (143) ; Differential (144) ; Notation (14."j) ; 
 To differentiate (14G) 151-153 
 
 Rules for Differentiating : 
 
 Rule 1. — To differentiate a single variable (147) 154 
 
 Rule 2.— Constant factors (148) 154 
 
 Rule 3. — Constant terms (149) 151 
 
 Rule 4. — The sum of several variables (150) 155 
 
 Rule 5. — The product of two variables (151) 155 
 
 Rule C. — The product of several variables (152) 155, 15G 
 
 Rule 7. — Of a fraction with a variable numerator and denomina- 
 tor (153) ; Cor. With constant numerator (151) ; Sch. Constant 
 
 denominator (155) 150 
 
 Rule 8 — Of a variable affected with an exponent (15G) ; Sch. 
 
 Rate of change (157) ; Examples 156-158 
 
 SECTION II. 
 
 Indeterminate Coefficients. 
 
 Definition (158) 159 
 
 Prop.— In A ^^ Bx -\- Gx^ + etc. = ^'+ B'x -i- C'x"^ + etc., coefficients 
 of like powers of x equal to each other (159) ; Cor. A, B, C, 
 etc., = (160) 150 
 
 Development of Functions (161) ; Examples 159-10^ 
 
 Decomposition of Fractions (163) ; Cme 1. When the denominator 
 is resolvable into re(jl and unequal factors of the firrit de 
 gree (164) ; Cane 2. Into real and equal factors of the first de- 
 gree (165) ; Ca«e 3. \\\io real and quadratic factor^ (lOG) ; Sch. 
 Forms combined (167) ; Examples 102-164 
 
XVlll CONTENTS. 
 
 SECTION III. 
 The Binomial Formula. 
 
 PAOB 
 
 Binomial Theorem (168) ; Cor. 1. The general term (169) ; Scale of 
 Relation (170) ; Cor. 2. Formula for scale of relation (171) ; 
 Examples 165-168 
 
 SECTION IV. 
 Logarithms, 
 
 Definitions.— Logarithms, Base (172) ; Cor. Logarithm of 1 (173) ; Sys- 
 tem of Logarithms (174) ; Two in use (175) ; Cor. Quantities 
 
 that cannot be used as a base (176) 168, 169 
 
 Prop. 1. — Logarithm of product equals sum of logarithms (178) 169 
 
 Prop, 2. — Logarithm of quotient equals difference of logarithms (179). 170 
 
 Prop, 3. — Logarithm of a power (180) 170 
 
 Prop, 4,— Logarithm of a root (181) 170 
 
 Loirarithras of most numbers not integral (182) 170 
 
 ("liaructeristic and Mantissa (183) 170 
 
 Prop, — Mantissa of decimal fraction or mixed number (184) ; Cor. 1. 
 Characteristic of any number (185) ; Cor. 2. Logarithm of 
 
 0(186) 170,171 
 
 Computation op Logarithms : 
 
 Modulus (187) ; Prop. Differential of a logarithm (188) 172 
 
 Prob. — To produce the logarithmic series (189) ; Cor.\. Loga- 
 rithms of same number in two different systems, as moduli (190); 
 Cor. 2, To find logarithm of a number in any system know- 
 ing the modulus, and also to find modulus (191) 173, 174 
 
 Prob. — To obtain series for computing Napierian logarithms (192) 175 
 Prob. — To compute Napierian logarithms of natural num- 
 bers (193) 175 
 
 Prop, — Modulus of common system (194) 1 76 
 
 Tables of Logarithms.— What (195) 177 
 
 Prob,— To find the logarithm of a number (196, 197) 177, 178 
 
 Prob. — To find a number corresponding to a logarithm (198) . . . 178 
 
 Prop.— The Napierian base (199) ; Examples 179-181 
 
 SECTION V. 
 Successive Differentiation and Differential Coefficients. 
 
 Prop. — Differentials not necessarily equal (200) ; Cor. dy a varia- 
 ble (201) ; Notation (202) ; dx constant (203) ; Second and Third 
 Differentials (204) ; Examples 181-183 
 
 Differential Coefficients : 
 
 First Differential Coefficient (205) ; Second Differential Coeffi- 
 cient (206) ; Examples ; Successive coefficients written by in- 
 spection (207) 183-185 
 
CONTENTS. XIX 
 
 SECTIC:^ VI. 
 Taylor's Formula. 
 
 PAGB 
 
 Definition (208) : Partial Differential Coefficients (209) : Lemma. 
 
 ^ and ~ equal(210) 185,186 
 
 Prob. — To produce Taylor's Formula (211) ; Sch. First, second, etc., 
 terms (212) ; To develop a function of a variable witli an in- 
 crement (213) ; Examples 186-189 
 
 SECTION VII. 
 
 Indeterminate Equations. 
 
 Definition, Nature, etc. (214-220) ; Examples 189-195 
 
 CHAPTER 11. 
 
 LOCI OF EQUATIONS. 
 
 Pkop. — Every equation between two variables may represent a line (221) 196 
 
 Definitions. — Axes of Reference, Abscissa, Ordinate, Co-ordinates 
 
 (222) ; Locus, Constructing Locus (223) ; Examples 198-202 
 
 Prob. — To construct real roots of equations witli one unknown quan- 
 tity (224) ; Examples 202, 203 
 
 CHAPTER HI. 
 HIGHER EQUATIONS. 
 
 SECTION I. 
 
 Solution op Numerical Higher Equations having Commensurable 
 OR Rational Roots. 
 
 No general method of solution (226) ; Real, commensurable roots found 
 
 with little difficulty (227) 203 
 
 Prop. — Transforming an equation into the form .t« -t- ^4.?"-' h- Bx!^-^ 
 
 + Cx^'3 X = (228) ; Examples 204, 205 
 
 Prop. — Roots of an equation factors of absolute term (230) ; If <^ is a 
 
 root, f{x) divisible by {x — a), and converse (231) 206 
 
 Prop.— Wliat equation can have no fractional root (232, 233) 206, 207 
 
XX CONTENTS. 
 
 TAan 
 Prop. — Equation of nth degree has n roots (334) ; Cor. 1. /(i) = (x—a) 
 
 {x — b){x — c) - • - • {x — n), when a, b, c - - - - n are roots of 
 f{x) - (SJJo) ; Cor. 2. f{x) can have equal roots (236) ; Cor. 3. 
 Imaginary roots enter in pairs (237) ; Cor. 4. Number of real 
 roots in e(iuations of odd and even degrees (238) ; Limits of 
 imaginary roots (239) ; Sch. 1. Proposition illustrated geome- 
 trically (240) ; Sc/i. 2. Imaginary roots entering in pairs illus- 
 trated (241) 207-209 
 
 Prop. — Method of finding equal roots (242) ; Sc7i. Sometimes conve- 
 nient to apply process several times (243) 209, 210 
 
 Prop.— Cliange of sign in f{x) (244) ; How illustrated by loci (245) 211 
 
 Prop.— Clianging signs of roots of f{x) (246) ; Coi: Another method (247) 212 
 
 Prob.— To evaluate f(x) for x = a (248) 212, 213 
 
 Prob. — To find commensurable roots of numerical higher equations 
 
 (249); Examples 213-216 
 
 To produce an equation from its roots (250) ; Examples 216 
 
 SECTION II. 
 
 Solution of Numerical Higher Equations having Real, Incommen- 
 surable, OR Irrational Roots. 
 
 Typical form of equation (251) ; Best general method (252) 216, 217 
 
 Sturm's Theorem and Method : 
 
 Definition and Object (253, 254) ; Sturmian Functions (255) ; No- 
 tation (256) ; Permanence and Variation (257) 217, 218 
 
 Sturm's Theorem (258) ; Cor. 1. To find the number of real roots 
 of f{x) (259) ; Cor. 2. To find the number of real roots of f{x) 
 between a and b (260) ; Sch. Number of imaginary roots known 
 by implication (261) 219-221 
 
 Prob. — To compute the numerical values of f{x), f\x), fi{x), etc. 
 (262) ; Sch. 2. Usually unnecessary to find /n(t") (263) ; Sch. 3. 
 Wlien the equation has equal roots (264) ; Sell. 4. Generally 
 the change of sign in f{x) enables us to determine situation of 
 roots more easily than Sturm's Theorem (265) ; Sch. 5. Not ne- 
 cessary that the coeflficients should be integral (266) ; Exam- 
 ples 221-228 
 
 Horner's Method of Solution : Object (267) 228 
 
 Prob. — To transform an equation into another with roots less by 
 
 a (268) ; Sch. Signification of result (269) 229 
 
 Prob. — To compute the numerical values of f{a), /'(a), hf"{a), 
 etc. (270) ; Examples 229-233 
 
CONTENTS. XXI 
 
 rXOB 
 
 Prop. — Value of a;,, when a + x^ is a root of /(«) = (271) 23^235 
 
 Horner's Rule (272) ; jScJioliums (273-278) ; Examples 235-247 
 
 SECTION III. 
 
 General Solution of Cubic and Biquadratic Equations. 
 
 Cardan's Solution of Cubic Equations : 
 
 Prob.— To resolve a;^ + px^ + ga; + r - (279) 248, 249 
 
 Prop. — Solution satisfactory and unsatisfactory, when (280) ; 
 
 Scholium. Apparently 9 roots (281) ; Examples 249-251 
 
 Descartes's Solution of Biquadratics : 
 
 Prob. -To resolve x^ + ax"" + bx^ + dx ■\- e = (282) ; 8ch. In- 
 volves solution of acubic(283) ... 251, 252 
 
 Recurring Equations : 
 
 Definition (284) 253 
 
 Prop. 1. — The roots reciprocals of each other (285) ; Sch. Recip- 
 rocal Equations (286) ; Cor. 1. Corresponding coefficients with 
 like or unlike signs (287) ; Cor. 2. Reduced to form having 
 
 first coefficient unity (288) 252, 253 
 
 Prop. 2. Of an odd degree have roots — 1, and + 1, when (289). 253, 254 
 
 Prop. 3. — Of an oven degree have same roots, when (290) 254 
 
 Prop. 4. — Of an even degree above second reduced to one of half 
 
 that degree (291) ; Examples 254, 255 
 
 Binomial Equations and the Roots of Unity : 
 
 Definition (292) ; Examples and Scholium (293) 255, 25G 
 
 lilXPONENTIAL EQUATIONS : 
 
 Definition (294) 256 
 
 Prob. 1.— To solve a"" = m (295) 256 
 
 Prob. 2.— To solve x^ = m (296) ; Examples 256-260 
 
 CHAPTEll IV. 
 DISCUSSION, OR INTERPRETATION, OF EQUATIONS. 
 
 Definition (297) 260 
 
 Prop.— Statement of Principles (298) 260-262 
 
 Real Number or Quantity (299) 262 
 
 Imaginary Number (300) ; Examples 263-266 
 
 Arithmetical Interpretation of Negative and Imaginary Re- 
 sults (301) ; Sch. Symbol ^ (302) ; Examples 267-271 
 
XXU CONTENTS. 
 
 APPENDIX, 
 
 SECTION I. 
 Series. 
 
 PAOB 
 
 Definitions.— Series, Tenn (303) ; Recurring Series, Scale of Rela- 
 tion (304) ; Infinite, Convergent, Divergent (305) ; To revert a 
 Series (306) ; Orders of Differences (307) ; Interpolation (308) ; 
 Enumeration of Problems (309) 273-374 
 
 Lemma. — First term of any order of differences (310) ; Cor. Number of 
 
 terms necessary (311) ; Examples 274, 275 
 
 Prob. 1. — To find scale of relation in recurring scries (312) ; Sch. De- 
 pendence on too many or too few terms (313) ; Examples 275-277 
 
 Prob. 2.— To find the nth term (314) ; Examples 277, 278 
 
 Prob. 3. — To determine whether a series is convergent or divergent 
 
 (315) ; Examples 278-280 
 
 Prob. 4.— To find sum of a series (316) ; Examples 280-285 
 
 Piling Balls and Shells (317) 285 
 
 Prop. — Number in triangular pile (318); Cor. Number of 
 
 courses (319) 285, 286 
 
 Prop. — Number in square pile (330); Cor. Number of courses (321) 286 
 
 Prop. — Number in oblong pile (332) ; Examples 286, 287 
 
 Reversion of Series : 
 
 Prob.— To revert a series (323) ; Examples 287, 288 
 
 Interpolation : 
 
 To interpolate between functions (324) ; Sch. 1. Result correct 
 when (325) ; Sell. 2. Another formula (326) ; Sch. 3. Used in 
 Astronomy (327) ; Examples 288-291 
 
 SECTION II. 
 
 Permutations. 
 
 Depinitions.— Combinations (328); Permutations (329); Arrange- 
 ments (330) 393 
 
 Prop.— Number of arrangements of m things n and n (331) ; Cor. 1. 
 Permutations of m things (333) ; Cor. 8. When p things are 
 alike, etc. (333); Cor. 3. Combinations of m things n and 
 n (334) ; Examples 292-294 
 
 Probabilities : . 
 
 Mathematical Probability and Improbability (335) ; Examples.. 294-398 
 
(3 
 
 SECTION L 
 GENERAL DEFINITIONS, AND THE ALGEBRAIC NOTATION. 
 
 BRANCHES OF PURE MATHEMATICS. 
 
 1. Pure Mathematics is a generjil term applied to several 
 branches of science, which have for their object the investigation of 
 the properties and relations of quantity — comprehending number, 
 and magnitude as the result of extension — and of form. 
 
 2. The Several Branches of Pure Mathematics are Arith- 
 metic, Algebra, Calculus, and Geometry. 
 
 3. Arithmetic, Algebra, and Calculus treat of number, and Geo- 
 metry treats of magnitude as the result of extension. 
 
 4. Quantity is the amount or extent of that which may be 
 measured; it comprehends number and magnitude. 
 
 The term quantity is also conventionally applied to symbols used 
 to represent quantity. Thus 25, m, xi, etc., are called quantities, 
 although, strictly speaking, they are only representatives of quantities. 
 
 5. Number is quantity conceived as made up of parts, and 
 answers to the question, " How many ? " 
 
 6. Number is of two kinds. Discontinuous and Continu- 
 ous, 
 
 7. Disco7itinuous Number is number conceived as made 
 up of finite parts; or it is number which passes from one state of 
 value to another by the successive additions or subtractions of finite 
 units ; i. e., units of appreciable magnitude. 
 
 8^ Continuous Number is number which is conceived as 
 composed of infinitesimal parts; or it is number which passes from 
 
 1 
 
j2«;| f^] t' INTRODUCTION. 
 
 one state of value to another by passing through all intermediate 
 values, or states. 
 
 0, Arithmetic treats of Discontinuotis Number,— oi 
 
 its nature and properties, of the various methods of combining and 
 resolving it, and of its application to practical affairs. 
 
 10. Alf/ebra treats of the Equatioiiy and is chiefly occupied in 
 explaining its nature and the methods of transforming and reducing 
 it, and in exhibiting the manner of using it as an instrument for 
 matliematical investigation.* 
 
 H. Calculus treats of Contijiuous Niimher, and is chiefly 
 occupied in deducing the relations of the infinitesimal elements of 
 such number from given relations between finite values, and the con- 
 verse process, and also in pointing out the nature of such infinites- 
 imals and the method of using them in mathematical investigation. 
 
 12, Geometry treats of magnitude and form as the result of 
 extension and position. 
 
 LOGICO-MATHEMATICAL TERMS. 
 
 13. A I^roposition is a statement of something to be con- 
 sidered or done. 
 
 14. Propositions are distinguished 2kS Axioms, Theorems, Lemmas, 
 Corollaries, Postulates, and Problems. 
 
 15. An Axiom is a proposition which states a principle that 
 is so simple, elementary, and evident as to require no proof. 
 
 10. A Tlieorem is a proposition which states a real or supposed 
 fact, whose truth or falsity we are to determine by reasoning. 
 
 17* A Danonstration is the course of reasoning by means 
 of which the truth or falsity of a theorem is made to appear. The 
 term is also applied to a logical statement of the reasons for the 
 processes of a rule. A solution tells hoiu a thing is done ; a demon- 
 stration tells why it is so done. A demonstration is often called proof. 
 
 * The common definition of Algebra, which makes its distin^nishing feature;* to be the literal 
 notation, and the me of ttie signs, is entirely at fault. When Algebra firct appeared in Europe, it 
 possessed neither of these features ! What was it then ? On the other hand, the signs are 
 common to all branches of mathematics, and the literal notation is as prominent in the Calculus 
 sm in Algebra, and is used, more or les?, in common Arithmetic and Geometry. 
 
LOGICO-MATHEMATICAL TERMS. 3 
 
 1S» A. Lemma is a theorem demonstrated for the purpose of 
 using it in the demonstration of another theorem. 
 
 19, A Corollary is a subordinate theorem which is sug- 
 gested, or the truth of which is made evident, in the course of the 
 demonstration of a more general theorem, or which is a direct 
 inference from a proposition. 
 
 20, A Postulate is a proposition which states that something 
 can be done, and which is so evidently true a^ to require no process 
 of reasoning to show that it is possible to be done. We may or may 
 not know how to perform the operation. 
 
 21, A Prohletn is a proposition to do some specified thing, 
 and is stated with, reference to developing the method of doing it. 
 
 22, A Rule is a formal statement of the method of solving a 
 general problem, and is designed for practical application in solving 
 special examples of the same class. Of course a rule requires a 
 demonstration. 
 
 23, A Solution is the process of performing a problem or an 
 example. It should usually be accompanied by a demonstration of 
 the process. 
 
 24, A Scholiuin is a remark made at the close of a discussion, 
 and designed to call attention to some particular feature or features 
 of it. 
 
PART L* 
 
 LITERAL ARITHMETICt 
 
 CHAPTER I. 
 
 FUNDAMENTAL, RULES. 
 
 SECTION L 
 
 NOTATION. 
 
 25. A System of Notation is a system of symbols by means 
 of whicli quantities, the relations between tliem, and the operations 
 to be performed upon them, can be more concisely expressed than 
 by the use of words. 
 
 Symbols of Quantity. 
 
 26, In Arithmetic, as usually studied, numbers are represented 
 by the characters, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, called Arabic figures, or, 
 simply, figures. 
 
 27* In other departments of mathematics than Arithmetic, num- 
 bers or quantities are more frequently represented by the common 
 letters of the alphabet, rt, &, c, . . . in, n, , . . x, y, z. These letters 
 may, however, be used in Arithmetic ; and the Arabic figures ai-e 
 used in all departments of mathematics. This method of represent- 
 
 * Parts I. and IT. are a compend of the elements of the science, designed as a review for 
 pupils who have studied some elementary treatise, or for the use of such teachers and classes as 
 desire a text-book which contains a condensed treatment of the subject, to be filled ont by them- 
 selves. In the author's Complete School Algebra, the topics here presented will be found 
 fully amplified, illustrated, and applied. All the elementary principles are here stated, and are 
 usually demonstrated. There are also numerous examples under every topic. The Key to the 
 Complete School Algebra will furnish !idditi<inal examples for use in connection with this part, 
 t Part I. treats of the familiar operations of Addition, Subtraction, Multiplication, Division, 
 Involution and Evolution, and the theory of Fractions. The only difference between the pro- 
 cesses here developed and the correspond in" ones in <xymmon Arithmetic grows out of the 
 notation. 
 
NOTATION. 5 
 
 ing quantities by letters is often called the Algebraic method, and 
 the method by the Arabic characters the Arithmetical, It would be 
 better to call the former the Literal method, and the latter the 
 Decimal. 
 
 28, Tlie Literal Notation has some very great advantages 
 over the decimal for purposes of mathematical reasoning. 1st, The 
 symbols are more general in their signification ; and 2d, We are 
 enabled to detect the same quantity anywhere in the process, and 
 even iu the result. Thus it happens that the processes become 
 general /(9rm?<7cp, or rules, instead of special solutions. 
 
 29. In using the decimal notation certain laius are established, in 
 accordance with which all numbers can be represented by the ten 
 figures. Thus, it is agreed tliat when several figures stand together 
 without any other mark, as 435, the right-hand figure shall signify 
 units, the second to the left, tens, the third, hundreds, etc. ; also that 
 the sum of the several values shall be taken. This number is, there- 
 fore, 4 hundreds + 3 tens + 5 (units). 
 
 In like manner, certain laws are observed in representing numbers 
 by letters. 
 
 First Law. 
 
 30, Known Quantities^ that is such as are given in a prob- 
 lem, are represented by letters taken from the first part of the 
 alphabet; while Unknown Quantities, or quantities whose 
 values are to be found, are represented by letters taken from the 
 latter part of the alphabet. 
 
 Accented letters, as a', a", a'", a"", etc., (read " a prime," " a sec- 
 ond," "a third," etc.,) and letters with subscripts, as a^, «,, a^, a^^ 
 etc., (read ^' a sub 1," "« sub 2," etc.,) are sometimes used. This 
 form of notation is used when there are several like quantities in the 
 same problem, but which have different numerical values. Thus, in 
 a problem in which several walls of different heights, breadths, and 
 lengths are considered, we may represent the several heights by a\ 
 a", a", etc., or a,, ct.^, a^, etc. ; the thicknesses by b', b", b'", etc., or b^, 
 i,, .^3, etc., and the lengths by I', I", I'", etc., or Z^, Z,, 4> etc. 
 
 The Greek letters are also often used both for known and unknown 
 quantities. 
 
 Second Law. 
 
 31. When letters are written in connection, without any sign 
 between them, their product is signified. Thus abc signifies that the 
 three numbers represented by a, h, and c are to be multiplied together. 
 
6 LITERAL ARITHMETIC. 
 
 32, A character like a figure 8 placed horizontally, oo , is used to 
 represent what is called Infinity, or a quantity larger than any 
 assignable quantity. 
 
 Symbols of Operation. 
 
 S3. The Symbols of Operation used in Algebra are the 
 same as those used in Arithmetic, or in any other branch of mathe- 
 matics, and need not be recapitulated here. 
 
 EXPONEIJ^TS. 
 
 34» An Exponent is a small figure, letter, or other symbol 
 of number, written at the right and a little above another figure, 
 letter, or symbol of number.* 
 
 35. A Positive Integral Exponent signifies that the 
 number affected by it is to be taken as a factor as many times as 
 there are units in the exponent. It is a kind of symbol of multipli- 
 cation. 
 
 36, A JPositive Fractional Exponent indicates a power 
 of a root, or a root of a power. The denominator specifies the root, 
 and the numerator the power of the number to which the exponent 
 is attached. 
 
 57. The Radical Sif/n^ >/, is also used to indicate the 
 square root of a quantity. When any other than the square root is 
 to be designated by this, a small figure specifying the root is placed 
 in the sign. 
 
 38, A Negative Exponent, i. e., one with the — sign before 
 it, either integral or fractional, signifies the reciprocal of what the 
 expression would be if the exponent were positive, ?. e., had the 
 + sign, or no sign at all before it. 
 
 Symbols of Kelatioi^. 
 
 39, The Sign of Geometrical Matio is two dots in the 
 form of a colon, : . 
 
 40, The Sign of Arithmetical Matio is two dots placed 
 horizontally, •• . 
 
 41, TJie Sign of Equality is two parallel horizontal lines, 
 = . Tlie double colon, : : , is the sign of equality between ratios. 
 
 * In giving this definition, be careful and r<r>/' add, "and indicates the power to whicli tho 
 number is to be raised." This is false : an exponent docs not necessarily indicate a power. 
 
NOTATION. 7 
 
 42, The Sign of Variation is somewhat like a figure 8 
 open at one end and placed horizontally, a . 
 
 43, The Sign of Tnequality is a character somewhat like 
 a capital V placed on its side, < , the opening being towards the 
 greater quantity. 
 
 Symbols of Aggregation?^. 
 
 44, A Vinculum is a horizontal line placed over several 
 terms, and indicates that they are to be taken together. The paren- 
 thesis, ( ), the brackets, [ ], and the brace, -j i , have the same 
 signification. 
 
 4:5, A vertical line after a column of quantities, each having its 
 
 own sign, signifies that the aggregate of the column is to be taken 
 
 as one quantity. Thus, + a 
 
 -b 
 
 + c 
 
 X is the same as (ft — ^ + c)x. 
 
 Symbols of Continuation. 
 
 46, A series of dots, , or of short dashes, , 
 
 written after a series of expressions, signifies " etc." Thus, a : ar 
 
 : nr"^ : ar^ nr" means that the series is to be extended 
 
 from ar^ to ar", whatever may be the value of w. 
 
 Symbols of Peduction. 
 
 47, Three dots, two being placed horizontally and the thiru 
 above and between, .*. , signify therefore, or some analogous expres- 
 sion. If the third dot is below the first two, •.* , the symbol is read 
 "since," "because," or by some equivalent expression. 
 
 Positive and Negative Qiiantities. 
 
 48, I^ositit^e and Negative are terms primarily applied to 
 concrete quantities which are, by the conditions of a problem, 
 opposed in character. 
 
 Ill, — A man's property may be called positive, and his debts negative. Dis- 
 tance up may be called positive, aiid distance dotcn, negative. Time before 
 a given period may be called positive, and after, negative. Degrees above on 
 the thermometer scale are called positive, and below, negative. 
 
 40, The signs -j- and — are used to indicate the character of 
 quantities as positive or negative, as Avell as for the purpose of indi- 
 cating addition and subtraction. 
 
8 LITERAL ARITHMETIC. 
 
 50, In problems in which the distinction of positive and negative 
 is made, each quantity in the fonnulw is to be considered as having 
 a sign of charade?' expressed or understood besides the plus or 
 minus sign, which latter indicates that it is to be added or sub- 
 tracted. The positive sign need not be written to indicate character, 
 as it is customary to consider quantities whose character is not 
 specified as positive. 
 
 III. 1. — In the expression a'') + m — e.r, let the problem out of which it arose 
 be such, that a, m, and x tend to a positive result, and 6 and c to an opposite, or 
 a negative result. Giving these quantities their signs of character, we have 
 ( + fl^) X (— &) + ( + wi) — (— c) X ( + 0-), which may be read, "positive a mult'- 
 plied by negative b, plus positive m, minus negative c multiplied by positive x." 
 Suppressing the positive sign, this may be written, a{-h) + m — { — c)x, by also 
 omitting the unnecessary sign of multiplication. 
 
 III. 2. — As this subject is one of fundamental importance, let careful atten- 
 tion be given to some further illustrations. We are to distinguish between dis- 
 cussions of the relations between mere abstract quantities, and problems in which 
 the quantities have some concrete signification. Thus, if it is desired to ascer- 
 tain the sum or difference of 4C8, or m, and 327, or n, as mere numbers, the 
 question is one concerning the relation of abstract numbers, or quantities. No 
 other idea is attached to the expressions than that each represents a certain num- 
 ber of units. But, if we ask how far a n:an is from his starting point, who has 
 gone, first, 468, or m miles directly east, a ul then 327, or n miles directly west ; 
 or if we ask what is the difference in time between 468, or m years B. C, and 
 327, or n years A. D., the numbers 468, or m, and 327, or n, take on, besides their 
 primary signification as quantities, the additional thought of opposition in direc- 
 tion. They therefore become, in this sense, concrete. 
 
 Again, a company of 5 l3oys are trying to move a wagon. Three of the boys 
 can pull 75, 85, and 100 pounds each ; and they exert their strength to move the 
 wagon east. The other two boys can pull 90 and 110 pounds each ; and they 
 exert their strength to move the wagon west. It is evident that the 75, 85, and 
 100 are quantities of an opposite character, in their relation to the problem, 
 from 90 and 110. Again, suppose a party rowing a boat up a river. Their 
 united strength would propel the boat 8 miles per hour if there were no cur- 
 rent ; but the force of the current is sufficient to carry the boat 2 miles per hour. 
 The 8 and 2 are quantities of opposite character in their relation to the problem. 
 Once more, in examining into a man's business, it is found that he has a farm 
 worth m dollars, personal property worth n dollars, and accounts due him worth 
 c dollars. There is a mortgage on his farm of h dollars, and he owes on account 
 a dollars. The m, n, and c are quantities opposite in their nature to h and a. 
 This apposition in cliaracter is indicated hy calling those quantities which con- 
 tribute to one result positive, and those which contribute to the opposite result 
 negative. 
 
 51, Purely abstract quantities have, properly, no distinction as 
 positive and negative; but, since in such problems the plus or 
 
NAMES OF DIFFERENT FORMS OF EXPRESSION. 9 
 
 additive, and the minus or subtractive terms stand in the same 
 relation to each other as positive and negative quantities, it is cus- 
 tomary to call them such. 
 
 III. — In the expression bac — Zed + %xy — 2ad, though the quantities, a, c, d, 
 X and y be merely abstract, and have no proper signs of character of their own, 
 the terms do stand in the same relation to each other and to the result, as do 
 positive and negative quantities. Thus, 5ac and %xy tend, as we may say, to 
 increase the result, while — 'dcd, and — 'iad tend to diminish it. Therefore the 
 former may be called positive terms, and the latter negative. 
 
 52. ScH. — Less than zero. Negative quantities are frequently spoken of 
 as ''less than zero." Though this language is not philosophically correct, 
 it is in such common use, and the thing signified is so sharply defined and easily 
 comprehended, that its use may possibly be allowed as a conventionalism. 
 To illustrate its meaning, suppose, in speaking of a man's pecuniary affairs, 
 it is said that lie is worth "less than nothing; " it is simply meant that his 
 debts exceed his assets. If this excess were $1000, it might be called nega- 
 tive $1000, or —$1000. So, again, if a man were attempting to row a boat 
 up a stream, but witli all his effort the current bore him down, his progress 
 might be said to be less than nothing, or negative. In short, in any case 
 where quantities are reckoned both ways from zero, if we call those 
 reckoned one way greater than zero, or positive, we may call those reckoned 
 the other way "less than zero," or negative. 
 
 53. The value of a Negative Quantity is conceived to increase as 
 its numerical value decreases. 
 
 III. — Thus —3 > —5, as a man who is in debt $3 is better off than one who is 
 in debt $5, other things l)eing equal. If a man is striving to row up stream, 
 and at first is borne down 5 miles an hour, but by practice comes to row so well 
 as only to be borne down 3 miles an hour, he is evidently gaining ; i. e., —3 is an 
 increase upon —5. Finally, consider the thermometer scale. If the mercury 
 stands at 20° below (marked —20°) at one hour, and at —10° the next hour, the 
 temperature is increasing ; and, if it increase suHiciently, will become 0, passiiu/ 
 ichich it will reach +1°, +2°, etc. In this illustration, the quantity passes from 
 negative to positive by passing through 0. 
 
 It appears in geometry, that a quantity may also change its sign in passing 
 through infinity. Thus the tangent of an arc less than 90° is positive ; but if 
 the arc continually increases, the tangent becomes infinity at 90°, passing which 
 it becomes negative. 
 
 Now, as we know of no other way in which a varying quantity can change its 
 sign, it is assumed as a fundamental principle in mathematics that, if a vary- 
 ing QUANTITY CHANGES ITS SIGN, IT PASSES THROUGH ZERO, OR INFINITY. 
 
 NAMES OF DIFFERENT FORMS OF EXPRESSION. 
 54:» A I^olyiioinial is an expression composed of two or more 
 
10 LITERAL ARITHMETIC. 
 
 parts connected by the signs plus and minus, each of which parts is 
 called a term. 
 
 5o. A 3Ionoinial is an expression consisting of one term ; a 
 Binomial has two terms; a Trinomial has three terms, etc. 
 
 06, A Coefficient of a term is that factor which is considered 
 as denoting the number of times the remainder of the term is taken. 
 The numerical factor, or the product of the known factors in a term, 
 is most commonly called the coefficient, though any factor, or the 
 product of any number of factors in a term may be considered as 
 coefficient to the other part of the term. 
 
 tT7. Similar Terms are such as consist of the same letters 
 affected with the same exponents. 
 
 SECTION IL 
 ADDITION. 
 
 58. Addition, is the process of combining several quantities, so 
 that the result sliall express the aggregate value in the fewest terms 
 consistent with the notation. 
 
 59* The Snm or Ainount is the aggregate value of several 
 quantities, expressed in the fewest terms consistent with the nota- 
 tion. 
 
 60, JProp, 1, Similar terms are united hy Addition into one. 
 
 Dem. — Let it be required to add 4rt€, hac, — 2ac, and — dac. Now 4ac is 4 
 times ac, and 5ac is 5 times the same quantity {ac). But 4 times and 5 times the 
 same quantity make 9 times that quantity. Hence, 4ac added to ryrtc make 9ac. 
 To add — 2ac to 9ac we have to consider that the negative quantity, — 2ac, is so 
 opposed in its character to the positive, 9ac, as to tend to destroy it when com- 
 bined (added) with it. Therefore, — 2ac destroys 2 of the 9 fmes nc, and gives, 
 when added to it, 7ac. In like manner, — Sac added to Tr^r, gives 4ac. Thus the 
 four similar terms, 4nc, tiac, — 2ac, and — Sac, have been combined (added) into 
 one term, 4ac ; and it is evident that any other group of similar terms can be 
 treated in tlie same manner. <j, E. D. 
 
 01, Cor. 1. — In adding similar termsy if the terms are all posi- 
 tive, the sum is positive ; if all negative, the sum is negative ; if 
 some are positive and some negative, the sum takes the sign of that 
 kind (positive or negative) loMch is in excess. 
 
 ScH. — The operation of adding positive and negative quantities may look 
 to the pupil like Subtraction. For example, we say +5 and —3 added make 
 
 i? 
 
ADDITION. 11 
 
 -f 2. This looks like Subtraction, and, in one view, it is Subtraction. But 
 why call it Addition ? The reason is, because it is simply putting the quanti- 
 ties together — aggregating them — not finding their difference. Thus, if one 
 boy pulls on his sleigh 5 pounds in one direction, while another boy pulls 3 
 pounds in the opposite direction, the combined (added) effect is 2 pounds in 
 the direction in which the first pulls. If we call the direction in which the 
 first pulls positive, and the opposite direction negative, we have +5 and —3 
 to add. This gives, as illustrated, +2. Hence we see, that the sum of +5 
 and —3 is +2. 
 
 But the difference of + 5 and — 3 is 8, as will appear from the following 
 illustration : Suppose one boy is trying to draw a sleigh in a certain direction, 
 and another is holding back 3 lbs. If it takes 10 lbs. to move the sleigh, the 
 first boy will have to pull 13 lbs. to get it on. But if, instead oi holding hach 
 3 lbs., the second boy jpt^Aes 5 lbs., the first boy will have to pull only 5 lbs. 
 Thus it appears, that the difference between pushing 5 lbs. (or + 5) and hold- 
 ing back 3 lbs. (—3) is 8 lbs. 
 
 In like manner the sum of $25 of property and $15 of debt, that is the 
 aggregate value when they are combined, is $10. +25 and —15 are +10. 
 But the difference between having $25 in pocket, and being $15 in debt, is 
 $40. The difference between +25 and —15 is 40. 
 
 G^, Cor. 2. — The sum of two qnantities, the one positive mid the 
 other negative, is the numerical difference, toith the sign of the greater 
 prefixed. 
 
 OS, Cor. 3.—/^ appears that addition in mathematics does not al- 
 ways imply increase. Whether a quantity is increased or diminished 
 hy adding another to it, depends upon the relative nature of the two 
 quantities. If they both tend to the same end, the result is an increase 
 in that direction. If they tend to ojjposite ends, the result is a dimi- 
 nution of the greater by the less. 
 
 64, I^rop, 2, Dissimilar terms are not united into 07ie by addi- 
 tion, but the operation of adding is expressed by writing them in 
 succession, with the positive terms preceded by the + sign, and the 
 negative by the — sign. 
 
 Dem. — Let it be required to add + 4<!y*, + 3«&, — 2icy, and — mn. ^cy^ is 4 
 times c«/', and 3rf6 is 3 times (th, a different quantity from cy'^ ; the sum will, 
 therefore, not be 7 times, nor, so far as we can tell, any number of times ry^ or 
 a}), or any other quantity, and we can only exprcHS the addition thus : 4cy^ i 3a6. 
 In like manner, to add to this Fum — S;?'// we can only (>xpress the addition, as 
 4ny2 -f ^(ih 4- (_ 2.ry). But since "2xy is nejjative, it tends to destroy the positive 
 quantities and will take out of them Ixy. Hence the result will be 4c?/ '^ + 3«& 
 — 2a;y. The effect of — mn will be the same in kind as that of — 2.r^, and 
 hence the total sum will be \cy^ + Zcih — 2xy — mn. As a similar course 
 
12 LITERAL ARITHMETIC. 
 
 of reasoning can be applied to any case, the truth of the proposition ap' 
 pears. 
 
 Sen. — In such an expression as 4cy ' + dab — %vy — w;?,the — sign before the 
 mil does not signify that it is to be taken from the immediately preceding 
 quantity ; nor is this the signification of any of the signs. But the quan- 
 tities having the — sign are considered as operating to destroy any which 
 may have the + sign, and vice versa. 
 
 Go, Cor. — Adding a negative quantity is the same as subtracting 
 a numerically equal positive quantity ; that is,m + (— n) is m — w, 
 shown as above. 
 
 Dem. — Since a negative quantity is one which tends to destroy a positive 
 quantity, — ii when added to /« (t. €. + m) destroys n of the units in m, and 
 lience gives as a result m — n. 
 
 60, I^rob, — To add polynomials, 
 
 RULE. — Combine each set of similar terms into one 
 
 TERM, AND CON'NECT THR RESULTS WITH THEIR OWN SIGNS. ThE 
 polynomial THUS FOUND IS THE SUM SOUGHT.* 
 
 Dem. — The purpose of addition being to combine the quantities so as to 
 express the aggregate (sum) in the fewest terms consistent with the notation, 
 the correctness of the rule is evident, as only similar terms can be united into 
 one (60, 64). 
 
 67, Prop, »?, Literal terms, which arc similar only toith respect 
 to part of their factors, may be united into one term ivith a polynomial 
 coefficient. 
 
 Dem. — Let it be required to add 5ax, — 2cx, and 2mx. These terms are 
 similar, only with respect to x, and we may say 5a times x and — 2c times x 
 make {5a — 2c) times x, or (5a — 2c)x. And then, 5a — 2c times x and 2m times 
 X make (o« — 2c + 2m) times x, or (5a — 2c + 2m)x. q. e. d. 
 
 68, Prop, 4, Compound terms which have a common compound, 
 or polynomial factor, may be regarded as similar and added with 
 respect to that factor. 
 
 Dem. 5{x^ — y^), 2{:X- — y-) and — 3(j;^ — y^) make, when added with re- 
 spect to {x^ — y-), 4(.r- — y-), for they are 5 + 2 — 3, or 4 times the same quan- 
 tity {x- — y^). In a similar manner we may reason on other cases. Q. E. D. 
 
 ♦ This is the proficient's rule, as exhibited on p^ge 45 of the Complete School Algebra, 
 ScH. 2. 
 
ADDITION. 13 
 
 ScH. — The object and process of addition, as now explained, will be 
 seen to be identical witli the same as the pupil has learned them in Arith- 
 metic, except what grows out of the notation, and the consideration of 
 positive and negative quantities. For example, in the decimal notation let 
 it be required to add 248, 10506, 5003, 81, and 106. The units in the several 
 numbers are similar terms, and hence are combined into one : so also of the 
 tens, and of the liundreds. The process of carrying has no analogy in the 
 literal notation, since the relative values of the terms are not supposed to be 
 known. Again, there is nothing usually found in the decimal addition like 
 positive and negative quantities. With these two exceptions the processes 
 are essentially the same. The same may be said of addition of compound 
 numbers. 
 
 Examples. 
 
 1. Find the sum of 2a —3x^, bx^ — la, — 3« + z^, and a — Sx^, 
 
 2. Find the sum of a^ - b^ -h Sa^b - 5ab^, 3a^ - 4a^b + Sb^ 
 
 - 3ab\ a^ +b^ + daH, 2a^ - U^ - 5ab^, 6aH + lOab^, and - (Ja^ 
 
 - la^b + ^ab^ + 2b^. 
 
 3. Find the sum of bca^x^ + Ua^x^ + mx^y^, and lOca'-x^ 
 
 — 2ba^x^ -f 67nx^y^. 
 
 4. Add 2x^ — 4:X^ + x^, bx^y — ab + x^, 4x^ - x^, and 2xi - 3 
 
 5. Add ^(x + y) and |(.t — y), 
 
 6. Add ax-\-2by + cz, ^x + Vy + Vz, Sy^'-2xi + Sx^, 4:cz - Sax 
 
 — 2by, and 2ax - Wy - 2A 
 
 7. Add cz — 2ay, 2az — Say, my — az, with respect to z and y. 
 
 8. Add {a-\-b)Vx-(2-hifi)\/y,^y^ + (a + c)x^,3uVy—{2d-e)x^, 
 —2nVx + 12a Vy, and (m + n)ij^-}-(b + 2c)\/x. 
 
 9. Add x^ + xy ■¥ y^, ax^ — axy + ay^, and — by^ + bxy + bx^, 
 
 10. Add a(x + y) + b(x — y), m{x i- y) — n{x — y). 
 
 11. Add 'SmVx — y + QnVx — y — ^^x -— y — Sn^Jx — y. 
 
 12. Add Sax'i -f hy~^ — 2c, — = 1- 8c, and — ^ax~^ — my^ 
 
 Vx y 
 
 -de. 
 
 13. Add i^/a^ - x^, -^^/^^^l^, and ^a^ - x^. 
 
14 LITERAL ARITHMETIC. 
 
 14. Add J^-tA.-^, iL_-_iJ_^, and - 2a{x^ - 1)4. 
 
 Vx^ - 1 {x- - ip 
 
 15. Add i(l/^2 - y-i + 2;-2), and |(a;t + -^- - :^). 
 
 16. Add (a - b + c)y/x^~Y^y {a + h - c) {x^ ~ ?/2)i and 
 (b-{-c- a) Vx^ - yK 
 
 17. Condense tlie polynomial ^ax^ — 3?/« + %cz — 4:mVx i 3)ny^ 
 ^2ax^ + Gcz, into 2(a — 2?7i)Vx +3(;?i — l)^^ + Scz. 
 
 SECT/ON IIL 
 SUBTRACTION. 
 
 CO. Subtraction is, primarily, the process of taking a less 
 quantity from a greater. In an enlarged sense, it comes to mean 
 taking one quantity from another, irrespective of their magnitudes. 
 It also comprehends all processes of finding the difference between 
 quantities. In all cases the result is to be expressed in the fewest 
 terms consistent with the notation used. 
 
 70, TJie Difference between two quantities is, in its primary 
 signification, the number of units which lie between them ; or, it is 
 what must be added to one in order to pi'odttce the other. When it is 
 required to take one quantity from another, the difference is what 
 must be added to the Subtrahend in order to produce the Minuend, 
 
 71. JProb, — To perform Subtraction, 
 
 RULE. — Change the signs of each term in the subtra- 
 hend FROM + TO — , OR FROM — TO +, OR CONCEIVE THEM TO 
 BE CHANGED, AND ADD THE RESULT TO THE MINUEND. 
 
 Dem. — Since the difference sought is what must be added to the subtrahend 
 to produce tlie minuend, we may consider this difference as made up of two 
 parts, one the subtrahend with its signs changed, and the other the minuend. 
 When the sum of these two parts is added to the subtrahend, it is evident that 
 the first part will destroy the subtrahend, and the other part, or minuend, will 
 be the sum. 
 
SUBTRACTION. 15 
 
 Thus, to perform the example : 
 From 5ax — 6^ — 3(f — 4m 
 
 Take 2ax + 2b — 5d + Sm^ 
 
 Subtrahend with signs changed, — 2ax — 2b -{- 5d — Sm 
 Minuend, 5ax — &> — Sd — 4m 
 
 If these three 
 quantities are 
 added together, 
 the sum will 
 
 Difference, Sax — 8& + 2d— 12m evidently be the 
 
 minuend. If, therefore, we add the second and third of them (that is, the sub- 
 trahend, with its signs changed, and the minuend) together, the sum will be 
 what is necessary to be added to the subtrahend to produce the minuend, and 
 hence is the difference sought. Q. E. D. 
 
 72, Con. 1. — When a parenthesis, or any symbol of like significa- 
 tion {4:4), occurs in a polynomial, preceded by a — sign, and the 
 parenthesis or equivalent symbol is removed, the signs of all the terms 
 which were within must be changed, since the — sign indicates that 
 the quantity tvithin the parenthesis is a subtrahend. 
 
 73, Cor. 2. — A7iy quantity can be placed ivithin a parenthesis, 
 preceded by the — sign, by changing all the signs. The reason of 
 this is evident, since by removing the parenthesis according to the 
 preceding corollary, the expression loould return to its original form. 
 
 Examples. 
 
 1. How much must be added to 8 to produce 12 ? What is the 
 difference between 8 and 12 ? How much must be added to ^ax^ 
 — 5^3 (the subtrahend) to produce ?>ax^ + 2^^ ? 
 
 Aiistuer. — ^To Zax^ we must add bax^ ; and to — 5^' we must 
 add + 7;y3. Hence in all we must add hax^ + '^y^- 
 
 2. From 3x^ - 2x^ - x - 7 take 2x^ - 3x^ + x + 1. 
 
 3. From a^ — x^ take a^ + 2ax -f x^. 
 
 4. From 1 + Sx^ + 3x + x^ take 1 - Sx^ ^-Sx- x^. 
 
 5. From x'^ + 2x^y'^ + y^ take x'^ — 2x'^y^ + y^. 
 
 6. From 7\/l + x^ - 3ay^ take - sVl + x^ + Say^. 
 
 7. From ay^ + 10 Vab take ay + x\/7ib. 
 
 8. From bx^ — 3\/m?i + 1 take b^x + (mn)^ — 1. 
 
 9. From a + b -h Va — b take b -\- a — (a — b)^ + VaK 
 
16 LITERAL ARITHMETIC. 
 
 10. Remove the parentheses from the following: 
 
 a- {{b-c) - d] ; la- j3a - [4a- (5« - "Za)]] ; 
 ^a - b) - c + d - {a - b —'Z (c - d)] ; 
 Z(U--b-c)-b {a- (U + c)\ + 3 \b-(c-a)\. 
 
 11. Include within brackets the 3d, 4th, and 5th terms of Zab 
 — x^-hax — lOby + 50. Also the 4th and 5th. Also the 2d and 3d. 
 
 Theory of Subtraction. — Subtraction is finding the difference between 
 quantities, that is, finding what must be added to one quantity to produce the 
 other. This difference may always be considered as consisting of two parts, one 
 of which destroys the subtraliend, and the other part is the minuend itself. 
 Hence, to perform subtraction, we change the signs of the subtrahend to get 
 that part of the difference which dertroys the subtrahend, and add this result to 
 the minuend, which is the other part of the difference. 
 
 ^«» 
 
 SECT/ON IV, 
 
 MULTIPLICATION. 
 
 74. Multiplication is the process of finding the simplest ex- 
 pression consistent with the notation nsed, for a quantity which 
 shall be as many times a specified quantity, or such a part of that 
 quantity, as is represented by a specified number. 
 
 7i>. CoK. 1. — The mtdtlplier must alioays be conceived as an ab- 
 stract number^ since it shoivs how many times the multiplicand is 
 to be taken. 
 
 76, Cor. 2. — The product is always of the same hind as the mul- 
 tiplicand. 
 
 77, Prop, 1, — Tlie product of several factors is the same in 
 whatever order they are taken. 
 
 Dem. — 1st. a -K &, is a taken 6 times, 0Ta-\-a + a + a + a to 6 terms. 
 
 Now, if we take 1 unit from each term (each a), we shall get & units ; and this 
 process can be repeated a times, giving a times b, or b x a. .'. a x b = b x a. 
 
 2d. Wlien there are more than two factors, as abc. We have shown that ab 
 = ba. Now call this product in, whence ftbc — mc. But by part 1st, mc = cm. 
 .'. abc = bac = cah = cba. In like manner we may show that the product of any 
 number of factors is the same in Avhatever order they are taken, q. e. d. 
 
 78, JProp, 2, — When two factors have the same sign their prod- 
 uct is positive: when they have different signs their product is neg- 
 ative. 
 
MULTIPLICATION. 17 
 
 Dem. — 1st. Let tlie factors be + a and + b. Considering a as the multiplier 
 we are to take + b, a times, which gives + ab, a being considered as abstract in 
 the operation, and the product, + «6, being of the same 'kind as the multipli- 
 cand ; that is, positive. Now, when the product, + ab, is taken in connection 
 with other quantities, the sign + of the multiplier, a, shows that it is to be 
 added ; that is, written with its sign unchanged. .'. {+ b) x {+ a) = + ab. 
 
 2d. Let the factors he — a and — b. Considering a as the multiplier, we are 
 to take — b,a times, which gives — «&, a being considered as abstract in the 
 operation, and the product, — ab, being of the same kind as the multiplicand ; 
 that is, negative. Now, when this product, — ab, is taken in connection with 
 other quantities, the sign — of the multiplier shows that it is to be subtracted ; 
 that is, written with its sign changed. :. {—b) x {— a) = + ab. 
 
 3d. Let the factors be — « and + &. Considering a as the multiplier, we are 
 to take ■{■ b, a times, which gives + ab, a being considered as abstract in the 
 operation, and the product, ■{- ab,heing ot the same kind as the multiplicand; 
 that is, positive. Now, when this product, + ab, is taken in connection with 
 other quantities, the sign — of the multiplier shows that it is to be subtracted ; 
 that is, written with its sign changed. .•.(+&) x {— a) = — ab. 
 
 4th. Let the factors he + a and — b. Considering a as the multiplier, we are 
 to take — b, a times, which gives — ab, a being considered as abstract in the 
 operation, and the product, — ab, being of the same kind as the multiplicand ; 
 that is, negative. Now, when this product, — ab, is taken in connection with 
 other quantities, the sign + of the multiplier shows that it is to be added ; that 
 is, written with its own sign. .*. {— b) x {+ a) = — ab. q. e. d. 
 
 70, Cor. 1. — The j^^'oduct of any riumher of positive factors is 
 positive. 
 
 SO, Cor 2. — The product of an even number of negative factors is 
 positive. 
 
 81, Cor. 3. — TJie product of an odd number of negative factors is 
 7iegative. 
 
 82, Prop, 3, — Tlie product of two or more factors consisting of 
 the same quantity affected with exponents, is the common quantity 
 with an exponent equal to the snni of the exponents of the factors. 
 That is a'" X a" = «'""^" ; or rt'"- a"- a* = «"*+"+*, etc., whether the expo- 
 nents are integral or fractional, positive or negative. 
 
 Dem. — 1st. Wlien the exponents are positive integers. Let it be required to 
 
 multiply a"* by a"" and a\ a'^ = aaaa to m factors, a" = aaaaa io n 
 
 factors, and a'' — aaaaa to a factors. Hence the i)roduct, being composed 
 
 of all the factors in the quantities to be multiplied together, contains m + n A- s 
 factors each flr, and hence is expressed «'"+ " + *. Since it is evident that this rea- 
 soning can be extended to any number of factors, as «"• x a" x a" x a"", etc., 
 etc., the proposition in this case is proved. 
 
 2 
 
18 LITERAL ARITHMETIC. 
 
 2d. WJien the exponents are positive fractions. Let it be required to multiply 
 
 mem 
 
 a" by a* . Now a" means m of the ii equal factors into which a is conceived to 
 be resolved. If each of these n factors be resolved into b factors, a will be re- 
 
 m 
 
 solved into bn factors. Then, since a" contains m of the n equal factors of a, 
 and each of these is resolved into b factors, m factors will contain bm of the bn 
 
 m bm c 
 
 equal factors of a. Hence a" = «'"' . In like manner a^ may be shown equal to 
 
 en m c t>m en 
 
 a''* ; and a* x a^ = a^'^ x a^". This now signifies that a is to be resolved into 
 
 m e ^ 
 
 bn factors, and bm + en of them taken to form the product. /. a" x a^ = a*"* 
 
 xa*" = a *" ,ora" *, which proves the proposition for positive fractional 
 exponents, since the same reasoning can be extended to any number of factors, 
 
 m e e 
 
 as o" X a'* X a'', etc. 
 
 3d. Whe7i the exponents are negative. Let it be required to multiply «-"' by 
 a—", m and w being either integral or fractional. By definition a—"^ x «— * = 
 
 — X — . Now, as fractions are multiplied by multiplying numerators together 
 a** a* 
 
 and denominators together, we have — x — = -—7- by part 1st of the demon- 
 ^ a"» a" «"• + " 
 
 stration. But this is the same as a- (»« + ") or a-"'-". .'. a—"" x a-" = a-"'-'* 
 
 Examples. 
 
 1. Prove as above that 81^ X 81* = 81^'" and that 81^ = 81^ 
 
 2. Prove that m" X vi"" = 7)1"+". 
 
 3. Prove that 16"* x 16"* = 16"*. 
 
 4. Prove that 25"* X 25* is 1. 
 
 5. Prove that a~' X a^ is a. 
 
 ScH.— The student must be careful to notice the difference between the 
 signification of a fraction used as an exponent, and its common signification. 
 Thus I tised as an exponent signifies that a number is resolved into 3 equal 
 factors, and the product of 2 of them taken •, whereas I used as a common 
 fraction signifies that a quantity is to be separated into 3 equal parts, and 
 the sum of two of them taken. 
 
 S3, Proh, — To muUiiyly monomiaU. 
 
 RULE. — Multiply the numerical coefficiej^ts as in the 
 
 DECIMAL NOTATION, AND TO THIS PRODUCT AFFIX THE LETTERS OF 
 ALL THE FACTORS, AFFECTING EACH WITH AN EXPONENT EQUAL TO 
 THE SUM OF ALL THE EXPONENTS OF THAT LETTER IN ALL THE 
 
MULTIPLICATION. 19 
 
 FACTORS. The SIGN^ of the product will be + EXCEPT WHEN^ 
 THERE IS AN ODD NUMBER OF NEGATIVE FACTORS; IN WHICH CASE 
 IT WILL BE — . 
 
 Dem. — This rule is but an application of the preceding principles. Since the 
 product is composed of all the factors of the given factors, and the order of ar- 
 rangement of the factors in the product does not affect its value, we can write 
 the product, putting the continued product of the numerical factors first, and 
 then grouping the literal factors, so that like letters shall come together. 
 Finally, performing the operations indicated, by multiplying the numerical 
 factors as in the decimal notation, and the like literal factors by adding the ex- 
 ponents, the product is completed. 
 
 84, I^rob, — To multiply two factors together token one or hoth 
 are polynomials, 
 
 R ULE. — Multiply each term of the multiplicand by each 
 
 TERM OF THE MULTIPLIER, AND ADD THE PRODUCTS. 
 
 Dem. — Thus, if any quantity is to be multiplied by a -|- & — c, if we take it a 
 times {i. e. multiply by a), then h times, and add the results, we have taken it 
 a + h times. But this is taking it c too many times, as the multiplier required 
 it to be taken a + h minus c times. Hence we must multiply by e, and subtract 
 this product from the sum of tlie other two. Now to subtract this product is 
 simply to add it with its signs changed {71). But, regarding the — sign of c 
 as we multiply, will change the signs of the product, and we can add the partial 
 products as they stand, even without first adding the products by a and h. 
 
 Q. E. D. 
 
 S5, Theo. — TJie square of the sum of two quantities is equal to 
 the square of the first^ plus twice the product of the two, plus the 
 square of the second. 
 
 80, Theo. — The square of the difference of two quantities is 
 equal to the square of the first, ininus twice the product of the two, 
 plus the square of the second. 
 
 87, Theo. — The product of the sum and difference of two quan- 
 tities is equal to the difference of their squares. 
 
 The demonstration of these three theorems consists in multiplying 
 2^ + y by ic + ^, a; — ^ by ic — y, and x + y hy x — y. 
 
20 UTERAL AlilTHMETIG. 
 
 Examples. 
 
 1. Multiply together Sax, — Sa^x^, 4:hij, — y^, and 2x*y*. 
 
 2. Multiply together 3x*, — mx^, 2m', x-% — 2, and 2a;"*. 
 
 3. Multiply together 40a;*, x^y and f A/^ ; also Sa^h^, and 
 
 « _i _I i 
 
 4. Multiply m^ by m ^, a~* by a", aH'"" by a^ft*, m " by w*", 
 
 V'rt by ^^, ^^3 by ^^. 
 
 5. Multiply '6a — 2hhy a + 4i. 
 
 6. Multiply a;2 + a;y + y^ by a;* — xy -{■ y^. 
 
 7. Multiply 7/1* + ?i* + 0* - 7)1^71^ — m^o^ — n^o^ by m^ + n» 
 
 8. Multiply a"' — «" + rt^ by ^T — a. 
 
 9. Multiply together z — a, z — b, z — c, z — d. 
 
 10. Multiply together x + y, x — y, x^ -\- xy + y^ and a;* — xy 
 
 SuG. — Try the factors in different orders, and compare the labor required. 
 
 m t m t_ nt. t 
 
 11. Multiply a^b' * - cl^f '' + 1 by a^'"^ + 1. 
 
 12. Multiply 2a^-''b^-'' 4- Sa^-^b"" by 10a''-^+^Z'" + » - oa'-'b-"^. 
 
 13. Square the following by the theorems (8S^ 86) : 
 
 1+a, x-2, 3/ +3^, a~i-a~^b^, a:" + a:, f±-, a;-* + .V, 
 }rt^ - i^r^' Z/.c-i//~ n — ay-^xX 2a2J-(3-p) _^ Ja^^r*. 
 
 14. Write the following products by (87) : 
 
 (1 + frt) X (1 - f«), (99ax 4- 9a/«^) X (99rta: - da^x^). 
 
 15. Expand (a + ^ + c) (a + b — c) {a — b + c) (— a + b + c). 
 
MULTIPLICATION. 21 
 
 MULTIPLICATIOIS' BY DeTACHED COEFFICIElsrTS. 
 
 88m lu cases in which the terms of both multiplicand and multi- 
 plier contain the same letters, and can be so arranged that the ex- 
 ponents of the same letters shall vary in the successive terms of 
 each according to the same law, a simiUir laAV ^vill liold good in the 
 product, and the multiplication can be efifected by using the co- 
 efficients alone, in the first instance, and then writing the literal 
 factors in the product according to the observed law. A few 
 examples will make this clear : 
 
 1. Multiply 2a^ - 3a^x + ^ax^ — x^ by 2a^ — ax + 7x^. 
 
 OPERATION. 
 
 2 - 3 + 5- 1 
 
 2-1+7 
 
 4 - 6 + 10 - 2 
 -2+3-5+1 
 
 + 14-21 + 35-7 
 
 4-8 + 27-28 + 36 
 
 Prod., 4«6 - 8a^x + 27a^x^ - 2Sa^x^ + SQax^ - 7a:« 
 2. Multiply a:3 + 2a; — 4 by rc2 — 1. 
 
 SuG. — By writing these polynomials thus, x^ + Ox^ + 2x — 4, x^ + Ox — 1, 
 the law of the exponents in each case becomes evident. Hence we have, 
 
 1 + + 2-4 
 1 + 0-1 
 
 1+0+2-4 
 
 _l_0-2 +4 
 
 1+0+1-4-2+4 
 
 Prod., a;' + Oa;* + x^ —4x^ — 2x + 4, or x'^ + x^ — 4x^ — 2a; + 4 
 
 3. Multiply 3«3 -j. 4:ax - ox^ by 2a^ - 6ax + 4x^. 
 
 4. Multiply 2^3 - 3ah^ + 5h^ by 2a^ - 6b^. 
 
 Bug.— The detached coefficients are 2 + — 3 + 5, and 2 + — 5. 
 
 5. Multiply «3. -\-aix + ax^ ■}■ x^ hy a — x. 
 
 6. Multiply x^ - 'dx^ + 3a; - 1 by x^ - 2x -h 1. 
 
22 LITERAL ARITHMETIC. 
 
 SECTION F. 
 
 DIVISION. 
 
 SO, T>ivision is the process of finding how many times one 
 quantity is contained in another. 
 
 00, The problem of division maybe stated: Given the product 
 of two factors and one of the factors^ to find the other ; and the siiffi- 
 cient reason for any quotient is, that midtiplied by the divisor it 
 gives the dividend. 
 
 01, Cor. 1. — Dividend and divisor may both be multiplied or 
 both be divided by the same number without affecting the quotient. 
 
 02, Cor. 2. — If the dividend be multiplied or divided by any 
 number, while the divisor remains the same, the quotient is multiplied 
 or divided by the same. 
 
 03, Cor. 3. — If the divisor be multiplied by any number while the 
 dividend remains the same, the quotient is divided by that number / 
 but if the divisor be divided, the quotient is midtiplied. 
 
 94:, Cor. 4. — The sum of the quotients of two or more quantities 
 divided by a common divisor, is the same as the quotient of the sum 
 of the quantities divided by the same divisor. 
 
 05, Cor. 5. — The difference of the quotients of two quantitie.\ 
 divided by a common divisor, is the same as the quotient of the dif- 
 ference divided by the same divisor. 
 
 These corollaries are direct consequences of the definition, and need no 
 demonstration ; but they should be amply illustrated. 
 
 96, Def. — Cancellation, is the striking out of a factor common to both 
 dividend and divisor, and does not affect the quotient, as appears from {01), 
 
 97, Lemma 1. — When the dividend is positive, the quotient has 
 the same sign as the divisor ; but when the dividend is negative, the 
 quotient has an o2yposite sign to the divisor. 
 
 08, Lemma 2. — When the dividend and divisor consist of the 
 same quantity affected by exponents, the quotient is the common 
 quantity with an exponent equal to the exponent in the dividend, 
 m.inus that in the divisor. 
 
DIVISION. 23 
 
 These lemmas are immediate cousequences of the law of the signs and 
 exponents in multiplication. 
 
 99, Cor. 1. — An^/ quantity rcith an exponent is 1, since it may 
 be considered as arising from dividing a quantity by itself. 
 
 Thus, X representing any quantity, and m any exponent, a;"' -5- a;"* =: a;° = 1. 
 
 100, Cor. 2. — Negative exponents arise from division whe^i 
 there are more factors of any number in the divisor than in the divi- 
 dend. 
 
 101, Cor. 3. — A factor may be transferred from dividend to 
 divisor (or from numerator to denominator of a fraction^ ichich is 
 the same thing), and vice versa, by changing the sign of its expo?ient. 
 
 102 • J^rob, 1, — To divide one monomicd by another, 
 
 RULE. — Divide the numerical coefficient of the divi- 
 dend BY THAT OF THE DIVISOR AND TO THE QUOTIENT ANNEX THE 
 LITERAL FACTORS, AFFECTING EACH WITH AN EXPONENT EQUAL TO 
 ITS EXPONENT IN THE DIVIDEND MINUS THAT IN THE DIVISOR, AND 
 SUPPRESSING ALL FACTORS WHOSE EXPONENTS ARE 0. ThE SIGN 
 OF THE QUOTIENT WILL BE + WHEN DIVIDEND AND DIVISOR HAVE 
 LIKE SIGNS, AND — WHEN THEY HAVE UNLIKE SIGNS. 
 
 Dem. — The dividend being the product of divisor and quotient, contains all 
 the factors of both ; hence the quotient consists of all the factors which are 
 found in the dividend and not in the divisor. 
 
 103, Fvoh, 2, — To divide a j^olynomial by a monomial. 
 RULE. — Divide each term of the polynomial dividend by 
 
 THE MONOMIAL DIVISOR, AND WRITE THE RESULTS IN CONNECTION 
 WITH THEIR OWN SIGNS. 
 
 Dem. — This rule is simply an application of the corollaries {94, 95), 
 
 104, Dep. — A polynomial is said to be arranged with reference to a certain 
 letter when the term containing the highest exponent of that letter is placed first 
 at the left or right, the term containing the next highest exponent next, etc., etc. 
 
24 LITERAL ARITHMETIC. 
 
 105, Prob. S, — To perform division when both dividend an^ 
 divisor are polynomials. 
 
 RULE. — Having arranged dividend and divisor with 
 
 REFERENCE TO THE SAME LETTER, DIVIDE THE FIRST TERM OF THE 
 dividend by the first TERM OF THE DIVISOR FOR THE FIRST 
 TERM OF THE QUOTIENT. ThEX SUBTRACT FROM THE DIVIDEND 
 THE PRODUCT OF THE DIVISOR INTO THIS TERM OF THE QUOTIENT, 
 AND BRING DOWN AS MANY TERMS TO THE REMAINDER AS MAY 
 BE NECESSARY TO FORM A NEW DIVIDEND. DiVIDE AS BEFORF, 
 AND CONTINUE THE PROCESS TILL THE WORK IS COMPLETE. 
 
 Dem. — The arrangement of dividend and divisor according to the same letter 
 enables us to find the term in the quotient containing the highest (or lowest if 
 we put the lowest power of the letter first in our arrangement) power of the 
 same letter, and so on for each succeeding term. 
 
 The other steps of the process are founded on the principle, that the product 
 of the divisor into the several parts of the quotient is equal to the dividend. 
 Now by the operation, the product of the divisor into the Jird term of the 
 quotient is subtracted from the dividend ; then the product of the divisor into the 
 second term of the quotient ; and so on, till the product of the divieor into each 
 term of the quotient, that is, the product of the divisor into the \cholc quotient, 
 is taken from the dividend. If there is no remainder, it is evident that this 
 product is equal to the dividend. If there w a remainder, the product of the 
 divisor and quotient is equal to the whole of the dividend except the remainder. 
 And this remainder is not included in the parts subtracted from the dividend, by 
 operating according to the rule. 
 
 ScH. — Tliis process of division is strictly analogous to " Long Division " 
 in common arithmetic. The arrangement of the terms corresponds to the 
 regular order of succession of the thousands, hundreds, tens, units, etc., 
 while the other processes are precisely the same in both. 
 
 Examples. 
 
 3 1 ^» - 
 
 1. Divide m^ by w^, /i" by n'^y (ab)^'^ by (ab)" , a^ hy a^, a~* 
 
 by «5, ;c 3 by x'^, x'^ by x~^, 
 
 „ ^ «-2^2 2ar^x~^y , bcd-^bx-^ ^ 
 
 2. Free -37—,, » o -1 o - and -5-3^ — —- from negative expo- 
 
 nents, and explain the process. 
 
 3. Divide 15ay« by 3ay, Sa^b^a^d by ^aH^c^, 3ah^ by a^b^, 
 — doa*'bx^hy1a^bx,-20aJbh by ~-4.0ab^c, y" -by y", -?/ by y-^ 
 na^b^-'y by -^a-'b'-Py-", -^a^b'^c^ by - 12a'^bc^-'', a'-'+^b'-'c^ 
 
 m n 
 
 by a*-^+i^''+Y'2, and xi>^j/~^ by a;«y"i. 
 
DIVISION BY DETACHED COEFFICIENTS. 7^ 
 
 L Divide Ha^k- - l'2a^k^ +3a'k^ by dak, Ux^y^a^b + Ulx^y^ 
 ~ -^ii^ifab- by Wx'^y^, Ibax^ — Iba^x -\- 'dax by — bax, i:a'^^ni^ 
 
 - 12ri-i»//i8 4- 5280 by - 12«-i% '^Q^x^y'"' - 2Vixy'''+' by l^xy, 
 y^ + 3^2/ - 2^^ by yK V^" - b'^'" - &'+'"- b'^*" by b'% ax^ 
 
 - 2fla;"« ' + 3«.r by «2.-'+\ 
 
 5. Divide 4a;2 - 28z^ + 4%2 by 2.t - '7y. 
 
 6. Divide G2* — Idax^ + ISa^T^ — 13<«3a:-5rt* by 2a;2— 3aa:— a^. 
 
 7. Divide a;^ 4- ?/3 _{_ 3^^ _ 1 by ^ 4. 1^ _ 1. 
 
 8. Divide ««^i2 _ 54 by ab^ — 2, x — 4:J by x^ — 2«^. 
 
 9. Divide xy- ahy x^y^ — a^, 243^' + 1024 by 4 + Sa. 
 
 10. Divide ^8 - 1^^* + Uy' - fe^ - W^ + | by ^2 _ | + 5. 
 
 11. Divide 1 + 2x^ — 7x^ - 16x^ by 1 + 2.?: + 3x^ + 4rc3. 
 
 12. Divide {x^ - y^f by (x - «/)^ a^ + ^,-3 by « -j- b'K 
 
 13. Divide ?/* i^J !/ • 
 
 14. Divide 1 by 1 — x^, also by 1 + ^^, 1 + x, and by 1 — a;. 
 
 15. Divide «'+" + a^b + fli" + b'+" by ft" + b\ 
 
 16. Divide a'"'-'"'b^''c - a"^+'-'Z>'-V + a-^b-'c"^ 4- a"""" Z»'''+V'^ 
 
 - «"«+«»- '^,3^.">-i + jp+x^.n.+«-i by «-»^-^-' + ^6''^-'. 
 
 17. Divide ?>i"'+' + a^w^i"" + n??!"" + aii'"'^'' by m + n. 
 
 18. Divide 7/m(a:« +l) + (w2 -\- m^) (x^+x) + {n^ +2nm){x^ +x^) 
 by «a:2 4- /?z:c 4- n. 
 
 19. Divide Ma;* 4- 2(h - k)x^ - {h^ 4- 4 - h^)x^ 4- 2 (7i 4- h)x 
 ^ Ilk by /l'a:2 — 7i 4- 2a7. 
 
 20. Divide x + y -{- z — 3 \/xyz by a;"^ 4- «/^ 4- ;z^ 
 
 Division by Detached Coefficients, 
 
 106, Division by detached coefficients can be effected in the same 
 cases as multiplication (88). The student will be able to trace the 
 process and see the reason ji'oni an exampje. 
 
26 LITERAL ARITHMETIC. 
 
 1. Divide 10a* — '^la^x + ^^^x^ - l%ax^ - 8a;* by 2a« - Soaj 
 
 OPERATION, 
 
 2 - 3 + 4) 10 - 27 + 34 - 18 - 8 1 5 -6 -2 
 
 10 - 15 + 20 I 5a^ — Qax — 2a;» Qiwt, 
 
 -12 + 
 
 -12 + 
 
 14- 
 
 18- 
 
 18 
 24 
 
 
 
 
 4 + 
 4 + 
 
 6- 
 6- 
 
 -8 
 -8 
 
 2. Divide re* - Zax^ - %a^x^ + 18«3:c - 8a* by x* + 2ax — 2aK 
 
 3. Divide 6a* - 96 by 3a - 6. 
 
 SuG.— The detached coefficients are 6 + + + - 96 and 3 — 6. 
 
 4. Divide 3y^ + 3xy^ — Ax^y —4:X^ hy x -\- y. 
 
 5. Divide x'^ + y'^ hj x -h y ; »ilso re* — y* by a;^ — y*. 
 
 Synthetic Division. 
 
 107, When division by detaclied coefficients is practicable, as in 
 tlie examples in the last article, the operation may be very much 
 condensed by an arrangement of terms first proposed by W. G. Hor- 
 ner, Esq., of Bath, Eng., which is hence called Horner's method of 
 synthetic division. A careful inspection of tlie operatiok under 
 Ex. 1, in the last article, will acquaint the student with the process. 
 
 Explanation of Operation. — Arrange the 
 coefficients of the divisor in a vertical column 
 at the left of the dividend, changing the signs of 
 all after the first. Draw a line underneath the 
 whole under which to write the coefficients of 
 5a'-^x-2x', Quot. the quotient. 
 
 The first coefficient of the quotient is found 
 evidently by dividing the first of the dividend by the first of the divisor, 
 and in this case is 5. As the first term of the dividend is always destroyed by 
 this operation, we need give it (10) no farther consideration. Now, multiplying 
 the other coefficients after the first (t. e. + 3 and — 4) icith their sig-ns changed, 
 by 5, we have + 15 and — 20, which are to be added (?) to — 27 and + 34. Hence 
 we write the former under the latter. The first term* of the second partial divi- 
 dend can be formed mentally by adding (?) + 15 to — 27, and the next term of 
 the quotient by dividing this sum (— 12) by 2. Hence — 6 is the second term of 
 
 * Strictly, the " coefficient of; " but tUis form is asedfor breyity. 
 
 
 operation. 
 
 2 
 
 10-27 + 34-18-8 
 
 + 3 
 
 + 15-20 + 24 + 8 
 
 -4 
 
 -18-6 
 
 
 5 -6 -2 
 
SYNTHETIC DIVISION. 
 
 27 
 
 the quotient. (We did not add (?) — 20 to + 34, because there is more to be 
 taken in before the first term of the next partial dividend is formed.) 
 
 Having found the second term of the quotient (— 6), we multiply the terms 
 of the divisor, except the first, (with their signs changed) by — 6, and write the 
 results, — 18 and + 24, under the third and fourth of the dividend, to which 
 they are to be added (?). Now we have all that is to be added"* to +34 (viz., 
 
 — 20 and — 18) in order to obtain the first term of the next partial dividend. 
 Hence, adding, we get — 4. which divided by 2 gives — 2 as the next term of 
 the quotient. Multiplying all the terms of the divisor except the first, as before, 
 we have — 6 and + 8, which fall under — 18 and — 8. Now adding + 24 and 
 
 — 6 to — 18, nothing remains. So also +8 — 8 = 0, and the work is complete, 
 as far as the coefficients of the quotient are concerned. 
 
 2. Divide x^ - bx'^ + 15a;* - Ux^ + 21x^ - 13a; + 5 by aj* - 2x^ 
 + 4a;2 - 2a; + 1. 
 
 OPERATION. 
 
 Quot, 
 
 1 
 
 + 2 
 -4 
 + 2 
 -1 
 
 1 
 
 -5 + 15 
 
 -24 + 
 
 27- 
 
 13 + 
 
 5 
 
 
 + 2- 
 
 4 
 
 + 2- 
 
 1 + 
 
 3- 
 
 •5 
 
 
 
 6 
 
 + 12- 
 + 10- 
 
 6 + 
 20 
 
 10 
 
 
 j_ 
 
 -3 + 
 
 _^ 
 
 
 
 
 
 
 
 ^ 
 
 a;' - 3a; + 5 
 
 3. Divide 4i/6 - Uy^ + 60y* - SOy' + my^ — 24y + 4 by %y^ 
 
 -4^ + 2. 
 
 4. Divide x'^ - y'^ hj x — y ', also 1 by 1 — a;. 
 
 5. Will a; + 2 divide a;* + 2a;' — 7a;' - 20a; 4- 12 without a re- 
 mainder? Willa;-3? 
 
 6. Will a; + 3, or a; — 3, divide a^ — 6a;* — 16a; + 21 without a re- 
 mainder ? Will a; + 7, or a; — 7 ? 
 
 ♦ The student will not fail to eee that this addition is equivalent to the ordinary subtraction 
 since the signs of the terms have been changed. 
 
28 LITERAL ARITHMETIC. 
 
 CHAPTER n. 
 
 FACTORING. 
 
 SECTION I. 
 FUNDAMENTAL PROPOSITIONS. 
 
 108, The Factors of a uumber are those numbers wliich mul- 
 tiplied together produce it. A Factor is, therefore, a Divisor. A 
 Factor is also frequently called a measure, a term ai'ising in Geome- 
 try. 
 
 109, A Common Divisor is a common integral factor of 
 two or more numbers. The Greatest Common Divisor of two or 
 more numbers is the greatest common integral factor, or the product 
 of all the common integral factors. Common Measure and Com- 
 mon Divisor are equivalent terms. 
 
 110, A Common Multiple of two or more numbers is an 
 integral number which contains each of them as a factor, or which 
 is divisible by each of them. The Least Common Multiple of two 
 or more numbers is the least integral number which is divisible by 
 each of them. 
 
 111, A Composite Number is one which is composed of 
 integral factors different from itself and unity. 
 
 112, A Prime N'umber is one which has no integral factor 
 other than itself and unity. 
 
 lis, Numbers are said to be Prime to each other when they have 
 no common integral factor other than unity. 
 
 ScH. 1. — The above definitions and distinctions have come into use from 
 considering Decimal Numbers. They are applicable to literal numbers only 
 in an accommodated sense. Thus, in the general view which the literal no- 
 tation requires, all numbers are composite in the sense that they can be fac- 
 
FACTORING. 29 
 
 tored ; but as to whether the factors are greater or less than unity, integral 
 or fractional, we cannot affirm. 
 
 114, Prop, 1, — A monomial viay be resolved into literal fac 
 tors by separating its letters hito any number of groups, so that the 
 sum of all the exponents of each letter shall fnake the exponent of 
 that letter in the given monomial. 
 
 1 15, Prop, 2, — Any factor which occurs in every term of a 
 polynomial can be removed by dividing each term of the poly^iomial 
 by it. 
 
 116, Proj}, 3, — If two terms of a trinomial are positive and 
 the third ter)a is twice the jyroduct of the square roots of these two, 
 and POSITIVE, the trinomial is the square of the SUM of these square 
 roots. If the third term is negative, the trinomial is the square of 
 the DIFFERENCE of the two roots. 
 
 117, Prop, 4, — The difference between two quantities is equal 
 to the product of the sum and difference of their square roots. 
 
 118, Prop, S, — When one of the factors of a quantity is givefi, 
 to find the other, divide the given quantity by the given factor, and 
 the quotient will be the other. 
 
 110, Prop, 6, — The difference between any two quantities is a 
 divisor of the difference between the same powers of the quan^ 
 titles. 
 
 The SUM of two quantities is a divisor of the difference of the 
 same EVEN jyowers, and the SUM of the same ODD powers of the quan- 
 tities. 
 
 DE\f. — Let X and y be any two quantities and n any positive integer. First, 
 x — y divides a;" — y". Second, if n is even, x + y divides a^ — y". Third, if n is 
 odd, X -\- y divides ic" + y*. 
 
30 LITERAL ARITHMETIC. 
 
 FIRST. 
 
 Taking the first case, we proceed in form with the division, till four 
 of the 
 
 terms of the x — y)a;" — y* (a;"-' + a^-gy + a;"-^y2 + g^-ys + etc, 
 
 quotient (enough to ^^^^-^^e^y_ 
 
 determine the law) are x'^-^y — y" 
 
 found. We find that each x^ '^y — x^-^y^ 
 
 remainder consists of two terms, x'^-^y'^ — y" 
 
 the second of which, — y", is the x*-^y^ — x'—'y^ 
 
 second term of the dividend constantly a;"- "y 3 _ y» 
 
 brought down unchanged; and the first x^-^y^ — x''-*y^ 
 
 contains x with an exponent decreasing by a?"-*y*— y* 
 
 unity in each successive remainder, and y with an 
 
 exponent increasing at the same rate that the exponent of x decrecbses. At this 
 rate the exponent of ar in the nth remainder becomes 0, and that of y, n. Hence 
 the Tith remainder is y" — y* or ; and the division is exact. 
 
 SECOND AND THIRD. 
 
 X + y>r" ± y» (a;"-' - ar"-«y + ^"-'y^ - xr-*yi 
 
 ,etc. 
 
 a?- + a--V 
 
 
 
 
 a--*y2 ± yn 
 Taking x + y ar^-'ys + a;"-'yS 
 
 
 for a divisor, we —a^-^y^±y^ 
 
 observe that the exponent — x^-'y^ — x^'-^y^ 
 
 of x in the successive re- a;"-*y4 ± y" 
 
 
 mainders decreases, and that of y increases 
 
 the same as before. But now we observe that the first term of the remainder is 
 — in the odd remainders, as the 1st, 3d, 5th, etc., and + in the ceen ones, as the 
 2d, 4th, 6th, etc. Hence if n is emn, and the second term of the dividend is — y", 
 the nth remainder is y"* — y" orO, and the division is exact. Again, if n is odd, 
 and the second term of the dividend is + y» , the nth remainder is — y" + y" , 
 or 0, and the division is exact, q. e. d. 
 
 120, Cor, — The last proposition applies equally to cases involv- 
 ing fractional or 7iegative exponents. 
 
 Dem. — Thus, x^—y^ divides x^—y^, since the latter is the difference between 
 the 4th powers of x^ and y*. So in general a;" «♦ — y ' divides x »* — y ^ , a 
 being any positive integer. This becomes evident by putting x «i=v, and 
 y^r —qff. whence x^'^ = v', and y ^ = vf*. But «« — zo« is divisible by v~w, 
 hence x~ » — y ~ is divisible by a; » — y ' . 
 
FACTORING. 31 
 
 121, JProp, 7, — A. trinomial can he resolved into two binomial 
 factors, when one of its terms is the product of the square root of 
 one of the other two, into the sum of the factors of the remaining term. 
 The two factors are respectively the algebraic sum of this square root, 
 and each of the factors of the third term. 
 
 III. — Thus, in a;* + 7aj + 10, we notice that Ix is the product of the square 
 root of x^, and 2 + 5 (the sum of the factors of 10). The factors of x- + Ix 
 + 10 are 2; + 2 and x 4 5. Again, x^ — ^ — 10, has for its factors .t; + 2 and 
 « — 5, — 3.C being the product of tlie square root of x- (or x), and the sum of 
 — 5 and 2, (or — 3), which are factors of - 10. Still again, x^ +^x — 10 
 = (a; — 2) {x + 5), determined in the same manner. 
 
 Dem. — Tlie trutli of this proposition appears from considering the product of 
 X + ahy X + b, which is x^ + (a + b) x + ab. In this i)roduct, considered as a 
 trinomial, we notice that the term (a + b)x is the product of fa;* and a + b, the 
 sum of the factors of ab. In like manner (x + a) (x — b) z=x' + (a— b)x — ab, 
 and (x — a) {x — b)=x^ — (a + b)x ■{■ ah, both of which results correspond to the 
 enunciation. Q. E. D. 
 
 [Note. — In application, this proposition requires the solution of the problem: 
 Given the sum and product of two numbers to find the numbers, the complete 
 solution of which cannot be given at this stage of the pupil's progress. It Avill 
 be best for him to rely, at present, simply upon inspection.] 
 
 122, I*i*op, S, — We can often detect a factor by separating 
 a polynomial hito parts. 
 
 Ex. Factor x^ + 12a; - 28. 
 
 Solution. — The form of this polynomial suggests that there may be a bino- 
 mial factor in it, or in a part of it. Now a;* — 4c + 4 is the square of a; — 2, 
 and (.c« - 4c + 4) + (16a;-32) makes a;« + 12aj - 28. But (a;'-4r + 4) + (16a;-32) 
 = {x- 2) (a; -2) + (a; - 2)16 = (.i; - 2) (a; - 2 + 16) = (a: - 2) (.c + 14). Whence 
 X — %, and aj + 14 are seen to be the factors of x^ + 12a; — 28. 
 
 Miscellaneous Examples. 
 
 1. Factor Ifg^y - 2Sf^gy^ 4- i2pgy, ^x^y^ - Hx^y^ + UxyK 
 
 2. Factor ?>?> - n^, 1 - 2V~v + x, 256«* -f 544^2 + 289, 1 - c\ 
 
 3. Factor x^- x - TZ,y^- z^,a^ -^ b^,^ + ^ _2, a^ +23« + 22. 
 
 0^ a^ 
 
 4. Factor ^ - ---^ + 15., c« - d^, c^ - d-\ c^ - d-\ 
 
 m* mx^ X*' ' ' 
 
S2 LITERAL ARITHMETIC. 
 
 i .J ...4 K..-A « 
 
 5. Factor a' - m % 4:t-* — 5?/--*, — — Ji o, a;S 4- 22a: - 7623. 
 
 6. Factor a;" - 1, 507?n* + 13267^2,^1 ^ 867w3, Vrt - V^. 
 
 7. Factor x^-2ax — a^, a"" dt U^VcT^" + U^c"", x^-\-\^+2J^. 
 
 8. Factor A«*" - ^W"^'"+' + A^*"^', 3« + 3^ - 61/^. 
 
 9. Resolve x into two equal factors ; also two unequal factors. 
 
 10. Resolve dSx^y^z'^ — 3Vy*z into two factors of which one is 
 2y^Vz. 
 
 11. Resolve 121a^&^c^ into two equal factors ; also into four equal 
 factors. 
 
 12. Remove the factor ^{ak^)^ from S^a^k*. 
 
 w* 7c-2 49rf» 
 
 13. Remove the factor — ^ + — ^^^om m®w~* — ~K7r' 
 
 14. Remove the factor a* — a^b + a^b^ — aZ>^ 4- b^ from «* 
 + b'. 
 
 15. Factor 15a + 5rta; — a; — 3, 21abccl—2%cdxy-{-\babmn—20mnxy, 
 21a:2 + 232:^ - 20^2, 12^20;* - 12rt2:2;} + 3a«. 
 
 16. Factor 3.c3 - 12^3^2 _ 4^2 + 1^ T2cd*m^ - Ucdhn* 
 + 9Gc2r^2;,j2, 
 
 17. The terms of a trinomial are ZOab, 9<t2 and 25^2, What sign 
 must be given to each that the trinomial may be factored ? 
 
 18. The terms of a trinomial are — 9rt, \)i>^/a and 4. What must 
 be the signs of the last two terms that the trinomial may be 
 factored ? 
 
 — 4 4- 
 
 19. Is « 5 — J'" exactly divisible hy a^ — b or dJ -\- b "i 
 
 20. Is m^ — n^ exactly divisible by Vm — Vn't by Vm + Vn'i 
 by \/m ± ^7i ? 
 
 21. Is a;!*'! + y^^^ exactly divisible by x + ?/ ? by a; — y ? 
 
 22. Is .-^2019 _|_ ^20 79 exactly divisible by x'^ - y"^ ? by .t^ + y'^ ? 
 
 23. What is the quotient of (%J + mz^) -^ {k^ Vy + ^/m z^) ? 
 
HIGHEST COMMON DIVISOR. 38 
 
 24. What is the quotient of {x^ + y^) -f- (a^^V + ^tV) ? 
 
 25. Write the following quotients : (a^ + b^) -^ (a^ + ^2) ; 
 (a;"" - ;2'''*) -^ (a; - z) ; {x"" - z'"") -f- (x + 2;) ; {x'"'+' + ^""+') 
 -f- (:r + 2), m being a positive integer. 
 
 1 100 
 
 26. Factor x^ + ax -{- x + ay 1 — a, 1 + a, -[^ — —[-^ and 
 
 x^ -X- 9900. 
 
 ^lU yj 
 
 27. Factor 10rJ^+ |^] - 20«, 4a; + 4a;^ + 1 and Sda'^ - 5b\ 
 
 28. x^ -x^ -2x + 2, 6a;3 - 7«a;2 - 20a^Xy x'"^ + 31af* — 32. 
 
 SECTION IL 
 GREATEST OR HIGHEST COMMON DIVISOR. 
 
 123, Def. — It is scarcely proper to apply the term Greatest Common Divisor 
 to literal quantities, for the values of the letters not being fixed, or specific, 
 (jreat or small cannot be affirmed of them. Thus, whether « ' is greater than a, 
 depends upon whether a is greater or less than 1, to say nothing of its character 
 as positive or negative. So, also, we cannot with propriety call a^ — y ' greater 
 than a — y. If a = i, and y =: \, a^ — y ' = ^4, and a —y = \ -^ .-.in this case 
 ^ ' — y ' < a — y. Again, if a and y are both greater than 1, but a <y,a^ — y* 
 though numerkaUy greater than a—y is absolutely less, since it is a greater 
 negative. 
 
 Instead of speaking of G, C. D. in case of literal quantities, wc should speak 
 of the Highest Common Dicisor, since what is meant is the divisor which is of 
 the highest degree with reference to the letter of arrangement. 
 
 [Note. — The general rule for finding the Greatest or Highest Common 
 Divisor is founded upon the four following lemmas.] 
 
 Jl^4, Lemm.a. 1. — The Greatest or Highest G. J), of two or more 
 numbers is the product of their common prime fa^ctors. 
 
 Dem. — Since a factor and a divisor are the same thing, all the common fac- 
 tors are all the common divisors. And, since the product of any number of fac- 
 tors of a number is a divisor of that number, the product of all the common prime 
 factors of two or more numbers is a common diiisor of those numbers. More- 
 over, this product is the Oreatest or Highest C. D , since no other factor can be in- 
 troduced into it without preventing its measuring (dividing), at least, one of tho 
 given numbers. Q. E. D. 3 
 
84 LITERAL ARITHMETIC. 
 
 Examples. 
 
 1. Whai is the G. C. D. of 72, 84, and 180 ? 
 
 Solution. — Resolve the numbers into their prime factors, and take the pro- 
 duct of those which are common to all. 
 
 2. Find the G. 0. D. of 48, 204, and 228. 
 
 3. Find the G. C. D. of 81, 123, and 315. 
 
 4. Find the Highest C. D. of %x^yz^ and Ihx^y. 
 
 Solution. — Here we see that x, x, and y are all the literal factors com- 
 mon to both ; and since 8 and 15 have no common factor, x x. x x y is the 
 Highest C. D. 
 
 5. Find the H. C. D. of UkH^m^ and ^()kHhn^n*. 
 
 6. Find the H. C. D. of SaHc, 18aH^, and 2(jaHhm, 
 
 7. Find the H. C. D. of Hx^i/'^z^ and 4xy-^zr- \ 
 
 8. Find the H. C. D. of ba^x^i/ — lOax^y + bax^y and Za'^x^y 
 
 - ^x^yK 
 
 9. Find the H. C. D. of a:^ - a: - 12 and x^ - x^ - ^x + 9. 
 
 Solution.— a;« - a? - 12 = (« - 4) (a? + 3) {121). x^-x^-^x + ^ = x^{x - 1) 
 
 - 9(a; - 1) = («* - 9) (« - 1) = (« - 3) {x + 3) (a: - 1). Now we see that a; +3 is 
 a common divisor of the two polynomials, and since it is the only divisor com- 
 mon to both, it is the H. C. D. 
 
 10. Find the H.C.D.of 4:h^x^ - Ub^x^ -f Ub^x-U^ and U^x^ 
 
 - 8b^x* - 4b*x + 8^2. 
 
 J2i>, ScH. — The difficulty of factoring renders this process impracticable 
 in many cases. There is a more general method. But, in order to demon- 
 strate the rule, we require three additional lemmas. 
 
 120. Lemma 2.— A poly?iGmial of the form Ax" + Bx"-' 
 + Cx""'- - - - Ex -f F, which has no common factor in every 
 term., has ow divisor of its own degree except itself 
 
 Dem. — 1st. Such a polynomial cannot have one factor of the n\\\ degree — its 
 own — with reference to the letter of arrangement, and another which contains 
 the letter of arrangement, for the product of two siicli factors would be of a 
 higher (or different) degree from the given polynomial. 
 
 2d, It cannot have a factor of the n\\\ degree with reference to the letter of 
 arrangement, and another factor which does not contain that letter, for this last 
 factor would appear as a common factor in every term, which is contrary to the 
 hypothesis. Q. E. D. 
 
HIGHEST COMMON DIVISOR. 35 
 
 127 • Lemma 3. — A divisor of any number is a divisor of any 
 multiple of that number. 
 
 III. — This is an axiom. If a goes into 6, q times, it is evident that it goes 
 into n times &, or n6, n times q, or nq times. 
 
 128, Lemma 4. — A common divisor of tioo numbers is a divisor 
 of their sum. and also of their difference, 
 
 Dem. — Let a be a C. D. of m and n, going into m, p times, and into n, q times. 
 Then {m ± n) -i- a = p ± q. Q. E. D. 
 
 120. I^rob, — To find the H. C. D. of two polynomials without 
 the necessity of resolving them hito their prime factors. 
 
 RULE. — 1st. Arrangikg the polykomials with reference 
 TO the same letter, and uniting into single terms the like 
 
 POWERS OF THAT LETTER, REMOVE ANY COMMON FACTOR OR FACTORS 
 which may appear in all the TERMS OF BOTH POLYNOMIALS, RE- 
 SERVING THEM AS FACTORS OF THE H. C. D. 
 
 2d. Reject from each polynomial all other factors which 
 
 APPEAR IN EACH TERM OF EITHER. 
 
 3d. Taking the polynomials, thus reduced, divide the one 
 
 WITH the greatest EXPONENT OF THE LETTER OF ARRANGEMENT, 
 BY THE OTHER, CONTINUING THE DIVISION TILL THE EXPONENT OF 
 TH1E LETTER OF ARRANGEMENT IS LESS IN THE REMAINDER THAN IN 
 THE DIVISOR. 
 
 4th. Reject any factor which occurs in every term of this 
 
 REMAINDER, AND DIVIDE THE DIVISOR BY THE REMAINDER AS THUS 
 reduced, treating THE REMAINDER AND LAST DIVISOR AS THE 
 FORMER POLYNOMIALS WERE. CONTINUE THIS PROCESS OF REJECT- 
 ING FACTORS FROM I:ACH TERM OF THE REMAINDER, AND DIVIDING 
 THE LAST DIVISOR BY THE LAST REMAINDER TILL NOTHING RE- 
 MAINS. 
 
 If, at any TIME, A FRACTION WOULD OCCUR IN THE QUOTIENT, 
 MULTIPLY THE DIVIDEND BY ANY NUMBER WHICH WILL AVOID THE 
 FRACTION. 
 
 The LAST DIVISOR MULTIPLIED BY ALL THE FIRST RESERVED COM- 
 MON FACTORS OF THE GIVEN POLYNOMIALS, WILL BE THE H. C. D. 
 SOUGHT. 
 
3G LITERAL ARITHMETIC. 
 
 Dem. — Let A and B represent any two polynomials whose H, C. D. is 
 sought. 
 
 1st. Arranging A and B with reference to the same letter, for convenience in 
 dividing, and also to render common factors more readily discernible, if any 
 common factors appear, they can be removed and reserved as factors of the H. 
 0. D., since the H. C. D. consists of all the common factors of A and B. 
 
 2d. Having removed these common factors, call the remaining factors C and 
 D. We are now to ascertain what common factors there are in C and D, or to 
 find their H, C. D. As this H. C. D. consists of only the common factors, we can 
 reject from each of the polynomials, C and D, any factors which are not common. 
 Having done this, call the remaining factors E and F. 
 
 3d. Suppose polynomial E to be of lower degree with respect to the letter of 
 arrangement than F. (If E and F are of the same degree, it is immaterial which 
 is made the divisor in the subsequent process.) Now, as E is its own only divisor 
 of ito own degree (Lem. 2), if it divides F, it is the H, C. D. of the two. If, in 
 attempting to divide F by E to ascertain whether it is a divisor, fractions arise, 
 F can be multiplied by any number not a factor in E (and E has no monomial 
 factor), since the common factors of E and F would not be affected by the opera- 
 tion. Call such a multiple of F, if necessary, F'. Then the H. C. D. of E and 
 F', is the H. C. D. of E and F. If, now, E divides F', it is the H. C. D. of E and 
 F. Trying it, suppose it goes Q times, with a remainder, R. 
 
 4th. Any divisor of E and F' is a divisor of R, since F' — QE = R, and any 
 divisor of a number divides any multiple of that number (Lem. 3), and a divisor 
 of two numbers divides their difference. The H. C. D. divides E, hence it di- 
 vides QE, and, as it also divides F', it divides the difference between F' and QE, 
 or R. Therefore the H. C. D. of E and F', is also a divisor of E and R, and can- 
 not be of higher degree than R. 
 
 5th. We now repeat the reasoning of the 3d and 4th paragraphs concerning 
 E and F, with reference to E and R. Thus, R is by hypothesis of-lower degree 
 than E ; hence, dividing E by it, rejecting any factor not common to both, or in- 
 troducing any one into E, which may be necessary to avoid fractions, we ascer- 
 tain whether R is a divisor of E. If it is, it divides P', since F' = R -f- ^^ (Lem. 
 8, 4), and hence id the H. C. D. of E and F'. 
 
 6th. Proceeding thus, till two numbers are found, one of which divides the 
 other, the last divisor is the H. C. D. of E and F, since at every step we sliow 
 that the H. C. D. is a divisor of the two numbers compared, and the last divisor 
 is its own H. D. 
 
 7th. Finally, we have thus found all the common factors of A and B, the pro- 
 duct of which is their H. C. D. Q. e. d. 
 
 Examples. 
 
 1. Find the H. C. D. of UaH^ + U^y^ - Ibah^y + 12^2^^ -f- Uy^ 
 
 - Ibahy"-, and QaH^ - QaH^y - U^y'^ -^^ab^y^ + QaHy - 6««%« 
 
 - 2hy* 4- 2aby\ 
 
HIGHEST COMMON DIVISOR. 37 
 
 OPERATION. 
 
 12a^b^ + db'y^ - l^db'-y + X'ia'hy + 36y ' - l^aby^ - - {A). 
 
 6ffl^6' - 6a-6^y - 26-.y » + 2ah'y- + Qa'by - 6g^-6y^ - 26y* + 2<i6y^* - - {B). 
 
 4a-'& + 6y- — Ort% + 4«/'?/ + y' - oay^ (C). 
 
 ^a^h - 3a^&y - ?>y^^ + ciby'- + Sa'y - 'da'y' -y' + ay' {D). 
 
 (46 + ^y)a-' - {5by + by') a + (6^=^ + y') {E). 
 
 (35 + ^y)a' - (Sby + dy')a' + {by' + y')a- {by^ + y*) {F), 
 
 (5) W 
 4a' - 5ya + y-) da^ - 3^^^ + y'a - y=* 
 ^ 
 
 (/)---- 12(1' - 12y^^ + 4y'a - 4yX3a 
 
 (70 - - - - 12fl^' - Wya' + 3y'a 
 
 {L) ------ - 'dya- + yVt - 4y* 
 
 £ 
 
 (Jf) 12ya^ + 4y'a-Wy^{Sy 
 
 (iV) ------ - 12yf<^ - 15y'fl^ -H by' 
 
 (0) - - Reject 19y' - - - 19y=^a - 19y ' (7 
 
 (P) - - a — y)4a'—5ya + y^{4a-'i/, 
 
 4yg — 4yflg 
 
 — ya + y' 
 
 :. The H. C. D. of (A) and (B) is (6) (b 4-y) {a - y) = ab' + aby - b^y - by^. 
 
 ScH. — It often occurs that one or more of the above steps are not required, 
 especially the removing of a compound factor from the given iwlynomials. 
 
 2. Find the H. C. D., with respect to ar, of x* - %x^ + 2\x^ - 20» 
 -h 4, and 2x^ - \%x^ + 21a; - 10. 
 
 OPERATIOK. 
 
 2ic=» - 12«* -I- 21ar - 10)aj* - Sr' + 21a?« - 20tJ + 4 
 
 (C) - 2a!-' - lftc=* + 42.1"' - 40a; -I- 8(« 
 
 2a?'' - 12.g=^ 4- 2U'* - lOx 
 
 - 4»'» -h 21^;'' - 3ac -f- 8 
 
 - 4g^ -f 24g^ - 42a; + 20 
 (D) Reject - 3 - ac'* + 12aj - 12 
 
 (^) 
 
 cpi I'P A. A. 
 
 
 x' - 
 
 Ax + 4)2.c^ - nx' + 2\x - 10(2a; - 4 
 
 2^3 _ 8,^2 + 8.C 
 
 
 - 4a;''^ + 13a; - 10 
 
 
 - 4a;2 + 16a; - 16 
 
 
 - Reject - 3 - - 3a; + 6 {E) 
 x-2)x' -4x + i(x-2 
 
 x^ -2x 
 
 -2a; + 4 
 
 
 -2a; 4- 4 
 
 Hence a; - 2 is the H, C. D. 
 
38 LITERAL ARITHMETIC. 
 
 3. Find the H. C. D. of 2x^ + 5 - 8.r + x^, and 42:c8 + 30 - nx. 
 
 4. Find the H. C. D. of 2ax^ + 2a -{- 4a«, and 7b + 14^>.r + ^bx^ 
 + UbxK 
 
 5. Find the H. C. D. of Ga^ + Hax - dx^, and 6a^ + llax + 3x^. 
 
 6. Find the H. C. D. of 4a3 _ 4«2 - ab* + b\ and 4^2 + 2ab 
 
 7. Find the H. C. D. of 12a;* - 2^x^ij + 12x*y^, and Sx^t/^ 
 
 - 24a;2^3 ^ 24a;y* - 8y«. 
 
