4 .. ^^J^^y^ ^' ^■, 2^' 'Ji ,Mm if% "i^- i hra^uL ,/^ e-^*-c/2v^^.^<^^^^- ^ UNIPLANAR ALGEBRA PART I OF A PROPEDEUTIC TO THE HIGHER MATHEMATICAL ANALYSIS IRVING STRINGHAM, Ph. D. Professor of Mathematics in the University of California San Francisco THE BERKELEY PRESS I So-! CCPVRIGHT, 1893 BY IRVING STRINGHAM Typography and Presswork by C. A, MURDOCK ^- Co , San Francisco CMS PREFACE. From the beginning, with rare exceptions, -^ a singular logical incompleteness has characterized our text-books in elementary algebra. By tradition algebra early became a mere technical device for turning out practical results, by careless reasoning inaccuracies crept into the explanation of its principles and, through compilers, are still perpetuated as current literature. Thus, instead of becoming a classic, like the geometry handed down to us from the Greeks, in the form of Euclid's Elemeiits, algebra has become a collec- tion of processes practically exemplified and of principles inadequately explained. The labors of the mathematicians of the nineteenth century — Argand, Gauss, Cauchy, Grassmann, Peirce, Cayley, Sylvester, Kronecker, Weierstrass, G. Cantor, Dedekind and others*-!^ — have rendered unjustifiable the longer continuance of this unsatisfactory state of algebraic science. We now know what an algebra is, and the prob- lem of its systematic unfolding into organic form is a definite and achievable one. The short treatise here presented, as the first part of a Propaedeutic to the Higher Analysis, endeavors to place concisely in connected sequence the argument required for its solution. * Notably, in English, Chrystal's Algebra, 2 vols., Edinburgh, 1886, 18S9. Oti the continent of Europe the deficiency has been compensated mainly in works on the Higher Analysis. **The literature through which algebra has been rehabilitated during the present century is extensive. See Stolz: AUgemeuie A^itlnnetik, Leipzig, 1885, for many valuable references. IV PREF-ACE. The first three chapters were made pubHc, substantially in their present form, in a course of University Extension Lectures in San Francisco during the winter of 1891-92, a synopsis of which was issued from the University Press in October, 1S91. At the close of these lectures the manu- script of the complete work was prepared for the press ; but unavoidable obstacles prevented its immediate publication and a consequent delay of somewhat more than a year has intervened. This delay, however, has made possible a revision of the original sketch and some additions to its subject-matter. The logical grounding of algebra may be attained by either of two methods, the one essentially arithmetical, the other geometrical, I have chosen the geometrical form of presentation and development, partly because of its simpler elegance, partly because this way lies the shortest path for the student who knows only the elements of geometry and algebra as taught in our schools and requires mathematical study only for its disciplinary value. The choice of method, therefore, is not to be interpreted to mean that the writer underestimates the value and the importance to the special mathematical student of the Number-System.* This system, however, has no appropriate place in the plan here presented. The point of departure is EucHd's doctrine of proportion, and the point of view is the one that Euclid himself, could he have anticipated the modern results of mathematical science, would naturally have taken. It is interesting to note that of logical necessity the development falls mainly into the historical order. For convenience of reference the fundamental propositions of proportion are enunciated and proved in an Introduction, in which I have followed the * Fine : The Nmnber'System of Algebra, Boston, 1S91. PREFACE. method recommended by the Association for the Impro\-e- ment of Geometrical Teaching, and pubhshed in its Syllabus of Plane Geometry. Except a few additions and omissions, the enunciations and numbering in Sections B and C of this Introduction are those of Hall and Stevens' admirable Text- Book of Euclid's Elejuents, Book V; and in Section D those of the Syllabus of Plane Geometry, Book IV, Section 2. The proofs vary in unessential par- ticulars from those of the two texts named. The subject-matter and treatment are such as to con- stitute, for the student already familiar with the elements of algebra and trigonometry, a rapid review of the underlying principles of those subjects, including in its most