 8. Find the H. C. D. of 62ax^ - 2^ax*^ - Uax* - 12a + Sax^ 
 + GOax, and Ua^b + 60a*bx* - 16a*bx^ + 2rt«Z>a;« - 74fl2^a; 
 
 - 2aHxK 
 
 ISO, I^vob, — To find the H. C. D. of three or more polynomials. 
 
 RULE. — FiN'D THE H. C. D. of any two of the given poly- 
 nomials JJY one of the foregoing methods, and then find 
 thk H. C. D. of this H. C. D. and one of the remaining poly- 
 nomials, and then again compare this last H. C. D. with 
 another of the polynomials, and find their H. C. D. Con- 
 tinue this process till all the polynomials have been 
 used. 
 
 Dem. — For brevity, call the several polynomials, A, B, C, D, etc. Let the H. C. 
 D. of A and B be represented by P, whence P contains all the factors common 
 to A and B. Finding the H. C. D. of P and C, let it be called P'. P', therefore, 
 contains all the common factors of P and C ; and as P contains all that are 
 common to A and B, P' contains all that are common to A, B, and C. In like 
 manner if P" is the H. C. D. of P' and D, it contains all the common factors of 
 A, B, C, and D, etc. Q. E. d. 
 
 Examples. 
 
 1. Find the H. C. D. of x^ + Ux + 30, 2x^ + 21a; + 54, and 9a;3 
 H- h'^x* -^x- 18. The H. C. D. is x + Q. 
 
 2. What is the H. C. D. of lOa;^ + 10:^3^2 + 20a;*y, 2x^ + 2y8, 
 and4y* + \2x^y^ + ^x^y + 12xy^ ? 
 
LOWEST COMMON MULTIPLE. 39 
 
 SECTION III. 
 
 LOWEST OR LEAST COMMON MULTIPLE. 
 
 131* Def. — In speaking of decimal numbers, the term Least Common 
 Multiple is correct, but not in speaking of literal numbers. For example, the 
 numbers (« + h)'^ and («* — h'^) are both contained in {a -f h)'- x {a — h), and in 
 any multiple of this product, as m{a + b)' (a — h). But whether 7?i{a + by- (a—h) 
 is greater or less than {a + b)'^ {a — b) depends upon whether a is greater or less 
 than b, and also whether m is greater or less than unity. In speaking of literal 
 numbers, we should say Lowest Common Multiple, meaning the multiple of low- 
 est degree with respect to some specified letter. 
 
 132, Pvoh, — To find the L. C. M, of two or more numbers. 
 
 RULE. — Take the literal number of the highest degree, 
 or the largest decimal number, and multiply it by all the 
 factors found in the next lower which are not in it. 
 Again, multiply this product by all the factors found in 
 
 THE NEXT LOWER NUMBER AND NOT IN IT, AND SO CONTINUE 
 TILL ALL THE NUMBERS ARE USED. ThE PRODUCT THUS OBTAINED 
 IS THE L. C. M. 
 
 Dem. — Let A, B, C, D, etc., represent any numbers arranged in the order of 
 their degrees, or values. Now, as A is its own L. M., the L. C. M, of all the 
 numbers must contain it as a factor. But, in order to contain B, the L. C. M. 
 must contain all the factors of B. Hence, if there are any factors in B which are 
 not found in A, these must be introduced. So, also, if C contains factors not 
 found in A and B, they must be introduced, in order that the product may con- 
 tain C, etc., etc. Now it is evident that the product so obtained, is the L. C. M. 
 of the several numbers, since it contains all the factors of any one of them, and 
 hence can be divided by any one of them, and if any factor were removed it 
 would cease to be a multiple of some one or more of the numbers. Q. E. d. 
 
 1. Find the L. C. M. of {x^ - 1), (x'^ - 1), and (x + 1). 
 
 Solution. — The L. C. M. must contain a;' — 1, and as it is its own L. M., if it 
 contains all the factors of the other two, it is the required L. C'. M. The factors 
 of a; ' — 1 are (x — l){x^ + x + 1). But this product does not contain the factors 
 of («' — 1). which are {x + 1) {x — 1). Hence, we must introduce the factor 
 {x + 1), giving (a;' — 1) (a; -I- 1), as the L. C. M. of .tr ' — 1 and x- — 1. Now, as 
 this product contains the third quantity, it is the L, C. M. of the three, 
 
 2. Find the L. CM. of (^+ by,a^ -b^, {a- b)^,3Lnda^ + 3a'-b 
 + dab^ -f ^. 
 
^ LITERAL ARITHMETIC. 
 
 3. Find the L. C. M. of (x^ - 4), (x^ + 2), and (x^ - 2). 
 
 4. Find the L. C. M. of («* - 2a^ + 1), (1 + a), (a - 1), and 4. 
 
 5. Find the L. C. M. oVda^b^xy, 57ax^, 87y3, and 9a«6i 
 
 G. Find the L. C. M. of (1 - 18a + 81^2), (3«2 + 1) (1 - SVa), 
 
 and (27r7l-9rt - 3a/« + 1). 
 
 Sen. — In applying this rule, if the common factors of the two numbers are 
 not readily discerned, apply the method of finding the H. C. D., in order to 
 discover them. 
 
 7. Find the L. C. M. of x^-2ax^ + 4a«a; - Sa^, x^ + 2ax^+ ia^x 
 + 8a 3, and x- — 4a «. 
 
 Solution. — The L. C. M. of these numbers must contain a;' — 2ax* + ^a^x 
 
 — 8«^ ; and as it is its own L. M., if it contains all the factors of x^ + 2ax^ 
 + Aa^x + 8rt ', it is the L. C. M. of thcsr two iK)lynomials. But as the common 
 factors of these numbers, if they have any, are not readily discerned, we apply 
 the nu'thod of II. V. D., and find that x^ + 4a* is the H. C. D. of the two. Since, 
 then,.T' — '2 > ' - 4// ' r — 8a' contains the factor x^ + 4a* of the second number, 
 it is only lu ( r-<aiy to introduce the other factor in order to have the L. C. M. of 
 the two. ^'ow, (.c ' + 2rt.i-' + 4rt«.c + 8a') -h {x^ + 4a*)=.r + 2a. Hence, (.c='—2a.T* 
 + Aa^x— 8a')(.r + 2a) or .r* — 16a^ is theL. C. M. of the first two numbers, 
 since it contains all the factors of each, and no more. Now, to find whether 
 X* — 16a ^ is a multiple of the remaining number, x^ — 4a', or, if it is not, what 
 factors must be introduced to make it so, we proceed in the same way as with 
 the first two numbers. But our first step (or 117) shows us that x* —16a'' is a 
 multiple of x^ — 4a'. .'. x* — 16a* is the L. C. M. of the three given numbers. 
 
 8. Find the L. C. M. of x^ - 3x - 70 and x^ - d9x + 70. 
 
 9. Find the L. C. M. o^ x^^ x - 2, x^-- x - 6, and x^ - 4x + 3. 
 
 10. Find the L. C. M. of a^- AaH + 9a^2_ io63 and a^-\- 2aH 
 -3a^>2-h 20^3. 
 
 11. Find the L. C. M. of x^- ^x^ + 2^x - 24, x^- Wx^ + 3lx 
 
 — 30, and x^ - Ux^ + SSx - 40. 
 
 12. Find the L. CM. of a:*-10a;2 + 9, rr* +10a;3 +20.T»-10a;~21, 
 and z* + 4a;3 - 22a;2 — 4a; + 21. 
 
FRACnONB. 41 
 
 OHAPTEE IIL 
 jPJB^ cti on s. 
 
 DEFINITIONS AND FUNDAMENTAL PRINCIPLES. 
 
 133. A Fraction, in the literal notation, is to be considered 
 as an indicated operation in Division. 
 
 134. ScH. — In the literal notation it becomes impracticable to consider 
 the denominator as indicating the number of equal parts into which unity is 
 divided, and the numerator as indicating the number of those parts repre- 
 sented by the fraction, since the very genius of this notation requires that 
 
 the letters be not restricted in their signification. Thus in -, it will not do 
 
 to say, b represents the number of equal parts into which unity is divided, 
 since the notation requires that whatever conception we take of these 
 quantities should be sufficiently comprehensive to include all values. 
 Hence h may be a mixed number. Now suppose ft = 4|. It is absurd to 
 speak of unity as divided into 4} equal parts. 
 
 135. Cor. 1. — Since numerator is dicidend and denominator 
 divisor, it follows from (01^ f)2, 03) that dirndincf or midtiply- 
 ing both terms of a fraction does not alter its value ; that midti- 
 plying or dividing the numerator multiplies or divides the value of 
 the fraction ; and that multiplying or dividing the denominator 
 divides or multiplies the fraction. 
 
 136. Cor. 2. — A fraction is midtiplied by its denominator by 
 simply removing it. 
 
 137. The terms Integer or Entire Number, Mixed Number, 
 Proper and Improper, are applied to literal numbers, but not with 
 strict propriety. 
 
 Whether m + n is an integer, a mixed number, or a fraction, depends upon 
 the values of m and n, which the genius of the literal notation requires to be 
 understood as perfectly general, until some restriction is imposed. 
 
 As a matter of convenience, we adopt the following definitions : 
 
42 LITERAL ARITHMETIC. 
 
 13S, A number not having the fractional /or»i is said to have 
 the Integral Form ; as m + n, 2c^d — Sa'^x + Sx^y*. 
 
 139. A polynomial having part of its terms in the fractional 
 and part in the integral form, is called a Mixed Wiunber, 
 
 140. A Proper Fraction, in the literal notation, is an ex- 
 pression wholly in the fractional form, and which cannot be expressed 
 in the integral form without negative exponents. 
 
 By calling such an expression a proper fraction, we do not assert anything 
 
 a 
 with reference to its value as compared with unity. Thus -r- is a proper frac- 
 tion, though it may be greater or less than unity. It may also be written 
 
 141. An Improper Fraction is an expression in the frac- 
 tional form, but which can be expressed in the integral or mixed 
 form without the use of negative exponents. 
 
 142. A Simple Fraction is a single fraction with both 
 terms in the integral form. 
 
 143. A Compound Fraction is two or more fractions con- 
 nected by the word of. 
 
 This term is not generally applicable in the literal notation. Thus we may 
 
 3 3 a . m , , ^ . , 
 
 write -^ of ^ with propriety, but not y of — , unless a and h are mtegral, so 
 
 that the fraction -j- may be considered as representing equal parts of unity, as ^ 
 
 does. If the word of is considered as simply an equivalent for x , the notation 
 is of course, always admissible. But it is scarcely a simple equivalent. 
 
 144. A Complex Fraction is a fraction having in one or 
 both its terms an expression of the fractional form. 
 
 145. A fraction is in its Lowest Terms when there is no com- 
 mon integral factor in both its terms. 
 
 146. I7ie Lowest Comtnon Denominator is the num- 
 ber of lowest degree, which can form the denominator of several 
 given fractions, giving fractions of the same values respectively, 
 while the numerators retain the integral form. 
 
 147. Heduction, in mathematics, is changing the form of an 
 expression without changing its value. 
 
FRACTIONS. 43 
 
 Reductions. 
 14:8. There are five principal reductions required in operating 
 with fractions, viz. : To Lowest Terms,— From Improper Fractions 
 to Integral or Mixed Forms,— Front Integral or Mixed Forms to hn- 
 iwoper Fractions, — To Forms haimig a Common Denominator,— 
 and from the Complex to the Simple Form.. 
 
 14:9 • JProb. 1. — To reduce a fraction to its lowest terms. 
 RULE. — Reject all common^ factors from both terms; or 
 
 DIVIDE both terms BY THEIR H. C. D. 
 
 Dem. — Since the numerator is the dividend and the denominator the divisor, 
 rejecting tlie same factors from each does not alter the value of the fraction 
 {fH). Having rejected all the common factors, or, what is the same thing, the 
 H. C. D. (which contains all the common factors), the fraction is in its lowest 
 terms {145). 
 
 ScH. 1. — Since the H. C. D. is the product of all the common factors 
 (109), the above process is equivalent to dividing both terms of the frac- 
 tion by their H. C. D. Whenever the common factors of the terms are not 
 readily discernible, the process for finding their H. C. D. (129) may be 
 resorted to. 
 
 ScH. 2. — The opposite process is sometimes serviceable, viz.: the intro- 
 duction of a factor into both terms of a fraction, which will give it a more 
 convenient form. It requires no special ingenuity to solve such problems, 
 since, if the factor does not readily appear, it can be found by dividing a 
 term of one fraction by the corresponding term of the other. 
 
 ISO, Prob, 2, — To reduce a fraction from an improper to an 
 integral or mixed form. 
 
 RULE.— Ferfoiui the division" indicated {133), 
 
 1S1» Cor. — Bg means of negative indices {expo7ie7its) any 
 fraction can he expressed in the integral form. 
 
 IS 2, Pvob, 3, — To reduce numbers from the integral or mixed 
 to the fractional form. 
 
 RULE. — Multiply the integral part by the given de- 
 nominator, and annexing the numerator of the frac- 
 tional part, if any, write the sum over the given de- 
 nominator. 
 
44 LITERAL ARITHMETIC. 
 
 Dem. — In the case of a number in the integral form, the process consists of 
 multiplying the given number by the given denominator and indicating the 
 division of the product by the same number, and hence is equivalent to multi- 
 plying and dividing by the same quantity, which does not change the value of 
 the number. The same is true as far as relates to the integral part of a mixed 
 form, after which the two fractional parts are to be added together. As they 
 have the same divisors, the dividends can be added upon the principle that the 
 sum of the quotients equals the quotient of the sum {94). 
 
 IS 3. Prob, 4,— To reduce fraction having differetU denomi- 
 nators to equivalent fractions having a common denominator. 
 
 RULE. — Multiply both terms of each fraction^ by the 
 
 DENOMINATORS OF ALL THE OTHER FRACTIONS. 
 
 Dem. — This gives a common denominator, because each denominator is the 
 product of all the denominators of the several fractions. The value of any one 
 of the fractions is not changed, because both numerator and denominator are 
 multiplied by the same number {135). 
 
 lo4, CoR. — To reduce fractions to equivalent ones having the 
 Lowest Common Denominator.^ find the L. C. M. of all the denomi- 
 nators for the new denominator. Then multiply both terms of each 
 fraction by the quotient of that L. C. M. divided by the denomhiator 
 of that fraction. 
 
 ISS, Pvob, S, — To reduce complex fractions to the form of 
 simple fractions. 
 
 RULE. — Multiply numerator and denominator of the com- 
 plex FRACTION BY THE PRODUCT OF ALL THE DENOMINATORS OF 
 the partial FRACTIONS FOUND IN THEM; OR, MULTIPLY BY THE 
 L. C. M. OF THE DENOMINATORS OF THE PARTIAL FRACTIONS.* 
 
 Dem. — This process removes the partial denominators, since each fraction is 
 multiplied by its own denominator, at least, and this is done by dropping the 
 denominator. It does not alter the value of the fraction, since it is multiplying 
 dividend and divisor by the same quantity. 
 
 Addition. 
 ISO, Pvob. — To add fractions. 
 
 R ULE. — Reduce them to forms having a common denomina- 
 tor, if they have not such forms, and then add the numera- 
 tors, AND write the SUM OVER THE COMMON DENOMINATOR. 
 
 * The pnpil Is snppoped to have obtained sufficient knowledge of fractions in common arith- 
 metic to perform these operations. 
 
FRACTIONS. 4^ 
 
 Dem. — The reduction of the several fractions to forms having a common denomi- 
 nator, if they have not such forms, does not alter their values {135), and hence 
 does not alter the sum. Then, when they have a common denominator (divisor), 
 the sum of the several quotients is equal to the quotient of the sum of the sev- 
 eral dividends divided by the common divisor, or denominator (f>i^). 
 
 1S7» Cor. — Expressio7is in the mixed form may either be reduced 
 to the improper form and then added, or the integral parts may be 
 added into one sum, and the fractional into another, and these restdts 
 added. 
 
 SUBTRACTIOK. 
 
 IS 8, JPvoh, — To subtract fractions. 
 
 RULE. — Reduce the fraction's to forms having a common 
 
 DENOMINATOR, IF THEY HAVE NOT SUCH FORMS, AND SUBTRACT THE 
 numerator of the subtrahend FROM THE NUMERATOR OF THE 
 MINUEND, AND PLACE THE REMAINDER OVER THE COMMON DENOMI- 
 NATOR. 
 
 Dem, — The value of the fractions not being altered by the reduction, their dif- 
 ference is not altered. After this reduction, we have the difference of two quo- 
 tients arising from dividing two numbers (the numerators) by the same divisor 
 (the common denominator). But this is the same as the quotient arising from 
 dividing the difference between the numbers by the common divisor {95). 
 
 ISO, Cor. — Mixed nmnbers may be subtracted by annexing the 
 mbtrahend with its signs changed, to the minuend, and then combining 
 the term^ as much as may be desired. The reason for the change of 
 signs is the same as in whole numbers (71)- 
 
 Multiplication. 
 
 160, I^rob. 1, — To midtiply a fraction by an integer. 
 
 E ULE.— M.\JLTITLY THE NUMERATOR Oil DIVIDE THE DENOMI- 
 NATOR. 
 
 Dem. — Since numerator is dividend and denominator divisor, and the value of 
 the fraction is the quotient, this rule is a direct consequence of {92, 93). 
 
 161, JProb, 2, — To multiply by a fraction. 
 
 RULE. — Multiply by the numerator and divide by the 
 
 DENOMINATOR.* 
 
 * It is assumed that the pupil knows how to divide a fraction by an integer, from his study 
 of arithmetic. Nevertheless the problem will b« introdaced hereafter for the purpose of famil- 
 iarizing the pupil with the literal operations. 
 
46 LITERAL ARITHMETIC. 
 
 Dem. — Let it be required to multiply m, which is either an integer or a fra<v 
 
 tion, by -. 
 
 1st. Suppose a and 6 are both integers. Multiplying m by a gives a product 
 
 6 times too large, since we were to multiply by only a 6th part of a ; hence we 
 
 am 
 divide the product, am, by 6, and have -r-. 
 
 2d. When either a or h, or both, are fractions. Let c be the factor by which 
 
 a a 
 
 numerator and denominator of r must be multiplied to make -, a simple frac- 
 tion (155). Then will ^ be a simple fraction, i. e., ac and he are each integral ; 
 and the multiplication is effected as in Case 1st, giving -r—. This reduced by 
 
 dividing both terms by c gives -rr-. Hence we see that in any case, to multiply 
 
 by a fraction, we have only to multiply the multiplicand by the numerator of 
 the multiplier, and divide this product by the denominator. It is also to be ob- 
 served that this reasoning applies equally well whether the multiplicand is inte- 
 gral or fractional. 
 
 162, Cor. — To multiply mixed numhers^ first reduce them to im" 
 proper fractions. 
 
 Division. 
 163. JProb, 1, — To divide a fraction hy an integer, 
 RULE. — Divide the numerator or multiply the denomi- 
 nator. 
 
 Dem. — Since numerator is dividend, and denominator divisor, and the value 
 of the fraction the quotient, this rule is a direct consequence of {92 f 93). 
 
 16 4 » Proh, 2, — To divide by a fractio7i. 
 
 RULE. — Divide by the numerator and multiply the quo- 
 tient BY THE DENOMINATOR. Or, WHAT IS THE SAME THING, 
 invert THE TERMS OF THE DIVISOR AND PROCEED AS IN MULTIPLI- 
 CATION. 
 
 Dem. — The correctness of the first process appears from the fact that division 
 is the reverse of multiplication, and, hence, as we multiply by the numerator 
 and divide by the denominator in order to multiply by a fraction, to divide by 
 one we must divide by the numerator and multiply by the denominator. 
 
 The process of inverting the divisor and then multiplying by it is seen to be 
 the same as the other, since this multiplies the dividend by the denominator of 
 the divisor and divides by the numerator. 
 
 Again, this process may be demonstrated thus: Inverting the divisor shows 
 
FRACTIONS. 47 
 
 liow many times it is contained in 1. Then if the given divisor is contained so 
 many times in l,it will be contained in 5, 5 times as many times ; in f , | as many 
 times ; in cu^ , ax^ times as many times ; or in any dividend as many times the 
 number of times it is contained in 1, as is expressed by that dividend, whether 
 it be integral, fractional, or mixed. (The author prefers this demonstration.) 
 
 Sen. 1. — Since to multiply one fraction by another we may multiply the 
 numerators together for the numerator and the denominators for the denomi- 
 nator, and since division is the reverse, we may perform division by dividing 
 the numerator of the dividend by the numerator of the divisor, and the de- 
 nominator of the dividend by the denominator of tlie divisor. 
 
 This method will coincide with the others when they are worked by per- 
 forming the operations by division as far as practicable, and this is worked 
 by performing the multiplications equivalent to the divisions when the latter 
 are not practicable. 
 
 16S, Cor. — The reciprocal of a quantify being 1 divided by that 
 quantity, the reciprocal of a fraction is the fraction inverted. 
 
 General Scholium. — In both multiplication and division of fractions, or by 
 fractions, all operations which can be performed by dividing should be so per- 
 formed, in order that the result may be in its lowest terms. 
 
 Signs of a Fraction. 
 
 1G6» In considering the signs of a fraction, we have to notice 
 three things, viz.: the sign of the numerator, the sign of the denomi- 
 nator, and the sign before the fraction as a whole. This latter sign 
 does not belong to either the numerator or denominator separately, 
 but to the whole expression. 
 
 Thus, in the expression — -—, in the numerator the sign of 4a is +, 
 
 and of tied is — . In the denominator, the sign of 2x is +,and of A^y^ 4- also, 
 'i he sign of the fraction is — . These are the signs of operation (50). 
 
 107. The essential character of a fraction, viS positive or 
 ncyativey can be determined only when the essential character of all 
 th(^ numbers entering into it is known. It may then be determined 
 by principles already given (78^ 97). 
 
48 literal arithmetic. 
 
 Examples. 
 1. Reduce the following fractions to their lowest terms : 
 
 UlOaH^i/^' bba'^x' 2^^^i^.„ ' 1 -x^' 12m^u^ + 2inm^ + 1271^' 
 
 a;~* - y-* a^b - 1 tg* — 3rg — 4 Sa + 'Saff 3«« + 12a; 4- 9 
 .r-i — y-i 1 + aV^' x^ — 4x - 5' 4^t__4^t^2' .r^ + 5x^ + 6 ' 
 
 63:8-3:?:-45 (l+a:)^ n''-''+'//-'c^ 2x^y^ + 2 ^x^ + 2x^ -^x 
 ^x*-\-Vdx^l{fJ\^¥f' „_p_„,,+,.|' 0:6^6-1' 9^-3 _i;^.c 'J _3o.c + 48' 
 
 2x*'-x^- 0.r2+ 133:-5 2^^^3 4.^^2_ 8^^+5r< a;^- 8:^^ +21a:-18 
 lx-^-\\yx-i-\-i^x-b ' 7^3_i2<^2+5^ ' 3a;3- 16.^2+ 21a; ' 
 16a;* — 53a-2 + 45a; + 6 
 
 8a;* - 30a;3 + Sla:^ _ 12 
 
 2. What factor will change 7 to ^^-r z-~- ? 
 
 ® a — b a^ — b^ 
 
 fl* + a^x-\- a*x* + fla;3 ■}■ ^-4 n^' - x^ .^ x^ + j^s + jg; 4- 1 
 
 a + x a^ - x^ ' i^' - i 
 
 4a;g -a- + 4 ^ 6j3* - 12;x/3 - 0/^3^ _^ 127* ^^ Gf;;^ _ 2y3] ^ 
 a; — 1 * 2)^ — q^ P + Q 
 
 ^ ^ , 1 a;* 4- 14a:* + 27 J^i* 4. ,^2 4. 2771?^ — x — y 
 
 3. Reduce , z -. , : -, 
 
 1—x a;2 + 4 m + n 
 
 1 _ ^1 2 a* + 4^3,^ 4_ Qa^x^ + 4rta;3 + a;* a;" 
 
 a + 1 ' 2 -a;' a^ _,. 3^^2^ 4. 3«^2 4. ^3 ' i_^- 
 
 and 
 
 -^-r: to inteorral or mixed forms. 
 
 ^' ^^P^'^'^ 7^^«' (^ + m)-»" ¥W^^ "^'^ a;-»(«-6)3 
 in the integral form. 
 
 X 
 
 5. Reduce the following mixed to fractional forms : 1 + :j , 
 
 14a26 + 12a2^,2 - ^ab ^ ^ 4a; - 4 
 
 — , 1 + 2a; -- 
 
 2ab ' ox 
 
 1 + 7a + 6«^» - -^— — ^; , 1 + 2a; - . , 
 
 ^ 3a86 4- 2ab^ - h^ _, b^ ^ c^ - a^ , , 1 - a; 
 
 "-^ + .2 - b^ — ' 1 + — Wc — '^^^"-^-rr^- 
 
FBACTIONS. 49 
 
 ^ „ , 3a 4:X ay 7y , 4^^ , 
 
 6. Reduce — , ;;;^, -^, ~, and - — - to forms having a 0. D. 
 
 7. Reduce , --, and -r ^ to forms having a C. D. 
 
 a — X a + X- a — x , 1 , - ^ , 
 
 8. Reduce , —r-^ rr, , and to equivalent 
 
 X x(a^ — x^y a i- X a — x ^ 
 
 fractions having the L. C. D. 
 
 X X 1'" 
 
 9. Reduce , ^ ~, and ; — - — - to equivalent frae- 
 
 tions having the L. C. D. 
 
 1 _|_ 2;3 (1 — x)^ 
 
 10. Reduce j—- — — and ^- ^ to forms having the L. C. D. 
 
 (I + x)^ 1 —x^ *^ 
 
 11. Reduce — ^, — r, — -— — -, and — r- to forms having the 
 
 L. C. D. 
 
 12. Reduce — , - — — -., and r—^ — to forms having the L. 0. D. 
 
 6x da; -t- 4 yrc^ — 16 
 
 13. Reduce the following complex fractious to simple ones : 
 
 a . fflg - ja^ 3 ^li^ /^ ~ "^ . 
 
 be ' ib^y^ + -2\c^x^ ' 3 ^ ' ^ ~7 ' u-^ -ni'^ ' 
 
 c 
 
 b -^ J e 
 
 ^ , . ^^ a a a ^ a a;— 7 ,7 — 0; 1 , 1 
 
 14. Add -, -, -, and - ; — ^- and -y-; --^^ and ^-j-- ; 
 
 a + h , a-b-%^ 2 , 3 
 
 : — and ; -^- — and 
 
 2 2 ' x^ + x^ +x-^\ x^ - x^ + x—i' 
 
 a — bc — a jJ~c 1 x + 1 ^x^ + x -{-l 1 
 
 1 , 2flj 
 
 r, and - — ^7^. 4 
 
 a + *' a^ - ^>8 
 
50 LITERAL AiarUMETIC. 
 
 lo. Add -5 ■ — J, — — -, and ^— — -. ; « — ( -7- + 4a*a;^ ) 
 
 2i^ ^ A J J 
 
 and Z> + — + 4a^a;^ 
 c 
 
 16. Add -, — -, and — t 7 ; 1 r and 
 
 a b~V b -\- I' I + X + X* 1 - X + x^ ' 
 
 .T + Sa:— 4 ^ X + o 
 — my Tyy and — ■— i-. 
 
 17. Add 7 r— , jr and 7 — ; — r-7 — . — ^v ; ^ — ^ — and 
 
 y - emp y* . x _y_ ^^^^ ^' 
 (3my« — a;)« ' y' a: 4- y' a;* + a;y' 
 
 18. Add -1^, ' _^-,andi^:^*i^4,i*-£^-±i£^^. 
 a — — cc — a {a — 0} {0 — c) [c — a) 
 
 19. From ^ take -^; from i^^ take i^i^ ; 
 a; — 3 a; + 3 ao ab 
 
 from take 
 
 x-7 a; -3* 
 
 20. From 7a; jj take x — ; from — 77^ take 
 
 3 2 ab{a — by 
 
 ff ^ c, 
 b a 
 
 21. From -n -r-^ — zttt-, tt take 
 
 2(a: + l) 10 (a:-!) 5 (2a: + 3)* 
 
 22. Multiply--- by --- ; -— -^ by -^-^ ; ^^^ 
 4a:2 - 12a: - 40 
 
 3a-2 - 18a: + 15 
 
F1UCTI0N8. 51 
 
 23. Multiply ^^— by 4c; - ^x't by Jo; 2; - ||l' by 
 
 10^-2 1 + X . ^ o . 2 «^ I. <^'"" ^'^"^ t. 2/'^" 
 
 a ^ "^ 
 
 ^'"t 2a:^ + 1 x^ 1 a"" -\- b'' 2c'* 
 
 24. Multiply -J by ; „_,, — by the 
 
 y^ ^yi^yt^l yi _ I « ^ 
 
 (^2._.y2)2 ^2-1 gS - 1 
 
 by ^^2 _ y'zyz + (^2 + ^2^2 ' (a + l)2 by ^^g _ ^y 
 
 25. Multiply together -^, — ;^|^, and 1 + ^^. 
 
 26. Multiply a:^ — a; + 1 by a;-* + a;-i + 1 ; 1 ~-t by 
 
 2 + 
 
 b 
 
 a + b 
 2b 
 a—b' 
 
 *' ax 
 
 27. Divide -r- by -—- : — ?— rbym^w*; ^ „ by m^n^ : -r^- 
 2 -^ 13 m^n^ ^ ' m^n^ ^ ' ^ax 
 
 7^37,T^i -8- i. i 1 
 
 28. Divide -^^^-^^-4-^ by H^'^ V ^^ J ^^ _ ,^^, by 1 + 9a» ; 
 
 — by a;-"- 
 
 29. Divide by 1 — a ; ^ by — ; ( ) by 
 
 1 + « ^ ' a +2b -^ 3a -h Qb ' \a x) ^ 
 
 (rt + x)J^ / X 1 — a: \ / a; _ 1 — a; \ /c— <^ c ^^^s x 
 
 ;^ ' vFT^ "^ "IT"; ^ \i + a; ic y ' Vc+* c^r^y 
 
 30. D,v,de ,«*-«-* by ,» + -; \^^^ by ^^^--p; a^-h^ 
 
 -cs- 
 
 24, by '^-tii ; f L+_% + f ) by {^^^ ^). 
 
 •'« + * — c'V a; + y y/ ' \ y x + yj 
 
52 LITEKAL ARITHMETIC. 
 
 31. Divide {LpiL by 
 
 1 + ^ 
 
 {^-y) 
 
 32. Divide — ^-^^ -r-ir^ ^^ "l ?• 
 
 « + * «* + J* 
 
 33. Divide ^ +.i + ^ - 3«-'^>-V-' by - + ^ + -. 
 
 rt3 ^' c' '' a c 
 
 34. Free . , -^-, , •' , !,-4 ■ ' ^ > ^^^ «^^ 
 + ^~*rt of negative exponents. 
 
 ^ ^r, , . ,, . , « 6' + ^/ /5a:3\^ ./m + n , 
 
 35. What IS the reciprocal of (^) ' '^^ _ > and 
 
 36. Is the fraction — ^-^ j-^ essentially positive, or negative, 
 
 when a, in, x, and y are each negative ? 
 
 Solution. — Since (— ay = n*, 4«' is essentially positive. Since (— m)(— a;) 
 = 7WJ*, the term Smx, in itself, is positive, and the numerator becomes 4a* 
 
 — (+ dmx), or 4a* — 3»w; (7^)- Now, whether 4a* — 3mx gives a + or a — 
 result, depends upon the numerical values of a, m, and a;. If 4a* > 2mx,4a^ 
 
 — Zmx is + ; but, if 4a' < 3inx, 4a' — Snix is — . Again, since (— a;)' = — i»^, 
 the first term of the denominator, 2x\ is essentially negative. And since 
 (— y)^ = y*, the term 4y- is essentially positive and the denominator becomes 
 
 — 2x^ 4- ( + 4y'), or — 2j;* + 4y'. Whether this is + or — , depends upon the 
 relative values of x and y. If we suppose 4a* > 3mx the numerator becomes + , 
 
 and if 2^;"' be greater than 4y* the denominator becomes — , and we have — — , 
 which gives a positive result. 
 
 37. What is the essential sign of r-^ j-, when «= —1, b=2, 
 
 ° abxy — 4 
 
 x= —3, and y = — 4 ? 
 
 1 
 
 3^^^ 3 7J"3'|/ 
 
 38. What is the essential sign of — __ . , — -y when a = — 3, 
 i = —B, 771 = —1, and y = I? 
 
FRACTIONS. 53 
 
 2a^x^ daJb 
 
 39. What is the essential sign of j j, when a= —32, 
 
 h = — 2, 771 = — S, and x = —2? 
 
 40. Simplify 
 
 X + 
 
 1 \ , 1 
 
 y{xyz + cc • 
 
 + ^)' 
 
 1 
 
 y 
 
 
 
 1 
 
 1 
 
 X a — y 
 
 + (^ 
 
 • X 
 
 J' 
 
 a — 
 
 - xY (a 
 
 - ?/)' 
 
 
 1 
 
 
 1 
 
 
 (a 
 
 3 
 
 »;)» 
 
 («-y)M« 
 
 -X) 
 
 1 
 
 1 1 
 
 3- 
 
 a 
 
 
 and 
 
 be ca (lb 
 
 
 
 
 a + 
 
 b 
 
 
 
 
 
 -b*) 
 
 1. 
 
 
 a — 
 
 -1 
 
 
 
 
64 LITERAL AlilTHMETIC. 
 
 CHAPTER IV. 
 
 POWERS AND BOOTS. 
 
 SECT/ ON L 
 INVOLUTION. 
 
 Definitions. 
 
 l^SS. A ^ower is a product arising from multiplying a number 
 by itself. The Degree of the power is indicated by the number 
 of factors taken. 
 
 ScH. — It will be seen that a power is a species of composite number in 
 which the component factors are equal. 
 
 i(>.9. A Root is one of the equal factors into which a number is 
 conceived to be resolved. The I^egree of the root is indicated 
 by the number of required factors. 
 
 170. An Exponent or Index is a number written a little 
 to the right and above another number, and 
 
 1st. If a Positive luteffpr. it indicates a Power of the number; 
 
 2d. If a Positive Fraction, the numerator indicates a Power, and 
 the denominator a Root of the number ; 
 
 3d. If a Negative Integer or Fraction, it indicates the Reciprocal 
 of what it would signify if positive. 
 
 ScH, — It is obviously incorrect to read 4% "the f power of 4." There is 
 no such thing as a 2-fifths power, as w^ill be seen by considering the defini- 
 
 m ni 
 
 2. - m — - 
 
 tion of a power. Read 4% "4 exponent | ; " also a" , "« exponent ^ ; " a " , 
 
 "« exponent — ^." These are abbreviated forms for, "« with an exponent 
 
 — ^, " etc. In this way any exponent, however comphcated, is read witliout 
 difficuay. 
 
POWERS AND ROOTS. 55 
 
 17 !• A Radical dumber is an indicated root of a number. 
 If the root can be extracted exactly, the quantity is cnlled Rational ; 
 if the root cannot be extracted exactly, the expression is called Irra- 
 tional, or Surd. 
 
 172, A Root is indicated either by the denominator of a frac- 
 tional exponent, or by the JRadlcal Sigti, V. This sign used 
 alone signifies square root. Any other root is indicated by writing 
 its index in the opening of the v part of the sign. 
 
 173, An Imaginary Quantity is an indicated even root 
 of a negative quantity, and is so called because no number, in the 
 ordinary sense, can be found, which, taken an even number of times 
 as a factor, produces a negative quantity. 
 
 Thus V — 4 is imaginary, because we cannot find any factor, in the ordinary 
 sense, which multiplied by itself once produces — 4. Neither + 2 nor — 2 pro- 
 duces — 4 when squared. For a like reason V — da"^, V — 5x,ot >y/— 14^xy* 
 are imaginaries. 
 
 174, All quantities not imaginary are called Real 
 
 17 5, Similar Radicals are like roots of like quantities. 
 
 Thus A>/5a, ^y/5a, and (a* — x^)V5a are similar radicals. 
 
 176, To nationalize an expression is to free it from radicals. 
 
 177, To affect a number with an Exponent is to per- 
 form upon it the operations indicated by that exponent. 
 
 178, Involution is the process of raising numbers to required 
 powers. 
 
 179, Evolution is the process of extracting roots of numbers. 
 
 180, Calculus of Radicals treats of the processes of re- 
 ducing, adding, subtracting, or performing any of the common 
 arithmetical operations upon radical quantities. 
 
 Involution. 
 
 181, I^rob, 1, — To raise a number to any required power, 
 RULE. — Multiply the number by itself as many times, less 
 
 ONE, AS THERE ARE UNITS IN THE DEGREE OF THE POWER. 
 
 182. Cor. — Since any nmnher of positive factors gives a positive 
 product, all powers of positive monomials are positive. Again, 
 
66 LITERAL AKITHMETIC. 
 
 since an even number of negative factors gives a positive product^ 
 and an odd number gives a negative product, it follows that even 
 powers of negative numbers are positive^ and odd powers negative. 
 
 183, JProb. 2, — To affect a monomial with any exponent. 
 RULE. — Perform upon the coefficient the operations 
 
 INDICATED BY THE EXPONENT, AND MULTIPLY THE EXPONENTS OF 
 THE LETTERS BY THE GIVEN EXPONENT. 
 
 Dem. — iBt. When the exponent by which the monomial is to be affected is a positive 
 
 n 
 
 integer. Let it be required to affect ia^b" x- • with the exponent p; or in other 
 words raise it to the pth power, p being a positive integer. The pi}\ power of 
 * " " " 
 
 ^"'b'^ x-^ is ia^b"^ x-" x 4a"'b' x-* x ^"^b"^ x-" to p factors. But as 
 
 the order of the arrangement of the factors does not affect the product (77), 
 this product may be considered as, p factors each 4, into p factors each a**, into p 
 
 factors each b% into p factors each x-*. Now p factors each 4 give 4'' by definition. 
 
 p factors each a"" are expressed a^'", since a* is m factors each a, and p factors con 
 
 taining m factors each, make in the whole pm factors, or a^'"». Again, p factors 
 
 £? * 1 
 
 each & ■■ are expressed b ^ , since 6 ' is n factors each b " , and p factors, containing n 
 
 - — 1 11 
 
 factors each, are pn factors each 6 ' , or 6 »■ . And since xr*— — , p factors, or — x — 
 
 Xf iff X* 
 
 X _ . . . to p factors make — , as fractions are multiplied by multiplying 
 numerators together for a new numerator and denominators for a new denomi- 
 nator, and J?' X ar* X «»- - - to » factors are xf"". But — = x-p". Hence collect- 
 
 n P* 
 
 ing the factors we find that (4a"'yx~'')P = 4''a^"*&'^ x-"'. q. E. D. 
 
 2d. When the exponent is a positive fraction. Let it be required to affect 
 
 4a'^b'^ x-', with the exponent — . This means that Aa'^b'^x-* is to be resolved 
 into q equal factors and p of them taken. Now, if we separate each of the fac- 
 
 n 
 
 tors of 4a"*6 ^ x-' into q equal factors, and then take p of each of these, we shall 
 
 have done what is signified bv the exponent ~. 
 
 <? 
 1 . 
 
 By definition, 4 ' represents one of the q equal factors of 4. 
 
 To obtain one of the q equal factors of a"s we take one of the q equal factors 
 
POWERS AND ROOTS. 57 
 
 of a from each of the m factors represented. But one of the q equal factors of 
 1 »' 
 
 a is represented hy a'' , and m of these is a' by definition. 
 
 n n 
 
 To separate }/ into q equal factors, we notice that l{ is ?«, of the r equal fac- 
 tors of h. Now, if we resolve each of these r factors into q equal factors, h is 
 resolved into rq equal factors ; doing the same with each of the n factors repre- 
 sented, and taking one from each set, we have h resolved into rq equal factors 
 
 and n of them taken ; that is 6*^* is one of the q equal factors oth^ . 
 
 1 
 To resolve «-»= — into q equal factors, we consider that a fraction is 
 
 resolved by resolving its numerator and denominator separately. But one of the 
 
 q equal factors of 1 is 1 ; and one of the q equal factors of -x^ is x^ as seen in the re- 
 
 1 1 -- 
 solution of a"*. Hence one of the q equal factors of x-i or jL is ~7 •= -j "^ . 
 
 n 
 
 Collecting these factors we find that one of the q equal factors of 4n!'»&'.c-'' is 
 
 1 m n _ ♦ 
 
 4'3'^'j'5<}r^ «. And finally j? of these being obtained according to Case 1st, gives 
 
 p pm Tt* p» n 
 
 41 ^7 ^TT^ « ^ ag the expression for 4a"'6'^ a;-* affected with the exponent ?; which 
 
 g 
 
 result agrees with the enunciation of the rule. 
 
 3d. When, tJie exponent is negative and either integral or fractional. Let 
 
 n 
 
 it be required to affect 4«'"6'"i;-« with the exponent —t. This by the definition 
 of negative exponents, signifies that we are to take the reciprocal of what the 
 
 J* 
 expression would be if t were positive. But Aa"'h^x-* affected with the exponent 
 
 t (positive) is 4'a'"'6'' «-'■"' by the preceding cases, whether t is integral or frac- 
 tional. The reciprocal of this is . But since these factors can be 
 
 A'a'"'hTx-*» 
 transferred to the numerator by changing the signs of their exponents, we have 
 
 '* * 
 
 4-'a-'"'6 "^'j^", as the result of affecting 4a"'b''x-' with the exponent —t, which 
 
 result agrees with the enunciation of the rule. 
 
 184, I^rob* 3» — To expand a binomial affected with any expo- 
 ne?it. 
 
 liULB.—TuiS RULE IS BEST STATED IN A FORMULA. ThUS, 
 LET rt, bf AND m BE ANY NUMBERS WHATEVER, POSITIVE OK 
 
68 LITERAL ARITHMETIC. 
 
 NEGATIVE, INTEGRAL OR FRACTIONAL, THEN WILL (flf-f *)"* REPRE- 
 SENT ANY BINOMIAL, AFFECTED WITH ANY EXPONENT, AND 
 
 (a + by = a*" + 7na'^-^ + ^ ^f^ ~ ^^ rt'»-2^>2 
 
 1 . /i 
 
 m {m - 1) { m - 2) 
 + 1 • 2 • 3"''' ^ 
 
 m(m-l)(m-2)(m-3) .._ 
 + 1 • 2 • 3 • 4 ^ ^ 
 
 m (m - 1) (m - 2) (m - 3) (;// - 4) 
 + 1 • 2 • 3 • 4 • 5 « ^, + etc. 
 
 This is the celebrated Binomial Formula, or Theorem. Its demonstra- 
 tion will be found in the subsequent part of the work. At this stage of his 
 progress the student should learn the formula and become expert in applying it. 
 
 18o, CoR. 1. — T/te exjXf9isio9i of a binomial terminates only when 
 the ex'ponent is a positive integer^ since only when m is a positive 
 integer will a factor of the form in(m — 1) (m — 2) (m — 3), etc., 
 become 0, as is evident by inspection. 
 
 186, Cor. 2. — lV7ie7i m is a positive integer, that is when a bino- 
 mial is raised to any power, there is one more term in the develop- 
 ment than xinits in the exponent. 
 
 Since the first coefficient is 1; the 2d, m\ the 3d, ^ ~ ; the 4th, 
 
 mim, —!)(»» — 2) ^, ^,, w(w — l)(m — 2) (w — 3) . ^. ^. ^ ^- 
 
 ^ — ^-^— ■' ; the 5th, — '— - , / . ; etc., we notice that the 
 
 3 * O i ' O ' '*■ 
 
 last factor is m — (the number of the term — 2) ; and the number of the term, 
 therefore, which has m — m as a factor is the (w + 2)th term. But this is 0. 
 Hence the {m + l)th term is the last. 
 
 187, Cor. 3. — When m is a2)ositive integer, the coefficients equally 
 distant from the extremes are equal. 
 
 Thus {a + ft)* = (6 + a)"*; the former of which gives a"' + ma'"-^b + 
 ffl(m - 1) ^^_2^8 _j_^ g^p^ ^^^ ^j^g j^^^^,j. j« ^ mb'"-'a + ^^^~ ^V "-V +, etc. 
 
 Whence it appears that the first half of the terms and the last half are exactly- 
 symmetrical. 
 
 188, Cor. 4:. — The sum of the exponents i7i each term is the same 
 as the expo7ient of the power. 
 
 Sen. — The last two corollaries- apply to the form {x + yy\ and not to such 
 forms as (2a' — 35-)'", after the latter is fully expanded. 
 
POWERS AND ROOTS. ^ 
 
 1S9, Cor. 5. — A convenient rule /or writing out the poioers of 
 binomials may be thus stated: 
 
 \st. There is one more term in the development than there are 
 units in the exponent of the power. 
 
 2d The FIRST contains only the first letter of the binomial^ and the 
 last term only the second^ while all the other terms contaiji both the 
 letters. 
 
 3d. T/ie exponent of the first letter of the binomial in the first term 
 of the development is the same as the exponent of the required power 
 and DIMINISHES by unity to the right, xchile the exponent of the 
 second letter begins at unity in the second term of the ex^^ansio?! and 
 INCREASES by unity to the right, becouiing, in the last term, the same 
 as the exponent of the power. 
 
 4tth. The coefficient of the first term of the expansio7i is unity ; of 
 the second, the exponent of the required poyner ; and that of any other 
 term may be found by mxdtiplying the coefficient of the preceding 
 term by the exponent of the first letter in that term, and dividing the 
 product by the exponent of the second letter + 1. 
 
 190, Cor. 6. — If the sign betweoi the terms of the binomial is 
 jninus, as (a — b)°*, the odd terms of the expansion are + and the 
 eve7i ones — . This arises from the fact that the odd terms involve 
 even powers of the second or negative term of the bhiomial, and the 
 even terms invclve the odd powers of the same. 
 
 Examples. 
 
 1. What is the square of 3^3 ? Of -2fAc ? Of ^x~^^ Of -^a^x ? 
 Oi^^/x'i OfiV2? Of-^? 
 
 2. What is the square of 1 - x + x'- ? Of 2a - 3x^ ? 
 
 2 3 , 3 
 
 3. Expand the following: (3-1x-x^) , {3x^ - 1) , {x-y + z) , 
 {l~x^), {x^-yh'- 
 
 4. Aifeofc 3rAr« with the exponent 4; ^a^x^ with tlie exponent 2; 
 
 a'^x with the exponent ~m,vi'\t\\ the exponent J, f ; bx^y with the 
 exponent |, ^, — 3. 
 
00 LITERAL ARITHMETIC. 
 
 5. Perform the following operations and explain each as a process 
 
 3 8 ^ 
 
 of factoring, according to (De^h.ISS): (12ba^x^)^, (Ua^xy, 
 
 -4 3 -4 4 1-4-1^ •*_'_" 
 
 6. Expand the following by the Binomial Formula: (x -\- yY, 
 {x-yY, (3«2-.t)3, {x + y)-^, (^-y)"*, (5 + :c2)* (a:^-^^)'^ 
 
 77(free results. 
 
 \/«* — a^e^= rtVl — «- = ^(1 —U^ — ,r--;g* — »- . ^ e«— etc.) 
 
 \ » 2*4 2 '4 '6 ' 
 
 (1 — .T«)'^ = 1 + ^a:2 +lx*' + A^« + i%a:8 ^^ etc. 
 
 (fl?« + hx^y =a + — — 4- z^—. -, etc. 
 
 7. Write out by Cou's. 5 and 6, the expansions of the follow- 
 ing: (a + l)y,{a-by,{a^'-b'^)\ (x^ -y^Y,(a^ ^y^)\ {x'^ -y^y. 
 
 SECTION II. 
 
 EVOLUTION. 
 
 191, I^rob, 1. — To extract any root of a perfect power of that 
 degree. 
 
 RULE. — Kesolve the number into its prime factors, and 
 
 SEPARATE THESE INTO AS MANY EQUAL GROUPS AS THERE ARE 
 UNITS IN THE DEGREE OF THE ROOT REQUIRED; THE PRODUCT OF 
 ONE OF THESE GROUPS IS THE ROOT SOUGHT. 
 
 192, Sen. — The sign of an even root of a positive number is ambiguous 
 (that is 4- or — ), since an even number of factors gives the same product 
 whether they are positive or negative (79 f 80). The sign of an odd root is 
 the same as that of the number itself, since an odd number of j^ositive factors 
 gives a positive product and an odd number of negative factors gives a 
 negative product (80, 81). 
 
 193, Cor. 1. — The roots of monomials can be extracted by 
 extracting the required root of the coefficient and dividing the expo- 
 nent of each letter by the index of tfie root^ since to extract the square 
 
POWERS AND ROOTS. ^1 
 
 root is to affect a number with the exponent \^ the cube root ^, the nth 
 root i, etc. (183). 
 
 104, Cor. 'Z. — 71ie root of the product of several numbers is the 
 same as the product of the roots. 
 
 Thus, "Vobcx = "Va ■ 'Vb Vc • V.T, since to extract the mih root of obex 
 Ave have but to divide the exponent of each letter by m, which gives, 
 
 ILil _ _ _ _ 
 
 a"'b^c^x"^, or Va ■ Vb ■ Vc ■ Wx. 
 
 IQS, Cor. 3. — The root of the quotient of two numbers is the same 
 as the quotient of the roots. 
 
 Thus, 4/' — is the same as _^, since to extract the rih. root of — we liave 
 
 but to extract the 7*th root of numerator and denominator, which operation 
 
 is performed by dividing their exponents by r. Hence a/'HI — — r ::= !^. 
 
 r n — W^ 
 
 nr >^^ 
 
 Examples. 
 
 1. Extract the square root of each of the following numbers by 
 resolving them into their factors, i. e. by (191) : 222784 ; 2131G ; 
 and 5499025. 
 
 2. Extract the square root of each of the following, as above : 
 81«*a:-'?/V% a^c^-\-2a^bc^ +a^b^c^, m^—2m*x + 7n*x^. 
 
 3. Extract as above : V^oa^b'*, V G4rr«ic% y49a;y% \^lUa^m^f 
 
 '49rt 
 
 ^, yHJn ^7/8, ^125m».Ti«, yiTZSx^y^, ^/ -'d'Za^^y-K 
 
 y 36m2/i 
 
 4. Solve exercises 2 and 3 also by (193)- 
 
 5. Show as in (194) that >v^8 x 27 = V^xv^27; also that 
 
 6/ \_ . 5/ 1_ 
 
 y a-'^b* = ya"" x -(/ i". 
 
 6. Is V«±^= V«±V^? Is i /^=-i? Is Vab=VaVb? 
 
 y ^ Vb 
 
 Why does the reasoning in the cases which are true not apply to the 
 others? State the true propositions ; also the false assumption. 
 
 ScH. — The extraction of roots by resolving numbers into their factors 
 according to ihi» rule, is limited in its application for several reasons. In 
 
62 LITERAL ARITHMETIC. 
 
 the case of decimal numbers we can always find the prime factors by trial, 
 and hence if the number is an exact power, can get its root. But in case 
 the number is not an exact power of the degree required, we have no method 
 of approximating to its exact root by this rule, as we have by the common 
 method already learned in arithmetic. In case of literal numbers the diffi- 
 culty of detecting the polynomial factors of a polynomial is usually insuper- 
 able. Hence we seek general rules which will not be subject to these 
 objections. 
 
 196 • JProb, 2, — To extract roots whose indices are composed of 
 the factors 2 and 3. 
 
 Solution. — To extract the 4th root, extract the square root of the square root. 
 Since the 4th root is one of the 4 equal factors into which a number is conceived 
 to be resolved, if we first resolve a number into 2 equal factors (that is, extract 
 the square root) and then resolve one of these factors into 2 equal factors (that 
 is, extract its square root) one of the last factors is one of the 4 equal factors 
 which compose the original number, and hence the 4th root. In like manner 
 the Gth root is the cube root of the square root, etc. 
 
 107. JProb. 3, — To extract the mth {any) root of a number. 
 
 Solution. — Instead of giving in detail the demonstrations of the processes for 
 the extraction of roots, we assume that the student is familiar with the subject 
 as presented in common arithmetic,* and propose here to show him how to see 
 a rule for the extraction of any root of a decimal number, and of a polynomial, 
 in the expansion of a binomial. Thus 
 
 For the 
 
 Square root (a + 6)* = a' + (2a + 6)6 gives the rule ; 
 
 Cube " (a + &)^'=«=' + (3a«+3a6 + 62)6 " " " 
 
 Fourth " (a + &)*=a* + (4a« + 6a«6 + 4<i6«+6')6 " " " 
 
 Fifth " (a + &)'=«'+ (5a* + lOa'ft + 10a*&« +5ofZ>'^ + 6*)6 " " " 
 etc., etc., etc. 
 
 In all cases a represents the part of the root already found, and h the next 
 figure or term of the root ; observing that in decimal numbers, a is tens with 
 reference to b. 
 
 The method of pointing off decimal numbers into periods, and the reason, 
 are shown for the square and cube root in common arithmetic ; and the same 
 reasoning extends to other roots. 
 
 A polynomial must be arranged as for division, since this is the form which, 
 a power takes when the root is similarly arranged. 
 
 The solution of a few examples will familiarize the student with this method. 
 ♦ The wh(rie subject is fully presented in the Complete School Aloebba. 
 
powers and roots. 
 
 Examples. 
 1. Extract the square root of 7284601. 
 
 SOLUTION. 
 
 The fonnula is (a -f &)* = n^ + {2a + b) b. 
 At first a^ = the greatest square in 7. .". a = 2. 
 
 I 7284601 12699 
 
 4 
 
 2f(, = 2(20) = the Trial Divisor 40 j 
 
 .•. 328 -f- 40 = 8 is the jJi'obaJde* second root figure, 
 (2ti + 6) = 40 + 8 is the True Divisor if 8 is the second root 
 figure. But 48 x 8 = 384. .-. 8 is too large. We will try 
 
 6 as the second root figure 6 
 
 Whence (2a + 6) = the True Divisor 46 
 
 328 
 
 276 
 
 J!^otc, 2a = 2 (260) = the 2Hal Divisor 520 j 5246 
 
 .'. 5246 -h 520 = the probable next root figure 9 I 
 
 {2a + 6) - 520 + 9 = the True Divisor 529 I 4761 
 
 Again, 2a = 2 (2690) = the Trial Divisor 5380 I 48501 
 
 .-. 48501 ■+■ 5380 = the probable next root figure 9 
 
 {2a + b) = 5380 + 9 = the True Divisor 5389 I 48501 
 
 2. Extract the cube root of 99252847. 
 
 SOLUTION. 
 
 The formula is {a + b)^ = a^ + (3a« + Sab + b*)b. 
 At first a' = the greatest cube in 99. .'. a — 4. 
 
 3a* = 3 (40)* = the THal Divisor 4800 
 
 .-. 35252 -4- 4800 = 7, Wxa probable next root figure. 
 (3a* + 3rt6 + &*) = 4800 + 840 + 49 = 5689, the True Divisor 
 if 7 is the next root figure. But, as this does not go 7 
 times in 35252, 7 is too large ; and we try 6. 
 Noic, the corrections to be added to the trial divisor to make 
 
 the true divisor, are 3a6 = 3 (40) 6 - 720 
 
 and b' = (6)^ - 86 
 
 992528471463 
 64 
 
 35252 
 
 Hence the true divisor is 5556 j 33336 
 
 New THal Divisor, 3a* -= 3 (460)* = G348(X) i 1916847 
 
 j3a6:=3(460)3 = 4140 ! 
 
 Corrections:^ &^ == (3)* = _ 9! 
 
 True Divisor 638919 I 1916847 
 
 * The new root figure cannot be larger than this quotient. It is often not eo large, and the 
 probability of its being considerably less increases with the degree of the root wc arc extracting,. 
 
^ LITERAL ARITHMETIC. 
 
 3. Extract the 5th root of 30036242722357. 
 
 SOLUTION. 
 
 Formula : {a + by = a' + 5a*b + 10«='6« + 10^*6=' + Bab* + ft» 
 
 = a^ + [5a* + lOa'^b + lOa'b' + Bab'' + b*]b. 
 
 At first a* = the greatest 5th power in 
 
 Trial Divisor : 5a* = 5 (50)* = 
 
 1st. lOa'ft = 10(50)' X 1 = . 
 2d. 10rt«6* = 10(50/ X 1^ = 
 3d. 5a*^ =5(50) x !=» = .... 
 4th. b* = V = 
 
 Corrections : - 
 Tnie Divisor 
 
 31250000 
 
 1250000 
 
 25000 
 
 250 
 
 1 
 
 ¥2525251 
 
 36936242722357|517 
 
 3125 
 
 56862427 
 
 32525251 
 
 Ti-ial Divisor : 5a* - 5 (510)* = 338260050000 
 
 1st. 10a '6 = 10 (510) ' X 7 = . . . 9285570000 
 2d. 10a«6« = 10 (510) « x 7' = . 127449000 
 
 3d. Bab' = 5 (510) x 7» = 874650 
 
 4th. 6* = 7* = 2401 
 
 Tnie Divisor : 847673946051 
 
 Corrections : - 
 
 2433717622357 
 
 2433717622357 
 
 4. What is the 7th root of 1231171548132409344 ? 
 
 SOLUTION. 
 
 Formula: (a + 6)^ = a'+7a«6 + 21a»6* + 9Ba*b' + SBa^'b* + 2U^b'' -h 7ah^ + b' 
 = a' + I7a« + 21a»6 4- dBa*b' + SBa'b^ + 21a*6* + 7a*' + b^]b. 
 
 1231171548132409344|384 
 
 2187 
 
 Trial Divisor : 7a6 = 7 (30)^ = 5103000000 
 
 .. fist. 21a»6 = 21(30)» X 8= 4082400000 
 
 2d. 35a*6* = 35(30)* x 8* = . 
 3d. SSrrVy^' = 35(30)-' x 8^ = . 
 4th. 2la^b* = 21 (30)* x 8* = , 
 
 5th. 7aA«=:7(30) x 8^ = 
 
 L6th. 66 = 8'^ = 
 
 , 1814400000 
 
 , 483840000 
 
 77414400 
 
 0881280 
 
 262144 
 
 11568197824 
 
 101247154813 
 
 92545582592 
 
 Trial Divisor : 7a«* = 7 (380)*^ = . . 
 
 .21076564688000000 
 
 87015722212409344 
 
 .. 
 
 f 1st. 21a'b = 21 (380)' x 4 = . . 
 
 . Ga557541 1200000 
 
 
 02 
 
 2d. Soa^y- = 35 (380)* x 4« = 
 
 . 11676761600000 
 
 
 *■? 
 
 3d. 35a^'6-^ = 35(380)=' x 4=' = 
 
 122913280000 
 
 
 g 
 
 4th. 21a'6* = 21 (380)« x 4* = 
 
 776294400 
 
 
 o 
 
 5th. 7a*' = 7 (380) x 4' = . . . 
 
 2723840 
 
 
 
 6th. 56 ^ 4<5 ~ 
 
 4096 
 2i7o3930o53102336 
 
 
 
 
 87015722212409344 
 
POWERS AND ROOTS. 6© 
 
 5. Extract the square root of each of the following numbers : 7225, 
 9801, 553536, 5764801, 345642, 2, .5, 3, 50, 1.25, 1.6. 
 
 6. Extract the cube root of each of the following numbers : 74088, 
 122097755681, 2936.493568, 61234, 12.5, .64, .08, 2, 5. 
 
 7. Extract the 4th root of 52764813. (See 196.) 
 
 8. Extract the 6th root of 2985984. (See 196.) 
 
 9. Extract the 8th root of 1679616. (See 196.) 
 
 10. Extract the 5th root of 5. \/5"= 1.37974 -. 
 
 11. Extract the 7th root of 2. -^^2 = 1.104 + . 
 
 12. Extract the square root of 49a:8|/« — 30a;3y + 16y*— 24a;y» 
 ■f 25a:*. 
 
 SOLUTION. 
 
 Formula : {a + hy =a' + {2a + h)b. 
 :. a=^x* 25«^ 
 
 2a= Trial Div. = 10;c'' -'dOx'y + AQx'y^ 
 b= -30^^^-f-lO.c' = -3.ry 
 .-. True Div.:=10.c- — Sxy 
 
 -30x'*p+ 9a;'.?/ 
 
 i^it 
 
 2a= Trial Div. = 10.c* — 6.r^ 
 
 4Qx'y'-24xy' + lQy* 
 
 .-. *=40j;V«-4-10.c*= 4y2 
 
 
 and True Div.=iac«-6a^ + 4y' 
 
 40;rV-24i:y^ + 16y* 
 
 CONDENSED SOLUTION. 
 
 25a;* -30a; 'y + 49a;'y« -24^2/=' + 16y* |5a;^-3a^ + 4y* 
 25x* " 
 
 10x^-3x1/ 
 
 ■30a;V + 49a;V' 
 -30.2; '.v+ 9a;«.v« 
 
 lOx'— 6a^ + 4^/• 
 
 40a;V'-34a^'+16y^ 
 40a;^y«-24a;y=» + 16y* 
 
66 
 
 LITERAL AlUTHMETIC. 
 
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POWERS AND ROOTS. 67 
 
 15. Extract the cube root of each of the following: a^ -- Sb^ 
 + VZab^ — 6a^b, 5^:3 — 1 - Sx^ + x^ — ^x, 66x* + 1 — 63a;3— 9a; 
 + Sx^ - 36a:« + 33a;2, mc^x^ + ^^cx^ - %lc^ + lO^o^x - '^{)c^x^ 
 + 8a;« — 806'3a;3, 204:C^x^ — lUc^x + 8a;6 - 366-^«- 171c^x^ + 64.a^ 
 
 + 102c^x^, 27a; - 8a;* - 36 + 362;^ + 12a;-^ - 54a;* + 9a;"* + 27a;* 
 
 4-a;-"— 6a;'*. 
 
 16. What is the 4th root of 16«* - 96^3:2; + 216«2a;2 - 216aa;3 
 4- 81a;* ? 
 
 17. What is the 6th root of 729 - 2916a;2 + 4860a;* - 4320a;« 
 + 2160a;8- 576a;i«+ 64a;i2? 
 
 [Note.— Solve the 16th and 17th both by {197) and {196)1 
 
 18. Find the fifth root of 32a;5- 80a;4 + 80a;3- 40a;2 + 10a; - 1 ; 
 also of a;-" + 15a;-"-5a;-'* + 90a;-"-60a;-'» + 280a;-'-270a;-« + 495a;-* 
 - 550a;-' + 513 - 465a;2 + 275a;* - 90a;« + 15a;8 -xAK 
 
 19. Find the 6th root of ««- 6a^b + 15a*^2- 20^3^3 4. 15^2^4 
 -6a*6+J«by (196). 
 
 SECTION IIL 
 CALCULUS OF RADICALS. 
 
 Reduction. 
 19S, Pvob, !• — To simplify a radical by removing a factor, 
 RULE. — Resolve the number under the radical sign into 
 
 TWO FACTORS, ONE OF WHICH SHALL BE A PERFECT POWER OF THE 
 DEGREE OF THE RADICAL. EXTRACT THE REQUIRED ROOT OF THIS 
 FACTOR AND PLACE IT BEFORE THE RADICAL SIGN AS A COEFFICIENT 
 TO THE OTHER FACTOR UNDER THE SIGN. 
 
 Dem, — This process is simply an application of CoR., Art. 194:, 
 
 199, Cor. — The denominator of a surd fraction can ahcays be 
 removed from under a radical sign by multiplying both terms of tha 
 fraction by some factor which icill make the denominator a perfect, 
 power of the degree required. 
 
BS LITERAL ARITHMETIC. 
 
 ScH. — A surd fraction is conceived to be in its simplest form when the 
 smallest possible wfiole nwnber is left under the radical sign. 
 
 200, I^rob, 2, — To simplify a radical^ or reduce it to its lowest 
 terms, when the index is a composite number, and the number under 
 the radical sign is a perfect poioer of the degree indicated by one of 
 the factors of the index. 
 
 RULE. — Extract that root of the number which corre- 
 sponds TO ONE OF THE FACTORS OF THE INDEX, AND WRITE THIS 
 ROOT AS A SURD OF THE DEGREE OF THE OTHER FACTOR OF THE 
 GIVEN INDEX. 
 
 Dem. — The mnXh. root is one of the mn equal factors of a number. If, now, 
 the number is resolved first into m equal factors, and then one of these m factors 
 is again resolved into n other equal factors, one of the latter is the mni\\ root of 
 the number. 
 
 201, JProb, 3, — To reduce any number to the form of a radical 
 of a given degree. 
 
 RULE. — Raise the number to a power of the same degree 
 AS the radical, and place this power under the radical sign 
 
 WITH THE REQUIRED INDEX, OR INDICATE THE SAME THING BY A 
 FRACTIONAL EXPONENT. 
 
 Dem. — That this process does not change the value of the expression is evi- 
 dent, since the number is first involved to a given power, and then the corre» 
 spending root of this power is indicated, the latter, or indicated oi^ersiimn, being 
 just the reverse of the former. 
 
 202, Cor. — To introduce the coefficient of a radical under the 
 radical sign, it is necessary to raise it to a power of the same degree 
 as the radical ; for the coefficient being reduced to the same form as 
 the radical by the last rule, we have the product of two like roots, 
 which is equal to the root of the product. 
 
 203, Proh, 4, — To reduce radicals of different degrees to equiv- 
 alent ones having a common index. 
 
 RULE. — Express the numbers by means of fractional in- 
 dices. Reduce the indices to a common denominator. Per- 
 form UPON THE NUMBERS THE OPERATIONS REPRESENTED BY THE 
 
CALCULUS OF RADICALS. 69 
 
 NUMERATORS, AND INDICATE THE OPERATION SIGNIFIED BY THE 
 DENOMINATOR. 
 
 Dem, — The only point in this rule needing further demonstration is, that mul- 
 tiplying numerator and denominator of a fractional index by the same number 
 
 a ma n 
 
 does not change the value of the expression, i. e., that x^ = x"'". Now, x'^ signi- 
 fies the product of a of the b equal factors into which x is conceived to be re- 
 solved. If we now resolve each of these b equal factors into m equal factors, a 
 of them will include ma of the mb equal factors into which x is conceived to be 
 resolved. Hence ma of the mb equal factors of x equals a of the b equal factors. 
 
 [The student should notice the analogy between this explanation and that 
 usually given in Arithmetic for reducing fractions to equivalent ones having a 
 common denominator. It is not an identity.] 
 
 204, JProb, S, — To reditce a fraction having a monomial radi- 
 cal denominator, or a monomial radical factor in its denominator^ 
 to a form having a rational denominator. 
 
 RULE. — Multiply both terms of the fraction by the radi- 
 cal IN THE DENOMINATOR WITH AN INDEX WHICH ADDED TO THE 
 GIVEN INDEX MAKES IT INTEGRAL. 
 
 20S» I^vob, 0. — To rationalize the denominator of a fraction 
 when it consists of a binomial, one or both of whose terms are radi- 
 cals of the second degree. 
 
 RULE. — Multiply both terms of the fraction by the de- 
 nominator WITH one of its signs CHANGED. 
 
 Dem. — In the last two cases the student should be able to show, 1st. That 
 the operation does not change the value of the expression ; and, 2d. That it 
 produces the required form, [This is the substance of all demonstrations in Me- 
 ductions.] 
 
 206, I^rop. 1, — A factor may be found which will rationalize 
 any binomial radical, 
 
 Dem. — If the binomial radical is of the form V(a -F fc)'", or {a + by , the fac- 
 
 n — m 
 
 tor is (a + &) " , according to {204). 
 
 L. ]L 1 
 
 If the binomial is of the form "VaF + Vft*^, or «"' + 6". Let a'" = x, and 
 1 LI 
 
 b'^=y; whence a"* = a;* , and b"* = y ^ ^jgo j^t p be the least common multiple 
 
 up rp 
 
 of m and n, whence x"" and yp are rational. But x^ = a"\ and y""" = b". If 
 now we can find a factor which will render x^ + y^, as»P ± y""", this will be a fac- 
 
Vo 
 
 LITEBAL ARITHMETIC. 
 
 L L 1!L ^ 
 tor which will render a" + &*, «"'± 6" which is rational. To find the factor 
 
 which multiplied by x^ + y" gives afp ± y'T, we have only to divide the latter by 
 the former. Now ^— = x^p-^) — x^p-2) «»• + x<p-*) v^r _a^(p-4) yir . 
 
 ;C* + y*" ^ ^ if T^ 
 
 - - - ± y^p-') (J), the + sign of the last term to be taken when;? is odd, and 
 the — sign when it is even (119). Therefore .t*<p-') — a'Kp-s)^'- + a^P-ny^r^ 
 «^''~*V + ± y''^'"^, is a factor which will render *'Va'+ Vibrational, 
 
 x' being understood to be a"*, and y = b* , and p tlie L. C. M. of m and n. 
 
 If the binomial is Va' — Vft^ , the factor is found in a similar manner, and is 
 
 af<-P-^) + x^P-^hf" + X<P-^*r ^ ^ y^P-^), 
 
 207. Prop, 2. — A trinomial of the form Va +\/b +Vc 
 may he transformed into an expression with but one radical term by 
 midtiplyiiig it by itself with one of the signs changed, a» \/ a + V b 
 
 — V c. The product thus arising may then be treated as a binomial 
 radical by coiundering the sum of the rational terms as 07ie term, 
 and the radical term as the other. 
 
 Thus, {Va +\/T + y/'c) (y/a + \/~b—\/~c) = a + h— c + 2Vab. Again, 
 [(a + 6 - c) -f- 2\/ab] x [(a + b - c) ^ 2v/a*] = a» + IP + c^ - 2ab - 2bc 
 
 — 2ac, a rational result. 
 
 Examples. 
 
 1. Reduce the following to their simplest forms: VlOSa*x^^, 
 
 j/2G46rAr«, (7047a:i«t/«)i, Vx^ - x'^ + a;*, ^(x^-y^) (x-y), 
 
 {xr-^tr ) % 7A/363a;y3, (a - b) [{a^ -b^)(a- b)Y, 5\rmx<^y^, 
 ab\/Q3a-'xH-\ 
 
 2. Reduce the following to their simplest forms (see ScH. 199) : 
 
 /!■ i/i- \r\- 1/|- fl- V|. D^T. 
 
 li/?. n/fs' 'fl- i/f' f?'-- 
 
 x^ - y ^ . / 3a;g + 6a;,y + dy^ 
 
CALCULUS OF RADICALS. 71 
 
 3. Reduce the following to their simplest forms (see 200) : 
 
 Vl^5a3^, V363«8.c^ Vll^^^V, V-1029a;i2, 
 Vl35a*^ — 405rt3:c2 + 405rt22;3 _ I'dbax^. 
 
 4. Reduce 5fa*3 to the form of tlie square root; also 7xt/; also J; 
 also Sa — 2. Reduce 2x^y^ to the form of the 3d root, — to the form 
 
 of the 5th root. Reduce -7- to the form of the 4th root, — to the 
 
 
 
 form of the cube root. Reduce \^i to the form of the cube root, — 
 to the form of the 4th root. 
 
 5. Introduce under the radical signs the coefficients in the follow- 
 ing expressions: 
 
 2v^» I v/ 3 , -iV 3 , i V25, 2xVx^ (x + y)Vx^ - 3x^y + dxy^ - y\ 
 (x + y)V^^, y ^/Ubx, «'(l--i) • 
 
 6. Reduce to equivalent forms having a common radical index, 
 
 \/2 and \/3 > ^^so ^3, Vo ; also \/2x, \^3x^f ^x, and ^/%x^ ; also 
 
 2\/c, 3v^, and jVo; also ^t's/hax, 2^2xy, \/\^x\ also a; — y and 
 
 {x + yy. Explain each operation upon the principles of factoring 
 as in {203). 
 
 7. Prove upon the principles of factoring that ^2 — VS ; also 
 that \/h = \/25 ; also that \/3 = ^21. 
 
 8. Reduce the following to equivalent forms having rational de- 
 
 . ^ 2a\/5l' 5 ^/a "s/ X \^x 1 1 
 
 nominators: ■ , --7==, —;=, -tj= ^ -Tr ^ -J-y "vh- ' 
 
 V^x 2Va^ Vb vy Vy V2 V2 
 
 Jl_ VI \^ \^ 
 
 V'o' V^' </l' 'V3' 
 
 9 Reduce the following to equivalent forms having rational de- 
 
 . , \/a;2 + a:w + V* ^^ ^ V^?^ — Vy 
 
 nominators: — , — 3 ^ 7=? -^= ^r, 
 
 V a; — y 3 V 3 — .t2 a; + V y yx + V ?/ 
 
 3 3 2 \/l2- VlO 3 4- 2a/2 
 
 VS + V2 a/3 + A/5 V5 - V4 a/6 + a/5 A^S - V3 
 
72 LITERAL AlilTHMETlO. 
 
 Va^ + :c2— a;* Va:« + 1 + a;' V^^^ - V-^Tl' 
 
 V.i'-- + ^i: + 1 + Vx^ + x-\ 8 ^ 3 
 
 and — =- 
 
 Vic* + a; + 1 - Va:^ + x - I ^/'d +\/^ + l' ' Vo + V3-a/3* 
 
 10. What factor will rationalize ^x— ^yt What ^/Ic'^ — ^/V^'i 
 What V8 + V3 + a/5? 
 
 11. By what must numerator and denominator of be 
 
 multiplied to reduce it to the form of a simple fraction ? By what 
 
 Milt 
 
 ' a;* 
 
 12. Introduce the coefficients of each of the following into the 
 parentheses: 8 (a* — x'^y, a^(a + a^xy, and a;«(l — x^y. 
 
 13. Show that r = ; also 
 
 a — bx -\- Vn* + b^x^ bx 
 
 that ^-J!^ = a/^^Lz21±. 
 
 SECT/ON IV. 
 COMBINATIONS OF RADICALS. 
 
 Addition and Subtraction. 
 208, IProh. 1. — To add or subtract radicals. 
 
 RULE.— It? the radicals are similar, the rules already 
 GIVEN (66 f 71) are sufficient. If they are not similar, make 
 
 THEM SO BY (198^ 203), AND COMBINE AS BEFORE. If THEY CAN- 
 NOT BE MADE SIMILAR, THE COMBINATIONS CAN ONLY BE INDICATED 
 
 BY^ CONNECTING THEM WITH THE PROPER SIGNS. 
 
 [Note. — The student is presumed to be able to give the demonstrations of 
 the problems and propositions in this section, as they are but a recapitulation of 
 what has preceded.] 
 
CALCULUS OF RADICALS. 73 
 
 Multiplication. 
 
 209, ^rop, !• — The product of the same root of two or more 
 quantities^ equals the like root of tlieir ^woduct. (See 104:*) 
 
 210. JProp, 2 •—Radicals of the same degree arc rniiltipUed ly 
 midtiplying the quantities U7ider the radical sign and wriiing the 
 product under the common sign. 
 
 Similar radicals are midtiplied by indicating the root hy fractional 
 indices, and, for the product, taking the common number with an 
 index equal to the sum of the indices of the factors. (See 82,) 
 
 211, X^vob, 2, — To multiply radicals. 
 
 RULE. — If the factors have not the same index, reduce 
 
 THEM TO A COMMON INDEX, AND THEN MULTIPLY THE NUMBERS 
 UNDER THE RADICAL SIGN, AND WRITE THE PRODUCT UNDER THE 
 COMMON SIGN. 
 
 Division. 
 
 212, I^rop, — TTie quotient of the same root of two quantities 
 equals the like root of tJieir quotient. 
 
 213, JProb,.3, — 2h divide radicals. 
 
 RULE. — If the radicals are of the same degree, divide 
 
 THE NUMBER UNDER THE SIGN IN THE DIVIDEND BY THAT UNDER 
 THE SIGN IN THE DIVISOR, AND AFFECT THE QUOTIENT WITH THE 
 COMMON RADICAL SIGN. 
 
 If the radicals are of different degrees, reduce THEM TO 
 THE SAME DEGREE BEFORE DIVIDING. 
 
 Involution. 
 
 214, JProb, 4, — To raise a radical to any power. 
 
 RULE. — Involve the coefficient to the required power, 
 
 AND ALSO THE QUANTITY UNDER THE RADICAL SIGN, WRITING THE 
 LATTER UNDER THE GIVEN SIGN. 
 
 215, Cor. — To raise a radical to a power lohose index is the in* 
 dex of the root., is simply to drop the radical sign. 
 
74 LITERAL ARITHMETIC. 
 
 Evolution. 
 2 to, I^rob, S, — 2h extract any required root of a monomial 
 radical. 
 RULE. — Extract the required root of the coefficient, 
 
 AND OF THE QUANTITY UNDER THE RADICAL SIGN SEPARATELY, 
 AFFECTING THE LATTER WITH THE GIVEN RADICAL SIGN. REDUCE 
 THE RESULT TO ITS SIMPLEST FORM. 
 
 [Note. — This problem should not be taken till after Quadratic Equations.] 
 
 217, I^vob, 0» — To extract the square root of a binomial, one 
 or both of whose terms are radicals of the second degree. 
 
 Solution. — Such binomials have either the form a ± nVb or mV a ± nV b 
 Now observing that {x ± j/Y = x^ ± 2xy + y*, we see that if we can separate 
 either term of any such binomial surd into two parts, the square root of the pro. 
 duct of which shall be \ the other term, these two parts may be made the first 
 and third terms of a trinomial (corresponding to x^ ± 2xy + y^), and the middle 
 term being the second term of the given binomial, the square root will be the 
 sum or difference of the square roots of the parts into which the first term is 
 separated. 
 
 [Note. — This process requires the solution of a quadratic equation. Thus to 
 extract the square root of 12 — t^IiO. Letting x and y represent the terms of 
 the binomial root, we have «* + y* = 12, and 2xy — — V'140. Whence x= Vb 
 or VY, and y — V'7 or V5, and the root is V o — V7. The sign between the terms 
 being detennined by the sign of the surd in the given binomial. On this ac- 
 count this subject should be reserved until after the student has studied quad- 
 ratic equations, or the solution effected by inspection. Thus, in this example 
 VliO = 2^35. Now v'35 = VS x Vl, and since the sum of the squares of these 
 factors is 12, we have Vl2 - V140 = \/E - V7.] 
 
 Examples. 
 
 1. Add a/50 and a/98. Add ^^n)tab^x iind \/^bUb^x. Add 
 ^yiZTla*'X^ and ^/bOOa^xK Add Vx^ and a/«V- ^"^^ A/n87j 
 and -a/1008^ Add 5A/f and 2\/^. Add a/| and ^^/U^. Add 
 y^l, y^l and iv'sT Add ^/l^aH^, VbOaJb^ and dVlSaHxK 
 
 2. Show that a:y 1 + (|) -^yf 1 + {^J= {xi-hyh (y* + a:^)* 
 
 Show that i A^!^f^ 
 
 y a'+2ax + x^ J/ a^—'Zax + x^ \a^—x^J 
 
CALCULUS OF llADICALS. 75 
 
 Add and ; also "^ ^ ^ ^ ^ and 
 
 a + V «^ — 2;2 a — V«^ —x^ Va + x — ya — x 
 
 V(i -{-x—Va- 
 
 ■x 
 
 \/a + x+ "^ a—x 
 
 3. From 3 Vf take 2VS- F^'^^^i ^/^^ take v^24. From V^i 
 take Vf From f Vf take -|A/f From \^a6^ take ^^y . From 
 ^'arb take aX^^ 
 
 4. Show that |/|i - 1/3^ = ttVe. 
 
 5. Show that A/ "Vf:^': - a/^ 
 
 -2rtZ>2 4_^3 4rt^,y<{, 
 
 2ab + b^ a'^ — b-' 
 
 6. Multiply v^3 by V^-* Multiply V^^" by v^a Multiply 
 VT by ^|, V^2a^ by \^ax^, 2\/^ by 3\^x^y, a/1 + ^ by 
 \/rTx^, vTby -v/j, 2A/i by 3a^2; 2^25 by 3V5, v^24 by e-^S, 
 
 7. Multiply 9 + 2\/r0 by - 2\/l0, v'^^ - A^iiJ^ + \/]f^ by 
 \/x+\/y, 3\/5 + 2a/6-2 by 2^5 + 18^/0, v^5-2v^6"by 
 3^4- v^36. 
 
 8. Divide ^v^byiv^, 8\/9 by2\^, a/6 by v^, i\/5by-jViO. 
 
 9. Divide 2 a/32 + 3 a/2" + 4 by 4a/8^ 4a/? by 2a/^, 
 6 + 2a/3 - ^8 byA/6, V^^^^a; - ^»2^-:r by Va'^, /</ f" by /d/^, 
 (rt + b)Va^^l- by (« - Z>) \/(a + 1)2, « + ^ - c + 2a/«^ by 
 
 V^+ V^- V,I I !^l±^^ _ f^W^S I by 4|/-^. 
 - lx^-\/x^-a^ x'^ + Vx^-a^ ) f x^-i-a^ 
 
 * It is of the utmost importance that the pupil be able to give a complete analysis of such 
 examples. Thus, \/ 2 = >/ S, since the former is oneoi the two equal factors of 2, and the 
 latter is three of the six equal factors of 2. In like manner VS = V 9. Consequently 
 v/2 X V'3= V^ X ■n/Q. Now since the jjroduct of the same root of two numbers is equal to 
 the like root of the product, VS x v/9 = V72. 
 
76 LITERAL ARITHMETIC. 
 
 10. Raise 3 v^2a;« to the second power. Raise ^ \/)lax^ to the 5th 
 power. Cube — fVf Square \/3 — V^. Cube 'dVa—x. Cube 
 
 11. Extract tlie square root of 27 \^VS6x^y^ ; the cube root of 
 ,».riP*Vy; the4throotof 25^*Vy; the 5th root of 224 v^3^; the 
 
 cube root of (1— a;)Vl— ^; the cube root of vi/-; the square 
 
 5 1/ X 
 
 root of iVi- 
 
 12. Extract the squai:e root of 49+12a/5'; of 57 + 12a/15"; of 
 {a^-ha)x-2axVa; x-2\/^x^; of VI8-4; of -^+-|Va«- c«. 
 (See ;^ir.) 
 
 SECT/ON F. 
 
 IMAGINARY QUANTITIES. 
 
 )^18. ^n Imaginary Quantity is an indicated even root 
 of a negative quantity, or any expression, taken as a whole, which 
 contains such a form either as a factor or a term. 
 
 Thus >/ —x,\/ —y* , S^—x*, 24^^^— 4, V— 6, 3— V—1, etc., are imaginary 
 quantities. 
 
 219, ScH. 1. — It is a mistake to suppose that such expressions are in any 
 proper sense more unreal than other symbols. The term Impossible Quantities 
 should not be applied to them : it conveys a wrong impression. The ques- 
 tion is not whetlier the symbols are symbols of real or unreal (imaginary) 
 quantities or operations, but what interpretation to put upon them, and how 
 to operate with them when they occur. 
 
 220. Sen. 2. — A curious property of these symbols, and one which for 
 some time puzzled mathematicians, appears when we attempt to multiply 
 
 y/—xhY\/—X' Now the square root of any quantity multiplied by itself 
 should, by definition, be the quantity itself ; hence V—xx \/—x= —x. 
 But if w^e apply the process of multiplying the quantities under the radicals, 
 we have v/^ x\/^ =Vx^ = + a; as well as — x. What then is the pro- 
 duct of v/3^ W^^ ? Is it —a-, or is it both + x and —x ? The true 
 product is —x ; and the explanation is, that v/a;* is, in general., +x and —x. 
 But when we know what factors were multiplied together to produce a;*, and 
 the nature of our discussion limits us to these, the sign of \/x^ is no longer 
 ambiguous : it is the same as was its root. 
 
CALCULUS OF IMAGINAKIES. 77 
 
 221. Prop. — Every imaginary monomial can be reduced to the 
 
 form ni V — J^j ^'^ which m is real (not imaginary), m may be 
 rational or surd. 
 
 p 
 
 Dem. X V —y, p being an even number, is the general symbol for an imagin- 
 ary monomial. Now if p is a power of 2, we may write at once p = 2», whence 
 
 p 2" 2" 2" 2" 2" - 
 
 .f V — i/ = :v V—i/ — X V(/{~1) =x Vy V— 1 = m V—1. If p contains other 
 
 factors than 2, let r represent their product, and 2» the product of all the factors 
 
 of 2 contained in p ; whence p = r2", in which r is odd, since the product of any 
 
 p. r2*!^ rS" 
 
 number of odd factors is odd. We then have x V—y = x V —y = x Vy{—V) 
 z= X V y V — 1 — X yy y V—1 = x Vy V—1, since any odd root of — 1 
 is — 1. Putting X V y = m, this becomes m V—1- 
 
 222, ScH. — When n—1^ i. e., when there is but one factor of 2 in the 
 index of the root, the form becomes m V—1. This form is called an imagin- 
 
 ary of the second degree ; m r — 1 is of the fourth degree, etc. In this dis- 
 cussion we shall confine our attention mainly to imaginaries of the second 
 degree. 
 
 223, I^vob, — To add and subtract imaginary mo7iomials of the 
 second degree^ or such as may be reduced to this degree. 
 
 RULE. — Reduce them to the form mV— 1? and then com- 
 bine THEM, CONSIDERING THE SYMBOL V— 1 AS A SYMBOL OF 
 
 character.* 
 
 Examples. 
 1. Add \/^=T and ^/~^. 
 
 Operation. V~^4: = V^{ - 1) = 2 V^^, and V^^ = 3 V^. 
 .-. V^^4: + V^^9 = 2 V^l + 3 V^^l = 5 V^^l. 
 
 Sen. — The last operation should not be looked upon as taking the sum of 
 2 times a certain quantity (represented by \/—l) and 3 times the same 
 quantity, but ns 2 of a certain character added to 3 of the same character. 
 
 ■* St-e (■!*, 49 f 50). Wc mean to pay that, as a qnantity (considered numerically), m and 
 m V— 1, are exactly tlie same, just as is the case in the expreseions + m and — m ; but that the 
 eynibol $/ — i gives* gome peculiar or concrete significance to m, as does the sign +, or — , or f. 
 What, this concrete pigniflcance is, we cannot here say. It has its clearest interpretation in the 
 Co-ordinate, or General Geometry. 
 
78 
 
 LITERAL ARITHMBTia 
 
 Thus the operation of reducing ^/ —^ and ^ —^Xa the forms %^^\ and 
 3^/— 1 is to be looked upon as rendering the expressions homogeneous. It 
 is, however, to be observed that the symbol ^ —\ is subject, also, to the 
 ordinary laws of number, and may be operated upon accordingly. Thus it 
 has a double significance. 
 
 2. Add 4V -%n and 3V - 16; also ^a^^^lib and %a^/~^^', 
 also ^V^^^ and c\/"^. V^f^^ + cv/^^ = {W + 3c)V^. 
 
 3. Add V- 1024 and V-729. 
 
 Opkration. ^-1024 = ^i^ \/^' and ^-729= ^729 aJ~^\. 
 .-. ^^^=^024 + >v/- 729 = 59 v^^. 
 
 4. Show that in general v^ — a; ifc "s/ — y = {\/ x dc \^)V^ — 1, 
 ■when ^t is any integer. 
 
 5. From V^^ take \/"^4. From 4a/^=^7 take 3V^^6. 
 From 3fl\/ — 25 take 2rt V — 4. Show that aV — 6 — cV — d 
 
 6. From V- ^090 take V - 9. i^cw. ClV— "L 
 
 8cH. — It would seem improper to onrft the 1 before the symbol /y/— 1 in 
 such a case as the last, thougli it has been customary to do so. If we arc to 
 consider \/— 1 as a sign of character ("affection," as some say), there is no 
 more reason for omitting the 1 in such a case, than there is in such as 4 — 5 
 = — 1. That is, if we write 4^^/^ — S^y/^ = 'y/^, we ought to write 
 4 — 5 = —, or $5 — $4 = I, to be consistent. 
 
 7. Show that V^S + V~-^l^ = 5 V^^ = bV^V - 1 ; also 
 that 4:\/~^^ 4- 2V'^^^^n = 16 a/3 V^^ ; also that a/~^=1g 
 — \/ — 1 = 1 \/ — 1 ; also that 2\/ — a — \/ — a = \^\^ — 1. 
 
 224, JPt*op, — Evert/ polynomial coiitainiiig some real, and 
 some imaginary terms of the second degree, or such as can he re- 
 duced to this degree, can he reduced to the form adbbv — 1» **'* 
 which a and b are real, a and b may he rational or surd. 
 
 Dem, — This is evident from the fact that all the real terms can be combined 
 into one (it may be a polynomial) and represented by a, and the imaginary terms 
 being reduced to the form m V^ 1 can also be combined into one term repre- 
 sented by ± & V^. 
 
CALCULUS OF IiMAGINARIES. 79 
 
 225, ScH. — The form a±h^ — l is considered the gcncnil form of an 
 imaginary quantity of the second degree. Wlien « = 0, it becomes tlie samo 
 as m^/—l. The two expressions a + h\^—l and a — h\/ — l arc called Con- 
 jugate Imaghmries. Hence tlie sum of two conjugate imaginaries is rcil 
 (3a). Also the product of two conjugate imaginaries is real [{a + h\/ — \) 
 
 X (a — J^— 1) =: a'^ + Jr^ as will appear hereafter]. The square root of the 
 product of two conjugate imaginaries, taken with the 4- sign, is called the 
 Modulus of each. Thus + /y/^"^ + b^ is the modulus of both a + 'b^^ —I and 
 a - h^/^. 
 
 Exs. — Fi lid the sum, and the difference of 2 + V^ and 3 — V— 04 ; 
 also of a + \/ — d and a+V—o. Last results^ 2a + ( V^ + Vc) V' — i , 
 
 Multiplication and Involution. 
 
 220, I* rob, — 7b determine the character of the product of sev- 
 eral imaginary monomiaX factors of the second degree. 
 
 Solution. — Letting n represent any integer, including 0, 47i, 4w4-l, 4n + 2, 
 and 4/2 + 3 may represent all integral exponents, so that all the forms to be treated 
 
 are (^"1^, (V^)'"^', (^=1)"''". ^"^ W~^T'" - Thus, if n = 0. 
 4// +1 =: 1, 4/1 + 2 = 2, and 4n4-3 - 3. If n = 1, 4m = 4, 4r.+ 1 = G, 4;^ + :? = C, 
 and 4n + 3 = 7. If n = 2, 4n = 8, etc. 
 
 We shall now show that (V~0^' = 1» 
 
 _ and _^ (.y/3i)*^+' ^ _^Z:i. 
 
 (v^ir = ( v^i v^i ^ v^i v^i)" = [(-1) X (-1)]" = i« = 1, 
 
 since y'— 1 ^/^l = —1 by {220\ (—1) (—1) = 1, and any power of 1 is 1. 
 (a/~1)'*^' = (V=ir X V^l - V=l = a/-1. since (y— 1^ = 1. 
 
 (V=i)"^' = (V~i)" X ( V-iy = 1 X (-1) = -1. 
 
 (y^— 1)^-+^ ^ {^~\)'^^ X V-^1 = (-1) V=^ = - V=T = - V~,. 
 
 III. — To find what (-\/— l)* is, we have but to observe that if 7i=l, 471 + 1=5, 
 
 and (vri/ = (y— l)^«^^y'i:i. ^ 
 
 Again, (-y/^)' = (V-^^""' = - a/-^. ^i^^cc, if 7i = 0, 4/i + 3 = 3. 
 
 Once more, {\^\Y^ = {\^~\Y"^'^ = —1, since, if ti = 2, 4n + 2 = 10. 
 
 In like manner, {^/—\Y = {\^—l) " = 1, since, if n = 1, 4//. = 4. 
 
 SCH. — In the above we have confined ourselves to the powers of the posi- 
 tive square root, sinca — \/ — 1 may be understood to be — 1 V — ^ ' ^^-^ ^^^^ factors 
 -1 bo treated separately. Thus, (- ^/'-^Y" r= (-l)'" (^/-lY" = ( V"' )** 
 =- 1. So also (- V~0'"'' = (-1)*"' X (V-iy-i = (-1)a/-1 = 
 
So LITERAL ARITHMETIC. 
 
 — \/^^, etc. Or, in other words, giving the — sign to ^/—l simplj changes 
 the signs of the odd powers. In examples we shall confine attention to the 
 
 + root of ^Z — 1. 
 
 Examples. 
 
 1. Multiply 4\/^=^ by 2V~^^. 
 
 Operation. 4 V^ = 4 Va V^, and 2 V^ = 2 V2" V^. 
 
 .'. 4 V^ X 2 V^ =aVs V^ X 2 V2 V^ = 8 V^{^^^= -SVQ. 
 
 2. Show Unit V—x^ X \/ — 7j^ = —x)/; also tliatSV— 5x4^— 3 
 = —12 Vl^ What is the product of ~2V~-^ by -3 V^^ ? 
 
 3. Show that V— 2 x V— 8 = W— 1; also that V— 256 x 
 V'^^27 = 48\/3 V^^^; also that \/^^ x W'^^ = W^ V^- 
 
 4. Show that 4a/— T + V— ^ multiplied by 2a/— 1 — V— 3 
 equals a/O + 4a/3 — 2\/2 — 8. 
 
 5. Show that J — iv^^^ squared equals — J(l + V— 3). 
 
 6. Showthat(3-2A/^^)x(5 + 3A/- 4)=39-2a/- 1 ; also that 
 
 (i+v^=nr)x(i-A/'^=n:)=2. 
 
 7. Sliow tliat 2 is the modulus of V2 -\- V~^^ and a/2 — a/^^- 
 What is the modulus on]i-2V'^^> and 3-2a/^^? Ofo-SV-i^ 
 
 8. Show that (a/^)''i= _ 75a/^=3; also that (V^^Y^^S^ 
 0. AVhat is the 5th power of 2V~-^? Of 3 a/^^^ ? 
 
 10. What is the product of a/— a;^, a/-^/-, a/— ^S and V—w^? 
 
 Division" of Imaginaries. 
 
 227, I* rob. — To divide one imaginary of the second degree by 
 another. 
 
 RULE. — Reduce tfie imaginary term, or terms, to the form 
 
 m\/ — 1. or mW — l)", AND DIVIDE AS IN DIVISION OF RADICALS, 
 OBSERVIXG THE PRINCIPLES OF {220) TO DETERMINE THE CHARAC- 
 TER OF THE QUOTIENT OF IM AG IN ARIES. 
 
CALCULUS OF IMAGINARIES. 81 
 
 Examples. 
 
 1. Divide \/- 16 by V - 4. 
 
 Operation. V— 16 = 41^^, and V^^ = 2V^; .-. V^^^ -^ V^IIl 
 = 44/31 -^ 2f^^ = 2( V=l)°. But by (226) {V^^lY = + 1 ; hence the 
 quotient is 2. 
 
 ScH. — A superficial view of the case might make the quotient ± 2. Thus, 
 as the radicals are similar it might be inferred that V^l -^ V^^ = 4/ 3_ 
 = \^ = ±1. (See 220,) 
 
 2. Show that 6V - 3 -4- 2 V"^^ = iV~S ; also that - V~^^ 
 -^ - 6\/~^^ = t^a/3; also that 1 -^ V - 1 = - V"^ ; also 
 that 6 ^ 2^^^ = - 3 V^^. 
 
 3. Divide 2^/- 1 by V -2; also 3V - 16 by - 12; also Va 
 by V^^' 
 
 SuG's. 2 v^ -^ V:i2 = V 16 VFi? - V^ V^i = V^ V^. 
 
 4. Show that sV^^H:^ -^ 2 v""^^ = 4v^2'>/^=l. 
 
 5. Show that (1 + V"^^) -^ (1 - V"^^) = V^^; also that 
 (4 4. ^/~Ir2) -^ (2 - V"^^) = 1 + V^a/""^ ; also that 
 
 1 ^ (3 _ 2V - 3) = — ^^^ — - — ; also that 1 -4 
 
 21 a + V - .^• 
 
 a^ — X -\- 2aV — X. , ^1 i. ^^ + V — ^ , a — V — 
 
 : ■ > also that 7== + y= 
 
 a'^ + X a — ^/ — h a ■\- \ —h 
 
 2ia^ - b) 
 
 '' a^ + b ' 
 
 6. Simplify 
 
 {a + bV -if + (g-^V-l)^ 
 
 [Note. — Here ends the subject of Literal Arithmetic. The student is now 
 prepared for the study of Algebra, properly so-called ; i. e., The Science of the 
 Equation.] 
 
PART IL 
 
 AN ELEMENTARY COURSE IN 
 ALGEBRA. 
 
 OHAPTEE I. 
 
 SIMPLE EQUATIONS. 
 
 SECT/ON L 
 
 EQUATIONS WITH ONE UNKNOWN QUANTITY. 
 
 Definitions. 
 
 1. An Equation is an expression in mathematical symbols, of 
 equality between two nnmbers or sets of numbers. 
 
 2. Algebra is that branch of Pure Mathematics wliich treats 
 of the nature and properties of the Equation and of its use as an 
 instrument for conducting mathematical investigations. 
 
 3. The First Member of an equation is the part on the left 
 hand of the sign of equality. The Second Member is the part 
 on the right. 
 
 4. A ^tnnerical Equation is one in which the knoivn 
 quantities are»represented by decimal numbers. 
 
 5. A Literal Equation is one in which some or all of the 
 known quantities are represented by letters. 
 
 6. The Degree of an Equation is determined by the liighest. 
 number of unknown factors occurring in any term, the equation 
 being freed of fractional or negative exponents, as affecting the un- 
 known quantity. 
 
 7. A Simple Equation is an equation of the first degree. 
 
 8. A Quadratic Equation is an equation of the second 
 degree. 
 
 9. A Cubic Equation is an equation of the third degree. 
 
TRANSFORMATION OF EQUATIONS. 8^' 
 
 10. Equations above the second degree are called Higher 
 
 Equations, Those of the fourth degree are sometimes called 
 Biquadratics. 
 
 Transformation of Equations. 
 
 11, To Transform an equation is to change its form without 
 destroying the equality of the members. 
 
 12, There are/owr principal transformations of simple equations 
 containing one unknown quantity, viz : Clearing of Fractions, Trans- 
 position, Collecting Terms, and Dividing by the coefficient of the 
 unknown quantity. 
 
 13, These transformations are based upon the following 
 
 Axioms. 
 
 Axiom 1. — Any operation may be jyerfornted upon any term or 
 upon either member^ which does not affect the value of that term or 
 member^ without destroying the equation. 
 
 Axiom 2. — If both members of an equation are increased or di- 
 rninished alike, the equality is not destroyed. 
 
 14, JProb, — To clear an equation of fractions. 
 
 RULE. — Multiply both members by the least or lowest 
 
 COMMON MULTIPLE OF ALL THE DENOMINATORS. 
 
 Dem. — This process clears the equation of fractions, since, in the process of 
 multiplying any particular fractional term, its denominator is one of the factors 
 of the L. C. M. by which we are multiplying ; hence dropping the denominator 
 multiplies by this factor, and then this product (the numerator) is multiplied by 
 the other factor of the L. C. M. 
 
 This process does not destroy the equation, since both members are increased 
 or diminished alike. 
 
 III. — An equation is aptly compared to a pair of scales with equal arms, kept 
 in balance by weights in the two pans. 
 
 Transposition. 
 
 IS, Transposing a term is clianging it from one member of 
 the equation to the other without destroying the equality of the 
 members. 
 
64 ELEMENTARY ALGEBRA. 
 
 16, Frob, — To tra?ispose a term, 
 
 RULE.— DllOV IT FROM THE MEMBER IN WHICH IT STANDS AKD 
 INSERT IT IN THE OTHER MEMBER WITH THE SIGN CHANGED. 
 
 Dem, — If the tenn to be transposed is +, dropping it from one membcir 
 diminishes that member by the amount of the term, and writing it with the — 
 sign in the other member, takes its amount from that member; hence both 
 members are diminished alike, and the equality is not destroyed. (Repeat 
 Axiom 2.) 
 
 2d. If the term to be transposed is — , dropping it increases the member from 
 Avhich it is dropped, and writing it in the other member with the + sign i7i- 
 creases that member by the same amount ; and hence the equality is preserved. 
 (Repeat Axiom 2.) 
 
 17, To Solve an eqiuitioii is to find the value of tlie unknown 
 quantity; that is, to find what yaliie it must have in order that the 
 equation be true. 
 
 18, An equation is said to be Satisfied for a value of tlie un- 
 known quantity which makes it a true equation ; i. e., which makes 
 its members equal. 
 
 19, To Verify an equation is to substitute the supposed value 
 of the unknown quantity and thus see if it satisfies the equation. 
 
 Sen. 2. — The pupil must not understand that the verili cation is at all 
 necessary to prove that the value found is the correct one. This is demon- 
 strated as Are go along, in obtaining it. Tlie object of the verification is to 
 give the pupil a clearer idea of the meaning of an equation, and to detect 
 errors in the work. 
 
 20, Proh, 1, — To solve a simple equation with one unknown 
 quantity. 
 
 R TILE. — 1. If the equation contains fractions, clear it of 
 THEM BY Art. 14. 
 
 2. Transpose all the terms involving the unknown quan- 
 tity to the first member, and the known terms to the second 
 member by Art. 16. 
 
 3. Unite all the terms containing the unknown quantity 
 into one by addition, and put the second member into its 
 simplest form. 
 
 4. Divide both members by the coefficient of the unknown 
 quantity. 
 
SOLUTION OF SIMPLE EQUATIONS. 86 
 
 Dem. — The first step, clearing of fractions, does not destroy the equation, 
 since both members are multiplied by the same quantity ^^AxiOM 2). 
 
 The second step does not destroy the equation, since it is adding the same 
 quantity to both members, or subtracting tiie same quantity from both members 
 (Axiom 2). 
 
 The third step does not destroy the equation, since it does not change the 
 value of the members (Axiom 1). 
 
 The fourth step does not destroy the equation, since it is dividing both mem 
 bers by the same quantity, and thus changes the members alike (Axiom 2). 
 
 Hence, after these several processes, we still have a true equation. But now 
 the first member is simply the unknown quantity, and the second member is all 
 known. Thus we have ichat the unknoicii quantity is equal to ; i. e. its value. 
 
 21, Sen. 1. — It must 1)6 fixed in the pupiVs mind that lie can make htt two 
 ■classes of chwiges upon an equation: viz., Such as do not affect the value 
 OF the members, or such as affect both members equally. Every opera- 
 tion must be seen to conform to these conditions. 
 
 22, Cor. 1. — All the si(/ns of the tenuis of both members of an 
 equation can be changed from + to —,or vice versa, without destroy- 
 ing the equality.^ since this is equivalent to midtiplying or dividing 
 by -1. 
 
 23, ScH. 2. — It is not always expedient to perforin the several trans- 
 formations in the same order as given in the rule. The pupil should bear in 
 mind that the ultimate object is to so transform the equation that the un- 
 known quantity will stand alone in the first member, taking care that, in 
 doing it, nothing is done which will destroy the equality of the members. 
 
 24, ScH. 3. — It often happens that an equation which involves the second 
 or even higher powers of the unknown quantity is still, virtually, a simple 
 equation, since these terms destroy each other in the reduction. 
 
 Simple Equation's containing Radicals. 
 
 25, Many equations containing radicals which involve the un- 
 known quantity, though not primarily appearing as simple equations, 
 become so after beinsr freed of such radicals. 
 
 26, JProb, 2, — To free an equation of radicals. 
 
 RULE. — The common method is so to transpose the terms 
 
 THAT the radical, IF THERE IS BUT ONE, OR THE MORE COMPLEX 
 radical, IF THERE ARE SEVERAL, SHALL CONSTITUTE ONE MEMBER, 
 
86 ELEMENTARY ALGEBRA. 
 
 AND THEN INVOLVE EACH MEMBER OF THE EQUATION TO A POWER 
 OF THE SAME DEGREE AS THE RADICAL. If A RADICAL STILL RE- 
 MAINS, REPEAT THE PROCESS, BEING CAREFUL TO KEEP THE MEM- 
 BERS IN THE MOST CONDENSED FORM AND LOWEST TERMS. 
 
 Dem. — That this process frees the equation of the radical whicli constitutes 
 one of its members is evident from the fact that a radical quantity is involved 
 to a power of the same degree as its indicated root by dropping the root sign. 
 
 That the process does not destroy the equality of the members is evident from 
 the fact that the like powers of equal quantities are equal. Both members are 
 increased or decreased alike. 
 
 Summary of Practical Suggestions. 
 
 27, In attempting to solve a simple equation, always consider, 
 
 1. Whetlier it is best to clear of fractions first 
 
 2. Zook out for compound negative terms. 
 
 3. If the nnmerators are polynomials and the denominators mono- 
 mials, it is often better to separate the fractions into parts. 
 
 4. It is often expedient, when some of the denominators are mono- 
 mial or simple, and others polynomial or more complex, to clear of 
 the most simple first, and after eacli step see that by transposition, 
 uniting terms, etc., the equation is kept in as simple a form as pos- 
 sible. 
 
 5. It is sometimes best to transpose and unite some of the terms 
 before clearing of fractions. 
 
 G. Be constantly on the lookout for a factor which can be divided 
 out of both members of the equation, or for terms Avhich destroy 
 cacli other. 
 
 7. It sometimes happens that by reducing fractions to mixed num- 
 bers the terms will unite or destroy each other, especially when there 
 are several polynomial denominators. 
 
 28. When the equation contains radicals, specially 
 consider, 
 
 1. If there is but one radical, by causing it to constitute one mem- 
 ber and the rational terms the other, the equation can be freed by 
 involving both members to the power denoted by the index of the 
 radical. 
 
SOLUTION OF SIMPLE EQUATIONS. 8? 
 
 2. If there are two radicals and other terms, make the more com- 
 plex radical constitute one me!uber, alone, before squaring. Such 
 cases usually require two involutions. 
 
 3. If there is a radical denominator, and radicals of a similar form 
 occur in the numerators or constitute other terms, it may be best to 
 clear effractions first, either in whole or part. 
 
 4. It is sometimes best to rationalize a radical denominator. 
 
 Examples for Practice in Solving Simple Equatioists. 
 
 1. Solve and verify the following : (1.) 40— 6a:— 16=120— l4;r. 
 
 X X x_ a: - 3 a; _ ^^'-19 /.w^+S a; 
 
 <^-) 2~3'^4-^^- (^'^ "2~'^3-^^~^- (^•)"T~'^3 
 
 . «— 5 /^x 9.T + 20 4a:— 12 x ,„. lOic + 17 12a: + 2 
 
 5a:— 4 ,^. ax—h a hx hx—a ,_ . aih^+x^) cix 
 
 (9.) 3:2^3 = ax + bx + ex. (10.) 2.04 - 0.68y - 0.02y = 0.01. 
 (11.) 8.4a: - 7.6 = 10 + 2.2a:. 
 
 o o 1 /i \ 1 , 1 1 tc,\ AQ .'i'2a:-.05 
 
 2. Solve (1.) -T \- 1 r- — • (^O 4:.8a; — 
 
 ^ ' ah— ax be— ox ac—ax ^ ' .5 
 
 = 1.6^ + 8.9. (3.) ^±^'^11^ = li^lZ^. (x = ^ifc^). (4.)^ 
 ^ ' p—q m \ m J ^ ' 2h—a 
 
 (Sbc + mVjx _ bah _ {Uc-a(l)x _ 6a(2b —a) / _ 5^h-a)\ 
 ~ 2ah(a + h) ~ 3c^ " 2ab(a-h) ^^~h^' V~ 'Sc-d J' 
 
 ax ^j _ ex ( _ ^ah^+4.h^~Ua^b \ Am(K^-bx') 
 
 ^""'^ ^{Zr^'^'^^-^da + b' V" '6a^+ab-ac + be)' ^^^ %x 
 
 -imp^ 4 ' V-^^p^-bg^y ^^'Hx^dx^fx^hx-"' 
 
 8.5 .2 1 - .Ix 2 -3a: 5a: 2a:-3_a:-2 
 
 (10.) 5-a:(3i-?)=ia:- 
 
 3a: - (4 - 5a:) 
 
^ ELEMENTARY ALGEBRA. 
 
 3. In solving the following be careful to observe the suggestions 
 in,.,,: (..,|H)-JH)*K-~i)-»- »•¥-' 
 
 _x—5 x—6 
 ~ x—(j x—7' 
 
 4. Solve the following, giving special heed to the suggestions in 
 
 ^ ^ it 
 
 (28) : (1.) V^^=^ = 16 - V^. (2.) ^ — = 6-. (3.) Vx 
 
 Vb^ +x—b 
 
 -^Va; — 7= — . (4.) — ==::= 1 = . (o.) ^ a-VV x 
 
 \^x-l \/5x4-3 2 
 
 + Vrt_V^=V^. (6.) V(l+rt)* + (l-a)a:4-V(l-«)' + (l+rt)i 
 
 =2fl. (7.) V!l3 + V'[7 + A/(3 + V.r)]!=4. (8.) ^^^lTV(3 -f V6a.-) 
 
 ... (9.) Vjng^^VO^-Q^ (10.) ^^^+^^\^=1G.-8V3^ 
 V(3;c + 2 4V««+6 16a;-3 
 
 + 3. (11.) ^^-^^"^^^ = ^. (12.) "f_-^ =4 + ^^'-' 
 
 f? 4- V«* — X* 's/ax +1 ^ 
 
 nq\ 3V^— 4 _ 15 + \/9^ a/^ + V^ _ Va 4- a/^ 
 
 2 + a/^ 40 + a/^ ' V«^ — V^ V^ 
 
 /i-\ a/« — '^rt— a/<^* — «a; ., ,.^. V in ^ \/ m — y 1 
 
 (lo). =r = 0, (lb.) — — = ;z=^ = — • 
 
 ^/liJ^W a-^ni^-ax V m - ^/ m - y ^ 
 
 (17.) ^ + ^+V'j^tf! = J. (18.) ^^^^-^^-^ = i. 
 
 rt + .T — A/2rta; + x^ yx + 1 + V a; — 1 
 
 /^ci\ J/~~r- . .V 7. ,^_ . l+a;+ a/2^+^ a/^ + sj+a/^ 
 
 (19.) A/rt + .y + Va—x = b. (20.) =rt — — . 
 
 1+x— V^'^x + x^ V^-hx—Vx 
 
 (^l.)^S±^=4. (22.) -^=3 - + J -. 
 
 \/ox-\-l—\^'6x Va — x + Va ^/ a — x — ^ a 
 
 _ A^ 
 
 ~ X 
 
 [Several of these equations cau be noro eb^aatly reduced by the method given on p. 138» 
 Ex. 47.] 
 
APPLICATIOKS OF SIMPLE EQUATIONS. 
 
 Applicatioks. 
 
 29, x^ccording to the definition (2), Algebra treats of, 1st, The 
 nature and properties of the Equation ; and 2d, the method of using 
 it as an instrument for mathematical mvestigation. 
 
 Having on the preceding pages explained the nature and proper- 
 ties of tlie equation, we now give a few examples to illustrate its 
 utility as an instrument for mathematical investigation. 
 
 30, The Alf/ebraic Solution of a problem consists of two 
 parts : 
 
 1st. Tlie Statement^ which consists in expressing by one or 
 more equations the conditions of the problem. 
 
 2d. The Solution of these equations so as to find the values of 
 the unknown quantities in known ones. This process has been 
 explained, in the case of Simple Equations, in the preceding articles. 
 
 31, The Slateinent of a problem requires some knowledge of the 
 subject about which the question is asked. Often it requires a great 
 deal of this kind of knowledge in order to "state a problem." This 
 is not Algebra ; but it is knowledge which it is more or less important 
 to Ixave according to the nature of the subject. 
 
 32, IPirections to guide the student in the Statement of Prob- 
 lems : 
 
 1st. Study the meaning of the problem, so that, \fyou had the answer given, 
 yon could prate it, noWcmg csLTGinWy just what operations you would have to 
 perform upon the answer in proving. This is called. Discovering the relations 
 between the quantities involved. 
 
 2d, Represent the unknown (required) quantities (the answer) by some one or 
 more of tlie final letters of the alphabet, as t, y, z, or w, and the known quan- 
 tities by the other letters, or, as given in the problem. 
 
 3d, Lastly, by combining the quantities involved, both knoiun and vnknown^ 
 according to the conditions given in the problem (as you would to prove it, if the 
 ahsAver were known) express these relations in the form of an equation. 
 
 mi. Sen. — It is not always expedient to use x to represent the number 
 sought. Tlie solution is often simplified by letting x be taken for some 
 number from which the one sought is readily found, or by letting 2x, Sa;, or 
 some multiple of x stand for the unknown quantity. The latter expedient 
 is often used to avoid fractions. 
 
90 ELEMENTARY ALGEBRA. 
 
 Problems. 
 
 1. A's age is double B's, B's is triple C's, and the sum of their 
 ages is 140. Required the age of eucli. 
 
 2. A's age is m times B's, B's is n limes C's, and the sum of their 
 ages is s. liequired the age of each. 
 
 3. The sum of two numbers is 48, and their difference 12. What 
 are the numbers? 
 
 4. The sum of two numbers is «, and their difference d. What 
 are the numbers ? 
 
 5. Having the sum and difference of two numbers given, how do 
 you find the numbers, arithmetically ? 
 
 G. A post is -Jth in the earth, ^ths in the water, and 13 feet in the 
 air. What is the length of the post ? 
 
 7. A post is -itli in the earth, ^ths in the water, and a feet in the 
 air. What is the length of the post ? 
 
 8. What fraction is that, whose numerator is less by 3 than its 
 denominator; and if 3 be taken from the numerator, the value of 
 the fraction will be J? 
 
 9. G'wQ i\\Q general &o\\\i\on of the last; /.c, the solution when 
 the numbers are all represented by letters. Then substitute the 
 above numbers and find the answer to that spetial problem. 
 
 SuG. — Letting the numerator be a less than the denominator, and -^ be the 
 
 am + bn 
 
 fraction after b is taken from the numerator, the fraction is t-it- 
 
 an + bn 
 
 10. A man sold a horse and chaise for 1200 ; J the price of the 
 horse was equal to \ the price of the chaise. Required, the price of 
 each. Chaise, $120 ; horse, $80. 
 
 Generalize and solve the last, and then by substituting the numbers given in 
 it find the special answers. Treat in like manner the next nine problems. 
 
 11. Out of a cask of wine which had leaked away a third part, 21 
 gallons were afterward drawn, when it was found that one- half re- 
 mained. How much did the cask hold ? J «.s., 126 galls. 
 
 12. A and B can do a piece of work in 12 days, but when A worked 
 alone he did the same work in 20. How long would it take B to do 
 the same work ? Ans.^ 30 days. 
 
APPLICATIONS OF SIMPLE EQUATIONS. 91 
 
 13. A cistern ctui be filled by 3 pipes; by the first in IJ hours, by 
 the second in 2J hours, and by the third in 5 hours. In what time 
 will the cistern be filled, when all are left open at once ? 
 
 14. Four merchants entered into a speculation, for which they 
 subscribed 4755 dollars; of which B paid tiiree times as much as A; 
 C paid as much as A and B ; and D paid as much as C and B. What 
 did each pay ? 
 
 15. A and B trade with equal stocks. In the first year A tripled 
 his stock and had $'27 to spare ; B doubled his stock, and had 1153 
 to spare. Now the amount of both their gains was five times the 
 stock of either. What was that ? 
 
 16. A and B began to trade with equal snms of money. In the 
 first year A gained 40 dollars, and B lost 40; but in the second A 
 lost one-third of what he tiien had, and B gained a sum less by 40 
 dollars than twice the sum that A had lost; when it appeared that 
 B had twice as much money as A. What money did each begin 
 with ? Ans., 320 dollars. 
 
 17. What number is that to which if 1, 5, and 13 be severally 
 added, the first sum divided by the second shall equal the second 
 divided by the third? 
 
 18. Divide 49 into two such parts that the greater increased by 6 
 divided by the less diminished by 11, shall be 4|^. 
 
 10. A cistern which contains 2400 gallons can be filled in 15 
 minutes by three pipes, the first of which lets in 10 gallons per 
 minute, and the second 4 gallons less than the third. How much 
 passes througli each pipe in a minute? 
 
 20. Find a number such that, if from the quotient of the number 
 increased by 5, divided by the number increased by 1, we subtract 
 the quotient of 3 diminished by the number, divided by the number 
 diminished by 2, the remainder shall be 2. 
 
 21. Divide a into two such parts, that one may be the ^th part of 
 the other. 
 
 22. Divide a into two such parts, that the sum of the quotients 
 which are obtained by dividing one part by m, and the other by n, 
 
 shall be equal to b. The parts are —^ , and — . 
 
 ^ ^ n—m 7)1— n 
 
92 ELEMENTARY ALGEBRA. 
 
 J33. Letting p represent the principal, i the interest for time 7, a 
 the amount, and r ihe per cent, for a unit of time, produce the fol- 
 lowing formnlcB, and give their meaning : 
 
 (2.) a=p^-i=p ^^^ 
 
 100' 
 
 100 + tr 
 
 rp 
 
 (5.) p = 
 
 (6.)p 
 
 tr ' 
 100^, 
 100 + tr 
 
 24. In what time will a given principal double, triple, or quadru- 
 ple itself, at 5^? at 6^? at 7^? 
 
 25. What is the worth of a note of 1500 Nov. 2d, 1872, which is 
 dated Feb. 23d, 1870, bears 12^ interest, and is due Jan. 1st, 1875, 
 money being worth 7^ ? A ns., $087.23 + 
 
 26. On a sum of money borrowed, annual interest is paid at 5^. 
 After a time 1200 are paid on the principal, and the interest on the 
 remainder is reduced to i^. By these changes the annual interest is 
 lessened one-third. What was the sum borrowed ? 
 
 27. An artesian well supplies a manufactory. The consumption 
 of water goes on each week-day from 3 a.m. to 6 p.m. at double the 
 rate at which the water flows into the well. If the well contained 
 2250 gallons of water when the consumption began on Monday 
 morning, and tlie well was just emptied at 6 p.m. on the next Tliurs- 
 day evening but one, how many gallons flowed into the well per 
 hour ? 
 
 28. The hind and fore wheels of a carriage have circumferences 
 16 and 14 feet respectively. How far has the carriage advanced 
 when the fore wheel has made 51 revolutions more tlian the other? 
 
 29. A merchant gains the first year 15^ on his capital; the second 
 year, 20^ on the capital at the close of the first; and the third year, 
 25^ on the capital at the close of the second ; when he finds that he 
 has cleared $1000.50. Required his capital. Capital |;1380. 
 
 30. A man had 12550 to invest. He invested part in certain 3^ 
 stocks, and part in R. R. shares of $25 each, wjiich pay annual divi- 
 dends of $1.00 per share. The stocks cost him $81 on a hundred, 
 and the R. R. shares $24 per share ; and his income from each source 
 is the same. Huw many R. R. shares did he buy ? 
 
SIMPLE EQUATIONS WITH TWO UNKNOWN QUANTITIES. 93 
 
 SECTION II, 
 
 INDEPENDENT, SIMULTANEOUS, SIMPLE EQUATIONS WITH TWO 
 UNKNOWN QUANTITIES. 
 
 Definitions. 
 
 34. Independent Equations are such as express different 
 conditions, and neither can be reduced to the other. 
 
 35. Siinultiineoiis JEquatlons are those wliich express dif- 
 ferent conditions of the same problem, and consequently the letters 
 representing the unknown quantities signify tlie same things in each. 
 All the equations of such a group are satisfied by the same values of 
 the unknown quantities. 
 
 36. Eli^ninatlon is the process of producing from a given 
 set of simulran^'ous equations containing two or more unknown 
 quantities, a new s^^t of equations in which one, at lejist, of the un- 
 known quantities shall not appear. The quantity thus disappearing 
 is said to be eliminated. (The word literally inean.s j^utting out of 
 doors. We use it as meaning causing to disappear.) 
 
 37. There are Five Methods of Elimination, viz., by 
 Cowpariso7i, by Substitution, by Addition or Subtraction, by Unde- 
 termitied Multipliers^ and by Division. 
 
 Elimination by Comparison. 
 
 38. I^VOb. 1. — Having given two inde]?endent, simidtaiieous^ 
 simple equations hetioeen ttoo tenknown quantities, to deduce therefrotn 
 by Comparison a new equation coiitaining only one of the unknoicn 
 quantities. 
 
 RULE. — 1st. Find expressions for the value of the same 
 
 UNKNOWN QUANTITY FROM EACH EQUATION, IN TERMS OF THE 
 OTHER UNKNOWN QUANTITY AND KNOWN QUANTITIES. 
 
 2d. Place these two values equal to each other, and the 
 
 RESULT WILL BE THE EQUATION SOUGHT. 
 
 Dem. — The first operations being performed according to the rules for simple 
 equations with one unknown quantity, need no further demonstration. 
 
94 ELEMENTAIiY ALGEBRA. 
 
 2d. Having formed expressions for the value of the same unknown quantity 
 in both equations, since the equations arc simultaneous this unknown quantity 
 means the same thing in the two equations, and hence the two expressions for 
 its value are equal. Q. e. d. 
 
 ScH. — The resulting equation can be solved by the rules already given. 
 
 Elimination by Substitution. 
 
 39^ JProh, 2, — Having given two independent^ simidtaneous, 
 simple equations, between two unknown quantities, to deduce there- 
 from by Substitution a single equation with btU one of the unknown 
 quantities. 
 
 RULE.— 1st. Find from one of the equations the value 
 OF the unknown quantity to be eliminated, in terms of the 
 other unknown quantity and known quantities. 
 
 2d. Substitute this value for the same unknown quan- 
 tity IN the other equation. 
 
 Dem. — The first process consists in the solution of a simple equation, and is 
 demonstrated in the same way. 
 
 The second process is self-evident, since, the equations being simultaneous, 
 the letters mean the same thing in both, and it does not destroy the equality of 
 the members to replace any quantity by its equal, q. e. d. 
 
 Elimination by Addition or Subtraction. 
 
 40, I^vob, 3, — Having given two independent^ simultaneous, 
 simple equations betwee^i two unknown quantities, to deduce therefrom 
 by Additioji or Subtraction a single equation with hut one unknown 
 quantity, 
 
 RULE. — 1st. Reduce the equations to the forms ax + by 
 = m, AND ex -[• dy = n. 
 
 2d. If the coefficients of the quantity to be eliminated 
 are not alike in both equations, make them so by finding 
 their L. C. M. and then multiplying each equation by this 
 
 L. C. M. exclusive of the factor which the TERil TO BE 
 
 eliminated already contains. 
 
 3d. If the signs of the terms containing the quantity to 
 be eliminated are alike in both equations, subtract onk 
 
SIMPLE EQUATIONS — ELIMINATION. 95 
 
 EQUATION FROM THE OTHER, MEMBER BY MEMBER. If THESE SIGNS 
 ARE UNLIKE, ADD THE EQUATIONS. 
 
 Dem. — The first operations are perfQrmed according to the rules already given 
 for clearing of fractions, transposition, and uniting terms, and hence do not viti- 
 ate the equations. The object of this reduction is to make the two subsequent 
 steps practicable. 
 
 The second step does not vitiate the equations, since in the case of either 
 equation, both its members are multiplied by the same number. 
 
 The third step eliminates the unknown quantity, since, as the terms containing 
 the quantity to be eliminated have the same numerical value, if they have the 
 saitie sign, by subtracting the equations one will destroy the other, and if they 
 have different signs, by adding the equations they will destroy each other. The 
 result is a true equation, since. If equals (the two members of one equation) are 
 added to equals (the two members of the other equation), the sums are equal. 
 Thus we have a new equation with but one unknown quantity, q. E. D. 
 
 Elimination by Undetermined Multipliers. 
 
 4:1. I*VOb, 4, — Having given two indepe7ident, simultaneous, 
 simple equations between, two unknown quantities^ to deduce therefrom 
 by Undetermined Midtipliers a single equation with but one unknown 
 quantity. 
 
 RULE. — 1st. Reduce the equations to the forms ax -f by 
 = m, AND ex + dy = n, 
 
 2(1. Multiply one of the equations by an undetermined 
 
 FACTOR, AS /, and FROM THE RESULT SUBTRACT THE OTHER EQUA- 
 TION, MEMBER BY MEMBER. 
 
 3d. In the resulting equation, place the coefficient of 
 
 THE UNKNOWN QUANTITY TO BE ELIMINATED EQUAL TO 0; FROM 
 this equation find the value of /, ANP SUBSTITUTE IT IN THE 
 OTHER TERMS OF THE EQUATION. 
 
 Dem. — [Reason for the first step, same as in the last method.] 
 
 Now multiply one of the equations, as «cj + by = m, by /, and subtract tlio 
 otuer, member by member, giving (af— c)x + (&/— d)y = mf— n. To eliminate 
 
 y, put bf—d = 0, giving f = -. This value of / substituted in {af — c).v 
 
 + (V— d)y = mf— n, will cause the term containing y to disappear by making 
 its coefficient 0, and there will result an equation containing only the unknown 
 quantity x, and known quantities. Q. E. d. 
 
96 ELExMENTAKY ALGEBRA. 
 
 Thus, given 3^ + 7^ — 33, and 2.f + 4y =20. 
 
 Multiply the 1st by/, 3^iP + 7/y = 33/ 
 
 Subtract the 2d, 2x' + Ay = 20 
 
 And we have (3/- 2)* + (7/- % = 33/- 20. 
 
 Putting 7/ - 4 = 0, / = f . Substituting, (3 x ^-2)j! = 33 x * - 20. Whence, 
 
 In like manner, putting 3/ - 2 :=i 0,/= f. And (7 x ^ - 4)y = 33 x ^ - 20- 
 Whence y = 3. 
 
 Elimination by Division. 
 
 42, Proh, S» — Havmg yioen t%co independent^ sinndtaneoiis, 
 equations of any decp'ee^ betireen two tmknown quaiitities^ to deduce 
 therefrom by Division a sinyle equation with but one unknown 
 quantity. 
 
 RULE. — Clear the equations of fractions, and transpose 
 
 ALL the terms TO ONE MEMBER. TrEAT THE POLYNOMIALS THUS 
 OBTAINED AS IN THE PROCESS FOR FINDING THE HIGHEST COM- 
 MON Divisor, continuing the process until one unknown 
 
 QUANTITY DISAPPEARS FROM THE REMAINDER. PUTTING THIS RE- 
 MAINDER EQUAL TO 0, WE HAVE THE EQUATION SOUGHT. 
 
 Dem. — Since each of the polynomials is equal to 0, any number of times one 
 subtracted from the other (t. c. any remainder) is 0. 
 
 Examples. 
 
 [Note. — Tlie pupil should solve the following by each of the preceding 
 methods, so as to make all familiar, and in each instance notice what method is 
 most expeditious.] 
 
 (1.) 2x4-7j/=41, (2.) a: + 152/ = 49, (3.) Qx+ 4y=236, 
 3a: + 4?/=42. 3a;+ ltj=n, 3.r + 15y=573. 
 
 (4.) %^x^\1b = Uy, (5.) 188-5.r-9!/=0, (6.) bx-A=Zy, 
 
 87a;-56y=497. 13a:=57 + 2y. 10 + 7a;- 12?/= 0. 
 
 (7.) 5i/-21= 2rc, (8.) 72/-3a:=139, (9.) ^^-\lx= 103, 
 13a;-4?/=120. 2x-Vhy= 91. Ux-l^=-4:l. 
 
 (10.) ?Z:?-l^- = ?^, (11.) ab. + ccly=% 
 
 %y+i_ix+y + 13 d-h 
 
 -3- 1 -. ax-cy=-^. 
 
SIMPLE EQUATIONS — ELIMINATION. 97 
 
 (12.) T^=u-^, (13.) {b + c){x+c-b)+a(y-\-a)=2a^, 
 
 b-\-y '6a + x 
 ax-\-%hy=d. 
 
 ay _ {h + cY 
 {b-c)x ~ fl2 • 
 
 (15.) 2.4.+.32,-:?^^p!^=.8. + ?:^±^, My^^^n^^ 
 
 "2x by ^x y 
 
 i^a^ "^~i^ T~3 ^ , x-y 1 
 
 (16.) — --T — = 2, and — -^ = -. 
 
 4 T 
 
 (17.) i + |- = 5, (18.)^+-=19, (19.) -^+^=m +71, 
 ^ ' ax by ^ ' X y ^ ' nx my ' 
 
 5 3 „ 8 3^ w ?;i „ . 
 
 ^=2. =7. - + - =m2+7z«. 
 
 ax by ^^ y ^ y 
 
 [Note. — Solve the following by {42).] 
 
 20. Eliminate x between the following: 5x-{-y=106 and x + Sy 
 =35 ; also i(2a: + 3y) = 8-^:1' and U+y=l(7y-dx) ; also iiy-2) 
 -i{^0-tj)={(z-10) and i(2z + A)-i{2y + z) = i(y+13); also 
 x^+6xy=U4: and 6a-?/ + 36y2=432 ; also a:3 4-?/3=:2728 and x^-xy 
 +^2 — 124; iilso x^+x^yi-x^y^ -\-xy^ -\-y^ = la,jid.x^ -{-y^=2; also 
 x-\-y + xy=M and a;2+?/2 = 52. 
 
 SOLUTION OP THE LAST. 
 
 x-{-y + xy-34: ) a;^ +i/g-52| a:4-34-y 
 
 (l+y)a: + y-34 ) (l+y).r2 + (l+i/)^2_52(i+y) 
 (1 -\-y)x^ -\-xy—'d4:X 
 
 (34-7/)a;+(:?/«-52)(l + i/) 
 (l+y)(U-y)x + {y^'-o2)(l+yY 
 (l+y){U-y)x-m-yy 
 Equation sought, (34-?/)2 + (;//2-52)(l + ?^)2=0. 
 
 [Note. — The equations resulting from the elimination in several of the above 
 cases are of degrees higher than the first, and hence their reeolutipn is not to 
 be expected at this stage of the student's progress.] 
 
 7 
 
98 ELEMENTARY ALGEBRA. 
 
 Applications. 
 
 1. A wine merchant has two kinds of wine, one worth 72 cents a 
 quart, and the other 40 cents. How. much of each must he put in a 
 mixture of 50 quarts, so that it shall be worth 60 cents a quart? 
 
 2. A crew that can pull at the rate of 12 miles an hour down the 
 stream, finds that it takes twice as long to row a given distance up 
 stream as down. What is the rate of the current ? 
 
 3. A man sculls a certain distance down a stream which runs at a 
 rate of 4 miles an hour, in 1 hour and 40 minutes. In returning it 
 takes him 4 hours and 15 minutes to reach a point 3 miles below his 
 starting place. How far did he scull down the stream, and at what 
 rate could he scull in still water? 
 
 4. A man puts out $10,000 in two investments. For the first he 
 gets 5^ and fur the second 4j^. The first yields annually $50 more 
 than the second. What is each investment? 
 
 [Note. — Generalize the statement and solution of the preceding problems.] 
 
 5. AVhat fraction is that whose numerator being doubled and de- 
 nominator increased by 7, the value becomes f; but the denomina- 
 tor being doubled, and the numerator increased by 2, the value be- 
 comes I ? 
 
 6. There is a number consisting of two digits, which is equal to 
 four times the sum of those digits ; and if 18 be added to it, the 
 digits will be inverted. What is the number? 
 
 7. A work is to be printed, so that each page may contain a cer- 
 tain number of lines, and each line a certain number of letters. If 
 Ave wished each page to contain 3 lines more, and each line 4 letters 
 more, then there would be 224 letters more in each page; but if we 
 wished to have 2 lines less in a page, and 3 letters less in each line, 
 then each page would contain 145 letters less. How many lines are 
 there in each page ? and how many letters in each line ? 
 
 8. A sum of money put out at simple interest amounted to $5250 
 in 10 months, and to $5450 in 18 months. What was the principal, 
 and what the rate ? 
 
SIMPLE EQUATIONS WITH TWO UNKNOWN QUANTITIES. 99 
 
 9. In an alloy of silver and copper, — of the whole + i? ounces 
 
 was silver^ and — of the whole — q onnces was copper. How many 
 ounces were there of each ? 
 
 10. When a is added to the greater of two numbers, it is m times 
 the less; but when h is added to the less, it is n times the greater. 
 What are the numbers ? 
 
 11. When 4 is added to the greater of two numbers, it is 3;^ times 
 the less ; but when 8 is added to the less, it is J the greater. What 
 are the numbers? Solve by substituting in the results of the pre- 
 ceding. 
 
 12. There is a cistern into which water is admitted by three cocks, 
 two of which are of exactly the same dimensions. When they are 
 all open, five-twelfths of the cistern is filled in 4 hours ; and if one 
 of the equal cocks be stopped, seven-ninths of the cistern is filled in 
 10 hours and 40 minutes. In how many hours woifld each cock fill 
 the cistern ? 
 
 13. A banker has two kinds of change ; there must be a pieces of 
 the first to make a crown, and h pieces of the second to make the 
 same : now a person wishes to have c pieces for a crown. How many 
 pieces of each kind must the banker give him ? 
 
 Ans., —, of the first kind, -\ of the second. 
 
 o—a h—a 
 
 14. An ingot of metal which weighs 7i pounds loses jy pounds when 
 weighed in water. This ingot is itself composed of two other metals, 
 which we may call M and M' ; now n pounds of M loses q pounds 
 when weighed in water, and 7i pounds of M' loses r pounds when 
 weighed in water. How much of each metal does the original ingot 
 contain ? 
 
 Ans., — ^ pounds of M, -^ — ^ pounds of M'. 
 
 r — q ^ r — q ^ 
 
100 ELEMENTARY ALGEBRA. 
 
 A 
 
 SECTION III. 
 
 INDEPENDENT, SIMULTANEOUS, SIMPLE EQUATIONS WITH MORE 
 THAN TWO UNKNOWN QUANTITIES. 
 
 43, Prob, — Having given several independent^ simultaneous, 
 simple equations^ involving as many unknown quantities as there are 
 equations, to find the values of the unknown quantities. 
 
 RULE. — Combine the equations two and two by any of 
 
 THE METHODS OF ELIMINATION, ELIMINATING BY EACH COMBINA- 
 TION THE SAME UNKNOWN QUANTITY, THUS PRODUCING A NEW SET 
 OF EQUATIONS, ONE LESS IN NUMBER, AND CONTAINING AT LEAST 
 ONE LESS UNKNOWN QUANTITY. COMBINE THIS NEW SET TWO AND 
 TWO IN LIKE MANNER, ELIMINATING ANOTHER OF THE UNKNOWN 
 QUANTITIES. REPEAT THE PROCESS UNTIL A SINGLE EQUATION IS 
 FOUND WITH BUT ONE UNKNOW^N QUANTITY. SOLVE THIS EQUATION 
 AND THEN SUBSTITUTE THE VALUE OF THIS UNKNOWN QUANTITY IN 
 ONE OF THE NEXT PRECEDING SET OF EQUATIONS, OF WHICH THERE 
 WILL BE BUT TWO, WITH TWO UNKNOWN QUANTITIES, AND THERE 
 WILL RESULT AN EQUATION CONTAINING ONLY ONE, AND THAT 
 
 another of the unknown quantities, the value of which 
 can therefore be found from it. substitute the two values 
 now found in one of the next preceding set, and find the 
 value of the remaining unknown quantity in this equation. 
 Continue this process till all the unknown quantities are 
 determined. 
 
 Dem. — 1. The combinations of the equations give true equations because they 
 are all made upon the methods of elimination already demonstrated. 
 
 2. That the number of equations can always be reduced to one by this pro. 
 cess, is evident, since, if we have n equations and combine any one of them with 
 each of the others, there will be ?i. — 1 new equations. Combining one of these 
 71 — 1 new equations with all the rest there wiU result n — 2. Hence n—\ 
 such combinations will produce a single equation; and as one unknown quan- 
 tity, at least, has disappeared from each set, there will be but one left. q. e. d. 
 
 ScH. 1. — If any equation of any set does not contain the quantity we are 
 seeking to eliminate, this equation can be Avritten at once in the next set, 
 and the remaining equations combined. 
 
 ScH. 2. — ^In eliminating any unknown quantity from a particular set of 
 
SIMPLE EQUATIONS WITH SEVERAL UNKNOWN QUA^TJTJ!ES. 101« 
 
 equations, any one of the equations may be combined with each of the 
 others, and the new set thus formed. But some other order may be prefer- 
 able as giving simpler results. 
 
 ScH. 3. — It is sometimes better to find the values of all the unknown 
 quantities in the same way as the first is found, rather than by substitution. 
 
 Examples. 
 1. 2. 3. 
 
 x + y + z = 31, X+ y + z = 9, 2x + dy -h 4:Z =. 29, 
 
 X + y — z = 25, X -i- 3y —dz= 7, dx + 2y + 5z = 33, 
 
 x — y — z= 9. ic — 4y + 83 = 8. 4^; + 3y + 2^ = 25. 
 
 5. 6. 
 
 \x + \y + iz=Q2, a;+i^=100, >! a:=64; ?^±?^ + 2^=8, 1 a:=3; 
 
 -^ I 
 
 ^^+iy + -i^=47, ;y + l^ = 100, 
 i-^+iy+i^=38. z-\-\x=\0^. 
 
 y=72', x + 2y-oz=2, > y=2; 
 
 7. 
 y + z 
 
 2 
 
 X -{- z 
 
 x-h- 
 
 y- 3 
 
 , + ^ = 85. 
 
 85, 
 85, 
 
 8. 
 
 ay + bx = c, 
 
 ex + az = b, 
 hz +cy = a. 
 
 2 1_3 
 
 X y~ ^ 
 
 y 
 
 i + i = *. 
 
 X z 3 
 
 10. 11. 
 
 y-{-z=2yz, 
 
 x-{-z=3xZf 
 
 x + y=4:xy. ^4-V=l. bcx -\- cay + abz=l. 
 
 13. xyz=a{yz—zx—xy)=b{zx—xy—yz)=c(xy—yz—zx). 
 
 12. 
 
 a b ^ 
 
 -+-=1, o;+y + z=0, 
 
 b c 
 
 - + -=1, (b-hc)x-\-(c-}-a)y + {a-{-b)z=0, 
 
 y ^ 
 
 Z X 
 
'ttfjt^'ti i •*A > ELEMENTABY ALGEBRA. 
 
 14 15. 16. 
 
 5x—'7z=n, dx—5y + 2z— 4:21 = 11, n-\-v-{-x + z = ll, 
 
 2a; + 3y=39, lOy-Sz -\-3u-2v=i 2, 2c + v + i/-\-z=12, 
 
 4y+32=41. 5z-{-4:U-\-2v—2x= 3, u-{-x-\-y + z=13, 
 
 6ic—3v+4:X—27/= 6. i; +.T + ?/ + ;?=: 14. 
 
 17. x+y + z=a-\-b + c, hx-\-cy + az=cx^-ay + hz=a^ +J)* -\-c^. 
 
 Applications. 
 
 1. Three persons, A, B, and C, were talking of their guineas; 
 says A to B and C, give me half of yours and I shall have 34; says 
 B to A and C, give me a third part of yours and I sliall have 34; 
 says C to A and B, give me a fourtii part of yours and I shall have 
 34. How many had each ? Ans., A 10, B 22, C 26. 
 
 2. For $8 I can buy 2 lbs. of tea, 10 lbs. of coffee, and 20 lbs. of 
 sugar, or 2 lbs. of tea, 5 lbs. of coffee, and 30 lbs. of sugar, or 3 lbs. 
 of tea, 5 lbs. of coffee, and 10 lbs. of sugar. What are the prices ? 
 
 3. A person possesses a certain capital which is invested at a certain 
 rate per cent. A second person has £1000 more capital than the 
 first and invests it at one per cent, more; tlius his income exceeds 
 that of the first person by £80. A third person has £500 more 
 capital than the second, and invests it one per cent, more advan- 
 tageously ; and thus receives £70 more income. Find the capital of 
 eacii and the rate of investment. 
 
 4. Find four nnmbers, such that the first with half the rest, the 
 second with a third the rest, the third with a fourth the rest, and 
 the fourth with a fifth of the rest shall each be equal to a, 
 
 5. A number is represented by 6 digits, of which the left-hand 
 digit is 1. If the 1 be removed to units place, the others remaining 
 in the same order as before, the new number is 3 times the original 
 number. Find the number. 
 
 6. A man has £22 14,<?. in crowns (55.), guineas (21^.), andmoidores 
 (275.) ; and he finds that if he had as many guineas as crowns, and 
 as many crowns as guineas, he would have £36 G.9. ; but if he had 
 as many crowns as moidores, and as many moidores as crowns, he 
 would have £45 lGcS\ How many of each has he? 
 
SIMPLE EQUATIONS WITH SEVERAL UNKNOWN QUANTITIES. 103 
 
 7. A person has four casks, the second of which being filled ft-om 
 the first, leaves the first -f full. The third being filled from the 
 second, leaves it J full; and when the third is emptied into the 
 fourth, it is found to fill only j\ of it. But the first will fill the third 
 and fourth and have fifteen quarts remaining. How many quarts 
 does each hold ? 
 
 8. A, B, C, and D, engage to do a certain work. A and B can do 
 it in 12 days, A and D in 15 days, and D and C in 18 days. B and C 
 commence the work together, after 3 days are joined by A, and after 
 4 days more by D. Then, all working together, they finish it in 
 2 days. How long would each have required to do the entire work? 
 Solve with one unknown quantity, as well as with four. 
 
 9. A person sculling in a thick fog. meets one tug and overtakes 
 another which is going at the same rate as the former ; show that if 
 a is thj greatest distance to which he can see, and b, V are the dis- 
 tances that he sculls between the times of his first seeing and of hid 
 
 2 11 
 passing the tugs, - = . + t-,- 
 
104 ELEMENTARY ALGEBRA. 
 
 CHAPTER II. 
 
 BATIO, PBOPORTION, AND mOGBESSION. 
 
 SECTION I. 
 RATIO. 
 
 44:. Ratio is the relative magnitude of one quantity as com- 
 pared with another of the same kind, and is expressed by the quotient 
 arising from dividing the first by the second.* Tlie first quantity 
 named is called tlie Antecedent^ and the second the Consequent. 
 Taken together they are called the Terms of the ratio, or a Covplet. 
 
 43, The Sif/n of ratio is the colon, : , the common sign of 
 dNision, -^, or the fractional form of indicating division. 
 
 The last form is coining into use quite generally, and is to be preferred. 
 
 46, Cor. — A ratio being merely a fraction, or an unexecuted 
 problem in Division, of which the antecedent is the numerator, or 
 dividend, and the consequent the denominator, oi' divisor, any changes 
 made upon the terms of a ratio produce the same effect upon its value, 
 as the like changes do upon the value of a fraction, when made upon 
 its corresponding terms. The priiicipal of these are, 
 
 1st. If both terms are mtdtiplied, or both divided by the same 
 number, the value of the ratio is not changed. 
 
 2d. A ratio is multiplied by multiplying the antecede^it, or by 
 dividing the consequent. 
 
 3d. A ratio is divided by dividing the antecedent, or by multiply- 
 ing the consequent. 
 
 * There is a common notion among us that the French exprcs* a ratio by dividing the con- 
 Fequent by the antecedent, uhilo the English express it as above. Such is not the fact. 
 French. German, and English writers agree ia the above definition. In fact, the Germans very 
 generally use the sign : instead of ^^ and by all, the two signs arc used vm exact equivalents. 
 
RATIO. 105 
 
 47 • A Direct Ratio is the quotient of the antecedent divided 
 by the consequent, as explained above, (44). An Indirect or 
 Reciprocal llatio is the quotient of the consequent divided by 
 the antecedent, i. e., the reciprocal of the direct ratio. A ratio is 
 always written as a direct ratio. 
 
 48, A ratio of Greater Inequality is a ratio which is 
 ^a-eater than unity, as 4 : 3. A ratio of Less Inequality is a 
 ratio which is less than unity, as 3:4. 
 
 49, A Compound liatio is the product of the corresponding 
 terms of several simple ratios. Thus, the compound ratio a : b, 
 c : d, m : n, is acm : bdn. This term corresponds to cotnpound frac- 
 tion. A compound ratio is the same in effect as a compound fraction. 
 
 50, A Duplicate Ratio is tlie ratio of the squares, a tri- 
 plicatCf of the cicbesy a snhduplicate^ of the square roots, and 
 a siibtriplicate, of the cube roots of two numbers. Thus, aJ^ : b^, 
 
 «3 ; ^3^ y^fi . .y/^^ r^ij^ y^^^ . ,^^ 
 
 Examples. 
 1. What is the-ratio of 8 to 4 ? of 4 to 8 ? of J to f ? of 6a^m 
 to3am? of x^-y^ tox-y? ofJto|? of - to | ? of ^'~^^ 
 
 91 X — X 
 
 . a + b^ 
 
 to- ? 
 
 1 — X 
 
 2. Write the inverse ratio in each case in the last paragraph. 
 
 3. Reduce the following to their lowest terms: 85 : 187, a^ — b^ 
 : a* - b*, n{a - xy : Q(a^ - x^). 
 
 4. Wliat is the duplicate ratio of 3:5, of rt:Z>? What the tripli- 
 cate ? What the subduplicate of 25 : IG ? of 3 : 7 ? of m : ^i ? What 
 the subtriplicate of 729 : 1728 ? of .r : y ? 
 
 5. AVhich is the greater, the compound ratio of |:f and 5:4, or 
 the inverse triplicate ratio of 3:2? 
 
 6. Prove that a ratio of greater inequality is diminished by adding 
 the same number to both its terms. How is it with a ratio of less 
 inequality? How with equality? 
 
 7. If 5 gold coins and 30 silver ones are worth as much as 10 gold 
 coins and 10 silver ones, what is the ratio of their values ? 
 
106 ELEMENTARY ALGEBRA. 
 
 8. Prove that a^ — x^ia^-hx^y a—x:a-\-x. Is x^ -\- y^ :x^ +y^ 
 greater, or less, than x* + i/^ : x -\- y? 
 
 9. Prove that 4:a^—3a^x — -kax^ + 3x^ : da^— 2a^x — 3nx^ + 2x^ 
 is equal to 4^ — 3x : 3a — 2x. 
 
 10. Prove that, if a: be to ^ in the duplicate ratio of a to J, and a 
 to b in the subduplicate ratio o^ a + x to a — y, then will 2x : a 
 ^x-y:y. 
 
 SECTION If. 
 
 PROPORTION. 
 
 51, Proportion is an equality of ratios, the terms of the ratios 
 being -expressed. The equality is indicated by the ordinary sign of 
 equality, =, or by the double colon, : : . 
 
 ScH. — The pupil should practice writing a proportion in the fonn r = ~,, 
 
 still reading it "a is to & as c is to rf." One form should be as familiar as 
 the other. He must accustom himself to the thought that a :b :: c : d means 
 
 .- = -- and nothing mare. 
 a 
 
 52, The Extremes (outside terms) of a i)roportion are the 
 first and fourtli terms. The Means (middle terms) are the second 
 and third terms. 
 
 53, A Mean Proportional between two quantities is a 
 quantity to which either of the othtr two bears the same ratio that 
 the mean does to the other of the two. 
 
 o4, A Tfiird Proportional to two quantities is such a 
 quantity that the first is to the second as the second is to this third 
 (proportional). 
 
 SS. A proportion is taken by Inversion when the terms of 
 each ratio are written in inverse order. 
 
 SO, A projwrtion is taken by Alternation when the means 
 are made to change places, or the extremes. 
 
 S7. A proportion is taken by Cotnjyosition when the sum of 
 the terms of eacli ratio is compared with eitlier term of that ratio, 
 the same order being observed in both ratios ; or when the sum of 
 
PROPORTION. 107 
 
 the antecedents and tlie sum of the consequents are compared with 
 either antecedent and its consequent. 
 
 S8. If the difference instead of the siim he taken in the last defi- 
 nition, the proportion is taken by Division. 
 
 SO. Four quantities are Inverseh/ or Reciprocally Proportional 
 wlien the first is to the second as the fourth is to the third, or as the 
 reciprocal of the third is to the reciprocal of the fourth. 
 
 60. A Continued Proportion is a succession of equal 
 ratios, in which each consequent is the antecedent of the next ratio. 
 Thus \i a-.b :\ b'.c '.: c. d '.: d: e, we have a continued proportion. 
 
 (il» Prop. 1. — In any proportion the product of the extremes 
 equals the product of the means. 
 
 Dem. — If a:b :: c:d then ad = be. For a:b :: c:d is the same as - = -, which 
 
 b d 
 cleared of fractions becomes ad = be. Q. E, d. 
 
 62, Cor. 1. — The square of a mean proportional equals the pro- 
 duct of its extremes, and hence a mean proportional itself equals the 
 square root of the product of its extremes. 
 
 If a:m :: m:d,\)y the proposition m* = ad. Whence extracting the square 
 root of both members, m ~ Vad. 
 
 63. Cor. 2. — Either extreme of a proportion equals the ^^roduct 
 of the means divided by the other extreme; and, in like manner, 
 either mean equals the product of the extremes divided by the other 
 mean. 
 
 64. Prop. 2, — Ty the product of two quantities equals the pro- 
 duct of two others, the two former may he made the extremes, or the 
 means of a p7'oportion, and the tioo latter the other terms. 
 
 Dem. — Suppose my = nx. Dividing both members by xy, we have — = -, 
 i. e.,m'.x :: n:y. In like manner dividing by w;i we have - = — , i. e., y.n 
 
 11 TTh 
 
 Deduce each of the following forms from the relation my = nx 
 
 1. m : X : ; ??. : y. 
 
 2. m : n :: X : y. 
 
 3. y : n •.: X : m. 
 
 4. X : y :: m: n. 
 
 5. 
 
 y 
 
 X : 
 
 : n 
 
 m 
 
 6. 
 
 X 
 
 m: 
 
 • y 
 
 n. 
 
 7. 
 
 n 
 
 m : 
 
 •• y 
 
 X. 
 
 8. 
 
 n 
 
 y- 
 
 : m 
 
 X. 
 
108 ELEMENTARY ALGEBRA. 
 
 6S» Cor. — If four qumitities are in proportion^ they are in pro- 
 portion by alternation and by inversion. 
 
 (jQ, Prop, 3, — If four qua7itities are i?i proportion, the propor- 
 tion is not destroyed by taking equal multiples of 
 
 1st. 77/e ie7')ns of the same couplet, 
 
 2d. 7%e antecedents, 
 
 3d. The co7isequents, 
 
 4th. All the tei-ms. 
 
 Demonstrate these facts from the nature of a proportion as an equality of 
 ratios. 
 
 07» Sen. — Observe that such changes, and only such, may be made upon 
 the tenns of a proportion without destroying it, jis 
 
 1st. Do not change the vdlues of the ration, 
 
 2d. Change loth ratios alike. 
 
 Query. — If the first term of a proportion be divided by any number, in what 
 ways may the operation be compensated for so as to preserve the proportion ? 
 
 08, JProj), 4, — The products or the quotients of t/ie corrtspond- 
 ing terms of two {or more) proi)ortions are proportional to each 
 other. 
 
 Demonstrated on the axioms that equals multiplied by equals give equal 
 products, and that equals divided by equals give equal quotients. 
 
 09. Cor. — Like powers, or roots, of proportionals are propor- 
 tional to each other. 
 
 How does this corollary grow out of the proposition ? 
 
 70, Prop. S. — If two proportions have a ratio in one equal to 
 a ratio in the other, the remaining 7'atlos are equal and may form a 
 p>ro2yo7'tion. 
 
 Demonstrated on the axiom that things which are equal to the same thing 
 are equal to each other. 
 
 Kl. Pvop. 0, — Afiy proportion may be takeii by composition, 
 or by division, or by both at once, without destroying it. 
 

 PROPORTION. 
 
 
 Dem.— If a:b'.:c.d, 
 
 
 
 a + h : h : : c + d : d, 
 
 (1) 
 
 We may write by 
 
 a + b : a : : c + d : c, 
 
 (3) 
 
 composition. 
 
 a + c : a : : h + d : b, 
 
 (3) 
 
 
 . a + c : c : : b + d : d. 
 
 (4) 
 
 10^ 
 
 By division, we may write the same forms with the — sign instead of the + . 
 
 By composition and f a + b : a - b : : c + d : c - d, 
 
 division at the same < 
 
 ., \ a + c : a — c : : b + d : b — d. 
 
 time, we may write, v. 
 
 These forms may all be verified by representing the ratio of « to 6 by r, 
 
 a 
 whence -r = r, or a = I'b, and since the ratio of c to d is the same as that of 
 
 a to b, - = r, orc= dr, and then substituting in each of the above forms these 
 d 
 
 values of a and c. Thus, the Ist becomes br + b : b :: di' + d : d, which ratios are 
 equal, smce each is r -i- 1 . Let the student verify the other forms in the same 
 way. 
 
 Queries. — If a : b : : c : d,\s a±b : a :: c±d :bt Is a-\-b : c + d : : a—c : b—dt 
 
 72, Cor. — If there be a series of equal ratios in the form of a 
 conti I med proportion^ the sum of all the antecedents is to the sum of 
 all the consequents, as any one antecedent is to its consequent, 
 
 Dexi. — If a :b : : c :d : : e :f: : g : h, etc., a + c + e + g + etc. : b {-d +/-I- h + etc, 
 : : a : &, or c : d,or e :f,OT g : h, etc. Substitute for a br, for c dr, for e fr, for 
 g hr, and we have 
 
 br + dr+fr + hr + etc. : b + d+f-\-7i + etc. : : h' : b, 
 in which the ratios are seen to be equal, since each is r. 
 
 73, ScH. — TJie method pursued in tlie demonstration of the preceding propo- 
 sition wUl be found sufficient in itself to test any projwsed transformation of a 
 propoHion. We will give a few examples : 
 
 1. l{ a : h : : c : d, prove as tibove that ad = be. 
 
 SUG. — By substituting as above we have the identity brd = bdr. 
 
 2. l^ (I : b : : c : d, prove as above that a :c :: b : d, and b : a:: d : c. 
 
 3. If a : b :: c : d, and m : n :: x : i/, prove as above that am : bn : : 
 ex : dy. 
 
 Sug's, — Let — = r, whence — = r; and — = r' , whence _ = /. Substitut- 
 b d n y 
 
 ing for a br, for c dr, for m nr' , and for x yr' , in the proportion to be tested, it 
 
 ie shown to be true. 
 
110 ELEMENTARY ALGEBRA. 
 
 4. If ^a — X : ^a + X : : b — 1/ : i) -\- J, show that 2x : y '.: a : b. 
 
 Sug's. — From! = r find x in terms of a and r, and from . — ^ = r find « 
 
 ia + a; & + y 
 
 in terms of & and r. 
 
 5. Ua:b::p: q, prove that a^ _|_ ^2 . _fL_^ ..2^2 + ^72 ; _P , 
 
 fl^ + ^^ p -^ q 
 
 6. Four given numbers are represented by a, b,Cyd; what quantity 
 added to each will make them proportionals? 
 
 . be — ad 
 
 A71S., -_ 
 
 a — — c -h d 
 
 7. If four numbers are proportionals, show that there is no num- 
 ber which, being added to each, will leave the resulting four num- 
 bers proportionals. 
 
 8. If a : b :: c : dy show that via :mb :: c:d; a :b :: mc : md', ma : 
 b :: mo : d; a : mb :: a : md ; and ma :71b:: mc : nd. 
 
 Applications. 
 
 [Note. — The first five of the following examples should be solved without 
 converting the proportions into equations.] 
 
 1. A merchant having mixed a certain number of gallons of brandy 
 and water, found that if he had mixed 6 gallons more of each, there 
 would have been 7 gallons of brandy to every 6 gallons of water, 
 but, if he had mixed 6 gallons less of each, there would have been 
 gallons of brandy to every 6 gallons of water. How much of each 
 did he mix ? 
 
 Solution, a; + 6 : y + 6 : : 7 : 6, and a; - 6 : y - 6 : : 6 : 5. 
 Ilencc « - y : y + 6 : : 1 : 6, and a; — y : y — 6 : : 1 : 5. 
 
 Hence y + 6 : y - 6 : : 6 : 5, or 2y : 12 : : 11 : 1, or y : 66 : : 1 : 1. 
 
 Substituting, « + 6 : 72 : : 7 : 6, or « + 6 : 6 : : 14 : 1, or .t : 6 : : 13 : 1, or a? : 
 1 : : 78 : 1. 
 
 2. Find two numbers, such, that their sum, difference, and pro- 
 duct, may be as the u umbers 5, d, and;j, respectively. 
 
 Solution, x + y : x — p :: s : d, and x — y:xy::d:p. 
 Hence <i^ ■ p ': s -i- d : s - d, and x : p :: dx + p : p. 
 
 Hence dx+p :p:: s + d: 8- d,ordx \ p :: 2d :a - d,OT X : p :: 2 :«-(!, 
 
 OTXil ■..2p:8-d,i.e.x = —^. 
 8 — d 
 
PROPORTION. HI 
 
 3. It is required to find a number, such, that the sum of its digits 
 is to the number itself as 4 to 13 ; and if the digits be inverted, their 
 difference will be to the number expressed as 2 to 31. 
 
 4. Divide the number 14 into two such parts, that the quotient 
 of the greater divided by the less shall be to the quotient of the less 
 divided by the greater, as 16 to 9. 
 
 5. Find two numbers whose difference is to the difference of their 
 squares as m : n, and wTiose sum is to the difference of their squares 
 RS a :b. 
 
 [Note. — In tlie following, use the proportion more or less, as is found ex- 
 pedient.] 
 
 6. The sides of a triangle are as 3 : 4 : 5, and the perimeter is 480 
 yards: find the sides. 
 
 7. A fox makes 4 leaps while a hound makes 3 ; but two of the 
 hound's leaps are equivalent to 3 of the fox's. What are their relative 
 rates of running? 
 
 8. A courier sets out from Trenton for Washington, and travels 
 at the rate of 8 miles an hour; two hours after his departure 
 another courier sets out after him from New York, supposed to be 
 68 miles distant from Trenton, and travels at the rate of 12 miles an 
 hour. How far must the second courier travel before he overtakes 
 the first ? 
 
 9. There are two places, 154 miles apart, from which two persons 
 set out at the same time to meet, one travelling at the rate of 3 miles 
 in two hours, and the other at the rate of 5 miles in four hours. How 
 long, and how far, did each travel before they met ? 
 
 10. A courier, who travels 60 miles a day, has been dispatched 
 five days, when a second is sent to overtake him, in order to do 
 which he must travel 75 miles a day. In what time will he overtake 
 the former? 
 
 11. Two travellers, A and B, set out at the same time from two 
 different places, C and D; A from C to D, and B from D to 0. 
 When they met, it appeared that A had gone 30 miles more than 
 B ; also, that A can reach D in 4 days, and B can reach C in 9 days. 
 Required the distance from C to D. 
 
 12. A hare, 50 of her leaps before a greyhound, takes 4 leaps to 
 the greyhound's 3; but two of the greyhound's leaps are as much as 
 
112 
 
 ELEMENTARY ALGEBnA. 
 
 three of the hare's. How many leaps must the greyhound take to 
 catch the hare ? 
 
 13. A runuer left this place n days ago, at the rate of a miles 
 daily. He is pursued by another, at the rate of b miles a day. In 
 how many days will the second overtake the first ? 
 
 . mi 
 
 Ans., -r . 
 
 — a 
 
 14. Find the time between 3 and 4 when the hands of a watch are 
 opposite each other. When they are at right angles to each other. 
 When they are together. 
 
 15. How often does the minute hand of a watch pass the hour 
 hand ? How often at right angles ? How often opposite? 
 
 16. Do the hands of a watch occupy the three relative positions 
 of opposite, at right angles, and together between each two hours of 
 the 12 ? If there are exceptions point them out, and show why they 
 occur. 
 
 17. Before noon, a clock which is too fast, and points to afternoon 
 time, is put back 5 hours and 40 minutes ; and it is observed that 
 the time before shown is to the true time as 29 to 105. Required 
 the true time. 
 
 18. Two bodies move uniformly around the 
 circumference of the same circle, which measures 
 8 feet. When they start, one is a feet before the 
 other; but the first moves m and the s^^cond M 
 feet in a second. AVhen will these bodies pass 
 each other the first time, when the second, when 
 the third, etc., supposing that tliey do not disturb 
 each other's motion ? When will they pass if 
 the first starts t seconds before the second, and M > m? When if 
 M < m ? When will they pass if the first starts / seconds later than 
 the second and M > m ? When if M < m ? When will they meet 
 if tliey start at the same time and move toward each other, or over 
 tlie distance a, first ? If they move from each other, or over the arc 
 s — a first ? When will they meet if ihi^ first starts / seconds before 
 the other, and they move toward each other, or over the distance a 
 first? If they move from each other, or over the arc s — a first? 
 If they move in opposite directions, and the first starts t seconds 
 later than the second? When they move over the arc a first? When 
 they move over the arc s — a first ? 
 
PROGRESSIONS. 113 
 
 19. The force of gravitation is inversely as the square of the dis- 
 tance from the centre of the earth. At the distance 1 from the 
 centre of the earth this force is expressed by the number 32.16. By 
 vrhat is it expressed at the distance 60 ? Ans., 0.0089. 
 
 20. If the velocity of one body moving around another is propor- 
 tional to unity divided by the duplicate of the distance, and the 
 velocity be represented by v when the distance is r, by what will it 
 be expressed when the distance is r' ? 
 
 Ans., -Tiv 
 
 SECTION III, 
 
 PROGRESSIONS. 
 
 74, A JProgvession is a series of terms which increase or de- 
 crease by a common difference, or by a common multiplier. The 
 former is called an Arithmetical, and the latter a Geometrical Pro- 
 gression. A Progression is Increasing or Decreasing according as 
 the terms increase or decrease in passing to the right. The terms 
 Ascending and Descending are used in the same sense as increasing 
 and decreasing, respectively. In an Arithmetical Progression the 
 common difference is added to any one term to produce the next term 
 to the right. If the progression is decreasing the common difference 
 is mimis. In an increasing Geometrical Progression the constant 
 multiplier by which each succeeding term to the riglit is produced 
 from the preceding is more than unity; and in a decreasing progres- 
 sion it is less than unity. This constant multiplier in a Geometrical 
 Progression is called the Ratio of the series. 
 
 t5. The character, ••, is used to separate the terms of an Arith- 
 metical Progression, and the colon, : , for a like purpose in a Geo- 
 metrical Progression. 
 
 ILLUSTRATIONS. 
 
 1 •• 3 •• 5 •• 7, etc., etc., is an increasing Arithmetical Progression with a common 
 
 difference 2, or + 3. 
 15-10-5 -O" — 5, etc., etc., is a Decreasing Aritnmetical Progression with a 
 common difference — 5. 
 a '• a ± d •• a ± 2d ■• a ± M, etc., etc., is the general form of an Arithmetical 
 Progression, ± d being the common difference. 
 
 2 : 4 : 8 : 16, etc., etc., is an increasing Geometrical Progression with ratio 2. 
 12 : 4 : ^ : ^ : jV, etc., etc., is a Decreasing Geometrical Progression with ratio i 
 
 8 
 
114 ELEMENTARY ALGEBRA. 
 
 ai ar : ar^ : ar^ : ar^, etc., etc., is the general form of a Geometrical Progres- 
 sion, r being the ratio, and greater or less than unity, 
 according as the series is increasing or decreasing. 
 
 76. When three quantities taken in order are in arithmetical pro- 
 gression, the second is t^e Arithmetical Mean between the other two, 
 and is equal to half their sum. 
 
 III. — If « •• & •• c, 6 is the arithmetical mean between a and c ; and since h — a 
 = c — b, b = \{a + c). 
 
 77, When three quantities taken in order are in geometrical pro- 
 gression, the second is the Geometric Mean between the other two, 
 and is equal to the square root of their product. 
 
 Let the student illustrate. 
 
 78. There are Five Tilings to be considered in any progression ; 
 Tiz., the first term, the last term, the common difference or the ratio, 
 the number of terms, and the sum of the series, any three of whicli 
 being given the other two can be found, as will appear from the sub- 
 sequent discussion. 
 
 Arithmetical Progression. 
 
 79, I^rop. 1, — The formula for finding the nth, or last term of 
 an Arithmetical Progressio7i ; or, more properly, the formula express- 
 ing the relation between the first term, the nth term, the common dif- 
 ference, and the 7iumber of terms of such a series, is 
 
 1 = a + (n - l)d, 
 
 in which a is the first term, d the common difference, n the number 
 of terms, and 1 the nth or last term, d being positive or negative 
 according as the seHes is increasing or decreasing. 
 
 Dem. — According to the notation, the series is 
 
 a '■ a + d ■' a + 2d •• a + dd - a -!- Ad •• a + M, etc., etc. 
 Hence we observe that as each succeeding term is produced by adding the com- 
 mon difference to the preceding, when we have reached the nth term, we shall 
 have added the common difference to the first term n—1 times ; that is, the nth 
 term, or I = a + {n — \)d. q. e. d. 
 
 Sen. — As this formula is a simple equation in terms of a, I, n, and <Z, any 
 ouc of them may be found in terms of the other three. 
 
ARITHMETICAL PROGRESSION. llo 
 
 SO, Pvop. 2, — 7%6 formulcL for the sum of an Arithmetical 
 Progression^ or expressing the relation between the sum of the series^ 
 the first term^ last tei'ni, and number of terms, is 
 
 s representing the sum of the series^ a the first term^ 1 the last term^ 
 ami 11 the number of terms. 
 
 Dem. — If I is the last term of the progression, the term before it is I —d, and 
 the one before that ^— 2rf, etc. Hence, as a-' a + d-'a + ^id'-a + 'dd I re- 
 presents tlie series, l-l—d-l—2d--l—Zd a represents the same series 
 
 reversed. Now the sum of the first series is 
 
 »=o4-(« + <Z) + (a + 2<^+ il—M) + {l—d)-k-l'y 
 
 and reversed 8=1 ■{■{l —d)-\-{l —'ii,d)-\- - - - {a + 2d) + {a + d)-{-a. 
 
 Adding 28={a + l) + {a + l) +{a-\-l)->r - - {a + l) + {a + l)-^{a + l). 
 
 If the number of terras in the series is n, there will be n terms in this sum, each 
 
 of which is ('fc + ^) ; hence 2« =(«4-0''. or « = "o" "• ^' ^' ^* 
 
 ScH. — This formula being a simple equation in terms of «, a, Z, and n, 
 any one of the four can be found in tenns of the other three. 
 
 51, Cor. 1. — Formulas 
 
 (1) l=a + (n-l)d, and 
 
 (2) s =1^*— ~Jn, being two equations between 
 
 the five quantities, a, 1, n, d, and s, any two of these five can be found 
 in terms of the other three. 
 
 52, Cor. 2. — The formula for inserting a given number of arith- 
 metical means between two given extremes is d=: -, in which m 
 
 represents the number of means. Froyn this d, the common differ- 
 ence, being found, the terms can readily be written. 
 
 Dem. — If a is the first term and I the last, and there are m terms between, 
 or m means, there are in all m. + 3 terms. Hence, substituting in the formula 
 
 I — a 
 l—a-\-Oi-l)d,ior ;?, m + 2, wc have l-.-a + {in+'l)d. From this <Z="— ^. Q. e. d. 
 
116 
 
 ELEMENTARY ALGEBRA. 
 
 S3. FoRMULiE IN Arithmetical Progression. 
 
 [It will afford a good exercise for the student to solve the following cases on 
 review, after having gone through Quadratics ; though no importance need be 
 attached to remembering the results, as the fundamental formulas 
 
 (1) l=a + {7i-l)d, and (2) «=r^Jw, 
 are sufficient to resolve all cases.] 
 
 NUMBEU. ! GIVEN. 
 
 I 
 
 BEQUIRSD. 
 
 FORMULAS. 
 
 1. 
 
 2. 
 3. 
 
 4. 
 
 a, d, n 
 a, d, S 
 
 a, n, S 
 d, n, S 
 
 I 
 
 . l=a + {n-\)d. 
 
 n 
 n 2 
 
 5. 
 6. 
 
 7. 
 8. 
 
 a, d, n 
 a, d, I 
 
 a, n, I 
 d, n, I 
 
 s 
 
 S=in{2a + (w-l>?}, 
 ^- 2 + 2d ' 
 
 S=\n{2l-{n'-l)d\. 
 
 9. 
 10. 
 11. 
 12. 
 
 d, n, I 
 d, n, S 
 d, I, S 
 n, I, S 
 
 a 
 
 a=l-{n-\)d, 
 ^_S_{n-l)d 
 n 2 ' 
 a-^d±^{l+id)^-2d^, 
 
 a= — —L 
 n 
 
 13. 
 14. 
 15. 
 16. 
 
 a, n, I 
 a, n, S 
 a, I, S 
 n, I, S 
 
 d 
 
 l-a 
 
 2(S-aw). 
 ^- n{n-\) 
 
 2(nl-S) 
 
 17. 
 18. 
 19. 
 
 20. 
 
 a, d, I 
 a, d, S 
 a, I, S 
 d, I, S 
 
 n 
 
 l-a ^ 
 
 ± \/{2a-dY + M^-2a + d 
 ""= 2d 
 2S 
 
 2l + d±^{2l + dY-M^ 
 ^- 2d 
 
GEOMETRICAL PROGRESSION. 117 
 
 Examples. 
 
 1. Find the 21st term of 3 •• 7 •• 11 •• etc., and the sum of these 
 terms. 
 
 2. Find the 24th term of 7--5--3-- etc., and the sum of these 
 terms. 
 
 3. Find the nth term of |..|..f.. etc., and the sum of the n 
 terms. 
 
 4. Find the ?ith term of ^^—~ •• -^^- -^^•. etc., and the sum 
 
 n n n 
 
 of the n terms. 
 
 5. Insert four uritlimetical means between 193 and 443. 
 
 6. Prove that tlie sum of 7i terms of 1 •• 3 •• 5 •• 7 •• etc., is to the 
 sum of m terms as n^ : 7)i^. 
 
 .7. What is the first term of an arithmetical progression whose 
 59th term is -:l\, aiidGOth -If? Whose 2d term is i, and 55th 5.8? 
 
 8. How many terms in the progression wliose common differ- 
 ence is 3, first term 5, and last term 302 ? 
 
 9. Insert three arithmetical means between w? and n. 
 
 10. Produce the formula for inserting m arithmetical means be- 
 tween a and h, viz., 
 
 am + h am — a -i- 2b bm — h-\-2a bin + a , 
 d . . . . _----_ . . . . . ^. 
 
 VI + 1 m + l VI + 1 m + 1 
 
 11. If a body falling to the earth descends a feet the first second, 
 Sa the second, 5a the third, and so on, how far will it fall during the 
 ^th second ? Ans., (2t — 1 )a. 
 
 12. If a body falling to the earth descends a feet the first second, 
 3a the second, 5a the third, and so on, how far will it fall in t 
 seconds? Aois-^at^. 
 
 Geometrical Progression. 
 
 84, JPrO]}, 1, — The formula for finding the nth, or last term 
 of a f/eowetrical prof/ressio7i ; or, more properly, the formula ex- 
 pressinr/ the relation between the first term^ the wth term, the ratio, 
 and the number of terms of such a. series, is 1 =:ar°~\ in which, 1 is 
 the last^ or nth term, a the first term, r the ratio ^ and n the number of 
 term^. 
 
118 KLEMENTAKY ALGEBRA. 
 
 Dem, — Letting a represent the first term and r the ratio, the series is 
 a : ar : «;•- : ar^ : ar^ : etc. Whence it appears that any term consists of the 
 first term multiplied iuto the ratio raised to a power whose exponent is one 
 less than the number of the term. Therefore the ni\\ term, or I =. ar'*~ \ Q. e. d. 
 
 83, J^i'Op. 2 ,— The formula for the sum of a geometrical j^ro- 
 gresslon, or exjjressvig the relation between the sum of the series, the 
 frst tei')n, the ratio, and the number of terms is 
 
 ar" — a 
 
 ^ = T3T' 
 
 in which s represents the sum, a the first term, r the ratio ^ and n the 
 number of terms. 
 
 Dem.— The sum of the series being found by adding all its terms, we have, 
 » = a -\- ar -\- at* ->r ar^ -v - - ar*-* + ar»-' + ar"-', and multiplying by r, 
 r8 = ar + ar* + ar' + . . ar"-» + gr"-* -f- ar*-^ + «?•*. Subtracting, 
 rs — s = ar* — a, or 
 {r — i)s = ar* — a, and s — — -. q. e. d. 
 
 T — 1 
 
 ^6*. Cor. 1. — Formulas 
 
 (1) 1 = ar»-*, and 
 
 ar" —" '1 
 
 (2) s = — ; -^ being two equations be- 
 tween the five quantities, a, 1, r, n, and s, are sufficient to determine 
 any two of them when the others are given. 
 
 87, Cor. 2. — Since 1 = ai-""*, lr=ar", which substituted in (2) 
 
 Ir — a 
 gives s — -; — -; which formula is often co7ivenient. 
 
 88, Cor. 3. — The formula for inserting m geometrical means 
 
 heticeen a and 1 is x = y -. 
 
 ^ a 
 
 89, Cor. 4. — The formula for the sum of an iiifinite decreasing 
 geometrical progression is s = . 
 
 Dem. — Since in a decreasing progression the ratio is less than unity, the last 
 term, ar"-^, is also less than the first term, and numerator and denominator of 
 
 the value of s, , become negative. Hence it is well enough to write the 
 
 formula for the sum of such a series s = zr— — , that is, change the signs of 
 
 1 — r 
 
 both terms of the fraction. Now, if the terms of a series are constantly decreas- 
 ing, and the number of terms is infinite, we can fix no value, however small, 
 which will not be greater tlmn the last, or than some term which may be 
 reached and passed. Hence we are compelled to call the last term of such a 
 
 series 0, which makes the formula s — . Q. E, D. 
 
GEOMETRICAL PROGllESSION. 
 
 119 
 
 90, Geometrical Formula. 
 
 [In a review, after the pupil has been through the hook, it will be a good exer- 
 cise for him to deduce the following formulas from the two fundamental ones?. 
 It is not necessary to memorize these,] 
 
 10. 
 
 11. 
 12. 
 
 GIVEN. 
 
 a, r, n 
 
 a, r, S 
 
 a, 11, S 
 
 r, n, S 
 
 a, r, 7i 
 
 a, r, 
 
 a, n, 
 
 r, n, 
 
 r, n, 
 
 r, n, S 
 
 r, I, S 
 
 n, I, S 
 
 REQUIRED. 
 
 I = «?•»-', 
 
 ^ a + (r-l)S 
 
 ~" r 
 
 I (s - 1)n-^ - «(S - a)"-' = 0, 
 ^^ (r-l)S7 -'>-' 
 r» — 1 
 
 S: 
 S: 
 
 a(r» — 1) 
 rl — a 
 
 r-1* 
 
 .-I - n-l 
 
 V ^" — V *** 
 ^r" — ; 
 
 
 (r - 1)S 
 r»-l ' 
 rl - (?• - 1)S, 
 
 = 0. 
 
 13. 
 14. 
 15. 
 16. 
 
 a, n, I 
 
 a, n, S 
 
 a, I, S 
 
 71, I. S 
 
 
 ?•«■ 
 
 — r + = 0, 
 
 a a 
 
 S-a 
 
 S ? 
 
 s - r ^ s-i 
 
 17. 
 18. 
 19. 
 20. 
 
 a, r, I 
 
 a, r, S 
 
 a, I, S 
 
 r, I, S 
 
 log r 
 log [g + (r — 1)S] — log a 
 log r 
 log I — log a 
 
 + 1, 
 
 log (S - a) - tog (S - /) 
 logl-log[lr-{r-\)S-] ^ ^ 
 log r 
 
120 ELEMENTARY ALGEBBA. 
 
 Examples. 
 
 1. In a geometrical progression the first term is 3, the ratio 5, and 
 the number of terms 7. What is the last term ? What the sum ? 
 
 2. Insert 5 geometrical means between 2 and 1458. 
 
 3. Find the llth term of -^ : ^^ : ^ : etc., and the sum of the 
 11 terms. 
 
 4. Find the 7th term of - | : J : ~ J : etc., and the sum of the 7 
 terms. 
 
 5. Insert 4 geometrical means between ^ and ^^. 
 
 6. Find the sum of 3 : J : ^ : etc., to infinity. Also of ^i —J 
 •.etc., to infinity. Also of .54. Also of .836. 
 
 7. If a body move 20 miles the first minute, 19 miles the second, 
 18^ the third, and so on in geometrical progression forever, what is 
 the utmost distance it can reach ? A?is., 400 miles 
 
 8. What is the distance passed through by a ball, before it comes 
 to rest, which falls from tiie height of 50 feet, and at every fall 
 rebounds half the distance, the time of ascent equalling the time cf 
 descent? Ans., 150. 
 
 9. In the preceding problem, suppose the body falls 16^ feet the 
 first second, 3 times as far the next second, and 5 times as far the 
 third second, and so on, how long will it be before it comes to rest ? 
 
 Ans., 3V^V'579(4 + 3\/2) = 10.27657 + seconds. 
 
 10. Find the sum of the following series : 
 
 i—i + i—-^+ etc., to n terms. 
 
 l+i + i + ^+ etc., to 10 terms. Also to infinity. 
 
 l|^ + .5+ etc., to 12 terms. Also to infinity. 
 
 11. To find what each payment must be in order to discharge a 
 given principal and interest in a given number of equal payments at 
 equal in tervals of time. 
 
 Solution. — Let p represent the principal, r tlie rate per cent., t one of the 
 equal intervals of time, n tlie number of payments (i. e., nt is the whole time), 
 and X one of the payments. 
 
 There will be as many solutions as there are different methods of computing 
 interest on notes upon which partial payments have been made. 
 
 1st. By the United States Court Mule. — As the payments must exceed the 
 
GEOMETRICAL PROGRESSION. 121 
 
 interest in order to discharge the principal, this rule requires that we find the 
 
 vt 
 amount of p, for time t, at r per cent. This is done by multiplying by 1 + ~ — , 
 
 100 
 
 and gives |>(1 + ■ — )• From this subtracting the payment r, the new prin- 
 cipal is j?{ 1 + — ) —X. Again, finding the amount of this for another period 
 of time, t, and subtracting the second payment, 
 
 ^\ 100/ V 100/ 
 
 In like manner, after the third payment there remains 
 
 After the 4th payment, the remainder is 
 
 ^\ 100/ \ 100/ V 100/ \ 100/ 
 
 Finally, after the ni\\ payment, we have 
 
 •^V 100/ V 100/ \ 100/ V 100/ 
 
 -x(l+lL\-x = 0. 
 \ 100/ 
 
 Whence 
 
 This denominator being the sum of a geometrical progression whose first term 
 
 (\ + ^V 
 
 / rt\ ^ 100/ - 1 
 
 is 1, ratio ( 1 + — ), and number of terms n, its sura is - 
 V 100/ 
 
 100 \ ^ 100/ 
 
 100 
 
 Hence x = 
 
 (-^r- 
 
 2d. By the Vermont Rule. — The amount of the principal for the whole time 
 
122 ELEMENTAKY ALGEBRA. 
 
 The amount of the 1st payment is xVl -f- _-(7j, — 1)1 
 
 " Sd " x[l+Jl{n-2)], 
 
 L 100 J 
 
 "3d « ........ x\l+ Ii(n-d)], 
 
 L 100^ U' 
 
 etc., etc., ........ etc. 
 
 The ?ith payment (with no interest) is x. 
 
 The sum of the amounts of these payments is 
 
 ''"' + ]^*[('^- 1) + 0^ - 2) + (^ - 3) 1]. 
 
 The series in the brackets being an arithmetical progression whose first term 
 is {n — 1), common difference — 1, last term 1, and number of terms {n — 1), its 
 
 )n. Hence the sum of the payments ie tix + ^^x (^ ~ iw.. 
 
 2 / ^ ^ 100 \ 2 / 
 
 sum IS 
 
 nrt 
 
 (n-l)- 
 
 r 100^ n 
 
 or a; L^ H o -• • ^^* ^^ *^® condition this sum equals the amount of 
 
 the principal ; consequently 
 
 ScH. — If the payments are made annually, t = l. And letting r'= — , 
 
 i. e., letting the rate per cent, be expressed decimally, the formulas become. 
 
 By the U.S. Rule, ^^?>r'(l + rr 
 
 (1 +r')"— 1 
 
 By the Vermont Rule, x = ^^(^ +/'^) . 
 
 2n + rn{n—t) 
 
 12. Whafc must be the annual payment in order to discharge a note 
 of $5000, bearing interest at 10^ per annum, in 5 equal payments ? 
 Ans., By the U. S. Rule, $1318.99 within a half cent. 
 By the Vermont Rule, $1250. 
 
 Query. — What occasions the great disparity between the payments required 
 by the different rules ! 
 
VARIATION. 123 
 
 SECTION IV. 
 
 VARIATION. 
 
 91, Variation is a term applied to the consideration of quan- 
 tities so related to each other that any change in one makes the 
 others change in the same ratio, 4ii*ect or inverse. 
 
 One quantity varies directly as another, when any change in the 
 latter makes the former change in the same {direct) ratio. 
 
 One quantity varies inversely as another, when any change in the 
 latter makes the former change in the corresponding inverse ri^tio. 
 
 Ill's.— The amount earned by a laborer in a given time varies directly as his 
 daily wages. The time required to earn a given amount varies inversely as the 
 daily wages. 
 
 92, One quantity wsines joiiitly as two others, when any change in 
 the product of the latter two makes the former change in the same 
 ratio as this product. 
 
 III. — The amount a laborer receives varies jointly as his daily wages and the 
 time of service. 
 
 93, One quantity varies directly as a second and inversely as a 
 third, when it varies as the quotient of the second divided by the 
 third. 
 
 III. — The time required to earn any amount varies directly as the amount, 
 and inversely as the daily wages. 
 
 94, The Sign of variation is a. 
 
 III. — If X varies directly as y, we write x cc y, and read " x varies as y." If x 
 varies inversely as y, we write a? a - , and read " x varies inversely as y" If x 
 
 y 
 
 Y&Ties jointly as y and z, we write x a yz, and read " x varies jointly as y and z." 
 If X varies directly as y and inversely as z, we write a; a — , and read ** x varies 
 directly as y, and inversely as z." 
 
 95^ Prop, — Variation may always be expressed in the form of a 
 proportion. 
 
 Dbm. — 1st. The expression x ex. y signifies that if x is doubled y is double 1, 
 if X is divided y is divided by the same number, etc. ; i. e., that the ratio of x to 
 
 y is constant. Let m be this ratio, so that — = m. Therefore x : y : : m : 1. 
 
124 ELEMENTAllY ALGEBRA. 
 
 2d. The expression a; a — signifies that if y is multiplied by any number, x is 
 
 divided by the same, and if y is divided by any number x is multiplied by the 
 same. Hence the product of x and y is constant. Let this product be in. Then 
 xy = m,ov x:l :: m:y. 
 
 3d. X (X yz signifies that the ratio of x to yz is constant. Let this be m. Then 
 — = m, OT x:yz :: mil, or x : y : : mz : 1, or x : z : : my :l,or x :y :: 2 : — . 
 
 yz Wi 
 
 4th. X cx^ signifies that the ratio of x to -^ is constant. Let this be m. Then 
 
 z z 
 
 x: ^::m:l, or x:y::m:z 
 
 z . 
 
 Exercises. 
 
 1. If a: a y. and y (x z, show that x a z. 
 
 Dem. — If X (X y, the ratio of a; to y is constant. Let this ratio be m. Then 
 X = my. In like manner let n be the ratio of y to z. Then y = nz. Hence 
 X = mnz. That is, the ratio of .r to z is constant, or x a z. 
 
 2. If a; a -, and y a -, show that x (x z. 
 
 y ^ z 
 
 Sug'8.— We may write a; = — , and y — —. Hence x-=''^z. That is, the ratio 
 of a; to 2 is constant, or a; a z. 
 
 3. If o; a z, and ?/ a -, show that x <x -. 
 
 ' -^ z y 
 
 SuG's. x = mz, y = —, .•.x = —,otxcc—. 
 
 z y y 
 
 X v 
 
 4. If a; a y, show that - a -^, and xz a yz. 
 
 X V 
 
 5. If a; a y, and z a n, show that xz ex. yu, and - oc -. 
 
 6. If a; a y, and y* <x z*, how does x vary in respect to z ? 
 
 7. If a; a ;/, and for x = 8, ?/ = 4, what is the value of y for 
 a; = 20 ? 
 
 Solution.— Since x ocy, and for a: = 8, y = 4, the ratio of a; to 2^ is 2. That 
 is, - = 2. Hence for x = 20, we have ?? = 2, or y = 10. 
 
 y y 
 
 8. If a: a - and for x = G, y =^2, what is the value of xfory =3? 
 
 SuG. x:— : : 6 : - . Hence for y = 3, .t = 4. Or we may reason thus, in 
 
 changing from 2 to 3, y increases | times. Then, as x changes in the reciprocal 
 ratio, cc = § of 6 = 4. 
 
HARMONIC PROPORTTOX AND PROGRESSION. 125 
 
 9. If a -}- b <x a — b, prove that a^ -\- b^cc ab. 
 
 10. If y = 2^ -{- q, in which p o: x and qcc-; and if when x = 1, 
 
 4 14 
 ?/ = G ; and when x = 2, y = 5; prove that y ^ -x -{- —. 
 
 11. The area of a triangle equals half the product of the base and 
 altitude. Show that if the base is constant the area varies as the 
 altitude ; if the altitude is constant the area varies as the base ; and 
 if the area is constant the altitude and base vary inversely. 
 
 12. The volume of a pyramid varies jointly as its base and alti- 
 tude. A pyramid whose base is 9 feet square, and height 10 feet, 
 contains 10 cubic yards. What must be the height of a pyramid 
 with a base 3 feet square in order that it may contain 2 cubic yards? 
 
 13. Given that .^ a t^, when / is constant; and s a/, when t is 
 constant; also, 2s ==/, when ^ = 1. Find the equation between /, s, 
 arid t. 
 
 SuG. — The first two conditions are equivalent to saying that s varies jointly as 
 t^ and /, i. e. 8 <x ft'^ ; since in this expression if / is constant s a t^, and if t is 
 constant « a /. 
 
 SECTION V, 
 
 HARMONIC PROPORTION AND PROGRESSION. 
 
 OG, Three quantities are in Harmonic Proportion when the dif- 
 ference between the first and second is to the difference between the 
 second and third (the differences being taken in the same order) as 
 the first is to the third. 
 
 III. 6,4, and 3 are in harmonic proportion, since 6 — 4 : 4 — 3 : : 6 : 3. \i a,h, 
 c are in harmonic proportion, a — b : b — c : : a : c. 
 
 07, Def. — When three quantities taken in order are in har- 
 jnonic proportion, the second is the Harmonic Mean between the 
 other two. 
 
 08, JProp, — If three quantities are in harmonic proportion^ their 
 reciprocals are in arithmetical proportion. 
 
 Dem.— If a, b, c are in harmonic proportion, a — b : b — c '. : a : c, and 
 
 ac — be — ab — ac. Dividing by abc, we have . = r . i- c- — .. . 
 
 ^ J ' b a c b abc 
 
126 ELEMENTARY ALGEBRA. 
 
 09, Def. — The reciprocals of the terms of an arithmetical pro- 
 gression form what is called a Harmonic Progression, 
 
 III.— Thus as 1, 2, 3, 4, 5, 6 is an arithmetical progression. 1, — , — , — , — , — 
 
 A o 4 5 6 
 is a harmonic progression. Also if a, b, c, d, etc., constitute a harmonic progres- 
 sion, — , -r, — , -:, etc., consfitute an a^rithmetical progression. 
 abed 
 
 100. ScH. — The term Harmonic is applied to such a series, since, if strings 
 of the same size, substance, and tension, be taken of the lengths 1, ^, i, i, i, ^ , 
 any two of them vibrating together produce harmony of sound. 
 
 Exercises. 
 
 1. If a, h, c, d are in harmonic progression, show that ab : ccl :: 
 a — h : c — d. 
 
 SuG 8. -J- •• — •• -J. Hence -j- = — ,or acd — bed = abc — abd. 
 
 abed bade 
 
 2. If a, h, e are in harmonic proportion, show that J (the harmonic 
 mean) = . 
 
 3. Show that the geometric mean between two numbers is a geo- 
 metric mean between their arifchmetio and harmonic means. 
 
 4. To insert n harmonic means between a r«nd h. 
 
 SuG. — First find the form of the terms for n arithmetical m«ans befn^een 
 -1 and i. See [82). The hftfmo^ic series i^ a "^'^ "^ ^^ ^^^ "^ ^^ 
 
 « b ' ' bn + a ' bn + 2a — b ' 
 
 db(n + 1) ^ 
 
 r— , b. 
 
 an + b 
 
 5. If a and b are the first two terms of a harmonic progression, 
 
 show that the 7ith term is — ,^ ,, -. 
 
 a(n — 1) — b(7i — 2) 
 
 6. Insert 3 harmonic means between \ and ■^. 
 
 ScH.— There is no method known for finding the sum of a harmonic 
 series. 
 
PURE QUADRATICS. 127 
 
 CHAPTER III. 
 
 QUADRATIC EQUATIONS. 
 
 SECTION I, 
 PURE QUADRATICS. 
 
 101, A Quadratic Equation is an equation of the second 
 degree {6, 8). 
 
 102, Quadratic Equations are distinguished as Pure (called also 
 Incomplete), and Affected (called also Complete), 
 
 103, A Pure Quadratic Equation is an equation which 
 contains no power of the unknown quantity but the second; as 
 ax^ -\-h = cd, x^ - U = 102. 
 
 104, An Affected Quadratic Equation is an equation 
 which contains terms of the second degree and also of the first, with 
 respect to the unknown quantity or quantities; as x^ — 4a: = 12, 
 bxy — X — y^ = 16a, mxy ■\- y = h. 
 
 105, A Root of an equation is a quantity which substituted for 
 the unknown quantity satisfies the equation. 
 
 103, Prob. — To solve a Pure Quadratic Equation. 
 
 PULE. — Transpose all the terms containing the unknown 
 
 QUANTITY INTO THE FIRST MEMBER, AND UNITE THEM INTO ONE, 
 CLEARING OF FRACTIONS IF NECESSARY. TRANSPOSE THE KNOWN 
 TERMS INTO THE SECOND MEMBER. DiVIDE BY THE COEFFICIENT OF 
 THE UNKNOWN QUANTITY. FINALLY, EXTRACT THE SQUARE ROOT 
 OF BOTH MEMBERS. 
 
 Dem. — According to the definition of a Pure Quadratic, all the terms contain- 
 ing the unknown quantity contain its square. Hence they can be transposed 
 And united into one by adding with reference to the square of the unknown 
 
128 ELEMENTARY ALGEBRA. 
 
 quant it}'. Tliat transposition, and division of both members by the same quan- 
 tity, do not destroy the equality has already been proved. Extracting the square 
 root of the first member gives the first power of the unknown quantity, i. e. the 
 quantity itself. And taking the square root of both members does not destroy 
 the equation, since like roots of equal quantities are equal. 
 
 107, COE. 1. — Ei:ery Pure Quadratic Equation has two roots 
 ^numericalhj equal but with ojyposite signs. 
 
 For every such equation, as the process of solution shows, can be reduced to 
 the form ar' = a {a representing any quantity whatever). Whence, extracting 
 the root, we have x = ± V~a ; as the square root of a quantity is both + , and 
 - {203, Part I). 
 
 108. Cor. 2. — The roots of a Pure Quadratic Equation may 
 both be imaginary, a^id both loill be if one is. 
 
 For if after having transposed and reduced to the form x^ = a, the second 
 member is negative, as a;' = — a, extracting the square root gives x = + V—a, 
 and X — — V—a, both imaginary. 
 
 Examples. 
 
 a \/(i*--x*__x . 45 _ 57 
 x'^ X ~b' ' 2a;2+3~4.r2_5* 
 
 5 1 1 _Va ^ x^ -V>_ x^-4: 
 
 Va^x+V~a Va-^x—Va x 3 4 
 
 7. x^—ax-hb=ax{x—l). 8. 8-^'Sx^ = 6 + 2x^, 
 
 '■ f1?^-|/?^=*- 
 
 10. * + ^ ' 
 
 4 + 9?j~2-.t' 
 
 11. 12 + 4(.'r2 + 12) = (2-a:)(2 + a;)-16. 12. xV6Tx^=l+x^. 
 
 ._ ax + l + Va^x^ — l ,_ _ a + x-{-\/2ax-j-x^ ^ 
 
 13. =:=ibx. 14. — =b' 
 
 ax-\-l — ya^x^ — l a-hx—V2ax + x^ 
 
 Applications. 
 
 1. Fiiul two numbers which shall be to each other as 3 to 5, and 
 the ditiercncc of whose squares shall be 25G. 
 
PURE QUADRATICS. 1'20 
 
 2. Find a number such that if the square root of LIio diiference 
 between the square of the number and a^, be successively subtracted 
 from and added to a, the difference of the reciprocals of these results 
 shall be equal to a divided by the square of the number. 
 
 3. Find three numbers which shall be to each other as m, 7i, and 
 p, and the sum of whose squares shall be s. 
 
 4. An army was drawn up with 5 more men in file than in rank, 
 but when the form was changed so that there were 845 more in 
 rank, there were but 5 ranks. How many men were there in the 
 army? 
 
 5. From two towns, m miles asunder, two persons, A and B, set 
 out at the same time, and met each other, after travelling as many 
 days as are equal to the difference of miles they travelled per day, 
 when it appeared that A had travelled n miles. How many miles 
 did each travel per day ? 
 
 6. For comparatively small distances above tlie earth's surface the 
 distances through vviiich bodies fall under the influence of gravity 
 are as the squares of the times. Thus, if one body is falling 2 
 seconds and another 3, the distances fallen through are as 4:9. A 
 l)ody falls 4 times as far in 2 seconds as in 1, and 9 times as far in 3 
 seconds. These facts are learned both by observation and theoret- 
 ically. It is also observed that a body falls 16^ feet in 1 second. 
 How long is a body in falling 500 feet ? One mile (5280 ft.) ? Five 
 miles ? 
 
 7. The mass of the earth is to the mass of the sun as 1 : 354930, 
 and attraction varies directly as the mass and inversely as the square 
 of the distance. The distance between the earth's centre and eund 
 centre being 91,430,000 miles, find the point between the earth and 
 sun where the attraction of the earth is equal to that of the sun. The 
 earth's radius being 3,962 miles, where is this point situated with 
 reference to the earth's surface ? 
 
 8. A certain sum of money is lent at 5^ per annum. If we multiply 
 the number of dollars in the principal by the number of dollars in 
 the interest for 3 months, the product is 720. What is the sum lent ? 
 
 9. The intensity of two lights, A and B, is as 7 : 17, and their dis- 
 tance apart 132 feet. Where in the line of the lights are the points 
 of equal illumination, assuming that the intensity varies inversely 
 as the square of the distance ? 
 
130 ELElrtfil^fAltt ALGfi^'TtA. 
 
 10. The loudness of one church bell is three times that of another 
 Now, supposing the strength of sound to be inyersely as the square 
 of the distance, at what place on the line of the two will the bells be 
 equally well heard, the distance between them being a ? 
 
 SECTION IT. 
 AFFECTED QUADRATICS. 
 
 109, An Affected Quadratic equation is an equation which 
 contains terms of the second degree and also of the first with respect 
 
 to the unknown quantity, x* —ox= 12, 4.^ + ^ax^ = - 
 
 5 
 
 2^8 
 
 a*x 
 
 and -j-^ — 4:ax + 3J* = are affected quadmtic equations. 
 110, Prob, — To solve an Affected Quadratic Equation. 
 
 RULE.— I. l^EDUCE THE EQUATION TO THE FORM X^ -^ OX = h, 
 THE CHARACTERISTICS OF WHICH ARE, THAT THE FIRST MEMBER CON- 
 SISTS OF TWO TERMS, THE FIRST OF WHICH IS POSITIVE AND SIMPLY 
 THE SQUARE OF THE UNKNOWN QUANTITY, ITS COEFFICIENT BEING 
 UNITY, WHILE THE SECOND HAS THE FIRST POWER OF THE UNKNOWN 
 QUANTITY, WITH ANY COEFFICIENT (a) POSITIVE OR NKGATIVE, 
 INTEGRAL OR FRACTIONAL; AND THE SECOND MEMBER CONSISTS OF 
 KNOWN TERMS {b). 
 
 2. Add the square of half the coefficient of the second 
 term to both members of the equation. 
 
 3. Extract the square root of each member, thus producing 
 
 A SIMPLE equation FROM WHICH THE VALUE OF THE UNKNO^V:?^^ 
 QUANTITY IS FOUND BY SIMPLE TRANSPOSITION. 
 
 Dem. — By definition an affected quadratic equation contains but three kinds 
 of terms, viz : terms containing the pquare of the unknown quantity, terms con- 
 taining the first power of the unknown quantity, and knoicn terms. Hence each 
 of tht three kinds of tenns may, by clearing of fractions, transposition, and 
 uniting, as the particular example may require, be united into one, and the 
 results arranged in the order given. If, then, the first term, i. e. the one con- 
 taining the square of the unknown quantity, has a coefficient other than unity, 
 or is negative, its coefficient can be rendered unity or positive without destroy- 
 ing the equation by dividing both the members by whatever coefficient this term 
 may chance to have after the first reductions. The equation will then take the 
 
AF^cfflS QUADRATICS. 131 
 
 form x^ ± ax = ± b. Now adding ( -^ ) to the first member makes it a perfect 
 
 square (the square ot x ± ~j , since a trinomial is a perfect square when one of 
 
 its terms (the middle one, ax, in this case) is ± twice the product of the square 
 roots of the other two, these two being both positive {110, Part I,). But, if we 
 add the square of half the coefficient of the second term to the first member iaZZZ 
 make it a complete square, we must add it to the second member to preserve the 
 equality of the members. Having extracted the square root of each member, 
 these roots are equal, since like roots of equals are equal. Now, since the first 
 
 term of the trinomial square is x^, and the last f — | does not contain x, its 
 
 square root is a binomial consisting ot x ± the square root of its third term, or 
 half the coefficient of the middle term, and hence a known quantity. The 
 square root of the second member can be taken exactly, approximately, or indi- 
 cated, as the case may be. Finally, as the first term of this resulting equation 
 is simply the unknown quantity, its value is found by transposing the second 
 tenn. 
 
 Sen. 1. — This process of adding the square of half the coefficient of tlie 
 first power of the unknown quantity to the first member, in order to make 
 it a perfect square, is called Completing the Square. There arc a variety 
 of other ways of completing the square of an affected quadratic, some of 
 which will be given as we proceed ; but tliis is the most important. This 
 method will solve all cases: others arc mere matters of convenience, in 
 special cases. 
 
 Ill, Cor. 1. — An affected quadratic equation has two roots. 
 These roots may both be positive, both be negative, or one positive and 
 the other negatiiie. They are both real^ or both imaginary, 
 
 Dem. — Let x"^ + j).r = q be any affected quadratic equation reduced to the form 
 for completing the square. In this form p and q may be either positive or 
 
 V) 
 
 negative, integral or fractional. Solving this equation we have a? = — ^ 
 
 ± \ -r- ■\- q. We will now observe what different forms this expression can 
 take, depending upon the signs and relative values of 'p and q. 
 
 Ist. Wke)i\} and q are both positiDC. The */////« will then stand as given ; i.f., 
 
 9^— —~ ± i/^ + q. Now, it is evident that 4/ ^'- + «/ > V' ^^^ V + 7 
 
 IS the square root of something more than ^-. Hence, as ^ < 4/ -r- + q, 
 
 4 ,0 ' 4 
 
 V / v'^ "0 / 'ft' 
 
 — ^ + y -^ + <? is positim ; but — ^ — y ^ + q '^^ negative, for both i)arts 
 
 are negative. Moreover the negative root is numerically greater than the 
 positive, since the former is the numerical sum of the two parts, and the latter 
 
132 ELEMENTARY ALGEBRA. 
 
 the numerical difference. .'. When jp and q are both + in the given fonn, one 
 root is positive and the other negative, and the negative root is numerically 
 greater than the positive one. 
 
 2d. Wien p is negative and q positive. We then have x = ~ ± y — ^ -f q 
 
 = ~ ± i/ ~ + q. If we take the plus sign of the radical, x is positive ; but if 
 
 /p* p 
 
 we take the — sign, x is negative, since y ^ + ? > o* Moreover, the positive 
 
 root is numerically the greater. .'. When p is negative and q positive, one root 
 is positive and the other negative ; but the positive root is numerically greater 
 than the negative. 
 
 3d. When p and q are both negative. We then have x— ^ ± y ^—?— + (— y) 
 
 =z ^± y — — q. In this \i ^ > q, y j^ — q '^s real, and as it is less than — , 
 3 ' 4 4 4 2 
 
 V' / P* 
 
 both values are positive. It ^ = q, y t — g = and there is but one value 
 
 of X, and this is positive. (It is customary to call this two equal positive roots 
 for the sake of analogy, and for other reasons which cannot now be appreciated 
 
 by the pupil.) If ^ < g, y L-. — q becomes the square root of a negative 
 quantity and hence imaginary. 
 
 4th. When n is positive and q negative. We then have x = — ~ ± y K — q. 
 
 As before, this gives two real roots when g < 4- ^^'^^^ ^^^Js is the case both 
 
 roots arc negative. [Let the pupil show how this is seen.] When q:=~, the 
 
 roots are equal and negative ; i. e., there is but one. When ^ <q both roots 
 arc imaginary. 
 
 112, Cor. 2. — An affected quadratic being reduced to the form 
 x2+ px = q, the valne of x can always be tcritten out without taking 
 the intermediate stejys of adding the square of half the coefficient of 
 the second term^ extracting the root., and transposing. Hie root in 
 such a case is half the coefficient of the second term taken with the 
 opposite sign, ± the square root of the sum of the square of this 
 half coefficient, and the know7i term of the equation. This is observed 
 
 directly from the form x= — 2±i/^+q^ and more in detail 
 in the demonstration of the preceding corollary. 
 
AFFECTED QUADRATICS. 133 
 
 H3» Cor. 3. — Upon tJie principle that the middle term of a tri- 
 nomial square is twice the product of the square roots of the other 
 two, we can often complete the square more advantageously than by 
 the regular rule. 
 
 Thus having 4j;- — Vlx = 10. Since 4^^ is a perfect square, and 12iC is diria- 
 ible by twice the square root of 4tx^ , i. e. by 4r, we see that the wanting third 
 term is 3', or 9. Adding this to both members, we have 4a;*— VZx + = 25, 
 
 Again, if the coetficient of X' is not a perfect square, it can be rendered such 
 by multiplying by itself (or often by some other factor). If then the second 
 term (the term in .r) is not divisible by twice the square root of this first term, we 
 may multiply both members of the equation by 4, and the first term will still 
 be a perfect square, and the second tenn divisible by twice its square root. 
 
 114* ScH. 2. — The method of Art. 110 is perfectly general, and will 
 solve all cases ; but some may prefer the more elegant methods indicated in 
 {1 13), in special cases. Same illustrations of these methods are given in 
 the examples following. 
 
 EXA3IPLE8. 
 
 1. x^-Qx=U. 2. 3.r2=24.'c-36. 3. x^-^ax-la^. 
 
 4. x^-lx + 2 = 10. 5. 3.^3+ 135=12.7:. 6. x^ + {a-i)x^a. 
 X 3 x—l 4x x—1 a^x^ 2ax b^ 
 
 = -:-| Zi — • t). ; — - — — r — 4. y. ■ , - - \-—r^=\J, 
 
 .-c - 1 2 2 x-\- 1 2x + 'd b^ a c 
 
 10. Solve 9:c2+12a;=32, 7x^-Ux= -of, and 3x^-V3x=-.lO, 
 by Art. li:i. 
 
 Bug's. — Dividing 12.t' by 2y^Qx', or Gx, we have 2 as the square root of the 
 third term. Hence 9.c' + 12.r + 4 = 86, is the equation with the square com- 
 pleted. 
 
 7x*— 14j; = — ;■>?, becomes, by multiplying by 7, 40.c'^— 98.r = — 40. Hence, 
 completing the square as in the last, 49x-— 9Sx + 49 = 9. 
 
 dx^ - 13.C = 10, multiplied by 3 and by 4 becomes mx^ - 156a; = 120. Hence, 
 completing the square as b(!fore, 36^''— 150.1' + (13)*= 289. 
 
 [Note. — Solve the following by any of the preceding methods, according to 
 taste or expediency.] 
 
 11. (2.r + 3)^x(3a;4-7)*=12. 12. 3:^:2+2^^=85. 
 13. a^l+b^x^) = b(2a^x^{-b). 14. 5x^ -~9x-^2i=0. 
 
 15. 3Vll2-8.2; = 19 + \/3:c + 7. 16. 7a;2-lU = 6. 
 
134 ELEMENTARY ALGEBRA. 
 
 17. (x-c)Vab-(a-b)\/(^=0. 18. 3x^+x=\L 
 
 19. 5(?£Z1) ^ l^=Wx, 20. ^^V-'-l^^_^ 
 
 l + sVa^ Va x—Vx^—a'^ ^ 
 
 22. ^-^^^ '^ 
 
 1 + Vl+^ l-Vl-a;* .r+V.'c+i 11 
 
 /,,_j ,\fa-U ,\ ^2 90 90 27 
 
 25. 
 
 4/4 + V2a-3+a:«=^. 26. 2V^+ -— ^5 
 
 ■} 5 x—'Za x + a 
 
 29. 
 
 31. 
 
 X + V a:^ — 9 
 
 2Vx + ^4a:+VV^T~2=l. 30. ;;'^ ^ ■" = (:. -2) »> 
 
 x^ , 4 ,4. 1 
 
 -(a^-b^)x= 
 
 a^ + h^ («*«)'* + (a«6)"i 
 
 SECTION IIL 
 
 EQUATIONS OP OTHER DEGREES WHICH MAY BE SOLVED AS 
 
 QUADRATICS. 
 
 115^ Vrop^ l*^Anii Pare Equation (i. e., one containing the 
 finknoicn quantiti/ affected i/^ith hxtt o?ie exponent) can be solved in 
 a manner sindkir to a Pure Quadratic. 
 
 Dem.— In any such equation we can find the vahie of the unknown quantity 
 affected by its exponent, as if it were a pimple equation. If then the unknown 
 quantity is affected witli a positive integral exponent it ran be freed of it by 
 evolution ; if its exponent be a positive fraction it can be freed of it by extract- 
 ing the root indicated by the numerator of the exponent, and involving this root 
 to the power indicated by the denominator. If the exponent of the unknown 
 quantity is negative it can be rendered positive by multiplying the equation by 
 the unknown quantity with a numerically equal positive exponent. Q. K. D. 
 
HIGH]- 11 EQUA'JIONS SOLVED AS QUADRATICS. 135 
 
 Hff, I*rop, 2\ — -^^'y equation containing one imknoicn quan- 
 tity affected with only tico different exponents, one of rohich is twic^ 
 the other, can be solved as a)i Aff'ected Quadratic. 
 
 Dem. — Let m represent any number, positive or negative, integral or frac- 
 tional ; then the two exponents will be represented by m and 2m ; and the 
 equation can be reduced to the form x^"' + px"^ = q. Now let y = x'", and y'^=x^", 
 
 whatever m may be. Substituting we have y^ + py — q, whence y = — ^ 
 
 ^_1 
 
 ir4/— + q. But y — a;"*; hence ^ = (— f ± y x "^ *^) ' Q- ^- ^ 
 
 117. Prop, fV. — Equations may frequently he put in the form 
 of a quadratic by a judicious grouping of terms containing the 
 unknown quantity, so that one group shall be the square root of the 
 other. 
 
 Dem. — This proposition will be established by a few examples, as it is not a 
 general truth, but only points out a special method. 
 
 lis. Cor. — The form of the compound iiY.wsi may sonieUmes 
 be found by transpositig all the terms to the first member, arranging 
 them with reference to the unknown quantity, and extracting the 
 square root. In trying this expedient, if the highest exponent is 
 not eve?i it must be made so by midtiplying the equation by the 
 unknown quantity. In like mariner the coefficient of this term is 
 to he made a perfect square. When the process of extracting the 
 root terminates, if the root found can be detected as a part, or factor, 
 or factor of a part of the remainder, the root may he the polynomial 
 term. 
 
 110, Prop, 4, — Wheji an equation is reduced to the form 
 
 x" + Ax"~* + Bx"~' + Cx"~' f- L = 0, the roots xcith their signs 
 
 changed are factors of the absolute {knoion) term L. 
 
 Dem. — 1st. The equation being in this form, if rt is a root, the equation is 
 divisible by x — a. For, suppose upon trial x — <i goes into the polynomial 
 «" + A.c""''-^, etc., Q times mth a remainder R. (Q represents any series of 
 terms which may arise from such a division, and R, any remainder.) Now, since 
 the quotient multiplied by the divisor, + the remainder, equals the dividend, we 
 have (^ — rt) Q + R = a;" + A.r«-' + Bij"-' + C^;"-" - - - + L. But this polyno- 
 mial = 0. Hence (.c — rt)Q + R =: 0. Now, by hypothesis a is a root, and conse- 
 quently X — a = 0. Whence R — 0, or there is no remainder, 
 
 2d. If now X — a exactly divides a;" + Ax"* ~ * + B.?;" ~ ' + Ct;" — ^ - - - + L, a 
 must exactly divide L, as readily appears from considering the process of 
 
136 ELEMENTARY ALGEBRA. 
 
 division. Hence — a is a factor of L, a being a root of the equation. 
 Q. E. D. 
 
 ±20* Many equations of other degrees than the second, and 
 which do not fall under the preceding cases, may still be solved as 
 quadratics by means of Special Artifices. For these artifices the 
 student must depend upon his own ingenuity, after having studied 
 some examples as specimens. These methods are so restricted 
 and special that it is not expedient to classify them ; in fact, 
 every expert algebraist is constantly developing new ones. See 
 Ex's. 47-57. The following principle is often of service in such 
 solutions : 
 
 121, Prop, 5, — When an eqtiation can he put in such a form 
 that the product of any number of factors equals 0, the equation is 
 satisfied by putting any 07ie of these factors equal to 0. 
 
 Dem. — This scarcely needs demonstration, but will appear evident if we 
 consider such an expression as {x^ + 1) («*—«' + 1) («—!) = 0. Now, on the 
 hypothesis that any factor, as a;' + 1, is 0, the equation is satisfied * So also, if 
 a;' — «' + 1 = 0, the equation is satisfied, etc. 
 
 122, ScH. — Ability to recognize a factor in a polynomial is of prime im- 
 portance in the . soltUion of such equations. It is the grand key to difficult 
 solutions. 
 
 Examples. 
 
 1. x^ = 81. 2. x^ = 32. 3. x^ = m» 
 
 4. y^ = 243. 5. z^ ^ 1331. 6. y^ = 4. 
 
 7. ic" = b. 8. x^+V^= - ^ . 9. x^ -f- ix^ = 12. 
 
 x^-V-'i 
 
 10. x''^ -hx"' -p, 11. x^ - x^- 56. 12. ax^ + hx^ = c. 
 
 13. a^» - 2ax^= b. 14. x^ + x^ = 756. 15. x^ 4- 6x^- 22 = 0, 
 
 16. ax^ -hx^ -^c^O. 17. .T* + -^ = 3f 
 
 2x^ 
 
 * In etrioti.css wc ehoald add " »ince this hypothesis cannot render any other factor «. 
 
HIGHER EQUATlONe SOLVED AS QUADRATICS. 137 
 
 18. Sx* -v/^ + T-=i = 16. 19. x^ - 2x + Wx^ -2x + b = 11.* 
 
 20. X -\-W- 7Vx + IG = 10 - Wx + 16. 
 
 21. a;2 - a; + 5 V2a;2 _ 5^ + 6 = ^(Sa; + 33). 
 
 22. Vx+l'Z + ^/x + 12 = 6. 23. V ^ -\-yi---x, 
 
 X '^ 
 
 24. 1 +i/7^- = l/l+-. 25. x^ +-, + 2/'a: + -) 
 X a x^ \ xj 
 
 x 
 
 142 
 ^9 * 
 
 26. 2a:2 _ o^. 4. 2V2a;2 - Ix + 6 = 5a; - 6. 
 
 27. 7(1 + :i:)2 - a/(i - ^)' = yi - ^'.t 
 
 28. :c* - 8a:3 4. 29a;2 - 52a: + 36 = 126. 
 
 Solution. — See {118). Transposing 126, and extracting the square 
 root ; when we have the two terms 'x^ —Ax of the root, we have a 
 remainder 13.C- — 52a; — 90. We now notice that, if we call 4 the next term 
 of the root, the next remainder will be 5^* — 20a; —106, which we may 
 write 5(a;* — 4t' + 4) — 126. Hence our equation may be put in the form 
 (a;2 _ 4p + 4)2 4. 5(a;« _ 4^. _,_ 4) ^ 126. 
 
 29. a;*-6x3+5a;2+12a;=60. 30. .T3-6a;8 +lla;=6. 
 31. 4.t* + |=4.t3+33. 32. a;^ 4-5a;2 +3a:-9=:0.j: 
 33. .^3-63:2 + 13.1-10 = 0. 34. a;3-13a:2 4-49.^-45 = 0. 
 35. a;3+8a;2 4-17a; + 10=0. 36. a;3 - 29.1-2 -f 198a: -360 = 0. 
 37. a:3-15.T2 + 74a:-120 = 0. 38. a:* + 2a;3-3a:2 -4.r+4=0. 
 
 *a;«-2a;+ 5 + 6 Va;2-2a; + 5 = 16. Pnttinga;*- 2a; + 5 = y^ y«+6y = 16. 
 Such pnb^tituHon is not absolutely necessary, as we may treat a;* — 2a; + 5 as the unknown 
 quantity without substituting. Solve the followinj? in like manner. 
 
 t Dividing by 1^\ _ x'i "c have i / I + ^ _ a/^ —^ _ \ Then, multiplyinir by 
 
 y \—x rl-i-a; 
 
 % By (i/9) we arc led to try + 1 or — 1, or + or — 3, as roots. The equation is divisible 
 by a: — 1, and a; + 3. 
 
138 ELEMENTARY ALGEBRA. 
 
 39. x*'-10x^ + 35x^-50x + 2i:=0. 40. x*-4:X^ +Sx^-8x=21. 
 41. a;*-2a:3-25a;2+26a:=-120. 42. 3x* + 13a;3-117a;=243. 
 ,^ X 30 VZ + ix 1 ... 
 
 U.x= ^^ + ^f . PntVx=i,. 
 
 X — D 
 
 Special Expedients. 
 
 45. To find the roots of a:« = =t 1, x^ = ±il, x* = =fc 1, x^ = ±l, 
 x^ = ±:lf and a;® = ± 1. 
 
 Sug's. a;' — 1 = 0. Factoring {x — 1) (a;* + a; ' + jc* + .r + 1) = 0. .*. x = l, 
 and also a;* + a;' +x* + x+ 1 = 0. Dividing by a;*, a;'+ a; + 1 +- + -^ =0,or 
 
 + 2 + ^ + *+i = i--(" + D'^(" + 5) = ^- 
 
 1 -f x^ 
 46. To find the roots of 7^— — rr = «. 
 (1 + xy 
 
 Sug's. 1 + a;* = a(l + a;)* = a(l + 4r + 6a;' 4- 4r=» 4- x*). Whence, dividing 
 
 . , . . ,1 4a / 1\ 6rt 
 by a;* and arranging terms, x* -\ — , — [x + - I = _ . 
 
 47. To solve -=; — a + J. 
 
 1 — .r + V 1 -j- x^ 
 
 Sug's. — This can be cleared of fractions, and tlien of radicals, in the ordinary 
 
 way. But the following expedient will be found elegant in this case, and 
 
 convenient in many. Dividing by 2a, treating the resulting equation as a 
 
 1— x a — b 1+a;*— 2a; 
 
 proportion, and taking it by division, we have . = .'. ; — - 
 
 ^ ^ Vl+x' a + b 1+x^ 
 
 = f ^^ ' , or -?^ = 1 - C^^) ' = ^ ,. Taking this again by com- 
 Va + 6/ ' 1 + a:^ \a + b/ (a + b)^ ^ ^ 
 
 ^ ,. . . , . (\+xY (a + b)'+^ab (a-b)^ +Sab 
 
 position and division, we obtain ^^— ^ = -__^.^^_^ =. _______ or 
 
 l+a; v/ (a — by + Sab .. -u^ . 
 
 ^— -;- =r — ^- . Again, by division and composition, we obtain 
 
 Via^^^Y +Sab-(a-b) 
 
 V{a -by + Sab + {a-b) 
 
HIGHER EQUATIONS SOLVED AS QUADRATICS. l3& 
 
 48. To solve (I + x -{-x^)^ = ^^- (1 -{- x^ -^ x^). 
 
 SuG's.— Dividing by 1 +« + a;^ 1 + a; + a;^ = --±i (1 - a; + x\ or l±^±f! 
 
 a — 1 X—x-^T? 
 
 = . :. = a. 
 
 a — \ X 
 
 49. To solve a = a;^ + (1 — «)*. 
 
 SuG's.— Since (1— a;)* = (a;— !)*,■ we may write a = (« — i + i)* + (a; — i — ^)*. 
 Now put X — ^ = 1/, substitute and expand. 
 
 50. To solve \/x-- - \/l-- = ^— i 
 
 XXX 
 
 j's.— Dividing by i/ 1 _ 1, a/.^ + 1 - 1 = ^'^.- • Squaring, etc., 2 y^a; + 1 
 f X y X 
 
 = 1 H h ar. Squaring, etc., again, (x j — 2(x J=: — 1. 
 
 51. Solve x^ -x-h SVTx^ - 3a; + 2 = - + 7. 
 
 9 
 
 52. Solve T — o — x — x^, 
 
 1 -^ X -\- x^ 
 
 53. Solve — ; — - = — . 
 
 a^ — ax + x^ x^ 
 
 54. Solve — - _ — = Vx^ — a* (Va;* + Oic — V^^ — ar). 
 
 |/a; + Va;2-«2 
 
 1 + rr3 13 
 
 55. Soif6 2iV 1 - a;^ = a(l + icM. Also ,, , _ . 
 
 56. Solve 6x^ — 5x^ -\- x = 0. Also .^3 + x^ — 4x — 4 = 0. 
 
 57. Solve 8^:3 + 16a; = 9. Also 3x^ + 8.^* — 8a;2 = 3. 
 
 SuG. — The solutions of the last four depend upon the recognition of a com- 
 mon factor. 
 
140 ELEMENTARY ALGEBRA. 
 
 SECTION IV. 
 
 SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE BETWEEN 
 TWO UNKNOWN QUANTITIES. 
 
 123. Prop, 1, — Two equations between two imknoion quanti- 
 ties^ otie of the second degree and the other of the Jirst, may always 
 he solved as a quadratic. 
 
 Dem.— The general form of a Quadratic Equation between two unknown 
 quantities is 
 
 ax^ + hxy + cy'^ -\- dx + ey ^f =0, 
 
 since in every such equation all the terms in a;* can be collected into one, and its 
 coefficient represented by a ; all those in xy can also be collected into one, and 
 its coefficient represented by 6, etc., etc. 
 
 The general form of an equation of the First Degree between two variables is 
 a'x + h'y + c' = 0. 
 
 Now, from the latter x = —, . which substituted in the former gives 
 
 a 
 
 no term containing a higher power of y than the second, and hence the resulting 
 
 equation is a quadratic. Q. E. D. 
 
 124:, JProj), 2, — In general^ the solution of two quadratics 
 between two xinknown quantities^ requires the solution of a biqua- 
 dratic ; that is, an equation of the fourth degree. 
 
 Dem. — Two General Equations between two unknown quantities have the 
 forms 
 
 (1) oar* + &j^ + cy» + da; + «y +/= 0, and 
 
 (2) a'x" + h'xy + c'y^ + d'x + e'y + /' = 0. 
 From (1), ^ = - ^ ± /^^ -^yl±^y±f. 
 
 Now, to substitute this value of x in equation (2), it must be squared, and 
 also, in another term, multiplied by y, either of which operations produces 
 rational terms containing y'^, and a radical of the second degree. Then, to free 
 the resulting equation of radicals will require the squaring of terms containing 
 y', which will give terms in y*, as well as other terms. Q. E. D. 
 
 12S, Def. — A Homogeneous Equation is one in which 
 each term contains the same number of factors of the unknown 
 
SIMULTANEOUS QUADRATIC EQUATIONS. 141 
 
 quantities. 2x" — oxy — ?/2 = 16 is homogeneous, ^x^ — ^^ + ^^ 
 = 10 is not homogeneous. 
 
 120, Prop, 3. — Two Homogeneous Quadratic Equations he- 
 tween two unkno^cn quantities can always be solved by the method 
 of quadratics, by substituting for one of the unknown qua?itities the 
 product of a new unknown quantity into the other. 
 
 Dem. — The truth of this proposition will be more readily apprehended by 
 
 means of a particular example. Take the two homogeneous equations x'^ 
 
 — xy + y* = 21, and y^ — 2xy + 15 = 0. Let x = vy, v being a new unknown 
 
 quantity, called an auxiliary, whose value is to be determined. Substituting in 
 
 the given equations, we have v^y^ — vy^ + y'^ = 31, and y'^ — 2vy^ = — 15. From 
 
 21 15 
 these we find y'^ = —. 7 , and y^ = r . Equating these values of y^, 
 
 21 15 
 
 — = ; whence 42« — 21 = 15^^ — 15i^ + 15. This latter equation 
 
 t)' — « + 12« — 1 
 
 is an affected quadratic, which solved for v, gives v = 'S, and ^. Knowing the 
 
 15 
 values of v we readily determine those of y from y^ = , and find y 
 
 = ± \/3 when « = 3, and y = ±5 when v = ^. Finally as x = vy, its values 
 are jc = ± ^\^'S, and ± 4. 
 
 By observing the substitution of vy for x in this solution, it is seen that it 
 brings the square of y in every term containing the unknown quantities, in each 
 equation, and hence enables us to find two values of y- in terms of v. It is easy 
 to see that this will be the case in any homogeneous quadratic with two 
 unknown quantities, for we have in fact, in the first of the given equations, all 
 the variety of terms which such an equation can contain. Again, that the equa- 
 tion in V will not be higher than the second degree is evident, since the values of 
 y'^ consist of known quantities for numerators, and can have denominators of 
 only the second, or second and first degrees with reference to v. Whence v can 
 always be determined by the method of quadratics ; and being determined, the 
 
 value of y is obtained from a pure quadratic (y^ = — , in this case), and that 
 
 2« — 1 
 of X from a simple equation {x= vy in this case). 
 
 127, JPvop, 4, — When the unknoicn quantities are similarly 
 involved in two quadratic, or even higher equations, the solution can 
 often be effected as a quadratic, by substituting for one of the un- 
 known quantities the sum of two others, and for the other unknovm 
 quantity the difference of these new quantities. 
 
 As this is only a special expedient, and not a general principle, its truth will 
 be rendered suflBciently evident by the solution of a few examples. See Ex's. 13, 
 14,15. 
 
142 ELEMENTAKY ALGEBRA. 
 
 Examples. 
 
 * (5a; + 2y=7. * t x-{-y=2, 
 
 (^+^=^' .^(x^+y^=65, 
 
 '' 1 + 1=1. 'nxy=2S. 
 
 ^x y 
 
 ^x^+xy=16, (x*+xy-\-^*=e, 
 
 j rr2+ xy + 2y^ = 7^, ^ (X*+xy=12, 
 
 i2x*+2xy-\- y*=7'd. ' \xy+y^=2. 
 
 4?/2 = 9, ^^ ix^+y^+l=Zxy, 
 
 11. 
 
 13. 
 
 15. 
 
 \xy + 2y^=^. ' \2{xy-^^)=^y*. 
 
 (a:2+a:?/ + ?/2 = 52, j a;«-2a:y-y8=31, 
 
 \xy-x^ = %. ' {ix^+2xy-y^=101, 
 
 U{x-\-y)=3xy, ^^ cx^+y*=ixy, 
 
 \x-\-y+x*-\-y^=2Q, * I aj—^^ =ixy. 
 
 (xy{x-\-y)=SO, 
 (a:3 +2/8=35. 
 
 SxjG.— The last three are readily solved by (127)- Thus, in the 15th, putting 
 jc r= 3 + T, and y = z — ti, the equations become 2z^— 2v'z = 30, and 2z^ + 6«*2 
 = 35. 
 
 il«+/=20. *-^+y^+-^=133. ^'i.-,* = 14560. 
 
 Special SoLUTioisfs. 
 
 19. yi - ^xy + 20a;« + dy - 264a;=0, 5y^ - 38a;^ + x^ - 12^^ 
 + 1056a; = 0. 
 Sua. — Add 4 times the first to the second. 
 
 * Two homogeneous quadratics can always be solved by {126), but special expedients are often 
 more ele;;ant. In this case by adding twice the second to the first, and extracting the square 
 root, we have x + y- ill. Subtracting twice the second from the first, and extracting the 
 iquarc root, we have x— y z=. ±3. 
 
SIMULTAKEOUS QUADJIATIG EQUATIONS. 143 
 
 20. X + y = x^, '6y — x — y^. 
 SuG. — Subtract the first from the second. 
 
 21. r-^=^' 22. ]^+^='' 23.1^+^=*' 
 
 Sug's.— To solve the 23d, square the first, writing the result ai^ + 2/* = 16 
 — 2a?y, and square again. Then for x^ + y* substitute 82. 
 
 24. To solve x — y = ^, and x^ — y^ =z 3093. 
 
 Sug's, — Divide the second by the first, and proceed in a manner similar to 
 that given for the last. 
 
 25. To solve x^ — xy + y^ = 7, and x* + x^y^ + 2/* = 133. 
 SuG. — Divide the second by the first. 
 
 26. To solve (s - -^IL\ \ (s + -^l-) ' = 82, and xy = 2. 
 
 \ X -\- y/ \ X — y) -^ 
 
 SuG's.-Write the first f?^zAn% (^^±^\ ^ 83. and put ^^^^ = ^. 
 \ x^y J \ x-y J ^ x + y 
 
 Whence 9^2 + -^ = 82. 
 
 27. To solve x^ + y (xy — 1) = 0, and y^ — x (xy + 1) = 0. 
 
 Sug's.— Write x* + a;V — ^V '— 0, and y* — a;V — xy=Q, and subtract the 
 second from the first. Whence x* — y* + 2x^y^ = 0, or x* +2x^y^ +y* = 2^*, and 
 
 a;* + y' = /y/2 y^, or - z= V\/2 — 1. From the given equations we get ~^ 
 
 xy 
 , . Hence . "" ^ = 3 - 2^/2. or xv = ^^72. 
 
 ^- Hence^-^ = 3-2V2.or.Ty = iV2- 
 
 28. Given xy = a{x + y), xz = J(a; 4- z)y and 1/2; = c{y + z). 
 
 SuG.— These i^re readily put into the forms - = - + -, - = - + -, and ^ = - 
 
 a y X b z X ^^ c z 
 1 
 + -. 
 
 y 
 
 29. Given x(x-{-y-\-z) = 18, y{x-hy+z) = 12, and z{x-\-y+z) = 6. 
 
 30. Given xtjz = ^S, ~ = ^, and ^ = t- 
 
 ^ yz 12 z 3 
 
 31. Given x + y + z = 6, 4:X + y = 2z, and x^ + ^8 + «« = 14. 
 
 32. Given Wx^ - y^ + a^y = 26, and 7 - | = ^0' 
 
144 ELEMENTARY ALGEBRA. 
 
 33. Given ^-^^ + 10^^^ = 1, and xy^ = 3. 
 
 X- y X + y 
 
 34. Given y{x^ + y^) z^ 4:{x + y)^, and xy = 4(x + y), 
 
 35. Given x + y = 10, and a/- + i /^ = -. 
 
 36. Given Vi — v^ = ^Vxy, and x + y — 20. 
 
 37. Given Vx^ + y^ + V^t^ - y* = 2y, and .t* - ^* = «*. 
 
 38. Given A/- + i/ '- = -— : + 1, and ^x^y + V^ = 78. 
 
 Y y y ^ yxy 
 
 39. Given y a: + y + 2Va: — V = \ ^ , and •— = — . 
 
 ^/x-y ^y 15 
 
 'y2_64=8a:iy, ,, ( a:* + ?/^=3a:, ._ ( 8a:4-7/i=14, 
 
 40. -^^ :^' 41. f \ '^, ' 42. . 
 
 ( y--4:=2y^x^, (x^-{-y'^=x. ( x^y^=2y*. 
 
 Applications. 
 
 1. The plate of a looking-glass is 18 inches by 12, and it is to be 
 surrounded by a plain frame of uniform width, and of surface equal 
 to that of the glass. Required the width of the frame. 
 
 2. A person bought some fine sheep for $360, and found that if he 
 had bought 6 more for the same money, he would have paid $5 less 
 for each. How many did he buy, and what was the price of each ? 
 
 3. A traveller sets out for a certain place, and travels one mile the 
 first day, two the second, three the third, and so on: in 5 days after- 
 ward another sets out, and travels 12 miles a day. How long and 
 how far must he travel before they will come together ? 
 
 4. Divide the number 48 into two such parts that their product 
 may be 432. 
 
 5. Divide the number 24 into two such parts that their product 
 may be equal to 35 times their difference. 
 
 6. For a journey of 108 miles, 6 hours less would have sufficed, 
 had the traveller gone 3 miles an hour faster. At what rate did he 
 travel ? 
 
APPLICATIONS OF QUADRATIC EQUATIONS. 145 
 
 T. The fore wheel of a, coach makes 6 revolutions more than the 
 hind wheel in going 120 yards; but, if the circumference of each 
 wheel be increased by 1 yard, the fore wheel will make only 4 revo- 
 lutions more than the hind wheel in the 120 yards. What is the cir- 
 cumference of each wheel ? 
 
 8. The product of two numbers isj!?; and the difference of their 
 cubes is equal to m times the cube of their difference. Find the 
 numbers. 
 
 9. Find two numbers whose product is equal to the difference of 
 their squares, and the sum of their squares equal to the difference of 
 their cubes. 
 
 10. There are 4 numbers in arithmetical progression. The sum of 
 the extremes is 8; and the product of the means is 15. What are 
 the numbers ? 
 
 SuG. — In solving examples involving several quantities in arithmetical pro 
 gression, it is usually expedient to represent the middle one of the series, when 
 the number of terms is odd, by x, and let y be the common difference. If the 
 number of terms is even, represent the two middle terms by x — y, and x + y, 
 making the common difference 2y. 
 
 11. Five persons undertake to reap a field of 87 acres. The five 
 terms of an arithmetical progression, whose sum is 20, will express 
 the times in which they can severally reap an acre, and they all 
 together can finish the job in 60 days. In how many days can each, 
 separately, reap an acre ? 
 
 12. There are three numbers in geometrical progression, the sum 
 of the first and second of which is 9, and the sum of the first and 
 third is 15. Required the numbers. ; 
 
 Sug's.— In solving examples involving several quantities in geometrical pro- 
 gression, it is sometimes expedient to represent the tirst by x, and the ratio by y, 
 so that the numbers will be x, xy, xy^, etc. In other cases it is expedient, if the 
 number of numbers sought is odd, to make xy the middle term of the series and 
 
 V x^ v^ 
 
 '- the ratio. Thus 5 terms will be represented - , x^, xy, y-, '— . When the 
 
 number of numbers sought is even, it is sometimes expedient to represent the 
 
 two means by x and v, and the ratio bv -. Thus 4 terms become — , x, y, '— . 
 ■'if' ' X y X 
 
 13. There are three numbers in geometrical progression wliose 
 continued product is 64, and the sum of their cubes is 584. Required 
 the numbers. :j.P 
 
146 ELEMENTARY ALGEBRA. 
 
 14. The sum of the first and second of four numbers in geometri- 
 cal i)rogression is 15, and the sum of the third and fourth is 60. 
 Required the numbers. 
 
 15. There are three numbers in geometrical progression, whose 
 product is 64, and sum 14. What are the numbers ? 
 
 16. It is required to find four numbers in arithmetical progression, 
 such that if they are increased by 3, 4, 8, and 15 respectively, the 
 sums shall be in geometrical progression. 
 
 17. It is required to find four numbers in geometrical progression 
 such, that their sum shall be 15, and the sum of their squares 85. 
 
 18. The sum of 700 dollars was divided among four persons. A, B, 
 C, and D, whose shares were in geometrical progression ; and the 
 difference between the greatest and least, was to the difference be- 
 tween tlie two means, as 37 to 12. What were the several shares? 
 
 19. The sum of three numbers in harmonica! proportion is 191, 
 and the product of the first and third is 4032 ; required the numbers. 
 
 20. The 2d and 6th terms of a geometrical progression are respec- 
 tively 21 and 1701. What is the first term, and what the ratio ? 
 
 21. A and B travel on the same road, at tlie same rate, and in the 
 same direction. When A is 50 miles from the town D, he overtakes 
 another traveller who goes at the rate of 3 miles in 2 hours ; and 
 two hours after, he meets a second traveller who goes at the rate of 
 9 miles in 4 hours. B overtakes the first traveller 45 miles from D, 
 and meets the second 40 minutes before he (B) reaches the 31st mile- 
 stone from D. How far are A and B apart ? 
 
 22. The joint stock of two partners, A and B, was $2080. A's 
 money was in trade 9 months, and B's 6 months, when they shared 
 stock and gain, A receiving $1140 and B $1260. What was each 
 man's stock ? 
 
 23. Tliere is a number consisting of three digits, the first of which 
 is to the second as tlie second is to tlie third; the number itself is 
 to the sum of its digits as 124 to 7; and if 594 be added to it the 
 digits will be inverted. What is the number ? 
 
 24. A person has $1300, which he divides into two portions, and 
 loans at different rates of interest, so that the two portions produce 
 
QUADRATIC EQUATIONS. 147 
 
 equal returns. If the first portion had been loaned at the second 
 rate of interest, it would have produced 136, and if the second por- 
 tion had been loaned at the first rate of interest, it would have pro- 
 duced $49. Required the rates of interest. 
 
 25. A person traveling from a certain place, goes 1 mile the first day, 
 2 the second, 3 the third, and so on ; and in six days after, another 
 sets out from the same place to overtake him, and travels uniformly 
 15 miles a day. How many days must elapse after the second starts 
 before they come together ? 
 
148 ELEMEiiTAliY ALGEBKA. 
 
 CHAPTER IV. 
 
 INEQUALITIES. 
 
 128, An Inequality is an expression in mathematical sym- 
 bols, of inequality between two numbers or sets of numbers. 
 
 Ill, — Thus a > h (read " a greater than h ") is an inequality ; also a^x — 3 
 < 5 + 2 (read " a^x - 3 less than 5 + 2 "). (See Part I., 43,) 
 
 129, Fnndamental Princijfle. — In comparing two posi- 
 tive numbers, that is called the greater which is numerically so. 
 Tlius 5 > 3. But, in comparing two negative numbers, that is 
 called the greater which is numerically the less. Thus — 5 < — 3. 
 Of course any negative number is less than any positive number. In 
 general, we call a>h when a — h \s positive, and a <h when a — h 
 is negative. 
 
 130, The part of an inequality at the left of the sign >, or <, 
 is called i\\^ first memher^ and the part at the right, the second mem- 
 ber of the inequality. 
 
 131, For the purposes of nuithematical investigation, inequali- 
 ties are subjected to the same transformations as equations, but with 
 certain characteristic diiferences in the results, which will be pointed 
 out in the following propositions. 
 
 132, If, in transforming an inequality, the same member that 
 was the greater before the transformation is the greater after, tlie 
 inequality is said to continue to exist in the same sense ; but, if tlie 
 transformation changes the general relation of the members, so that 
 the member which was the greater before the transformation is the 
 less after, tlie inequality is said to exist ill an opposite sense in the 
 two inequalities. 
 
 133, JPvop, — The sense in which an inequality exists is not 
 changed., 
 
 1st. By adding equals to both member Sy or subtracting equals from 
 both; 
 
INEQUALITIES. 149 
 
 2d. By multiplying or dividing the members by equal positive 
 numbers ; 
 
 3d. By adding or multiplying the corresponding members of two 
 inequalities which exist in the same sense, if all the members are 
 essentially positive / 
 
 4tli. By raising both tnembers to any power whose index is an odd 
 number ; 
 
 5th. £y raising both members to any power, if both members are 
 essentially ]?ositive ; 
 
 6 th. By extracting the same root of both members, if when the de- 
 gree of the root is even, only the positive roots be compared. 
 
 III. and Dem. — The 1st is, in general, an axiom. Thus if a > h, it is evi- 
 dent that a ± c > h ± e. When c> a, <i — c'ls, negative, but since b < a, b — c 
 is also negative and numerically greater than a — c. Therefore, in this case, 
 a-ob -c {120). 
 
 2d. This is wholly axiomatic. If a > & it is evident that ma > mb, and that 
 
 m m' . 
 
 3d. This, too, is an axiom. \i a >b, and c> d, a, b, c, and d being each + , 
 it is evident that a + c > b + d; and that ae > bd. 
 
 4th. This becomes evident by considering that if a > b, raising both members 
 to any power whose degree is odd will leave the sigiis of the members as at the 
 first, and also the sense of the numerical inequality the same. 
 
 5th. This appears from the fact that neither the signs nor the sense of the 
 numerical inequality of the members is changed by the process. 
 
 Cth. This is evident from the fact that the greater number has the greater 
 root, if only positive roots are considered. 
 
 134, Prop, — The 8e?i8e in %ohich an inequality exists is changed^ 
 
 1st. By changing the signs of both members ; 
 
 2d. By mxdtiplying or dividing both members by the same negative 
 quantity ; 
 
 3d. By raising both members to the same even power, if the members 
 are both negative in the first instance / 
 
150 ELEMENTARY ALGEBRA. 
 
 4th. By comparing the negative even roots (the members^ in the first 
 instance, being both essentially positive), 
 
 III, and Dem.— The first is evident, since Si a>h, —aK — hhy {129). 
 That is, of two negative quantities the numerically greater is really the less. 
 
 2d. These operations do not change the numerical relation of the members, 
 Ijut do change the signs of the members ; hence it falls under the preceding. 
 
 3d. and 4th. Essentially the same reasoning as in the last. 
 
 Exercises. 
 
 1. When a and b are unequal, show that «* + b*>2ab. 
 
 Solution. — Let a>h; whence a—b>0, or a*—2ab + b^>0,oTa* + b*>2ab. 
 Similarly if a <b. 
 
 2. Prove that the arithmetical mean between two quantities is, 
 in general, greater than the geometrical. Hoav if the quantities are 
 equal ? 
 
 3. If «, b, c, are such that the sum of any two is greater than the 
 third, show that a^ -\- b^ + c^ <2(ab -\- ac -h be). 
 
 4. If o* + J* + c^= 1, and m* + n^ + r^= 1, show that am ^-bn 
 + cr < 1. How \fa = b = c = m = n = r^ 
 
 5. Show that, in general, {a-\-b-c)? + {a-\-c—b)^-\- (b + c— a)^ 
 > ab •{■ be -\- ac. How if a=b—c? 
 
 G. Which is greater, 2x^ or .c + 1 ? 
 
 Solution.— 1st. If x>l, a;*> 1(?), 2x^>2x{1); but 2aj>« + l(?). 
 ,-. 3a;' > X + 1. 
 
 If X <1, a similar process shows 2a;'* < a; + 1. 
 
 7. From 5a; — 6 < 3a: + 8, and 2a; 4- 1< 3a; — 3, show that x 
 may have any value between 7 and 4 ; i. e., that the limiting values 
 are 7 and 4. 
 
 8. What are tlio limiting values of x determined from the con- 
 ditions 3a; —2 > ^x — ^, and J — Ja; < 8 — 2a; ? 
 
 9. The double of a iiuml)er diminished by o is greater than 25, 
 and triple of the number diminished by 7 is less than tlie double 
 increased by 13. What numbers will satisfy the conditions? 
 
PART III. 
 
 AN ADVANCED COURSE IN 
 ALGEBRA. 
 
 CHAPTER I. 
 
 INFINITESIMAL ANALYSIS. 
 
 SECTION I. 
 
 DIFFERENTIATION. 
 
 135, In certain classes of problems and discnssions the quantities 
 involved are distinguished as Constant and Variable. 
 
 136, A Constant quantity is oric wliich maintains the same 
 value throughout the same discussion, and is represented in the 
 notation by one of the leading letters of the alphabet. 
 
 137, Variable quantities are such as may assume in the same 
 discussion any value within certain limits determined by the nature 
 of the problem, and are represented by the final letters of the 
 alphabet. 
 
 III.— If X is the radius of a circle and j/ is its area, p = 7tx^, as we learn from 
 Geometry, 7t being about 3.1410. Now if a', the radius, varies, y, the area, will 
 vary ; but it remains the same for all values of x and y. In this case x and y 
 are the variables, and ;r is a constant. 
 
 Again, if y is the distance a body falls in time x, it is evident that the greater 
 X is, the greater is y, i. e., that as x varies y varies. We learn from Physics that 
 y — lQ^fX'^, for comparatively small distances above the surface of the earth. 
 In the expression y = 16,^^3;'^, x and y are the variables, and 16,\- is a constant. 
 
 Once more, suppose we have y^ = 25x^ — 3.^;* — 5, as an expressed relation 
 between x and y, and that this is the only relation which is required to exist 
 
152 ADVANCED COURSE IN ALGEBRA. 
 
 between them ; it is evident that we may give values to x at pleasure, and thus 
 obtain corresponding vahies for p. Thus if x = l, y = ± V\l, if x = %, y 
 = ± Vi88, etc., etc. In such a case x and y are called variables. But we notice 
 that if we give to x such a value as to make ^x^ + 5 > 25^'^ (as, for example, ^, 
 \, etc.), y will be imaginary. This is the kind of limitation referred to in our 
 definition of variables.* 
 
 138, ScH. — The pupil needs to guard against the notion that the terms 
 constant and vnriaUe arc synonyms for known and unknown^ and the more so 
 as the notation might lead him into this error. Tlie quantities he has been 
 accustomed to consider in Arithmetic and Elementary Algebra have all been 
 constant. The distinction here made is a new one to him, and pertains to a 
 new class of problems and discussions. 
 
 139, A Function is a quuntit}', or a mathematical expression, 
 conceived as depending for its valne upon some other quantity or 
 quantities. 
 
 III. — A man's Avages/o?" a given time is a function of the amount received per 
 day, or, in general, his wages is a function of the time he works and the amount 
 he receives per day. In the expression y = 16,V-c* {Vi7), second illustration, 
 y is a function of x, i. e., the space fallen through is a function of the time. The 
 expression 2«x* — 3a; -:- 5&, or any expression containing x, may be spoken of as 
 a function of x. 
 
 140, When we wisli to indicate that one variable, as y, is a func- 
 tion of another, as x, and do not care to be more specific, we write 
 9/ =f(x), and read "?/ equals (or is) a function of x" This means 
 nothing more than that i/ is equal to some expression containing the 
 variable x, and which may contain any constants. If we wish to 
 indicate several different expressions each of which contains a:, we 
 >vrite/(.T), (p{x)y orf'{x), etc., and read "the /function of x,'* "the 
 (p function of .r,"' or " the/' function of a*." 
 
 III. — The exprestion /(«) may stand for x^ —2x + 5, or for 3(a' — x^), or for 
 any expression containing x combined in any way with itself or with constants. 
 But in the same discussion f{x) will mean the same thing throughout. So again, 
 if in a particular discussion we have a certain expression containing x (e. </., 
 3a!* — ax + 2ab), it may be represented by /(.t), while some other function of x 
 (e. g., n{a^ — x"^) + 2x'-) might be represented by /'(«), or q>{x). 
 
 ., 141, In equations expressing the relation between two variables, 
 as in ?/2 =: 3ax^ — x^, it is customnry to sjuak of one of the variables, 
 as ?/, as a function of the other .r. Moreover, it i^ convenient lo think 
 
 * The limits of this vohime do not permit the interpretation of imaginnrics a«^ other than im- 
 po?siblo quantities, i. e., inconsistent with tiic restricted view talicn of the particular problem 
 which may be under conbidcration. 
 
DIFFERENTIATION. 153 
 
 of X as varying and thus producing change in y. When so con- 
 sidered, X is called the Independent and y the Dependent variable. 
 Or we may speak of ^ as a function of the variable x, 
 
 142, An Infinitesimal is a quantity conceived under such 
 a form, or law, as to be necessarily less than any assignable quantity. 
 
 Infinitesimals are the increments by. which continuous number, or 
 quantity (S), may be conceived to change value, or grow. 
 
 III. — Time affords a good illustration of continuous quantity, or number. 
 Thus a period of time, as 5 hours, increases, or grows, to another period, as 7 
 hours, by infinitesimal increments, i. c, not by liours, minutes, or even seconds, 
 but by elements which are less than any assignable quantity. In this way we 
 may conceive any continuous, variable quantity to change value, or grow, by 
 infinitesimal increments. 
 
 14:3, Consecutive Values of a function, or variable, are 
 values which differ from each other by less than any assignable 
 quantity, /. e., by an infinitesimal part of either. 
 
 144=, A Differential of a function, or variable, is the difier- 
 ence between two consecutive states of the function, or variable. It 
 is the same as an infinitesimal. 
 
 III. — Resuming the illustration y=r»6-,Vc' (IST)* let x be thought of an 
 some particular period of time (as 5 seconds), and y as the distance through 
 which the body falls in that time. Also, let x' represent a period of time infini- 
 tesimally greater than x, and y' the distance through which the body falls in time 
 x'. Then x and x' are consecutive values of .r, and y and y' are consecutive 
 values of y. Again, the difference between x and x', as x' — .?■, i., a differential 
 of the variable x, and y'— y is n differential of the function y. 
 
 145, W^otation,— A differential of x is expressed by writing the 
 letter d before x, thus dx. Also, dy means, and is read ^* differen- 
 tial yr 
 
 Caution. — Do not read dx by naming the letters as you do ax ; but read it 
 •' differential x." The d is not a factor, but an abbreviation for the word differ- 
 ential. 
 
 146, To lyifferentiate a function is to find an expression 
 for the increment of the function due to an infinitesimal increment 
 of the variable; or it is the process of finding the relation between 
 the infinitesimal increment of the variable and the corresponding 
 increment of the function. 
 
154 ADVANCED COURSE IN ALGEBRA. 
 
 Rules for Differentiating. 
 
 147, RULE 1. — To differentiate a single variable, sim- 
 ply WRITE the letter cl BEFORE IT. 
 
 This is merely doing what the notation requires. Thus if x and x' are conse- 
 cutive states of the variable x, i. e., if x' is what x becomes when it has taken an 
 infinitesimal increment, «'— x is the differential of x, and is to be written ci«. In 
 like manner, y'— y is to be written dy, y' and y being consecutive values. 
 
 148. RULE 2. — Constant factors or divisors appear in 
 
 THE DIFFERENTIAL THE SAME AS IN THE FUNCTION. 
 
 Dem, — Let us take the function y = ax, in which a is any constant, integral 
 or fractional. Let x take an infinitesimal increment d.r, becoming x -f dx ; and 
 let dy be the corresponding * increment of y, so that when x becomes x + dx, y 
 becomes y + dy. We then have 
 
 1st state of the function - - - - y = ax ; 
 
 2d, or consecutive state y + dy = a(x + dx) = €ix + adx. 
 
 Subtracting the 1st from the 2d dy = adx, 
 
 which result being the difference between two consecutive states of the function, 
 is its differential {144:), Now a appears in the differential just as it was in the 
 
 function. This would evidently be the same if a were a fraction, as — , We 
 
 should then have, in like manner, dy — —dx as the differential oi y = —x. 
 Q. E. D. 
 
 149, RULE 3. — Constant terms disappear in differen- 
 tiating; OR THE DIFFERENTIAL OF A CONSTANT IS 0. 
 
 Dem. — Let us take the function y =z ax + h, in which a and h arc constant. 
 Let X take an infinitesimal increment and become x + dx ; and let dy be the 
 increment which y takes in consequence of this change in x, to that when x 
 becomes x + dx, y becomes y + dy. W^e then have 
 
 1st state of the function y = flW! + &; 
 
 2d, or consecutive state - - . . y + dy = a{x + dr) + & = ax + adx + h 
 
 Subtracting the 1st from the 2d - - - - dy z= adx, 
 
 which being the difference between two consecutive states of the function, is its 
 differential {144). Now from this differential the cons^tant h has disappeared. 
 
 We may also say that as a constant retains the same value, tlu ro is no differ- 
 
 * The Word "contemporaneous"' is often ueed iu this cunncction. 
 
DIFFERENTIATION. 155 
 
 ence between its consecutive states (properly it has no consecutive states). 
 Hence the differential of a constant may be spoken of (though with some lati- 
 tude) as 0. q. E. D. 
 
 150, RULE 4. — To differentiate the algebraic sum of 
 
 SEVERAL VARIABLES, DIFFERENTIATE EACH TERM SEPARATELY AND 
 CONNECT THE DIFFERENTIALS WITH THE SAME SIGNS AS THE TERMS. 
 
 Dem. — Let u=ix-'fy — Zy u representing the algebraic sum of the variables 
 X, y, and —z. Then is du = dx ■\- dy — dz. For let dx, dy, and dz be infinitesimal 
 increments of x, y, and z ; and let du be the increment which u takes in conse- 
 quence of the infinitesimal changes in x, y, and z. We then have 
 
 1st state of the function w = a; + y — 2; 
 
 2d, or consecutive state u ->t du — x -v dx ^- y -v dy — {z -\- dz), 
 
 Or u + du — x + dx+y+dy—z — dz. 
 
 Subtracting the 1st state from the 2d - - - du = dx + dy — dz. q. e. d. 
 
 ISl, RULE 5. — The differential of the product of two 
 
 VARIABLES IS THE DIFFERENTIAL OF THE FIRST INTO THE SECOND, 
 PLUS THE DIFFERENTIAL OF THE SECOND INTO THE FIRST. 
 
 Dem. — Let u = xy be the first state of the function. The consecutive state is 
 u + du = {x 4- dx){y + dy) = xy + ydx + xdy + dx dy. Subtracting the 1st state 
 from the consecutive state we have the differential, i. e., du — ydx + xdy + dx dy. 
 But, &s dxdy is the product of two infinitesimals, it is infinitely less than the 
 other terms (ydx and xdy), and hence, having no value as compared with them, is 
 to be dropped.* Therefore du = ydx + xdy. q. e. d. 
 
 1S2, RULE G. — The differential of the product of sev- 
 eral VARIABLES IS THE SUM OF THE PRODUCTS OF THE DIFFER- 
 ENTIAL OF EACH INTO THE PRODUCT OF ALL THE OTHERS. 
 
 Dem. — Let u = xyz ; then du — yzdx + xzdy + xydz. For the 1st state of the 
 function is w — xyz, and the 2d, or consecutive state, u + du = (x + dx) {y + dy) 
 (z + dz), or u + du = xyz + y^dx + xzdy + xydz + xdydz + ydxdz + zdxdy 
 
 * It will doubtle«K appear to the pnpil, at first, as if this gave a rcsuU. only approodinaleJy rnr- 
 rcct. Such is not tli? fdct. The result is absoMdy correct. No error is introduced by dropping 
 dx dy. In fact this term mt'sf. be dropped according to the nature of infinitesimals. Notice 
 that by definition a quantity which is iiidni^c^invil with respect to another is one which has 710 
 assignable magnihidc with reference to that other. Hence wo must so treat it in our re isoiiing. 
 Now dxdy ie an iufinitesimal of an infinitcs'mnl (t. e.. two infinitesimals multipl'el together), 
 and hence is infinitesimnl with reference to ijdx and x "y. and mu^t be treated as having no as- 
 signable v,ilnc with respect to them ; that is. it must bo dropped. 
 
I5(j ADVANCED COURSE IN ALGERRA. 
 
 ■\- (Udydz. Subtracting, and (Iroi)ping all infinite»im-ila of infinitesimala (see 
 preceding rule and f(x>t-note), we have du = yzdx + xzdy + a-ydz. 
 
 In a similar manner the rule can be demonstrated for any number of varia- 
 bles. Q. E. D. 
 
 153. RULE 7.— The differential of a fraction having 
 
 A VARIABLE NUMERATOR AND DENOMINATOR IS THE DIFFEREN- 
 TIAL OF THE NUMERATOR MULTIPLIED BY THE DENOMINATOR, 
 MINUS THE DIFFERENTIAL OF THE DENOMINATOR MULTIPLIED BY 
 THE NUMERATOR, DIVIDED BY THE SQUARE OF THE DENOMINATOR. 
 
 X vdx xdv 
 
 Dem. — Let u = - ; then is du = :t — ~ . For, clearing of fractions. 
 
 y y* ^ 
 
 yu = X. Differentiating this by Rule 5th, we have udy + ydu = d.r. Substi- 
 tuting for u its value '-, this becomes' — - + ydu = dx. Finding the value of 
 
 - , _ ydc — xdy 
 du, we have dn = , -. o. e. d. 
 
 154, Con. — Tlie differential of a fraction having a constant 
 numerator and a variable denominator is the product of ths numera- 
 tor with its sign changed into the differential of the denominator, di- 
 vided by the square of the denominator. 
 
 Let u = ■-. Differentiating this by the rule and calling the differential of 
 the constant (a) 0, wo have du = -— — ^^— — . o v: D 
 
 liio. Sen. — If the numerator is variable and tlie denominator constant, 
 it falls under Rule 2. 
 
 150. RULE 8.— The differential of a variable affected 
 
 WITH AN exponent IS THE CONTINUED PRODUCT OF THE EXPO- 
 NENT, THE VARIABLE WITH ITS EXPONENT DIMINISHED BY 1, AND 
 THE DIFFERENTIAL OF THE VARIABLE. 
 
 Dem.— 1st. When the exponent is a positive integer. Let y = of", m being a 
 positive integer ; then dy = ntV'-^dx. For y = x"^ - xxx-x- to m factors. Now, 
 differentiating this by Rule G, we have dy = (xxx - - to m - 1 factors) dx 
 + (xxx . . to m — 1 factors) dv + etc., to m terms ; or dy = .T^'-'dr + x'^-^dx 
 + x"'-^dx + etc., to m terms. Therefore dy = mx^-^dx. 
 
 m 
 
 2d. When the cxjyoncnt i^ a positive fraction. Let y = x'^,'^ being a positive 
 
 7)1 
 
 fraction ; then dy = ^.i" dx. For involving both members to tlie nth power wo 
 have y" = x'". Differentiating this as just shown, we Ijave ny^-'^dy == mx"i~'^dx. 
 
DIFFERENTIATION. 157 
 
 ^ win— m 
 
 Now from y = ic» we have y^-' = x~^ . Substituting this in the last it be- 
 comes wic " dy = mx"'-Hx ; vfhence dy = '^x" n dx = ^x^ dx. q. e. d. 
 
 • 3d. When the exponent is negative. Let y = .'c-'», n being integral or 
 
 fractional ; then dy — — nx-'*-^dx. For y = a;-'* = — , which differentiated by 
 
 X" •' 
 
 Rule 7, Cor., gives dy = — — = — nx-^'-^dx. q. e. d. 
 
 Examples. 
 
 1. Differentiate // = 'Sx- — 2x -{- A. 
 
 Solution.— The result is dy = 6vdx - 2dx. Which is thus obtained : By 
 Rule 1, the differential of y is dy. To differentiate the second member we dif- 
 ferentiate each term separately according to Rule 4. In differentiating 3.i;^ we 
 observe that the factor 3 is retained in the differential, Rule 2, and the differen- 
 tial of .1* is, by Rule 8, 2xd.r. Hence, the differential of dx'^ is 6xdx. The differ- 
 ential of — Zv is — 2dx. By Rule 3, the constant 4 disappears from the differen- 
 tial, or its differential is 0, 
 
 'Z. Differentiate y = 2ax^ + 4:Clv^ — x + m. 
 
 Result, dy = ^axdx + VZax^dx — dx> 
 
 3. Differentiate y = Ur? — 30.?:' + 4.t. 
 
 4. Differentiate y = Ax' + Bx' + Cx\ 
 
 157. 'ScH. — It is desirable that the pupil not only become expert in writ- 
 ing out the differentials of such expressions as the above, but that lie know 
 what the operation signifies. Thus, suppose we have the equation y = ^x. 
 This expresses a relation bctAvcen x and y. Now, if x changes value, y must 
 change also in order to keep the equation true. In this simple case it is easy 
 to see that y must change 5 times as fast as x in order to keep the equation 
 true. This is what differentiation shows. Thus, differentiating, we have dy 
 = bdx. That is, if x takes an infinitesimal increment, y takes an infinitesi- 
 mal increment equal to 5 times that which x takes; or, in other words, y 
 increases 5 times as fast as x. 
 
 Now let us take a case which is not so simple. Let y=Bx'^—2x+4:, and let 
 it be required to find the relative rate of change of x and y. Differentiating, 
 we have dy = 6xdx — 2dx — (6x — 2)dx. This shows that, if x takes an infini- 
 tesimal increment represented by dx, y takes one (represented by dy) which 
 is 6a; — 2 times as large ; /. e., that y increases Qx — 2 times as fast as x. 
 Notice that in this case the relative rate of increase of x and y depends on 
 the value of x. Thus, when x=l, y is increasing 4 times as fast as x \ when 
 x=2, y\s increasing 10 times as fast as a;; when a; =3, y is increasing Iff 
 times as fast as x ; etc. 
 
158 ADVANCED COURSE IN ALGEBRA. 
 
 5. Differentiate ^ = r* — ic^, and explain the significance of the 
 result as above. Result, dij — (5.r* — Zx^)dx, 
 
 6. In order to keep the relation "ly — 3.c* true as x varies, how- 
 must y vary in relation to a; ? What is the relative rate of change 
 when a; = 4 ? When a; = 2 ? When a; = 1 ? When a; = J ? When 
 
 Anstvers. When a: = 4, y increases 13 times as fast as x. AVhen 
 X = i, y increases at the same rate as x. In general // increases 3x 
 times as fast as x. Wlien x is less than |, y increases slower than x. 
 
 2x^ X* — 1 
 7 to 12. Differentiate the following: u = -— ; u = ^ ; 
 
 3y x^ -\- I ^ 
 
 y = x^z^] n = xhf 4- 6a;; ^ = .-c* - 3a^ + 4a:» - a;' + 1 ; and 
 2/ = K - K + ^• 
 
 13 to 17. Differentiate y=(«2 4-a;3)5 ; ^=(a + a;«)*; 2^=(3a;-2)^; 
 2/ = (2 - .r2)-2 ; and ?/ = (1 + x)~^. 
 
 Sug's. — Such examples should be solved by considering the entire quantity 
 
 within the parenthesis as the variable. This is evidently admissible, since any 
 
 expression which contains a variable is variable when taken as a whole. Thus 
 
 a. 
 to differentiate y — {a + cc*) , we take the continued product of the exponent {%), 
 
 the variable {a + x^) with its exponent diminished by 1, [t. e., {a + x^)~^], and 
 the differential of the variable {i. e., the differential of a + x*, which is 'Zxdx). 
 
 _i _i 4xdx 
 
 This gives us dy = §(a + x') ^2xdx, or dy = ^iX(a + x') ^dx = — ^ 
 
 'dVa 
 
 + x^ 
 
 18 to 22. Differentiate^ ; ^^ r:; : — -: —m- 
 
 i + x' (1 + xY' (1 +xy' "\i -i-xY 
 
 and — in —— — r^ . 
 (1 -f xY 
 
 23. In the expression Ga:^, when x is greater than 1 does the func- 
 tion (Ga;^) change faster or slower than x ? How, when ,r is less 
 than ^? What does the process of differentiating eaj* signify 'f 
 
 Answer to the Inst. Finding the relative rate of change of 6.c ' and x, or find- 
 ing what increment 6.r' takes when x takes the increment dx. 
 
 Or, in still other words, finding the difference between two consecutive states 
 of 6a;^ and hence the relation between an inflnitesimal increment of x and the 
 corresponding increment of Qx\ 
 
INDETERMINATE COEFFICIENTS. 159 
 
 SECTION IT. 
 
 INDETERMINATE COEFFICIENTS. 
 
 158, Indeterminate Coefficients are coefficients assumed 
 in the demonstration of a theorem or the solution of a problem, 
 whose values are not known at the outset, but are to be determined 
 by subsequent processes. 
 
 159. Prop— If A + Bx + Cx'+ Dx^ + etc. = A' + B'x + CV 
 -f D'x'+ e^c, in which x is a variable* and the coefficients A, B, 
 A', B', etc. are constants, the coefficients of the like poicers of x are 
 equal to each other. That is, A = A' (these being the coefficients ofx^), 
 B = B', C = C, etc. 
 
 Dem.— Since the equation is true for any value of w, it is true for x=0. Substi- 
 tuting tliis value, we liave^ = ^'. Now as A and A' are constant, they have the 
 same values whatever the value assigned to x. Hence for any value of x, A — A'. 
 Again, dropping A and A', we have Bx + Cx^+ I>x^+ etc. = B'x + C'x^-j- D'x'^ 
 •f etc., which is true for any value of x. Dividing by x, Ave obtain B+C.v+Dx'' 
 + etc.=B' + C'x + D'x^+ etc., likewise true for any value of x. Making x = 0, 
 JB = ^', as before. In this manner we may proceed, and show that C=C 
 B = D' , etc. Q. E. D. 
 
 160. Qo^.—If A + Bx -I- Cx'^ + Dx' + etc. - 0, is true for all 
 values of x, each of the coefficients A, B, 0, etc.^ is 0. 
 
 For we may write A + Bx + Cx^ + Bx^ + Ex* + Fx^ ■+- etc. = + 0.c + Ox- 
 + Ox^+ Ox*+ Oir'+ etc. Whence by the proposition ^ = 0, ^ = 0, C= 0, etc 
 
 Development of Functions by means of Indeterminate 
 Coefficients. 
 
 101, A Function is said to be Developed when the indicated 
 operations are performed; or, more properlv, when it is transformed 
 into an equivalent series of terms following some general law. 
 
 Ill's, — Division affords a method of developing some forms of functions, 
 
 ♦ Saying that r it a variable, is eqnivalent to saying that tlie equation must be true for any 
 value of X. Thii? it? an cssontial thing in this discussion. The members of such an equation 
 are somsiimos said to be Identically equal. 
 
160 
 
 ADVANCED COURSE IN ALGEBRA. 
 
 Thus y = 
 
 when developed by division becomes y = \ + x + x^ + x^-^ etc. 
 
 The binomial formula (Complete School Algebra, 195, or 168 of this 
 treatise) is a formula for developing a binomial. Thus y — (a-\- xf when devel- 
 oped becomes y = a^ -\- ba\T -\-\Oa^x- -^-Kki^x^ -^ ^nx* + x^ . The subject is one 
 of great importance in matliematics, and the method of Indeterminate Coeffi- 
 cients forms the basis of most that is valuable upon it. 
 
 Examples. 
 
 \-x 
 
 1. Develop r- into a series by the method of Indeterminate 
 
 ^ l-{-x-\-ar 
 
 Coefficients. 
 
 1-3 
 
 Solution. — Assume 
 of fractions, 
 
 \-^x-\-x^ 
 
 A + Bx + Cx^-ir i).c ' + Ex^± etc. Clearing 
 
 A+n 
 
 ■¥A 
 
 +B 
 
 ^A 
 
 x'+B 
 
 + C 
 
 x' + E 
 +B 
 
 4-6' 
 
 etc. 
 etc. 
 etc. 
 
 Equating the coefficients of the corresponding powers of x by {l/>0),xve have 
 the following equations from Avhich to find the values oi A, B, C,'B, etc. : 
 A=l', A+B=-l; A-hB+C==0; B-hC+B = 0; C-\-B+E=0. Solving 
 these, we have ^ = 1, B=—2, C=l, D=l, and i^= — 3. 
 
 Substituting these in the assumed development, we have 
 
 \—x 
 
 ^ = l—2x+x^+x^~2x*+ etc. 
 
 1+x + x* 
 
 This can readily be verified by actual division. 
 
 2. Develop, or expand into a series (a'— x^y by means of Inde- 
 terminate Coefficients. 
 
 Solution. — Assume 
 
 {a- - x^)^= A+Bx+Cx* +Bx^-hEx* +Fx'^ -^Gx'^ + etc. 
 Squaring both members and expanding {a^—x^y, we have 
 
 a' 
 
 -da'x'-+daKi*'-x'^A^+AB 
 
 x+ACx''\^AD 
 
 .v*+AEx*'{-AF,c' + AG 
 
 A"" 4- etc. 
 
 
 +AB 
 
 +-B* '¥BC -^BD +BjJ -\-BF 
 
 -fete. 
 
 
 
 +AC -\BC 
 
 +C' i -hcn +CE 
 
 +etc. 
 
 \ 
 
 ■¥AD 
 
 +BD\ +CD -f/;* 
 
 4- etc. 
 
 
 ■\-ABi\ -^BE -^VE 
 
 -l-etc. 
 
 
 -^Ali', +BF 
 
 +etc. 
 
 
 
 
 +A0 
 
 +etc. 
 
 Equating the cOefficiefits of the cOri-espOndhig powers of .t, wc find A^ — a^, 
 OtA- tt ' ; 2AB s 0, whence ^ =: ; 2^C -h -B^ = - 3a^ whence f = - ^n ; 
 
 2Ui) ^ SO) « 0, whence 2) = 
 
 2(AS -h BB) + C^ = 3a ^ . whence E^^\ 
 
, indeter:minate coefficients. 161 
 
 2[AF-irBE-\- CD)=0, whence F—0 ; and in like manner G = -„ etc. (If the 
 
 feipansion of the second member had hern carried farther, each of the succeeding co- 
 efficients would be equated with 0, as there are no terms in the first member contain- 
 ing higher powers of a; than the Ctli.) Substitutingthe values of y1, B, C, jD^ etc, as 
 
 now found, we have (a^—x^)'=a^— ^ax-+ -^ — f- v;r-5 + etc. 
 
 3. Expand, or develop (1 —x^)^ by meciiis of Indetenniuate Coeffi- 
 cients. Also — -, .; , and 
 
 x + V b-ax' (i_.c)i* 
 
 SuG. — To expand the last, put the expression equal to the usual series, square 
 both members, and then clear of fractions. 
 
 102, ScH. — In using the method of Indeterminate Coefficients, as the 
 series A + Bx+ Cx' -\- etc., is merely hypothetical at the outset, we must 
 carefully observe whether the subsequent processes develop any inconsist- 
 ency. For example, perhaps a particular expression will not develop in the 
 form assumed. If so, some inconsistency will appear in the process. Thus, 
 
 2 2 
 were we to attempt to develop -j~ — -^ by assuming ~, .^ — A -\- Bx -}- Cx'^ 
 
 + Dx'*+ etc., we should find, after clearing of fractions, that the first mem- 
 ber had only tl:e term 2, which is 2x" ; and as there would be no correspond- 
 ing term in the second member, we should have to write 2 = 0, which is 
 absurd. In general, we observe that, when we equate the coefficients, the 
 second, or assumed member, must have a term containing as low a power of 
 the variable as the lowest in the first member. This may be secured either 
 by putting the expression to be developed into a proper form before assum- 
 ing the scries, or by assuming a series of proper form. Thus, in the above 
 
 2 12 2 
 
 case, we may write for — , — - • , and then develop by 
 
 x^ — x^x*l—x 1—x 
 
 2 
 assuming ■ -- = A + Bx + Cx^+Bx"^ + etc., and finally multiplying by 
 
 1 2 
 
 — ; or it may be developed by assuming — ^ = Ax~^+Bx-^+Cx^+ Bx 
 
 -f Ex^ -+- etc. 
 
 lT — "ix •+- Z 
 
 4. Expand — , — r- by the method of Indeterminate CoefR- 
 
 'X — ~ iC ~r X 
 
 .- 1-1- 2a: ., l-x 
 
 cients. Also ^r^. . Also 
 
 3:r* 2x' + ^x^ 
 
 5. Expand y"! -x.. Also (1 + x)^. 
 
 11 
 
162 ADVANCED COUKSE IN ALGEBKA. 
 
 Decomposition of Fractions by means of Indeterminate 
 
 Coefficients. 
 
 163. For certain purposes, especially in the Integral Calculus, 
 it is often necessary to decompose a fraction into partial fractions. 
 There are three principal cases. 
 
 104:, Case 1. — A fraction which U a function of a single varia- 
 ble^ whose numerator is of lower dimensions than its denominator, 
 and whose denominator is resolvable into n REAL and unequal fac- 
 toi's of the first degree^ can be decomposed into n partial fractions of 
 
 , ^ A B C N 
 the form 1 r ^ • > x + a, x + b, 
 
 '' x + ax + bx + c x+n 
 
 X + c, - - - - - x+n being the factors of the denominator. 
 
 M* ABC N . 
 Dem.— Assume ---^ = + r + , in 
 
 q){u-) X + a X + b x + c x + n 
 
 which f{x) is of lower dimensions f than q){z), and <p{x) = {x + a){x + b) {x + c) 
 
 (x + n). I Reducing the partial fractions , r , etc., to 
 
 forms having a common denominator, this denominator will be the product of all 
 the denominators x + a, x + b, x + c, etc., and hence will be (p{x), and each 
 numerator will contain one less of these factors than the common denominator, 
 and hence will be of the {n — l)th degree, the denominator being of the wth de- 
 gree.g Then, as the denominators of both members will be equal, the numera- 
 tors will also be equal. Placing them so, we can find the values of the indeter- 
 minate coefficients A, By C, etc., by the principle in {139). The necessity for 
 having /(a-) of lower dimensions than (p{x) is the same as is pointed out in {102). 
 Thus, if j\x) contained a term like 5^;* while (p{x) contained none higher than 
 2a;*, we should be required to write 5 = 0, as there would be no term in the sec- 
 ond member having an «' in it. Finally, having obtained the values of A, B, C, 
 
 ABC 
 
 etc., we can substitute them in , , , etc., and have the 
 
 X + a X + b X + c 
 partial fractions sought. 
 
 lOS, Case 2. — A fraction which is a function of a si?igle varia^ 
 ble, whose numerator is of lower dime?isions tha?t its de?iominator, 
 and whose denominator is resolvable into n real a)id TiQUALfactoi's 
 
 * See (139, 140). 
 
 t That i:?, does not contain so high a power of x. 
 
 X The proposition aspumes that cp{x) is resolvable into n real and unequal factors of the 
 fir<t degree. 
 
 § That is, containing x to the nth power, and no higher power. 
 
PBCOMPOSITION OF FRACTIONS. 163 
 
 of the first degree^ can be decomposed into ii partial fractions of the 
 . A ]^ C N 
 
 form J—- r-, + — — r-3y + 
 
 (x + a)' (X + a y-' (x + a)"-=^ x + a' 
 
 X + a being one of the equal factors of the denominator. 
 
 Dem.— Assume ^, = ; ^+^ — ; — ;— f 4 7- ^ — ; - - - - -=^— , 
 
 in which J\.v) is of lower dimensions than q){x), and (p[x) = {x + a)"". Reducing 
 the partial fractions to forms having the common denominator {x + a)" (i. e. <p(,x)), 
 and placing the numerators of the members equal, we Lnd tiiat thu tecoiid mem- 
 ber is not of lower dimensions with respect to the variable x, than the first mem- 
 
 N 
 ber, since tlie niiinerator of the fraction -; will contain the highest power of 
 
 X of any of the terms, and this will have no higher power than 2?"-^ as (a? + «)*"* 
 is the factor by which the terms of the fraction will be multiplied in the re- 
 duction. Hence, we can find the values of A, B, C, etc., by {15i)), and these 
 substituted in the assumed series will give the required partial fractions. 
 
 100, Case 3. — A fraction which is a functioii of a single varia- 
 ble, whose numerator is of lower ditnensions than its denominator, 
 and whose denominator is resolvable into n real and equal quad- 
 ratic yac^ors, can be decomposed into n vartial fractions of the form 
 Ax + B Cx + D Ex + F 
 
 [(x + a)" + b'^]- ' L(x + ^)' -r V]"-' [(X + a)^ + b^t 
 
 Mx + N 
 
 (X + a)' + b^ 
 factors of the denominator. 
 
 (x + a)'' + b' being one of the equal 
 
 Dem. — Assume 
 
 f{x) _ A x + B Cx + B ^ Ex + F 
 
 ^) " [{x + ay + 6=^]" "*" [(3 + ay + &'^]"-» ^ [:- -r af + Z»=^]»-« 
 
 Mx + N 
 {X + a)* + 6* ' 
 
 Bringing the terms of the secoBid member to a common denominator q){x), or 
 [{x -f ay + &*]", we find that the highest power of x involved in the numerators 
 is «*"-', which will ari.e in multiplying Mx -{• N\)y \{x + ay + ft^]"-'. But, as 
 f{x) IS of lower dimensions than cp{x), and (p{x) is of 2n dimensions, the numera- 
 tor of the second member will not be of lower dimensions than j\x), and hence 
 equating theni, the values of A, B, G, etc., can be determined and substituted in 
 the assumed series of parti?,! fractions. 
 
 107. Sen.— When the dcnomir.ator of the fraction to be decomposed is 
 composed of factors of two or more of the forms referred to in the three given 
 
164 
 
 ADVANCED COURSE IN ALGEBRA. 
 
 cases, the forms of the assumed partial fractions must be made to correspond. 
 Thus were it required to decompose ^j._i,^^^^ayix' +a-')* ' *^® assumed par- 
 
 tial fractions would be — + -, + ; r^ + 7 rr + + ^ , -; 
 
 X x — b {X + ay {X + ay x + a {X* + a*y 
 
 Hx + I 
 
 '^ x*+a*' 
 
 Examples. 
 1. Decompose 3 into partial fractions. 
 
 » ^ x* -2 A B C , ^.v. 
 
 Solution.— Assume = — + + , x,l — x, and 1 + a; bemg 
 
 x — x*xl — xl + x ^ 
 
 the unequal factors of 2; — x'^ {115, 117). Bringing the terms of the second 
 member to a common denominator, we have 
 
 x' -2 _ A- Ax* + Bx + Bx* + Cx - Cx* 
 x-x^~ x{X — x)(X + x) 
 
 Hence x* — 2 — A ^■ {B -^t C)x ^ {B — A — C)x* ; from which we get ^ = - 2, 
 B + C = 0, and B — A — C =1. Solving these equations we find A = — 2, 
 B = — ^, and C = i. These values inserted in the assumed forms give 
 
 «^--2 _ -2 i_ ^_ ^ _ 2 1 1 
 
 x^x^~ X i-a:'^l+a; x 2(1 - i-) "^ 2(1 + a;)* 
 
 «i«T^ .i*.ii- X -{■ d X -{• 1 
 2 to 6. Decompose the following : -, ; —, -rr ; 
 
 X^ — X — -v XKX ~'~ /C) 
 
 a; + 1 3a; — 5 , x* 
 
 -^; and 
 
 x" - Ix + 12' a? - Qx + %' a;' + 6a;' + 11a: + 6 * 
 
 Suo. — In case the factors of the denominator are not readily discerned, place 
 the denominator equal to and resolve the equation. Thus the last example 
 ^ves «' + 6x* 4- 11a; +6 = 0. From which we have a; = — 1, — 2, and — 3 
 {119), and the factors are a; + 1, a; + 2, and a; + 3. 
 
 7 to 11. Decompose the fractions -(^3^)3- 5 ^^^x^+^zu + 2r 
 
 and 
 
 a:^(l - a;')(l + a;) ' a;* - 1' (x - 2)> 4- 3)'-'* 
 
 a:*— 2a; + 3 3ar^— a^- 10.^*+ 15a;»4- 2a; — 8 
 
 12 to 18. Decompose 
 
 (a;« + l)' ' a-(a;^-2)'(a;-l) 
 
 a;'-a; + l 1 1 1 ^ 6a;2-4a;-6 
 
 and 
 
 x'ix + 1)' ^- 1' a'-a;*' x'- (a+b)x-{-ay ar^- 6a;H lla:-6 
 
THE BIKOMIAL FOKMULA. 165 
 
 SECTION IIL 
 
 THE BINOMIAL FORMULA. 
 
 108, Theove^n, — Letting x and y repi'esent any quantitiea 
 whatever (i. e. he variables) and m any constant, 
 
 Dem. — We may write (r+y)"* = a;"* ( 1 + - ) . Now put '- = s and assume 
 
 (1 + 2)» = .4 + Z?z + Cfe' + Dz^ + ^2^ + Fs** + etc., (1) 
 
 in which A, B, C, etc., are indeterminate coefficients independent of z {i. e. con- 
 stants), and are to be determined. To determine these coefficients we proceed 
 as follows : 
 
 Differentiating (1), we have 
 
 m{\ +z)'''-^dz=Bdz+2Czdz-\-SDz'^dz+iEz^dz+5Fz*(k-\- etc. 
 
 Dividing by dz, we have 
 
 m(l+zr-'=B+2Cz + 3i)2= + 4Ez'+5FY+ etc. (2) 
 
 Differentiating (2) and dividing by dz, we have 
 
 mim-l){l+zr''=2C+ 2 - 32)2 + 3 • AEz'-\- 4 - 5i^='4- etc. (3) 
 
 Differentiating (3) and dividing by dz, we have 
 
 7n(7/i-l)(m-2Xl+2)'"-'=2 ■ 32) + 2 • 3 • 4£fe + 3 - 4 - 5^'' + etc. (4) 
 
 Differentiating (4) and dividing by dz, we have 
 
 m(m-lXw-2)(w-3Xl+2)"-^=2 • 3 - 4^ + 2 - 3 - 4 Si^^s + etc. (5) 
 
 Differentiating (5) and dividing by dz, we have 
 
 w(m-lXw-2Xm-3Xm-4Xl+2)'"-'=2 - 3 ■ 4 • 5F+ etc. (0) 
 
 We have now gone far enough to enable us to determine the coefficients A, 
 B, C, D, E, and F, and doubtless to determine the law of the series. 
 
 As all the above equations are to be true for all values of z, and as the coeffi- 
 
 * This form is read " factorial 3," " factorial 4," etc. ; and eignifiee the product of tlic nat- 
 ural numbers from 1 to 3, 1 to 4, etc. 
 
100 ADVANCED COURSE IN ALGEBRA. 
 
 cients A,Ii, C, etc., are constants,/, e., have the same values for one value of z 
 as for another, if we can determine their values for one value of z, these will 
 be their values in all cases. Now, making 2=0, we have from (1) A = l ; from 
 
 (2), B = 711 ; from (3), C = — (the factor 1 being introduced into the de- 
 
 nominator for the sake of symmetry) ; from (4), D = -p ; from (5), 
 
 11 
 
 IL = p ; from (6), F= ^ . 
 
 These values substituted in (1) give 
 
 m{ m — 1) , w(m - \){m — 2) , m(m— l)(m— 2)(m— 3) ^ 
 
 -j2— ^ + [3 ' + ji ' 
 
 yn(m-l)(7n-2)(7n-3)(7n-4) _, 
 -\ j^j z -+- eic. 
 
 Finally, replacing e by its value -, we have 
 
 m{m — \)(m - 2){m — 3) y* m(,m - \\m — 2)(m — 3)(m - 4) y' ) 
 
 -*■ [4 ^^ + j5^ ^ +•^^^•1 
 
 =af' +rnx-'2,+ ^^^i)r"-y+ m(m-lXm-2)^.3^_^m (m-1) (m-2)(m-8) 
 
 LI I?. li 
 
 ^ ^ rw(m-lKm-2Xwi-3)(m-4) 
 af-V + -^^ ^ -^aJ— V + etc. 
 
 109, Cor. 1. — Tlte nth, or general term of the series is 
 m(m — 1) (//; — 2) {in — n + 2) ^,_ 
 
 \n-l 
 
 af'-*+^y*-\ 
 
 For we observe that the last factor in the numerator of the coefficient of any 
 particular term is m — the number of the term less 2, i. e., for the nth term, 
 m — {n—2), or 771 — n + 2 ; and the last factor in the denominator is the number 
 of the term — 1, i. e., for the r<th term, n — 1. The exponent of x in any par- 
 ticular term is m — the number of the term less 1, i. e., for the wth term, 
 7n—{n — 1), or 7n — n 4- 1 ; and the exponent of y in any term is one less than 
 the number of the term, i. e., for the nth term, n — \. 
 
 170, Def. — In a series the Scale of delation is the relation 
 which exists between any term or set of terms and the next term or 
 set of terms. 
 
 171, CoR. 2. — The scale of relation in the binomial series is 
 
 I — 1 )-» since the nth term multijMed hy this produces the 
 
 (n + 1 )th term. 
 
TKi: BINOMIAL FORMULA. 167 
 
 This is readily seen by inspecting the series, or by writing the (a + l)th term 
 
 and dividing it by the nth. 'J'iius, substituting in the general term as given 
 
 . . , m(w— l)(m— 2) (m—n+1) 
 
 above, n + 1 for n, we have ; ^^ -V'-" v», as the 
 
 (rt + l)th term. This divided by the nth, or preceding term,* gives — , 
 
 Examples. 
 
 1 to 6. Expand the following : (a — b)^ ; (x — y)'^ ; (a — a;)"* ; 
 {l-\-xy; {l-yy; (1 - y)". 
 
 7 to 11. Expand {x + y)-' ; (x - y)-' ; {a - x)-' ; ^^ ^ ^^, ; 
 , or {x-\- y)-\ 
 
 x + y 
 
 [Note. — For practical suggestions in the use of this theorem, see Complete 
 School Algebra, pages 148-154, or Part I. of this volume, pages 58, 59.] 
 
 12. Expand {a + xy by using the scale of relation. 
 
 Solution. — The scale of relation { 1 )- becomes in this case 
 
 V n Jx 
 
 ( 1 )- . Now the first term is a^ . To obtain the next ti = 1, whence 
 
 \ n J a 
 
 the scale of relation 5 - . Multiplying a' by this scale of relation, we find the 
 
 a 
 
 X 
 
 second term 5a*x. For the next the scale of relation is 2 - . Hence tlie 3d 
 
 a 
 
 term is lOa'a;*. For the next the scale of relation is -, giving for the 4th 
 
 a 
 
 (6 \x X 
 
 T — 1 )- or A— , 
 4 /a a 
 
 giving for this term 5ax*. For the Gth term the scale of relation equals 
 
 /6 \ X X /6 \ X 
 
 I - — 1 I— or i-, giving x^. For the 7th term the scale of relation is ( ^ — 1 ) - 
 
 or 0. Hence the series terminates, 
 
 13. Expand {m — n)~^ by nsing the scale of relation, and also 
 by the general formula. 
 
 14 to 17. Expand (1 - a^)^ ; (3 + x^)^; {x - y)'^ ; (a + x)'^. 
 
 * The numerator of the coefficient of the preceding, ornth term, contains all the factors of 
 the numerator of the (rt + l)th except m - n+1, as the factor in the (n+l)th preceding m - n + 1 
 is m - n + 2, etc. Similarly in the denominator. 
 
168 ADVANCED COURSE IN ALGEBRA. 
 
 18 to 20. Expand (a^-x^-)^ ; (U - x^)-' ; (a^ + c*)*. 
 
 SuG's.— In cases in which the terms of the binomial are not single letters or 
 figures, it will be best to substitute single letters, expand and then replace the 
 
 values. Thus, to expand (a; ' — 3«)"='", put x-^= y, and 3rt = h, and expand (y— 5)"^; 
 and in this expansion restore the values of y and h. In like manner the for- 
 mula may be applied to any polynomial. Thus, to expand (1 — ar*4- 3y)', put 
 (1 — «*) = e, and 3y = u, expand (« + «)*, and then restore the values. 
 
 21. Expand - into a series. 
 
 SuG'8. . ^ =a(6'-c^Jr')~^ Put 6*= -p, and c»^«= y, and expand 
 
 (« — y)~*, etc. The result is 
 
 , c»a^ 1 3 cV 13 5 c^^• 1 -3-5-7 cV ^ 
 
 &»"^2.4" i.*"^2 4-6" i." "^2-4 6-8 ■ "fe^"^ ®*^*f 
 
 22. What is the 4th term of the development of (a^+z)*? 
 (See 169>i 
 
 SuG.-The general term is ^^ - D (m-n + 1) ^.„_.+, ,_, i^ ^his 
 
 [n — 1 
 
 case m = i, n. = 4, a; = a*, y — z. Whence the 4th term is — 
 
 16a' 
 
 23. What is the 7th term of (a'^-h^)^'t The 10th term? 
 
 SECTION IV, 
 
 LOGARITHMS. 
 
 ^7^. A Logarithm is the exponent by which a fixed number 
 is to be affected in order to produce any required number. The 
 fixed number is called the Base of the System. 
 
 III.— Let the Bate be 3: then the logarithm of 9 is 2 ; of 27, 3 ; of 81, 4 ; 
 
 of 19683, 9 ; for 3*= 9 ; 3=^= 27 ; 3*= 81 ; and 3^= 19683. Again, if 64 is the 
 
 base, the logarithm of 8 is i, or .5, since 64% or 64''= 8; i.e., |, or .5 is the 
 
 exponent by which 64, the base, is to be affected in order to produce the num- 
 
 j. 
 ber 8, So, also, 64 being the base, i, or .333+ is the logarithm of 4, since 64^, 
 
LOGARITHMS. 169 
 
 or 04'"^— 4 ; i. e., }i, or .333+ is the exponent by which 64, the base, is to be 
 
 affected in order to produce the number 4. Once more, since 64^, or 64*^^^+=16, 
 
 §, or .666-1- is the logarithm of 16, if the base is 64. Finally, 64~^ or 64—^ 
 = -}, or .125 ; hence — ^, or —.5 is the logarithm of \, or .125, when the base is 
 64. In like manner, with the same base, — ^, or —.333+ is the logarithm of \, 
 or .25. 
 
 173, COH. — Since any number icith for its €xpo7ient is 1, the 
 logarithm of 1 is 0, in all systems. Thus 10^= 1, ichence in the 
 logarithm of 1, ^V^ a system in \chich the base is 10. 
 
 174, A System of Lof/aritluns is a scheme by which all 
 numbers can be represented, either exactly or approximately, by 
 exponents by which a fixed number (the base) can be affected. 
 
 175, There are 7\vo Systems of Logaritlims in common use, 
 called, respectively, the Briggean or Common System, and the Na- 
 pierian or Hyperbolic System.* The base of the former is 10, and of 
 the latter 2.71828 +. In the present treatise we shall confine our 
 attention to systems whose bases are greater than 1. 
 
 176, Cor. 1. — Neither 1 7ior any iiegative number can he used 
 as the base of a system of logarithms. 
 
 For all numbers cannot be represented either exactly or approximately by ex- 
 ponents of such numbers. Thus with 1 as a base we can represent no other 
 number than 1 by its exponents, for 1 with rt?i</ exponent is 1. Moreover, with a 
 negative base the logarithms which were odd numbers would represent negative 
 numbers, and those which were even numbers would represent positive numbers. 
 For example, with —2 as a base, 3 might be considered as the logarithm of —8, 
 since (— 2) '= - 8 ; but no number could be found as a logarithm to correspond 
 to 8 (*. e. +8), since —2 cannot be affected with any exponent which will pro- 
 duce 8. 
 
 177 , One of the most important uses of logarithms is to facilitate 
 the multiplication, division, involution, and the extraction of roots 
 of large numbers. These processes are performed upon the following 
 principles : 
 
 17 H, I^rop. 1, — The logarithm of the product of two 7iumbers 
 is the sum of their logarithms, 
 
 Dem. — Let a be the base of the system. Let m and n be any two numbers 
 whose logarithms are x and y respectively. Then by definition a'^m, and 00=71. 
 
 * The common syBtem is the one used for practical p^rRoges. a»d the only one of which 
 there are table? in com -non use, Napierian Ipgarithnis j^re HStially in^plied in abstj-act mathe- 
 matical difciipgion. 
 
170 ADVANCKI) COLKSE l.\ ALGKBRA. 
 
 Multiplying tUo corresponding members of these equations together we Lave 
 «' '—ma. Whence x + y \» the logarith of mn. Q. e. d. 
 
 170, J*l'op, 2* — The logarithm of the quotient of two jrumbers 
 is the lof/arithm of the dividend minus the logarithm of the divisor. 
 
 Dem. — Let a be the base of the system, and m and n any two numbers whose 
 logarithms are, respectively, ot, and y. Then by definition we have a'=zm, 
 
 and W — n. Dividing, we have a""-" = — . Whence x — yia the logarithm 
 
 - m 
 of — . Q. E. D. 
 
 ISO, Prop, 3, — T7ie logarithm of a power of a number is the 
 logarithm of the number nndtiplied by the index of the j^ower. 
 
 Dem. — Let a be the base, and x the logarithm of m. Then n^=m ; and raising 
 both to any power, as the 2tli, we have a*-=m-. Whence xz is the logarithm of 
 the 2th power of m. Q. E. D 
 
 IHl, Prop, 4, — 77ie logarithm of any root of a number is the 
 logarithm of the number divided by the number expressing the degree 
 of the root, 
 
 Dem. — Let a be the base, and x the logarithm of m. Then a'=m. Ex- 
 
 X 
 
 tracting the 2th root we have a== /^/m. Whence - is the logarithm of -y^m. 
 
 z 
 
 Q. E. D. 
 
 182, It is evident that in any system, the logarithms of most 
 numbers will not be expressed in integers. Thus in the common 
 system the logarithm of 100 is 2, and of 1000 3 ; hence the loga- 
 rithm of any number between 100 and 1000 is between 2 and 3, i. e. 
 2 and some fraction. This fraction is usually written as a decimal 
 fraction, and, as we shall see more clearly hereafter, can in general be 
 expressed only approximately. 
 
 ISS, Tlie Integral Part of a logarithm is called the Character^ 
 istiCf and the decimal part the Mantissa, 
 
 184, JProp. — TTie Mantissa of the logarithm of a decimal frac- 
 tioriy or of a mixed number^ is the same flw the mantissa of the num- 
 ber considered as integral^ 
 
 * Usaally, in speakinj? of logarithms, if no particular syetem is mentioned, tlie common 
 system is to be understood as meant, especially when practical numerical operations are 
 referred to. 
 
JQ4.4641 86 
 
 28456.73, 
 
 ^Q3>454l 86 
 
 2845.673, 
 
 102.454185 _ 
 
 384.5673, 
 
 101.464185 _ 
 
 38.45673, 
 
 JO**-***' 85 __ 
 
 3.845673, 
 
 384567.3 
 
 = 5.454185, 
 
 38456.73 
 
 = 4.454185, 
 
 3845.673 
 
 = 3.454185, 
 
 384.5673 
 
 = 3.454185, 
 
 38.45673 
 
 = 1.454185, 
 
 LOGAHITHMS. 171 
 
 Dem.— It will be found hereafter that log 3845673=6.454185. Now this 
 means that 10^ • ^ ^ 4 1 8 s -2845673. Dividing by 10 successively we have 
 10«-45 4i85 ^ 284567.3, or log 
 
 or log 
 
 or log 
 
 or log 
 
 or log 
 
 or log 3.845673 = 0.454185. 
 
 Now if we continue the operation of division, only writing 0.454185 — 1, 
 1.454185, meaning by this that the characteristic is negative and the mantissa 
 positive, and the subtraction not performed, we have 
 
 IOT.45 4 1 85 _ 2845673, or log .3845673 =1.454185, 
 
 10^.454 186 _ 02845673, or log .03845673 =3.454185, 
 
 107.46 4 185 ^ 003845672, or log .003845673 = 3.454185, 
 etc., etc. Q. E. D. 
 
 185. Cor. 1. — TTie characteristic of the logarithm of an integral 
 number, or of a mixed integral and decimal fractional yiumher^ is one 
 less than the number of integral places in the number. 
 
 The characteristic of the logarithm of a number entirely decimal 
 fractional is negative and numericaUy one greater than the number 
 of O'j immediately following the decimcd point. 
 
 Thus the characteristic of the logarithm of any number between 1 and 10 
 is 0, between 10 and 100 1, between 100 and 1000 3, etc. Or let it be asked, 
 " What is the characteristic of the logarithm of 5136? " Now this number lies 
 between 1000 and 10000, hence its logarithm lies between 3 and 4, and is, there- 
 fore, 3 and some fraction. 
 
 Again, as to the numerical value of the characteristic of the logarithm of a 
 number wholly decimal fractional, consider that 10~* = /j=.l ; 10~2=yiij=.01 ; 
 10-3 = j-ij- = .001. Thus it appears that any number between 1 and .1, i. e., any 
 number expressed by a decimal fraction having a significant figure in tenth's 
 place, as .3564, .846, .1305, etc., will have its logarithm between (the logarithm 
 of 1) and —1 (the logarithm of .1), Hence such a logarithm will be —1 4- some 
 fraction (the mantissa). In like manner, any number between .1 and .01. *. e., 
 any decimal fraction whose first significant figure is in lOOth's place, as .03568, 
 .0956, .01303, etc., will have for its logarithm —3 + some fraction. 
 
 186, Cor. 3. — The common logarithm ofO is — oo. 
 
 Since a number less than unity has a negative characteristic, and this char- 
 acteristic increases numerically as the number decreases, when the number 
 decreases to 0, the logarithm increases numerically to oo. Hence log 0=— oo. 
 To illustrate, log .1=1, log .01= 3, log .001 = 3, log .0001=4. Hence when the 
 number of O's becomes infinite, and the number therefore 0, we have log 
 
 = —00. •■ 
 
172 ADVANCED COUIISE IN ALGEBRA. 
 
 Computation of Logarithms. 
 
 187 » Tlie Modulus of a system of logarithms is a constant 
 factor which depends upon the base of the system and characterizes 
 the system. 
 
 188 » IProp* — The differential of the logarithm of a numher is 
 the differential of the number mtdtiplied by the modulus of the system, 
 divided by the number ; 
 
 Or, in the Napierian system, the modulus being 1, the differential 
 of the logarithm of a number is the differential of the number divided 
 by the number. 
 
 Dem. — Let X represent any number, i. e. be a variable, and n be a constant 
 such that y=oe*. Then log y=n log x {180). Differentiating y=x'*, we have 
 dy=nx'*~^dx; whence 
 
 dy 
 
 n=-^—= ^ = ^=JL /IN 
 
 ^*"'^^ "^dx y-dx '^' 
 
 XXX 
 
 Again, whatever the differentials of log y and log x are, n being a constant 
 factor, we shall have the differential of log y equal to n times the differential of 
 log X, which may be written 
 
 rfdog y)=/i . ff(log X), whence n = |}^. (3) 
 
 Now equating the values of n as represented in (1) and (2), we have "; ^ ' 
 dy 
 = -~. Whence d{log y) bears the same ratio to — , as d(log x) does to — . Let 
 
 X 
 
 ni be this ratio. Then <f(log y)— — -, and <^(log x)— . 
 
 y ^ 
 
 We are now to show that m is constant and depends on the base of the 
 system. 
 
 To do this, take y—Z'"', from which we can find as above ?i'=r ,^, ' 
 
 d(log 2) 
 
 dy 
 
 =-— — . Now as m is the ratio of <?(log y) to — , it is also the ratio of (Z(log z) to 
 
 z 
 
 — ; and <?(log 2)= . Thus we see that in any case the same ratio exists be- 
 
 z z 
 
 tween the differential of the logarithm of a number and the differential of the 
 
 number divided by the number. Therefore m is a constant factor. 
 
THE LOGARITHMIC SERIES. 173 
 
 That m depends upon the base of the system is evident, since in a system of 
 logarithms the only quantities involved are the number, its logarithm, and the 
 base. Of these the two former are variables ; whence, as the base is the only 
 constant involved in the scheme, m is a function of the base.* 
 
 189, JProb. — 2'o produce the logarithmic series. 
 
 Solution.— The logarithmic series, which is the foundation of the usual 
 method of computing logarithms, and of much of the theory of logarithms, ia 
 the development of log (1 + x). To develop log (1 + x), assume 
 
 log {1 i-x) = A + Bx+ Cx^ + Bx^ + Mc"- + Fx^ + etc., (1) 
 
 in which x is a variable, and A, B, C, etc., are constants. 
 
 Differentiating (1), we have 
 
 ?^ = Bdx + 2Cxdx + ^Dx^dx + ^Ex^dx + ^Fx^dx + etc. 
 
 \ + X 
 
 Dividing by dx, 
 
 -^^ =B + 2Cx + ZDx'' + 4^3 + ^Fx' + etc. (2) 
 
 Differentiathig (2), and dividing by dx, we have 
 
 - m y—^— = 2C7 + 2 • ZDx + Z A:Ex^ + 4. bFx^ + etc. (3) 
 
 (1 -\-xy 
 
 Differentiating (3), and dividing by 2 and by dx, we have 
 
 m — ^ — = 37> + 3 4i^.c + 2 3 5F«2 + etc. (4) 
 
 (1 + xY 
 
 Differentiating (4), and dividing by 3 and dx, we have 
 
 _ m ^ — - --=: 4JS; + 4 5iP!c + etc. (5) 
 
 (1 ^-xY ^ ' 
 
 Differentiating (5), and dividing by 4 and dx, we have 
 
 We have now gone far enough to enable us to determine the coefficients A, 
 B, C, D, E, and F, and these will probably reveal the law of the series. 
 
 As all the above equations are to be true for all values of x, and as the coeffi- 
 cients A, B, C, etc., are constant, t. e., have the same values for one value of x as 
 for another, if we can determine their values for one value of x, these will be 
 their values in all cases. Now, making x = 0, we have, from (1), A — log 1=0; 
 
 * Whftt the relation of the modiiln? to the base is, we are not now concerned to know ; it 
 will be determined liereaftcr. 
 
 t The numb<'r is 1 4 a; ; hence tiie differential is m times tlie differential of 1 + a^ divided by 
 the number \-^x. 
 
 X Of course the student will observe what forms the sncceedinir t- rms in this and the other 
 similar cases would have. Thus here we should have 5-F + 6 • 6C?iC 4 3 5 • T/Zr' + etc. 
 
174 ADVANCED COURSE IN ALGEBRA. 
 
 from (2), B — m\ from (3), C= — \m ; from (4), D — ^m ; from (5), E= ~ \m; 
 from (6), F= im. These values substituted in (1) give 
 
 ^^ 0*y^ Ct^ 3?'' 
 
 log (l + x)= m{x — ^ + -__ + -_ etc.), 
 
 the law of which is evident. This is the Logarithmic Senes, and should be fixed 
 in memory. 
 
 ScH. — The Napierian system of logarithms is characterized by the modu- 
 lus being 1 (m = 1). Hence the Napierian logarithmic series is 
 
 .... x^ X* x^ a-' 
 
 log(l+ar)=«-- +-3-J+5- etc. 
 
 190, Cor. 1. — The logarithms of the same number in differoit 
 systems are to each other as the moduli of those sy steins. 
 
 This is evident from the general logarithmic series. Thus the logarithm of 
 1 + « in a system whose modulus is m, is expressed 
 
 log„(l +x)« = m(;r-^ + ^'-?j? +^-etc.): 
 Ss o 4 a 
 
 and the logarithm of the same number in a system whose modulus is m' is ex- 
 pressed 
 
 log«.(l + X)* = m'{x -^ + ^-^4-^- etc.). 
 
 Now, as the number (1 + x) is, by hypothesis, the same in both cases, x is the 
 same. Hence, dividmg the members of the first by the corresponding members 
 
 of the second, we have logmd + x) ^ _m ^ 
 
 logaiXl + «) m' 
 
 101, Cor. 2.-^JIavin(/ the logarithm of a number in the Napierian 
 system, ice have but to multiply it by the modulus of any other system 
 to obtain the logarithm of the same number in the latter system. 
 
 Or, the logarithm of a number in any system divided by the loga- 
 rithm of the same number in the Napierian system, gives the modidus 
 of the former system. 
 
 102, I^voh, — To adapt the Napierian logarithmic series to nu- 
 merical computation so that it can be conveniently used for comjyuting 
 the logarithms of numbers. 
 
 /J.8 ^3 ^4 /j»5 
 
 Sol. — That log(l -v x) = x— — + — — t'^~F — ^^^-> ^^ ^°* ^" ^ practica- 
 
 2 o 4 o 
 
 ble form for computing the logarithms of numbers will be evident if we make 
 
 the attempt. Thus, suppose we wish to compute the logarithm of 3. Making 
 
 * The subscripts m and m' are need to distinsrnish between the system?, as log (1 -t O") is not 
 the sann in one syptem as in the oihcr. Ilead log»»(l + a;), "logarithm of \-^x in a system 
 whose modulus is m.^'' etc. 
 
COMrUTATION OF LOGARITHMS. 
 
 175 
 
 2* 2'* 2* 2' 
 a: = 2, we have ]og(l + 2) = log 3 = 2 — — + tt — r + ^ — etc., a series 
 
 ,6 o 4 5 
 
 in which the terms are growing larger and larger (a diverging series). 
 
 We wish a series in which the terms will grow smaller as we extend 
 
 it (a converging series). Then the farther we extend the series, the more 
 
 nearly shall we approximate the logarithm sought. To obtain such a series, 
 
 substitute —x iorx in the Napierian logarithmic series, and we have 
 
 , ,. . x'^ a?' X* x^ 
 
 logil-x)=-x--- ------ etc. 
 
 Subtracting this from the former series, we have 
 
 '1+x 
 
 (1 +3'\ 
 — — j =2{x + ^x^ + W +^x^ + etc.). 
 
 Now put x = - -, whence l+ic=l + ir 
 
 ^ 224-1 2s+l 
 
 2g + 2 
 
 22+1 
 
 l-x = 
 
 2z 
 
 22 + 1 
 
 , and 
 
 posmg, 
 
 l0g(l+2) = l0g2 + 2(^ 
 
 = . Hence, as log ( j = log (1 + 2) — log 2, substituting, and trans- 
 
 This series converges quite rapidly, especially for large values of 2, and is 
 convenient for use in computing logarithms. 
 
 103, I^rob, — To compute the N'apierian logarithms of the natural 
 numbers 1,2, 3, 4, etc.y ad libitum. 
 
 Solution. — In the first place wo remark that it is necessary to compute the 
 logarithms of prime numbers only, since th3 logarithm of a composite number 
 is equal to the sum of the logarithms of its factors {17 S). 
 
 Therefore beginning with 1, we know that log 1=0 (173). 
 
 To compute the logarithm of 2, make 2=1, in series (A), and we have log (1 + 1) 
 
 - log 1 = log2 = 2(g + 3^.+5^> + 74'' + 9ir" + TT^- + i3^ + i5V'+^''')- 
 
 The numerical operations are conveniently performed as follows : 
 3 2.00000000 
 
 .GG6G6G67 
 
 1 
 
 .07407407 
 
 3 
 
 .00823045 
 
 5 
 
 .00091449 
 
 7 
 
 .00010161 
 
 9 
 
 .00001129 
 
 11 
 
 .00000125 
 
 13 
 
 .00000014 
 
 15 
 
 .66660667* 
 
 .02469136 
 
 .00164609 
 
 .00013064 
 
 .00001129 
 
 .00000103 
 
 .00000009 
 
 .00000001 
 
 log 2 = .69314718* 
 
 * Thongh the decimal part of a logarithm U generally not exact, it is not customary ta 
 annex the + «»ign. 
 
176 
 
 ADVANCED COURSE IN ALGEBRA. 
 
 Second. To find lo^ 3, make s = 2, whence 
 
 log 3 = log 2+2(1+ Jg-.+gl^.+Jj,+ J5j+ etc). 
 
 Computation. 5 
 
 25 
 25 
 25 
 25 
 
 2.00000000 
 
 .40000000 
 
 1 
 
 .01600000 
 
 3 
 
 .00064000 
 
 5 
 
 .00002560 
 
 7 
 
 .00000102 
 
 a 
 
 .40000000 
 .00533333 
 .00012800 
 .00000366 
 .00000011 
 
 .40546510 
 log 2 = .69314718 
 
 /. log 3 = 1.09861228 
 
 Third. To find log 4. Leg 4 = 2 log 2 = 2 x .69314718 = 1.38629436 
 Fourih. To find log 5. Lei z = 4, whence 
 
 log o = log 4 + 2(1 + ^. + ^, + ^,+ etc.). 
 
 Computation. 9 
 
 81 
 81 
 81 
 
 2.00000000 
 
 .22222222 
 .00274348 
 .00003387 
 .00000042 
 
 .22222222 
 .00091449 
 .00000677 
 .00000006 
 
 .22314354 
 log 4= 1.38629436 
 
 .-. log 5 = 1.60943790 
 
 In like manner we may proceed to compute the logarithms of the prime num. 
 bers from the formula, and obtain those of the composite numbers on the prin- 
 ciple that the Ic^rithm of the product eqaals the sum of the logarithms of the 
 factors. 
 
 Thus, the Napierian Ic^rithm of the base of the common system, 10, = log 5 
 + log 2 = 2.30258508. 
 
 194, Prop, — The modithts of the common system is .43429448 + . 
 
 Dem. — Since the logarithm of a number, in any system, divided by the Na- 
 pierian logarithm of the same number is equal to the modulus of that system 
 (191), we have 
 
 — — '■ — ?- — = modulus of common system. 
 Nap. log 10 
 
TABLES OF LOGARITHMS. 177 
 
 But com. log 10 = 1, aucl Nap. log 10 = 2.30258508, as found above. Hence, 
 ModtUus of common system = -^r^f^^^-^iTx = .43429448. 
 
 Tables of Logarithms. 
 
 19S, As one of the most important uses of logarithms is to 
 facilitate the performance of multiplication, division, involution, and 
 evolution, when the numbers are large, according to (178-lSl), 
 it is necessary to have at hand a table containing the logarithms 
 of numbers. Such a table of common logarithms is usually found 
 in treatises on trigonometry and on surveying, or in a separate 
 volume of tables.* These tables usually contain the common loga- 
 rithms of numbers from 1 to 10000, Avith provision for ascertaining 
 therefrom the logarithms of other numbers with sufficient accuracy 
 for practical purposes. Four pages of such a table will be found 
 at the close of this volume. 
 
 196, J^vob, — Tofiiid the logarithm of a number from the table. 
 
 Solution. — The logarithm of any number from 1 to 100 inclusive can be 
 taken directly from the first page of the table. Thus log 2 = 0.301030, and 
 log 21 = 1.322219.t 
 
 To find the logarithm of any number from 100 to 999 inclusive, look for the 
 number in the column headed N, and opposite the number in the first column at 
 the right is the mantissa of the logarithm. The characteristic is known by 
 {185). Thus log 182 = 2.260071 ; log 135 = 2.130334. 
 
 To find the logarithm of any number rej^resented by 4 figures, find the first 3 
 left-hand figures in column N, and opposite this at the right in the column which 
 has the fourth figure at its head, will be found the last four figures of the niau- 
 The other two figures of the mantissa will be found in the column, oppo- 
 
 * Mathematicians and practical computers generally use more complete and extended table:* 
 than those found in connection with such elementary treatises. The common tables give five 
 places of decimals* in the mantissa. Those in connection with this series give six. Callet's 
 tables edited by Haslcr are standard eight-place logarithms. Vega's tiiblcs are among the best. 
 Dr. Bremiker's edition, translated by Prof. Fischer, is a favorite. KOiilcr's edition of Vega's 
 contains Gaussian logarithms. Vega's tables are iseven-place. Ten-place logarithms arc neces- 
 sary for the more nccurnto astronomical calculations. Prof. J. Mills Peirce, of Harvard, has re- 
 cently issued nn elegant little folio edition of tables containing among other tilings a table of 
 three-place logarithms whicli is very convenient for most uses. 
 
 t This page is really unnecessary, since nothing ran be found from it which cannot be found 
 with equal case from \\\^ succeeding part of the table. Thus, the mantissa of log %i& the eamo 
 as the mantissa of log 200 ; and the mantissa of log 21 is the same as thai of lo^ 210. 
 
 12 
 
178 ADVANCED COURSE IN ALGEBRA. 
 
 Bite the first three figures of the number or just above, unlesd heavy dots have 
 been passed or reached in running across the page to the right, in which case the 
 first two figures of the mantissa will be found in the column just below the 
 number. The places of the heavy dots must be supplied witli 0'.-;. The charac- 
 teristic is determined by {183). Thus log 1316=3.119250 ; log 2012=3.310056 ; 
 log 1868 = 3.271377. 
 
 To find the logarithm of a number represented by more than 4 fignres. Let 
 it be required to find the logarithm of 1934261. Finding the mantissa correspond- 
 ing to the first four figures (1934) as before, we find it to be .286456. Now in the 
 pame horizontal line and in the column marked I), we find 225, which is called 
 the Tabular Difference. This is the difference between the logarithms of two 
 consecutive numbers at this }X)int in the table. Thus 225 (millionths) is the 
 difference between the logarithms of 1934 and 1935, or, ns we are using it, 
 between the logarithms of 1934000 and 1935000, which differences are the same. 
 Now, assuming that, if an increase of 1000 in the number makes an increase of 
 225 (millionths) in the logarithm, an increase of 261 in the number will make an 
 increase of -|2i|i,V,, or, .261, »)f 225 (millionths) in the logarithm,* we have .261 
 X 225 (millionths) = 59 (millionths), omitting lower orders, as the amount to be 
 added to the logarithm of 1934000 to produce the logarithm of 1934261. Adding 
 this and writing the characteristic {185) we have log 1934261 = 6.28G515. In 
 like manner the logarithm of any other number expressed by more than four 
 figures may be found. 
 
 197, Sen. — As the mantissa of a mixed integral and decimal fractional 
 number, or of a number entirely decimal fractional, is the same as that of an 
 integral number expressed by the same figures {184), we can find the man- 
 tissa of the logarithm of such a number as if the number were wholly inte- 
 gral, and determine the characteristic by {185). 
 
 198, I^rob. — To find the number corresponding to a given 
 logarithm. 
 
 Solution. — Let it be required to find the number corresponding to the log- 
 arithm 4.2vJ4567. Ix)oking in the table for the next kfss mantissa, we find .234517, 
 the number corresponding to which is 1716 (no account being taken as to 
 whether it is integral, fractional, or mixed ; as in any case, the figures will be the 
 same). Now, from the tabular difference, in column D, we find that an increase 
 of 253 (millionths) upon this logarithm, would make an increase of 1 in the 
 number, making it 1717. But the given logarithm is only 50 greater than the 
 logarithm of 1716 ; hence, it is assumed (though only approximately correct) 
 that the increase of the number is -/s^ of 1, or .1976 f . This added (the figures 
 annexed) to 1716, gives 17161976 -\- . The characteristic of the given logarithm 
 being 4, the number lies between the 4th and 5th powers of 10, and hence has 5 
 integral places, .*. 4.234567 = log 17161.976 +. In like manner the number 
 corresponding to any logarithm can be found. 
 
 * This atsir.i ption, thotig'i not et iitly correct, is snfficluntly accurate for all ordinary 
 purposes. 
 
COMPUTATION BY LOGARITHMS. 179 
 
 199. JProp.— The rnqyierlan base is 2.718281828. 
 
 Dem. — Let e represent the base of the Napierian system. Then by {190) 
 com. log e : Nap. log e : : .43439448 : 1. 
 But the logarithm of the base of a system, taken in that system is 1, since 
 a' = a. Hence, Nap. log e = l, and com. log e = .43429448. Now finding from 
 a table of common logarithms the number corresponding to the logarithm 
 .43429448, we have e = 2.718281828. 
 
 Examples. 
 
 1. If 3 were the base of a system of logarithms, what would be the 
 logarithm of 81 ? Of 729 ? If 5 were the base, of what number would 
 3 be the logarithm ? Of what 2 ? Of what 4 ? 
 
 2. If 2 were the base, what would be the logarithm of J ? Of ^ ? 
 
 3. If 16 were the base, of what number would .5 be tlic logarithm ? 
 Of what .25 ? 
 
 4. Ill the common system we find that log 156=2.193125. Show 
 
 I.JLJL3126 
 
 that this siguifies that 10^«"«^" =156. 
 
 5. Log 1955=3.291147. To what power does this indicate that 
 10 is to be raised, and what root extracted to make 1955 ? 
 
 6. Find from the table at the close of the volume what root of 
 what power of 10 equals 2598. 
 
 7. Multiply 1482 by 136 by means of logarithms, using the table 
 at the close of the volume. (See 178,) 
 
 8. Perform the following operations by means of logarithms: 
 116.8 X 1879; 2769 -r 187; 15.13 x 1.3476; 257.16 -^ 18.5134; 
 .126 H- 6.1413; .11257 x .00126; (1278.6)'^; (112.37)'. 
 
 9. Perform the following operations by means of logarithms: ^2 
 to 5 places of decimals ; y 5 to 3 places of decimals ; y 2341564273 
 to two places of decimals ; a/3015618 to 4 places of decimals. 
 
 10. Perform the following operations by means of logarithms: 
 V^.01234 to 4 places of decimals; a/.03125 to 5 places of decimals* 
 V^0002137 to 5 places of decimals. 
 
 SuG's.— Log .01234=2.091315. Now to divide this by 3. we have to remember 
 that the characteristic alone is negative, i. e. that 2.091315 =- 2 I-. 091315, ox 
 
180 ADVA^'CED COUKSE IN ALGEBUA. 
 
 — 1.908685, whidi is all negative. Dividing this by 3, we have —.636228, or 
 0— .636228=1.363772. But a more convenient way to effect the division is to 
 write 2.091315 = 3 + 1.091315, and dividing the latter by three we obtain 
 1.363772, in which the characteristic alone is negative, thus conforming to the 
 tables. 
 
 To divide 13.341652 by 4, we write for 13.341652, -16+3.341652, and dividing 
 the latter obtain 4.835413. 
 
 11. Divide as above 11.348256 by 3; 17.135421 by 5; 1.341263 
 by 6. 
 
 12. Given the following to compute x by logarithms : 
 201.56 : 134.201 : : 18.654 : x; 2350.64 : .212 : : 1.1123 : x ; 
 X : 234.008 : : 15.738 : 200.56 ; 123 : a: : : 2.01 : .03. 
 
 /fit /pt 
 
 13. Having i/ = A/ — — — to express the equivalent operations 
 
 in losrarithms. 
 
 — rt) (« — h) (s — c) 
 
 SuQ'8. y = V{a - X) {a + .t)-i-(1 + x). .-. log y= \ [log {a - x) + log {a+x) 
 -\og{l+x)l 
 
 2 1 
 
 14. Given y=x^ (l—.v^)^ to express the equivalent operations in 
 logarithms. Also // = A/ j-. Also // = A/ ■ 
 
 Also 1/ = rir— • Also ^ = i / — - . Also giveu Ti •• -7 : : 
 
 \/m^—X' : y to express log y. 
 
 15. Differentiate y = \og{a^ —x*), 
 
 Sug's. — Write y =± log (a + a^) + log (a — x). Then differentiating, we have 
 
 mdx mdx _ ,._ ^. ^. .^i x^ _^ • t j d{a*~x^)* 
 
 dy = . Or differentiating without factoring, we have ay =—^-5 5- 
 
 (ti -k' X a — X a — X 
 
 iimxdx 
 
 = — --i 5. WHien reduced the results are the same, but the former is usualh' 
 
 «'— a;* 
 
 the more elegant method. 
 
 16. Differentiate the following: y = log (1 — a:) ; y — log ax; 
 
 y =z log x^ ; y = \o^ ; // = log v^l + .r. 
 
 ♦ This form signifies that a' -x^ is to be differentiated. The operation is only indicated, r.ot 
 performed. 
 
SUCCESSIVE DIFFEKENTIATION. 181 
 
 Sug' 8.— Remember that log x"^ = 3 log a; ; and also that log /y/l -\- x — 
 Hog(l+a). 
 
 17. Find from the table at the close of the volume that Kap. log 
 1564=7.3550018. Find in like manner the Napierian logarithms of 
 5, 120, and 2154372. 
 
 18. Knowing that the Napierian logarithm of 22 is 3.0910425, how 
 would you find the common logarithm of 23 from the logarithmic 
 series {192) ? 
 
 19. The common logarithm of 25 is 1.39794. What is the modu- 
 lus, and what the base of a system which makes the logarithm of 
 25 2.14285? 
 
 Query. — How do you see at a glance that the required base is a little less 
 than 5 ? 
 
 SECTION V. 
 
 SUCCESSIVE DIFFERENTIATION, AND DIFFERENTIAL 
 COEFFICIENTS. 
 
 200, JProp, — Differentials^ though itifinitesimah, are not neces- 
 sarily equal to each other. 
 
 Dem. — ThuB, let 11— ^x"^. Then dt/=6x^dx. Now, for all finite values of x, 
 dy is an infinitceinial, Binco no finite number of limes ibe infinitcf^imal dx 
 can make a finite quantity, and dy is 6x'* times dx. But for ,^'=l, dy is G times 
 dx ; for x=2, dy is 24 times dx; for a;=8, dy is 54 times dx. 
 
 201* Coil. — When y = f(x), dy 18 got er ally a variable, and hence 
 can he differentiated as any other variable. 
 
 202. NoTATiO]!?. — The differential of dy is written d^y, and read 
 " second differential of y." The differential of d^y is written d'^y, and 
 read "third differential of y," etc. The superiors 2 and 3 in such 
 cases are not of the nature of exponents, as the t? is not a symbol of 
 number. 
 
 203, In differentiating y=:f(x) Bticcessively, it is customary to 
 regard dx as constant. Tliis is conceiving x to change (grow) by 
 equal infinitesimal increments, and thence ascertaining liow y raries. 
 In general, y will not vary by equal inciements when x does, as 
 appears from the demonstration above. 
 
182 ADVANCED COURSE IN ALGEBRA. 
 
 204, A Second differential is the difference between two 
 consecutive states of n first differential. — A Third Differential 
 
 is the difference between two consecutive states of a second differ- 
 ential, etc. 
 
 III.— In the function y-2.c',if x passes to the next state, we have dy—Qx^dx. 
 Now dy, though an infinitesimal, is still a variable, for it is equal to Qdx times 
 a-*, and ^ is a variable. Hence if ,r takes an infinitesimal increment, di/ will pass 
 to a consecutive state. In other words, we can differentiate di/=Gdx.v^, just as 
 we could It = j?ix^, dy being a variable function. M.v a constant factor, and x the 
 variable. Representing the differential of dy by d-y, wo have d'^y = 6d?; 2.idx, 
 or d^y=l'ixdx', dx- being the square of dx, not the differential of x\ To indi- 
 cate the latter we would write rf(.r*). 
 
 JEXAMPLES. 
 
 1. Given y = Sj^ — 2a^ to find the third differential of i/, or (Py. 
 
 Solution. — Differentiating y=3.r'— 2.r*, we have rfy=15.c*rfx— 4r(f.c. Now, 
 regarding dx as constant, and differentiating again, we have d^y=60x*dx' 
 —Adx^* Differentiating again in like manner, we obtain d*y=180x^dx^, the 
 second term disappearing, since 4dx* is constant. 
 
 2. Given y = 2a^ — Sx -{- 5 to find the second differential of ?/, 
 t. c. d^y. 
 
 3. Given y = [x — af to find the third differential of ^. 
 SuQ'8. dy=3(x-aydx. d*y=6(x-a)dx', d^y=Ux^. 
 
 4. Given y = Ax + Bx" + Cct" + Dx\ to find the 4th differential 
 of y, J, B, C, and D, being constiint. (ty = 4 • 3 • 2 Ddx*. 
 
 5. Differentiate y =: A + Bx + Cj^ -{- Dx" + Ea^ -\- Fa^ -\- etc., 5 
 times in succession. 
 
 G. Differentiate y = {x — \){x — 2) (a: — 3) (a: — 4) twice in suc- 
 cession without expanding. 
 
 SoG'8. dy = {x-%){x-Z){x-^)dx + {x-\){x-S){x-^)dx+{x-l){x-2){x-A) 
 dx-{-{x-\){x-2){x-'S)dx. 
 
 = [(.r-2) {x-2) (3--4) + {x-\) ix—Z) {x-i) + ix-\) {x-2){x-'i) 
 ^{x-\){x-2){x-2,)]djc. 
 
 d'^y = \{x-Z){x-\)dx+{x-2){x-^)dc^{x-'^){x-Z)dx + {x-Z){x-^)dx + {x-\) 
 (x-^)dx + {x-\) (a;-3) dx-^{x-2) (.r-4) dx+{x-\) (.r-4) dx+{x-\){x-2) 
 dx +{x-^) {x-'S) (ic+(.c-l) (.r-3) dxH-t-X) (.c-2)dc]dr. 
 
 * To differentiate \bx*dx. calling dx constant, we may write 15cto x*. Now 15dx is con- 
 stant. Hence differentiating a;4,we have Ax^dx, which multiplied by the constant 15tfx, gives, ai 
 above, BOx'ctr^. The dx^ is " the sqnarc of rfr," not t'ac cliflfereniial of x"^. 
 
DIFFERENTIAL COEFFICIENTS. 183 
 
 = [(^-3) (^-4) + {.v-2) (,i--4) + (.c-2) (x-'S) + (x-d) (x-i) + (x-l) {x-4) 
 + (^-1) (^-3) + {x-2) (x-i) + (x-l) (x-i) + (x-l) {x-2) + {x-2) {x-3) 
 + (x-l) {x-'S) + {x-l){x-2)]dx'. 
 
 7. As above, differentiate y = (x — a){x — b)(x — c) twice in suc- 
 cession without expanding. 
 
 Differential Coefficients. 
 
 205. The First Differential Coefficient is the ratio of 
 the differential of a function to the differential of its variable. Thus, 
 
 if y=f(x), and (lij=f'{x)dx, ^-^ = f'(x), and ^, or its equivalent 
 
 f'{x), is the first differential coefficient of ;y, oy f(x). 
 
 III. — The meaning of this is simple. Thus, if y = 2x^, -~- = 8x^ ; that is, if 
 
 X takes an infinitesimal increment dx, y takes an infinitesimal increment dy, 
 which is to dx, as 8.i' ' is to 1, or the ratio of dy to dx is 8.f '. In still other words, 
 y increases 8.c ' times as fast as x. The reason for calling this a differential 
 coefficient, is that it is the coefficient by which the increment {dx) of the variable 
 must be multiplied to give the increment {dy) of the function. 
 
 206, The Second Differential Coefficient is the ratio of 
 the second differential of a function to the square of the differential 
 of the variable. Thus, if y—f{x)y dy=f{x)dx, and d'^i/=f"{x)dx'^, 
 
 — ^=/"(rc), -y^ or its equivalent /"(a:), is the second differential coef- 
 ficient of //, or f(x). In like manner Third, Fourth, etc., differential 
 coefficients are the ratios respectively of the third, fourth, etc., dif- 
 ferentials of a function, to the cube, fourth power, etc , of the dif- 
 ferential of the variable. Thus, if y=f(x), dy=f\x)dx, dhj—f'ix rZr 2, 
 d^y=^f"'(x)dx^, and d'^y =/'" {x)dx^, the successive difCercntitil coclli- 
 
 cientsare f=} {x), J=/ (x), ^=f ix), and ^,=/'\.r). 
 
 III. — Too much pains cannot be taken by the student in order to get a clear 
 conception of the meaning of the various symbols f{x), f'{-i'), /"(•')» f"'i-^)' ^^^- 
 
 To illustrate, suppose we have y = 2.c^— a;'H-C, whence -p = 8.i;'—C.t;'-, - -— 
 
 * To produce the pnccessive differential coefBcients wc maj' produce the corrcspnndin;? guc- 
 cessive differentials as in the preceding example?, or wc may proc.;c 1 thn^*: ■_='\r3-3x2 can 
 be differentiated, remembering that dy is variable and dx constant, and it gives ^— =24x2(/a? 
 
 (fiu 
 
 -%xdx, whence ;i-^'=24a:'-6a:. 
 
184 ADVANCED COUKSE IN ALGEBRA. 
 
 d ii d^ 7/ 
 
 = 24c*— dr, -r-^ = 48a'— 6, and -j— =48. Now in this case y =f(x), i. e., v is a 
 
 dv 
 
 function of .r ; so -r^ is also a function of x, being equal to 8^''- 3a'' ; but, as it 
 
 is not the same function of r that y is, we call it the / prime function, and write 
 
 diJ ft i/ ft^ fj 
 
 ~ = f'U). In like manner -~^ =f"(x) means that -~ is some function of ar, 
 
 dx ^ ^ ' dx* '' ^ ' dx* 
 
 but a different one from either y,oT~ . It may be observed that, in t?iis example, 
 
 d*y 
 
 3-j is not a function of x, and hence the inquiry arises as to the propriety of the 
 
 d*u 
 notation ^-^ =/'* (x). It must be remembered that this form of notation is the 
 
 d*v 
 general form, and it is the general fact that -j-^ is a function cf .r, though in 
 
 special cases it may not be. 
 
 Examples. 
 
 1. Produce the 1st, 2cl, 3(1, and 4th differential coeflRcients of 
 y=a:»— 3a:» + .T— 10. 
 
 Operation, dy = 5x*dx — Qx'dx + dx, whence ■— = 5x* — 9a;* + 1. Differ- 
 
 dx 
 
 entiating the latter* -r^ = 2(te^da5 - 18a-da?, whence -^ = 20x^ - ISa-. Again 
 ax (tx 
 
 differentiating, ^ = (60a;* - 18)dc, whence ^ = OOc* - 18. Finally, ^ 
 ax ax' ax 
 
 = 12ar. 
 
 2. If y =00:*— 3a;, what is the ratio of the increase of y to that of a;, 
 in general ? What is it when x=-\ ? When a:=2 ? AVhen a:=3 ? 
 
 Ans, In general, y increases 10a:— 3 times as fast as x. When 
 a:=l, y is increasing 7 times as fast as x. When a:=2, y is increas- 
 ing 17 times as fast as x, 
 
 3. If 7/=x*4-2a:*— a: + 10, what is the ratio of the 3d differential of 
 ?/ to the cube of the differential of .^• ? WHiat is it when .^=1? 
 When x=^ ? ATlien a:=| ? What is the name of this ratio ? 
 
 4. If ?/=(rt + a:)'% what is the 1st differential coefficient of tlie func- 
 tion? What the 2d? What the 3d? AVhiit the 5th? What 
 the 11th? 
 
 ^ = j»(M-l)(?;i-2)(m-3)(m-4)(^<-f-a:)'"-». 
 * See foot-uote on preccdliis paje. 
 
Taylor's formula. 185 
 
 5. Produce the first 5 successive differential coefficients of 
 
 207, Sen. — The successive differential coefficients of a function of the 
 form A+Bx+Cx--\-Dx^-\- etc., or .^«^-^a;«-^ +^a;"-2+ etc., are readily writ- 
 ten by inspection. Thus, CfxW x* —2x^+bx^ +x—\2,f{x). Let /'(.!•) mean the 
 first differential coefficient, f"{x) the second, f"'{x) the third, etc. We have 
 
 f{x) = x^ - 2^=' + 5.r^ + X - 13. 
 /'(.r) = 4c' - 6a;2 + lOi; + 1. 
 f"{x) = 12.C2 _ 12^ ^ 10. 
 /'"(.y) = 242; - 12. 
 /'^(i) = 24. 
 /v(.r) = 0. Here the processes terminate. 
 
 Each of the above is produced from the preceding by multiplying the 
 coefficient of x in each term by the exponent of x in that term and diminish- 
 ing the exponent by 1. 
 
 6. According to the method indicated in the last scholium, write 
 out the successive differential coefficients of the function 2x^-\-Sx* 
 -5a:' + 10. Also of 2^-32:'"' + ^''. Also of 3 + 2:c-4a;H3^. 
 
 SECT/ON VL 
 TAYLOR'S FORMULA. 
 
 208. Def. — Taylor^s Formula is a formula for developing 
 a function of the sum of two variables in terms of the ascending 
 powers of one of the variables, and finite coefficients which depend 
 upon the otlier variable, the form of the function, and its constants. 
 
 209, Def. — If n —fix + y), i- e., if u is a function of the sum 
 
 of the two variables x and y, and we differentiate as though one of 
 
 the variables, as x or y, was constant, the differential coefficients thus 
 
 formed are coW^di. Partial Differential Coefficients, The 
 
 partial differential coefficients of u, when x is considered variable 
 
 ^ ^ ^, du cl^u cfu (fn 
 
 and y constant, are represented thus: t-, ^, -r^, -7^4, etc. 
 
 When y is considered variable and x constant, we write the coeffi- 
 
 . , du d\i d^u dSi , 
 cients v", 3—2, -7-1, -J-., etc. 
 dy dy'' df dy* 
 
186 ADVANCED COUllSE IN ALGEBRA. 
 
 210. Lemma. — If u = f(x + y), the partial differential coeffi- 
 cients T- and T- are equal. 
 dx dy 
 
 Dem. — Having u =:J{x + y), if x take an increment, we have u + dxU* 
 =J[z + dx + y) = /[(.c + y) + dx] ; whence d^u = f [{x + y) + dx] - f{x + y), 
 eince a differential is the difference between two consecutive states of the func- 
 tion. Again, if y take an increment, we have u + dyU = J\x + y + dy) 
 =/[(« + y) + ^y] ; whence d„w =/[(-» + y) + dy] - f{x + y). Now the foi'm of 
 the values of drU and dyU, as regards the way in wliich x and y are involved, is 
 the same ; hence, if it were not for dx and dy, they would be absolutely equal. 
 Passing to the differential coefficients by dividing the first by dx and the second 
 
 by dy. we have "-^ =/[<^-*-y'+'^]-«^+y) . and 'Jl! =M^+y)+dl/]-Al±^) , 
 
 '' " dx dx dy dy 
 
 But, in differentiating, the differential of the variable enters into every term ; 
 hence /[(j; + y) + dx] —f[x + y), as it would appear in application, would have 
 & dxin each term which would be cancelled by the dx in the denominator in the 
 
 coeflBcient, and — would be independent of dx. In like manner — is independ- 
 ent of dy. Hence, finally, as these values of the partial differential coefficients 
 are simply functions of ( c + y), of the same form, and not involving dx or dy, 
 they are equal, q. e. d. 
 
 III. — To make this clear, let u = (x + yy. Then dxU = 3(.c + yfdx, or 
 
 — =S{x + y)*. Again, dyU —8(x -\- y)*dy, or ^ — 3(.r+ y)". Hence we see that 
 dx dy 
 
 _ ... , , . du 1 . du 1 - 
 
 So, agam, \t u = log (x + y), -r — . ^^^ t- = 5 hence 
 
 ' ^ ^^ '^^' dx x+y dy x + y' 
 
 du 
 dx'~ 
 
 du 
 ~dy- 
 
 du 
 dx 
 
 du 
 ~ dy' 
 
 211. I*rob. — To produce Taylor^ s Formida, 
 
 SOLIJTION. — Let u =f{x + p) be iHie function to be developed. It ie proposed 
 to discover the law of the development when the function can be developed in 
 the form 
 
 u =J{x + y) = A + By + Cy'' + Dy' + Ey* + etc., (1) 
 
 in which A,B, C, etc., are independent of y, and dependent on x, the form of the 
 function, and its constants. 
 
 Supposing X constant and differentiating with reference to y as variable, re- 
 membering that, as A, B, (7,-etc., are functions of x, and not of y, they will be 
 considered constant, we liave 
 
 ^ =B-^2Cy + SDy' + 4^» + etc. (2) 
 ay 
 
 * As we are to consider the effect produced npon u by an increment in x. and al?o by an in- 
 crement in y, we adopt a form of notation to distinguish between the increments of //. Thus 
 rfj-»/ means the increment which w takes in consequence of a; having talvcn the increment dx, 
 whMc y remained constant. So cfy?< represents the increment of m consequent upon the incre- 
 ment dy of y. 
 
Taylor's formula. 187 
 
 Again, differentiating (1) with respect to x, y being supposed constant, and re- 
 membering that A, B, C, etc., are functions of x, wc have 
 
 dx dx dx dx dx dx 
 
 Hence by {210) 
 B+2Cy + dDy' + ^W\ etc. = ^ + ^^ + ^2/' + ^2/^ + ~P'+ etc. (4) 
 
 Now, by the theory of indeterminate coefficients, the coefficients of tiie like 
 powers of y are equal, and we have 
 
 B = ^, 2(7=^, 3i)=^. 4:E=^, etc. 
 dx dx dx dx 
 
 But as (1) is true for all values of y, we may make y = 0; whence 
 A =f(x) = u' ; letting u' represent the value of the function u, when y = 0. 
 Now, as A is independent of y, it will have the same value for one value of y as 
 for another ; hence A =f{x) = u' is the general value of A. 
 
 dA du' 
 
 Aeain, B — — . But as J. = w', a function of x, dA — du' , and B = —-. 
 ^ dx dx 
 
 In like manner 2C = — . But as B = —, dB = dl-—) = --—, and 
 dx dx \dx/ dx 
 
 So, also,as3D= -. and^(7 = ^^(-,) = -^^,,i) = j^^. 
 
 1 d^u' 
 Similarly we find E = — ^ > and the law of the series is apparent. 
 
 Finally, substituting the values of A, B, C, etc., in (1), we obtain 
 
 , du y d^u'y"^ d'^u' y'^ d*u' y* ^ ,^, 
 „=/(. + y)=«+^f+^| + ^|+^^+etc., (5) 
 
 which is Taylor's Formula. 
 
 212, SCH. — Taylor's Formula develops u — f (x + y) into a series in 
 which the jirst term is the value of the function when y = ; the second 
 term is the first differential coefficient of the function when ^ = 0, into y ; 
 the third term is the second differential coefficient of the function when 
 
 y = 0, into ^ ; etc., etc. 
 
 If. 
 
 As u' is f(x+y) when y = 0, we may write /(a;) for u', and for — , f'{x) ; for 
 ^^» f"{x) ; for ^» /'"(«); etc., as before explained. The formula then be- 
 
 ClX (IX 
 
 * Tne>«e forms arc indicated operations. Thus, as ^ is a function of x. when we difTerentiatc 
 With re?pcct to x wc write dA, and to pass to the differential coefficient have to divide by dx. 
 
188 ADVANCED COURSE IN ALGEBRA. 
 
 u=f{x.^y) =/(.r)+/'(.r) | + f\x) '^ ^f"\x) ^' +/- {x) 'j^' + etc. (6) 
 
 This is a very important method of writing Taylor'a Formula, and should be 
 clearly understood, and firmly fixed in memory. 
 
 Examples. 
 
 1. Develop (x-\-yY by Taylor's Formula. 
 
 Solution. — Putting u = {x+yf, we have u' = .t', —t- = 5x*, -^ = 20a;', 
 
 — ^ = 60^ — — = 120a;, and — - = 120. Here the coefficients terminate, as 
 dx> ' dx* dj* 
 
 the differential of a constant is 0. 
 
 Substituting these values in (5) {211), or {%).{212), we have 
 
 u = (a;+y)»= a;»+ 5x*y + 10a;='y«+ 10a;*y'+ 5ay*-}-y«. 
 
 The same as by the Binomial Formula. 
 
 2. Develop (x—yY by Taylor's Formula, and compare the result 
 with that obtained by means of the Binomial Formula. Also {x + y) ^. 
 Also {x—y)~\ Also (x-^y) '. 
 
 3. Show that 
 
 n = log(. + y)=loga:+|-£ + g-£ + etc. 
 
 4. Develop (x+y)" by Taylor's Formula, thus deducing the Bi- 
 nomial Formula. 
 
 213» Taylor's Formula is much used for developing a function 
 of a single variable after the variable has taken an increment. When 
 so used the increment may be conceived as finite or infinitesimal, 
 only so that it be regarded as a variable. 
 
 Ex. 1. Given y = 2x^ — x* -\- 6x — 11, to find y', which represents 
 the value of the function after x has taken the increment Ji. 
 
 Solution. — In the function as given, we have p = fix), and are to develop 
 ^'=zf{x + //). By Taylor's Formula we have 
 
 dv d^v 
 
 From y = 2x^—X' + 5a; — 11, we have -f- — (ix- — 2x + 5, ~ = 12a; — 2, 
 
 dx ux 
 
INDETERMINATE EQUATIONS. 189 
 
 — ^ = 12, aud subsequent differential coefficients 0. Substituting these valuei 
 in the formula, we obtain 
 
 2 2 3 
 
 = 2^''-.i'2+5«-ll +(6x2-2^+5)7i+(6a; -l)h^+2h\ 
 
 This result is easily verified by substituting x+h for x in the value of y, as 
 given in the example. Thus, 
 
 y'=%i:+hf-{x+hf-^b{x+h)-n; 
 a result which will reduce to the same form as the other. 
 
 2. Given y=^x^—2x^, to develop y\ the value of y when x takes 
 the increment h. 
 
 SECTION VII, 
 INDETERMINATE EQUATIONS. 
 
 214. An Indeterminate Equation between two quan- 
 tities, as X and y, is an equation which expresses the only relation 
 which is required to exist between the two quun titles. 
 
 III. — Suppose we have 2a;+8y=7, and that this is the only relation which is 
 7-equired to exist between x and y. Then is %x-\-Zy=l an indeterminate cqua- 
 
 X 
 
 Cion. So also, if 6=:cy is the only relation required to exist between x and y, 
 
 0/ 
 
 this is an indeterminate equation. In like manner y- = 2x-^ — 'dx is an in- 
 determinate equation if it expresses the only relation which is required to exist 
 between x and y. 
 
 The propriety of the term indeterminate is seen if we observe that such an 
 equation does not fix the values of x and y, but only their relation. Thus, in the 
 equation 2.7; + % = 7, ar may be 2, and y 1, and the equation be satisfied. So x 
 may be 3, and y ^, and the equation be satisfied. In fact, any value may be 
 assigned to one of the quantities and a corresponding value found for the other. 
 Hence the equation does not determine the values of the quantities. 
 
 2 15. An equation between three quantities is indeterminate if it 
 expresses the only required relation between the quantities, or if 
 there is but one otlier relation required to exist. 
 
 III. — Thus, if 2x + Sy—5z—10 is the only relation which is required to exist 
 between x, y, and z, it is evident that the equation does not determine particular, 
 definite values for x, y, and z. So also if, in addition to the relation expressed 
 by this equation, it is required that 2x shall equal fSy, or 2a;— 6y, these two 
 
190 ADVANCED COURSE IN ALGEBRA. 
 
 equations will not fix tlie values of .i; y, and z. For if 2.r=6y, the former 
 equation becomes 9^—52=10, which may be satisfied for any value of z, and 
 a corresponding value of y, as shown above. 
 
 216* In general, if there are n quantities involved in any 
 number of equations less than n, and these are the only relations 
 required to exist between the n quantities, the equations are in- 
 determinate. 
 
 217> In indeterminate equations the quantities between which 
 the relation or relations are expressed are properly variables, i. e., 
 they arc capable of having any and all values.* 
 
 III. — Thus in the indeterminate equation 5y — Sx= 12, any value may be 
 assigned to x, and a corresponding value found for y; or any value may be 
 assigned to y, and a corresponding value found for x. 
 
 218, There are, however, many classes of problems which give 
 rise to equations which are called indeterminate, although they are 
 not absolutely so: in such problems there is some other condition 
 imposed than the one expressed by the equation, but which con- 
 dition is not of such a character as to give rise to an independent, 
 simultaneous equation. Such an equation may have a number of 
 values for the variables, or unknown quantities, involved, but not an 
 unlimited number. 
 
 III. — Let it be required to find the podtite, integral values of x and y which 
 will satisfy the equation 2x +3y = 35. Now, if 2a; + 3y = 35 were the only rela- 
 tion required to exist between x and y, there would be an infinite number of 
 values of each which would satisfy the equation, as shown above. But there is 
 the added condition that x and y shall be positive integers. Tliis greatly re- 
 stricts the number of values, but does not furnish another equation between x 
 and y. We may usually solve such a problem by simple inspection. Thus, in 
 
 this case, we have y = — ~ . Now, trying the integral values of x till 2x be- 
 3 
 
 comes greater than 35, t. e. till x = 18, we can determine what integral values 
 
 of X give positive integral values for y. For x=l, y = ll. For x = 2, 
 
 y = lOi ; hence a; = 2 is to be rejected. For a? = 3, y = OJ, and a; = 3 is to be 
 
 rejected. For a; = 4, y = 9 ; hence aj = 4 and y = 9 are admissible; etc, 
 
 [Note. — This subject is not of sufficient importance to justify our going into 
 a general discussion of it. We sliall content ourselves with a few practical 
 examples concerning simple indeterminate equations between two or three 
 quantities, and these restricted to positive integral solutions. The chief thing of 
 importance is t?uit tlie student comprehend the nature of an indeterminate equa- 
 tion.l 
 
 * This statement requires us to iuclnde imaginary values. 
 
INDETERMINATE EQUATIONS. 191 
 
 Examples. 
 
 1. What positive, integral values of x and // will satisfy the equa- 
 tion 5:i;+ 7//= 29 ? 
 
 Solution.— We inuy write x = — -— ^ = 5—^4- —~ -— 5— y+ -^-- . Now 
 
 O i) O 
 
 to make a; positive we must have 7y<29; and as y is to be an integer it can 
 
 2 .. 
 
 only have values less than 5. Again, to render x integral — — ^ must be integral, 
 
 
 
 2—v 
 or 0. Finally, as no value for y less than 5 will render — — - integral or 0, ex- 
 
 o 
 
 cept y=^2, this is the only value of y wiiich fulfills the conditions. Hence the 
 
 answer is y=2, x='d. 
 
 2. What positive, integral values of x and y will satisfy the equa- 
 tion ll:?:-17y=5? 
 
 Solution. — We have x = — j^ — =y-r -^^ — . From this we see that any pos- 
 itive value of y which will render -^ — integral, will meet the conditions. Put 
 
 6y+5 / . X ^ 1 llw— 5 ^m—1 _, i ^i . , 
 
 -^ — =7U (an mteger) ; whence y = — - — =m-f 5 — - — . To make this value 
 11 Go 
 
 of y integral — — must be integral. Put — ^p- = s (an integer) ; whence m 
 
 =Qs+l. Now any positive integral value for s will fulfill the conditions. Thus, 
 put 8 = 0;* whence m=l, y — \, and x = 2. Again, put 5 = 1; whence m = 7, 
 y=12, and a;=19. For « = 2, m:=13, y— 23, and j^^: 36, etc. Hence there is an 
 infinite number of positive, integral values of x and y which satisfy the equation. 
 
 3. What positive, integral values of x and y will satisfy the equa- 
 tion 21a: +17^^=2000? 
 
 ^ , 2000-17y . ,,^ , . 2000-17iy ^^ 5-17y 
 SuG's. x = — ^ . .'. y is < 118. Again x = ^ — - =95h g— , 
 
 J 5— 17y . ^. , 5— 4m ,,„ 5— Am 
 
 and — — — = m. .'. m is negative, and y= — m H p=— . Whence — ^n — ~ **' 
 
 and m = — z — = 1 — 4« H — 7- . .*. « is +, and any value of » which renders 
 4 4 
 
 - — , or integral, and gives y < 118, will meet the conditions. 
 
 8= 1, gives w=— 3, y= 4 and a; = 92. 
 
 8=5, " m= —20, y= 25 and a; = 75. 
 
 * = 9, " m= —S7, y= 46 and x = 58. 
 
 » = 13, " m= —54, y= 67 and x — 41. 
 
 » = 17, " m = -71, y= 88 and x = 24. 
 
 4 = 21, " m=-88, y=109and.r= 7. 
 
 * is considered an iulcger. 
 
192 ADVANCED COURSE IN ALGEBRA. 
 
 Since any greater value of s makes y > 118, these are all the values of x and y 
 which fulfill the conditions. 
 
 4. Find the positive, integral values of x and y which satisfy the 
 following : 
 
 (a) 5.r + Uy = 254; (b) 7a: + 13?/ =71; 
 
 (c) dx + 13y = 2000; {d) ITx = 542 - 11^; 
 
 (e) Ux + S5y = 500; (/) 19:c - 117.V = 11; 
 
 (g) 117.T - 128^ = 95; (//) 39.r + 29?/'= G50; 
 
 (i) 5x ± 9y = 40 ; (k) 6x ± Oy = 37. 
 
 Applications. 
 
 1. In how many ways can I pay a debt of $2 with 3-cent and 
 5-cent pieces ? 
 
 Sug's, — Let X = the number of 3-cent pieces and y = the number of 5-cent 
 pieces required. Then we are to determine in how many ways the equation 
 3iF-|-5y=200 can be satisfied for positive, integral values of x and y. 
 
 We find it to be in 13 ways, as follows : 
 
 y = 11 4 1 7 I 10 I 13 I 16 I 19 I 22 I 25 I 28 I 31 I 34 I 87 
 2; = 65 I 60 I 55 I 50 I 45 I 40 I 35 I 30 I 25 I 20 I 15 I 10 I 5 
 
 This means that 1 5-cent piece and 65 3-cent pieces will pay the debt, or 4 
 5-cent and 60 3-cent, or 7 5-cent and 55 3-cent, etc. 
 
 2. A man hands his grocer 15 and tells him to put up the worth 
 of it in 11-cent and 3-cent sugars. Can the grocer do it in even 
 pounds? If so, in how many ways? What is the greatest number 
 of pounds of the poorer sugar that he can use ? AVhat the least ? 
 
 3. In how many ways can a debt of £50 be discharged with guineas 
 and 3-shilling pieces? ^1 ??.<?., Not at all. 
 
 4. If my creditor has only 3-6hilling pieces and I only guineas, can 
 he so make change with me that I can pay liim £50 ? Can I pay him 
 £201 ? In how many different ways ? What is the least number of 
 guineas and 3-shilling pieces? How is it if I have crowns instead of 
 guineas? How if I have guineas and my creditor crowns ? How if 
 I have crowns and my creditor pounds ? 
 
 5. In how many ways can a debt of £1000 be paid in crowns and 
 guineas ? 
 
 SuQ. — Having obtained a few of the possible values of x and y, the law will 
 become evident. 
 
indeterminate equations. l93 
 
 219. Indeterminate Equations between Three Quantities. 
 
 1. What are the positive integral values of Xy y, and z which 
 Siitisfy '6x+by -h7z= 100 ? 
 
 Solution. — We have x = ~ — ; whence as 1 is the least value that y 
 
 or z can have, x cannot be greater than 29. Also y = • '^- ; whence y 
 
 cannot be greater than 18. Also z= — - ; whence z cannot bo greater 
 
 than 14* 
 
 __, .^ 100-5.y-72 ^.^ ^ l-2p-z „ l-2v-z 
 
 Write X = ^ =33_y_23H ^ — . Hence ,j must be an 
 
 integer. Put ^ — - =m ; whence y=z —m -\ . From this we see 
 
 that m is negative. 
 
 Let us now proceed to examine in succession for 2=1, 2=2, 2=3, etc. 
 
 For 2=1.— For this value of z, a;=31— y— ~ , and y= — m — — . From the 
 
 o -* 
 
 latter we see that m must be an even negative number ; and from the former, 
 
 that y must be a multiple of 3. Hence the following computation : 
 
 For m= 0, 1/ = 0, which is inadmissible. 
 
 For w = - 2, y = 3, and x - 26. 
 
 For m— — 4, y — C, and a; = 21. 
 
 For m = — 6, y= 9, and a; = 16. 
 
 For m = — 8, y = 12, and x = 11. 
 
 For wi = — 10, y — 15, and x — 6. 
 
 For ?» = — 12, y = 18, and a; = 1. 
 
 Since the values of x decrease as m increases numerically, and 1 is the least 
 admissible value of x, we have all the values of y and x which correspond to 
 « = 1. 
 
 For 2 = 2.— For this value of z, a; = 28 — y 4- -^^— » *^<1 y——m 
 
 — . From the latter we see that m must be a negative oM number ; 
 
 i 
 and from the former, that y must be 1, or a unit more than a multiple of 3. 
 
 Hence the following computation : 
 
 For ?7i = — 1 , y — 1 , and a; = 27. 
 
 For m = — 3, y — 4, and x — 22. 
 
 For m— — 5, y = 7, and x — 17, 
 
 For w = — 7, y = 10, and x = 12. 
 
 For m = — 9, y = 13, and a; = 7. 
 
 For m = — 11, y = 16, and x = 2. 
 Henco, these are all the values of y and x wliich correspond tq 2 = 2, 
 
 Of coarse the quantities uecd not come up to ^ese limits. 
 
194 ADVANCED COURSE IN ALGEBRA. 
 The other values are as follows : 
 
 '-** U=23 I 18 I la I 81 8 I • "'* U=19 I 14 I 9 I 4 
 
 . (y= 1 I 41 7 1 101 „ iy= 2 1 5 I 8 I 11 
 
 '='u=2oll5llol sl' ^=«U=16lll 6 1 
 
 
 :5 
 
 2. What positive, integral values of x, y, and z satisfy 17a; + lOjf 
 + 21;? = 400? 
 
 SuG. — There are 10 sets of values. 
 
 3. AVhat positive, integral values of x, y, and z satisfy 5x + "71/ 
 + 11;? = 224? 
 
 4. What positive, integral values of a;, y, and z satisfy Gx + Sy 
 + 6z = 12 ? Also 2a; + 3// + 52 = 41 ? 
 
 220, If tlic conditions of a problem furnish less equations than 
 unknown quantities, the problem is mdetermmate, and in general 
 can have an infinite number of solutions. But if the solution be 
 limited to positive, integral values, it can be effected as above. Thus, 
 if there are two equations and three unknown quantities, one of 
 the unknown quantities can be eliminated and the resulting equation 
 solved as heretofore. In like manner if there are three equations 
 and four unknown quantities, a single equation between two may be 
 found and solved ; or if four unknown quantities and but two equa- 
 tions, a single equation between three unknown quantities may be 
 found and solved. 
 
 Examples. 
 
 1. Given 2a; + 5y + 3^ = 51, and 10a; + 3^ + 2^ = 120, to find all 
 the positive, integral values of a;, y, and z. 
 
 2. Given 3.?; + 5?/ + 7z = 5G0, and \)x + 25?/ + 495; = 2920, to find 
 all the positive, integral values of x, y, and z. 
 
 3. Given 2a; 4- H// -Zz= 10, and 3a; - 2y + 3z = 30, to find all 
 the i)ositive, integral values of a*, y, and z. 
 
INDETERMINATE EQUATIONS. 19^ 
 
 Application's. 
 
 1. I wish to expend $100 in the purchase of three grades of sheep, 
 worth respectively $3, 17, and $17 per head. How many of each kind 
 can I buy ? In how many different ways can I make the purchase ? 
 How many of the first two kinds must I take in order to get the least 
 possible number of the third kind ? 
 
 2. A merchant has three kinds of goods. The value of 20 yards 
 of the first, less the value of 21 yards of the second, is $38 ; while 
 the value of 3 yards of the second and 4 yards of the third is $34. 
 What is the price per yard of each kind, the question being restricted 
 to even dollars ? AVhat if the latter restriction be removed ? 
 
 3. In how many ways can I pay a debt of $171 with $20, $15, and 
 $6 notes? What is the least number of $20 notes that I can use? 
 Of $15 notes ? What the greatest number of $6 notes ? 
 
 4. A farmer has calves worth $10, $11, and $13 per head. What 
 relative number of each must he take and sell them at the uniform 
 rate of $12, without gain or loss ? If he is to sell only 15 animals, 
 how must ]ie select them ? 
 
 5. A man bouglit 124 head of cattle, viz., pigs, goats, and sheep, 
 for $400. Each pig cost $4^, each goat $3|, and each sheep $1^. 
 How many were there of each kind ? 
 
 G. A grocer has an order for 150 pounds of tea at 90 cents per pound, 
 but having none at that price, he would mix some at 75 cents, some 
 at 87^^ cents, and some at $1.00 per pound. How much of each sort 
 must he take ? 
 
 Sug's.— The nature of the 4th and 5tli problems restricts their solutions to 
 positive integers. The 6th is, however, only restricted by its nature to positive 
 numbers ; they may be fractional as well as integral. 
 
 [See Complete School Algebra, subject Alligation.] 
 
 7. What quantity of raisins, at 10 cents, 18 cents, and 20 cents per 
 pound, must be mixed together to fill a cask containing 150 pounds, 
 and to be worth .19 cents a pound ? 
 
 8. A wheel in 36 revolutions passes over 29 yards ; and in x of 
 these revolutions it describes z yds., y ft, and 5 in. What are the 
 values of x, ?/, and z ? 
 
196 
 
 ADVANCED COURSE IN ALGEBRA. 
 
 CHAPTER IL 
 
 LOCI OF EQUATIONS. 
 
 [Note. — This subject, though properly geometrical, is introduced here for the 
 purpose of the elegant and clear illustrations which it affords of the abstract 
 principles of the subject of Higher Equations. It is thought that the aid which 
 it will afford the pupil in comprehending the principles of the succeeding chapter 
 will more than compensate for the time required to master this. Moreover, the 
 subject of Loci of Elquations is of prime importance in a mathematical course, 
 and is always pursued with pleasure by the pupil. No geometrical knowledge 
 is required in reading this chapter, farther than the ability to draw and measure 
 straight lines.] 
 
 22 !• JPvop» — Every equation between two variables,'^ having 
 real root8^\ may be interpreted as representing some line either 
 straight or curved. 
 
 This proposition will be made sufficiently evident for our present purpose, if 
 we show how such equations can be made to represent lines. This we shall do 
 by means of particular examples. 
 
 Examples. 
 
 1. Draw the line represented by the equa- 
 tion 1/ = 2x ■{■ 6. 
 
 Solution. — First, in all cases, draw two straight 
 lines, as X'X and YY' ,at right angles to each other, 
 as in the figure. Then, in the equation y = 2x + Q, 
 assign values (arbitrarily) to x, and find the corre- 
 sponding values of j/. Thus, 
 
 Also, if x=—l, 
 
 " x=-2, 
 " x=-Z, 
 
 -5-4 h 
 
 ■»-t-iAI 
 
 If a;=0, 
 
 y= 6. 
 
 " x=l. 
 
 P= 8, 
 
 " x=2, 
 
 y=io. 
 
 " ar=3. 
 
 y=i2, 
 
 " x=4, 
 
 y=i4. 
 
 etc.. 
 
 etc. 
 
 x=-4, 
 x=—5, 
 etc.. 
 
 y= 4, 
 y= 2, 
 y= 0. 
 
 y=-4, 
 etc. 
 
 Y 
 
 Fio. 1. 
 
 Having computed a few corresponding values 
 of X and y in this way, we proceed with the figure, 
 as follows : Measure off a distance Al to the light 
 
 * This means eimply, " having two variables, and only two, in it." 
 
 t The geometrical interpretation of imaginary loci does not come within our present 
 purpose. 
 
LOCI OF EQUATIONS. 
 
 197 
 
 of A, of some convenient length, and call it the unit of distances. Draw 6 1, 
 
 at 1, perpendicular to AX, and make it 8 units long (t. e., 8 times as long as Al). 
 
 Now, 6 is at a distance 1 to the right of the line YY'. and 8 above the line X'X, 
 
 and is hence a point in the line which our equation represents. In like manner, 
 
 find the point c, 2 to the right of YY', and 10 above 
 
 X'X ; and c is another point in the line represented 
 
 by our equation. Again, when a; = 3, y = 13. 
 
 Hence, lay off three units to the right, as to 3 in the 
 
 figure, and draw d 3 perpendicular to X'X and 12 in 
 
 length. Then is d another point in the line we 
 
 seek. When a; = 4, y = 14. Hence e is a point in 
 
 the line ; since it is 4 from YY', and 14 from X'X 
 
 When a; = 0, y = 6 ; whence « is a point in the 
 
 line, as it is distance from YY', and 6 from X'X. 
 
 For negative values of a;, we have, when x =z — 1, 
 y = 4. Now, laying off negative values of x to the 
 left from A, since we laid off positive values to the 
 right, we measure from A to —1, the unit's distance, 
 take /I equal to 4 units, and thus find the point /. 
 When X = — 2, y = 2, and g is the corresponding 
 point. When a; = — 3, y = ; whence h is a point 
 in the line, as it is 3 to the left of YY' and above 
 X'X. When a; = — 4, y = — 2. As this value of y 
 is negative, we lay it off below X'X. Thus, taking 
 from A to —4, a distance of 4 units, and from — 4 to e, a distance of 2 units, * is 
 a point in the line. Thus also A: is a point in the line, since when a; = — 5, 
 y = — 4, and k is taken 5 to the left of YY' and 4 
 below X'X. 
 
 This process might be continued indefinitely, 
 both for y)ositive and negative values of x. We 
 might also use fractional values of a*, as a; = \, 
 x = ^,x — 2^, etc., and, finding the corresponding 
 values of y, locate points between those found by 
 taking integral values. 
 
 Finally, joining the points e, d, c, b, a,f, g, h, i, 
 k, we have the line MN, which is represented by 
 the equation y=:2x-{-Q. This line does not stop 
 at M and N, of course, since we might produce 
 it indefinitely either way, by continuing to take 
 larger and larger values of x (numerically). In 
 this case it is easy to see that the line is an indefi- 
 nite straight line. 
 
 2. What line is represented by the eqiia- f^°- s. 
 
 tiony=Sx-Q? (^ee Fir/.d.) 
 
 SuG's.— First compute a table of corresponding values of x and y, as in the 
 preceding example ; and then locate the points thus designated. 
 
108 
 
 ADVANCED COURSE IN ALGEBRA. 
 
 3. What line is represented by the equation y = —2a; + 4? (See 
 Fig. 4.) 
 
 222. Definitions.— The assumed fixed 
 lines X'X and YY' are called the Axes of 
 Meferencey or simply the Axes. A is 
 called the Origin. X'X is the Axis of Ab- 
 scissas, and YY' the Ax'is of Ordinates. 
 The distance of a point from the axis of ab- 
 scissas is called the Ordinate of the point ; 
 and the distance of a point from the axis of 
 ordinates is called its Abscissa. The ordi- 
 nate and abscissa of a p)int taken together 
 are called its Co-ordinates. 
 
 Abscissas measured to the right from the axis of ordinates are +, 
 and those to the left — . Ordinates measured above the axis of 
 abscissas are +, and those below — . 
 
 4 to 13. Draw as above the lines represented by the following equa- 
 tions: i/=x-\-5; y=x—5; y= — .t + 5; y= —x—5; t/=ix-{-(j; 
 
 y=4a;-6; y= -4x + 6; y= -ix-G; 2a;-3?/=-5;^i^ =2. 
 
 Sua. — Put such equations as the last two into the same form as the others 
 before proceeding with the solution as above. 
 
 OiOf 
 
 V, Def. — The line which is represented by an equation is 
 called the Locus of the equation; and drawing the line in the 
 manner indicated, is called Constrtictbiy ilva Locus of the equation. 
 
 X 
 
 14. Construct the locus of the equation y = 
 
 l+a'^' 
 
 Solution.— For x = 0, y = 0, 
 
 " x = i, 
 
 y = -1*7, 
 
 " x=A, 
 
 y = h 
 
 *' x = l. 
 
 y = h 
 
 " X=:2, 
 
 y = h 
 
 " x = d, 
 
 y=h. 
 
 " x = A, 
 
 y = A. 
 
 etc. 
 
 etc. 
 
 For a; = - i. y = - %, 
 
 " x= -1, y= -h 
 
 " x= -2, y= -h 
 
 " .-c - - 3, y= — -,%, 
 
 " x= -A, y= — -iV» 
 etc. etc. 
 
LOCI OF EQUATIONS. 
 
 199 
 
 Now, laying off on the axis of abscissas to the right distances equal to \, J, 
 1, 2, 3, and 4, on some convenient scale, and at these points erecting ordinates 
 equal respectively to -,\, |, i, f, ,%, and A of the same scale, we find the points a, 
 b, c, d, e, and / of the locus. We also see that if we continued to give x greater 
 and greater values, y would continually grow less, but would only become when 
 
 « = 00, for then we should have y — = — —-— — = 0. 
 
 In like manner laying off the negative values of a;, and the corresponding 
 values of y, we find the points a\ b' , c' , d', e', and/', and also find that y dimin- 
 ishes numerically as x increases numerically, and that for x negative y is 
 always negativ^e, and only becomes when x= — oo. Hence, the curve ap- 
 proaches the axis of abscissas to the left from below, as it does to the right 
 from above, reaching it in either direction only at an infinite distance from the 
 origin. 
 
 A line sketched through the points found represents the locus sought. 
 
 15 to 18. Construct the loci of the following equations: i/=x^ 
 + a:--6t (see Fig. 6) ; y=3-{-x—ix^ (see Fir/. 7); i/=x'—4:X + 4: 
 (see Fig, 8) ; g=x^—3x-{-5 (see Fig. 9). 
 
 Y 
 N| ,M 
 
 ii\ 
 
 
 nr, 
 
 X' -2-W 12346 X 
 
 Y 
 
 Fig. 9. 
 
 • Dropping the finite quantity 1, as prodiicinj;: no effect when added to the infinite ar*. 
 
 t In dctorininin-j points in tlio Iociib. it is often neceeeary to attribute fractional valiics to x. 
 Thue, in tiiis cnso. to sketch the curve from a to c\ we need an intermediate point. If (here is 
 any doubt about the character of the curve between two points, resolve the doubt in thi:* way. 
 
200 
 
 ADVANCED COURSE IN ALGEBRA. 
 
 19 to 23. Consfcrnct the loci of the following equations: %j=y* 
 -Ja;' + 2aj-f 2 (see Fig. 10) ; y-^-(^j? 
 + 13ar-10 (see Fig. 11) ; y=x'-2x-5 (see 
 Fig. 12); y=a^(o-x) (see Fig. 13); and 
 y=o^-Q:xf-^llx-6 (see F'ig. 14). 
 
 Fie. 16. 
 
loci OF EQUATIONS. 
 
 201 
 
 24 to 28. Construct the loci of the following equations: y^a"^ 
 ~5ic2+4 (see Fig. 15); yz=x'-\-'23?-'6x'-ix + 4. (see i^7^.'l6); 
 y=zx'-^x' + ^x-^n (see Fig. 17); y=i}^-2a^-7x'-Sxi-l() (see 
 Fig. 18) ; and g=a^+a^+x'-{-x-\-l (see Fig. 19). 
 
 X' A 
 
 X' » 
 
 ' X 
 
 Y 
 
 Fig. 18. 
 
 29. Construct the locus of the equation i/=x^ + 4:X*—14:X^—17x—6. 
 
 Sug's. — In attempting to construct this 
 locus, it is necessary to give x values from 
 —3 to +2, including these values, and also 
 to observe the character of the locus beyond 
 these limits. But it will be found that for 
 some values of x between these limits, y is in- 
 conveniently large. In sketching the figure, 
 we may use one scale for laying off values of 
 X, and another for laying off values of y. 
 Thus in the figure given, the unit used for x 
 is 6 times as great as that used for y. This 
 is equivalent to constructing the locus Gy=x'^ 
 + 4x* - Ux^- llx - 6, or y = ^x' + W-W 
 — Va; — 1. This locus has all the peculiar- 
 ities of the one required (that is, all the turns, 
 flexures, or bends), but is not of the same pro- 
 portions. The portion represented is G times 
 as wide in relation to its length as the re- 
 quired locus would have been. 
 
 30 to 41. Construct the loci of the 'y. 
 following equations, using a smaller 
 scale for /y than for x, as explained in 
 
ADVANCED COURSE IN ALGEBllA. 
 
 the suggestions above, when more convenient : y = x^— 2x — 15 ; 
 y = x' + 2x + 2; y = x''-10x+26'y y = x' - 3x' - Hx- 10; 
 y = 23?- 12a:» + 13a; - 15 ; y = x" - x"" - '^x ■{■ 12 ; y = o? - 2x^ 
 - 25a: + 50; y= ir*— 2r'+ 8a; - 16 ; y= a;'+ 2%''— Zx"- 4a; + 4; 
 y = a;*- 6a;^ + 5a;' + 2a; - 10 ; y = ar^ + 5a;^ + a;^- 16a;''- 20a; - 16 ; 
 and y = 5a:'-4a;^ + 3a;'-3a;'+4a;-5. 
 
 224, J^vob, — To construct the real roots of an equation contain- 
 ing only one unknoio}i quantity. 
 
 Solution. — Put the equation in the form /(a) = 0, tlien write y = f{x). 
 Construct this equation, and the abscissas of the points where the locus cuts the 
 axis of abscissas are the nwts of the equation /(.r) = 0. This is evident, since 
 for these points, and for these only, y = 0, and we have f{x) = 0. 
 
 Examples. 
 1. Construct the real roots of the equation 
 
 3a; -2 = 0. 
 
 Solution.— We will first write y z=zx*—^ — 2 
 
 for x=\, y - 
 
 X' 
 
 Fio. 21. 
 
 Now, for a; = 0, y — — 2\ 
 — 4 ; for x — 2, y = — 4 ; 
 for a; = 3, y — — ^\ and for a; = 4, y=2. 
 Hence we see that the locus of the equa- 
 tion y = ic* — 3x — 2, cuts the axis of ab- 
 scissas between a* = 3, and x = 4, since it 
 passes from below the axis of abscissas 
 (where y is — ) to above this axis (where 
 y is -h). There is therefore a root of a;* 
 — 3a: — 2 = between 3 and 4. To con- 
 struct this root, we sketch the curve be- 
 tween a* — 3 and x = 4, by finding the 
 values of y for a few intermediate values 
 of .T, and then sketching the curve. Thus 
 for a; = 3^, y = — J4 ; for a* = 35, y = \h Sketching the curve mil through 
 these points, we find by measurement Art = 3.50, as an api)roximate value of x 
 in the equation a;*— 3.r — 2 — 0. (Verify by solving the equation.) To construct 
 the other root, we notice that* for x = 0, y = — 2, and the curve cuts the axis of 
 abscissas again at the left of the origin (probably, as it certainly does not cut it 
 again at the right). Now, for x = — 1, y =2 ; whence we see that the locus 
 cuts the axis between a; = 0, and x = — 1. For x——i,y=—\; and for 
 x= — }, y = If . Sketching the curve through these points, we have m'n' ; 
 and measuring A a', we find the other value of x to be —.56. 
 
 SuG, — For constructing the approximate roots in this manner, as we only 
 need to sketch a small portion of the locus, in the vicinity of its intersection with 
 the axis, we can use a much larger scale than would otherwise be practicable, 
 and thus obtain a nearer approximation. With good instruments and some care, 
 
NUMERICAL HIGHEH EQUATIO:s'S. 203 
 
 we can usually construct the root with tolerable accuracy to hundredths. When 
 the locus cuts the axis quite obliquely, the approximation cannot be made as 
 accurate. 
 
 2 to 7. As above, construct the real roots of the following: 
 x^-8x=U', a^- 12a:'-' + 36:c - 7 = ; a:^ - x' - lOx + 6 = ; 
 a^ - 7a: + 7 = ; x* - 12a:3 + 60x' - 84a: + 49 = 0; and 
 
 2x' - 7x' + 10a: = 9. 
 
 225, Sen. — This method of approximating the roots of equations geo- 
 metrically is not given as a good practical method ; but simply to assist tha 
 learner in comprehending some subsequent processes, and for its geometrical 
 importance. 
 
 CHAPTER III. 
 
 HIGHER EQUATIONS. 
 
 SECTION I. 
 
 SOLUTION OF NUMERICAL HIGHER EQUATIONS HAVING COMMEN. 
 SURABLE (OR RATIONAL) ROOTS.* 
 
 226 » Equations of higher degrees than the second are called 
 Higher Equations {6-10 , or same in Complete School Algebra). 
 Ko general, practicable method of resolving such equations is known. 
 Theoretical solutions of equations of the third and fourth degrees 
 (cubics and biquadratics) are known ; but these solutions are 
 attended with practical difficulties in many cases, which render 
 them nearly or quite useless. We are, however, able to obtain the 
 real roots of Niimerical Higher Equations, in all cases, either exactly, 
 or to any required degree of approximate accuracy. 
 
 227, The Real, Covimensnrable Roots of numerical equations are 
 usually found with little difficulty by the methods given in this 
 section. 
 
 * A commensurable root (or number) is one which can be exactly expressed in the decimal 
 notation, either in an integral, fractional, or mixed form. Thus, 4, ^l, r25, etc., are com- 
 mcnvurable. But V2\, VXH, etc., arc incomirenpurablc. 
 
204 ADVANCED COURSE IN ALGEBRA. 
 
 228, I^VOJ), — fJcery equation with oite nnknoicn quantity^ 
 having real and rational coefficients^ cafi be transfortned into an 
 equatio)i of the form 
 
 x"+ Ax"-'-f Bx"-'+ Cx"-^' - - - - L = 0, 
 
 in which the exponents are all positive inteyers^ the coefficient of x" is 
 1, and the coefficients of the other terms^ A, B, C, etc., and also the 
 absolute term L, are integers. 
 
 Dem. — 1st, If the unknown quantity occurs in any of the denominators in the 
 given equation, remove it to the numerator by clearing of fractious. If there 
 are tlien any negative exponents, multiply each term by the unknown quantity 
 with a i>ositive exponent equal to the numerically largest negative exponent. 
 Then unite the terms with reference to the unknown quantity, and write them 
 in order with the term containing the highest exponent, at the left, and so that 
 the exponents shall diminish toward the right, transposing all the terms to the 
 first member. The most complicated form which can then occur is 
 
 m r 
 
 ^y" +^y' +yy' - - • • i = o, (i) 
 
 Tfi r 
 in which any or all of the exponents may be fractions ; and —>-><, etc. 
 
 n 8 
 is supposed. 
 
 2d. To free the equation of fractional exponents, substitute for the unknown 
 quantity a new unknown quantity with an exponent equal to the least common 
 multiple of the denominators of the exponents in the equation. Thus, in (1) 
 
 m r 
 
 put y — z"' , whence y " = af", y ' = z"', and y' = 2"*'. These values substituted 
 in the equation, will evidently give an equation of the form 
 
 !^r' + -«"->+ 42*-''' - - - - ^ = 0, (2) 
 
 0(1 J 
 
 in which all that is essential concerning the exponents is that they should be 
 
 all positive integers, decreasing in value from left to right, since iji (1) 
 
 m r 
 
 — > - > ^ etc. 
 
 n H 
 
 3d, Now divide by the coefficient of z*, and let the resulting equation be 
 represented by 
 
 2"+ ^,2-' +-6--^ . . . . ;'=0. (3) 
 
 d f 
 
 Finally, put z = — , and substitute in (3), thus obtaining 
 
 r-^^nF---'^fi^^ ■ ■ ■ ■ ' = »• (*) 
 
 Multiplying (4) by k", and representing the absolute term by L, we hare 
 
NUMERICAL HIGHER EQUATIONS. 205 
 
 If now k be so taken that these numerators will be divisible by the denomina- 
 tors, and the quotients represented b^ A,B, C, etc., we have 
 
 a;»4-^a;~-i-|-i?a;'»-2-f Caj^-a .... Z=0, 
 the form required. 
 
 Examples. 
 
 1 1 2 K Q 
 
 1. Transform - + ^x-' + ^x^ = 2x^ + iz~^ i-~-2, into a form 
 
 having positive integral exponents and coefficients, and having the 
 coefficient of the highest power 1. 
 
 Solution. — Multiplying by x^, we have 
 
 x+%v-^-\-^x^=2J-hhic~^+d-2xK (1) 
 
 Multiplying (1) by x"^, we have 
 
 x* + i+^^x^^z=2x^+{x^+3x'-2j*. (2) 
 
 Putting x=y^ , there results 
 
 y''+h+^f'=^'*+\y''^^t'-2f\ 
 
 Arranging with reference to the highest power of y, 
 
 23/"-^y"-2/"-/*+3y"'+iy"-J=0, or 
 
 z 
 Finally, put y = -r, whence 
 
 Now, if A; be made 12, this equation will be of the required form * 
 
 Z 2* 
 
 Notice that as ic = y', and y — :r^,x = -r^ ; so that, if the value of z could 
 be found, the value of x would be known by implication. 
 
 2. Show as above how to transform the following: 
 
 {a) Sy-"^ + iy-i + ^ - if^ = ^ + if- ^Vi 
 (b) --Zx + ix^-l = l; 
 
 X 
 
 (e) Vl-ar'=l~3a:i; (/) ^2^ 3? -- a: = a/T^. 
 
 ♦ This substltntion would be tedious, and as it is our present purpose simply to show the 
 poefibUify of the traneformation, and the method of making it, the enbetitutlon 1» iinnfce*fary. 
 
206 ADVANCED COURSE IX ALGEBRA. 
 
 ^*?.9. — Since every equjiiion with one im known quantity, and real 
 and rational coefficients, can be transibrmed into one of tlie Ibrm 
 
 ar-^Ax^-' + Bx^-'-j-Cx"^-' L=0, (I) 
 
 this will be taken as the typical numerical equation whose solution 
 we sliall seek in this and the succeeding sections; and we shall 
 frequently represent it by/(.r) = 0, read *' function x equals 0." The 
 notation/(a:) signifies in general, as has been before explained, simply 
 any expression involving x. Here we use it for this particular form 
 of expression. We shall also use f'{x) as the symbol for the first 
 differential coefficient of this function. 
 
 230, J*V02>» — W/t€7i an efjuafion is reduced to the form x" 
 
 4- Ax"-' -\- Bx-' + Cx°-' L = 0, t/ie roots, with their slgiis 
 
 changed, are factors of the absolute {knotcn) term, L. 
 
 Dkm. — 1st. The equation Ix-inrr in this form, it a is a root, the function is 
 divisible by ar—a. For, sui)Jkks«.' ui>on trial x—n goes into the polynomial xn 
 + J.r"-' + , etc., Q times \cith a remainder 11. {Q represents any series of terms 
 wliicli may arise from such a division, and JR any remainder.) Now, since the 
 quotient multiplied by the divisor + the remainder, equals the dividend, we 
 
 liave (t— a) Q + R=je* + .4.r"-> + 2?.i'"-'^ + Cx'-^ L. But this polynomial = 0. 
 
 Hence {x—a) Q + R—O. Now, by hyjwthesis a is a root, and consequently a;— a 
 =0. Whence B=0, or there is no remainder. 
 
 2d. If now z—a exactly divides x" + A.r"-'^ + Bx'*-^ + Cz"-^ L, a must 
 
 exactly divide Z, as readily appears from considering- the process of division. 
 Hence —a is a factor of L, a being a root of the equation, q. e. d. 
 
 23 !• Cor. 1.— T/'a is a root o/f(x) = 0, f(x) is divisible by x— a/ 
 and, converselt/j if f(x) is divisible by x — a, a is a root of f(x)=0. 
 
 Dem. — The first statement is demonstrated in the proposition, and the second 
 is evident, since as^a*) is divisible by;r— r/, let the quotient be q>{x)\ whence 
 (r— «) ^.r)=0. Now x—a will satisfy this equation, since it renders a*— a=0, 
 and does not render <p{x) infinity, since by hyi)othesis x docs not occur in the 
 denominator.* 
 
 232 • JProj). — If the coefficients and absolute term in x"" + Ax""^ 
 -f Bx"-'-f Cx""' - - - - L=0, are all inteycrs, the equation can leave 
 it o fractional root. 
 
 * Could thert' be n term of the form in <?> (r), x=a would render it oo, and (x-ay^P (a-) would 
 
 bv X ae. which is iiuleterminatc. since x«=Ox ^^J}. 
 
NUMERICAL HIGHER EQUATIONS. • 207 
 
 Dem.— Suppose in tliis equation x = p ^ being a simple fraction in its lowest 
 terms. Substituting this value of (v, we have 
 
 gn ,^.n-l *"-2 on- 3 
 
 Multiplying by <"-' we obtain 
 
 Now, by hypothesis, all the terms except the first are integral, and the fiist is a 
 simple fraction in its lowest terms, as by hypothesis s and t are prime to eacli 
 other. But the sum of a simple fraction in its lowest terms and a series of in- 
 tegers cannot be 0. Therefore x cannot equal -, a fraction. 
 
 t 
 
 233, Scu. — Tliis proposition does not preclude the possibility of surd 
 roots in this form of equation. These are possible. 
 
 234. ProiK—An equation f (x) = (229) of the nth degree, 
 has n roots {if it has any *), and no more. 
 
 Dem. — Let rt be a root of f{x) = 0, which is of the 7ith degree. Dividing 
 f(x) hy X — a {231), we have <p(j) = 0, an equation of the (/i— l)th degree. 
 
 Let 5 be a root of <^.r) = 0,and divide <p(.r) by x—h {231). Call the quotient 
 <p' {x), whence cp' {x) = 0, an equation of the («— 2)th degree. In this way the 
 degree of the equation can be diminished by division until, after n — 1 divisions, 
 there results (p" {x) f of the first degree, and the equation is x— l—O. Therefore,. 
 
 f{x) = {x - a) (p(,v) - {x-a) {x - b) cp' {.i) = {.v - a) {x - h) {x - c) q>" {x) 
 
 = {x- a) {x-h){x - c) (^ - ^) - ; 
 
 %. e., f{x) is resolvable into n factors, of the form x — m. 
 
 * Wc shall assume that every equation has a root real or imasjinary ; i. e., that there is some 
 form of exprcpgion which substituted for the unknown quantity will satisfy the equation. It 
 is shown in works treating more largely upon the theory of equations, that the general form of 
 a root is a + /? |/ — 1. When /^ = 0, the root is real. The general demonstration of this propo- 
 sition is loo abstnisc for an elementary treatise. That every equation of the form a-"+ Ax^'-^ 
 + Bx"^~'i+ Cz"-3 - - - - Z=0 {229) has a real root when n is an odd number, and also 
 when n is an even number if L be negative, is very simidc. Thus, if n is odd, and L +, when x 
 is made - oo the value of the first memb'T is - ; and when x is 0. tlie value is -t . Hence while 
 X passes from - oo to 0, the function changes sign, and hence must pass through : i. e., for 
 Bome value of x between - oo and 0, the equation is satisfied. In like manner, if L is -, when 
 sc - 0, the function is -, and when ar = ^ oo the function is + . Hence some value of x between 
 -0 and + (X, satisfies the equation. It follows from this that in an equation of an odd degree, 
 If the absolute term is +, there is at least one real, negative root ; and if the absolute term is -, 
 there is at least one real, jwsidve root. 
 
 If n is even and Z, -, cc = makes the function - , and x= ±ra makes it +. Hence while x 
 passes from - oo toO, the function changes sign from 4- to -, and there is at least one real, 
 negative root; also, while x passes from to + 00, the function changes sign from - to + , and 
 there is at least one real, positive root. Therefore every equation of an even degree in which 
 the absolute term is -, has at least two real roots, one negative, and one positive. 
 
 The difficulty occurs in proving that an equation of an even degree has a root when Z, is + . 
 The roots of such an equation may be all imaginary. 
 
 t This i? read " the nth cp function of x." 
 
208 ADVANCED COURSE IN ALGEBRA. 
 
 Now, as x= a, or x = b, or .r — any one of tlu' qmintitu*s a,b, e - - - - I, 
 will render /(j) equal to 0, each one of the.se will ^utit^fy the equation /(.r)=0. 
 Therefore this equation has n roots. 
 
 Again, since it is evident that we have resolved /(.r) into its j^rimc factors 
 with respect to x, there can be no other factor of the form x — m in J\.v), hence 
 no other root of /(t)=0, and this whether m is equal to one or more of the roots 
 a,b,c - ■ - - n, or not. Therefore /(.r) = has only n roots. 
 
 233. Cor. 1. — T/ie j^oli/nomial x°+ Ax"-'+ 13x"-'+ Cx""' 
 
 L, or f (x), = (x — a) (x — b) (x - c) - - - - (x — 1), in which 
 a, b, c - - - - 1 (fre the I'oofs of f (x) = 0. 
 
 23H, Cor. :2. — The equation f (x) = may have 2, 3, or even ii 
 equal roots, as there is no inconsistence/ in supposing a = b, iv = b 
 = c, or a = b = c = - - - - \^in the above demonstration. 
 
 237* Cor. 3. — Imaginary roots enter into equations having only 
 real coefficients, in conjugate pairs (22a ^ Part I.)y that is, (/'f(x)=0 
 has only real coefficients, if it has one root of the form a + /JV— 1, 
 it has another of the form a — ^V — 1 ; or, if it has one of the form 
 /^V— 1, it has another of the form — /SV — 1- 
 
 Tliia ia evident, since only thus can /(ar)=(ar— a) (a:— &) (.T—c) - - - - {x—n); 
 that is, if one root, a for example, is a—/iV—\, there must be another of the 
 
 form a+fi V—\, in order that the product of these two factors shall not involve 
 an imaginary. Thus, [T-{ix+ft V^)] x [x-{a-/S \^-l)]=x*-2ax+{a^+ ft'), 
 a real quantity. So also (x — ft V'— 1) {x +ft V—l) = x*+ ft*, a real quantity. 
 But if the assumed imaginary roots be not in conjugate pairs, the product of the 
 factors (x — a) (.c —&) (i^ — c) • - - - {x — /) will involve imaginaries. 
 
 238, Cor. 4. — Hence an equation of an odd degree must have at 
 least one real root ^ but an equation of an even degree does not 9ieces- 
 sarily have any real root. 
 
 230* Cor. 5. — If an equation has a pair of imaginary roots, the 
 known quantities entering into the equation mag be so varied that the 
 two imaginary roots shall first give place to two equal roots, and then 
 these to two real and unequal roots 
 
 As shown above, imaginary roots arise from real quadratic factors in /(.p). 
 l^t x^—'Zax + & be such a quadratic factor, whence .t*— 2ax + 6 =: satisfies 
 fix) = 0, and a ± Va* — b are the corresponding roots of f{x) = 0. Now, if 
 h > n-, these roots are imaginary. If, however, b diminishes or a increases (or 
 lH)th change thus together), when h = a^ the two imaginary roots disa])pear and 
 we have in their place two real roots, each a. If the same change in a and b 
 
NUMERICAL HIGHEU EQUATIONS. 203 
 
 continues, so that a^ becomes greater than h, the two real, equal roots in turn 
 give place to two real, unequal roots. Now as a and b are functions of the known 
 quantities of the equation f{.i) = 0, such changes are evidently possible. 
 
 240, Sen. 1. — That an equation has a number of roots equal to its 
 degree, is illustrated geometrically by the fact, that, if we write y =f{x) and 
 construct the locus, we shall always find that a straight line can be drawn 
 so as to cut the locus in 1 point and only 1, if /(a.) is of the 1st degree 
 (Ex's. 1-13, CiiAr. II.); in 2 and only 2 points, \i f{x) is of the 2d degree 
 (Ex's. 15-18, Chap. II.); in 3 and only o points, if f{x) is of the 3d degree 
 (Ex's. 19-23, Chap. II.); in 4 and only 4 points, if /(.<) is of the 4th degree 
 (Ex's. 24-28, CuAP. II.), and specially illustrated by the line X/ X„ {Fig. 20), 
 etc. 
 
 241, Scir. 2. — The fact that imaginary roots enter real equations in 
 pairs is also beautifully illustrated by the loci of equations. Thus the equa- 
 tion ^^— 3a; + 5=0 has two imaginary roots, and no real roots. Now, by ref- 
 erence to Fig. 9 of the preceding chapter, we see that the locus of y='x'^—'dx 
 + 5 does not cut the axis of abscissas at all ; i. «., that no real value of x will 
 giveyi;a-)=0. But, if the equation were so modified as to make each ordinate 
 only "V" less than it now is, /. ^., if y=x'^—Zx+"i^ we should 
 have the same locus, but changed in position so as just to 
 touch the axis of a*, as in c, thus giving f{x)—0 two real and 
 equal roots. If, again, we wrote y—x:^—^x—"d, we should have 
 the locus referred to the axis A"X", and /(ic) = would have 
 two real and unequal roots. Thus we see, conversely, how 
 two real, unequal roots can pass into two real and equal roots 
 by a proper change in the equation, and how by a further 
 change ttco equal real roots dlHajypear at a time, passing into two 
 imaginary roots as the equation changes form. All that is 
 necessary in this change in the form of the equation is a pro- 
 jier change in the absolute term. 
 
 Fig 32 
 Again, consider Fig. 14, and the corresponding equation 
 
 y=a;''— 6.r*+ll.T— G. First we observe that as this locus cuts the axis of x 
 
 three times, there are three real roots. Now cliange the absolute term —6 
 
 by allowing it to increase gradually, becoming —51, — 5^, —5, etc. We shall 
 
 find that the axis of x moves down, and the two roots A d and A/ approach 
 
 equality, first becoming equal Avhen the axis just touches the lowest point e 
 
 of the curve, and tlien hoth hecoming imaginary together. 
 
 Or, in conclusion, this matter is illustrated by the fact that whatever the 
 degree of the equation /(.r)=0, if we construct the locus of y=f{x), we shall 
 find that we can draw a straight line which will cut the curve in a number 
 of points equal to the degree of the equation, and that if tlie line gradually 
 moves from this positi(m so as to cut the curve in any less number of points, 
 it will always be found first to run two intersections together, corresponding 
 
 
 Y 
 
 •\ 
 
 J 
 
 A 
 
 X 
 
 A 
 
 X 
 
 Y' 
 
210 ADVANCED COURSE IN ALGEBRA. 
 
 to a change of two unequal roots into two equal roots, and then drop out 
 hoth these intersections, corresponding to the introduction of two imaginary- 
 roots at a time. 
 
 24:2, I^rop. — If the equation f(x)=:0 has equal roots, the highest 
 common divisor of f(x) and its differential coefficient* f (x), being 
 2nit equal to 0, co?istitutes an equation which has for its roots these 
 equal roots, and ?io othe^ roots.\ 
 
 Dem. — Let a be one of the m equal roots of f{x)=0, and let the other roots be 
 
 b,c I; then/(a-)=(a'-rt)'" ix-b){x^c) . • . . {x- l){235). Differentiating 
 
 (152) and dividing by dx, we have 
 
 f\x)=m{x-aY'-^{x-b){x-c) - - - - {x-l) + {x-n)"' (x-c) (x- I) + - • 
 
 . . . - + (x—a)" (x—b) (x—c) - - . . + etc. 
 Now (a?— a)*-' is evidently the highest common divisor of f{x) and f\x), and 
 {x—a)r~^ =0 is an equation having a for its root, and having no other. 
 
 In a similar manner, if f{x)=0 has two sets of equal roots, so that 
 
 f{x)=(r-a)'"{x-hyix~c){x-d) (x-l), 
 
 diflFerentiating and dividing by dx, we have 
 
 f'{x)=m{T-a)'--^x-by{x~c){x-d) {x-l)+{x-a)'^r{x-^by-\x-c) (x-d) 
 
 (x-l) 
 
 +(x-a)'^ (x--by (x-d) - ' ' - («— n)+(a;-rt)*(ar-6)''(«-c) - - - - {x-l)+-- 
 
 • ^{x—ay"{x—by{x—c){x—d)- - - - 4- etc. 
 
 Now the highest common divisor of /(x) and /'(a?) is evidently (x—a)"'-^{x—by-^. 
 Putting this equal to 0, we have {x—a)''-^{x—by-^=0, an equation which is sat- 
 isfied by x=a and x=b, and by no other values. 
 
 Thus we may proceed in the case of any number of sets of equal roots. 
 
 243, ScH. — In searching for the equal roots of equations of high degree, 
 it may be convenient to apply the process of the proposition several times. 
 Thus, suppose that /(.r)=0 has m roots each equal to a, and r roots each 
 equal to b. Then the highest common divisor of /(r) and f\x) is of the form 
 {x—a)'*~^{;t^by~^\ whence (jr_a)'"~'(j?— &)'-'=0 is an equation having the 
 equal roots sought. Therefore we can find the highest common divisor of 
 {x—aT'^^x—bY'^ and its differential coefficient which will be of the form 
 {x—a)''~^{x—by~^i and write {x—nY'~^(x—by-^z=0, as an equation containing 
 the roots sought. This process continued will cause one of the factors (x—a) 
 or (x—b) to disappear and leave ix—a)'*~'=Q, when m>r; {x—by~"'—0, 
 when r> m ; or {x—a)(x—b)=0, when m—r. From any one of these forms 
 we can readily determine a root. 
 
 * The differential coefficient of a fnnrtion is sometimes called its first derived polynomial. 
 
 t The student most not suppose that the roots o{J\x)=0, and its first differential coefficient 
 /'(j')=0, are necessarily alike. f'(x)=& series of temu some of which may be + and some - , and 
 which may destroy each other, so as to render/'(T)=0, Ibr other values of x than such as render 
 /{z)=0, and not necessarily for any which do render /(aj)=0, except the equal roots of the latter. 
 
NUMERICAL HIGHER EQUATION?. 211 
 
 , 244, JProp. — In an equation f(x)=:0, f(x) icill change sigyi when 
 X passes through any real root, if there is but one such root, or if there 
 is an odd number of such roots ; but if there is an even number of 
 such roots, f(x) will not change sign. 
 
 Let a, b, r . • . . e hv the roots of f{T)=0, so that f{.i)={.v—n) (x—b) (x—c) 
 
 (x—e)=0 {23t>). Conceive .i- to start with some value less than the least 
 
 root, and continuously increase till it becomes greater than the greatest root. 
 As long as .r is less than the least root, all the factors .v—a, x—b, etc., are nega- 
 tive ; but w^hen <ic passes the value of the least root, the sign of the factor con- 
 taining that root will become +,* and if there is no other equal root, this factor 
 will be the only one which will change sign. Hence the product of the factors 
 will change sign. But if there is an even number of roots, each equal to this, 
 an even number of factors will change sign ; whence there will be no change in 
 the sigu af the function. If, however, there is an odd number of equal roots, 
 tUe passage of x through the value of this root will cause a change of sign in an 
 odd number of factors, and hence will change the sign of the function. 
 
 Finally, as it ia evident that the signs of the factors, and hence of the function, 
 will remain the same while x passes from one root to another, and in all cases 
 changes or does not change as above when x passes through a root, tlie j^roposi- 
 tion ia established. 
 
 245, ScH. — This pro])osition is illustrated by putting y—f{:i) and con- 
 structing the locus, as in the preceding chapter. Thus, Ex. 15, Fig. 6. In 
 this case y=f{x)=x'-^rX—iS. The least root is —3. When x is less than —3, 
 as —4, or — 3.^ {anything less than —3), y, or/(.r), is +. When x is —3, ?/, 
 or/(.r)=0, and the equation /(.r)=0 is satisfied, and —3 is a root of the equa- 
 tion. When X becomes greater than —3, as —2, y, or /(^), becomes nega- 
 tive, changing sign when x passes through the value of tlie root —3. As x 
 increases, y, or /(.r), remains — , till x reaches -f 2, at which value of .r, 
 y,-/(2-)=r:0, and the equation /(.r)=0 is satisfied. When x passes this value, 
 becoming anytliing greater than 2, y, or /(.r), becomes +, i. c, changes sign 
 as x passes through the root 2. The same thing is illustrated by the loci 
 in Figs. 7, 11, 13, 14, 15, and 18, with their corresponding equations. 
 
 That/(.r) does not change sign upon .t's passing through the value of one 
 of two equal roots of /(.r)=0, is illustrated in Fig. 8 and its corresponding 
 equation, Ex. 17. In this case yz^f\x)-x^ -L%+^, and the equation 
 a;2_4^+4=0 has two roots each equal to 2. Now when x is anything less 
 than 2, y, i. e. f{x\ is + ; when x=2, y, or f{x\ is 0, and the equation /(.r)=rO 
 is satisfied. But when x passes the value 2, /(.t) does not change sign ; it 
 remains +. The same truth is illustrated by the loci in Figs. 10, 13, 16, 
 and 17, and their corresponding equations. Fig. 16 illustrates the case in 
 which there arc two pairs of equal roots. 
 
 ♦ SnppoBC c be the Kast root, and that c' is the next state of x greater than c; then c'- 
 
212 ADVANCED COUKSE IN ALGEBRA. 
 
 Ex. 29 will be found very instructive. The locus in Fig. 20 illustrates the 
 case of 3 equal roots. Here y =/(.r) = r'*-f- 4a!*— 14a;-— ITa; — 6. The least 
 root is — 3. When a; < — 3, f{x) is — ; when x— — 3,/(.r) = ; when x passes 
 — 3, increasing, /(.r) changes from — to +, and remains + till x = — 1, when it 
 becomes 0, and changes sign as x passes— 1, noUcithstanding there are equal 
 roots. But there is an odd number of such roots, viz., three. 
 
 Thus, if X,'X, were to revolve about ^;' until it took the position X'X, the 
 intersections b' and d' would run into c, the three intersections becoming one. 
 
 246. Prop. — Changing the signs of the terms of an equation 
 containing the odd powers of the xinhnown quantity changes the signs 
 of the roots, 
 
 De.m. — If x = a satisfies the equation x^— Ax^-\- Bx^— Ca; + i> = 0, we have 
 a^—Aa*-\- Ba^— Cu-\- 1) = 0. Now changing the signs of the terms containing 
 the odd powers of x, we have x^— Ax*— Bx^-\- Cx -\- D — 0. This is satisfied 
 by a; = — a, if the former is by x = a. For, substituting — a for x, we have 
 a^—Aa*+Ba^— Ca + D = 0, the same as in the first instance. 
 
 247* Cor. — Changing the signs of the terms containing the eve?i 
 powers will answer equally well, since it amounts to the same thing; 
 and if tee are careful to put the equation in the complete fortn, 
 changing the signs of the alternate terms xciU accomplish the purpose, 
 
 III. — The negative roots of a;^ — 7a; 4- 6 = 0, are the positive roots of — a;' 
 4- 7a; + 6 = 0, or of a;'— 7a; — 6 = (0 being considered an even exponent) ; or, 
 writing the equation a;' ± Ox*— 7a; + 6 =0, changing the signs of alternate 
 terms, and then dropping the term with its coeflBcient 0, we obtain the same 
 result. 
 
 Again, the negative roots of x*"— Ix^—rtx^-h 8a;'— 132a;* + 508a; — 240 = 0, 
 are the positive roots of x'^+ Ix^—^x*— Sx^— 132a;*— 508a; — 240 = 0, or of 
 - x^- 7x' + 5a;*+ 8a;'+ 132a;*+ 508a; + 240 = 0. 
 
 248. Prob. — To evaluate * f (x)/or any particular value of x, 
 cw X = a, more expeditiously than by direct substitution. 
 
 Solution.— As f{x) is of the form a?» + ^a?»-' + 5a;"-' + Ca;«-« - - - - L, 
 let it l>e required to evaluate x* + Ax^-if Bx^+ Ox 4- D for x = a. Write the 
 detached coefficients as below, with a at the right in the form of a divisor : thus 
 
 1 +^ +B +C +D \a_ 
 
 a a*+Aa a^+Aa^+Ba a*+Aa^+Ba'^ + Ca 
 
 a+A a* + Aa+B a*-\-Aa* -{-Ba+C a*+Aa^-]-Ba^ + Ca+l) 
 
 * This means to find the value of. Thus, snppose we want to find the value of a;6-53?'' 
 + 8iC*-8a?»+ 6a;«-x- 18, fora; = .5. We might pubptitnte 5 forr, of conrpc, and accomplish the 
 end. Bui there is a more expeditious way, as the solution of ihis probletn ^ill show. 
 
NUMERICAL HIGHER EQUATIONS. 213 
 
 Having written the detaclied coefficients, and the quantity a for which f{x) is 
 to be eviiluated as directed, multiply the first coefficient 1 by a, write the 
 result under the second, and add, giving a + A. Multiply this sum by a, write 
 the product under the third coefficient B, and add, giving a^ + Aa + B. In like 
 manner continue till all the coefficients (including the absolute term, which is 
 the coefficient of x^) have been used, and we obtain a^+ Aa-^+ Ba^+ Ca + D, 
 which is the value of f{x) for x = a. 
 
 Illustration.— To evaluate x^—5x'^+ 2x^— 3x^+ Qx^ — x — 12, for x = 5t 
 
 -5 +2 
 
 -3 
 
 +6 -1 
 
 -12 1 5 
 
 5 
 
 10 
 
 35 205 
 
 1020 
 
 2 
 
 7 
 
 41 204 
 
 1008 
 
 he value of x^ — 
 
 ■5x'+2x*- 
 
 .'dx'+Qx^-x- 
 
 12, for a; =.5; and it 
 
 is? easy to see that much labor is saved by this process. 
 
 "We arc now prepared for the solution of the following important 
 practical problem : 
 
 240, I^vob, — To find the commeiimirahle roots of numerical 
 higher equations. 
 
 The solution of this problem we will illustrate by practical examples. 
 
 Examples. 
 1. Find the commensurable roots of re*— 2a;''— 15.r^+ 8a:'+ 68a; 
 + 48 = 0, if it has any. 
 
 Solution. — By (232), if this equation has any commensurable roots they 
 are integral : — it can have no fractional roots. 
 
 Again, by (230), the roots of this equation with their signs changed are fac- 
 tors of 48. Now, the integral factors of 48 are 1, 2, 8, 4, C, 8, 12, 16, 24, 48. 
 Hence, if the equation has commensurable roots, they are some of these num- 
 bers, with either the -f or — sign. We will, therefore, proceed to evaluate 
 fix) (t. e., in this case x'^-2x*- 15.r '-l- Sx^+ (iSx + 48), ior x= + l,x= — 1, 
 x= -\-2,x= —2, etc., by {248), as follows : 
 
 1 -2 -15 + 8 +68 +48 [ +1 
 
 1 _ 1 _16 - 8 60 
 
 -1 -16 - 8 60 108 
 
 Hence we see that for x= + 1, f{x) = 108, and +1 is not a root of f{x) = 0. 
 Trying x = — 1, we have 
 
 1 -2 -15 + 8 +68 +48 | -1_ 
 
 -1 3 12 -20 -48 
 
 -3 -12 20 48 
 
 Thus we see that for a; = — l,f{x) — 0, and hence that — 1 is a root of our equ*. 
 tion. 
 
214 ADVANCED COURSE IN ALGEBRA. 
 
 We might now divide /(a*) by .r-t-1 {2:il) and reduce the degree of the equa- 
 tion by unity. But it will be more expeditious to proceed with our trial. Let 
 us therefore evaluate/(.r) for .v= +2. Thus : 
 
 1 _2 -15 +8 +C8 +48 I +2 
 
 2 -30 -44 +48 
 
 -15 -22 24 9G 
 
 Hence for x= +2, f{x)=dQ, and +2 is not a root. Trying x=—2, we have 
 
 1 -2 -15 + 8 +08 +48 I -2 
 
 -2 8 14 -44 -48 
 
 -4-7 22 24 
 
 Hence for x- —2,f{x)=0, and —2 is a root. Trying x=+S, we have 
 
 1 _2 -15 + 8 
 
 +08 
 
 +48 1 +3 
 
 8 3-30 
 
 -84 
 
 -48 
 
 1 -12 -28 
 
 -10 
 
 
 
 Hence for x = +o, f{x)=0, and +3 is a root. 
 
 Trying x= 
 
 —3, we have 
 
 1 -2 -15 + 8 
 
 +08 
 
 + 48 1 -3 
 
 -3 15 
 
 -24 
 
 -132 
 
 -5 8 
 
 44 
 
 - 84 
 
 Hence for x= -S,f{x)zz-S4, and -3 is not i 
 
 !i root. Tryi 
 
 ing ar=4, we 1 
 
 1 _2 -15 + 8 
 
 +08 
 
 + 48 1 4» 
 
 4 8 -28 
 
 -80 
 
 -48 
 
 2 - 7 -20 -12 
 
 Hence for x=4,f{x)=0, and 4 is a root. 
 
 We have now found four of the roots, viz., —1, —2, 3, and 4. Their product 
 with their signs changed is 24. Hence, by {2:i0) 48-s-24=2 is the otlier root 
 with its sign changed, i. e. there are two roots —2. 
 
 That our equation had equal roots could have been ascertained by the princi- 
 ple in {242) ; but as the process of finding the H. C. D. is tedious, it is generally 
 best to avoid it in practice. 
 
 2 to 12. Find the roots of tlie following : 
 
 (2.) 0^ - rc» - 39.7-' + Ux + 180 = ; 
 
 (3.) x" + ox' - 9x- - 45 = ; 
 
 (4.) a:* + 2a:* - 23.r - 00 = ; 
 
 (5.) a^ - 3.4-^ - Ux' 4- 48.T - 32 = ; 
 
 (6.) x' -Sx' + 13.7:- G = 0; 
 
 * Of course it it* not iieccHsary to relaui the -f- sign, as we have done in the preceding opera- 
 tions: it has been done simply for emphasis. 
 
NUMERICAL HIGHEIi EQUATIONS. 215 
 
 (7.) a;^ - Ux' + 18a; - 8 = ; * 
 
 (8.) x^ - Sx' + 6.r^ -3x^ — Sx -\- 2 = 0; 
 
 (9.) x' — I'dx* + Glx"" - 171a;' + 21Ga; — 108 = ; 
 
 (10.) x' - 45ar^ - 40a; + 84 = ; 
 
 (11.) x^ — 3:c* - 9a;^ + 'ilx' - 10a; + 24 = ; 
 
 (12.) x'-W + 11a;* - Tx' + 14a;^ - 28a; + 40 = 0. 
 
 13 to 20. Apply the process for finding equal roots (242 ^ 243) to 
 the following : 
 
 (13.) x"" + S.c' + 20a; + 16 = 0; 
 
 (14.) x" - a;'- 8a; + 12 = ; 
 
 (15.) x' - ox' - 8a; + 48 = ; 
 
 (16.) u^ - lla;^ + 18a; - 8 = •, 
 
 (17.) X* + 13x^ + 3dx' + 31a; + 10 = ; 
 
 (18.) x' - 13x* + 67a.'=* - 171a;^ + 216a; - 108 = 0; 
 
 (19.) x' + 3ar^ - 6x* - Gx^ + 9a;^ + 3a; - 4 = ; 
 
 (20.) a;'+ 5a;''4- 6ar^- Qx'- lox'- 3x'+ Sx + 4 = 0. (See 243.) 
 
 21 to 27. Having found all but two of the roots of each of the fol- 
 lowing by (248), reduce the equation to a quadratic by (231), and 
 from this quadratic find the remaining roots : 
 
 (21.) u^ - Qx' + lOa; - 8 = ; 
 
 (22.) a.-* - 4a;^ - 8a; + 32 = ; 
 
 (23.) ar' - 3.a;2 + a; + 2 = ; 
 
 (24.) x' - (jx^ + 24a; -16 = 0; 
 
 (25.) x' - 12ar' + 50:c^ - 84a; + 49 = ; f 
 
 (26.) a-* - 9ar» + 17a;^ + 27a; - 60 = ; 
 
 (27.) x' - 4a;* - 16:6-^ + 112ar' - 208a; + 128 = 0. 
 
 28 to 34. Apply the processes of (228) to reduce the following io 
 
 the form x* + Jia;""* + Bx""^ + (7a;"~^ L = 0, before searchmg 
 
 for roots : 
 
 {^^^j 2ar^ - 3a;' + 2a; - 3 = ; t 
 
 * In order to apply the process of evaluation, the coofflcieuts of the missing powers must be 
 Bupplicd. Thus we have 1 +0 - 11 + 18 -S. 
 
 t Apply the method for fliiding equal roots. The method of trial based upon {230) as applied 
 by {248) is likely to lead to much unnecessary work when there are several equal roots, and all 
 the others incommensurable. 
 
 $ Wehavea:»-|ar2+ar-^=0. Putar=|, whence l^-gj^y* + ^y-g=0,orya_ — y2 +^2y 
 
 - — = 0. If now i=2, we have y3 -3y«-f 4y-12=0, which can be solved as before, for one value, 
 of y.and the equation then reduced to a quadratic and solved for the other values. Finally, 
 remembering that a:= , J y, we have the values of a; required. 
 
216 
 
 ADVANCED COUKSE IN ALGEBUA. 
 
 (29.) 3a:* — 2ar' _ 6a; + 4 = ; 
 
 (30.) 8^ - 2Gx' + 11a; + 10 = 0; 
 
 (31.) .r* — ia; + 3^ = ; (Look out for equal roots.) 
 
 (32.) x'-63^+ OJa;'' - 3a: + 4i = ; 
 
 (33.) X 
 
 19a; 
 
 (34.) 
 
 / 
 
 ■y-?- 
 
 403a;- 
 
 a;' - 3a; + 22 
 
 ~i = n- + ^) 
 
 250. By means of the ])roperty exhibited in (235) produce the 
 equations whose roots are given in the following ex^jimples : 
 
 1. Roots 1, -3, 4. 
 
 2. Roots ^2, —a/2, —1, 3. 
 
 3. Roots 1, 2, 2, -3, 4. 
 
 4. Roots -3, 2 + \/^, 2-a/^1. 
 
 5. Roots 3, —2, —2, —2, 1. 
 
 6. Roots |, J, -|. 
 
 7. Roots l=b\/i:2, 2=fcA/^. 
 
 8. Roots li, 2, ^3, -V3. 
 
 9. Roots a/— ^, — a/^, a/5, 
 
 10. Roots 10, -13, i, 1. 
 
 11. Roots 3-2a/3, 3 + 2a/3", 
 
 2-3a/^, 2 + 3a/^, 1, 
 -1. 
 
 SECTION IL 
 
 SOLUTION OF NUMERICAL HIGHER EQUATIONS HAVING REAL, 
 INCOMMENSURABLE, OR IRRATIONAL ROOTS. 
 
 251. As all equations having real roots have real coefficients* 
 (237), and as all such can be reduced to the form a;" + ^af"* 
 + i5af "' + Ca;""' - - - - 2/ = 0, which we represent by f(x) = 
 (229), we shall consider this as the typical form. Moreover, since, 
 if an equation of this character has equal roots, they can be deter- 
 mined by (242, 243), and the degree of the equation depressed 
 by (231), we need only to consider the case in which /(a*) = has 
 no equal roots. 
 
 * This Is evident from the fact that f(x)={x-a) (x-b) (x—c) - - - - (a*-n)=0. In which if 
 a, 6, c, - • - - n are real, no ima]^nary quantity will be fonnd in the product of the binomials. 
 
217 
 
 2S2, The best general method of approximating the real, incom- 
 mensurable roots of such equations, is : 
 
 1st. To lind the number and situation of such roots by Sturm's 
 Theorem and the method based on it. 
 
 2d. Having found the first figure or figures of such a root by 
 Sturm's method, to carry forward the approximation to any re- 
 quired degree of accuracy by Horner's method of approxima- 
 tion. 
 
 These methods we will now proceed to develop. 
 
 Sturm's Theorem and Method. 
 
 253, Sturm's Theorem is a theorem by means of which we 
 are enabled to find the 7iumher and sitiiatwn of the real roots of any 
 numerical equation with a single unknown quantity, real and 
 rational coefficients, and without equal roots.* 
 
 III. — Thus, if we have the equation a;^— 7a; + 7 = 0, Sturm's Theorem 
 enables us to determine that it has three real roots, i. e., that all its roots are 
 real. It also enables us to ascertain that one root lies between 1.3 and 1.4, 
 anotlier between 1.6 and 1.7, and the third between —3 and —4. Hence it shows 
 us that the roots are 1.3+ , 1.G+, and —3. with a decimal fraction. 
 
 254, ScH. — Of course it follows from the above that if the equation has 
 commensurable (227) roots, Sturm's Theorem will enable us to find them, 
 or even when the roots are not commensurable, it will enable us to find any 
 number of initial figures. Thus in the equation x^ — Ix + 7 = 0, we might 
 by Sturm's Theorem find that the first root is 1.35689+ ; but it would be 
 too tedious an operation to be of any practical utility, as will appear hereaf- 
 ter. Wc use this theorem only to find one or two of the initial figures, or, 
 enough of the figures to enable us to distinguish between (separate) the roots. 
 Thus, if wc had an equation /(a) = 0, of which two roots were 2.356873+ and 
 2.3569564, we might use Sturm's Theorem to find the first five figures of each 
 root, i. €., to distinguish between (separate) the roots; but this is not the 
 l>e8t practical method, as will appear hereafter, 
 
 255, The Sturmian Functions of /(.r) = (v/hich has 
 no equal roots) are functions obtained by treating f{x) and its first 
 differential coefficient f'{x), as in the process of finding their H. C. 
 D., except that -in the process we must not multiply or divide by a 
 negative quantity, and the signs of the several remainders must be 
 
 * If the equation which we wish to solve has equal roots, Ihoy can be discpverpfl by 
 {242, 34itj, and the degree of the cqiiatiou reduced by (liyi.>^ipji. 
 
HM advanced coukse in algebra. 
 
 changed before they are used as divisors. These remainders with 
 their signs changed are the Sturmian Functions.''^ 
 
 III.— Let the equation f{x) = be *^ — 4x* — a; + 4 = 0. The first differential 
 coefficient of ;c^ — 4.«* — .c + 4 is 3x* — 8^- — 1. Dividing x^ — Ax'^ — .r -f 4 by 
 3.g«_ 8i^ _ ]^ first multiplying the former by 3 to avoid fractions,! exactly as in 
 the process of finding the H. C D., we find the first remainder of lower degree 
 tlian our divisor to be — XSix + 16, Hence 19jj — 16 is the first Sturmian Fhinc- 
 tion of x^ — 4j:* — x-\-A. Again, dividing '6x^— 8aj + 1 by 19a; — 16 (introducing 
 puch constant factors as necessary), we find the next remainder to be — 2025. 
 Hence 2025 is the second Sturmian Function of x^— Ax* — x + 4. 
 
 2oG, Wofation, — As the function which constitutes the first 
 member of our equation is represented by f(x), and its first differ- 
 ential coefficient by f'{x), we shall represent the Sturmian Func- 
 tio7is by fi(x)f /i(:i')> /3(^)> ^tc, read "/sub 1 function of x" "/sub 
 2 function of ar," etc., or simply " function sub 1," " function sub 2," 
 etc. 
 
 2S7> In any series of quantities distinguished as + and — , a 
 succession of two like signs is called a Permanence of signs, and a 
 succession of two unlike signs a Variation. 
 
 III. — In the function x^ — 3j''' — 2j-* + a'' + x* -\- fix — A, the signs of the terms 
 are 
 
 + -- + + + -. 
 
 The first and second constitute a variation ; the second and third a perma- 
 nence ; the third and fourth a variation ; the fourth and fifth a permanence ; the 
 fifth and sixth a permanence ; and the sixth and seventh a variation. Thus, in 
 this case, there are three permanences and three variations of signs. 
 
 So also if we have 
 
 f{x) = a;' - Ix*' + 13ar3+ ^* _ i^jj + 4, 
 
 fix) = 5a;* - 28a;-»+ 39a;«+ 2a; - 16, 
 
 /,(ar) = 1 l.c^ - 4ac« + 51a; + 2, 
 
 /,(a;) = 3a;='-8a: + 4, 
 
 /3(a-) = a;-2, 
 
 A{x)=0. 
 For X = 0, fix) = + 4, or f{x) is + ; f\x) is - ; /.(a-) is + ; /,(a;) is + ; 
 fi{x) is — ; and f^{x) being 0, its sign is not considered. Hence the series of 
 signs of these functions, for a; = 0, is + — + + — ; and has three variations 
 and one permanence. 
 
 * I have thought it best not to include /(x) and f'{x) under the term- Sturmian Functions. 
 There seems to be no propriety in iiidudini,' them, ina:*much as they are not peculiar to 
 Stnnn's method ; and by excluding them an important distinction is marked. 
 
 t We introduce or reject constant factors, just as in finding the H. C. D., only vvc may not 
 introduce or reject n^^o^tu^ factors, since the su/ns are an essential thinj; in these functions, and 
 to multiply or divide by a negative number would change the signs of the functions. 
 
STURM S THEOREM AND METHOD. 219 
 
 For ar = 1, we find f{x), - ; f\x), + ; f,{x), + ; f,{x), - ; and f^{x), - ; the 
 series of signs being - + h . This gives two variations and two per- 
 manences. 
 
 258. Prop.—Li the series of functions f(x), f'(x), f,(x), f,{x), 
 
 f3(x), f4(x), f5(x) f„(x), when f (x) — has no equal roots, if 
 
 X be conceived to pass through all possible real values, that is, to vary 
 continuously, from — oo to +qo , there loill be no change in the number 
 of variations and permanences in the signs of the functions, except 
 when X jxJ^ses through a root of f{x) =0; and when it does pass 
 through such a root, there loiU be a loss of one variation, and only 
 ojie,* 
 
 Dem. — 1st. Any change in x which does not cause some one of the functions 
 to vanish, cannot cause any change in the signs of the functions ; for no function 
 can change its sign without passing through or oo , and from the form of the 
 functions which we are considering, they cannot be go for any finite value of x. 
 (These functions are all of the form Ax'' + Hx"-'^ + Cx~'^ L.) 
 
 2d. No tijoo consemitive functions can vanish, i. e., become 0, for the same value 
 of J.. For, in the process of producing the Sturmian functions f rom /(.t) and 
 f\x), let the several quotients be represented by q, q', q", q"\ q^^, etc. ; whence, 
 by the principles of division, we have 
 
 f{^)=f'{^)q -Mx), (1) 
 
 f'{x)^f,{x)q' -f,{a^ (2) 
 
 A(^)=A{-i')q" ~f.(^), (3) 
 
 A{^)=fA^yr-A{x), (4) 
 
 Ux)=f\{x)q^--f,{x), (5) 
 etc., etc., etc. 
 
 Now, if possible, suppose that some value of x, sls x = a, renders two consecutive 
 functions, as fii-i-) and fi{x) each ; that is, that they vanish simultaneously. 
 Then, since from (4) we have f^ix) =fAx)q"' — fi{x), fi{x) = 0. So, also, from 
 (5), fsix) =f*{x)q''' — fi{x), and f^ix) = 0. Thus, as a consequence of the simul- 
 taneous vanishing of any two consecutive functions, we could show that all the 
 functions would vanish. But as, by hypothesis, f{x) and /'(•'') have no common 
 divisor containing x, the last remainder found by the process of finding the 
 H. C. D. cannot contain x, and hence cannot vanish for any value of x. It is 
 therefore impossible that any two consecutive functions of the series should 
 vanish for the same value of x {i. e., simultaneously). 
 
 3d. }V7i€n any one of the functions, except f (x), vanishes for a particular value 
 
 * This is the sabstancc, though not the exact form, of the celebrated theorem discovered by 
 M. Sturm in 1829, and for which he received the mathematical prize of the French Academy of 
 Sciences in 1S34. It is certainly one of the most elegant discoveries in algebraic anal}>is made 
 iu modern times. It is a masterpiece of logic, and a monument to tlie sagacity of its di>coverer. 
 The original memoir containing this theorem is found in the "Memoires prescntes par divert 
 Bftvants a 1' Academic des Sciences," Tom. VI., 1835. 
 
220 ADVANCED COUKSE IN ALGEBRA. 
 
 of X, the adjacent functions have opposite signs for this value. Thus, if f^ix) is 
 for X = b, we have, from {i),fs{i) = — /4(-f). *• «•» the adjacent functions, neither 
 of which can vanisli for tliis value (2d), liave opposite signs. 
 
 4th. When any value of x, as x = c, causes any function except f (x) to vanish, 
 the number of variations and peiinanences of the signs of the functions is the satne 
 for the jtreceding and the succeeding values ofs.,!. e.,for x = c — h and x = c + h, 
 h being an infinitesimal. Tlius, let x = c render fi{^) — ; then, since the adja- 
 cent functions liave opposite signs for this value of x, we have either +fi{x), 0, 
 — /4W. or —fi{x), 0, +f4{x), i.e., +, 0, — , or — , 0, -t- (3d). Again, as neither 
 of tliese adjacent functions vanishes for x = c (3d), neither of them can change 
 eign as x passes through e (1st). But fzix) may or may not change sign as x 
 passes through c {244) ; hence its signs may be 4^, =, ± , or T , the upper sign 
 representing the sign of /3(.e) just before x reaches c, and the lower its sign just 
 after it passes, i. c., for x = c — h, and x=z e+ h, re8i>ectively. Hence all tlio 
 clianges in signs which can occur are represented thus: + ^ —, -\- = —, 
 + ± — , 4- T — , — ^ +, — = +, — ± +, and — T +. These taken in 
 any way give simply one permanence and one variation. Jlenee there can he no 
 change in the number of variations and permanences of the signs of t?ie functions, 
 consequent upon the vanishing of any intermediate function. 
 
 5th. We are ixott to examine what changes, if any, arc; produced in the num- 
 ber of variations and permanences by tlie vanishing of an extreme function. 
 And in the first place we repeat that the last function cannot vanish for any 
 value of X, as it does not contain x. We have then to examine only the case in 
 which /(r) vanishes, i. e., when x passes through any root of f{x) = 0. For this 
 purpose let us develop f(x + h) by Taylor's Formula, considering 7i an infinitesi- 
 mal. Thus, 
 
 fix + h) =f{x) +fXc)h +/"(^) Y 4- /'"(.r) ^^ 4- etc. 
 
 Now let r be any root of f{x) = 0, and substitute in this development r for x ; 
 whence 
 
 /(r + h) =/(r) +r{r)h ^f'\r) ^ + f"\r) ^ + etc. 
 
 As r is a root of f{x) = 0, /(?•) = ; and as h is an infinitesimal, the terms con- 
 taining its higher powers may be dropped {lUlf and foot-note). Thur we have 
 /(r 4- A) = f'{r)h. Hence, as A is 4-, we see that f{r 4- h), that is the function 
 just after x passes a root, has the same sign as f'{r), i. e. f'(x) when a; is at a 
 root. But as /'(r) does not vanish when x = r (3d), f'{r — h), f'{r), and 
 fir 4- A) have the same signs.* Again, since, by hypothesis, f{x) = has no 
 equal roots, it changes sign wlien x passes through a root {244), i. e., f{r — h) 
 and f{r 4- /<) have different signs. Thus, as f{x) and f'{v) have like signs just 
 after x has passed a root, and /(/) changes sign in passing, while f'{x) does not, 
 these functions have unlike signs just l^efore x reaches a root,f and what was a 
 variation in signs becomes a permanence ; that is, a variation is lost. 
 
 * That 18, the first differential coefficient of /(r) docs not change sign when x pasecs through 
 arootof /(x)=0. 
 
 t From this we see that the roots of /'(x)=0 are intermediate between those of /(a;)=0, 
 BJncc if a, 6, and c arc roots of fix)—'), in the order of their niaguitudcs, just before x reaches^* 
 
Sturm's theorem and method. 221 
 
 Finally, as we have before shown that as x passes through all values from 
 
 — 00 to 4- oo , there can be no change in any of the functions except /(.i) which 
 will affect the number of variations and permanences in the signs of the func- 
 tions, there is only one variation lost when x passes through any root of /(.*■).— 0. 
 
 250. Cor. 1. — To ascertain the number of real roots of the equa- 
 tion f (x) = 0, ice substitute in f (x), f '(x), f,(x), i,{x) f„(x^),* 
 
 — 00 for X, and note the number of variations of sig^is. Then sub- 
 stitute + CO for X, and note the number of variations. The excess 
 of the 7iumber of variations in the former case over that in the latter 
 indicates the number of real roots of the equation. 
 
 This is a direct consequence of the proposition, since as x increases from — oo , 
 there is no change in the number of variations of the signs of the functions ex- 
 cept when X passes through a root ; and every time that it does pass through a 
 rrmt one variation is lost, and only one. But in passing from — oo to + x> , x 
 l)asses through all real values. Hence the excess of the number of variations 
 for a; = — 00 over the number for a; = -+- oo is equal to the total number of 
 real roots. 
 
 200* Cor. 2. — To ascertahi how many real roots of f (x) = lie 
 between any two numbers as a and b, substitute the less of the two 
 numbers in f(x), i'{\), f)(x), f.(x), etc., and note the number of vari- 
 ations of siyns. Then substitute the greater and note the nxmiber of 
 variations. The excess of the number of variations in the former 
 case over that in the latter indicates the number of real roots betioeen 
 the numbers a and b. 
 
 This appears from the proposition in the same manner as Cor. 1. 
 
 201, Sen. — Since the total number of roots of an equation corresponds 
 to the degree of the equation (234), if we ascertain as above the number of 
 rml roots in any given equation, the number of imaginary roots is known by 
 implication. 
 
 262, JProb. — To compute the numerical values of f (x), f '(x), 
 fj(x), f2(x), etc.^ i.e., of any function of x for any particxdar value 
 of X, when the function is of the form Ax" + Bx""* + Cx""' 
 4- Dx"-' P. 
 
 Solution. — Of course this can be done by merely substituting the proposed 
 value of X in the function. But there is a more elegant and expeditious way, 
 which we proceed to exhibit. 
 
 root rt, /(J?) and /'(x) have different signs, and just ajttr, they have like signs. But just before 
 X reaches 6, /(x) and /'(r) have unlike t?igns, and as /(x) cannot have changed sign, the sign 
 of /'(r) must have changed; i.e., x must have passed through a root of /'(a;)=0, in passing 
 from a to h. In like manner it may be shown that a root of fm lies between each two com- 
 secntive roots of /(j-)=0. This makes f'{x)^^ have one root les>^ than /(a;)=0, as it should. 
 
 * By this notation Is meant the wth or last of the Stnrmian Juncticms, in which x does not 
 appear; or, what is the same thintr, that in which the exponent of x \> 0. 
 
ADVANCED COURSE IN ALGEBRA. 
 
 Thus, let it be required to evaluate Ax'^ -\- Bx* -\- Cx^ + Bx^ -\- Ux + F for 
 X = a. Multiply A hy a and add the product to B. Multiply this sum by a 
 and add the product to C. Multiply this sum by a and add the product to J). 
 Continue this operation till all the coefficients have been involved and the abso- 
 lute term added. The last sum is the value of the function when a is substi- 
 tuted for X, as will appear from considering the following : 
 
 a 
 Aa 
 
 + B 
 a 
 
 
 
 Aa* 
 
 + Ba+ ... 
 a 
 
 
 Aa' 
 
 + Ba* jfVa + D 
 a 
 
 
 Aa* 
 
 + Ba* + C'a* 
 
 -\-lJa 
 
 + B 
 a 
 
 Aa"- + Ba* + (Ja^ + Ba* -\- Ea + F. 
 This is evidently the value of the function when a is substituted for x. 
 
 N. B. — 1. If the function is not complete, i. e., if it lacks any of the succes- 
 sive powers of r, rare must be taken to supply the lacking coefficients with O's. 
 Thus the coefficients of x* — 2j** -+- 5 are to be considered aa 1 , 0, — 2, 0, and 
 5 (which may be called the coefficient of a*"). 
 
 2. When the numbers involved are small the operation can be performed 
 mentally. 
 
 Ex. 1. Evaluate 'ZbW - 312a:» + 1553x - 5247865 for x = 342. 
 
 OPERATION. 
 
 175164 
 
 350328 
 262746 
 29953044 
 1553 
 29954597 
 
 342 
 
 59909194 
 
 119818388 
 
 8980 3791 
 
 10244472174 
 
 - 524786.5 
 
 10239224309 The value required. 
 
Sturm's method. 223 
 
 Ex. 2. Evaluate a^ — Sx" + bx — 20 for x = 2, performing the 
 operation mentally. 
 
 Examples of the Use of Sturm's Method. 
 
 1. Find the number and situation of the real roots of a;' — 4:X* 
 - 6a; + 8 = 0. 
 
 Sug's. — If the student has attended carefully to what precedes, he will have 
 no difficulty in determining that 
 
 f(x) = x^ — 4x^ -Qx + S; 
 fix) = Sx^ -Sx-Q; 
 /.(rc) = 172^-12; 
 and /j(a;") = 1467. 
 
 Now, for ;r = - 00 , we have /(a?) -, f'(x) +, /,(.t) -, and fi(x^) +; i. c.,the 
 signs of the functions are 1 h. There are therefore three variations. 
 
 Again, when x= + oo , the signs are + + + +, giving no variations. Hence 
 the number of real roots is 3 — = 3 ; i. e., they are all real. 
 
 To find the situation of tfiese roots we observe that for x = 0, the signs of the 
 
 functions are H h, giving two variations, or one less than — c- gives. 
 
 Hence there is one root between — oo and ; i. e., one negative root. The other 
 two must of course be positive. We will first seek the situation of this negative 
 root. Evaluate by {2G2). 
 
 For X = 0, the signs of the functions are H 1-. 
 
 " x= -\, « " " " " -f. 4- _ 4-. 
 
 " a; = - 2, " " " " " - H +.* 
 
 Hence, as one variation is lost when x passes from — 2 to — 1, there is one root 
 between — 1 and — 2 ; i. c, the negative root is — 1 and a fraction. 
 
 In like manner seeking the situation of the positive roots, evaluating the 
 functions by (202), we have 
 
 For X = 0, the signs -\ (- , 2 variations. 
 
 '* x=l, " " + +, 1 
 
 « x = 2, " " 4- +, 1 
 
 « a; = 3, " " + +, 1 
 
 " a; = 4, " " — + + +, 1 
 
 " .r = 5, " " -f- + + +, 
 
 * The evaluation of thcpc functions is most elegantly and oxpeditiouply cfTected by (262). 
 Thus for x=-2 we have 
 
 1 -4 -0 + 8 1^ 3 - S -6 I -2 
 
 -2 Vi -U - 6 28 
 
 -6 6 - 4=f(x) -14 a2=/'(a;) 
 
 When the value of x for which we arc cvalnatina is small, and the coefficients also small, this 
 process can be carried on mentally without writing, and should be so done. 
 
224 ADVANCED COURSE IN AL(;EEKA. 
 
 Therefore, as ono variation is lost wlien x passes from to 1 , there is one root 
 between and 1, ?*. e., an incommensurable decimal. Again, one variation is lost 
 when X passes from 4 to 5 ; lience the other root lies between 4 and 5, or is 4 
 and an incommensurable decimal. 
 
 2(iS, Sen. 2. — It is usually unnecessary to find fn{x") (the last of tlie 
 Stunniun functions), since its sign, which is all that is important, can be 
 detennined by inspection from tlie next to the last function and the pre- 
 ceding divisor. Thus, if we were to divide ar' -h 22.C — 102 by 122x — 393, 
 first multiplying the former by 122, it would be clear that the remainder 
 would be —, .without going through the operation. Hence /«(«") would 
 be +. 
 
 2 to 7. Find the number and situation of the real roots of the 
 following : 
 
 (2.) a:' -f 6a:' + 10a; - 1 = 0; (5.) ar* - 2a;^ + a:^ - 8a; + 6 = 0; 
 (3.) ar' - 6a;' + 8a; 4- 40 = 0; (6.) .a;^ - 4ar' + a;' + 6a; + 2 = 0; 
 (4.) a;* _ 4a;» - 3a; + 23 = 0; (7.) a;* + 2a;' + 17a;' - 20a; + 100 = 0. 
 
 264, ScH. 3. — In case the equation has equal roots, we shall detect them 
 in the process of producing the Stumiian functions, since in such a case the 
 division will become exact at some stage of the process, and the last Stur- 
 mian function will be 0. Having thus discovered that the equation has 
 equal roots, we might divide out the factors containing them, and then ope- 
 rate on the depressed equation as above for the unequal roots. But it is 
 not necessary to depress the degree of the equation, since the several Stur- 
 mian functions will have the same variations of signs in either case for any 
 particular value of a*. This arises from the fact that the common divisor of 
 f{x) and f'{x)y which contains the equal roots, is a factor of each of the 
 Sturmian functions, and hence its presence or absence will not affect their 
 signs for any particular value of x if the common factor is -H for this value, 
 and will change the signs of all if it is — ; but in either case the variations 
 of signs will not be affected. 
 
 8. Find the number and situation of the unequal real roots of 
 ar* — 6a;* -f 7.r* + 22a;' — GOx -f- 40 = 0, without depressing the equa- 
 tion. 
 
 Bug's. — Forming the required functions, we have 
 
 f{x) = ar» - (xc* + Zv^ + 22j;* - OO.r + 40 ; 
 / '{x) = 5x* - 24e=» H- 21«* + 44c - CO ; 
 fi{x) = dlx^ - 228aJ« + 4C8a; - 320 ; 
 fi{x) = a;* - 4a? + 4 ; 
 
 Now fii^x) is a factor of f{x), /'(«), and fi{x), and removing it from fUl, we 
 
STURM'S METHOD. 225 
 
 shall have the following functions, which nxay be used instead of the Sturmian 
 functions derived from the depressed equation : 
 
 f\a:) = 6x'' - 4x-15; 
 /,(«) = 37a; -80; 
 
 Hence, since the signs of these two sets of functions evaluated for any particular 
 value of X will be the same, either set may be used at pleasure. 
 
 Thus either set gives 
 
 For X z= — CO , 1 \-; 
 
 and for a* = + 00 , + -f- + -)_. 
 
 Therefore there are two unequal real roots of f{x) = ; and from the existence 
 of the factor {x —2)* mf{x) and /'(a;), we know that there are three equal roots, 
 each 2. 
 
 The situation of the unequal roots can now be found as before. 
 
 9 to 12. Find th6 number and situation of the real roots of the 
 following : 
 
 (9.) x^ - "^x* + ISar' + Ux' _ 66a; + 72 = 0; 
 (10.) a^ - ISx" - 28a;'^ + 2^x -f 48 = 0; 
 (11.) a^ - 4.0^ -h x' + 'ZOx + Id = 0; 
 (12.) a^ - lOa^ + 6x -{- 1 = 0. 
 
 265, ScH. 4. — Elegant as the method of Sturm is, and perfectly as it 
 accomplishes its object, the labor of producing the functions required and 
 evaluating them, especially wlien the roots are large and widely separated, 
 is so great as to deter us from its use when less laborious methods will serve 
 the purpose. In a great majority of practical cases in which there are no equal 
 roots^ the 2>rinciple that f (x) changes sign when x passes through a root of f (x) = 
 wiU enahle ns to determine the situation of the roots with far less bdior than Stui^i^s 
 Theorem. Often a simple inspection of the equation will determine the near 
 value of a root. Methods are usually given for ascertaining the limits (as 
 they are improperly called) of the roots of an equation, from the coefficients. 
 But these are of little practical value.* 
 
 ♦ For example, the two following, which are most frequently given : 
 
 1. In any equation the greatest negative coefficient •i'Vh Hs dan changed and ina^ased by unity 
 is a auPERion limit of (he roots. 
 
 2. In any equation unity added to that root of the greatest v>egatlve coefficient with its sign 
 changed, whose index is equal to the difference of the expomnts of the first term, and the first nega- 
 tive term is a pupkuior limit. 
 
 Now consider tho equation x''^ j- a?' -500=0. By the fir.-'t rule the superior limit of a root is 
 501, and by the second v/50b + 1, or 23 + . Now the fact is, the greatest root is 7.6 + . Again, by 
 1, the superior limit of the roots of a;* -3a;« -48x-72=0 is 73 ; and by 2 it is the same. But the 
 greatest root is 9. 
 
4-1 
 
 7 
 
 
 56 
 
 -500 
 392 
 
 ij: 
 
 8 
 
 56 
 
 -103, 
 
 i.e., fix) is-. 
 
 +1 
 8 
 
 
 
 72 
 
 -500 j 
 576 
 
 _8 
 
 9 
 
 72 
 
 76, 
 
 i.e., /(aj)is-f. 
 
 22*5 ADVANCED COURSE IN ALGKBIIA. 
 
 13. Find by inspection, and also by Sturm's method, ilie situation 
 of tiie roots of the equation a:* + a;* — 500 = 0. 
 
 Sug's. — Ijet the student apply Sturm's method. The following is a solution 
 by inspection : 
 
 3 3. 
 
 Since x = r 500 — x^, there is a + root less than r 500, or less than 8. Now, 
 trying 7, we have 
 
 Trying 8, 
 
 There is therefore a root between 7 and 8. 
 
 Also from the "elation t = r 500— .«*, or from the operations above, we see 
 that there is no other positive root ; since fix) evaluated for any positive quan- 
 tity less than 7 would certainly be — , and for anything greater than 8, -f. 
 
 Finally, that there can be no negative root is evident, since r 500— «' cannot 
 
 be negative until sc' > 500, but then r 500— «• < y—x*, and v—x* is always 
 
 < X. Hence for x negative we can never have x = y 500 — «*. Tlierefore 
 our equation has one real and two imaginary roots. 
 
 Note.— The advantage of this method of inspection over Sturm's method, in 
 this case, will not be fully seen unless the student observes that all this can be 
 done mentally, without writing a single figure. 
 
 14. Find by inspection, and also by Sturm's method, the number 
 and situation of the real roots of a:' + a;' + rr — 100 = 0. 
 
 Suo's. — A mere glance should show that there can be but one positive root, 
 and that that is less than 5. In like manner writing x-^ — x* + x + 100 = 0, or 
 X* + X + 100 = ar*, we see that no positive value of x can satisfy the equation; 
 for when x is less than 1, of course the first member is greater than the second, 
 and when x is greater than 1, x* itself is greater than «*. 
 
 15. Find, by inspecting the changes of sign of f(x) for varying 
 viilues of X, the situation of the roots of ar' — 3:r — 1 = 0, and also 
 by Sturm's method. 
 
 16. Find by inspection the situation of the roots of ar' — 22a; 
 - 24 = 0. 
 
 Sug's. — Writing x{x^ — 22) = 24, we see that any positive value of x which 
 satisfies this must make x^ > 22, that is, must be greater than 4. But 5 makes 
 x(x* — 22) = 15, and 6 makes it 84. Moreover, it is evident that no number 
 
STURM'S METHOD. 227 
 
 greater than C will satisfy the o<] nation. Seeking for negative roots, we write 
 x^-22x + 24: = 0; and then x{.ir - 22) = - 24. To satisfy this, x^ must be less 
 than 22, or ^ < 5. For x = 0, f(x) is +; for « = 1, /(x) is +; for a; = 2, /(.r) 
 is — . Hence a root of the given equation between — 1 and —2. Finally, for 
 x — 'd, /(.r) is — ; but for x = 4, f{x) = 0. Hence a root of the given equation 
 is —4. 
 
 17. Determine the situation of the roots of x'— 10a;' -f- C)x + 1 = 0, 
 by examining the changes of sign of f{x). 
 
 SuG's.— For a; r= 0, f(x) is +; for a; = 1, f{x) is -; for a; = 2, f{x) is -; for 
 ^ — 'S, f{x)ia — ; for a; = 4, f(x) is +; and will evidently remain +, as x ad- 
 vances beyond 4. This is seen from the following : 
 
 1 -10 +6 +1 I 4 
 
 4 16 24 +96 408 
 
 4 ^ 24 102 409 
 
 Now any positive number greater than 4 would destroy the —10 in this pro- 
 cess, and give the sum at that point greater than 6, and hence the aggregate 
 would rapidly increase. Thus notice, when 3 is substituted, we have 
 1 -10 +0 +1 |_3^ 
 
 n 9 -3 -9 -9 
 
 3 _ 1 _3 __:] _8 
 
 Now 3 is not large enough to destroy the —10; but every number larger than 
 4 will destroy it. 
 
 To examine for negative roots we write x^— 10a; '4- 6a; — 1 = 0. In this, for 
 x = 0, f{x) is -; for x= 1, f(x) is — ; for x = 2, f{x) is -; for a; = 3, /(a-) is 
 — ; but for a; = 4, and all numbers greater than 4,/(.r) is +. 
 
 We have now found that there are certainly three roots between — 4 and 
 -4- 4, and none beyond these limits either way. But it is not safe to conclude that 
 the other tioo roots are imaginary. The fact is, they are not. How, then, are we 
 to find them ? Sturm's method is thought to possess particular advantage in 
 saving us from such erroneous conclusions, and enabling us to find the eituation 
 of all the real roots with infallible certainty. And certainly it does do this ; but 
 let us see if we cannot do it, in this instance at least, as readily without that 
 method. It will be observed that wo know only that — 3 is the initial figure 
 of one root, and + 3 of another. The initial digit of the root between 
 and -(- 1 we have not found. Let us seek it. For a; = 0, f{x) is + ; and 
 by trying a; = .1, x = .2, we should at once see that /(a;) changes very slowly, 
 and as when a; = 1, f{x) is only — 2, we should be led to try numbers near 1. 
 Trying x =.8, we would find that f{x) is +, and for x =.9, /(a?) is — . Hence .8 
 is the initial figure of the root lying between and + 1. 
 
 We now know the initial figures of three of the roots. But where are the 
 other two roots ? If they are real we know that they lie between — 4 and + 4. 
 as we have seen above that no root can lie beyond these limits. Moreover, aa 
 the function changes value rapidly beyond 1, and slowly between — 1 and 1. i« 
 
228 ADVANCED COURSE IN ALGEBRA. 
 
 would naturally Ik^ suggested that there may be two changes of sign between 
 and + 1, or and — 1. Evaluating f{x) = x'^ — 10^ ' -t- Oc + 1 for .1, .2, .3, etc.. 
 we soon see that it will not change sign for values of x between and + i. 
 Evaluating f{x) = x^ — 10.f' + ftr — 1 for .1, .2, .3, etc., we find that the other 
 roots are between and — 1, and that their initial digits are —.1 and —.6. 
 
 18 to 23. Find by inspection, by the change in sign of /(x), or by 
 Sturm's method, the number and situation of the real roots of the 
 following : 
 
 (18.) .r'- ^x'- 4tx + 11 = 0; 
 
 (19.) ar'-2a:-5 = 0; 
 
 (20.) X* - 4.r' - 3.r + 23 = ; 
 
 (21.) .r'' + 11a;' - 102.c + 181 = 0; 
 
 (22.) x'-llx' -^ 54.r = 350 ; 
 
 (23.) x^ + 2x' + 3.r - 13089030 = 0.* 
 
 206. Sen. 5. — If we have an equation in which, when cleared of frac- 
 tions, the coefficient of the highest power of ac is not unity, it may be trans- 
 formed by {22S) into one having such coefficient. But thi^ it not necessary 
 in order to the application of Sturm's method^ as it is not required by anything in 
 the demonstration of that theorem that the coefficients should be integral, 
 
 24 to 31. Find l)y Sturm's method the number and situutiou of 
 the real roots of the following: 
 
 (24.) 2.r' + 3a:'- 4a: - 10 = ; (28.) 3.*:*- 4.c' + 2x - 1000 r^ ; 
 
 (25.) x"- 18^.c + 20,5^3^ = ; t (29.) Ix'- 83:c + 187 = ; 
 (26.) %3^- 36a:' + 4G.c -15 = 0; (30.) x"- lf.6'- Ifa: = 440 ; 
 (27.) 4a:'- 12a:' -h 1 l.r - 3 = ; (31.) x^- \x'- |a: = 312. 
 
 Horner's Method of Solution.^ 
 
 207, Horner's method of solving numerical equations is a method 
 of finding the incommensurable roots of such equations to any re- 
 
 • Observe that neglecting the termei 2a:' + Sx, which, since x is larjjc, arc small as compared 
 with x', we liavu x3 = 1.3089030, or x lics> between 200 and 20k) probab'y. 
 
 t Clear of fractions first. 
 
 X Among the manj methodp dis^covcred. and ("loiibtlc.-s to be discovered, for (his purpose, it 
 is scarcely pos?-il)le that Homer's should be Hiperceded, since the solution of pucli an equation 
 will certainly require the extraction of a root corresponding to the defrree of the equation ; iind 
 the labor required by Horner's method is not greater than that required lo extract this root. 
 Nor is this merely a method of approximation, except as any mcthfti for inrommensurafjle roots 
 is necessarily a method of approximation. If tlie nK>t can be expressed exactly in the decimal 
 notation, or by means of a repealing decimal, this process tffects it. The method was first 
 published by W. G. H<»rncr. Etq., « f Bath, England, in 1819, about fifteen years before Sturm'* 
 Theorem wis published. 
 
HOKNEIIS METHOD. 
 
 quired degree of approximate accuracy. It is based upon the two 
 following problems and proposition : 
 
 208, JProb, — 7h transform mi equation, as f (x) = 0, into another 
 whose roots shall be a less than those of tJie given equation. 
 
 Solution. — Let x=a-\-Xi, whence a?, —x—a, and we hare f{x)=f{a+Xi)=0, 
 or 0=f{a + Xi). Deyeloping the latter by Taylor's Formula, we have 
 
 0=f{a + x,) = fia)+ f'{a)x^+f"{a)~ + /"'{«)p + /*M«)^ + etc.. or 
 
 =/(«) +f'{a)Xx +f"{a)'^ +/'"(«) j3 +/''^(«) i^ > etc., as the required equa- 
 tion. 
 
 209, Sen. — The meaning of this may be stated thus : The absolute term 
 of the transformed equation is the value of /(r) when a is substituted for 
 a?; the coefficient of the first power of the unknown quantity, ;r,, in the 
 new equation is the first differential coefficient of f{x), when a is substituted for 
 X in this coefficient ; the coefficient of the second power of X\ is ^ the second dif 
 ferential coefficient of f(x), when a is substituted for x ; etc. 
 
 Ex. — From 5x* — 12.c^ + 3aj' + 4x + 5 = deduce a new equation 
 whose roots shall be each less by 2 than the roots of this. 
 
 SOLUTION. 
 
 f(x) = 5x* - nx^ + 3a;2 + 4a? + 5 =9 =f{a). 
 
 x=a=2* 
 
 f\x) = 20a;' - C().r ' + G.c + 4 =83 =f\a). 
 
 x=Z 
 
 f%r) = 60x' - 73 r + 6 =102 =f"a. /. \f'\a) = 51. 
 a:=2 
 
 f'"(x) = 120^ - 73 =108 =f"\a). .'. X f"'{a) = 28. 
 f^r^T) = 130 = 120 =/•%). .'. ,-i /'%) = 6. 
 
 x=2 
 
 Hence = + JJ3r, + 51.?,*+ 2a»t'+ 5.p,\ or Sv,^ + 28.c,^-t-51a;i''+32a;i+ = 0, 
 is an equation wluwc roots are 2 less than the roots of the given equation, 
 since a;, = j; — 2. 
 
 270* IPvob, — To compute the numerical values of f (a), f '(u), 
 if "(a), jj-f '"(a), jf f^(a), etc., from f(x), when f(x) has the form 
 Ax"+ Bx"-' -f Cx'"-'+ Bx"-^ P. 
 
 Solution -—Let /(.t) = Ax^+ Bx^+ Cx^ -i- Dx + E; whence, forming f'{x), 
 f'\x), f"'{x), and /""(.f), and substituting a for x, we have 
 
 ♦ TTi" meaning of lhi» Hotatlon ii» Ihat X i* made eqnal to 2 In the function whence rei^«lt» 
 the following value. 
 
2ae 
 
 ADVANCED COURSE IN ALGEBRA. 
 
 /(«) = Aa* + Ba^ + Ca^ + Da + E; 
 f'(a) = 4Aa^ + 3Ba' + 2Ca + I) ; 
 if"{a) = QAa^ + 'dBa+ C; 
 ^hf"'{a) = ^Aa + B; 
 ,i/-(a) = A. 
 
 Now, we may compute these as follows 
 
 + + 
 
 w 
 
 
 c^ 
 
 
 + 
 
 + 
 
 
 M 
 
 
 
 1 
 
 5 
 
 3 
 
 + 
 
 + 
 
 + 
 
 •4 
 
 M 
 
 •« 
 
 e 
 
 « 
 
 e 
 
 ^ 
 
 O) 
 
 
 
 + 
 
 + 
 
 
 rt 
 
 •0 
 
 
 « 
 
 e 
 
 
 ^ 
 
 CO 
 
 t 
 
 H 
 
 3 
 k 
 
 6 .-S 6 
 
 3 !». « 
 
 Si = 
 
 5 -a '5 
 
 ^ ^ be 
 
 ca 
 
 "^ ''^ p s 
 
 c - = "^ 
 
 c ^ -^ '^ 
 
 C ac a 
 
 '^ ^ 
 
 '^ z: 
 '§ % 
 
 I i 2 
 I ^ S 
 
 s S;'i 
 
 -- .2 H 
 
 H '3 r.^ 
 
 fi o s 
 
 C-'S 
 
 ho 
 
 ■^ S5 
 
 ■a 
 
 3 
 
 c 
 o 
 
 ^ § S 
 
 I I i' 
 
 7 ® 
 + c 
 
 O + cS 
 
 ri '^ O .^ ♦* 
 
 be 5 
 
 I-? 
 
 
 
 O n 
 
 XI o 
 
 Examples. 
 
 1. Transform 3a^— 42:* + Ta;*^ + 8a:— 12=0 into another, equation 
 each of whose roots shall be 3 less than the roots of this. 
 
 Solution. — Arranging the coeflBcients and proceeding as in the above solu 
 tion, we ha i the following : 
 
OPERATION. 
 
 
 
 + 7 +8 
 
 -12 
 
 1 3 
 
 15 66 
 
 222 
 
 
 22 74 
 
 210 = 
 
 = /(3) 
 
 42 192 
 
 
 
 64 266 = 
 
 = /'(3) 
 
 
 69 
 
 
 
 133 = i/"(3) 
 
 
 
 Horner's method. 231 
 
 s -4 
 _9 
 
 5 
 
 _9 
 14 
 
 Jl 
 
 23 
 _9 
 
 32 = \hf"'{S). 
 
 Hence the transformed equation is 
 
 dxi^+^zi' + 133a; ,2 + 266a;, +210 = 0. 
 
 2. Transform Sa^ — 13.^' + 7a:' — 8a: — 9 = into another equation 
 wliose roots shall be less by 3 than the roots of this. 
 
 The new equation is 3a:' + 23^'" + 52a:'' + 7a:— 78=0.* 
 
 3. Transform x^ + 2x'' — 6a:'' — 10a: + 8 = into another equation 
 whose roots shall be 2 less than the roots of this. 
 
 I 2 
 
 
 PROCESS. 
 
 
 
 
 
 + 2 
 
 -6 
 
 -10 
 
 + 8 
 
 2 
 
 4 
 
 12 
 
 12 
 
 4 
 
 2 
 
 6 
 
 6 
 
 2 
 
 12 
 
 2 
 
 8 
 
 28 
 
 68 
 
 
 4 
 
 14 
 
 34 
 
 70 
 
 
 2 
 
 12 
 
 52 
 
 
 
 6 
 
 26 
 
 86 
 
 
 
 2 
 
 i? 
 
 
 
 
 8 
 
 42 
 
 
 
 
 _2 
 
 10 
 .'. The equation is aj" + 10a;* + 42a;^ + 86a;* + 70a; + 12=0. 
 
 4. Transform x' - Gx" + 7.4a:' + 7.92a:' - 17.872a:-.79232 = into 
 another equation whose roots shall be each less by 1.2 than the roots 
 of this. 
 
 5. Transform a:^— 2a:' + 3a: + 4=0 into another equation whose roots 
 shall be 1.7 less than the roots of this. 
 
 6. Transform a^H Ha:'— 102a: + 181=0 into another equation whose 
 roots shall he 3 less than the roots of this equation : transform the 
 
 ♦ For convenience in reading and writing, it is customary to omit the subecripts which dis- 
 llnjfuis'h <hc nnlcnown quantity in the transformed equation from that in the given equation. 
 But it should be borne in mind that the iinlcnown quantities are different. 
 
232 
 
 ADVANCED COUllSE IN ALGEBRA. 
 
 resulting equation into another whose roots shall be .2 less than the 
 roots of the last: transform this equation into another whose roots 
 shall be .01 less than those of the last : transform this into another 
 whose roots shall be .003 less than its roots. 
 
 OPERATION. 
 
 + 11 
 
 14 
 
 _8 
 
 17 
 
 3 
 
 20* 
 .2 
 
 ao.2 
 
 20.4 
 
 ao.6t 
 
 .01 
 
 20.61 
 
 .01 
 
 20.G3 
 .01 
 
 20.63t 
 .008 
 
 -102 
 
 J? 
 
 -60 
 
 _51 
 
 4.04 
 
 -4.9G 
 
 4.08 
 
 + 181 
 -180 
 
 1* 
 -.992 
 
 .098+ 
 -.000739 
 
 .001201$ 
 -.001217403 
 
 .01 
 
 .003 
 
 .2061 
 .6739 
 
 -.4677 J 
 . 061899 
 
 -.405801 
 .061908 
 -.343893^ 
 
 EXPLANATION. 
 
 * These, together with the first, 
 are the coefficienta of the cquiition 
 whose roots are 3 less tlum those 
 of the given equation. The equa- 
 tion written out is a;' + 20x*— 9.C 
 + 1=0. (.1). But, instead of re- 
 writing these coefficients for the 
 second transformation, we operate upon them just as 
 they stand. 
 
 f These, together with the first, are the coefficients 
 of the equation whose roots arc .2 less than those of 
 {A), and consequently 3.2 less than those of the given 
 equation. This equation written out is :v* + 20.(yX'- ~ 
 .88.C+. 008=0. {B). But instead of rewriting these co- 
 efficients we effect the next transformation upon them 
 just as they stand. 
 I These, together with the first (which remains the same in all), are the co- 
 efficients of the e(iuation whose roots are .01 less than the roots of (B), .21 less 
 than the roots of {A), and 3.21 less than the roots of the given equation. This 
 equation is a:' +20.63J;- -.4677.r + .001261=0. (C). 
 
 ^ These are the coefficients of the equation whose roots are .OOo Ics.'J than 
 those of {€), .013 less than those of {B), .213 less than those of {A), and 3.213 
 less than those of the given equation. The last transformed equation is 
 a? • + 20.639^;' - .343893.C + .000043597=0. 
 
 7. Transform, as above, the equation x^—l^x^-hVix—S — O, suc- 
 cessively, into equations whose roots shall be 2 less, 2.8 less, and 2.85 
 less than the roots of the given equation. 
 
HORNER S METHOD. 
 
 233 
 
 6 
 _2 
 
 8» 
 
 8.8 
 _.S 
 9.6 
 
 10.4 
 .8 
 11.2t 
 
 m 
 
 
 OPERATION. 
 
 
 -13 
 
 + 12 
 
 -3 |! 
 
 4 
 
 -16 
 
 -8 
 
 -8 
 
 -4 
 
 -11* 
 
 8 
 
 
 
 8.9856 
 
 
 
 -4* 
 
 -2.0144t 
 
 12 
 
 15.232 
 
 1.71940625 
 
 12* 
 
 11.232 
 
 -.294993751 
 
 7.04 
 
 21.376 
 
 
 19.04 
 
 32.608t 
 
 
 7.68 
 
 1.780125 
 
 
 26.72 
 
 84.388125 
 
 
 8.32 
 
 l.e08375 
 
 
 35.04f 
 
 o6.1'JG500t 
 
 
 .5625 
 
 
 
 35.6025 
 
 
 
 .5650 
 
 
 
 J:6.1675 
 
 
 
 .5675 
 
 
 
 I 2.85 
 
 36.7850$ 
 
 {B). 
 
 11.25 
 .0 5 
 
 11.30 
 .05 
 ll.;i5 
 .ft5 
 11.40t 
 Hence the successive equations are. 
 The Primitive, ^^ -12.c« + 12a; -3-0 ; 
 
 One whose roots are 2 less than those of (.4), 
 
 ar^+8c' +-12.c^-4,c-ll=0 
 One whose roots are .8 less than those of {B), or 2.8 than those of {A), 
 
 X* + 11.2.C ' + 35.04c2 + 32.608.^-2.0144=0 ; {C). 
 One whose roots are .05 less than those of (C), .85 less than those of {B), oi 
 2.85 less than those of (.4), 
 
 X* + 11.4ar' + 36.735x2 + 36.1965a;-.29499375=0. 
 
 8. Transform, as above, the equation a.-^— 7a; + 7=0, successively, 
 into equations whose roots shall be 1 less, 1.3 less, 1.35 less, and 1.356 
 less than the roots of the given equation. 
 
 271, JPvop, — l/'a + Xi is a root of /*(ic)=0, andxx is sufficiently 
 
 small with reference to a, x^ 
 
 /(«) 
 
 , approximately. 
 
234 ADVANCED COUr^SE IN ALGEBRA. 
 
 Dem. — If a+Xi is a root of / (x) =0, f{a + a; , ) = 0. Developing tliis by Taylor's 
 Fbrmola, we have 
 
 /{a + x,)=f{a)+f(a)x,+f"{a)^ + f"(a)^-^ etc.=0. 
 
 Now, to determine a?, approximately, which is all the proposition proposes, when 
 Xx is quite small with reference to a, all the terms in the development involving 
 higher powers of a? I than the first maybe neglected; whence we have /(a) 4- 
 
 /■(a)^,=0.or«, = --^j. 
 
 Ex. — Knowing that 4.+ some decimal fraction which we will call 
 Xx is a root of ar'' + a;' + a; — 100=0, required the approximate value of 
 the decimal fmction x^. 
 
 Solution. — Finding /(a), i.e., in this case/ (4)* in the ordinary way, we have 
 
 1 +1 +1 -100 |_4_ 
 
 _4 JO _84 
 
 6 21 -16=/(a),or/(4)* 
 
 4 _36 
 9 57 =/'{«), or/'(4)» 
 
 _4 
 13 
 
 Hence — ——-•= -— - =.28+ \^ approximately the decimal part of the root. 
 
 / \^) o7 
 
 In fact, 2 is the tenths figure of the decimal part of the root, the root being 
 
 (as we shall find hereafter) 4.2644 + . 
 
 We thus have a^,*+ 13^*,'+ 57a?i —16=0, an equation whose roots are 4 less 
 
 than the roots of the given equation. We will now transform this into another 
 
 equation whose roots shall be .2 less than the roots of this equation, or 4.2 less 
 
 than the roots of the given equation. Thus 
 
 1 
 
 + 13 
 
 + 57 
 
 
 -16 
 
 1 -2 
 
 
 .2 
 
 2.64 
 
 
 11.928 
 
 
 
 13.2 
 
 59.64 
 
 
 -4.072 = 
 
 = /(4.2)t 
 
 
 .2 
 
 2.68 
 
 
 
 
 
 13.4 
 
 62.32 : 
 
 =/'(4.2)t 
 
 
 
 
 2 
 
 
 
 
 
 
 13.6 
 
 
 
 
 
 and the transformed equation 
 
 lis 
 
 
 
 
 
 a.,» + 
 
 13.6.rj« + 62.32«i -4.072=0, 
 
 
 * This notation means, the valne ot/(x) when 4 ie snbstitnted for x therein. 
 
 t That these are the values of f(x) (the first member of the {jiven eqnation) and/'(a;), when 
 4.'2 i!« substituted for x, will be evident if it is considered that they are the same results as 
 would have been obtained by transforming the given equation immediately (by one procei^e) 
 into another whose roots are 4.2 less. 
 
Horner's method. 285 
 
 which is an equation whose roots are 4.3 less than those of the given equation, 
 Hence by the proposition (Tg = — "looo —065 approximately. In fact, it will 
 
 O/ft.O,* 
 
 be seen that 6 is the hundredths figure of the root. 
 
 Writing both portions of the above work together, it stands thus : 
 
 +1 
 
 + 1 
 
 -100 
 
 [4.2 
 
 4 
 
 20 
 
 84 
 
 
 5 
 4 
 
 21 
 36 
 
 -16* 
 11.928 
 
 *.-. -16=/(«).or/{4) 
 
 9 
 4 
 
 13* 
 .2 
 
 57* 
 2.64 
 
 59.64 
 2.68 
 
 -4.072f 
 
 * .'. 57=/'(«), or/(4) 
 t .-. -4.072=/(4.2) 
 
 13.2 
 .2 
 
 13.4 
 
 62.32t 
 
 f.-. 62.32=/'(4.2) 
 
 .2 
 
 13.6t 
 
 
 
 
 HoRiq^ER's Rule. 
 
 272. RULE. — 1. Put the equatioi?^ i^ the for^i 
 Ax^ + Bsf-^ + 6'af-* Mx + L=Oy 
 
 1^ WHICH the coefficients A, B, C X, IF NOT INTEGRAL, 
 
 are expressed exactly in decimal fractions. 
 
 2. Find the number and situation of the positive real 
 ROOTS BY Sturm's Theorem, determining one or jiore (usually' 
 two) of the initial figures. (See Sen. 1.) 
 
 3. Write the coefficients in order with their propei^ 
 signs, being careful to supply with O's the places of co- 
 efficients OF missing terms, if the equation is not complete. 
 Taking the initial figures of one of these roots as thus 
 found, operate on these coefficients so as to obtain the co- 
 efficients OP the transformed equation whose roots shall 
 
 BE less by the portion OF THIS ROOT ALREADY FOUND. 
 
 4. Having found these coefficients, if the coefficient of 
 the first power of the unknown quantity in this trans- 
 
236 ADVANCED COURSE IN ALGEBRA. 
 
 FORMED EQUATION AND THE ABSOLUTE TERM, /'(«) AND /(«), HAVE 
 UNLIKE SIGNS, DIVIDE THE LATTER BY THE FORMER, AND THE FIRST 
 FIGURE OF THIS QUOTIENT WILL BE (APPROXIMATELY) THE NEXT 
 FIGURE OF THE ROOT. (See ScH. 2.) If THESE FUNCTIONS HAVE 
 LIKE SIGNS, MORE FIGURES OF THE ROOT MUST BE FOUND BY StURM'S 
 
 Theorem or by trial, before proceeding to apply this pro- 
 cess OF transformation. 
 
 5. Having found a figure of the root by dividing f{a) by 
 
 /'(a), ANNEX it to the ROOT AND OPERATE ON THE COEFFICIENTS 
 
 of the last (transformed) equation as they stand, to pro- 
 duce the coefficients of the next transformed equation, i. <?., 
 the one whose roots shall be less than those of the last, 
 by the last figure of the root, and less than those of the 
 given equation by' the entire portion of the root now found. 
 Having found these coefficients, divide the absolute term 
 
 BY the coefficient OF THE FIRST POWER OF THE UNKNOW|<^ 
 QUANTITY, IF THEIR SIGNS ARE UNLIKE, AND THE FIRST FIGURE 
 OF THIS QUOTIENT WILL BE (APPROXIMATELY) THE NEXT FIGURE 
 OF THE ROOT. If THESE SIGNS ARE ALIKE, THE LAST ASSUMED 
 FIGURE OF THE ROOT IS TOO LARGE AND MUST BE DIMINISHED. 
 
 (See ScH. 3.) 
 G. Proceed in this manner until the root is obtained; 
 
 OR, IF THE ROOT IS INCOMMENSURABLE, UNTIL AS MANY FIGURES 
 OF THE DECIMAL FRACTION ARE OBTAINED AS ARE DESIRED. (See 
 SCH. 4.) 
 
 7. In LIKE MANNER ALL THE POSITIVE REAL ROOTS, OR THEIR AP- 
 PROXIMATE VALUES, MAY BE FOUND. To OBTAIN THE NEGATIVE 
 ROOTS, CHANGE THE SIGNS OF ALL THE TERMS CONTAINING ODD POW- 
 ERS OF THE UNKNOWN QUANTITY, OR ALL OF THOSE CONTAINING THE 
 EVEN POWERS ; OR, IF THE EQUATION IS COMPLETE, EACH ALTERNATE 
 SIGN, AND PROCEED TO FIND THE POSITIVE ROOTS OF THIS EQUATION 
 AS BEFORE. ThE VALUES THUS FOUND WILL BE THE NUMERICAL 
 VALUES OF THE NEGATIVE ROOTS (246)* 
 
 This rule is based upon previously demonstrated principles, and needs no 
 special demonstration. 
 
 273. ScH. 1.— By means of (244, 24=5) we can usually find the initial 
 figure or figures of the roots with less labor than l>y Sturirv • Tlicoreni. 
 
HOliNERS METHOD. 237 
 
 274, Sen. 2.— Since hy {271)x,= - J^, if both /(«) and f\a) have 
 
 the same sign at amj time^ this quotient will be — , and hence the value 
 thus found for a-, will not be the amount to be added (annexed) to the por- 
 tion of the root already found, for the assumption is that this portion is 1er» 
 than the root of the equation which we are seeking. 
 
 27''>. Sen. 3.— That the figure of the root found by dividing/(«) by/'{«) 
 is liable to be too large is readily seen when we consider that instead of 
 /'(«)^i= — /(^) (in Dem. of 271)1 we should have, if no terms were 
 omitted, 
 
 /'(«>», +i/"(«)^,' + i/"'(r«).c,'+etc. = -/(«). 
 
 Now a value of .r, which satisfies the former may evidently be quite too 
 large to satisfy the latter. Thus consider .t;^ + 10.?^ -i-5j;— 2600=0. Neglect- 
 ing x^ and lO-c*, we have 5ar=2600, or a=520. But this will by no means 
 satisfy the equation when x^ and 10a;* are not neglected. 
 
 Again, the figure found by dividing/(a) hy f'{a) may be too small. Thus, 
 if we have .c^ — lO-c^ +12.^—3=0, and neglect .c*', and — 12.6% we have 12.r-3 
 =0, or x=\. But this is too small a value to satisfy the equation, since for 
 x=\, — 12j;* will be numerically much larger than x*^ and hence retaining 
 these terms will diminish the function, thus making \ too small to satisfy 
 the equation. 
 
 270, Sen. 4. — From Sen. 2 it appears that f{(i) cannot change sign in 
 the process unless f'{a) also changes sign. But when f{a) changes sign, we 
 know by (244) that we have passed a root of the equation ; if, however, f'{a) 
 also changes at t!ic same time, our work may still be right. In such a case 
 there are two roots having tlicir initial figure or figures alike, e. </., one may 
 be 23.56 + , and tlie other, 23.59 + . To obtain the less of the two roots, take 
 the largest figure which will not cause cither f (a) or f'(a) to change sign; 
 and for the larrrcr of the two roots take the smallest figure which will cause 
 both f(a) and f\a) to change sign. 
 
 [Note. — Those scholiums, as also the rule, will be better understood in con- 
 nection with their applications in the following examples. But in review, after 
 the solution of the examples, they should bo carefully learned.] 
 
 Examples. 
 
 1. Kequired the roots of. r''—4.r- — C.r-f8=0. 
 
 Soi.iTTiox. — By Sturm's method wo find that there are 3 real roots, one nega- 
 tive, and two positive (see Ex. 1, page 223), and also that the negative root is 
 — 1. and an incommensurable decimal, that one positive root is an incommen- 
 surable decimal, and that the other positive root is 4. + an incommensurable 
 decimal. We will seek the latter first. 
 
238 ADVANCED COUltSE IN ALGEBRA. 
 
 -4 -6 ^- 8 I 4.892+ 
 
 _4 
 
 
 _4 
 
 4 
 
 4 
 
 8.8 
 .8 
 
 9.6 
 .8 
 
 10.49 
 .0 
 
 10.58 
 .09 
 
 10.673 
 .002 
 
 10.674 
 .003 
 
 10.676 
 
 UfKKATlON. 
 
 -6 
 
 ^- 8 
 
 
 
 -24 
 
 -6 
 
 -16...' 
 
 16 
 
 13.632 
 
 10.. 
 
 -2.308... 
 
 7.04 
 
 2.309760 
 
 17.04 
 
 -.058231... 
 
 7.68 
 
 .053275288 
 
 24.72.. 
 
 -.004955712 
 
 .9441 
 
 
 25.6641 
 
 
 .9522 
 
 
 ca.6163.. 
 
 
 .021844 
 
 
 23.037044 
 
 
 .021348 
 
 
 26.658992 
 
 
 Remarks. — The general features of the process, being the same as heretofore 
 given {270, Example, need no further explanation than they have already re- 
 ceived. Each decimal figure of the root is added the first time in the first 
 column simply by annexing it. 
 
 In finding the second figure of the root, we have — ^,^ - = tk"= ^-G- ^^^ 
 
 this cannot be the proper addition, since we know that the root lies betwefiii 4 
 and 5 ; hence this trial fails to give the second figure in the root. (See 273*) 
 But as we know that this figure cannot be greater than 9, we try 9, and find 
 that it makes the absolute term change sign so that /(«) and /'(«) have the 
 same sign, and consequently .9 is too much to add. (See 270fa.nd also consider 
 that f{x) would thus be shown to change sign as x passed from 4 to 4.9, and 
 hence that a root lies between 4 and 4.0, 244.) We therefore try .8, and find 
 that it is the correct addition. We know that .8 is right, since we know that 
 as X passes from 4.8 to 4.9, f{x) changes sign. 
 
 In finding the third figure we have for trial —^77— J = ' = .00. Try- 
 
 / (a) M.la 
 
 ing 9 as the third figure of the root, we find that the absolute term does not 
 
 change sign, and hence we ktwio that 9 is the next figure, i. e., we know that a 
 
 root lies between 4.89 and 4.9. 
 
 The process may be thus continued indefinitely, and as many figures found as 
 
 we may desire. 
 
 277» N. B. — It will be observed that this process is simply one of substitu- 
 tion in f{x) of values for x which come nearer and nearer to making f{x) — 0. 
 
HOIINERS METHOD. 239 
 
 Thus in this example 4, substituted in a- ' — 4a:* — 6a; + 8, gives x^ — 4x^ — Qx 
 + 8 = — 16 ; 4.8 substituted for x, gives a;'* — 4a;* — 6a; + 8 = — 2.368 ; 4.89 
 gives «' - 4x^ - Qx+S= -.058231 ; 4.892 gives a;^-4a;2-6a-+8= -.004955712. 
 Thus we are coming nearer and nearer to the number which substituted for x 
 would make x^ — 4a;* — 6a; + 8 = 0, or would satisfy the equation. 
 
 2. To FIND THE ROOT WHICH LIES BETWEEN —1 AND —2, we take the equa- 
 tion x'^ + 4a;* — 6a; — 8 = (changing the signs of the terms containing the even 
 powers of x), and find the root of this equation which lies between 1 and 2 
 
 (246\ 
 
 OPERATION. 
 
 h4 
 
 -6 
 
 -8 1 1.8004H 
 
 1 
 
 5 
 
 -1 
 
 5 
 
 -1 
 
 -9... 
 
 1 
 
 6 
 
 8.992 
 
 6 
 
 1 
 
 5... 
 6.24 
 
 008 ••• 
 
 .007249504064 
 
 7.8 
 
 11.24 
 
 -.000750495936 
 
 .8 
 
 6.88 
 
 
 8.6 
 
 18.12 
 
 
 .8 
 
 .00376016 
 18.12376016 
 
 
 9.4004 
 
 
 .0004 
 
 .00376032 
 
 
 9.4008 
 
 18.12752048 
 
 
 .0004 
 
 
 
 9.4012 
 
 
 
 o. To FIND THE ROOT WHICH LIES BETWEEN AND 1. We first find thc 
 initial figure either by evaluating f{x) successively for .1, .2, .3, etc., and no- 
 ticing when it changes sign (244) ; or by Sturm's method. The former is much 
 the less laborious, and is to be preferred (2^S). In fact, to use Sturm's method 
 involves exactly the same work as the former method, w\i\\ considerable additional 
 work. Moreover, the former method can be applied mentally till the proper 
 initial figure is determined, and no other writing will need to be involved than 
 just what Horner's method requires. No figures will need to be written but 
 those in the following 
 
 OPERATION. 
 
 -4.. 
 
 -6... 
 -2.79 
 
 +8...' 1 .9082-f 
 
 .9 
 
 -7.911 
 
 -3.1 
 
 -8.79 
 
 .089 
 
 .9 
 
 -1.98 
 
 -086242688 
 
 -2.2 
 
 -10.77.... 
 
 .002757313 
 
 .9 
 
 .010336 
 
 
 -1.3.. 
 
 -10.780336 
 
 
 .003 
 
 
 
 -1.292 
 
 
 
240 ADVANCKD COUKSK IN ALGEBIIA. 
 
 It irt so evident that the last figure is 2, that the operation for verifying it i» 
 unnecessary. 
 
 2. Find the roots of x' — 13a;* + 53.'c' — 49a;' — 110.?; + 150 = 0, 
 extending the decimals to the 5th phice. 
 
 Suo. — Apply Sturm's method. If there arc equal roots, depress the equa- 
 tion. 
 
 3 to 5. Find all the real roots of the following, extending the 
 decimals to 4 or 5 places: 
 
 (3.) 7? + lOar* + bx - 260 = ; 
 
 (4.) ^4- 3a;* + 5^= 178; \ 
 
 (5.) ^+ 23;* = 23a; 4- TO. 
 
 Tiie cuhic equations on pages 223, 224, 22G, 228, will afford further 
 exercise. 
 
 G. Find the roots of the equation a^ — SOa;* + lOOSrc* — 14937:*; 
 + 5000 = 0. 
 
 Sug's. — Oi course we may always find the number and situation of the real 
 roots by Sturm's method. But as tlie labor of t^ubstituting in aU the functions 
 used in this method is frequently great, we avoid it when we can. Howecer, it 
 is generaUy heM to free tJie equation from equal roots, and find the number of 
 posUioe, and the number of negaticc roots by Sturm's method. But the situation 
 of the roots is almost mlways more readily found by inspection based mainly on 
 the change in sign of f{x) {244). We will solve this example in this way. 
 
 1 . By Sturm's method we find vhat our equation has no equal roots, and that 
 it has 4 positive roots, and no negative root (see 264). 
 
 2. We now^ proceed to find the least root. Observing that for x=0, 
 f{x) is -h, and for x=\, f{x) is — , we know that at least one real root lies 
 between these limits. To find it we have the following (see next page) : 
 
HORNER S METHOD. 
 
 2^ 
 
 FIRST OPERATION. 
 
 1 
 
 -80 
 .1 
 
 +1998 
 - 7.99 
 
 -14937 
 199.001 
 
 +5000 
 -1473.7999 
 
 .1 
 
 .2 
 
 -79.9 
 .1 
 
 -79.8 
 .1 
 
 1990.01 
 
 - 7.98 
 1982.03 
 
 - 7.97 
 
 -14737.999 
 198.203 
 
 -14539.796* 
 391.636 
 
 3526.2001* 
 -2829.6320 
 
 696. 5681.... t 
 - 274.4238,5424 
 
 .3 
 .02 
 
 .02 
 .01 
 
 -79.7 
 .1 
 
 1974.06* 
 
 - 15.88 
 
 -14148.160 
 
 388.468 
 
 422.14424576} 
 - 272.88640240 
 
 .35 
 
 -79.6* 
 .2 
 
 1958.18 
 ~ 15.84 
 
 -13759. 692.. .f 
 38.499288 
 
 149.25784336§ 
 - 135.86783711 
 
 
 -79.4 
 .2 
 
 1942.34 
 - 15.80 
 
 -13721.192712 
 38.467784 
 
 13.390006251 
 
 
 -79.2 
 .2 
 
 1926.54-. t 
 - 1.5756 
 
 -13682. 724928t 
 
 38.404808 
 
 
 
 -79.0 
 .2 
 
 19249644 
 - 1.5752 
 
 -13644.320120 
 38.373336 
 
 
 
 -78.8-f 
 .02 
 
 1923.3892 
 - 1.5748 
 
 -13605.946784§ 
 19.163073 
 
 
 
 -78.78 
 .02 
 
 1921.8144t 
 - 1.5740 
 
 -13586.783711 
 19.155211 
 
 
 
 -78.76 
 .02 
 
 1920.240 1 
 - 1.5736 
 
 -13567.6285001 
 
 
 
 -78.74 
 .02 
 
 1918.6668 
 - 1.5732 
 
 
 
 
 -78.72t 
 .02 
 
 1917.09361 
 - .7863 
 
 
 
 
 -78.70 
 .02 
 
 -78.68 
 .02 
 
 1916.3073 
 
 - .7862 
 
 1915.5211 
 
 - .7861 
 
 
 
 
 -78.66 
 .02 
 
 1914.7350t 
 
 
 
 
 -78.64§ 
 .01 
 
 
 
 
 
 -78.63 
 .01 
 
 -78.62 
 .01 
 
 
 
 • 
 
 
 -78.61 
 
 .01 
 
 -78.60 
 
 
 
 
 
249 ADVANCED COURSE IN ALGEBRA. 
 
 Remarks.— This work is given to show how we may proceed to find the first 
 twj figures of the root by successive simple approximations. If the student is 
 familiar with the principles heretofore developed and applied, he will have no 
 difficulty ill seeing the reasons for the operations above. We are simply adding 
 to the value of x substituted in /(r), so as steadily to diminish the absolute 
 term, being careful not to add so great an amount to x as to make this term 
 change its sign; and when we can add no more of one order (as of tenths), we 
 pass to the next lower order (hundreths) and proceed in the same manner. On 
 this process we make two remarks, viz. : 
 
 (a.) It is not sure to succeed. Thus, if there were two roots between .34 and 
 .35, for example, the absolute term would not change sign when we passed from 
 .34 to .35, although we would have passed both roots ; and it might occur that 
 no root lay beyoud .35, in which case our method would be fruitless. But such 
 cases are rare. It is in such cases, and in such only, that Sturm's method is 
 well-nigh indispensable for finding the situation of roots. 
 
 (6.) In most cases the ex{tct figure of any order can be told without such an 
 
 approximation as the above ; or, what is equivalent, without trying a figure, and 
 
 when it is found incorrect, erasing the work and trying another, and so on till 
 
 the right figure is found. In this particular case, the first figure in the root being 
 
 a small fra/'tion, the higher powers of x might be neglected (and more especially 
 
 us they differ in signs), and — 14937j + 5000 = would give the first figuro 
 
 5000 
 In the root at once. Thus x = ;, ,^.,- , =.3 +. So, in this case, for the second 
 
 14937 
 
 f (a) 696.5681 " ^^ i . , . ., . ^ ^ .,. 
 
 figure Tn~~ — fj?t;Q «Q» ~-^ "•"' ^^"icli gives the next figure of the 
 
 J Kfi) — Io7o9.o9i* 
 
 root. 
 
 3. To FIND THE NEXT GREATER ROOT. By Substituting l,we find, as on the 
 next page,/(ar) = — 8018 ; and when 1 is added to this, /(a*) = —17506. Now it 
 is evident that any slight addition, as of 2, 3, or 4, to the value of x, will only 
 make /(j;) increase negatively. This is seen by inspecting the coefficients 1, 
 —72, +1542, —7873, —17506. We therefore make a considerably larger addi- 
 tion to X, as 10. From this explanation the student should be able to see the 
 significance of the following (see next page :) 
 
HORNER S METHOD. ^4S 
 
 SECOND OPERATION. 
 
 -80 
 
 +1998 
 
 -14937 
 
 + 5000 
 
 1 
 
 1 
 
 - 79 
 
 1919 
 
 -13018 
 
 1 
 
 -79 
 
 1919 
 
 -13018 
 
 - 8018 
 
 10 
 
 1 
 
 - 78 
 
 1841 
 
 - 9488 
 
 12.7 
 
 -78 
 
 1841 
 
 -11177 
 
 -17503 
 
 
 1 
 
 .- 77 
 
 1689 
 
 1;M70 
 
 
 
 1764 
 
 
 - 403G .... 
 
 
 -77 
 
 - 9488 
 
 
 1 
 
 - 75 
 
 1615 
 
 3737.3441 
 
 
 -76 
 
 1689 
 
 - 7873 
 
 - 298.655d 
 
 
 1 
 
 - 74 
 
 9220 
 
 
 
 -75 
 
 1615 
 
 1347 
 
 
 
 1 
 
 - 73 
 
 4020 
 
 
 
 -74 
 
 1542 
 
 5367... 
 
 
 
 1 
 
 - 620 
 
 - 27.937 
 
 
 
 -73 
 
 922 
 
 5339.063 
 
 
 
 1 
 
 - 520 
 
 - 42.931 
 
 
 
 -72 
 
 402 
 
 5296.133 
 
 
 
 10 
 
 - 420 
 
 
 
 
 -62 
 
 - 18.. 
 
 
 
 
 10 
 
 - 21.91 
 
 
 
 
 ^52 
 
 - 39.91 
 
 
 
 
 10 
 
 - 21.42 
 
 
 
 
 ^42 
 
 - 61.33 
 
 
 
 
 10 
 
 - 20.93 
 
 
 
 
 -32. 
 
 - 82.20 
 
 
 
 
 -31.3 
 
 
 
 
 
 .7 
 
 
 
 
 
 -30.6 
 
 
 
 
 
 .7 
 
 
 
 
 
 ^d~A) 
 
 
 
 
 
 .7 
 
 
 
 
 
 As now f{a) and / '{a) have opposite signs, and tlie remainder of the root is 
 quite small as compared with that already found, the approximation can be 
 
 .,.,,. m, , /(«) -298.6559 ^, 
 earned on m the ordmary way. Thus we have — • = ^- —.05+, 
 
 and the next figure of the root is 5. 
 
 4. To FIND THE NEXT GREATER ROOT we resume the coeiRcients after the 
 roots had bee i diminished by 12. Then adding 1 to the value of cc, we find that 
 
244 
 
 ADVANCED COURSE IN ALGEBRA. 
 
 for .c =■ 13, /(ar) = 1282, having changed sign, as it should. Now as f\x), i. e. 
 52^9, aud/(jr) are both positive, and the other coeflBcients, though negative, are 
 comparatively small, it will take considerable increase in x to change the sign 
 of f{x). We therefore add 10. Now f\x) has changed sign, and by inspecting 
 the coefficients, 1, +12, —348, —1321, and 24872, it is evident that x cannot in- 
 crease another 10 without changing the sign of f{x). Hence we try 5. For 
 a similar reason we add 4 next. 
 
 TIIIUD OPERATION. 
 
 -82 
 
 - 18 
 
 +5367 
 
 - 4036 
 
 12 
 
 1 
 
 - 31 
 
 - 49 
 
 5318 
 
 1 
 
 -31 
 
 - 49 
 
 5318 
 
 1282* 
 
 10 
 
 1 
 
 - 30 
 
 - 79 
 
 23590 
 
 5 
 
 -30 
 
 - 79 
 
 - 29 
 
 5239* 
 
 -2880 
 
 24872t 
 -13180 
 
 4 
 
 1 
 
 32.+ 
 
 -29 
 
 -108* 
 
 2359 
 
 11692t 
 
 
 1 
 
 -180 
 
 -3680 
 
 -11588 
 
 
 -28* 
 
 -288 
 
 -13211 
 
 104§ 
 
 
 10 
 
 -80 
 
 -1315 
 
 
 
 -18 
 
 -368 
 
 -2636 
 
 
 
 10 
 
 20 
 
 - 765 
 
 
 
 - 8 
 
 -348t 
 
 -3401t 
 
 
 
 10 
 
 85 
 
 504 
 
 
 
 2 
 
 -263 
 
 -2897 
 
 
 
 10 
 
 110 
 
 1144 
 
 
 
 12t 
 
 -153 
 
 -1753§ 
 
 
 
 5 
 
 135 
 
 
 
 
 17 
 
 - 18t 
 
 
 
 
 5 
 
 144 
 
 
 ; 
 
 
 22 
 
 126 
 
 
 
 
 5 
 
 160 
 
 
 
 
 27 
 
 286 
 
 
 
 
 5 
 
 176 
 
 
 
 
 m 
 
 462§ 
 
 
 
 
 4 
 
 
 
 
 
 86 
 
 
 
 
 
 4 
 
 
 
 
 
 40 
 
 
 
 
 
 4 
 
 
 
 
 
 44 
 
 .^ 
 
 
 
 
 4 
 
 
 
 
 
 4^ 
 
 
 
 
 
 f'i^') 
 
 
 But as the 
 
 coefficients preceding — 
 
 1753 ar 
 
hornek's method. 246 
 
 all 4-, they will diminish it somewhat in the operation, and hence it is probable 
 that .06 is the proper addition to make to the root. The process can now be 
 continued to any extent desired. 
 
 5. To FIND THE NEXT GREATEST (in this casc the greatest) root, we have the 
 following operation, which we leave the student to trace: 
 
 fourth operation. 
 
 32 
 
 ^-48 
 
 +462 
 
 1 
 
 49 
 
 49 
 
 511 
 
 1 
 
 50 
 
 50 
 
 561 
 
 1 
 
 51 
 
 51 
 
 612* 
 
 1 
 
 53 
 
 52* 
 
 665 
 
 1 
 
 54 
 
 53 
 
 719 
 
 1 
 
 55 
 
 54 
 
 774. . . 
 
 1 
 
 45.44 
 
 55 
 
 819.44 
 
 1 
 
 46.08 
 
 56.8 
 
 865.52 
 
 .8 
 
 46.72 
 
 57.6 
 
 912.24 
 
 .8 
 
 
 58.4 
 
 
 .8 
 
 
 -1753 
 511 
 
 + 104 
 -1242 
 -1138* 
 16 
 
 1 
 1 
 
 -1242 
 561 
 
 34.8 + 
 
 - 681* 
 665 
 
 -1154..... 
 
 1086.8416 
 
 - 67.1584 
 
 
 - 16 
 719 
 
 
 703.... 
 655.552 
 
 
 
 1358.552 
 092.416 
 
 
 
 2050.968 
 
 59.2 
 
 The student should extend these solutions 2 or 8 figures farther. 
 
 7 to 12. Solve the following: 
 
 (7.) ^ + ^^x" - 800a; = 60000. 
 
 (8.) ar' + 2a;^ + 3a;^ + 4:^:^ + 5a; = 64321. 
 
 (9.) a;* 4- 4a;' - 4a:* - llrr + 4 = 0. 
 (10.) a^ _ 27a:' + ir)2a:« + 356a: = 1200. 
 (11.) x" - Zx' = 48654231721. 
 (12.) .a:' + 2a:' + 3a: = 13089030. 
 (13.) of - 10.r^ -f 6a: = 1. 
 (14.) .^•' + 173a: = 14760038046. 
 
246 ADVANCED COURSE IN ALGEBRA. 
 
 (15.) re* - 7035a:* + 15262754^ = 10000730880. 
 
 (16.) x' -H Ux = 35.4025. (Solve by Horner's method.) 
 
 (17.) ic^ + 4a;« - 9a; = 57.623625. 
 
 (18.) 2a;* + ba^-{- 4a:' 4- 3a; = 8002. (Observe that it is not neces- 
 sary to make the first coefficient unity. See examples in the firsu 
 part of the section.) 
 
 (19.) 3a;* - 4a;* + 2a; = 1000. 
 
 (20.) 5a;^ - 3.2a; == 41278.216. 
 
 Note. — The roots of several of the above are commensurable ; and their solu- 
 tion shows that Horner's method is adapted to such caseg. 
 
 21 to 25. Extract the roots of the following numbers by Horner's 
 metliod : 
 
 (21.) The cube root of 119736852154. 
 
 (22.) The square root of 5126485. 
 
 (23.) The fifth root of 2. 
 
 (24.) The fourth root of 35718271002567691. 
 
 (25.) The cube root of 3. 
 
 Sug'8. — To solve the 21st, write a;' — 119736852154, and solve as usual, being 
 careful to remember that the coefficients are 1,0,0, —119736852154. To find 
 the initial figure, point off as in the ordinary method of extracting roots. The 
 following exhibits the first steps of the process : 
 
 [ 49 
 
 
 
 
 
 -11973C852154 
 
 4 
 
 16 
 
 64 
 
 4 
 
 16 
 
 - 5573G 
 
 4 
 
 82 
 
 53649 
 
 8 
 
 48- 
 
 - 2087852 
 
 4 
 
 1161 
 
 
 129 
 
 5961 
 
 
 9 
 
 1242 
 
 
 138 
 
 7208 
 
 
 9 
 
 
 
 147 
 
 
 
 26 to 29. Solve the following by first eliminating, and then solving 
 the resulting equation by Horner's method: 
 
 (26.) 2.7;' — 5a; + 3y = 2a-// — 4a;* + 12, and 4?/' — 3a; = 2y + 5. 
 (27.) '2if - \xy + 2a;' - 3// - 2a;-8 ^- 0, and 4/ + 4a;-^ =r 11. 
 (28.) If - ^xy + 2a;' - 3// - 2a; =8,andi/2 x 2«/ + a;2— 6a;= — 6. 
 (29.) 2/ - ^xy + 2a;'— 3y -.2.c = 8, and ?/2^6y + .c'-4a; + 9 = 0. 
 
HORNER S RULE. 
 
 247 
 
 Sug's.— From the 2d of (26) we have y = \± ^ySx + V- Substituting this 
 in the 1st, we obtain Qx* — \^x - a/. = {x-i)^dx + \^, whence 36a;* - 69a;' 
 
 - 101a;2 + ^^x + H^=0. And dividing by 36, we have x* - 
 4- 3,687a; -|- 3.188 = 0, carrying the fractions to three places. 
 
 1.917a;=' - 2.806a;« 
 
 27 S, ScH. — There are various methods by which Honier's process may 
 be abridged, especially when a large number of decimals is required ; but 
 we have thought it better to exhibit fully and clearly the principles essentia) 
 to the process, than to spend time and distract attention by giving these 
 arithmetical abridgments. The most simple of these are : {a) the omission 
 of the decimal point ; (h) the writing of the sums only in the several working 
 columns, performing the various multiplications and additions mentally; 
 (c) after several decimals have been obtained, instead of annexing 0' ^ 
 (or ••• 's) to the working columns, d?'opping off ^giires from the right iw 
 each new operation, as one from next to the last right-hand column, two from 
 the next to the left, three from the next to the left, etc. ; {d) and, tinally, 
 when all the working columns but the last two have disappeared, continu- 
 ing the operation as a process of simple division, only dropping off a figure 
 from the right of the divisor at each step instead of annexing a to the 
 dividend. We condense an example from Todhunter as an illustration. 
 
 Ex. — To compute to 16 decimal places the root ot a? + dx^ 
 — 5 = 0, which lies between 1 and 2. 
 
 2x 
 
 
 OPERATION. 
 
 3 
 
 -3 
 
 -5 j 1.3300587305679821 
 
 4 
 
 2 
 
 -3000 
 
 5 
 
 700 
 
 -333000 
 
 CO 
 
 889 
 
 -663000000000 
 
 63 
 
 108700 
 
 -98647524875 
 
 66 
 
 110779 
 
 -8347885443 
 
 690 
 
 112867000000 
 
 -446624425 
 
 693 
 
 112870495025 
 
 -107998801 
 
 696 
 
 112873990075 
 
 -6411112 
 
 699000 
 
 11287454929 
 
 -767351 
 
 699005 
 
 11287510850 
 
 -90100 
 
 699010 
 
 1128751574 
 
 -11087 
 
 699015 
 
 1128752063 
 
 -929 
 
 
 112875208 
 
 -27 
 
 
 112875210 
 
 -4 
 
218 ADVANCED COUllSE IN ALQEBliA, 
 
 SECTION III. 
 GENERAL SOLUTION OF CUBIC AND BIQUADRATIC EQUATIONS. 
 
 CaKDAN'S SOLUTIOif OF CuBIC EQUATIONS. 
 
 ;S?7.9, Prob* — To resolve the general cubic equation x' +px' 
 + qx + r = 0. 
 
 Solution. — This solution consists of three steps: 1. To transform the equa- 
 tion into one of the form y* -f- my -\- n = Q, tliat is, an incomplete cubic lacking 
 the square of the unknown quantity. To effect this, we put x = y -\- z, and 
 substituting, have 
 
 y^ + dy^z + Syz^ + 2' + py^ + 2pyz + pz* -\- qy +'qz + r = 0, 
 ar,y'^ -\-{3z+p)y* + (^z* +2pz+q)y + z^ -j-pz"^ +qz + r = 0. (1) 
 
 Now as we have only one condition expressed between y and z, viz., y+z—x, 
 we are at liberty to impose another. Let us put 82 + 1> = 0, whence z = — ^p. 
 Tlien will this value of z substituted in (1) give 
 
 y' + (7 - ip')y + (rrP' -ipq + r) = 0. (2) 
 
 2. Since the above transformation can always be effected, a solution of 
 
 y"" + my + 11=0 (3) 
 
 will include the solution of all cubic equations. Our second step is to trans- 
 form this equation into one which can be solved as a quadratic. To do this we 
 put y = u + V, which gives (3) the form 
 
 u* 4- Su*v -f 3mo' -f- v^ -4- m{u -+- 1;) + 72, = 0, 
 or, u* + Sut{u + t) + v'-^ -h v\{u + t?) 4- 71 = 0, 
 or, u* -k-v* -\- (3mp -h mXu + t) + n = 0. (4) 
 
 Now, as we have but one condition expressed between u and v, viz., u+v=y, 
 we are at liberty to impose another. Let us put Zuv + m = (i, whence v = 
 
 — — ; and (4) becomes u^ ->t- v^ + n =z 0, 
 
 ou 
 
 or by substituting the value of t, 
 
 - m^ 
 
 » -27^ + '' = "*' 
 
 whence we have u^ -\- nu* = /rm\ (5) 
 
 3. Solving this quadratic we obtain 
 
 and as tj' = — {\l^ 4- w), v = 4/—^^ T -v/jV^*^ + 4^*. 
 
CARDAk's SOLUTION' OF CUBIC EQUATIONS. 249 
 
 Finally, taking the square root as + for the value of u, and - for the value 
 of V, since these are to -responding values, we have 
 
 y = 1^ - i^ + V-^7^' + in^ + ^ ~\n- A^-hm^ + \nK 
 
 (6) 
 
 280. Vrop. — 1. In the equation y^ + my + n = 0, when m ii 
 positive, and when m is negative and ^m^ < ^^n'^ the equation has 
 one real and two imaginary roots, and GardarCs formula (6) gives 
 a satisfactory solutloti, 
 
 2. When m is negative and ^m^ = Jn', tico of the roots are equal, 
 and Cardan^s method is satisfactory.* 
 
 3. But, when m is negative and ^m^ > Jn^ all the roots are real 
 and unequal, while Carda?i*s method makes them apparently imagi- 
 nary, and the solution, is unsatisfactory. 
 
 Dem. — A cubic equation must have at least one real root {238). Let this be 
 a. Now conceive the equation reduced to a quadratic by dividing /(.i) by x—a, 
 and let 6 + /y/ c, and b — ^^ c be the roots of this quadratic, these being the 
 general forms of the roots of a quadratic, in which if c is + the roots are real, 
 if c is — they are imaginary, and if c is these two roots are equal. 
 
 Now, a, b + ^Z c, and b — \/ c being the roots of the equation, we have 
 by (235) 
 
 (x-a) (x-[b+'^c]) {x-[b-^c])=x^-{a+2b)x^+{2ab+b^-c)x-a{b^-c)=0. 
 
 To transform this into the form y^ + my + n = 0, we must put a + 2b = 0; 
 whence a = —2b, and we have 
 
 y-« - (3^2 + c)y + 2b{b^ - c) = 0. 
 
 Comparing this with Cardan's formula, we see that 
 
 Hence we see that if c is +, that is, if all the roots of a cubic etiuation aro 
 real and unequal, Cardan's method gives a result apparently imaginary. But if c is 
 — , that is, if two of the roots are imaginary, Cardan's method gives s^real form. 
 Also when c = 0, that is, when the roots are a, 6, and 6, the form is real, since 
 
 Now by inspecting the quantity -y/ -.lim' + W we see that it is real when 
 m is positive ; and also when m is negative if ^m^ < {n"^. Hence in these 
 
 » If flW the roots aro equal, the equation talcos the form {x - a)3 =x^ - 3a.r;2 ^ Za"'X - a^ = 0, 
 a being the value of one of the equal iO'>tB C-iSn). In Miis case the transformation which makes 
 the term m z"^ disappear gives y' = 0, since a; = y - ip = y + a, and y = x - a = 0. 
 
25© Anxk^cKV coyr^SE in ^lqecra. 
 
 cases there are one real &ix(\ two imaginary roots, and Cardan's inethod,-giving 
 a real form, enables us to determine one of them, and hence to solve the 
 equation. 
 
 2d. We have also seen above that when c = 0, that is, when two of the roots 
 
 are equal (and not all three), 'y/Aw* + \n* = 0, in which case m must be nega- 
 tive and -^rm^ = in*. 
 
 3d It has also appeared above that when all the roots are recU and unequal, Car- 
 dan's method gives an apparently imaginary result. But this can only be the case 
 when rfi is negative, and -^Sm^ > \n*. 
 
 28 1, ScH. — Cardan's method would seem to give a cubic equation nine 
 roots instead of three, since as there are three cube roots of any number, 
 
 a/ — in+ \/-A»»"* + i"-* would liave three values, and 4/-7W- -y/ A"*^ + \n* 
 
 would have three other values. Now combining each of the former, in 
 turn, with each of the latter, we should have vitte results. In order to ex- 
 plain this seeming paradox, let us find the form of the three cube roots of a 
 number, as of a'. To do this we have but to solve the equation x'^ =a^. 
 Thus x^ —a^ = {x — a){x*+ax-{- a*) = 0. Whence x — a = 0, and x'+ax+a* 
 =0. From these we have x = a, —^aCi + 's/— 3), and — ^a (1 — /y/ — 8). 
 Now let the roots of a/ ^\n-\- ^J^-m* + \n^ be r, — ir(l-f \/— 3), and 
 
 — ir(l-V^^); and the roots of k/ -\n - V^Vw* + \n^ be r' , -\r' 
 
 (1 + -y/ — 3), and — \r' (1 — y'--3). It will be remembered that we assumed 
 
 «« = — —; that is, the products of the admissible roots must be real. 
 o 
 
 Therefore we can use for the parts of the root r and ?•', — \r{\ + >y/ — 8) and 
 
 — \t' iX — 'v/ — 3), and — \r{\ — ^ — \) and — ir'(l 4- -y/ — 8); and we can 
 
 use these parts in no other combination, as any other would not give a real 
 
 quantity. Thus we cannot have y —n -k- t — r -\r {\ 4- y' — 3), since wo 
 
 would then be — r[|r(l -f- y^ — 3)], which is an imaginary quantity, and hence 
 
 fix 
 
 not equal to — — , as it should be. 
 
 o 
 
 We will give a few examples to which the student may apply Cardan's pro 
 
 Examples. 
 
 Solve the following, finding one of the roots by Cardan's process, 
 and then depressing the equation by division, solve the resulting 
 quadratic. 
 
DESCARTES* S SOLUTION OF BIQUADRATICS. 25X 
 
 1. a^ - 9x + 28 = 0.* 
 
 2. a:f^ — 3x^ + 4: = 0. (See first step in general solution.) 
 
 3. a^ — Gx + 4: = 0. 
 4:, x" + 6x~2 = 0. 
 
 6. X + b + 3 \/^ = a. 
 Q. x" + 3x' + dx - 13 = 0. 
 Z a^-9x' + 6x-2 = 0. 
 
 8. x' — 6x' + 13a; - 10 = 0. 
 
 9. x^ - 48a; = 128. 
 
 10. x' + 2z = 12. 
 
 11. z' -3:^- 2z' - 8 = O.f 
 
 12. 
 13. 
 
 2:^ 
 
 7? - 
 
 - Qy' + 13?/ = 
 
 - 12a:^ + 36a: 
 
 12. 
 = 44. 
 
 
 11 
 
 1 + X ^ a + X 
 
 ^/x 
 
 15. 
 
 a '^ X 
 - '^x' + \^X ~ 
 
 12 = 
 
 c 
 
 : 0. 
 
 SuG. — An attempt to solve the last by Cardan's process will give roots 
 apparently imaginary, although it is easy to see that the roots are all real, and 
 commensurable. 
 
 Descartes's Solutioj^ of Biquadratics. 
 
 282, Prob. — To resolve the general biquadratic equation x* + 
 
 ax' + bx'' + dx + e = 0. 
 
 Solution. — The first step in the process is to transform the equation into one 
 wanting the 3d power of the unknown quantity. This is done in the usual way 
 (see Cardan's method of resolving cubics) ; i. e., by putting x=y -{-z, substituting, 
 collecting the coefficients with reference to y, and, putting the coefficient of y ' 
 equal to 0, finding the value of z. This value of z substituted in the given 
 equation will give the form 
 
 y* + my"^ + iiy + r = 0. 
 2. Assume y* + my^ + ny + r = (y^ + cy + /) (y^ + ey + g), and deter- 
 
 * It is bttlcr for the ptudent to use Cardan's processXhan to substitute in the formula. Tliii« 
 
 for JC» - Ox + 28 = 0, we have, by putting a = y + s, y' + 8^ + (%2 - 9) (2 + y) + 2S -.r o ; 
 
 3 27 3 
 
 and making; Zyz - 9 = 0, or 2 =-, i/S + -? + 28=0. Whence y= -1, and -3, and 2 = - = -3, and 
 y y* y 
 
 - 1. .'. X = y + 2 = - 4. Then {x^ - 9x + 28) ■+■ (a:f 4) .- a;^ - 4a; + 7 = ; whence a-=\!± |/^. 
 
 t An equation of the form x'^"^ + ax'i"^ -\ bx'» +c = can be reduced to a cubic of the form 
 y»+my+n=0, by putting x»t=y- ^<z. 
 
252 ADVANCED COURSE IN ALGEBIIA. 
 
 mine the quantities r, c,f, and g, ^o that they will fulfill the required conditions 
 Thu."^, expanding we have 
 
 y^ + my^ + ny + r = y*+ c y^ + f y^ + ef y + fg ; 
 
 + c 
 
 y'+f 
 
 r+ef 
 
 + e 
 
 + CC 
 
 + /7 
 
 + rg 
 
 whence, as the members are identical, 
 
 c + e = 0, f + ce + g = m, ef + eg = n, and fg = r. 
 From the first we see that c = — e. Substituting this value, we have 
 
 (1) f-e* + g = m; (2) e(f-g)=n; and (3) fg = r. 
 
 From (1) and (2) wt- have g = Ue^ — ~ + m\ and /= i^e' + - -f- mj; 
 
 which substituted in (3) give (e^ + - + m) (e^ _ ?- + mj = c* + 2me" 
 
 n* 
 ; + wi* = 4?-, or 
 
 6* 
 
 e^ + 2me* + {in' - Ar)e^ - n^ = 0. (4) 
 
 Now (4) can be reduced to a cubic in terms of c, by putting «'= ^, — Jm (see 
 foot-note on preceding page). Tliis cubic equation will have at least one real 
 mot {2'i8), and this will give real values to e,,and hence to «, r,/, and g. 
 ^VlK■refo^e, */ Cardan'8 method gives a practical solution of (4), we can resolve the 
 biquadratic. 
 
 283, Scir. — It will be observed that this resolution of a biquadratic in- 
 volves the resolution of a cubic, and hence is subject to tlie difficulty attend- 
 ing the irreducible case of cubics. We will give a sin'_,d'.; example, to wliich 
 the student can apply the process of Descartes. 
 
 Ex.— Find by Descartes's method the roots of x'— lOx"— 20a; — 16 
 = 0. 
 
 Recurring Equations. 
 
 284:» A jRecurring Equation is an equation such tliat the 
 coefficients equidistant from the first and last are numerically equal, 
 when the equation is in the complete form JaT-f Bx''~'^-\- Cx""'^ . - . - 
 iy = ; and the signs of the corresponding terms are either all alike, 
 or all unlike ; i, c, the coefficients of the first half recur in an inverse 
 order in the second half of the function. 
 
 III. 12a;' -I- Zx*^ — hx^ — 5^* + 3j; + 12 = is a recurring equation. 
 Ax^ + Bx^-^ + 0"-5 .... Cx' + Pj; -h ^ — is the type of such equations. 
 
 28S. Prop, 1, — The roots of (i recurring equation are recipro- 
 
 cols of each other ; i. e., ^/' a is a rooty - is also ^ and so of en ch 
 
 a 
 
 of the roots. 
 
RECURRING EQUATIONS. 2^53 
 
 Dem. — If a satisfiea the equation 
 
 Ax" + i?ar»-» + Oc"-' ... - Cx^ + Bx + A = 0, 
 ~ will also satisfy it, for the former when substituted gives 
 
 Aa^ 4- \Sa~-i + (7a— « - - • - Ca^ ^ Ba + Az^d-, 
 and the latter gives 
 
 A B ^ G OB, 
 
 which, by multiplying by «» becomes 
 
 A + Ba + Ca^ . - - - Ca""-^ + ^a"-' + ^a« = 0, 
 a result identical with that obtained when a is substituted. 
 
 280, ScH. — From this relation among the roots of recurring equations, 
 they are often called Recijwocal Equations. 
 
 287, Cor. 1. — If the degree of the equation is odd the correspond- 
 ing coefficients may all have like, or all unlike, signs ; hut, if the 
 degree is EVEX they rnttst have like signs sinless the middle term is 
 wanting, in which case they may have unlike sig7is, and the roots 
 still be recijyrocal. 
 
 That the signs may be unlike in the cases specified is evident since, if in such 
 
 cases a is a root, and we substitute - instead of a, clear of fractions, and change 
 
 all the signs, we shall have the same result as if a had been substituted. Thus, 
 
 if substituting rt gives ^a" + 5a'*—C'a^ + (7(r«-—J?«—yI=0, substituting - will 
 
 . A B C C B , ^ , , . ,^. , /* . 
 
 give -r+ — : 3 + — 7 A = (i: whence clearing of fractions and changing 
 
 all the figns we have — A — Ba + Ca- — Ca^ + Ba* + Aa^ = 0, a result iden- 
 tical with the former. The fact concerning the equation of an even degree ia 
 shown in a similar manner. Notice that all the corresponding coeiiicients must 
 have like signs or nil unlike signs. 
 
 2S8, Cor. 2. — A recurring equation may always he reduced to a 
 form having the coefficient of the highest poioer of the unknoicn 
 quantity, and the absolute term, each 1, since by definition these are 
 nnmerically equal. 
 
 28f)» Prop, 2. — A recurring equation of an odd degree has one 
 of its roots —1 if the signs of the corresponding terms are alike, and 
 + 1 if they are unlike. 
 
 Dem.— Having a;«±yl.c«-i ± Bx''-^± Gc"-' . . . . ± Cx'^ ± Bx- ±^^±1=0,* 
 
 * The ?ij?n of x* can always be made +. The ambiguous signs are to bo taken ■♦■ or — , aO" 
 cording to the hypo'heiia. 
 
2^ ADVANCED COURSE IN ALGEBRA. 
 
 taking the signs of the corresponding terms alike we can write 
 
 (a^ + 1) ± Ax{x*-^ + 1) ± Bx^x"-^ + 1) ± Cx'^ix*-^ + 1) + etc. = 0, 
 which is divisible by a; + 1 (Part I., 119), wherefore — 1 is a root {231). 
 Taking the signs of the corresponding terms unlike, we can write 
 
 (if - 1) ± Ax(x^-^ - 1) ± Bx'ix'^-* - 1) ± Qc^ix'^-^ - 1) + etc. = 0. 
 which is divisible by a; — 1 (Part I., 119), wherefore + 1 is a root {231). 
 
 290, I*r02^» 3, — A recurring equation of an even degree^ xohose 
 corresponding terms have opposite signs, has o?ie root + 1, a?id one 
 root —1. 
 
 Dem.— Having a'" ± Ax^"-^ ± J5a**"-' ± Cx^*-^ . . . . zf Cx'3 T Bx* T Ax 
 — 1=0, taking the signs of the corresponding terms unlike, and remembering 
 that the middle term, which would have no corresponding term, is wanting 
 {287), we can write 
 
 (a:^" - 1) ± Ax (a-'— » - 1) ± jBa;«(^"— ' - 1) ± Cx\x^''-^ - 1) + etc. = 0, 
 which is divisible by a:* — 1 (Part I., 119); wherefore a;* — 1 = 0, and 
 a- = + 1 and — 1. 
 
 29 1, ^rop, 4. — A recurring equation of an even degree above 
 the serondy mag be reduced to an equation of half that degree, when 
 the signs of the corresponding terms are alike. 
 
 Dem.— Having a^" J: ^a-»—'±i?a^"-«± Ca:^—5 .... ^j^^* . . . ±Ca?±Bx^±Ax 
 + 1=0, taking the signs of the corresponding terms alike, we can write 
 
 (a-*" + 1) ± ^(a;»— ' + x) ± B{x**-^ + x*) ± C{x^*-^ + x^) + etc. = ; 
 whence, dividing by a?*, we have 
 
 - - - - l(x + ^ ±M=0. 
 
 Now putting a; + - = y, we can write Ix + -) = a;' + 2 4- -j = y', 
 
 1 /1\^ 111 1 
 
 whence x* -\ r = y'— 2. [x + -) =a;'+3a;'- + dx -^ + —.. = x^ -\ 
 
 x* \ x/ XX* a;"* x* 
 
 + six +-) = y\ whence a;' + — ^ = y' — 3y. 
 
 (x* -I- ^2)'= «* + 2 + - = (y« - 2)S whence x' + ^^ = {y^ - 2)» - 2. 
 
 ( a; + - ) = a;« + 5.c* - + lOa;-* -, + 10a;* -. + Sa- — + - = a;* + -. 
 \ x/ XX* a;* a;* a;" a;^ 
 
 + 5 (a;^ + -J j + 10 I X + - j = y', whence x^ + -^ = y' — 5(y' — 3y) — lOy. 
 
BINOMIAL EQUATIONS. — ROOTS OF UNITY. 25^ 
 
 (^' ^ ^0 = ^' + ^ + ^6 = (y^- W, whence ^' + ^, = {y'- 33^)^-2. 
 Whence we see that any term of the fonn a^+ — may be expreesed in terms 
 
 of p, and will involve no higher power than y". Therefore the original equa- 
 tion, which is of the 2?2th degree, can by this substitution be transformed into 
 an equation in y, of the 7ith degree. 
 
 Examples. 
 
 Solve the following recurring equations by applying the foregoing 
 principles : 
 
 1. a^-5af+ 6x' — 5x +1 = 0. 
 
 2. ar^ - lla;^ + ITr' + 17x' - ll:c + 1 = 0. 
 
 3. 6x' - 11a;* - S^x" + 33a;' + 11a: - 6 = 0. 
 
 4. 1 +x^=a(l +xY, 
 
 5. x' -2x^ -^ x' + x"- 2x' +1 = 0. 
 
 6. Sx^ - 16.7.^ - 1h7? - I62:' +8 = 0. 
 
 7. ^x' - 24:0^ + 57a;* - 732;' + 57a;' - 24a; + 4 = 0. 
 
 8. X* + 4«a.'' - 19aV + 4«'a; + a* = 0. 
 
 9. ar* + a;3 + a;' + a; + 1 = 0. 
 10. 1 + a;* = i(l + xy. 
 
 Binomial Equations and the Roots of Unity. 
 
 292. A Binomial Equation is one of the form x" d= « = o. 
 Such equations may be considered as recurring equations and solved 
 accordingly. 
 
 III. — Having ic* ± a = 0, put x^ = ay" ; whence ay^ ± a = 0, or y" ±1=0, 
 which is recurring. 
 
 Examples. 
 
 1. ar' =t 5 = 0. 
 
 3. a;» ± 2 = 
 
 5. a;''± 11 = 0. 
 
 2. a;* d= 3 = 0. 
 
 4. a;« ± 7 = 0. 
 
 e. x' ± 1 = 0. 
 
 7. What are the two square roots of 1? The three cube roots of 
 1 ? The four fourth-roots of 1 ? The five fifth-roots of 1 ? The 
 six sixth-roots of 1 ? 
 
 SuG. — The solution of these questions consists in resolving a!^ — 1 = 0, 
 j;3 — 1 = 0, a;* — 1 = 0, etc. The five fifth-roots of 1 are 
 
 1, ^(i^5-l ±V- 10-2 1^5), and -i(t'5 + l ±V-10 +2 V5). 
 
 293, Sen. — It will be observed that the form x^ ±1=0 is omitted 
 above. Now x^ — 1 = has one root 1. The equation can therefore Ixj 
 
256 ADVANCED COURSR IN ALGEBRA. 
 
 depressed to a recurring equation of the 6th degree, having all its signs + . 
 This can be reduced to a cubic by {291). a;' + 1 = has one root rf- = — 1, 
 and can be reduced to a recurring equation of the Cth degree having its 
 signs alternately + and — . This can be resolved into one of theS'rd degree 
 by {291). Hence the complete resolution of a:^ ± 1 = depends on the 
 resolution of a cubic. 
 
 ^9 ± a = can be resolved by putting x^ = p, whence we have y^±a= 0. 
 Solving this for y we have 3 roots. Call them a,, Of, a^. Hence to com- 
 plete the solution we have to resolve the three cubics x^±ai=0, x* ±a2=0, 
 
 X^ ± «:, = 0. 
 
 Exponential Equations. 
 
 294, ^Exponential Equations are equations in which the 
 unknown quantity or quantities are involved in tlie exponents. 
 
 1 
 III. a^ + by- e, a' = d, 2' = 43, 3*' = 2, y = 256, «* = 100, and 
 xy — y = in are exponential equations. 
 
 2f)o» JProb* 1, — To soloe an exponential equation of the form 
 
 a' — ni. 
 
 Solution. — Taking the logarithms of both members we have x log a — log m 
 {180 f 181) ; whence x = ^ — . Therefore finding the logarithms of m and 
 a from a table of logarithms, and dividing the former by the latter, we find x. 
 
 200, JPvob, 2, — To solve an exponential eqx(,ation of the forin 
 X* = m. 
 
 Solution.— Taking the logarithms of both members we have x log « = log w. 
 Then find log m from the table, and determine x by inspection from the table so 
 that X X log X shall equal log m exactly or approximately.* 
 
 Examples. 
 1. Find the value of x in the equation 3' = 2546. 
 80LUTI0K. .log8 = log2546. .. = ^-^^i? . ?^f = 7.138 ^ . 
 
 2 to 6. Solve the following: (24)''=18r42; 2'=2673; (11)''=2681; 
 2^=10; 5'=:1; (12)" = !. 
 
 7. Find the value of x in the equation r" = 3561. 
 
 ♦ The meth(»d of solving such equations by Double Position is entirely useless, since a table 
 of lo;:arithniB ia necessary for that method, and having such a table at hand, the approximations 
 can be made to any extent likely to be desired, more readily by simple inspccion than by com- 
 l)nting the errors by Double Position. Moreover, the method here given affords a:i excellent 
 exercise in the use of the table?. 
 
 \ 
 
EXPONENTIAL EQUATIONS. 257 
 
 Solution.— We have x log x = log 3561 = 3.551572. . Now looking in a table 
 of logarithms, we soon see that x must be near 5, since 5 log 5 = 5 x .698970 
 = 3.494850. Thus Ave see that a- >5. Trying 5.1 we have 5.1 log 5.1=3.608607, 
 .-. X < 5.1. Therefore we try 5.05. 5.05 log 5.05 = 3.55161955, which coincides so 
 nearly with the required value of x log x, that undoubtedly the lOOths figure is 
 4. Again, for a nearer approximation try 5.049, as the value of x is very near 
 5.05. 5.049 log 5.049 = 3.550482. Hence we see that x = 5.049 + . 
 
 8 to 15. Solve the following as above : af =100; of = 7 ; of = 21; 
 
 ar^ - 402f = 200; 3^^ + 3^ = 100; a^ -.% = 2b; a'^-' = c; tr^b'^' 
 
 16 to 21. Solve the following: x^ = f, and ^=y^; x" = if, and 
 af^zy"; m'-' = n, 2i\\ii x + ij = q; 2^ 3" =: 500, and 2x = dy', 
 h'"-' = 256 ; (a' - 2a'¥ + h'y-' = {a - b)'' {a + b)-\ 
 
 VIZ 
 
 22. Given the fundamental formiilse of Geometrical Progression, 
 
 7 
 
 I = ar''-\ and 8 = t^:^^ , to find the following : 
 
 n = ^Qg ^ -• ^Q? ^ ^ 1 . ^ ^ log [a -]- (r - 1)S] - log a ^ 
 
 log /• ' log r ^ 
 
 '*-log(^^-«)-log(.V-0 ^^'^""^ ^- -]^ -^1- 
 
 23. Given the two fundamental formulas of Compound Interest, 
 viz., a = p{l + r )',* and i = a — p, to find the following : 
 
 _ log(;j + t)-log/? . _ log ^ -log J?? ^ 
 
 ^ - l^fiTTf) ' ^ - log (1 + r) ' ^^^ (^ + ^) 
 
 ^ log(;7 + 0- log;7, ]^g (1 ^ ^) ^ log ^ - log p , 
 
 ? t 
 
 _ log »! - log {a - i) . _ log « - log {a ~ i) 
 
 '- log (1 + r) ' log(l + »)- ^ . 
 
 Note. — Many problems in Compound Interest, Annuities, and kindred sub- 
 jects are most expeditiously solved by means of logarithms. The student who 
 has not a table of logarithms at hand may either omit the following examples 
 in this section, or content himself with selecting the proper formula and telling 
 how it is applied to the solution of the particular example. 
 
 24. What is the amount of $100 at Hfo annual compound interest 
 
 * This formula is obtained thus : letting r represent the rate for time 1, expressed decimally, 
 i. e., if the rate is 7 per ct., r=.OT, or — - , we have for time 1 (as 1 year), a=:p+pr=p{i-\-r) ; 
 
 fortimeS, a=;>(l + r)+7?r(t + r)=i)(l+r)z ; for time 3, a-p(l jr)2 +pril '^r)2=:p{l-\-ryi \ there- 
 fore for time <, a=p(l+r)'. 
 
258 ADVANCED COUHSE IN ALGEBIIA. 
 
 for 10 years? What if the interest is compounded semi-annually? 
 What if quarterly ? What in each case if the rate is 10^ ? If 6^ ? 
 If 3^ ? 
 
 Sug's, — We have a =p{l + rY, whence log a = log p + t log (1 + r) = log 
 100 + 20 log 1,035, for interest at 7% compounded semi-annually. 
 
 25. In what time will a sum of money double itself at 10^ com- 
 pounded semi-annually? At 7^ compounded annually? In what 
 time triple ? Quadruple ? 
 
 SUG. a=z2p=p{l +ry, whence 2 = (1 -H r)', and ^ = —i^— . 
 
 log(l+r) 
 
 26. In what time will 110 amount to $100 at 8^ compounded 
 annually ? 
 
 '27. What is the present worth of $2000 due 3 years hence, without 
 interest, if money is worth 10^ compound interest ? 
 
 SuG. — The present worth is a sum which, put at compound interest at 10^, 
 will amount to $2000 in 3 years. Hence 2000 = p (1.1) *, p standing for present 
 worth. Whence log p = log 2000 — 3 log (1.1). 
 
 28. A soldiers pension of 1350 per annum is 5 years in arrears. 
 Allowing o^ compound interest, what is now due him ? 
 
 Sug's. — The 5th, or last year's unpaid pension has no interest on it, as it is 
 just due. The 4th, or next to the last, has 1 year's interest due, and hence 
 amounts to 350 (1.05) . The 3d year's pension has 2 years' interest due, and hence 
 amounts to 350 (1.05)*. Thus the total is found to be 350 +350 (1.05) + 350 (1.05)- 
 + 350(1.05)' + 350(1.05)\ or 350 { 1 + (1.05) + (1.05)^ + (1.05) ' + (1.05)* \ 
 
 29. Letting S represent the amount of an annuity a, in arrears 
 for t years, compound interest being allowed, at r^, show that 
 
 r 
 
 30. What is the present worth of an annuity of $200 for 7 years, 
 money being worth 5^ compound interest ? 
 
 SuG. — Evidently, a sum which, put at 5% compound interest, will amount to 
 the same sum in 7 years, as the annuity will. 
 
 31. Letting P be the present worth of an annuity «, for time /, at 
 r^ compound interest, show that P= -» ^— - — --^f — . Also, that if 
 
 the annuity is perpetual (runs forever), P = -. 
 
EXPONENTIAL EQUATIONS. 259 
 
 ScG.-men * = oc , P =: - . -^.^^ = - . ^j^ = -. a* it evident!,. 
 
 should, since such an annuity is worth a present sum which will yield aii 
 annual interest equal to the annuity. 
 
 32. What is the present worth of a perpetual annuity of $350, 
 money being worth Ty^^ compound interest ? If money is worth 
 10^ compound interest ? 
 
 33. What is the present worth of an annual pension of $125, 
 which commences 3 years hence * (first payment to be made 4 years 
 hence), and runs 10 years, money being worth 10^ compound 
 interest.^ 
 
 SuG. — Evidently, the difference between the present worth of such a pension 
 for 13 years, and for 3 years. 
 
 34. An annuity a, which commences T years hence, and runs / 
 years at r^ compound interest, gives 
 
 ^ a j (l+rr--l _(l_+_r)^l ) « j .^ . ._._ a + ,)-<..o I 
 
 AVhen the annuity is perpetual after the time T, we have 
 P = ^ (1 + r)- ''. Student give proof 
 
 35. Two sons are left, one with the immediate possession of an 
 estate worth $12000, and the other with a perpetual annuity of 1800 
 in reversion after 7 years: money being worth 5^ compound in- 
 terest, which has the more valuable inheritance, and how mucli ? 
 
 3G. What annual payment will meet principal and interest of a 
 debt of $2000 at 8^ compound interest in 5 years? 
 
 Sug's. — The amount of $3000 at 8^ compound interest for 5 years = the 
 amount of the annuity a for the same rate and time. 
 
 37. Show that if Z) is a debt at compound interest at rfo, h an 
 annual payment, and i the number of years required to liquidate 
 
 thedebt,^^ ^^g^-\^fi^-^^) . 
 log(l + r) 
 
 38. The debt of a certain State is $20,000,000, bearing annual 
 interest at 4^^. A sinking fund of $2,000,000 annually is set apart 
 to meet it. How long will it require to extinguish the debt ? How 
 long if instead of paying the $2,000,000 annually on the debt, it is 
 invested at 6^ compound interest? 
 
 * An annuity which commences after some epecificd time is said to be in reversion. 
 
260 ADVANCED COURSE IN ALGEBRA. 
 
 39. A fanner lius paid !&10 per annum for newspapers, whicli he 
 considers liave increased his net annual income at least ^. For 10 
 years during which his net income has been $500 annually, money 
 lias been worth lOj^ compound interest. What is the total net gain 
 to be credited to his investment in neAvspapers? 
 
 40. A boy commenced smoking when 15 years old. For the first 
 5 years he smoked 2 5-cent cigars each day. For tlie next 20 years, 
 3 10-cent cigars per day. Now had he abstained from smoking and 
 invested at the end of each six months the amount thus saved, at 10^^ 
 annual compound interest, how much would he have accumulated from 
 this source at the age of 40 ? 
 
 41. A man pays a premium of 45104 per annum on a life policy of 
 *4200 for 20 years before his death. Money being worth 10^ com- 
 })ound interest, does the insurance company gain or lose, and how 
 much ? 
 
 CHAPTEK IV. 
 
 DISCUSSION, OR INTERPRETATION, OF EQUATIONS. 
 
 207, To DisciittSf or Interpret , an Equation or 
 an Algebraic Exiyression, is to determine its significance for 
 the various values, absolute or relative, which may be attributed to 
 the quantities entering into it, with special reference to noting any 
 changes of values which give changes in the general significance. 
 
 Such discussions may be divided into two classes : 1st. The dis- 
 cussion of equations or expressions with reference to their constants ; 
 and 2d. The discussion of equations or expressions Avith reference to 
 their variables. 
 
 The following principles are of constant use in such discussions : * 
 
 208, JProp, — A fraction^ when comjyared with a finite quantity^ 
 becomes : 
 
 * Those principles, and in fact most of this chapter, have been considered previously, but 
 »re collected here for review and connected study. 
 
INTERPRETATION OF EQUATIONS. 261 
 
 1. Equal to 0, lohen its numerator is and its denominator finite, 
 and when its numerator is finite and its denominator oo. 
 
 2. Equal to od , when its numerator is finite and its denominator 0, 
 and when its numerator is go and its denominator finite. 
 
 3. It assumes an indeterminate form when numerator and dejunn- 
 inator are both 0, and when they are both co .* 
 
 Dem, — These facts appear when we consider that the value of a fraction de- 
 pends upon the relative magnitudes of numerator and denominator. 
 
 1. Let a be any constant and x a variable, then the fraction - diminishes as 
 
 a 
 
 X diminishes, and becomes when x is 0. Again, the fraction - diminishes as 
 
 X 
 X increases, and when x becomes oo , i. c, greater than any assignable magni- 
 
 a 
 tude, — becomes less than any assignable magnitude or infinitesimal, and is to 
 
 X 
 
 be regarded as in comparison with finite quantities. (See 14:2 and 151^ Dem., 
 and foot-note.) 
 
 3. As X increases, the fraction - increases, and hence when x becomes infinite 
 
 a 
 
 the value of the fraction is infinite. Also as x diminishes the value of - in- 
 
 x 
 
 creases ; hence when x becomes infinitely small, or 0, the value of the fraction 
 exceeds any assignable limits, and is therefore oo . 
 
 X 
 
 3. Finallv, if x and y are variables, - diminishes as x diminishes, and increases 
 
 y 
 
 as y diminishes. What then does it become when x = 0, and y = ? i. e., what is 
 
 
 
 the value of - ? Simple arithmetic would lead us to suppose that - was abso- 
 lutely indeterminate, i. e., that it might have any value whatever assigned to it, 
 
 
 
 for - = 5, since = 5x0 = 0; - —7, since = 7 x = 0, etc. But a closer 
 
 
 
 
 inspection will enable us to see that the symbol - is not necessarily indetermi- 
 nate, or rather that the expression which takes this form |or particular values of 
 its components, has not necessarily an indefinite number of values for these 
 
 X 
 
 values of its components. Thus, what the value of — will be when x and y each 
 
 y 
 
 diminish to will evidently depend upon the relative values of x and y at 
 first, and which diminishes the faster. Suppose, for example, that y — Tix; 
 
 X X 
 
 then - = — . Now, suppose x to diminish ; the denominator will diminish 5 
 y 5x 
 
 * By this is meant that ;: and — may have a variety of values, not that they necessarily 
 00 
 
 do have. 
 
262 
 
 ADVANCED COURSE IN ALGEBRA. 
 
 X 
 or - = - 
 
 y 
 
 7x 7' 
 
 or 
 
 times as fast as the liuiitterAtor, Arid whatever the value of x, the value of the 
 
 fraction will be i. So if y = Tar, - = — , which is \ for any value of x. Hence 
 
 y 7^ 
 
 x X \ 
 when .T = 0, and y = 0, we have - = - = — = - 
 
 y 5^ 5 
 X a ... 
 
 - = - = any other value depending upon the relative values of x and y. So, 
 
 a; 00 iC 00 a; 
 
 also, if a; = 00 , and y = oo , - = — ; but if y = Qx, we have - = — = — 
 
 y <» y 00 6a: 
 
 1 a; 00 a; 1 
 
 = - . And so if y = 10^, we have - = — = -— - = — . Thus we see that the 
 6 y 00 l(te 10 
 
 mere fact that numerator and denominator become 0, or become oo , does not de- 
 termine the value of the fraction, i. e., gives it an indeterminate form, 
 
 299. A Meal dumber or Quantity is one which may be 
 conceived as lying somewhere in the series of numbers or quantities 
 between — oo and + oo inclusive. 
 
 III. — Thus, if we conceive a series of numbers varying both ways from 0, i.e. 
 positively and negatively to oo , we have 
 
 -4,-3,-2,-1, 0, +1, +2, +3, + 4, 
 
 — 00 - - - 
 
 + 00. 
 
 Now a real number is one which may be conceived as situated somewhere 
 within these limits; it maybe +, — , integral, fractional, commensurable, or 
 incommensurable. Thus + 15624 and — 15624 will evidently be found in this 
 series. + ^i- may be conceived as somewhere between + 5 and + 6, though iti 
 exact locality could not be fixed by the arithmetical conception of discontinuous 
 number. So, also, — ^3^ is somewhere between — 5 and — 6. Again 4- Vs is 
 somewhere between 4- 2 and + 3, though, &s above, we cannot locate it exactly 
 by the arithmetical conception. 
 
 The following Geometrical Illustration is more complete than the arithmetical. 
 Thus let two indefinite lines, as CD and AB, intersect (cross) each other, as at 0, 
 Now let parallel, equidistant lines be drawn between them. Call the one at a 
 
 """-^ f' 
 
 t-»-T -•-»-•» -Is -1 
 
 »--^ 
 
 + 1, that at 6 will be +2, at <; +3, etc. So, also, the line at a' being —1, that at 
 b' will be —2, at c' —3, etc. Xow conceive one of these lines to start from an 
 infinite distance at the left and move toward the right. When at an infinite 
 
INTERrEETATION OF EQUATIONS. 203 
 
 distance to the left of its value would be — go , and in passing to it would 
 pass through all possible negative values. In passing it becomes at O, 
 changes sign to + as it passes, and moving on to infinity to the right, passes 
 through all possible positive values. Hence we see how all real values are em- 
 braced between — oo and + oo inclusive * 
 
 300. An Imaginary Wuniher or Quantity is one 
 
 which cannot be conceived as lying- anywhere between the limits of 
 — GO and + 00 , as explained above. The algebraic form of such a 
 quantity is an expression involving an even root of a negative quan- 
 tity.f (See Part L, 218.) 
 
 Examples. 
 
 1. What are the values of x and y in the expressions x = — ^^^ , 
 
 a — a 
 aV — a'h , , ,, , , , 
 
 y — 3~"r- i when b = o and a and a are unequal ? When h^V 
 
 and a — a!'i When a — a and I and b' are unequal ? What are the 
 8\(jns of X and ?/ when b>h' and a > a!, the essential signs of «, a\ 
 h, and b' being + ? When h> b' and a <a'? If a' and b are essen- 
 tially negative, and a = a', and b = b', what are the values of x and 
 y ? If rt' and b' are each ? 
 
 f 
 
 2. Wliat qeneral relation between a and a' renders ; = ? 
 
 •^ 1+ aa 
 
 What renders it oo r 
 
 SoucTiON. — To render z ,= 0, we must have a' — a = Q, and 1 + aa' 
 
 1 + aa 
 
 finite or infinite ; or else we must have 1 + a«' = oo , while «' — « is finite or 
 
 (208). Now rtt'— rt = gives «' = a ; whence ; = :: — which is for 
 
 * \ + aa' \-^a^' 
 
 any value of a finite or infinite. Hence the relation a' = a fulfills the first re- 
 quirement. Let us now see if l-|-a«'=oo will also fulfill this requirement. This 
 gives aa' = oo , since subtracting 1 from oo would not make it other than oo . 
 
 Thus we have a' = — . Hence for all finite values of a (including 0) a' is oo , 
 a 
 
 * For example, the Ptndcnt who is acquainted with the elements of geometry knows how to 
 construct a line which ia exactly equal to >/5 (Geom., Part I., 110). This line he can locate 
 between +2 and + 3, and also between - 2 and - 3, since y/b is both + and -. 
 
 t Tran?cendental functions afford other forms of imaginary expressions ; for example, 
 gin~^ 2, i«cc~' }4, log (-130), log (-m), etc. But our limits forbid the consideration of the iii- 
 tcrprctarioi: of imaginaric!*, except in the most restricted sense, as indicating incompatibility 
 with the arithmetical sen^c of the problem. 
 
264 APVANCED COURSE IN ALGEBRA. 
 
 riid . = — ; = -, wliich can only be when a = oo. Therefore the 
 
 1 + aa aa a 
 
 particulnr values a = co = a = cc , render = ; but no genend values 
 
 do. 
 
 a! — (i 
 Again, in order that j = cc , wo must have 1 -+- aa' = 0, and a' — a 
 
 finite or infinite ; or else we must have a' — ^ = oo , and 1 + aa' finite or 0. 
 
 a'+i 
 T.T . . , n • la—a a' a'^ +1 «'« + 1 
 
 ^low 1 + aa =0 gives a = ; ; r = = — = = oo 
 
 a 1 + aa ^ "' a — a 
 
 a' 
 
 for any value of a! finite or infinite. Therefore the general relation a= 
 
 a' — a 
 
 between a and a' renders ;; , =oo .+ Let us now see if the relation a'— « = 00 
 
 1 + aa ' 
 
 will do the same. Now if «' — a = 00 , one or the other (a' or a) must be 00 . 
 
 Let a'= 00 . We then have , = — > = -, which can only be 00 when a=0. 
 
 \ -\- a^i aa a 
 
 Hence the particular values a'= oo and « = render ; = 00 , but no gen- 
 eral values meet the requirement unless a = ;. 
 
 3. What general relation between a and a renders — ; = 0? 
 
 ° a -\- a 
 
 AVhat renders it 00 ? 
 
 4. In the expression y = — 2a: + 4 ± ^x^ — 4.r — 5, how many 
 values has y, in general, for any particular value of x ? For what 
 value or values of x has y but one value ? For what values of x is y 
 real? For what imaginary? For what values of x is y iH)sitive? 
 For what negative? 
 
 SonjTiON. — Writing the expression thus, y — — (2j; — 4) ± ^/x'^ — Ax — 5, 
 we see that the value of y is made up of two parts, viz., a rational part — (2.t— 4), 
 and a radical part \/x^ — 4x — 5. But the radical part may be taken with 
 either the + or the — sign. Hence, in general, for any particular value of x 
 there are two values of y. 2d. But if such a value is given to x as to render the 
 radical part 0, for this value of x, y will have but one value, viz., the rational 
 part. But the condition /y/aj' — 4a: — 5 = gives x — ^ and — 1. Thus for 
 
 * This redaction i? made by dropping a and 1, since the subtraction of a finite from an in- 
 finite, or the addition of a finite to an infinite, does not change the character of the infinite. 
 Thus, in this case, to assume that dropping a and 1 aflTected the relation between numerator and 
 denominator, would be to assign to a and 1 some values with respect to tlic infinite a'. But 
 this is contrary to the definition of an infinite. 
 
 t It is to be observed th^t the relation a = - — requires that a and a' have difterent essen- 
 tial signs; while the relation a' =a requires that they have the same essential signs. 
 
INTERPRETATION OF EQUATIONS. 265 
 
 i* = 5, y = — C, but one value ; and for a; = — 1, y = + C, also but one value, 
 od. To ascertain for what values of x, y is real, we observe that y is real v/hen 
 x^ — 4» — 5 is positive, and imaginary when x^ — 4.t; — 5 is negative. Now 
 for X positive x^ — (4c + 5) is -f- when x^ > 4j + 5 ; and for x negative, we 
 have 05* -H 4e — 5, which is positive when x^ 4- 4.c > 5. The former inequality 
 gives a?' — 4« -F 4 > 9, or x > ^ \ and the latter gives a;^ + 4i» + 4 > 9, or .^ > 1. 
 Hence for positive values of x greater than 5, y is real, and for negative values 
 of X numerically greater than 1, y is real. The 4tli inquiry is answered by this: 
 y is imaginary for all values of x between —1 and +5. 5th. To ascertain what 
 4- values of x render y +, and what — , we observe that — (3a;— 4)± y^a?^^^^^ 
 can only be 4- when the + sign of the radical part is taken and when 
 ^x^ — 4aj — 5 > 3a; — 4. This gives a; < 3 ± ^— 3, t. e., an imaginary 
 quantity. Hence y is never + for a;+. Taking the negative sign of tlit; 
 radical we see that both parts of the value of y are — , and consequently y is ^ 
 real and negative for all + values of x which render y real, i. e., for values 
 greater than 5. Finally, for x — we have y = 3a; + 4 ± y^a;^ -f- 4a; — 5. Now 
 when we take the + sign of the radical both parts are + ; hence this value of 
 y is always +. When we take the — sign of the radical y is negative if 
 2a; + 4 < \/x^ + 4a; — 5. But this gives a; < — 3 ± '\/— 3. Hence y is never 
 negative for any negative value of x. Therefore both values of y are positive 
 and real for all negative values of x numerically greater than 1. 
 
 5 to 22. Discuss as above the values of ?/ iu the following ; i. e., 
 Ist Show how many values y has i)i geiieral, and whether they are 
 equal or unequal ; 2(1. For what particular value or values of x, y 
 has but one value ; 3d. For what values of x, y is real, and for what 
 imaginary ; 4th. For what values oi x^y \B +, and for what — ; 5th, 
 Also determine what values of x render y infinite ; 
 
 (5.) y' + 2xy - 2:r'- - 4^ - a; + 10 = 0; * 
 
 (6.) y — 'Zxy + 2:c' — 2?/ 4- 2.r = ; 
 
 (7.) y' + 2xy + x' - 6?^ + 9 = ; 
 
 (8.) y"" + 2xy + 3./;* - Ax ^ ; 
 
 (9.) y' - 2xy + 3.6-^ + 2y -^ Ax --3 = 0; 
 (10.) y' + 2xy -^ 3^-^ - 4.c - ; 
 (U.) y' — 2xy + x^ -^ xz:zO; 
 (12.) y' - 2xy + x' - 4y ^ x -^ A - 0-, 
 (13.) / -^ 2xy + ar* -f 2.y 4- 1 = ; 
 (14.) f - 2x' - 2y + 6a: - 3 = ; 
 (15.) f - 2xy - ix' - 2?/ + 7a; -- 1 = 0; 
 (IG.) y' ^2xy-2^0; 
 (17.) y' - 2xy + 2?/ 4- 4.?; -8 = 0; 
 
 ♦ In all cases solve the eqnation for y in the first place. In this exaropl© 
 
SI66 ADVANCED COURSE IN ALGEBllA. 
 
 (18.) Af + 4:x' -h 2y - 3a; + 12 = 0; 
 
 (19.) Sy'-S3^=12; 
 
 (20.) 12/ + 4a:* = 20; 
 
 (21.) x'-^/=16; 
 
 (22.) x' -y' = 20. 
 
 23. Discuss the equation ay^ — x? + {h — c) a:' + hex = 0, as above, 
 ■when ^ > c ; also when a > b. 
 
 SUG 
 
 '8. y = ± -7 \^x'^ — {b — c)x*— hex. Whence we see that y has two values 
 a* 
 
 for every value of x, numerically equal, but with opposite signs, y is 0, when 
 ar' — (& — e)x* — bcx = ; t. e., when a; = 0, ar = 6, and — c. Again 1/ is real for 
 z +, when a;-* > (6 — c)a;* + ftca;, or a:* > (b —c)x +bc; which gives x > b. For 
 
 a?—, we have y = ± —r^^— x^ — {b — c)x* -\-bcx, which gives y real when 
 
 a?' + (6 — c) a;- < bcx, which gives x numerically leas than c, i. c, greater than 
 
 — c. Hence y is imaginary for all values of x between and + b, and real for 
 all values of x from + 6 to +qo . So also y is real for all values of x from to 
 
 — c, and imaginary for all values of x from — c to — oo . 
 
 X — b 
 
 24. Discuss us above y^ = {x — a)* , showing that in general 
 
 X 
 
 y has two values numerically equal but with opposite signs ; that it 
 is for X = «, and x= b\ is imaginary from x = Q io x=.h (except 
 when x=z a, b being greater than a) ; real from x = b to a:= + <», 
 aud rcal for all negative values of x, i. e., from a: = Otoa;= — 00; 
 and that for a: = 0, y = ± go , and for x=. -^cOf y = zh 00 ; also for 
 x= — 00, y = -±1 cc. 
 
 25. Show from the equation y + x^y = x, that y = when a; = 0, 
 + 00, and —00 ; also that y has but one value for any particular 
 value of a:; that it is -f when x is +, and — when a: is — ; and that 
 y increases numerically as x pjisses from to +l,and from to —1, 
 but that it diminishes numerically as x passes from + 1 to -f 00 , 
 and filso from — 1 to — oo . 
 
 26. Discuss y^x = 4a' (2a — x) with reference to y as a function 
 of X, as above. 
 
 27. Show that in the equation y^ — 3axy + x^ = 0, y has tliree 
 real values between the limits x = 0, and x = a\/^, and only one 
 real value between the limits x = a\/ 4: and a; = + qo , and also be- 
 tween the limits a: = and a; = — 00 . 
 
 Sue. — This is done by means of Cardan's formula. (See 280, ) 
 
INTERPllETATION OF EQUATIONS. 267 
 
 301, Arithmetical Interpretation's of Negative axj) 
 Imaginary Solutions. 
 
 1. A is 20 years old, and B 16. When will A bo twice as old 
 asB? 
 
 SuG's.— We have 20 + ic = 2 (16 + «) ; whence x — — 12. The arithmetical 
 interpretation of this result is that A will never be twice as old as B,but that he 
 teas twice as old 12 years ago, i. e., when he was 8 and B 4. 
 
 2. A is rt years old, and B,J. When will A be n times as old as 
 B ? For 71 > 1 what are the possible relative values of a and h con- 
 sistently with the arithmetical sense of the problem? Interpret for 
 a > nh, a = nh, a < nh when n > 1. Also for ?i = 1, « > nb, a < nh, 
 and a = nh. 
 
 3. Two couriers, A and B, are traveling the same road in the 
 same direction, the former at rate «, the latter at rate K They are 
 at two places c miles apart at the same time. Where and when are 
 they together ? 
 
 Solution and Discussion. — Let XY represent the road which the couriers 
 are traveling in the direction from X to Y, and A and B the stations which they 
 
 t 
 
 pass at the same time, A being at A when B is at B, and D or D' the place at 
 which they are together. Call the distance from B to the place at which they 
 are together ±x, + x when D is beyond B, and —x when it is on the hither 
 side of A and B, as at D'. Then the distance from A to the point at which they 
 are together is c 4- (± a;). Now disregarding the essential sign of x, and leaving 
 it to be determined in the sequel, we have 
 
 Distance A travels from kr=c -\- x, 
 
 Distance B travels from B = x; 
 
 Time from passing A and B to the time they are together —^ and - . 
 
 But these are equal. Hence we are to discuss the equation 
 
 C A- X X ^c , . oc 
 ^ ^ = _ , or a; = r , and c-\-x= 
 
 a b' a-b a -b 
 
 The points to be noticed in the discussion are, (1) when a>b, (2) when a <h, 
 (8) when a = b,c being greater than in each case but not co . Also the like 
 cases wlien c = 0. 
 
 W?ien c > but riot oo . 
 
 We have, for a > b,x positive, which shows that the point at which they ar# 
 
i:68 ADVANCED COtRSE IN ALGEBRA. 
 
 together is at the right of B, i. e., in the direction which they are travelmg. 
 The time, r ( or — ^ )> ^^ positive, which shows that they are together qfter 
 passing A and B. 
 
 For a <b, x is negative, and c -h x, which equals j , is also negatire. 
 
 This shows that they were together at a point at the left of A, that is, hefore 
 they reached the stations A and B. With this the expressions for the time also 
 
 agree. Thus r becomes — j- , and is also negative, since in this case x>c. 
 
 d 
 
 MTi ». he he -, ae ac ,.,.,. 
 
 When a = 0,x— r = — =oo , and c + a? = j- = -77- =ao ; which mdi. 
 
 a—b - a—ft 
 
 cates that they are never together. 
 
 When c = 0. 
 
 In this case x = r = 0, and c + a? = r = 0, for a and 6 unequal, indi- 
 
 a — b a — b * ^ 
 
 eating that they are together when they are at A and B. This is evidently cor- 
 
 bc 
 
 rect, since A and B coincide in this case. When a =zb, x — = - , and 
 
 a — b 
 
 c -f a; = — , which shows that they are always together, - being a symbol of iu- 
 
 determination which in this instance may have any value whatever, as we see 
 from the nature of the problem. 
 
 302, ScH. — The student should not understand that the symbol - 
 
 dlway» indicates that the quantity which takes this form has an indefinite 
 number of values. It is frequently so, but not necessarily. The indeter- 
 mination may be only apparent^ and what the value of the expression is 
 must be determined from other considerations. The Calculus affords the 
 most elegant general methods of evaluating such expressions. But the 
 
 simple processes of Algebra will often suffice. Thus for a; = 1, -r— - — = -. 
 
 1 — a;^ \ — x^ 
 
 But -zr = 1+ a; + a;*, which, for a; = 1, is 3. Hence -z = 3, for a;=l. 
 
 1 —X ' ' ' 1 — a; ' 
 
 Here the apparent indetermination arises from the fact that the particular 
 
 assumption (that a; = 1) causes the two quantities between which we wish 
 
 the ratio, viz., the numerator and denominator, to disappear. Let the 
 
 1 _ 3.5 
 
 student find that z ^ j = 2^ for x = \, (See also 298, 3d part of 
 
 A — aj-j-a? "~"a5 
 
 demonstration.) 
 
 4. Two couriers starting at the same time from the two points 
 A au(l B, c miles apart, travel toward each other at the rates a 
 
INTERPRETATION OF EQUATIONS. 269 
 
 and h respectively. Discuss the problem with reference to the place 
 and time of meeting. (Consider when a> h,a<h, and a = J.) 
 
 5. Two couriers, A and B, are traveling the same road in the 
 same direction, the former at rate «, and the latter n times as fast. 
 They are at two places c miles apart at the same time. Discuss the 
 problem with reference to place and time of meeting as in Ex. 3, 
 adding the considerations, w > 1, n <1, n =1, n — 0. 
 
 6. Divide 10 into two parts whose product shall be 40. 
 
 Solution and Discussion. — Let x and y be the parts, then a; + y = 10, 
 xy = 40, and x = 5 ± -y/— 15, y = 5 T \/— 15. These results we find to be 
 imaginary. This signifies that the problem in its arithmetical signification is 
 impossible : this indeed is evident on the face of it. But, although impossible 
 in the arithmetical sense, the values thus found do satisfy the formal, or alge- 
 braic sense. Thus the sum of 5 + /y/— 15 and 5 — ^y/— 15 is 10, and the 
 product 40. 
 
 7. Tlie sum of two numbers is required to be a, and the product 
 b: what is the maximum value of b which will render the problem 
 possible in the arithmetical sense? What are the parts for tliis 
 value of Z* ? 
 
 8. Divide a into two parts, such that the sum of their squares 
 shall be a minimum. 
 
 Suq's. — Let X and a—x be the parts, and m the minimum sum. Then 
 x* + (a — xY = 2x^ — 2ax + a^ =m; 
 
 whence x = ^a ± ^ '\/ 2m — a^. From this we see that if 2m < a^, x is 
 imaginary. Hence the least value which we can have is 2m = a^^ or m = \a^. 
 
 9. Divide a into two parts, such that the sum of the square roots 
 shall be a maximum. 
 
 10. Let d be the difference between two numbers : required that 
 the square of the greater, divided by the less, shall be a minimum. 
 
 11. Let a and h be two numbers of which a is the greater, to find 
 a number such that if a be added to this number, and h be sub- 
 tracted from it, the product of this sum and this difference, divided 
 by the square of the number, shall be a maximum. 
 
 Sug's.— Let n be the number, and m the required maximum quotient. Then 
 
 n^ -\-(a — b)n — ah . ^ , 
 
 by the conditions ^ ^^^ - m, whence we fand 
 
 n' 
 
 a-b \/a^ -h 2«* + 6* — ^<tbm 
 
 2(1 - w) 2(1 - m) 
 
2T0 ADYANCED COURSE IN ALGEBRA. 
 
 From this we see tliat tLe greatest value which m can have and render n real 
 
 iam = — J—- . This gives n = — — ; r = r . 
 
 4ab ^ 2(1— m) a — b 
 
 12. To find the point on a line passing through two lights at 
 which the illumination will be the same from each light. 
 
 Solution. — Let A and B be the two lights, and XY the line passing through 
 
 X if 6 ^1 f -V 
 
 them. Let a be the intensity of the light A at a unit's distance from it, 6 the 
 intensity of B at a unit's distance from it, c the distance l)etween the two lights, 
 as AB, and x the distance of the point of equal illumination from the light A, as 
 AD (or AD). Then, as we learn from Physics that the illuminating effect of a light 
 varies inversely as the square of the distance from it, we have for the illumina- 
 tion of the x)oint D by light A -^ , and for the illumination of the same point 
 
 by light B, r, . But by the conditions of the problem these effects are 
 
 (c — X) 
 
 equal ; hence we have the equation to be discussed ; viz.. 
 
 {c - X)* 
 This gives ^ j — = -; or 
 
 (c — x)* b e — x .fb ± a/& 
 
 c ^ ± \fb c \/ a ± \/~b 
 or 1 = ^ ; or - = ~ ^ — ; 
 
 or, finally, x=.c — z=^ r, and a? =6 — z^ -, 
 
 wliich are the values of ar to be discussed. 
 Discussion. — I. Let c be finite and > 0. 
 
 1. When a>b, x=e J — >ic, since "^ ^ > ^ for ^ > &. Tliis 
 
 is as it should be, since for a> b the point of equal illumination will evidently 
 be nearer to B than to A. Again, the other value of x gives x = c — ^ r > c, 
 
 -' — is + and > 1, when a> b. Hence we learn that there is a 
 
 point beyond B, as at D', where the illumination is the same from each light. 
 
 If we assume y^=2>^/T, AD = ^ c, and AD' = 2c. 
 
 2. It is evidently unnecessary to consider the case when rt < ft, since this Avould 
 only situate the points of equal illumination with reference to A as the j)reced- 
 ing discussion does with reference to B. 
 
INTERPRETATION OF EQUATIONS. 271 
 
 3. When a = b,x = c-^^^^^^ = j^c, since 'S^-^= ^ =l This 
 
 is as it should be, since it is evident that in this case the point of equal illumina- 
 tion is midway between the lights. Again, for tlie second value of w, we have 
 
 A/a 
 
 x = c — — — = GO , This is also evidently correct ; for when the lights ar« 
 
 ^J a— ^ b 
 of equal intensity there can i>e no point beyond B, for example, at which the illu- 
 mination from A will be equal to that from B, except wlien a; = co , for which 
 the illumination is for each light, [Let the student give the reason,] 
 
 II. When c = In this case the original equation —x = ;; ^-r becomes 
 
 -J = -^ , whence a—b. We then have x=c — ^ — = ; and x =c — ~— — :::. 
 
 ^ ^ ^a+^b \^a—^/b 
 
 c-v/ a 
 
 = — ^rr^ — = -. The former shows that there is a point of equal illuniina- 
 
 ^ a-^yb 
 tion where the lights are (when c = they are together), and the latter shows 
 that any point in the line is equally illuminated by each light. Both these con- 
 clusions are evidently correct * 
 
 • In discussing this problem, some have committed the error of considering Hint, 8ince fo 
 
 c = and a and b unequal, x = c — — r = 0, therefore there i? a point of equal illumi nation 
 
 \^ a± \^b 
 at the i)oint where the lights are situated ! This is evidently absurd, since the hypothesis is 
 that the lights are of uneqiiul intensitj'. The error consists in not perceiving that the 
 
 hypothesis, c -0, excludes the hypothesis, a and h unequal. That the hypotheses a ^ b are 
 excluded by the hypotheses <^= and that there ia a point of equal illumination, is self-evident. 
 Perhaps the student may think that these conditions are no more inconsistent than those in I. %, 
 above, viz,, c finite, a=ft, and a point of equal illumination ; and that, if in the former case we in- 
 terpret a = c '^-- = 00 as indicating a point of equal illumination at a^ = oo, we should in 
 
 ^ a- ^ b 
 
 this interpret x = M— -r = as indicating a point of equal illumination at the place 
 
 where the light? are situated. But the closing remark in I, 3 will clear up this difficulty. 
 
ilr, ^ 
 
 PPJ^IVDIX 
 
 SECTION L 
 
 SERIES. 
 
 303 • A Series is a succession of related quantities each of 
 which, except the first or a certain number of the first, depends upon 
 the next preceding, or a certain number of the next preceding, 
 according to a common law. Each of the quantities is called a 
 Term of the Series. 
 
 III. — A Progression, as 1, 3, 5, 7, etc., or 3, 6, 12, 24, etc., is a series in which 
 each tenn after the first depends upon the next preceding according to a common 
 law. The numbers 1, 3, 7, 11, 21, 39, 71, 131, etc., constitute a series in which 
 oach term after the third is the »um of the three next preceding. The numlK;rs 
 2, 3, 5, 17, 88, 1513, etc., constitute a series in which each term after the first 
 three is the product of the two next preceding + the third preceding. 
 
 304. A Mecurring Series is a series in which each term 
 after the first n is equal to the sum of the products of each of the n 
 preceding terms multiplied resiTCctively by certain quantities which 
 remain the same throughout the series. These multipliers with 
 their respective signs constitute the Scale of Relation. 
 
 III. 1, 4c, 9j;*, 16j;', etc., is a recurring series whose scale of relation is 
 x'\ -3jj*, 3.r, since {I x x*) + (4e x [ - 3:r*]) + (9x* x 3.r) = lCx^ The next 
 term after 16.i; ' would be (4c x z*) + (9x* x [ - 3a;']) + (16a;' x ar) = 25a;\ 
 The next would be 36a;' . 
 
 305. An Infinite Series is one which has an infinite number 
 of terms. Such a series is said to be Convergent when the successive 
 terms decrease according to such a law as to make the sum finite ; 
 otherwise it is called Divergent. 
 
 III. -^, T*^, j-^a, jji^-fra, etc., to infinity, is an infinite, converging series 
 whose sura is ^, That -?,-, + joff + to^itt + ToJinr + etc., to infinity *= i is evi- 
 dent, since by division we liave ^ =.3333 + = -i% + too + ro^oo + etc. 
 
 * The expression " to infinity " is usually omitted, as being sufficiently indicated by " etc.; ** 
 and. In fact, either the + sign at the end or the " etc." may be omitted. 
 
SERIES. 273 
 
 306. To Hevevt a Series involving an unknown quaufcity 
 is to express tlie value of that unknown quantity in terms of 
 another quantity which is assumed as the sum of the first series, or 
 as involved in that sum. Thus the general problem is, having 
 given /(y) = ax-\- hx" + ex" + etc., to express x in terms of y, i, e., 
 to find ic =/(?/). 
 
 III.— Tims to revert the series x 4- Saj^ + ^x^ + 7a;* + Oa;' + etc., is to express 
 the value of x in another series involving y when g^ = x + 3aj* + 5a;^ + 7a;' 
 + 9a;« + etc., or when 1 - 2y + 5y-' = a; + 3a;=* + 5.c» + 7a;* + 9a;'* + etc., etc. 
 
 307. TJie First Order of Differences of a series is the 
 series of terms obtained by subtracting the 1st term of the given 
 scries from the 2d, the 2d from the 3d, the 3d from the 4tli, etc. 
 The Second Order is obtained from the first as tlie first is from the 
 primitive series. The Third Order is obtained in like manner from 
 the second ; etc. 
 
 These several series are called the Successive Orders of Differences. 
 
 III. — Having the series 
 
 1, 8, 27, 64, 125, etc., we obtain 
 
 1st order of diff's, 7, 19, 37, 61, etc., 
 
 2d " " " 12, 18, 24, etc., 
 
 Cd " " *' 6, 6, etc., 
 
 4th " " " 0, etc. 
 
 308. Interpolation is the process of finding intermediate 
 functions between given non-consecutive functions of a series, 
 Avithout the labor of computing them from the fundamental formula 
 of the series. 
 
 III. — The logarithms of the natural numbers 1, 2, 3, 4, 5, 0, 7, 8, etc., con- 
 stitute a series of functions. Now knowing these, interpolation teaches how to 
 find intermediate logarithms, as log 4.3, 4.5, 4.6, etc., or 2.7, 2.72, 3.102, 7.025, 
 etc., without the labor of computing them from the fundamental formula of 
 the series {192). 
 
 [Note. — The student must guard against the notion that every series is a 
 recurring series. Any snccesnon of numbers related to each other by a common 
 law, as, for example, the logarithms of the natural numbers, is a series, as well 
 as the more simple arithmetical, geometrical, and other recurring successions.] 
 
 300. Some of the more important problems concerning infinite 
 series are : To find the scale of relation of a series; To find the nth 
 (any) term of a series ; To determine whether a scries is convergent 
 or divergent ; To find the sum of a convergent series, or of n terms 
 of any series ; To revert a series ; and, To interpolate terms between 
 
274 APPENDIX. 
 
 given terms. To these problems we shall give attention after 
 having demonstrated the following lemm*, which is of use in the 
 solution of several of them. 
 
 310, Lemma, — The first term of the nth order of differences 
 
 , n(n-l) n(n-l)(n-2) ^ , , . 
 
 tsa — nb -\ — ^—- -c ^ -.^^^ ^ d + etc, ir/ten n is even, and 
 
 n(n — 1) n(n — l)(n — 2), , , . ,, 
 
 — a -f nb ^c-l — ^^ ,^ -a— etc., when n u odd; a, 
 
 b, c, d, etc., being successive terms of the series. 
 
 Dem. — Letting a, b, c, d, e, f, etc., be the series, we have 
 
 1st Order of diflf's, h — a, c — h, d — c, e — d, f— e, etc., 
 
 2d " " " c-2b + a, d— 2c ■hh,e — M->rC,f—2e + d, etc., 
 
 3d " " " d — 3c + 36 — a, e — 3(i + 3c — 6,/— 3e + 3rf— c, etc., 
 
 4th " " " g _4^ + 6c — 46 + a,/— 4c + 6rf — 4c + &, etc., 
 
 5th " " " /- 5c + \M - 10c + 56 - a, etc. 
 
 Now by inspection we observe that, numerically, the coefficients in these 
 terms follow the law of the coefficients in the development of a binomial. Thus 
 the coefficients in any term of the 2d order of differences, as in c~2b + a, are 
 the same as in the square of a binomial ; those in any term of the 3d order, as 
 in d—3c -f 36— a, are the same as in the cube of a binomial, etc. Hence, revers- 
 ing the order of the simple terms in the first terms of the successive orders, and 
 representing the first term of the first order by i), , the first term of the 2d order 
 by Di, the 1st term of the 3d order by Dj, etc., we have, for the even orders, 
 
 Bi =a — 2b + c, 
 
 D4 = a — 4b + Qc — 4d + e. 
 
 Hence, by induction, we have, for the 1st term of the nth order, when n is even, 
 
 ^ , n(n — 1) n{n — l){n — 2) ^ 
 
 Bn =a-nb + -^ — '-c p^ ' d + etc. 
 
 Again, for the odd orders, we have 
 
 Bx = — a + b, 
 
 i>3 =z - a 4- 36 - 3c h d, 
 
 B^ = - a + hb - 10c + lOrf - 5c + /. 
 
 Hence, by induction, when n is odd, the first term of the 7ah order is 
 
 -r. . n(n-\) 7i(n — 1) {n - 2) ^ 
 
 Bn= — a + nb —- — - c + -^ 4 d - etc.* 
 
 * The author does not deem it expedient to take the time and ej)ace to demonHtrate more 
 rigorously thii? law ; nor does he fully sympathize with the idea thai induction is in no case n 
 eatiefactory mathematical argument. 
 
SERIES. 275 
 
 311, Cov, — It will be observed that in order to find the \st tenn 
 of the first order of differences, we must have 2 terms of the series 
 given ; to find the \st term of the 2d order, 3 terms ; to find the \st 
 term of the 3c? order, 4 terms ; and, in general, to find the \st term 
 of the nth order we must know n + 1 terms of the series. 
 
 Examples. 
 
 1. Find the 1st term of the 3d order of differences in the series 
 7, 12, 21, 36, 62, etc. Also the 1st term of the 4th order. 
 
 Sug's. — For the 3d order we have 
 
 i)3 = - a + 3& - 3c + <? = - 7 + 3 12 - 3 • 31 + 30 = 2. 
 For the 4th order, 
 
 D4 = a - 46 + 6c - 4rZ + g =. 7 - 4 • 12 + 6 21 - 4 • 36 + 62 = 3. 
 
 2 to 6. Find the first terms of the orders of differences specified in 
 tlie following : 
 
 (2.) 2d, 3d, and 4th, in 1, 8, 27, 64, 125, etc. 
 
 (3.) 3d, and 5th, in 1, 3, 3'^ 3^ 3^ 3', etc. 
 
 (4.) 5th, in 1, ^, i, J, ^\, ^\, etc. 
 
 (5.) 5tli, in 1, 6, 21, 56, 126, 252, etc. 
 
 (6.) 6th, in 3, 6, 11, 17, 24, 36, 50, etc. 
 
 312, Proh, 1, — To fiiid the Scale of Relation in a recurring 
 infinite series when a sufficient number of terms is given. 
 
 Solution. — 1st. When each term after tJie first depends on the next preceding. 
 
 — Let m represent the scale of relation. Then h — ma (,304). Whence 7n = -. 
 
 a 
 
 2d. When, each term after the first two depends on the two terms next preceding 
 it. — Letting m, n be the scale of relation, we have c=ma + nh, and d=7nb + nc 
 
 {304). Whence m = z^, and n = r-^. 
 
 ac — b^ ac — h^ 
 
 3d. When each term after the first tJiree depends on the three terms next preced- 
 ing it. — Letting m, n, r represent the scale, we liave d = ma + nh + re, e—mb 
 + nc + rd, and/= mc -\- nd -^^ re. From these three equations the values of 
 m, n, and r can be found. 
 
 4th. We can evidently proceed in a similar manner when the dependence h 
 upon any number of preceding terms. 
 
 313, Sen. — In applying this method, if wc assume tliat tlie dcpendeneo 
 is upon more terms than it really is, one or more of the terms of the scale 
 will reduce to 0. If we assume the dependence to be on too few terms, the 
 
2fl5 APPENDiy. 
 
 error will appear in attempting to apply the scale when found. If we 
 attempt to apply the method to a series which is not recurring, the error 
 will appear in the form which the scale assumes, or when we attempt to 
 apply it. 
 
 When the dependence is upon two terms, any two equations of the scries 
 c = vin + Tib, d = mh + Jic, e = mc -h nd, f— md 4- nc, etc., will give the mme, 
 cnlues for m and u. So also if the dependence is upon three terms, any three 
 equations of the series d = ma + nb + rc^ e = mh + nc + rd, f= mc + nd + ?<;, 
 g = md + ne + rj\ etc., will give the same values to m, n, and r\ etc., etc. 
 
 There is no general method of determining that a series is absolutely vot 
 recurring. The best practical method of procedure is to assume jfird that 
 the dependence is upon tico terms: if this does not give a scale which will 
 extend the series, try whether the dependence is not upo'i three terms, then 
 upon four, etc. Of course, applying this process to an infinite series would 
 not determine that the series was absolutely not recurring. 
 
 Examples. 
 
 1. Find the scale of relation in the series 1, 12, 48, 384, 1920, etc. 
 
 Suo's. — Assuming that the dept^ndenco is upon two terms, we have 48 = in 
 + 12h, and 384 = \2m + A'Sti ; whence m = 24, and n = 2. Now since 1J)20 
 = 24- 48 4- 2 384, we conclude that + 24, + 2, is the scale. 
 
 2. Find the scale of relation in tlic series 1, O.r, IS.c^ 48.r\ 120;/^ 
 etc 
 
 Suo's. — We have 12.t' = m + Qxn,&nd 48^;' = (j.rm> + 12.c*;t ; whence m—(ix*, 
 iind w=.T. Now, as 120c* = Gx* ■ 12x* + x- 48x*, we conclude that the scale of 
 relation is + 6.c*, + t. 
 
 3. Find the scale of relation in the series 1, 4.c, C)x\ lla:*, 28:c*, 63.T*, 
 and extend the series two terms. 
 
 Scale of relation, +3ar', — .r^ +2.c 
 Next two terms, 131.r*, 283.r'. 
 
 4 to 11. Find the scale of relation in the following, and extend 
 each series 2 or 3 terms : 
 
 (4.) 1, X, 2x% 2x\ S:c*, 3.c\ 4x\ Ax\ etc. 
 
 (5.) 1, 3, 18, 54. 243, 720, 201G, 8748, etc. 
 
 (6.) 1, X, bx\ 13:r\ 41.^^ 121.r', d(jbx\ etc. 
 
 (7.) 1, 4, 12, 32, 80, etc. 
 
 (8.) 3, bx, l7^, \^T?, 23 A 45a;', etc. 
 
 ,^. a ac ac? ., a& , . 
 
SERIES. 277 
 
 (10.) 1, 4, 10, 20, 35, 56 84, 120, etc. 
 (11.) 1, 4, 8, 13, 10, 26, 34, etc. 
 
 814, JProb, 2, — To find the nth tenn of a series when a svf- 
 ficient number of terms is given, 
 
 Soi.UTiON. — Tlie best method of doing this depends upon the chara<'ter of 
 the series. We give the following : 
 
 1st. Tlie formula I z=: a -\- {ii — \)d, and I = ar^-^, resolve the problem for 
 arithmetical and geometrical series, I being any tenn. 
 
 2d. Tlie scale of relation may be determined by Puou. 1, and the series 
 extended to the nth term by means of it. 
 
 3d. But the first tenns of the successive orders of differences afford one of 
 the most elegant and general methods. Thus from {310) we have 
 
 J)j——a+b; .'. b=a+I>i ; 
 
 2),— n—2b+c; /. c=a+2D,+T)z ;* 
 
 i?3 = -rH-;3Z»-3c+rf ; .-. d=a+SDi -hlWz+Di', f 
 
 Dj= n+Ab-C)c+4d—c; .'. e=a+4n, +01)2+41) ,+1)^ ; 
 
 i>, = -/H-;V>-10c+10ff-5«+/; .*. f=a+5Di +10i>2 + 102>,,+5i)4 + Z?5. 
 etc., etc., etc. 
 
 Whence, by induction, we have, in general, the 72th term ~ a + {n — \)D, 
 
 (n - IXw - 2) ^ (n - l)(7i - 2){7i - 3) ^ .„ , 
 
 4- ^^ S ^2 + ... ' D3 + etc., till the term containing 
 
 « I o 
 
 D,_i is reached, or till an order of differences is reached of which each term 
 is 0. It is only in the latter case that the method is practically useful, since to 
 determine the first terms of the n — 1 successive orders of differences, requires 
 that n terms of the series be known. 
 
 Examples. 
 
 1 to 5. Solve the following by means of the scale of relation : 
 
 (1.) Find the 8th term of 1, 2x, Sx\ 28r', 100.9.^ etc. 
 
 (2.) Find the 9th term of 1, 3«, 6x% W, W, XW, etc 
 
 (3.) Find tlie 10th term of 1, 3.r, 2.r', - .r', - 3.r*, — 2r?;», etc. 
 
 (4.) Find the 12th term of 3, 5, 7, 13, 23, 45, etc. 
 
 (5.) Find the 11th term of 1, 1, 5, 13, 41, 121, etc. 
 
 6 to 12. Solve the following by means of the successive orders of 
 differences : 
 
 (G.) The 12th term of 1, 5, 15, 35, 70, 126, etc. 
 
 * c = - tt + 2ft + 2)3 = - a f 2(a 4 X>,) + 7)2 = a + 22), -t /),. 
 
 t d = a - 3ft f ;ic + /), = o - 3<a )- /),) ♦- 3(a + %D^ + D,) f />, =* a ^ 3/), \ 3/J, t />,. 
 
278 APPENDIX. 
 
 (7.) The 15th term of 1, 3, 6, 10, 15, 21, etc. Also the nth, 
 (8.) The ?^th term of 1-2, 2-3, 3-4, 4-5, etc. 
 (9.) The 12th term of 1, 4a:, 6x\ ll:r^, 28a;*, GSx\ etc.* 
 (10.) Solve the first five given above by this method, when it 
 will apply. Also determine the scale of relation in (6) to (9) in cases 
 in which the series is recurring. 
 
 (11.) Find the wth term of 1, 2*, 3', 4^ etc. 
 
 (12.) Find the 9th term of 70, 252, 594, 1144, 1950, etc. 
 
 13. Extend the following to 10 terms by the metliod of differences*. 
 1, 4, 8, 13, 19, etc Also x% 4x\ Sx\ ISa^, ldx'% etc. Also 1, 6, 20, 
 50, 105, 196, etc, 
 
 SIS, JProb, 3» — To determine whether a series is convergent or 
 divergent. 
 
 Solution. — 1st. When theternu are nil -\- . If the series is not decreasing, 
 of course it cannot be convergent. Thus a^-b + c + d + e + etc., \i n <h 
 < c < d < e, etc. , is > a oo . I^t us then consider the case wlieu the terms are 
 all + , and a > b > c> d> e, etc. We have 
 
 S=a + b + c + d + e + etc. = a(l+-+ -+- + - + etc. ) 
 
 \ a a a a / 
 
 (b cb deb edcb \ 
 
 l + - + i-+^-+ :r-r + etc. ). 
 a ba cba dcba / 
 
 Now if -, -, -, -, etc.<.p, S < n{\ + p + p^ + p'^ + p* + etc.), which, if 
 a b G a 
 
 ;><!,= . Therefore, An injinite series of positive terms is always conter- 
 
 1 —p 
 
 gent, if tlie ratio of each term to the preceding term is less than some assignable 
 
 quantity which is itself less than 1. 
 
 2d. When the terms are alternately f and — , and decreasing. Let the series 
 he a, —b, + c, — d, + <", — etc. Now we may write 
 
 5 = (« — Z») + (c — rf) + (« -/) + etc. ; 
 and also 8= a — {b — e) — {d — e) — etc. 
 
 Since the terms are decreasing {c — d), («—/), «tc., are +, and S>a — b. 
 Again, (6 — c), {d — e), etc., are + , and 8 < a. Therefore, Any series of decreas- 
 ing terms, which ttrms are alternately + and — , w convergent. 
 
 3d. When the terms are alternatdy + and — , and increasing, we have 
 8=ia — b-\-c — d + e —f + g — etc. = a — {b — c) — {d — e)— {f — g) — etc. 
 Now, since the terms are increasing, b — c, d — e, f — g, etc., are essentially 
 negative. Representing these differences by — d, —d^, —d^, etc., we have 
 
 * It is evident that the 12th term involves x to the 11th power, or contains a:". I.cncc we 
 have only to find the coefficient, or the 13th term of the scries 1, 4, 6, 11, 28, 63, etc. 
 
SERIES. 279 
 
 S = a + d + d^ + di + etc., a series which can be examined by the first process 
 given above, 
 
 4th. The process of grouping the terms and thus forming a new series, as in 
 the last case, is frequently serviceable in other cases than that there specified.* 
 
 Examples. 
 
 1. Determine whether 1 -f - + — - + _-_ + + etc., is 
 a convergent series. 
 
 Sug's. — Here - = 1, t = ^, - = o> -t= t> etc. : whence we see that each 
 a b 2 c d d 4: 
 
 of the ratios after the second is less than ^, which is itself less than 1. Hence 
 
 the series is converging. 
 
 2. Determine whether l+J + ^ + j^- etc., is a converging series. 
 3 to 6. Determine which of the following are converging : 
 
 (3.) i + i + i + iV + etc. 
 
 (4.) 1 + - + — 2 + -3 + etc., r being > 1, i. e., any decreasing 
 geometrical progression. 
 
 . . _3_ 4 5 
 
 ^ ^^ 1.2.2 "^ 2.3.2'^ "^ 3:4:2^ ^ ^*^- 
 
 2^' iC^ (K^ S^ 3^ 
 
 7. For what values of a; is x r + — — — -f-^ 4- etc., 
 
 2 3 4 5 6 
 convergent, and for what divergent ? 
 
 Bug's.— For a; ~ 1 we have a series with the terms alternately 4- and —, 
 and decreasing. Hence, by (t/iJ, 2d), the series is convergent, /gain, to 
 
 examine the series for «> 1, it may be written x — — + x^\- ~) 
 
 2 \ 8 4 / 
 
 + a^ |— —I + x^ I— —j + etc. Now, for ic > 1, some one of the 
 
 factors (-Q-~x)' \6~~q)' (t"^)' ^*^"' ^"^ ^^^ following it 
 
 8 \ X 
 
 will become negative. Thus, if a? = ^, all following ^ — 5 will be negative. 
 
 7 7 8 
 
 * This is confessedly quite an imperfect presentation of this prohlem ; but it is eufficient for 
 most purposes, and i» as fUIl as our limits will allow. 
 
280 APFKNDTX. 
 
 The sum of that portion of the series preceding this first negative factor will be 
 finite, since it will be composed of a finite number of finite terms. Let us now 
 examine the infinite series which is composed of negative terms. Let a be 
 the value of x for which we are examining the series, and y the exponent of x 
 
 in the first negative term. This term is therefore (ffy \ . Now this 
 
 may be taken as the general term of this portion of the series if we understand 
 that a is constant and y variable. As y increases by 2 in each successive term, 
 
 the first two terms of this series are api ; ) , «"+'( r •— ., | ; and 
 
 the ratio of the second to the first is a^ \l±lr^y:Z^ x J^^-^lJ- 
 
 ( (y + 2) (y + 3) y + \-(iyS 
 
 — ^ ] 7V^ — r-7^ /o .. ^ * /i/ — n ^ n [ > the limit of which, as y in- 
 
 creases to infinity, is a*. But as /7 > 1 , n^ > 1, and this negative series is diver- 
 gent and its sum is infinite. Hence the given series is convergent for « ~ 1, and 
 for all values of a; > 1 it is divergent. 
 
 316» JProb, 4. — To find the sum of n terms of a series. 
 
 This problem, like many others concerning series, does not admit of a general 
 solution. We specify the following cases : 
 
 ar convergent, for n finite, 8=in[2a + (n — l) d], or S = __ — . For an 
 
 Case 1. — W^ien the series is Arithmetical or Geometrical, either divergent 
 convergent, for n finite, 8 = in[2a + {n — 1) d] 
 
 infinite geometrical convergent series we have S = 7~~' • 
 
 Case 2.— When the series is an infinite, decreasing, recurring series, to find 
 the sum of the series (t. e., n being oo). Let the series be a + b + c-¥d + e+ etc., 
 and m, n the scale of relation, the dependence being upon two terms. Whence 
 we have 
 
 a = a, 
 
 b =b, 
 
 c = am + bn, 
 
 d =: bm + en, 
 
 e = ofn + dn, 
 
 f = dm + en, 
 
 Putting S=a + b + e + d^ etc., 
 
 and adding, this gives 8=a + b + 8m + {8— a)n. 
 
 Solving for 8, we hava 8 = ^'^^"'^ . (1) 
 
SERIES. 2S1 
 
 When the scale of relation consists of three terms, as m, n, r, '.vo i:av« 
 
 a := n, 
 
 b =b, 
 
 c = c, 
 
 d = am + bn + cr, 
 
 e = bm + en + dr, 
 
 f z= cm + dn + er, 
 
 g = dm + en + fr. 
 
 Whence 8= a ^-b -v c -v Sm + {S — a)n + {S — a — b)r. 
 
 .,,..„ ^ a + b + c — an — {a + b)r 
 
 And solving for S, S = :; — . (2^ 
 
 1 —m — n — r ^ ' 
 
 When the scale of relation consists of four terms, as m, n, r, », we can write 
 
 from analogy, 
 
 n ->rb->r e ■¥ d — an — {a + byr — {a ^-b + e)8 ,_. 
 
 ^ — z . (o) 
 
 l—m—n—r—s 
 
 Case 3. — To find the sum of n terms of a series by the method of differ- 
 ences. — Let the series be a, b, c, d, e,f, etc., which we will call {A). 
 Now if we write the series 
 
 (B) 0,a,a+b, a + b + c, a + b + e + d, a + b + c+d + e, etc., 
 
 of which the scries (A) is the first order of differences, it is evident that the 
 (71 + l)th term of (B) is the sum of 71 terms of the given series (.4). By tho 
 formula for the nth term {314:, 3d), which is 
 
 The nth term = a + („-l)Z), + f!^!^ D, + („-lX»-2)(n-3) ^^^ ^^ ^ 
 
 A jo 
 
 noticing that a, the first term, in series {B) is 0, that D^ of series (J5) is a of 
 series {A), D^ of series {B) is Di of series (-4), etc., we have, for the sum oi 
 n terms of (A) 
 
 n(n - 1) _. n{n — Vj{n - 2) ^ 
 S=na + -^-- — -' Dy + -5^ 1^^ ' I>i+ etc. 
 
 On this formula we observe that when the orders of differences do not vanish, 
 if the series is extended to the (;i + l)th term the coeflBcient of that term will 
 become 0, and the series will terminate. 
 
 Moreover, in cases in which the nth order of differences vanishes, the same 
 number of terms of this formula will give the sum of any number of terms of 
 the series above the nth. 
 
 Case 4. — Upon the principle that any fraction of the form — — ^ — c 
 
 n{n+p) 
 
 = - ( ~ — ) ,* many series of fractional terms of the form — — - — r 
 
 p\n n + p/ n{n + p] 
 
 may be summed. 
 
 * This is evident since ^ - -^ = ^ ^^ " "^ = -^l~. 
 n n+p nin-i-if) n{n + p) 
 
^Bj^ appendix. 
 
 Also many series of fractional terms of the form — f r— ; may be 
 
 ^ n{n + p)(n + 2p) 
 
 summed from the fact that 
 
 n{n + p){n + 2p) 
 
 = ii_5 ? I. 
 
 2p ( n{n + p) (n + p){n + 2p) ) 
 
 When the fractional terms are of the form — ^ ^ .^ -—., the 
 
 n{n + p){n + 2p){n + dp) 
 
 summation may often be effected upon the principle that 
 9 _ 1 i 9 
 
 —1 L 
 
 in + 2o)(n + 3») ) 
 
 n{n + ^X^ + 2;))(w 4- 3p) Sp ( 7i{n + p){n + 2p) (n + p){n + 2p\n + '6p) 
 
 The practicability of this method depends upon our ability to find the differ- 
 ence between two series. Tims, when the terms of the given series are of the 
 
 form , — r- , if we can find the difference between two series whose terms 
 n{n + p) 
 
 are of the form - , and — - — respectively, we can find the sum of the given 
 
 n n ■¥ p ^ 
 
 series. But the method will be more readily comprehended in connection 
 with its application. (See Ex's 15-30, ) 
 
 Examples. 
 1 to 7. Find the sum of the following recurring series : 
 
 (1.) 1 + 2a; + 8a:» + 28x^ -f lOOx^ + etc. 
 
 (2.) 1 + 2a: + 3ar 4- Sa:" 4- 8ar* 4- etc. 
 
 (3.) 1 + 3^; + 5ar« -H T-c* + etc. 
 
 (4.) 3 4- 5a; + 7a:' + 133:^ + 23a:< -f- 45a:5 + etc. 
 
 (5,) 1 + 1 + 5 4- 13 + 41 + 121 4- etc. 
 
 (6.) 1 4- a: + 2a;» 4- 22^* 4- 32:* 4- 32:* 4- 4.c' 4- 4a:' 4- etc. 
 
 ,^. a ac a(? , ^' , 
 
 (7.) j-jr^+-jr^--jr^ + etc. 
 
 8 to 14. Find the sum of the following by the method of differences : 
 
 (8.) 1 4- 3 4- 5 4- 7 4- etc., to 20 terms ; to n terms. 
 
 (9.) 14-24-3 + 44-5 + etc., to 50 terms; to n terms. 
 (10.) 14- 5 4- 15 4- 35 4- 70 4- 126 + etc., to 30 terms; to w terms. 
 (11.) 70 + 252 + 594 + 1144 + 1 950 etc., to 25 terms; to ^ terms. 
 (12.) 1 + 2* + 3* + 4* + etc., to 12 terms; to n terms. 
 (13.) 1 + 2' + 3' + 4' + etc. , to n terms. 
 (14.) 1 + 2' + 3' + 43 + etc., to n terms. 
 
 15. Find the sum of — ^ + ^r-r + r-j + — ^ + etc., by the method 
 
 x'tii Z'O 0''± 4'0 
 
 given in Case 4. 
 
SERIES. 283 
 
 Sug's. — If we put p = l, q = l, and n = 1, 2, 3, 4, etc., successively, the 
 
 general form of the term in this series is , — . Thus we have 
 
 n{n+p) 
 
 For the Ut term. ^_ = j^_ = -^ (j - ^1--)*= 1 (l - ^ ) 
 
 For the 2d term. ^^^^ = ^-^- = ^ (1 - ^-3)*= 1 (i - 1 ) 
 
 For the 3d term. ^-^ = __L_ = ^ (^ _ ^).= 1 (^ _ 1 ) 
 
 For the 4th term, -— ^^^ = ,,/ ,, = 1(1- 7"^-?)*= ^ (^7 - ^ ) 
 w(;i + p) 4(4 + 1) 1 \4 4 + 1/ \4 5 / 
 
 etc. , etc. , etc. 
 Putting S for the sum of the series and adding, we have 
 
 _ (l+i + i + i + etc. ) _ 
 ~i -i-^-i-etcf ~ ^• 
 
 Note, — It will be seen that this method is only an ingenious device for de- 
 composing the given infinite series into two infinite series, one of which destroys 
 all but a finite portion of the other. 
 
 16. Find the sum of — x + r-r + ;=-^ + l^r-^ + ^^c- 
 
 l«o O'D 0*7 7*y 
 
 17. Find the sum of 7i terms of each of the two preceding series. 
 SuG. — We have for the 7ith term of the last series q = l, p = 2, 7h = 2n — l, 
 
 since 2n — 1 is the nth odd number. Hence for the nth term 
 
 = 77 1 s T — s 7 ) • ^^ therefore have 
 
 3 \2n — 1 271+1/ 
 
 -^ 
 
 .111 1 
 
 1 + ^ 4- -;r + ^ 
 
 3 5 ■ 7 2n-l 
 
 111 11 
 
 8 6 7 2n - 1 2n + 1 
 
 _ ^ 
 
 ~ 2n + 1' 
 
 2\ 2n + lJ 
 
 18. Find the sum of— 7 + --^ + 7— ; + — -+ etc. Also of n terms 
 
 1'4 2'0 d-6 4-7 
 
 of the same. 
 
 19. Find the sum of— r — 773 + — — — - + etc., to n terms. 
 
 15 60 63 99 
 
 Since by Case 4, -^^— =l.(t- -^'^-). 
 n{n + p) p \,i n . pj 
 
284 APPENDIX. 
 
 SuG's.— It will be seen that this series is the same as r— — — - + =-r- — ^—rr 
 
 + etc. Hence by making q=2,S, 4, 5, etc., successively, n = S, 6, 7, 9, etc., 
 successively, and p = 2,vre have 
 
 ii(5 - f) - a - f) + (f - I) - (f - A) + etc.}, or 
 i!l - a + f) + (f + ^) - (^ + t) + W + etc.} 
 = ili-l + l-l+-A- + etc.}. 
 
 Now the form of this last term is -r ; and if an even number of terms of 
 
 2n + 6 
 
 1(2 n + 1 ) 
 
 the criven series is taken, we have ;r ■{ t — 1 + s r^ c . ^^^ *l^e intermediate 
 
 ^ 2 id 2n + 6) 
 
 terms destroying each other. But if an odd number is taken, we have 
 
 l/2n + l\^. „ n + 1 1 1 , . 
 
 — ( — — ). Fmallv, as ^ = -^ — ^-7^7 77, , we have for an even 
 
 2\ 3 2/1 + 3/ • 2/1 + 3 2 2(2/i + 3) 
 
 number of terms i || - [ - ^^^^ [ > <- ^ - J^"?, = -<» for an odd 
 
 number. ||| - ^ + 2^-3,} - <>' jV Sca^T^T)" '"''''^ "=*•'"' ''*'" 
 = ; whence the sum is 
 
 4(2/1 + 3) 12 
 
 20. Find the sum of :j-^ + — r + 5— + etc 
 
 1*0 Z'^ iJ'O 
 
 21. Find the sum of ^r-s — -t-t + jr-;^ — etc. 
 
 I'O Z-i O'O 
 
 22. Find tlie sum of -— + — — + — — + etc. 
 
 O'O D'l>i y«l0 
 
 8uo.-Tbi8 equals i(j-L + _!- + _L + etc.). 
 
 23. Find the 8um of ji + ^ + ^ + jgi-^ + etc. 
 
 24. Find the sum of — — - + ^r-5-7 + 5-7-^ + etc. 
 
 I'ii'O 4-6-4: 6'4:-0 
 
 Sdg's. — By putting p = 1, q = 4, 5, 6, etc. , successively, and 71 = 1 , 2, 3, etc., 
 
 successively, these terms take the form —7 ^7 7.-^, , and since 
 
 71 {n + p) {n + 2p) 
 
 -^ ^-— = ;r- ^ ^ — -, V . we may write the given 
 
 //(/I + p) {)i + 2p) 2p in{n + p) (w + p){n + 2p)) 
 
 series thus : 
 
 1 1 (ri - 2^3) + (2^3 - 3^4) + (3-4 - 4-5) + '=*^- \ 
 
PILING BALLS AND SHELLS. 
 
 m 
 
 4 5 6 
 
 2 3 3-4 
 
 etc. 
 
 = l(^ + 2^ + 5^4 + «*"•) - 1(2 + i) (^- E'^- 1^ = li- 
 
 25. Find the sum of 
 
 15 
 
 5.8.11 8-11.14 ' 11.14.17 
 
 + etc. 
 
 36. Find the sum of ^2^^ + ^^ + _A- + etc. 
 
 1 4 7 10 
 
 27. Find the sum of ^^ + ^^^ + .-^^ + ^:^^ + etc. 
 
 28. Find the sum of ^-:^.t-. + :rJ^— + .-r-rV-r + etc. 
 
 1.2.3.4 2.3.4.5 ' 3.4.5. G 
 
 SuG. — Consider that 
 
 I 
 
 ? I 
 
 {n + p) {n + 2p) {n + 'Sp) f 
 
 n (n + p) {11 + 2p) {n + 3^) 'dp ( n {n + p)(n + 2p) 
 
 29. Find the sum of 
 
 + - 
 
 1.3.5.7 3.5.7.9 5.7.9.11 
 
 -f etc. 
 
 30. Find the sum of 
 
 r^ + 
 
 3.6.9.12 6-9.12.15 y. 12. 15-18 
 
 + etc. 
 
 Note. — The above examples are taken from Young's Algebra, an excellent 
 »>ld Englisli work to which American editors are much indebted. 
 
 Piling Balls an^d Shells. 
 
 317* In arsenals and navy-yards, cannon-balls and shells are 
 piled on a level surface in neat and orderly piles of three different 
 forms, viz., triangular, square, and ohlong. The figures below will 
 sufficiently illustrate these forms : 
 
 _? ^^ (2)® {?!)"<§ <S) (D @ #' ® t) ® »; 
 
 a ^' ® # ® #® ® ® <® ®. ® # ® 1 
 ^■(i)®.®?i)#®j)®<®< 
 
 *) (fS) (S •'1^ ® ® B>fiM'&. 
 
 OBItOKG PILB. 
 
 TRIANGULAR PILE. SQUARE PILK. 
 
 318, I^rop. — The formula for the number of halls or shells in a 
 triangular pile having n balls or shells on a side of its loioest 
 Jn (n + 1) (n + 2). 
 
 course IS 
 
286 APPENDIX. 
 
 Dem. — The student will be able to discover that, beginning at the top, the 
 number of balls or sheila in each course is as follows : 
 
 ], 1+2, 1+2 + 3, 1 + 2 + 3 + 4, etc., 
 
 or 1, 3, 0, 10, 15, 21, etc. 
 
 Summing this series to n terms by the method of differences he will obtain the 
 formula. 
 
 3 If), Cor. — The number of courses in a triangular pile is equal 
 to the number of balls or shells in one side of the lowest course ; and 
 the number of balls or shells i7i the lowest course is 1 + 2 + 3-1-4 
 n, or J(n' + n). 
 
 320, Prop, — The formxda for the number of balls or shells in 
 a square j^Hc having n balls or shells on a side of its lowest course is 
 
 |n(n + l)(2n + l). 
 
 The student should be able to demonstrate tliis as above. 
 
 321, Cor. — 7'he number of courses in a square pile is equal to the 
 number (f balls or shells in one side of the lowest course ; and the 
 number of balls or shells in the lowest course is 1 + 3 + 5 + 7 + 9 
 2n — 1, or n*. 
 
 322, Prop, — Tfie formula for the number of balls or shells in 
 an oblmig jnle having m balls or shells in the length of the base and 
 n in the width is 
 
 in(n + 1) (3m -11 + 1). 
 
 Dem. — Observe that there are as many courses as there are balls in the width 
 of the base. Let m be the number in the top row, whence we have for the 
 number in the successive rows from the top downward, 
 
 w', 2<w' +- 1), 3(m' + 2), 4(m' + 3), 5(m' + 4), etc. 
 Taking the successive differences, we find i), = ?»' + 2, Dg = 2, and Dg =: 0. 
 Substituting in 
 
 8=^^ 2 ^^^ 2T3 ^*' 
 
 nin — 1) , , -, n{n — l)(n — 2) ... ... 
 
 we have S = m'n -+ -^^-^ — - (m' +- 2) H , which readily 
 
 « o 
 
 reduces to 8 - \n\{n -^ \) {^m' -\- 2n — 2)} . 
 
 Now m being the number of balls or shells in the length of the base, wo observe 
 
 that m! ■=m — n+\, which substituted in the previous equation gives 
 
 ScH. — If we make m = n, this gives the formula for the square pile, as it 
 should. 
 
EEVERSION OF SERIES. 287 
 
 Examples. 
 
 1. Find the number of balls in a triangular pile of 20 courses. 
 In a triangular pile with 42 balls on one side of the lowest course. 
 How many balls in the bottom course ? How many in one of the 
 faces ? 
 
 2. Find the number of shells in a square pile with 30 courses. 
 With 23 balls in one side of the lowest course. With 2209 in the 
 bottom course. How many balls in one face of each pile ? 
 
 3. Find the number of balls in an oblong pile whose bottom 
 course is 42 balls by 20. Whose top course contains 23 balls, and 
 which has fifteen courses. 
 
 4. How many shells remain in an incomplete triangular pile whose 
 top course contains 28 shells, and whose bottom course has 15 shells 
 on a side ? 
 
 5. How many balls in an incomplete square pile wliose top course 
 is 8 balls on a side, and whose bottom course is 20 balls on a side ? 
 
 8. How many shells in an incomplete oblong pile whose top 
 course is 12 by 20, and whose bottom course is 62 shells in length ? 
 
 Reversion- of Series. 
 
 323. Prob, — To revert a /Series. 
 
 Solution. — The problem is, having given 
 
 /(y) = ax+bx* -^ cx^ + dx*- + etc., {A) 
 
 to express a; as a function of y, i. e., to obtain 
 
 x = Ay + By^ + Cy* + Dy* + etc., (B) 
 
 the essential thing in the solution being to find the values of the indeterminate 
 coefficients A, B, C, D, etc. To do this, we form x^ , x^, x*, etc., from (B) in 
 terms of y, and substitute in the second member of (A). Whence we have 
 f(y) =/' (y)* From this relation we can obtain the values of the indeterminates 
 A, B, C, D, etc., in the ordinary way. 
 
 Examples. 
 
 1. Given y = x -\- ^x^ + \oi^ + ^x"^ ■{■ etc., to revert the series, i. e., 
 to express the value of a; in a series involving y. 
 
 * This notation mean? that both members are functions of y, bat that they are not the same 
 function : one is the /function, and the other the /' function. 
 
288 
 
 APPENDIX. 
 
 SuGs.— Assume x = Ay + By* + Cy^ + By* + etc. 
 Whence x' =A*y* + 2ABy^ + 2AC I y* + etc., 
 
 + i?M 
 
 and X* = A*y* + etc., these developments being extended 
 
 as far as is necessary in order to determine four terms of the reverted 
 eeries. 
 
 Substituting these values in the given series we have 
 
 y = Ay + B y* + C 
 
 y' + D 
 
 y* + etc. 
 
 + M* + AB 
 
 + AG 
 
 
 + U' 
 
 + \B' 
 
 
 +A'B 
 
 
 -4- U* 
 
 
 Whence ^ = 1, B + iA*=0,C + AB-\-^A' = 0, and B+AC+^B'+A'B 
 
 + iA*=0. These give ^ = 1, B= -^ C=^, and i? = - -/,-. Therefore 
 
 the reverted series is ^ = y— ■Ky'+-^ 
 
 i [9 
 
 y^-iy^ + eta 
 
 2 to 6. Revert the following : 
 (2.) y = X + x' + x" + a^ + etc. 
 (3.) yzzzx + Sx' + bxf'-^lx'-hdr^-h etc. 
 (4.) i/ = x-ijr'-\-ix^~\x' + etc.* 
 (5.) i/ = 2x + Sx" -^ 4a^ -^ 6.r' + etc. 
 (6.) y = 1 + a: + K + ,K + i^ + etc.f 
 
 7. Required to express the value of y in tei'ms of x from the 
 relation 
 
 y •\- ay^ ■\- by^ -\- cy* + etc. = mx + ut^ -\- px^ + qx^ -\- etc. 
 
 Interpolation. 
 
 S24. Prob, — Having given a series of functions a, b, c, d, e, 
 etc., to find a function intermediate between any two of this series, 
 which function shall conform to the laio of the series. 
 
 III.— Let the series of functions be the logarithms of 232,233, 234, 235, etc., 
 viz., 2.365488, 2.367356, 2.339216, and 2.371068 ; let it be required to find the 
 logarithm of 233.4, i. e., the function I of the way from 2.367356 to 2.369216. 
 
 Solution. — The solution of this problem is simply an application of the 
 
 * In this example it will be more expeditioas to assume xz= Ay-\- Bjfi + Cy^ ^ etc , though it 
 is not essential. 
 
 t Tranepose the 1, put z.= y-1, and then revert the series z = x + ^t^ + .\-xi+.L.x* + iiic. 
 
 This i^ necessary, since the theory of Indeterminate Coefficients assumes that both variables 
 hecome at the same time ; i. «., that z=0, makes z=Q, 
 
INTERPOLATION OF SERIES. 
 
 289 
 
 formula for finding the nth term of a series by the Method of Differences 
 {:il4); viz., 
 
 The fith term = a + {n — l)i>i + ^^ ^ B^ 4- ^^ , ^ • i>3+etc. 
 
 But for our present purpose it is more convenient to replace the {n—1) of the 
 
 formula, where n represents the number of the term sought, by - , a fraction 
 
 which indicates the distance of the term sought, from the first term used, 
 this distance being measured by calling the distance between any two given 
 terms 1. Thus in the series a, b, c, d, e, etc., a term i^ of the way from & to c, 
 
 would be reckoned at a distance \%, or '3 from a, i. c, - would be § in this case. 
 
 <l 
 Now by this method of reckoning it is evident that the {n—1) of the formula must 
 
 be replaced by - , for w stands for the number of the term, which is one more ' 
 
 than the number of intervals between it and the first term. Thus the 4th 
 
 P 
 term is 3 intervals from the first term. Making this substitution of - for w — 1 
 
 the formula becomes 
 
 ,+etc. 
 
 Termtobeinterpolated=a+ ^i>, + ^^(^-1\D2+^ ^(^ -l) (^-2\d 
 
 325. ScH. 1. — On this formula we observe that when the series of func- 
 tions is such that the differences vanish, i. e., D^, D^, D^, or some order 
 becomes 0, the formula gives an absolutely correct result. But when the 
 differences do not vanish, the result is only an approximation. However, 
 such is the closeness of approximation, that for practical purposes only 
 second differences are usually needed, although sometimes third and fourth 
 become necessary. 
 
 Examples. 
 
 1. Finding from the tables the logarithms of 232, 233, 234, 235, 
 to be 2.365488, 2.367356, 2.369216, and 2.371068, required to inter- 
 polate the logarithm of 233.4. 
 
 SOLUTION. 
 
 ARGUMENTS.* 
 
 FUNCTIONS. 
 
 1st diff's. 
 
 2d diff's. 
 
 3d diff's. 
 .000000 
 
 232 
 
 233 
 234 
 235 
 
 2.365488 
 2.367356 
 2.369216 
 2.371068 
 
 .001868 
 .001860 
 .001852 
 
 -.000008 
 -.000008 
 
 * In such acas« the number is called the -4 rflrt»we»e, a^4 \X» logaritl^m Ihp A^nptipn. This 
 means simply that the logarithm is f^ fuuctjon of ^lie number (or argumppl). 
 
290 
 
 APPENDIX. 
 
 In this case a = 2.3G5488, Bi = .001808, B^ = -.000008, D^ = 0, and ^ = p 
 
 q 5 
 
 Ilcnce we have 
 
 log 232 = a = 2.365488 
 
 ^ i>, = I (.001868) = .002615 
 q 5 
 
 .'. log 233.4 = 
 Mctly as it is in the tables. 
 
 2.368101, which is ex- 
 
 2. Finding from the table the logarithms of 61, 62, etc., interpo- 
 late the logarithm of 62.23. 
 
 S26, ScH. 2. — When second differences only are to be used, and four 
 
 functions of the series are known, a convenient and excellent formula is 
 
 ol>tainecl thus: Let the four functions be a, 6, c, d^ and let it be required to 
 
 v' 
 interpolate between 6 and c. Let —, be the interval from h to the place of the 
 
 term to be interjiolated. Now if we compute from ft, instead of from a, the 
 ])receding formula will become 
 
 The interpolated function =b + ^\ Di +2W~^)^M' 
 
 in which Z), is the second of the first differences, i. «., the one which falls 
 between b and c ; or, in general, if we tabulate the differences as above, it 
 is the first difference which falls in the same horizontal line with the func- 
 tion to be interpolated. Again, as the second differences are supposed to be 
 different, it is best to take the arithmetical mean of the two, which mean 
 will also fall in the same horizontal line with the interpolated function. 
 
 3. Find by (326) the logarithm of 68.53 from the logarithms of 
 67, 68, 69, 70. (See table.) 
 
 ARGUMENTS. 
 
 FXJNCTIONS. 
 
 IST DIFF'S. 
 
 2d diff'b. 
 
 MEAN OF 
 
 2d diff's. 
 
 67 
 
 68 
 
 * 
 
 1.826075 
 1.832509 
 
 .006434 
 .006340 
 .006249 
 
 -.000090 
 -.000091 
 
 - .0000905 
 
 69 
 70 
 
 1.838849 
 1.845098 
 
 Z), = .006340, and 
 
 Here we have & = log 68 = 1.833509, ^ = ^ 
 
 Df = — .0000905. The student should make the substitutions and compare 
 with the table. 
 
INTEUPOLATION OF SERIES. 
 
 291 
 
 327 - ScH. 3. — But it is not for interpolating logarithms that this 
 method is chiefly used. For this purpose the method given in {196) is 
 preferable. The student will readily discover that the method of {106} 
 is identical with that just given if only first differences are used. When 
 great accuracy is required, and the tables used give the logarithms to 8 or 
 10 places, it sometimes becomes necessary to use mean second differences, 
 as above. It is, however, in Astronomy that Interpolation has its most im- 
 portant applications. Thus, suppose the Right Ascension (analogous to 
 terrestrial longitude) of a planet has been observed four times at intervals of, 
 say one day. By interpolation we may find its Right Ascension at each interme- 
 diate hour, or point of time. In this problem the Right Ascension is t^e function, 
 and the time ia the argument. 
 
 4. Tlie Right Ascension of Jupiter to-day, July 1st, at noon, is 
 10h.5m. 38.6s.; July 2d, at noon, it will be lOh. 6m. 18.86s.; on July 
 3d, lOh. 6m. 59.41s., and July 4th, lOh. 7m. 40.24s. What will it be 
 July 2d, at midnight ? 
 
 SOLUTION. 
 
 ARGUMENTS.* 
 
 FUNCTIONS.* 
 
 1st diff's. 
 
 2d diff's. 
 
 MEAN 
 
 2d diff's. 
 
 July 1. 
 July 2. 
 July 3. 
 July 4. 
 
 lOh. 5 m. 38.6?. 
 lOh. 6 m. 18.86 s. 
 lOh. 6 m. 59.418. 
 lOh. 7 m. 40.24 s. 
 
 40.26 s. 
 40.55 s. 
 40.83 s. 
 
 0.29 s. 
 0.28 s. 
 
 0.285s. 
 
 »' 1 
 In this case -, = x , & = lOh. 6m. 18.86s., Di = 40.55s., and Dj = 0.285s. 
 q Z 
 
 The answer is lOh. 6m. 39.1s. 
 
 5. To-day, July 1st, at noon, the moon's declination (distance 
 from the celestial equator) is 6° 38' 10".8 north ; at 4 o'clock it will 
 be 5° 45' 51".3 ; at 8 o'clock, 4° 53' 7".8 ; at midnight, 4° 0' 2".8 ; and 
 at 4 o'clock in the morning it will be 3° 6' 38".7. Interpolate for 
 the intermediate hours. 
 
 ♦ In tliis oxamplethe argument is the time, and the function is the Right Ascension, i. «., 
 the Right Ascension id a fuuction of the time. 
 
292 APPENDIX. 
 
 SECTION 11. 
 
 PERMUTATIONS. 
 
 328. Combinations ure the different groups which can be 
 made of m things taken n in a group, n being less than m. 
 
 III. — Taking the 5 letters a, h, c, d, e, we have the 10 following cojnbi nations 
 when the letters are taken 3 in a group, or, as it is usually expressed, taKen 3 
 and 3 : abc, abd, ahe, acd, ace, ode, bed, bee, bde, ede. Taken 2 and 2, we have the 
 following 10 combinations : ab, ac, ad, ae, be, bd, be, cd, ee, de. It is to be notieed 
 that no tico combinations eontain tlie same letters; t. e., they are different groups. 
 
 329, J^ermutations are the different orders in which things 
 can succeed each other. 
 
 III. — Thus the two letters a, b have the two permutations ab, ba. The three 
 letters a, b, c have the 6 permutations abc, acb, cab, bac, bcu, cba. 
 
 330. Arrauffenients are permutations of combinations. 
 
 III. — Taking the 10 combinations of 5 letters taken 3 and 3, and permuting 
 each combination, we get the arrangements of 5 letters taken 3 and 3. Thus 
 the combination abc gives the C arrangements abc, acb, cab, bac, bca, cba. In like 
 manner each of the 10 combinations of 5 letters taken 3 and 3 will give 6 arrange- 
 ments ; whence, in all, 5 letters taken 3 and 3 have 60 arrangements. 
 
 331, I^roiJ. — The number of Arraiigements of m things taken 
 n and n is 
 
 m (m - 1) (m -■ 2) (m - 3) (m - n +1). 
 
 Dem. — Let us consider the number of arrangements which can be made of the 
 m letters a, b, c, d, etc., taken 2 and 2. Letting a stand first, we can have ah, ac, 
 ad, etc., to w — 1 arrangements. Letting b stand first, we can have ba, be, bd, 
 etc., to w — 1 arrangements. Thus taking each of tlie m letters in turn we can 
 have m — 1 arrangements in each case, or m (m — 1) arrangements in all. 
 
 Again, eac7i of these m(m — 1)2 and 2 arrangements will give m — 2 arrange- 
 ments 3 and 3, by placmg before it each of the letters not involved in it. Thus 
 we have m{m — l){m — 2) arrangements of m letters taken 3 and 3. 
 
 Once more, ea^ih of these m{m — 1) {m — 2) 3 and 3 arrangements will give 
 m — 3 arrangements 4 and 4, by placing before it each of the letters not involved 
 in it. Thus we have m{m — 1) {m — 2) {m — 3) arrangements of m letters taken 
 4 and 4. 
 
 Finally, we observe the law ; i. e., the number of arrangements is equal to 
 
PERMUTATIONS. 293 
 
 the continued product of m{m — 1) {m — 2) {m — S) - - - - {m — (n — l)} or 
 m(m — 1) (m — 2) (m — B) - - - - {jn^n + 1). 
 
 332, Cor. 1. — The number of Permutations o/m. things is 
 
 1.2.3.4 m. 
 
 This is evident since arrangements become permutations when the number Tn 
 a group is equal to the whole number considered ; i. e., when n = m. 
 
 333, Cor. 2. — If' p of the m letters are alike {as each a), q otKefs 
 alike, r others alike, etc., the number of permutations is 
 
 1.2.3-4 m 
 
 |p X [q X [r X etc. * 
 
 Thus consider the permutations of a, b, c, d, viz., abed, bacd, acdb, bcda, acbd, 
 bead, aMc, bade, adcb, bdca, etc. Suppose b to become a, then since for any par- 
 ticular position of c and d, as in abed, there are as many permutations of the foifr 
 letters as there can be permutations of the two letters a and b, viz., 1 x 3 ; if & 
 becomes a there will be 1 x 2 fewer permutations when these two letters are 
 
 alike than when they are different, i. e., ~' — . 
 
 So, in general, if p of the letters are alike, there will be 1-3 -3 - - - - jp, or 
 [p fewer permutations than if they are all different, etc. 
 
 334, Cor. 3. — T7ie number of Com,binations of m things taken 
 
 n and n is 
 
 m (m — 1 ) (m — 2) (m — 3) (m — n + 1) 
 
 _________ . 
 
 Since arrangements are permutations of combinations, the number of ar- 
 rangements of w things taken n and n is equal to the number of combinations 
 of ni things taken n and n multiplied by the number of permutations of n 
 things. Hence the number of combinations is equal to the number of arrange- 
 ments of m things taken n and n divided by the number of permutations of n 
 things. 
 
 Examples. 
 
 1. How many permutations can be made of the letters in the word 
 marble? Of those in A o m e .^ 0^ iho^Qmlog arithms? 
 
 2. How many arrangements can be made of 10 colors taken 3 and 
 3 ? Of 7 colors taken 2 and 2 ? Taken 3 and 3 ? 4 and 4 ? 5 and 
 5 ? 6 and 6 ? 7 and 7 ? How many mixtures in each case, irre- 
 spective of proportions ? 
 
 3. How many different products can be made from the 9 digits 
 taken 2 and 2 ? 3 and 3 ? 4 and 4 ? 5 and 5 ? 6 and 6 ? 7 and 
 
 ? 8 and 8 ? 9 and 9 ? 
 
294 APPENDIX. 
 
 4. How many different numbers can be represented by the 9 digits 
 taken 2 and 2 ? 3 and 3 ? 4 and 4 ? etc. 
 
 5. In a certain district 3 representatives are to be elected, and tliere 
 are 6 candidates. In how many different ways may a ticket be made 
 up? 
 
 6. There are 7 chemical elements which will unite with each other. 
 How many ternary compounds can be made from them ? How many 
 binary ? 
 
 7. How many different sums of money can be paid with 1 cent, 1 
 3-cent piece, 1 5-cent piece, 1 dime, 1 15-cent piece, 1 25-cent piece, 
 and 1 50-cent piece ? 
 
 SuQ. — If taken 1 and 1, how many ? If 2 and 2, how many ? If 3 and 3, etc.? 
 How many in all ? 
 
 8. In how many ways can 13 ladies and 12 gentlemen arrange 
 themselves in couples ? 
 
 9. If you are to select 7 articles out of 12, how many different 
 choices have you ? 
 
 10. How many different sums can be made from 1, 2, 3, 4, 5, 6, 
 taken 2 and 2 ? 
 
 11. How many permutations can be made from the letters in the 
 word possessions? (See 333,) How many from the letters 
 in the word consistencies? 
 
 12. How many different signals can be made with 10 different- 
 colored flags, by disi:)laying them 1 at a time, 2 at a time, 3 at a time, 
 etc., the relative positions of the flags with reference to each other 
 not being taken into account ? 
 
 Probabilities. 
 
 335. TJie Mathematical I^rohdbility of an event is the 
 number of favorable opportunities divided by the whole number of 
 opportunities. TJie Mathematical Iniprohahility is the number of un- 
 favorable opportunities divided by the whole number of opportunities. 
 
 III. — A man draws a ball from a bag containing 5 white and 2 black balls ; 
 the opportunities favorable to drawing a white ball are five, and the whole num- 
 ber of opportunities is seven ; hence the mathematical probability of drawing 
 a white ball is \. The mathematical improbability of drawing a white ball is \. 
 
PROBABILITIES. 295 
 
 Examples. 
 
 1. I learn that from a vessel on which my friend had taken pass- 
 age, one person has been lost overboard. There were 40 passengers, 
 and 20 in the crew. What is the probability that my friend is safe ? 
 "What the improbability ? If I learn that a passenger is lost, what 
 then is the probability that my friend is safe? What that he is 
 lost? 
 
 2. A man fires into a flock of birds of which 6 are white, 4 black, 
 5 slate-colored, and 3 piebald. If he kills one, what is the probability 
 of its being a black bird ? What the improbability of its being pie- 
 bald ? How much more probable is it that he will kill a white than 
 a piebald bird ? A black than a piebald ? 
 
 3. Twenty- three persons sit around a table. What is the proba- 
 bility of any given couple sitting together ? 
 
 III.— Call the two persons A and B. Then wherever A may sit, there are 22 
 others who may sit beside him in one of two places (on his right or left). There 
 are therefore 2 favorable and 20 unfavorable opportunities. 
 
 4. What are the odds against the fourth of July coming on Sun- 
 day in any year taken at random ? 
 
 SUG, — The odds against an event is the ratio of the unfavorable to the favor- 
 able opportunities. 
 
 5. The moon changes about once in 7 days. What is the proba- 
 bility that a change of weather will come within 3 days of a change 
 in the moon ? 
 
 6. The letters a, e, m, n, can be arranged so as to form four words, 
 viz., mane, mean, name, amen. If they are arranged at random, 
 what is the probabihty of their forming a word? What the "odds 
 against " their forming a word ? 
 
 7. Show that the probability that a leap-year will contain 53 Sun- 
 days is f . 
 
 8. Three balls are to be drawn from an urn which contains 5 ])lack, 
 3 red, and 2 white balls. What is the probability of drawing 2 
 black balls and 1 red ? 
 
 SuG's.— The first question is, How many opportunities in all ? That is, how 
 many different groups {combinations) can be made of 10 balls taken 3 and 3. 
 Second, How many opportunities favorable to drawing two black balls and one 
 
296 APPENDIX. 
 
 red at the same time ? There are 5 black balls, and these can be combined 2 and 
 
 5 4 
 
 2 in r-s , or 10, ways ; and as one of the three red balls can be obtained in 3 
 1 • /* 
 
 ways, each one of these combined with one of the 10 ways of obtaining the 
 
 black balls will give 10 x 3, or 30, favorable opportunities for selecting the balls 
 
 as desired. The probability is therefore -,^^o> or *• 
 
 9. If from a lottery of 30 tickets, marked 1, 2, 3, etc., 4 tickets are 
 drawn, what is the probability that 3 and 5 are among them ? What 
 are the odds against it ? 
 
 Sug's. — From 30 how many combinations of 4 and 4? From 28 how many 
 combinations of 2 and 2 ? Odds against drawing 3 and 5, 143 to 2. 
 
 10. A bag contains a $5 bill, $10 bill, and 6 blanks. What is the 
 expectation of one drawing ? That is, what is one drawing worth ? 
 
 SUG.— The probability that one draught will take the $5 bill is \, and hence 
 is worth $^. The probability that the $10 note will be drawn is also i, and 
 hence this expectation is %^^. The entire expectation is therefore $\-, or 
 $1.87^. Hence a gambler who should sell such chances at $2 each, would in 
 the long run make money. 
 
 11. What is the expectation of a draught from a bag containing 5 
 $2 bills, 4 $5 bills, 'Z $10 bills, 1 $100 bill, and 50 blanks? 
 
 12. In a given bag are 5 $2 bills, 3 $5 bills, and 6 blanks. What 
 is the expectation of 2 draughts ? 
 
 Suo's. — There are - ' ■ = 91 opportunities, or ways in which 2 things can 
 1 • »j 
 be drawn from 14. 
 
 5-4 . ' 
 
 There are :j — - ways in which $2 bills rnay be drawn. Hence the probability 
 
 of drawing 2 $2 bills is ^", and this expectancy is $if. 
 
 In like manner the probability of drawing 2 $5 bills is /, , and this expect, 
 ancy is $3?. 
 
 The probability of drawing 2 blanks is ^f , and this expectancy 0. 
 
 The probability of drawing 1 $2 and 1 $5 bill is ^\, and this expectancy is 
 
 $W- 
 
 The probability of drawing 1 $2 bill and 1 blank is ^, and this expectancy 
 i8$l?- 
 
 The probability of drawing 1 $5 bill and 1 blank is ^ ?, and this expectancy 
 is $1^. 
 
 The entire expectancy, or worth, of 2 draughts is therefore ^1 + l^+^^-i- + f?- 
 -f 1^ dollars, or $3.57f . 
 
 Observe that the sum of all the probabilities, i. 6., \^ + g^ -H i } + i.f + y^ + at, 
 is 1, as it should be. 
 
PROBABILITIES. 297 
 
 That the probabilit}^ of drawing 1 $3 bill and 1 $5 is ^f, is seen thus : There 
 are 5 opportunities favorable to drawing 1 $2 bill, and with each of these there 
 arc 3 opportunities favorable to drawing 1 $5 bill. 
 
 13. There are 4 white balls and 3 black ones in one bag, and 2 
 white ones and 7 black ones in another bag. What is the probability 
 of drawing a white ball from each bag at the first draught from 
 each ? 
 
 Solution.— There arc in all 7 opportunities of drawing a ball from the first 
 bag, and with each one of these there are 9 opportunities from the second 
 bag ; hence there are 7 x 9, or C3 opportunities in all. Again, there are 4 favor- 
 able opportunities for drawing a white ball from the first bag, and with each of 
 these there are 2 favorable opiwrtunities for drawing a white ball from the 
 Eccond bag; t. e., there are in all 4 x 2, or 8, favorable opportunities. Hence 
 the probability is -^.i. Notice that this compound probability is the product of the 
 two simple probabilities. 
 
 14. The probability that A can solve a problem is f, and that B 
 can do the same is \, what is the joint probability ? 
 
 Sug's. — The student will observe that there are 4 possible results, viz. : 
 1. Both may succeed, of which the probability is -^- ; 2. A may succeed and B fail, 
 of which the probability is f,! ; 3. 5 may succeed and A fail, of which the prob- 
 ability is -/a-; and 4. Both may fail, of which the probability is \%. Now either 
 the first, second, or the third result will give a solution. Hence the probability 
 of success is ^ + H + a*s = §f . or \. 
 
 This result may be more expeditiously obtained by considering that they 
 will succeed if both do not fail. The probability of J.'« failure is |, and of B's y. 
 Hence the probability that both will fail is ^ x f , or ?^; and the probability of 
 success is 1— f, or y. 
 
 15. It may be said that on an average 10 persons will die during 
 the next 10 years 
 
 Out of every G2 whose present age is 30, 
 
 « a 45 a a 40, 
 
 «« « 35 " '' 50, 
 
 « « 25 " " 60. 
 
 What is the probability Unit a person who is 30 will live till he is 
 60 ? What that a person who is 40 will live till he is 70 ? 
 
 StjG's.— Let ua examine the probability that the man who is 30 will die before 
 he if OJ. The probability that ho dies before 40 is |i^, and that he lives to 40 
 f |. Now the probability that a man who is 40 dies before 50 is ii Hence the; 
 probability i.i -^^ of §i that this man lives to 40 and dies between 40 and 50, or 
 it Is 5i of ^^1 that ho lives to 50. Finally, ^le probability that he dies between 
 
298 ATPLNDIX. 
 
 50 and 60 is ^ of ^ of H, or »t is U of H of B ^liat he lives from 50 to 6a 
 Hence the probability that a man who is 30 will die before he is 60 is 
 
 i^ + B X -L^ + B X ^^ X H,ori^g; 
 
 and, consequently, the probability that he will live is 1 — ^^, or i^§; i. e., it is 
 a little more probable tliat a man who is 30 will die before he is 60, than that 
 he will live to 60. 
 
 16. What is the probability that two persons, A and B, aged re- 
 spectively 30 and 40, will be alive 10 years lience ? 
 
 SUG'S.— Chance of A's being alive ^, of B'a U, of both ^ x J^, or ^^. 
 
LOGARITHMS OF NUMBERS. 
 
 N. 
 
 Log. 1 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 1 
 
 O'OOOOOO 
 
 26 
 
 I. 414973 
 
 51 
 
 1.707570 
 
 7(] 
 
 1-880814 
 
 2 
 
 0'3oio3o 
 
 27 
 
 i.43i364 
 
 52 
 
 1.716003 
 
 U 
 
 1-886491 
 
 8 
 
 0.477121 
 
 23 
 
 1.447158 
 
 53 
 
 1-724276 
 
 73 
 
 1-892095 
 
 4 
 
 o.6o2o6o 
 
 2'J 
 
 1.462393 
 
 54 
 
 1.732394 
 
 79 
 
 1.897627 
 
 5 
 
 0-698970 
 
 80 
 
 i-477'2i 
 
 55 
 
 1-740363 
 
 SO 
 
 1-903090 
 
 6 
 
 0.778151 
 0.845098 
 
 81 
 
 1.491362 
 
 56 
 
 I -748188 
 
 81 
 
 1-908485 
 
 7 
 
 82 
 
 i.5o5i5o 
 
 57 
 
 1.755875 
 
 82 
 
 i-9i38i4 
 
 8 
 
 0.903090 
 
 83 
 
 I.5i85i4 
 
 58 
 
 1-763428 
 
 83 
 
 1-919078 
 
 9 
 
 0.954243 
 
 84 
 
 1.531479 
 
 59 
 
 1.770852 
 
 84 
 
 1-924279 
 
 10 
 
 I'OOOOOO 
 
 85 
 
 1.544068 
 
 60 
 
 i-778i5i 
 
 85 
 
 I -929419 
 
 11 
 
 i.o4i3o3 
 1-079181 
 I- 1 13943 
 
 86 
 
 I -556303 
 
 61 
 
 1-785330 
 
 86 
 
 1-934498 
 
 12 
 
 37 
 
 1.568202 
 
 62 
 
 1-792392 
 
 87 
 
 1-939519 
 
 13 
 
 88 
 
 1-579784 
 
 63 
 
 I -799341 
 
 88 
 
 I-9444C3 
 
 14 
 
 1.146128 
 
 89 
 
 1-591065 
 
 64 
 
 1-806180 
 
 89 
 
 1-949390 
 
 15 
 
 1-176091 
 
 40 
 
 1-602060 
 
 1 g:> 
 
 1-812913 
 
 90 
 
 1-954243 
 
 IG 
 
 1-204120 
 
 41 
 
 1-612784 
 
 I 66 
 
 1-819544 
 
 91 
 
 1-9590:1 
 
 17 
 
 1-23044Q 
 
 1-255273 
 1-278754 
 
 42 
 
 1-623249 
 
 1 67 
 
 1-826073 
 
 92 
 
 1.963783 
 
 18 
 
 43 
 
 1 -633468 
 
 i 68 
 
 1-832509 
 
 !3 
 
 1 .968483 
 
 19 
 
 44 
 
 1-643453 
 
 i 69 
 
 1-838849 
 1-845098 
 
 94 
 
 1.973128 
 
 20 
 
 I -301030 
 
 45 
 
 1-653213 
 
 70^ 
 
 95 
 
 1.977724 
 
 21 
 
 1-322219 
 
 46 
 
 1-662758 
 
 71 
 
 1-851258 
 
 96 
 
 I -982271 
 
 22 
 
 1-342423 
 
 47 
 
 1-672098- 
 
 7-J 
 
 1-857333 
 
 97 
 
 1.986772 
 
 23 
 
 1.361728 
 
 43 
 
 1-681241 
 
 73 
 
 1.863323 
 
 9^ 
 
 I -991226 
 
 24 
 
 1.3802 1 1 
 
 49 
 
 1-690196 
 
 74 
 
 1-869232 
 
 99 
 
 1-995635 
 
 25 
 
 1.397940 
 
 50 
 
 1.698970 
 
 75 
 
 1 
 
 1-875061 
 
 100 
 
 2-000000 
 
 Remark. — In the following Table, the frst two fgurea^ in the first column of 
 Logarithms, are to be prefixed to each of the numbers, in the same horizontal 
 line, in the next nine columns; but when a point (•) occurs, a is to bo put 
 in its place, and the two initial fjurcs are to be taken from the next line below. 
 
SCO 
 
 LOGARirnMS cir numbers. 
 
 K. 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 I). 
 
 100 
 
 101 
 102 
 
 103 
 104 
 105 
 106 
 107 
 108 
 109 
 
 4321 
 86oo 
 
 012837 
 7o33 
 
 021180 
 53o6 
 o384 
 
 033424 
 7426 
 
 0434 
 4751 
 9026 
 3259 
 745 1 
 i6o3 
 57.5 
 
 nit 
 7S25 
 
 0868 
 5i8i 
 9451 
 368o 
 7868 
 2016 
 6i25 
 •195 
 4227 
 8223 
 
 i3oi 
 
 98^ 
 4100 
 8284 
 2428 
 6533 
 •600 
 4628 
 8620 
 
 1734 
 6o38 
 •3oo 
 452. 
 8700 
 2841 
 6942 
 1004 
 5029 
 9017 
 
 2166 
 6466 
 
 •724 
 4940 
 91 16 
 
 3252 
 
 7350 
 1408 
 
 5430 
 9414 
 
 2598 
 6894 
 1147 
 536o 
 95J2 
 3664 
 
 1812 
 583o 
 981 1 
 
 3029 
 7321 
 1570 
 5779 
 9947 
 4075 
 8164 
 2216 
 623o 
 •207 
 
 3461 
 
 7748 
 1993 
 6197 
 •36 1 
 4486 
 8571 
 2619 
 6629 
 •602 
 
 3891 
 8174 
 24i5 
 6616 
 
 8978 
 
 3o2i 
 
 7028 
 
 •998 
 
 4.12 
 428 
 424 
 419 
 4.6 
 412 
 4o3 
 404 
 400 
 396 
 
 110 
 111 
 112 
 lid 
 114 
 115 
 116 
 117 
 
 lis 
 
 119 
 
 041393 
 5323 
 9218 
 
 053078 
 6905 
 
 060698 
 44^8 
 8186 
 
 071882 
 5547 
 
 1787 
 5714 
 0606 
 
 3463 
 
 7286 
 
 1075 
 4832 
 8557 
 225o 
 59.2 
 
 2182 
 oio5 
 
 7666 
 
 1452 
 5206 
 8928 
 2617 
 6276 
 
 2576 
 6495 
 •380 
 423o 
 8046 
 1829 
 558o 
 9298 
 2983 
 6640 
 
 688? 
 •766 
 46i3 
 8426 
 2206 
 5953 
 
 7004 
 
 3362 
 
 ml 
 
 2582 
 
 6326 
 ••38 
 3718 
 7368 
 
 3755 
 7664 
 1538 
 5378 
 9185 
 2g58 
 6699 
 •407 
 4o8d 
 7731 
 
 414S 
 8o53 
 1924 
 5760 
 9563 
 3333 
 7071 
 
 2t 
 
 8094 
 
 4540 
 8442 
 2309 
 6142 
 0942 
 3709 
 7443 
 1145 
 4816 
 8457 
 
 4932 
 
 8b3o 
 2C94 
 6524 
 
 •320 
 
 4o83 
 7.S.5 
 i5.4 
 5.82 
 8819 
 
 393 
 389 
 386 
 
 382 
 
 l]l 
 32 
 369 
 366 
 363 
 
 120 
 121 
 122 
 123 
 124 
 125 
 126 
 127 
 128 
 129 
 
 130 
 131 
 132 
 138 
 134 
 135 
 106 
 137 
 133 
 139 
 
 140 
 141 
 142 
 143 
 
 1:4 
 U5 
 1-16 
 147 
 143 
 U9 
 
 079 1 81 
 
 082785 
 636o 
 0905 
 
 093422 
 6910 
 
 100J71 
 38o4 
 7210 
 
 110590 
 
 o543 
 3i44 
 6716 
 •258 
 3772 
 7257 
 0715 
 4146 
 7549 
 0926 
 
 7071 
 •611 
 4122 
 7604 
 1059 
 4487 
 7^88 
 1263 
 
 •266 
 3861 
 7426 
 •963 
 4471 
 79^1 
 i4o3 
 4828 
 
 b2 27 
 
 1599 
 
 •626 
 4219 
 
 4B20 
 82 >8 
 
 1747 
 5169 
 
 8563 
 
 1934 
 
 XI 
 
 8i36 
 1667 
 5169 
 8644 
 2091 
 55io 
 8903 
 2270 
 
 1347 
 4934 
 8490 
 2018 
 65i8 
 8990 
 2434 
 585, 
 9241 
 26o5 
 
 1707 
 5291 
 8845 
 2370 
 5866 
 9335 
 
 2777 
 6191 
 9579 
 2940 
 
 2067 
 5647 
 9198 
 2721 
 62i5 
 0681 
 3119 
 653i 
 C916 
 3275 
 
 2426 
 6004 
 9552 
 
 3071 
 6:62 
 ••26 
 3462 
 6871 
 •253 
 3609 
 
 36o 
 357 
 355 
 35i 
 
 34? 
 343 
 340 
 338 
 335 
 
 113943 
 
 7271 
 120574 
 3852 
 nio5 
 i3o334 
 3339 
 6721 
 
 i43oi5 
 
 7603 
 0903 
 4178 
 
 0655 
 3858 
 7037 
 •194 
 3327 
 
 4611 
 
 7934 
 
 I23l 
 
 45o4 
 7753 
 0977 
 4177 
 -354 
 •5o8 
 3639 
 
 4944 
 
 i265 
 
 1:60 
 
 41 3o 
 8076 
 .298 
 4496 
 7671 
 •822 
 3951 
 
 5278 
 
 8:95 
 
 18.^8 
 
 5i56 
 fc3v9 
 1619 
 4814 
 7987 
 ii36 
 4263 
 
 56ii 
 8926 
 2216 
 5481 
 8722 
 
 83o3 
 1450 
 
 4574 
 
 5,43 
 9256 
 
 2)44 
 5bo6 
 9045 
 2260 
 5451 
 8618 
 1763 
 4885 
 
 6276 
 9586 
 287I 
 6i3i 
 9368 
 258o 
 5769 
 8934 
 2076 
 5196 
 
 6608 
 9915 
 
 6456 
 9690 
 2900 
 6086^ 
 9249 
 2389 
 5507 
 
 6940 
 •245 
 3525 
 6781 
 ••.2 
 3219 
 64o3 
 9564 
 2702 
 58i8 
 
 333 
 33o 
 328 
 325 
 323 
 
 321 
 
 3i8 
 3i5 
 3i4 
 3u 
 
 146128 
 
 152288 
 5336 
 
 n^a 
 
 i6i3o8 
 4353 
 7317 
 
 170262 
 3ifc6 
 
 6438 
 9527 
 2594 
 6640 
 8664 
 1667 
 465o 
 7^1 3 
 0^35 
 3473 
 
 6748 
 9S35 
 2900 
 6943 
 8965 
 1967 
 
 4947 
 7908 
 
 0^48 
 3769 
 
 7o58 
 •142 
 32o5 
 6246 
 9266 
 2266 
 5244 
 8203 
 1141 
 4060 
 
 7367 
 •449 
 35io 
 6049 
 9:67 
 2564 
 5541 
 8497 
 1434 
 435i 
 
 76:6 
 •7^^6 
 38i5 
 6S52 
 9^-68 
 i863 
 583S 
 8792 
 1726 
 4641 
 
 79S5 
 io63 
 4120 
 7'54 
 •i63 
 3i6i 
 6.34 
 9086 
 2019 
 4932 
 
 IJ70 
 4424 
 
 7457 
 0469 
 3460 
 
 643o 
 9380 
 
 23 II 
 5222 
 
 86o3 
 
 1676 
 4728 
 
 7759 
 •769 
 3758 
 6726 
 9674 
 26o3 
 55i2 
 
 891 1 
 1982 
 5o32 
 8061 
 1068 
 4o55 
 7022 
 9968 
 2895 
 58o2 
 
 309 
 307 
 3o5 
 3o3 
 3oi 
 299 
 297 
 295 
 293 
 291 
 
 150 
 151 
 152 
 153 
 154 
 1.-5 
 156 
 157 
 158 
 159 
 
 176091 
 8977 
 
 181844 
 4691 
 7521 
 
 190332 
 3.25 
 
 5899 
 
 8657 
 
 201397 
 
 633i 
 9264 
 2129 
 4975 
 7fco3 
 0612 
 34o3 
 6176 
 8932 
 1670 
 
 6670 
 9552 
 240 
 5259 
 8084 
 0802 
 368 1 
 64^3 
 9206 
 1943 
 
 6959 
 9S39 
 2700 
 5542 
 8366 
 
 39^9 
 6729 
 9481 
 2216 
 
 7248 
 •126 
 2985 
 5825 
 8647 
 I45l 
 4237 
 7co5 
 9755 
 2488 
 
 7536 
 •4i3 
 3270 
 6108 
 8928 
 1730 
 4314 
 7281 
 •C29 
 2761 
 
 7825 
 «-699 
 3555 
 tlgl 
 9209 
 2010 
 
 •3o3 
 3o33 
 
 8m3 
 •985 
 3839 
 6674 
 9490 
 2289 
 5069 
 7832 
 •577 
 33o5 
 
 8401 
 1272 
 4.23 
 
 6936 
 
 977 • 
 2D67 
 5346 
 8107 
 •85o 
 3577 
 
 8689 
 1558 
 4407 
 7239 
 ••5 1 
 2846 
 5623 
 8382 
 1 1 24 
 3846 
 
 2P9 
 
 287 
 285 
 283 
 281 
 279 
 
 276 
 274 
 
 272 
 
 N. 
 
 
 
 1 1 2 
 
 8 
 
 4 
 
 5 
 
 6 7 
 
 8 
 
 9 
 
 D. 
 
LOGATJTmrS OF KTTMBERS. 
 
 301 
 
 N. 
 
 
 
 1 
 
 ?. 
 
 3 
 
 4 
 
 5 
 
 6 
 
 y 
 
 8 
 
 9 
 
 D. 
 
 160 
 
 304130 
 
 4391 
 
 4663 
 
 4934 
 
 5204 
 
 5475 
 
 5746 6016 
 
 6286 
 
 6556 
 
 271 
 
 161 
 
 6826 
 
 7096 7J65 i 76J4 
 
 7904 
 
 8173 
 
 8441 8710 
 1121 i383 
 
 8979 
 
 9247 
 
 269 
 
 162 
 
 95i5 
 
 978J ••5i I •J 19 
 
 •586 
 
 •853 
 
 ;654 
 
 1921 
 
 267 
 
 163 
 
 312188 
 
 2454 ! 2720 29% 
 
 3252 
 
 35iS 3783 1 4049 i 43i4 1 
 
 4379 
 
 266 
 
 164 
 
 4844 5109 ; 5373 1 
 
 5b J 3 
 
 5902 
 
 6166 
 
 6430 6694 
 
 6957 
 
 7221 
 
 264 
 
 165 
 
 7484 
 
 7747 
 
 8010 
 
 8273 
 
 8d36 
 
 8793 
 
 9060 9323 
 
 9585 
 
 9846 
 
 262 
 
 166 
 
 320108 
 
 0370 
 
 o63i 
 
 0892 
 
 n53 
 
 1414 
 
 1675 1936 
 
 2196 
 
 2456 
 
 261 
 
 107 
 
 2716 
 
 5309 
 
 ^9"^^ 
 
 3236 
 
 3496 
 
 3755 
 
 4oi5 
 
 4274 
 
 4533 
 
 4792 
 7372 
 
 5o5i 
 
 259 
 
 168 
 
 5D68 
 
 5826 
 
 6084 i 
 
 6342 
 
 6600 
 
 6858 
 
 7.15 
 
 763o 
 
 258 
 
 169 
 
 7887 
 
 8144 
 
 8400 
 
 8657 
 
 8913 
 
 9170 
 
 9426 
 
 9682 
 
 9938 
 
 •193 
 
 256 
 
 170 
 
 230449 
 
 0704 
 
 0960 
 
 I2l5 
 
 1470 
 
 1724 
 
 :?;? 
 
 2234 
 
 2488 
 
 2742 
 
 254 
 
 171 
 
 325o 
 
 35o4 
 
 3757 
 
 4011 
 
 4264 
 
 4770 
 
 5o23 
 
 5276 
 
 253 
 
 173 
 
 5d28 
 
 5781 
 
 6o33 
 
 6285 
 
 6537 
 
 6789 
 
 7041 
 
 7292 
 
 7544 
 
 7795 
 
 252 
 
 173 
 
 8046 
 
 8297 
 
 8548 
 
 8799 
 
 T5tl 
 
 9299 
 
 9550 
 
 9800 
 
 ••5o 
 
 •3oo 
 
 25o 
 
 17t 
 
 340349 
 
 0799 
 
 1048 
 
 1297 
 
 1795 
 
 2044 
 
 2293 
 
 2541 
 
 2790 
 
 1% 
 
 175 
 
 3o38 
 
 3286 
 
 3534 
 
 3782 
 
 4o3o 
 
 4277 
 
 4525 
 
 4772 
 
 5019 
 
 5266 
 
 176 
 
 55i3 
 
 5759 
 
 6006 
 
 6252 
 
 6499 
 8934 
 1395 
 
 6745 
 
 6991 
 
 7237 
 
 7482 
 
 772S 
 
 246 
 
 177 
 
 173 
 
 7973 
 
 25o420 
 
 8219 
 0664 
 
 8464 
 0903 
 
 8709 
 
 Ii5i 
 
 Vefs 
 
 9443 
 
 1881 
 
 9687 
 
 2125 
 
 l^ll 
 
 •176 
 2610 
 
 243 
 
 243 
 
 179 
 
 2853 
 
 3096 
 
 3333 
 
 358o 
 
 3822 
 
 4064 
 
 43o6 
 
 4548 
 
 4790 
 
 5o3i 
 
 242 
 
 180 
 
 255273 
 
 55i4 
 
 5755 
 
 5996 
 
 6237 
 
 6477 
 
 6718 
 
 6958 
 9355 
 
 7198 
 
 7439 
 
 241 
 
 ISl 
 
 7679 
 
 70.8 
 o3io 
 
 8i58 
 
 83o3 
 
 8637 
 
 8877 
 
 1263 
 
 91 16 
 
 9594 
 
 9833 
 
 239 
 
 183 
 
 260071 
 
 0343 
 
 0787 
 
 1025 
 
 i5oi 
 
 1739 
 
 1976 
 4346 
 
 2214 
 
 233 
 
 183 
 
 245i 
 
 26S8 
 
 2925 
 
 3i62 
 
 3399 
 
 3636 
 
 3873 
 
 4109 
 
 4582 
 
 237 
 
 1S4 
 
 43i8 
 
 5o54 
 
 6290 
 
 5525 
 
 5761 
 
 ^t 
 
 6232 
 
 6467 
 
 6702 
 
 6937 
 
 235 
 
 185 
 
 7172 
 
 7406 
 
 7641 
 
 7875 
 
 8110 
 
 8578 
 
 8812 
 
 9046 
 
 9279 
 
 234 
 
 1S6 
 
 95i3 
 
 9746 
 
 9980 
 
 •2l3 
 
 •446 
 
 •679 
 
 •912 
 
 1144 
 
 1377 
 
 1609 
 
 233 
 
 187 
 
 271842 
 
 2074 
 
 23o6 
 
 2533 
 
 2-'70 
 
 3ooi 
 
 3233 
 
 3464 
 
 360 
 
 3927 
 
 232 
 
 183 
 
 4.58 
 
 4389 
 
 4620 
 
 485o 
 
 5o8i 
 
 53n 
 
 5542 
 
 5772 
 
 6002 
 
 6232 
 
 23o 
 
 189 
 
 6462 
 
 6692 
 
 6921 
 
 7i5i 
 
 7330 
 
 7609 
 
 7838 
 
 8067 
 
 820 
 
 8525 
 
 229 
 
 H'O 
 
 278754 
 
 8982 
 
 0211 
 
 9439 
 
 9667 
 
 9895 
 
 •123 
 
 •35i 
 
 0578 
 
 •806 
 
 223 
 
 191 
 
 28io33 
 
 1261 
 
 14S8 
 
 1715 
 
 1942 
 
 2169 
 
 2396 
 
 2622 
 
 2849 
 
 3075 
 
 227 
 226 
 
 193 
 
 33oi 
 
 3327 
 
 3753 
 
 3979 
 
 42o5 
 
 443i 
 
 46D6 
 
 4882 
 
 5 1 07 
 
 5332 
 
 103 
 
 5557 
 
 5782 
 
 6007 
 
 6232 
 
 6456 
 
 663 1 
 
 6905 
 
 7i3o 
 
 7354 
 
 7573 
 
 225 
 
 194 
 
 7802 
 
 8026 
 
 8249 
 
 8473 
 
 8696 
 
 8920 
 
 9143 
 
 9366 
 
 f,l 
 
 9812 
 
 223 
 
 105 
 
 2QOo35 
 
 0257 
 
 2478 
 
 0480 
 
 0702 
 
 0925 
 
 1147 
 3363 
 
 1369 
 
 i59t 
 
 2o34 
 
 222 
 
 106 
 
 2256 
 
 2699 
 
 2920 
 
 3i4i 
 
 3584 
 
 3jo4 
 
 4025 
 
 4246 
 
 221 
 
 107 
 
 4466 
 
 46S7 
 
 4907 
 
 6127 
 7323 
 
 5347 
 
 5567 
 
 5787 
 
 8198 
 
 6226 
 
 6446 
 
 220 
 
 103 
 
 6665 
 
 68S4 
 
 7104 
 
 7542 
 
 7761 
 
 7979 
 
 8416 
 
 8633 
 
 219 
 
 199 
 
 8853 
 
 9071 
 
 9289 
 
 9507 
 
 9725 
 
 9943 
 
 •161 
 
 •378 
 
 •595 
 
 •81 3 
 
 218 
 217 
 
 200 
 
 3oio3o 
 
 1247 
 
 1464 
 
 1681 
 
 1898 
 
 2114 
 
 233i 
 
 2547 
 4706 
 
 2764 
 
 2980 
 
 201 
 
 It 
 
 34. i 
 
 3623 
 
 3 84 4 
 
 4059 
 
 4275 
 
 4491 
 
 4921 
 
 5i36 
 
 216 
 
 203 
 
 5566 
 
 5781 
 
 5996 
 
 6211 
 
 6425 
 
 6wJ9 
 
 8778 
 
 6854 
 
 7068 
 
 72S2 
 
 2l5 
 
 203 
 
 7496 
 
 7710 
 
 792 i 
 
 8137 
 
 835i 
 
 8364 
 
 8991 
 
 9204 
 
 9417 
 
 2l3 
 
 204 
 
 030 
 
 9843 
 
 ••56 
 
 •263 
 
 04S. 
 
 •693 
 
 •906 
 
 1118 
 
 i33o 
 
 1 542 
 
 212 
 
 205 
 
 311754 
 
 1966 
 
 2177 
 
 2389 
 
 2600 
 
 2812 
 
 3o23 
 
 3234 
 
 34 ',5 
 
 3656 
 
 211 
 
 206 
 
 3J67 
 
 4078 
 
 4289 
 
 4499 
 
 47«o 
 
 4920 
 
 5i3o 
 
 5340 
 
 555i 
 
 5760 
 7854 
 
 210 
 
 207 
 
 5970 
 
 6180 
 
 6390 
 
 nv. 
 
 6809 
 8893 
 
 7018 
 
 7227 
 
 7436- 
 
 764^ 
 
 209 
 
 203 
 
 8o63 
 
 8272 
 
 8481 
 
 0106 
 
 9314 
 
 9522 
 
 973 d 
 
 9933 
 
 203 
 
 209 
 
 320146 
 
 o354 
 
 o562 
 
 0769 
 
 0977 
 
 Ti84 
 
 1391 
 
 1598 
 
 i8o5 
 
 2012 
 
 207 
 
 210 
 
 322?I9 
 
 24^6 
 
 2633 
 
 2839 
 
 3046 
 
 3252 
 
 3458 
 
 3665 
 
 f^l 
 
 j'^V 
 
 306 
 
 £11 
 
 42';2 
 
 44.3 
 
 4694 
 
 '^?o 
 
 5io5 
 
 53io 
 
 53i6 
 
 5721 
 
 5926 
 
 6i3i 
 
 2o5 
 
 £12 
 
 6336 
 
 6541 
 
 6745 
 
 7155 
 
 If^ 
 
 7563 
 
 7767 
 
 7972 
 
 8176 
 
 204 
 
 213 
 
 833o 
 
 85:^3 
 
 8787 
 
 8991 
 
 9194 
 
 9601 
 
 9805 
 
 •••3 
 
 •211 
 
 203 
 
 214 
 
 33o4i4 
 
 0617 
 
 0819 
 
 1022 
 
 1225 
 
 1427 
 
 i63o 
 
 i832 
 
 2o34 
 
 2235 
 
 202 
 
 215 
 
 2438 
 
 2640 
 
 2842 
 
 3o44 
 
 3246 
 
 3447 
 
 3649 
 
 385o 
 
 4o5i 
 
 4253 
 
 202 
 
 216 
 
 4434 
 
 4655 
 
 4856 
 
 5o57 
 
 52 57 
 
 5453 
 
 5658 
 
 58 "9 
 
 6059 
 8o53 
 
 6260 
 
 201 
 
 217 
 
 6460 
 
 6660 
 
 6860 
 
 7060 
 
 7260 
 
 7459 
 
 7659 
 
 73:3 
 
 8257 
 
 200 
 
 213 
 
 8456 
 
 8656 
 
 8855 
 
 9054 
 
 9253 
 
 945i 
 
 96:50 
 
 9849 
 
 ••47 
 
 •24'^ 
 
 199 
 
 219 
 
 340444 
 
 0642 
 
 0841 
 
 io39 
 
 i[237 
 
 1435 
 
 1632 
 
 i83o 
 
 2023 
 
 2225 
 
 193 
 
 N. 
 
 i ^ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 G 
 
 7 
 
 8 
 
 9 
 
 H 
 
3C2 
 
 LOGARITHMS OF NUMBERS. 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 I). 
 
 220 
 221 
 222 
 228 
 224 
 225 
 226 
 227 
 £28 
 229 
 
 342423 
 
 63?3 
 83o^ 
 35o24S 
 2i83 
 4ioS 
 6o25 
 
 '^1 
 
 2620 
 4589 
 5549 
 85oo 
 0442 
 2375 
 43oi 
 6217 
 
 8l2D 
 ••25 
 
 2817 
 47M 
 6744 
 869i 
 06J6 
 2568 
 4493 
 6408 
 83i6 
 
 •2l5 
 
 3oi4 
 4981 
 
 ^l 
 
 0829 
 
 2761 
 4685 
 
 Its 
 
 •404 
 
 3212 
 
 5178 
 
 7t35 
 9083 
 
 1023 
 
 2q54 
 4S76 
 6790 
 8696 
 •593 
 
 3409 
 5374 
 7330 
 9278 
 1216 
 3i47 
 5o68 
 6981 
 8886 
 •783 
 
 36o6 
 5570 
 7525 
 9472 
 1410 
 3339 
 5260 
 7172 
 9076 
 •972 
 
 3802 
 5766 
 7720 
 9666 
 i6o3 
 3532 
 
 5432 
 
 7363 
 9266 
 1161 
 
 3999 
 59^2 
 
 9860 
 1796 
 3724 
 5643 
 V5i 
 94j6 
 i35o 
 
 8110 
 ••54 
 ,980 
 3916 
 
 5834 
 774 i 
 9646 
 I539 
 
 ;^ 
 
 193 
 194 
 193 
 193 
 192 
 19. 
 
 :?; 
 
 230 1 
 
 231 1 
 
 232 ! 
 253 
 2.i4 i 
 235 
 23ti 
 237 
 238 
 239 
 
 361728 
 36i2 
 
 5488 
 7356 
 
 Q2l6 
 
 37io68 
 2912 
 
 4748 
 
 5675 
 7542 
 0401 
 1253 
 3oq6 
 4932 
 
 2io5 
 
 3988 
 5862 
 7729 
 9587 
 1437 
 3280 
 5ii5 
 6942 
 8761 
 
 2294 
 
 4176 
 6040 
 79i5 
 9772 
 1622 
 3464 
 5298 
 
 8943 
 
 2482 
 4363 
 6236 
 8roi 
 9958 
 1806 
 3647 
 5481 
 7306 
 9124 
 
 2671 
 455 1 
 6423 
 
 8287 
 •143 
 
 3?3i 
 5664 
 7488 
 9306 
 
 2859 
 4739 
 6610 
 8473 
 •323 
 2.75 
 401 5 
 5846 
 7670 
 9487 
 
 3048 
 4926 
 6796 
 
 236o 
 4198 
 6029 
 7832 
 9668 
 
 3236 
 5. .3 
 
 6983 
 8845 
 •698 
 
 2344 
 
 4382 
 62.2 
 8o34 
 9849 
 
 3424 
 53oi 
 7169 
 9o3o 
 •883 
 2728 
 4565 
 6394 
 8216 
 ••3o 
 
 .837 
 3636 
 5428 
 72.2 
 8989 
 
 :?^? 
 4277 
 6025 
 7766 
 
 188 
 
 ii 
 
 .S5 
 
 .84 
 184 
 .83 
 182 
 181 
 
 240 t 
 211 i 
 242 
 243 
 
 2u ; 
 
 24-, ! 
 246 
 247 1 
 24^ ; 
 249 
 
 380211 
 2017 
 3Si5 
 56o6 
 
 9166 
 390935 
 
 :% 
 
 6199 
 
 0392 
 
 ^ 
 
 5785 
 7568 
 9343 
 1112 
 2873 
 4627 
 6374 
 
 0573 
 2377 
 
 5^64 
 7746 
 9520 
 1288 
 3048 
 4802 
 6548 
 
 III', 
 
 4353 
 6142 
 
 9^>98 
 1464 
 
 3224 
 
 4977 
 
 6722 
 
 0934 
 2737 
 4533 
 6321 
 8101 
 9875 
 1641 
 3400 
 5i52 
 6896 
 
 iii5 
 
 2917 
 4712 
 6499 
 
 "^? 
 
 53a6 
 
 7071 
 
 1296 
 3097 
 4891 
 6677 
 8456 
 •228 
 
 It 
 55o. 
 7245 
 
 1476 
 3277 
 D070 
 6856 
 8634 
 •4o5 
 2169 
 3926 
 5676 
 7419 
 
 1 656 
 3436 
 5249 
 7034 
 8^1. 
 •582 
 
 2343 
 
 4101 
 585o 
 7592 
 
 .81 
 
 .80 
 '79 
 
 178 
 .78 
 
 177 
 .76 
 .76 
 175 
 174 
 
 250 ' 
 
 251 
 
 252 
 
 258 
 
 2.54 
 
 256 
 
 256 
 
 257 ! 
 
 2--.a ; 
 
 25D 
 
 397940 
 
 9674 
 
 401401 
 
 3l2I 
 
 4834 
 6540 
 8240 
 9933 
 411620 
 3300 
 
 8114 
 
 is? 
 
 6710 
 8410 
 •102 
 1788 
 3467 
 
 8287 
 ••20 
 1745 
 3464 
 5176 
 6«8i 
 8579 
 •271 
 
 8461 
 
 •192 
 
 Zl 
 
 5346 
 7o5i 
 
 8749 
 •440 
 2124 
 .'Wo3 
 
 8634 
 •365 
 2089 
 3807 
 55.-! 
 7221 
 89.8 
 •600 
 2293 
 3970 
 
 8808 
 •538 
 2261 
 3978 
 5688 
 7301 
 
 2* 
 
 2461 
 4137 
 
 8981 
 •711 
 
 2433 
 4149 
 
 5858 
 756. 
 9257 
 •946 
 
 9154 
 •883 
 26o5 
 4320 
 6029 
 7731 
 9426 
 1114 
 2796 
 4472 
 
 9328 
 .056 
 
 2777 
 4492 
 6.99 
 
 5i 
 
 2964 
 4639 
 
 95oi 
 .228 
 2949 
 4663 
 6370 
 8070 
 9764 
 145. 
 3.32 
 4806 
 
 .72 
 171 
 .71 
 
 .70 
 169 
 
 !^ 
 167 
 
 260 i 
 2»il 
 262 
 2«8 
 
 264 ; 
 
 265 
 265 
 267 
 263 
 269 
 
 414973 
 6641 
 83oi 
 9956 
 
 421604 
 3246 
 4882 
 65ii 
 8i35 
 9752 
 
 5i4o 
 6807 
 8467 
 •121 
 1768 
 3410 
 5o45 
 6674 
 8297 
 9914 
 
 5307 
 6973 
 8633 
 •286 
 
 52o3 
 6836 
 
 2i?? 
 
 5474 
 
 9. 
 Si 
 
 •236 
 
 5641 
 73o6 
 
 2261 
 
 7.61 
 6783 
 •398 
 
 58nR 
 7472 
 
 2426 
 4o65 
 5697 
 7324 
 8944 
 •i59 
 
 5974 
 7638 
 9295 
 
 4228 
 
 586o 
 7486 
 9106 
 •720 
 
 6141 
 
 7804 
 9460 
 mo 
 2754 
 4392 
 6023 
 7648 
 9268 
 288. 
 
 63o8 
 
 lilt 
 .275 
 2918 
 4555 
 6186 
 78.. 
 9429 
 1042 
 
 6474 
 8i35 
 
 979' 
 1439 
 3o82 
 4718 
 6340 
 7973 
 9591 
 
 I203 
 
 .65 
 .65 
 .64 
 .64 
 i63 
 162 
 .62 
 161 
 
 270 , 
 
 271 
 
 £72 
 
 278 
 
 274 
 
 275 
 
 276 
 
 277 
 
 278 
 
 ' 279 
 
 1 
 
 43 1 364 
 616? 
 
 4045 
 
 .56o4 
 
 i525 
 
 3i3o 
 4729 
 
 6322 
 
 7909 
 9491 
 1066 
 2637 
 4201 
 5760 
 
 i685 
 3290 
 4888 
 6481 
 8067 
 9648 
 1224 
 2793 
 4357 
 59.5 
 
 1846 
 345o 
 5048 
 6640 
 8226 
 9806 
 T381 
 2950 
 4Di3 
 6071 
 
 2007 
 36io 
 5207 
 
 nt 
 
 3io6 
 4669 
 6226 
 
 2167 
 3770 
 5367 
 6957 
 8542 
 •122 
 1695 
 3263 
 4825 
 6382 
 
 2328 
 
 7II6 
 
 8701 
 •279 
 
 i852 
 3419 
 49B1 
 6537 
 
 2488 
 4090 
 5685 
 
 &59 
 •437 
 2009 
 3576 
 5.37 
 6692 
 
 2649 
 4249 
 
 5844 
 7433 
 
 2166 
 3732 
 5293 
 6848 
 
 2809 
 4409 
 6004 
 7592 
 9.75 
 
 •732 
 2323 
 3889 
 5449 
 7003 
 
 .6. 
 160 
 
 ,59 
 .59 
 i58 
 i58 
 
 ■H 
 
 i55 
 
 |. 
 
 
 
 1 
 
 2 
 
 8 
 
 ' 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1). ' 
 
ANSWERS. 
 
 PARTI. 
 
 [Note.— The full-faced figures in connection with the nnniber of the page refer to theArti- 
 eles in the text. The numbers in parenthesis in the paragraph refer to the particular Example. 
 * * * indicate that it is not thought expedient to give the anewer.] 
 
 ADDITION. 
 
 (Page 13, 68.) 
 (1.) -la. (2.) Aa^-b--^a'^b^ab'-^'Zb'-^a\ (3.) 15ca^x^-^2ba^x^-h 
 
 Irnx'^y^. (4.) Az^-x^-^-bx^+Tyx^y-db-x^-^. (5.) x. (6.) hcz~-x^-\-2z^. 
 (7.) {a^c)z-\-{m-ba)y (8.) (2rt+2&+8c-2^+^-2»>zj^+(13rt-4-4;i-f-2^yi 
 
 (9.) {a-^b+\)x*-¥(J>-a-^\)xy+{a-b+\)y\ (10.) {a+m){x-^y)+{h-n){x'-y). 
 (11.) 8(m+n-2)V^^. (12.) «.T;"^-H(3-m)^-i + 3c. (13.) ^Va'-x'. 
 
 (U.) 0. (15.) lx^+\y'^-h^--'. (10.) ia+b+c) V^^^^. (17.) 2(«-2m).r^ 
 +3(m— l)y*+8c2. 
 
 SUBTRACTION. 
 
 (Page 15, 7^.) 
 
 (2.) ic3+a;«-2aj-a (S.) -2(a;«+flw). (4.) 6.#+2;t5^. (o.) 4»V^. 
 
 (0.) 10 ^T+x^-QayK (7.) rt(^«-^)+(10-;r) 4^. (8.) bx{X'-h)-4: ^^-f 2. 
 (9.) 2 4/«j^-Va6. (10). a-^b+c-fd', 5(t; a-'b+c-d-, and 8rt+9A, 
 
 (11.) 
 
 « # # 
 
30 1 ANSWERS. 
 
 M ULTIBLICATION. 
 
 (Page 20, 87,) 
 (1.) na^bx'^y^. (2.) 24»i'+>r''^"^--\ (3.) lOOx^^'^y^ ; and - ^ah^ 
 
 (4.) w" ; 1; a« ; w"^^; rt^^; c''^*. (5.) 3a*-+-10a6-86*. (6.) a;*+a;'y«+y^ 
 (7.) m^' — rw'^t'o' + ^i^+o^. (8.) rt^— «"'+»+«"•+«- a^+'+r^^+'—tt*. (9.) «* — 
 ('^+6+c+f02'' + (nh+ac+lK-\-ad+hd-{-cd)z'' — {abc + abd + acd + 6crf)2 + abed. 
 (10.) x^-ys. {\\.) a^b i*+\. (12.) 20«6«+30a''--^^»&'^+»+> 
 
 — 10a^-*+r-»y-«-'--15a*+/'-t-'. (18.) * * ♦. (14.) * * * 
 
 (15.) -a*+2rt'6*-&<+2(a«+6«)c'-6*. 
 
 (Page 21, 88.) 
 (3.) 0«^-10a='a;-22^*«x«+46aiJ»-20t*. (4.) 4^«-16a'»6* + 10«^6'-f 15«6* 
 
 -C5&*. (5.) a*-.T*. (6.) a;»-r,x*+10a;»--10j;«+5a;-l. 
 
 DIVISION. 
 (Page 24, J05.) 
 (1.) mvV /-^; (^)^; 1,; 1,; .J; J,. (2.) ^; -^^^ ; ^!^. 
 
 (3.)***. (1.) Last two, //-«"~;/-''-/,-/>i+'';a:^~'^'-2--i--%-3,t.-.. 
 
 (5 to II.) * •:> ». (12.) (.r+y)«. (18.) * •" «. (14.) * * * (15.) a + b. 
 (16.) te'-- ••^r>+'f-rt««+2"-'^*o''+6''6"«. (17.) ?;»*+««•*. (18.) m^'4-??.a;*.-H 
 »/i- + m. (10.) ;i,B«— 2.T + A-. (20.) a>^— «'y^+y^— ajV-yM-f 3^. 
 
 (Page 20, 100.) 
 
 (2.) a;«-5^.r+4ri«. (3.) 2rt''4-4rt«4-8«+16. (4.) %y*^AxK (5.) aj"-*"^ 
 
 (Page 27, i07.) 
 
 (8.) 2y^~8y^'4-12v«-8y+2. (4.) :f.'^-^x'y'¥x*i/+:thf^x^y^+i'y''+y^\ 
 
 1 + ar -f x« -f- .r ' + .e^ 4- »^ 4- x^ -I- a;^ +0?^ ct-. (5.) * ♦ * ''-' > * * *. 
 
 (7.) • ♦ ♦. 
 
ANSWERS. 305 
 
 FACTOniNG. 
 
 (Page 31, 122. 
 
 Examples in factoring are, in general, of such a nature that the answers can- 
 not be giveu without destroying the utility of the problem ; hence only the fol- 
 
 l.»wing are given: (23.) k'y^—l^m^yh^+k'm^yK^-k^m^y^^-k^m^y^z^ 
 
 -k'myH + mz". (24.) aj^— a;' V^^ +^' V— -^'V'^ + ^ V— ^' V'^* +y^ • 
 
 (26.)***; l-a=(l+ i/«)(l-|/«); l+«isdivisiblebyl+ t'a, 1+ '^, 
 
 GREATEST OR HIGHEST COMMOX DIVISOR, 
 
 (Page 34, 124,) 
 
 (1.) 12. (2.) 12. (3.) 3. (5.) 2kH'mK (6.) 2a'b. (7.) x^Y^z\ 
 (S.) Q?y. (10.) Ah^x-AbK 
 
 (Page 38, 120.) 
 
 (3.) x-1. (4.) x+\. (5.) 2a+3x. (G.) 2«-6. (7.) 4(ar»-2ary+y'). 
 (8.) 2aa;=»-6ax*+10«a;-2«. 
 
 (Page 38, i50.) 
 
 (1.) 2+6. (2.) 2(a;+y). 
 
 LOWEST OR LEAST COMMOX MULTIPLE. 
 
 (Page 39, 132,) 
 
 (2.) («+&)Ma-&)^ (3.) a-'^-4. (4.) 4(a^-2a^+l). (5.) 49r)9«-'&^'j:V^. 
 (6.) 1-I8a4-81«^ (8.) (a;^-39x+70)(.i:-10). (0.) (x-l)(.i-+2)(a:-3). 
 
 (10.) («='-4^^6+9aft*-10&')(a+4^). (11.) ir^-Ur'+71a:«-154a;+123. 
 
 (12.) 2?''+7x*-10x'-70x*4-9.i- + 03. 
 
306 ANSWERS. 
 
 FRACTIONS. 
 (Page 48, 167.) 
 (1.) ♦ * * (2.) ♦ * *. (3.) \+x+x^ +ir'4- etc. ; iC^+lO- ~, ; 
 
 m-\-n 2 4 8 16 
 
 4- etc.; a+x\ x»+l+ar-*+iC-'»H-a;-'"-h etc. ; 1— wa-''»+w2a-4»— r^-^a-S" 
 + /i*a-8»-;t»a-i""+ etc. (4.) 3 • 7-»a- V ; (m + ^iy+s . c'^rf-'-z*'' ; 
 
 \+x ' ^^^ ' ^*'^ ' ^^-^ x{a*-x') ' x(a'-x')' 
 
 T(a+x) {x-yY xix-y) a;' (\+x^)(\-x^) 
 
 x{a*-x*) ' ^•'•^ {x-yy ' {x-y)^ ' {x-y)' ' ^ ^^ {l+xV {i-x') ' 
 
 2(0.7;' -1 6) fl.r«(ar-4) 3T(2+3i ,) ^^ a+6. 
 
 ec ^*-^ ar(9j;«-l(>)' ik(9j;^-l«)' 32(9j;'-16) * ^ ' abc ' 
 
 10a* X* adfh—hcfh m'^+mn*—m*n afd+ae 
 
 20b*y^+7c*x* ' 9 + Sxy^ ' bdeh + bdfg' m-ii ' bdf+be+cf 
 
 ,,., 886a 6(x-7) 2 t5x + l . 
 
 <l*)li55' —49-' nT- «~^^ i^; ^^ 
 
 ar«-f2ar4-13 ^ ,^i. . 2(a«+a;') ^ 2rt!»a;-3c.r2 ,^^^Sb'-2aV)-2a-S 
 
 -V^T27- '' ^' <*^-) -^^ ^ ^-^^-^ &r-- • ^^^-^ a(6^^--rT) — • 
 
 2 3a;=»+20x'-32g-235 ^ . 2«+(7+e y-px -'^mpy* 
 
 a:*+a;*-fl' a;^+8a;*-5x-84 * ^ ^rt + c)(a+d)(«+ c) ' (3wy-'-.r)^ ' 
 
 ?±.^ „8.)0. (19.)^,= 4, ^-^^-^,. CiOO^.^^i; 
 
 4fl& r21 ^ 2^- 3 ^oo \ ?^ . ^' + ^' . 1 . a''-(v c+ny-xy 
 
 (a_6)«* ^ ■'(x«-l)(2x+3)' ^—^3' a ' ' b^-bx+by-z-y' 
 
 a■>c4■&^c-2c"-^» 1 1.2 1_1. 1 .;^*_i3;s_a..5. 
 
 (,^A«-«)ft?(«-/<))(2a'«_36)' ' i*' a2"*"ac'^ c* &' ' ' ''" ' 
 
 91a« 1 Ta^i^ 1 J}_ 
 
 ^^'•^ "66^ ' ^;r^ ' ^ ' *• ^^^-^ 121^ ' 1^81^"" ' a:'^'*" * 
 
ANSWERS. 307 
 
 (29.) ! + ..' + «- 3(a + J); ^Z^; -J_; bcic-W_ 
 
 -mhi-^^mn-^^n-^; -^f^-y a^-2ac + c^-b^ ; 1. (31.) 1. 
 
 (32.) 1. (33.) a-^-a-^b-^+b-^-a-^c-^-b-^c-^ + c-^. (34 ) ^'^'^' +^''''^' . 
 
 , /m—n (wi + ;?)« 
 
 INVOLUTION. 
 
 (Page 59, J»0.) 
 
 ,^ X « « . ? , 9 9 ,25 1 7»* 
 
 (1.) 9a6; 4a«a;2; ^^-;j7^ ; ly^*' ; J-^; g^ "f- (2.) l-2a; + 3a;«-2a5» + a5*; 
 
 4a2-12^za;i + 9x'5. (3.) 9 - 1.2a; -2.^2 ^ 4c' + a?*; 27a;6 -27a;* +9x^-1 ; l-2a;^ 
 
 + ar; a;^— 3x^^^-3a;-y-y^ (4.) Sla^a;^ ; IGaV^ ; a-^"'x-'^', aW ; a-a;^'; 
 
 125a; V^* «"'^" (1681) V 
 
 T^' -^r- (<J-) a;^+7a;6y+21x»y2^35a!V+35a;V+31a;V+7a'y6+2/^ 
 
 a;*-42;»y-f-6a;«y«-4a'y»+y* , 27rt'5-27a*a;+9a'^a;2-a;=* ; a!-«-5a;-«2^+15.r-V'^ — 
 35a;-8y=' + 70a;-»y4 - etc. ; X'^ + 4a;-5y + lO.^;-^^* + 20.a;-''y=' + 35a-8^* + etc. ; 
 
 A ^* a;* a;^ 5a;» 1 ^ , 3«^ , ^^^1 ?:'>^ 
 
 '■'2"T5i 8":?^;^~l"^r^^^^ i.^2.^+8.-'' ^10.' +128.3 
 
 + etc.; l-4.^ + 6.-4.c^.«; -*; l^{l^^^^^^^^ 
 
 4- etc. ) • * * * • * * *^ (7 \ 4f * * * *^ 
 
 (Page 61, W.'j,) 
 
 (1.) 2 - 2 - 3 . 59 = 472 ; 2 ■ 73=146 ; 5 • 7 • 67=2345 ; * * *; ± {a - c • (»+&)} 
 = ±(«'c+a/>c) ; * *. (3 to 6.) * * * * 
 
308 
 
 ANSWEIIS. 
 
 (Page Co, J97.) 
 
 To give the roots in problems in evolution would be to destroy the benefit of 
 the exercise ; hence they are omitted. 
 
 REDUCTION OF RADICALS, 
 
 (Page 70, 20^,) 
 (1.) *•*». (2.) *; ^Vl8; *; *; *; Vl ; —{a^-h^)^ ; **»***. 
 
 a—b 
 
 — rT—^— T — Vl^x*—y*). In such examples be careful to leave only integral 
 forms under the radical sign, in the reduced expression. (3 and 4.) * * * *, 
 
 V\ = V\. (5.) * * *, a'A -^y=ia*-b*)^. (6.) V^ and Vd ; 
 
 2« 4/T« . • * ♦ • . ^"'^' 
 
 »*»♦. 4/^:^11^ and V7+]^. (7.) *♦*. (S.) ^VTS;**** 
 
 o if 
 
 aj-y ' 2 . ' 
 
 g«+a; a' + a;« ^^ 2a;« + l-2^ i^a;« + l ; -(«+ VS^^l) ; a;*+a;+ VJ^+W+aj«--l; 
 a* 
 
 2(_ V6+ V2 + 2) ; 2»^ 3^i^30- .^>^2 . (lo.) a^V^/Zy^+^^/^/ay 
 
 + j; =* 2/'^ + .r 'y + x 'y^ + x^y^ + .c*y' + .-c^y' + ar'y* + ^y ^ + ar 'y * + a;^y " + y '^ ; ,r ^ 
 
 ia 1 1 14 ^ 11 Ji IXt g 1 21 . ^ 4 21 2 15. 
 
 + X ^ y* 4- aj^'y* + .t ' y* + a; * y ' + a;^y * + a; 'y* + x^i/ * + x^y^ + x-^y * + .x'y * 
 +y"; (i/8+ V^- Vs) (3-24/c). (11 to 18.) » * * 
 
 COMBINATION OF RADICALS. 
 
 (Page 74, ;^77.) 
 
 (1.) * * *. (2.) * * *; ^ ; — . (3, 4, 5.) * * *. (6.) 4^864; 
 
 X X 
 
 fi 19, 10 f, 
 
 iV ^151875; .V^'x''; QxVxy' -, Vl+5a;« + l(te* + 10a;«+5a.' + 2;W 
 
ANSWERS. 309 
 
 i'i'W^:^; dVWZ; 30; 12 Vi. (7.) 41 ; x+y; 246+581^ 
 
 -11/^-36 i^d; 3'1^30-12V'3 - V180 + 12. {S.)lViO; AVd; 
 
 Vdi^; IV'IO. {d.)\^~i-kV2; 2xi^x; iVQ + V2; b Vx ; f-' 
 
 (a-/*) (a -hi) 
 
 ^V4«*x; ^A.Vl5; 5-2 VQ; 27{a-x)Va-x; J-3ab^+dah-b^. 
 
 (11.) 9^ V5P ; ia;2 VJ ; by 5 \^y ; 2 j/7a; V'sT ; Vl^x; i 1^3125^ ; 
 i^3; (12.) 2+31/5"; 2V3+34^; ^V^-i^^; ***. 
 
 IMAGINARY QUANTITIES. 
 
 (Page 78, ;^;?,!/.) 
 
 (2.) ISC^^+l)^^^; 19a V^; (463+3c)|/rT. (4.) (4^± V"^) ^^. 
 (5.) 1 V^ ; 12( V3~- 1) V~^ ; 11a V^ ; {a Vb - c Vd) V^. 
 
 (6.)1^^- (7.) 5^2 1^^; IQV^V^', \'V^', P7ii^^. 
 
 (Page 79, 225.) 
 5-7 V^, and 9 V^-1 ; 2a+( Vb+ V7) i^^, and ( VT- V7) i^^. 
 
 MULTIPLICATION ANB INVOLUTION OF 
 IM AGIN ARIES. 
 
 (Page 80, 226.) 
 
 (2.) *, *, -6>v/6. (7.) *, 3, V^. (9.) 278'v/^, or 278^3 V"^ 
 972 V^, or 972v'2 y/^, (10.) icy^M'. 
 
 DIVISION OF IMAGINARIES. 
 
 (Page 81, 227.) 
 
 (3.) ^iV4 4/~l; -i^aV-l. («•> -TI3)^- 
 
310 ANSWEllS. 
 
 PART IL 
 
 SIMPLE EQUATIONS. 
 
 (Page 87, 28.) 
 
 3& b a+h+c 
 
 (l.)12; 24; 23i; 3.S; 8; 4; -; -; —7^; 2.9; 2|f 
 
 ^Q>> J 6c-&*_ ^. n(9'-;)) 5rt(26-«) 8«6*+46»-12a2i 2k« 
 
 a m "6c— d Sa'^-hab—ac+bc ' 2Sp+6g'^ 
 
 4; 4i ( 
 
 2«Vli 81 
 
 ^p^^ ; 8; 0; 3. (Ji) --, 4; 4; 4^ (4.) 81 
 
 c«-2/>c; 16; 5; 4(a-l); 8;1;6;6;3; ± 
 
 6+1 a 
 
 \ a ) \ \ft + l/ j ' (w+l)«' 26^ ^ ' 
 
 r ^ \ 36 / ' 2a ' 16 ' 4 • 
 
 APPLICATIONS OF SIMPLE EQUATIONS. 
 (Page 90, S3.) 
 
 (1.) ^'8 84, J5's 42, (Ts 14. (2.) ^'s^— ^^-, 5's ""* 
 
 1 + n + mn' 1 + n + mn' 
 
 (7.) :rr^-. (8.) i (9.)?-"^|?. (18.) m of an hour ; '""* 
 
 mn—m—ns ^ an + bn ns + ms+mn 
 
 (14.) 317.951,1268,2219; ^V- ?^ • f-T™^ ^'-- ^S-) 90; 
 ^ ' 3 + 4m 3 + 4w 3+4w 3 + 4m ^ 
 
 g + 6 .- ^ V ^^^ mpa—pa + mb—na + mc ^ ^ 
 
 b'—ac 
 
 p + 2—m—n' ' m + n—mp+p ' ' ' a—2b + c' 
 
 (18.) 19, 30; 2±^,^, 2!!t^i. (19.) 73, 77; ^=^ , 
 
 ^ ' m + 1 7W + 1 2m 
 
 2m ^ m + n m + n 
 
 33i 50; 14f, 28i, 42?. (26.) 1200. (27.) 50. (28.) 5712. 
 (30.) 50. 
 
ANSWERS. 311 
 
 SIMPLE EQUATIONS WITH TWO UNKNOWN 
 QUANTITIES. 
 
 (Page 96, 4=2.) 
 
 (1.) aj=10,y=3. (2.)19,2.t (3.) 16, 35. (4.) 7, 2. (5.) 7. 17. 
 
 (6.) 2, 2. (7.) 12,9, (8.) -2, 19. (9.) -2, 1. (10.) ^J, ^. 
 
 („.)i, i (12,)?^^+^, 2^^!+!'. (13.) ,-21, ''1-^. 
 
 ab cd da * '6b ^ ' b+c a 
 
 (14.) (a+by, {a-h)\ (15.) 10, 5. (16.) 18, 12. (17.) i , - . 
 
 a 
 
 ^^^•^%' WV ^^^■'^\' li- (20.) 20, 5; 6.8; 7.10; 8^^-72=0; 
 
 y«-22y+120=0; y^-Ay^ ^ 14^4 _. 203,2 +9=0. 
 
 APPLICATIONS. 
 (Page 98, 42,) 
 (1.) 18f , 31^. (2.) 3. (3.) 20, 8. (4.) 5000, 5000. (5.) ^. (6.) 24. 
 
 (7.) 29, 32. (8.) 5000.0. pm^gn-qmn pmn-gn-pm ^ 
 
 mn—m—n mn—m—n 
 
 (10.) ^±^-, ^t?. (IX.) 48, 16. (12.) 24, 32. 
 mn—\ mn—1 \ / » \ 1 > 
 
 SIMPLE EQUATIONS WITH MORE THAN TWO 
 UNKNOWN QUANTITIES. 
 
 (Page 101, 43.) 
 
 (2.) 4, 3, 2. (3.) 2, 3, 4. (4.) 24, 60, 120. (5.) 64, 72, 84. (6.) 3, 2, 1. 
 
 (7.) 2.), 5.,. 65. (8.) 26c— ^ 2ac ' 2a6 ' ^^^ -Q ' 
 
 -h W ^l^-^l' I' ^' (n.)2a,2b,2c. (12.)^^!^, 
 
 1 1 (13.) -f-^. -^. -M. (14.) 12, 5, 
 
 {b—a){b—c)' {c—a){c—b)' ^ '' 6-f-c' a+c' a+&* 
 
 7. 4. (15.) 2, 1, 3, -1, -2. (16.) 3, 4. 5, 1. 2. (17.) h }-c-a, a-hc-b, 
 
 a + b—c. 
 
 t The valacB of the unknown quantities are given in tlie order x, y, Zy etc. 
 
312 ANSWEUS. 
 
 APPLICATIONS. 
 
 (Page 102.) 
 
 (2.) |2, 20 cents, 10 cents. (3.) £3000 at 4%, etc. (4.) ^ , ^, ^, 
 
 o7 ot o7 
 
 2Sa 
 
 — . (5.) 142857. (6.) 26, 9. o. (7.) 140, 00, 45, 80. (8.) 18,«^j, 34f, 
 
 23/i , 80. 
 
 BATIO, 
 
 
 
 (Page 105, 50.) 
 
 
 
 "•'^=^ 1= 7=--r„^ S— <■■ ^^ 
 
 4 
 3 ' 
 
 3 
 5a' 
 
 .W- 9= Z' .i*- (*.)5:n;l:«'+6.;2(a x):(««). 
 
 (4.) 9 
 
 :25; 
 
 a«:6«; 27:125; a^zft^ 5:4; V^iVl; Vm.Vn; 3:4; 
 
 ^7: 
 
 ^-y. 
 
 (5.) The former. (7.) 4:1. 
 
 
 
 rROPORTION.—APPUCATlONS. 
 (Page 111.) 
 
 (8.) 13,26. 39. (4.) 8,0. (5.) ^^ + |-^ , 2^-|i. (G.) 120, 160, 200. 
 
 (7.) 8:9. (8.) 252. (9.) 56,84, 70. (10.) 20. (11.) 150. (12.) 300. 
 (14.) 3h. 49j\m., 3h. 32i\m., • 3h. lOAm. (15.) Every h\ hours, A hours, 
 and 1 ,\ ; or 11 times in 12h., 22 times, and 11 times. (1(».) No ; since it talces 
 the minute hand 1-^- hours to gain a round, and ^ to gain half a round. 
 
 (17.) 8:45 A.M. (18.) 1st. ^-^. -^. etc.; 2d. -i=-^, |^-^, etc.; 
 M—m M—m m—M m—M 
 
 a+mt s+a + m t 2s+a+int s—n—mt 2s—a—mt 
 
 9—a+Mt 2s—a -\-Ml 
 M—m ' M—m 
 
 etc. 
 
 ABITHMETICAL BROGBESSION. 
 
 (Page 117, 8:i,) 
 
 (1.) 83, 903. (2.) -39, -384. (3.) ^-^ , ^^^ . (4.) 0, "^ 
 
ANSWERS. 313 
 
 (5.) 193 243 . • 293 • 343 • • 393 • • 443. (7.) -46, | . (8.) 100. (9.) ^^^, 
 tn+n m+Sn 
 
 GEOMETRICAL PROGRESSION, 
 (Page 120, 90.) 
 (1.) 46875, 58593. (2.) 6, 18, 54, 162, 486. (3.) 16384, 21845^. (4.) —h, 
 -hh (5.)il.n,§i (6.)¥; .3; ^g; If. (10.) -A[(-^3)»-iJ ; 
 nm; I; Wo\%^; i 
 
 VARIATION. 
 
 (Paoe 124, 95.) 
 (6.) «az. (12.) 18. (13.) «=i/i*. 
 
 HARMONIC PROPORTION AND PROGRESSION. 
 (Page 126, 100.) 
 
 (6.)i,TH-,T»J. 
 
 PURE QUADRATICS. 
 (Page 128, iO«.) 
 (1) ±4. (2.) ±5. (3.) ± V2a^ - &». (4.) ± V6. (5.) ±i^»V3. 
 (6.) ± 6. (7.) ± j/^. (8.) ± V:^. (9.) ± ^ V5. (10.) ± 3 V=^. 
 
 ai.) ^|V-io. (12.) ±f (13.) vffi^p. (1^-) - -S;:;^""- 
 
 APPLICATIONS. 
 
 ± w l^i 
 
 (1.) 12, 20. (2.) ± ia VS, (3.) -^^====^^^ . V^nJT^^' ' 
 
 ±pV7 /4X4K^ /K^ ■ - — _^^-IL-. (6.) 5.57+, 
 
 18,12_, 40.51 + . (7.) 149,247.2 + miles from the surface of the earth (8.) 240. 
 132 VL from A. (10.) -^ from the louder bell. 
 
314 ANSWERS. 
 
 AFFECTED QUADRATICS. 
 
 (Page 133, 114.) 
 
 (1.) 8, -2. (2.) 6, 2. (8.) o(2± Vn). (4.) 8, -1. (5.) 2 ± V^^. 
 
 (6.)1. -a. (7.) 2.-1. (8.)7,i (9.)-,-. (11.) 3, -6g. 
 
 (12.) 5. -^h (13.) ^^ . (14.) t, A. (16.) Wm, 6. (16.) 2, -?. 
 
 (17.) ~. ^ . (18.) i(-l ± i^l33). (19.) 4. i. (20.) ia(-3 ± V-?)- 
 
 (21.) ± k ^^. (22.) 8, -%. (23.) ^^^ . &. (24.) 4, -i 
 
 (25.) 12, 4. (26.) 4, i (27.) 7.12+, -5.73+. (28.) ia(l±3 V^). 
 
 (29.) 1, gV. (80.) 5, 3. 
 
 HIGHER EQUATIONS SOLVED AS QUADRATICS. 
 
 (Page 136, 122.) 
 (1.) ± 3, ±3 V^. (2.) 2 ; the other four roots not required. (3.) ± m^. 
 (4.) 27. (5.) 121. (6.) 64. (7.) b^. (8.) ± 8. (9.) ± V^, ± V2. 
 (10.) V'K-I ± Vl^i). (11.) 4, M (12.) I ^ (-^ =t V4^+T«) [ ^• 
 
 (18.) V'C/z ± V}^^T^\ (14.) 243, (-28)1 (15.) 16, ^^^^i. 
 
 (16.) |^(J± i/ft^Ti^i;;)} '. (17.) 8,^. (18.) ^,|/I^. (19.) 1. 
 1, 1 ± 2 1^15. (20.) 0, -12. (21.) 3. -i, i(5 ± Vl329). (22.) 4, 69. 
 
 (28.) i(l ± V5). (24.) |(l ± i/5). (25.) 3, h U-8 ± i^55). (26.) 2, H. 
 
 i(7 ± Vm. (27.) |7^^- (28-) 5, -1, 2 ± i^^H: (29.) 5. -2, 
 i(3 ± /=15). (30.) 2, 3, 1. (31.) 2. -|, i(l ± V^Z). (32.) 1, -3, -3. 
 •(33.) 2, 2 ± V^. (34.) 5, 4 ± V^ (35.) -2, -1, -5. (36.) 6, 20, 3. 
 (37.) 6, 4, 5. (38.) 1, 1, -2, -2. (39.) 4, 1, 3, 2. (40.) 3, -1, 1 ± i^^. 
 (41.) 5, -4, 3, -2. (42.) 3, -3, i(-13 ± V^ISS). (43.) 4, 3, i(7± V69). 
 (44.) 9, 4. i(-3± V^l (45.) + 1, -1, ± V^^i: ; -1, Kl ± ^^) ; ^ 
 
±1, 
 
 v^. 
 
 ANSWERS. 
 
 V2 
 
 315 
 
 i(± Va ± i^2 ± V-2 ± V2). 
 
 (46.) 
 
 2a±V2{l 
 
 2(1 
 
 ;?(1±^) ± A/( ^^^^a+<^) Y_r (48.) i(a ± V7^-4\ 
 
 -a) ^ \ 2(l-«) J 
 
 (49.)^± iV-3±i^8(^). (50.) i(l ± VE). (51.) 2, -K i(3 ± I^SOS)- 
 (52.) 1, 1, -2, -2. (53.) I (-1 ± V~^). (54.) ± -^ i -^^^^ + V^6t^T2 [ , 
 
 V a ( ^ 4 , . \ , 
 
 (55.) ±-|(Vl + a8-l)(Vl-a2 + l)'f^ 4,i. 
 
 (56.) 0,i,f -1, +2, -2. (57.) h i(-l± ^-35), ±1, ±Vi(-ll±v«5). 
 
 SIMULTAJSTEOUS QUADRATICS. 
 
 (Page 142, J;37.) 
 
 (1.) x=%, -I? ; y=-4, W. (2.) a;=± t'f ; ?/=2t V i (3.) a!=2 ; 
 
 y=2. (4.) «=±7, ±4; y=±4,±7. (5.) a?=±3, ±11^2; y=±2,±\V2, 
 
 (6.) a;=±2, ±i 4^10 ; y=±i, =Ff i^. (7.) .x-±3, t8 ; 2/=±5. 
 
 (8.) a;=±f i^li; y^ilVTi. (9.)a;=±f4^; y=±|4/2i. (10.) a-=±l. 
 ±V-4^^; y=±2, +^4/^=5. (ll.)a;=±2, i^i^jT; y=±6, ± ^i^ i^. 
 (12.) a; = ± 10, ± H 4^"=^ ; y= ± 3, =F ^ 4^^=47. (13.) aj = 4, 2, 
 
 i(-13 ± V377); 2^ = 2, 4, i(-13 T I^^TT). (I4.)a;=4, -2,0; y=2, -4, 0. 
 (15.)a; = 2,3; y = 3, 2. (16.) a; = ± 3 ^2 ; y = ± V2. (17.)a; = 9, 4; 
 y = 4,9. (18.) a; = 11, i(l ± V^^l) ; y = 3, i(- 15 ± V=^l). 
 
 (19.) a; = 15, ; y=45, 0. (20.)a;=±V2; y = 2^V2. (21.) a; = 0,2; 
 
 ^ = -2,0. (22.) a; = 1,4; y = 4, 1. (23.) ar = 1, 3, 2 T 5 f'^; y = 3. 
 1,2±5V^. (24.) a; = 5, -2, K3±V^^67); y = 2, -5, K-3 ± V'^^)' 
 (25.) a;=±3, ±2; y=±2. ±3. (26.) x= ±2, ±1, T 2 4^^, ± V^ ; 
 
 y=±l, ±2, ±4/31, ^2^^=!. (27.)a; = \^K^2-l); y= ^ 2(V2-^ ' 
 
 2abc 2€ihc 2ahc 
 
 (28.) X = 
 
 y = 
 
 7 7, y--j— 7 , ---r T-- (29.)a;=±3, 
 
 ac+bc—ab ab+bc—ac ab+ac—bc 
 
 y=±2, s = ± 1. (30.) a; = ± 2, y = ± 4, 2 = ± 6. (31.) a; = 1, 2/ = 2 
 2 = 3. (32.)a;=±V, T %^ ^^, ±5, ^4^/^; y = ± ^sS ± ¥ ^^ 
 ± 4, ± 5 i^^. (33.) a; = ± f 1^2, ±3, ±3 V^ ; y = ± '1^2, ± 1, 
 
 ± /Zn:. (34.) a;=8,0;y=8,0. (35.) a;=2, 8 ; y=8, 2. (36.) x=10T4.Vn, 
 
 (37.) a;= ±||/±15, ±^V±3a: 
 
 *0 T I Vl5 ; y = 10 ± 4 V6, 1*0 ± | VI5. 
 
316 ANSWERS. 
 
 ±aVWl', y=:±yV±3,0. (38.) ^ = 4, 9, -3T44^iO; y = 9, 4, 
 
 -3±4iC:iO. (39.) a? = 2^(15 ±6 1^^,5,1; y = M^S ± 10 V^\ 3, I 
 (40.) «= f ; y=16. (41.) a;=4, 1.0; y=8,0. (42.) 25=2744, 8 ; ?/=9r>04, 1. 
 
 APPLICATIONS. 
 (1.) 3. (2.) 18, $20. (3.) 10 and 3 days, 120 and 36 miles. (4.) 12, CC. 
 
 (5.) 14. 10. (6.) 6 miles an hour. (7.) 4 and 5. (8.) J ^(^^-l)P _^jy\ 
 
 \ Vm—1 / 
 
 VVj^P^Tj^Y ,9)j^.^(5^^) (10.) 1,3,5,7. (11.) 2, 
 \ Vm—1 / 
 
 3, 4, 5, 6. (12.) 3, 6, 12. (13.) 2, 4, 8. (14.) 5, 10, 20. 40. (15.) 2, 4, 
 
 8. (16.) 6, 8, 10, 12. (17.) 1, 2, 4, 8. (18.) 108, 144, 192, 256. (19.) 72, 
 
 63, 66. (20.) 7, 3. (21.) 25. (22.) $960, $1120. (23.) 248. 
 (24.) 6 and 7 per cent. (26.) 3 and 14. 
 
 IJS^EQ UA LI TIES, 
 (Page 150, 134.) 
 (8.) H and V- (9.) Any number between 15 and 20. 
 
 PART III. 
 
 DIFFERENTIA TION. 
 (Page 157, 156,) 
 (3.) l^bx^dx - mxdx -H 4rfj-. (4 ) 2Axdx + ZBx^dx + ^Cx'dx. 
 
 +Qdx; (5.7;*-12.c='+12a;*-2a;)rf«; {x-2x^ + l)dx. ilSton.)15{a^+x')'x'dx ; 
 
 ^{Zx-2)^dx ; i{2-x'y^xdx ; -—. . (18 to 22.) - -r-^ . 
 
 2(1 +a;)^ ^^"^^'^ ' 
 
 — t:, T^i : — 75 rr; ts rs .* Ti a • (23.) When x > 1, faster; also 
 
 {1+xy (!+«)* (l+ar)« ' (l+x)* 
 
 faster when x<i and > . When x = = , they both change at the same 
 
 dV2 SV2 
 
 nte. When x < , y changes slower than -a;. 
 
 3^2 
 
ANSWERS. 317 
 
 INDETERMINATE COEFFICIENTS.— DEVEhOPMENT OF 
 
 FUNCTIONS. 
 
 (Page 161, 161,) 
 
 (3.) l-ia;«-ia5*-etc. ; x-^^+x^-x^+etc, ; ^ + ^x+^x^ + ^x^ 
 
 -»-etc. ; l4-ia;+t«2_|_.i.a.3^.;^4^et(,^ (4.) S+Saj-ac^-Saj^'-aj^-etc; 
 
 l + 2x + a.» + te'+9.V4-etc.; ^ - |^ + ^ _ g, + ^,._et. 
 
 (5.) 1— 4iC— ^a;2— 8^5^— etc.; l+^x—^^+-^^x^—eic. 
 
 DECOMPOSITION OF FRACTIONS. 
 (Page 164, 167.) 
 5 3 3 15 4 7 
 
 (2 to 6.) 
 
 3(a;-2) 3(a;+l) ' 2(a;-2) 2a; ' a;-4 a;-3 ' 2(a;-4) 
 
 _ 1 ^ _J ^ 9 _2 1__ _3_ 
 
 2(aj-2) ' 2(a;+l) a;+2"^ 2(a;+3) * ^ ** ^^ (aj-l)^ (a;-l)=^ "^ a;-l ' 
 2 3__ _1_ . 1 _ J^ 2 _J 1 !___ 
 
 {x+ZY (a?+3)* "^ aj+3' a;* jc* + 3. + /^\-x) 2{l+x)'^ 4(l+a!)' 
 1 1 \__ , 1 2__ 1__ 2__ 
 
 4(a;-l) 4(a;+l) 2(a;«+l)' 25(a;-2)2 125(aj-2) "^ 25(a5+3)2 "^ 125(a;+3)* 
 
 ft^ 1 2^-2 ! 3(a; + 4) 1 1 J^ _ 2 _3_ 
 
 (12 to 18.) ^e:^^ (a;2+l)2' a; "^ (aj'^-2)=^ "^ a!2-2 "^ a;-l ' a;=^ a; "^ a;+l ' 
 
 1 (_2_ _ _1_ g-2 _ a;+2 j 1 1__ 
 
 elaj-l a;+l "^ a;*-a!+l x'+x+l) ' Aa\a+x) 4ta\a-x) 
 
 1_ 1 !____. J.? ? 12_ 
 
 ■^2a«(a*+a;«)' (a-6)(a;-a) (a-6)(a;-6)' a;-3 a;-l a;-2* 
 
 tjtje: binomial formula. 
 
 (Page 167, i7J.) 
 
 (1 to 6.) a^-Qa'h-'rl^a^h^-'ZOcL'h'-^X^a^h^-Qab'+h^ ; a;^-7a;6y+21a;«2^« 
 
 - 35ajV* + 35a;'y* - 2\x^y' + Ixy^ - y' ; a» - na**"* aj + ^^^^— ^"~' «^ 
 
 ^ M^-l)(^-g)^n-3^. ^^(^-l)(^-2)(^zi)^.-4^4_etc.; l4-4.+6.^-f4.^ 
 ^ if 
 
 W^_l) w(7l-l)(7l-2) 
 
 +«*; 1-5^4- 10y2_l0yJ+5y4-2^« ; l-ny+ —3 — 2/' jg y' 
 
 _^ ^^-l)(n 2)(7^- j)^^_^^^^ (7 to 11.)*****. (13, 14 to 17.)**** 
 
 fl8to20.) a^-\a H^-^a ^x'-ha ^a;6-etc.; ^-5 + ^^ + gj-^ 
 
818 ANSWERS. 
 
 i(V|.6 i a _3, 
 
 + i^29^ + etc. ; a'+4a^c''+6a*c+'ia'c'-\-c\ (28.) -^Siia ^ b'\ 
 
 LOGARITHMS, 
 
 (Page 179, 199.) 
 
 (1.) 4, 6, * * * (2.) -2, * * *. (3.) 4, * *. (5.) To the 3291147th 
 
 power, and the 1000000th root extracted. (G.) The 1000000th root of the 
 
 3414639th power, (7 to 9.) * * * *. (10.) .23108, .17677, * *. (11.) 4.449419, 
 4.637084,1.890210. (12.) 12.42, .00010031, 18.3625, 1.8358. (14.) flog aj 
 4-i[log (l+ir)+log (l-«)], Klog rt+log .^■- log 6-log y), ^[log («-«)+log {s-b) 
 
 +log («— c)— log *], ^[log ar+log (1— .r)] — J log y; - {m log a+p log 6- Hog c), 
 
 i log c— - log (f-H - [log(7/t-har)+log (w— .t)]— w log a+n log 6. (16.) — z 
 
 r; a; x 2{l + ir) 
 
 SUCCESSIVE DIFFERENTIATION, 
 (Page 182, 204,) 
 (2.) 12a;da;«. (6.) * * ♦. (7.) 2[{x-h)-\-{x-c)+(z-a)]dx*. 
 
 DIFFERENTIAL COEFFICIENTS, 
 
 (Page 185, 207.) 
 (6.) 10**4-12aJ*-l(te, 40«»-f36x«~10, 120x«+72ar, 2405+72, 240; ****. 
 
 TAYLOR'S FORMULA. 
 (Page 188, J^i;?.) 
 
 4-2aJ-»y+3ar-V*+4ar»y3^.5a^6y4^({,,-7y5^_e^c. ; a;~^-p"^y+t«"V 
 
 _|^^-V'y3+^a;*'^2/*-^f«"^^'^^/' +etc. Page 189, (2.) 3aj«-2aj2+(15aj4-4c)A 
 +(30ic3-2)A2 + SOar^A' + l&r^^ + ZhK 
 
ANSWERS. 319 
 
 INDETERMINATE EQUATIONS. 
 
 (Page 192, 218.) 
 
 Sy= 5, 14, 23, 32, 41, 50, 59, 68, 77,86,95,104,113,122,131,140,149. 
 
 U=215, 202, 189, 176, 163, 150, 137, 124, 111, 98, 85, 72, 59, 46, 33, 20, 7. 
 . Sy=2^, Q. ,^. (y= 8. (2^= 9, 28. 47, etc. (y =2, 119, 236, etc. 
 
 ^ i x=Vll, 28. ^ ' ( a;=20. ^-^ ^ { x=ZQ, 173, 290, etc. ^^^ \ a;=3, 131, 259, etc. 
 
 i^)\ll\ »5.+9.=40. None. 5.-9.=40, |fl,?; ^J J -• 
 
 APPLICATIONS. 
 (2.) Yes ; 15, 163, 9. (4.) No ; yes, in an infinite number of ways ; 4 3-shil- 
 Img pieces and 192 guineas ; possible ; possible ; possible. (5.) 190. 
 
 INDETEBMINATE EQUATIONS BETWEEN THREE 
 QUANTITIES. 
 
 (Page 194, 219,) 
 
 (z= 1, 2, 3, 4, 5, 0, 11, 12, 13, 14. 
 (2.)]y=ll, 9, 7, 5, 3, 1, 8, 6, 4, 2. (3.) 59 sets of values. 
 
 (a;=10, 11, 12, 13, 14, 15, 1, 2, 3. 4. 
 
 (A)z-l\^=^' 4' ^'8' 10-^ 2-2^^^='^' 3,5,7,9.) (y= 2,4,6,8. 
 
 ^*-^ '-^ U=15,12,9,6, 3.) <.r=rl4, 11, 8, 5, 2. f U-=; 
 
 (a;=r9,6,3.f f;r=5, 2.1 ]a;=4, l.f 
 
 :10,7,4, 1. 
 
 (Page 194, 220.) 
 (1.) «=7,y=2,a;=10. (2.) e=15, 30, ^^=82, 40, a; =15, 50. (8.) None. 
 
 APPLICATIONS. 
 
 (1.) 8 of 1st, 6 of 2d, 2 of 3d, and in 9 other ways ; 23 and 2, 10 and 5, 9 and 
 8, 2 and 11. (2.) $4, $2, $7 ; infinite variety of prices. (3.) 6, 3, 1, 16. 
 
 (4.) Nuniber of the 3d kind equals twice the number of the 1st kind, plus tlie 
 number of the 2d kind ; 1 of 1st, 6 of 2d, 8 of 3d kind. (5.) 40, 60, 24. 
 (6.) 55, 10, 85 is one result in integers. There are an infinite number of other 
 ways. (7.) * *. (8.) s :=. 10, y = 1, x = 13. 
 
320 ANSWERS. 
 
 LOCI OF EQUATIONS. 
 
 A LAROE number of these constructions are exhibited in the text, and to give 
 taore would be to destroy the possibility of the student's deriving any benefit 
 from the exercise. 
 
 HIGHER ^0 17^ TJOJV^S.— TRANSFORMATION. 
 
 (Page 205, 228.) 
 (2.) Multiply by y*, and then put y=x^°. Finally put x= — , etc. 
 It is not deemed expedient to give farther explanations. 
 
 (Page 214, 249.) 
 
 (2 to 84.) To give the roots of these equations would destroy the practical 
 Talue of the examples. 
 
 (Page 216, 250.) 
 
 {1.) x^-2x*-Ux+12=0. (2.) x*-2x^-5x'+4x+Q=0. (3.)***. 
 
 (4.) x^-x*-7x+15=0. (5.) » * * (6.) d0x'-nx'-l]x+6=0. 
 
 (7 to 10.) *****♦. (11.) x^-\0x''+dSx*-5Qx^-7Sx^+66x~\-S9=0. 
 
 EQUATIONS WITH INCOMMENSURABLE ROOTS. 
 
 (Pages 216-247.) 
 
 To give the answers to these examples would be to destroy their value to the 
 student. 
 
 CARDAN'S rROCESS. 
 
 (Page 251, 281.) 
 
 (2.) -1, 2, 2. (3.) 2, -1 ± Vs. (4.) V'i - '^'2. and the roots of 
 w^ -^.{i^-l^)x + {i^i-i^)\e.= 0, which are -^{i^i-h) 
 
 (0.) 1, -2 ± 3 V^. (7.) * * *. (8.) 2, 2 ± V^. (9.) 8. -4, -4. 
 
 (10.) * * *.• (11.) * * *. (12.) * * *. (13.) One root is 2.32748+ 
 
AN8WEKS. 321 
 
 VESCABTES'S FMOCESS. 
 
 (Page 252, 283.) 
 Ex. 4, -2, -1 + V^, and -1 - V^. 
 
 MECUBBING EQUATIONS. 
 (Page 255, 291.) 
 
 (1.) 2 ± V3, ^ (1 ± V^). (2.) - 1, ^ (9 ± V77), ^ (3 ± V5). (3.) 1, 3, 
 
 l+4a±V5 + 20a 
 
 i — 2, — f (4.) — 1, ^m ± V|^m2 — 1, in which m = „ .. . 
 
 ^ (.1— «) 
 
 (5.) -1, 1, I, -1, -1, 1 (1 ± V:^). (6.) 2, ^, ^m ± V^m^ - 1, in which 
 
 m = 1 (- 5 ± V5). (7.) 2, ^, 2, i, ^ (1 ± V-"3. (8.) |(3 ± VS), | (-7±3 VS). 
 
 (9.) i(V5- 1 ± V- 10-2 \/5), _ ^ (V5 + 1 T V- 10 + 2 V5). (10.) ^rti 
 ± \/^rn^ _ 1^ in which m = 2 (1 ± V3). 
 
 BINOMIAL EQUATIONS. 
 
 (Page 255, 292.) 
 (1 to 6.) See answe^' to (45), page 138, and multiply them respectively by 
 \^5, \^^, fe, ^, V'n. (7.) See as above. 
 
 EXPONENTIAL EQUATIONS. 
 (Page 256, 206.) 
 (2 to 6.) 3.0957+, 11.384+, 3.292+, 0, 0. (8.) 3.597+. (9.) 2.316+ 
 
 (10.) 2.879+. (11.) 3.233+. (12.) 2.001+. (13.) ^"^ i^g/'"^ 
 
 / 2Jo|a+log_c^ lo|c (ie.)2i,3|. (17.) (^)""^^ 
 
 ^ ^ Mogflj ^ ' w log a+7i log 6 \o/ 
 
 ( jj . (18.) ^[q + 1^^;. .|? - i^^j. (19.) 3l^^_p3 i„g 3 
 
 2(2+log5)_ 243 _^ 42^ J I'^S- (24.) $196.71 
 
 2 log 2+3 log 3 Mog(r?+6) 
 
 $198.98, $200.17, $259.37, $265.33, $268.51, $180.61, $181.43, $134.83 
 
322 ANSWERS. 
 
 (25.) 7.13, 10.24, 16.23, 20.48 years. (26.) 29.91 years. (27.) $1502.6a 
 
 (28.) 11933.97. (30.) $1157.28. (32.) $4794.52, $3500. (33.) $577.06- 
 
 (35.) The former by $629.03. (36.) $500.91. (38.) 13.58 years. (39.) $796.87. 
 (40.) $8229.70. (41.) Gains $1756.60. 
 
 APPENDIX. 
 
 SJEBIES. 
 (Page 275, 311.) 
 
 (2.) 12, 6, 0. (3.) 8, 32. (4.) -3^. (5.) 1. (6.) -14. 
 
 (Page 276, 313.) 
 
 (4.) -ar', +««, +x. 5x\ 5x^. (5.) -27, +9, +3. 32805, 98415. 
 
 m +3x\+2x. 1093aJ^ 3281a;«. (7.) -4, +4. 192,448. (8.) -2.7;*, +.r^ 
 
 +2x. 87aj», 173x^. (9.) -jx. ^x\-^ x\ (10.) -1. +4, -6, +4. 
 
 56, 84, 120. (11.) +1, -3, +3. 26, 34, 43. 
 
 (Page 277, 314.) 
 (1.) 4516a;'. (2.) 17a;». (3.) -x\ (4.) 2733. (5.) 29525. (6.) 1365. 
 (7.) 20. '^1\ (8.) n(n + l). (9.) 6396a;" (10.) +1, -5, 
 
 + 10, -10, +5. +1, -3, +3. +1, -3, +3. +3a;', -x^ , +2x. (11.) »«. 
 
 (12.) 8694. (13.) 26. 34, 43, 53, 64. 26a;", 34a;", 43a;», 53a;", 64a;". 196, 336, 
 540, 8:5, 1210, 1716. 
 
 (Page 279, 315.) 
 (2.) No. (8 to 6.) Yes. 
 
 (Page 282, 316.) 
 
 ^'-f l_3a;_2a;2 • ^-'' l-x-x"^ ' ^^'^ {\-xY ' ^ '' l-2a;-a;«+2a 
 
 5fctl) . (10.) S78256 ; «°+10"'+35";'+50»'+jjg . („., 103155O ; 
 
 i 120 
 
 3 . (12.) 60710; _+--+_- -^. 
 
ANSWERS. 323 
 
 KV^-^-^-^)- (20.) i (21.) i. (22.)-A-. (23.) 1. 
 (25.) ^i^, (26.) i. (27.) A-. (28.) -h (29.) f,. (30.) r^,-^. 
 
 PILING BALLS AND SHELLS. 
 
 (Page 287, 322.) 
 
 (i.) 1540, 13244, 903. (2.) 9455, 4324, 35720, 465, 276, 1128. (3.) 7490, 
 3880. (4.) 624. (5.) 2730. (6.) 36256. 
 
 REVERSION OF SERIES. 
 (Page 288, ^323.) 
 
 (2.) a; = y - y« + y' - y* + etc. (3.) a; = y -^ 3y« + 13y^ - 67^/* + etc. 
 
 (4.) a; = y + 4y' + hy^ + -j^sV'' + etc. (5.) « = iy - Ay"'' + f Ay*" - etc. 
 
 (6.) a;=(y-l)-i(y-l)2 4-^(y-l)-'-i(y-l)* + etc. (7.) x^^ + {am^-r>) y^ 
 
 [hm * — mp —2in{am * — n)']y ^ 
 
 + . — + etc. 
 
 INTERPOLATION. 
 (Page 290, 325.) 
 (2.) 1.794. (5.) (♦ * *). 
 
 PEJRMUTATIONS. 
 
 (Page 293, 334.) 
 
 (1.) 720, 24, 3,628,800. (2.) 720, 42, 210, 840, 2520, 5040, 5040. 120, 21, 
 
 35, 35, 21, 7, 1. (3.) 36, 84. 126, 126, 84, 36, 1. (4.) 72, 504, 3024. (5.) 20. 
 
 (6.) 35, 21. (7.) 127. (8.) 479,001,600. (9.) 792. (10.) 15. 
 :il.)166,320. 64,864,800. (12.) 1023. 
 
 PROBABILITIES. 
 (Page 295, 335.) 
 (1.) U, -h ; n, 4^. (2.) f, f, 2:1, 4:3. (4.) 6 to 1. (6.) (* * *> 
 
 (6.) i 5toi. (11.) mh 
 
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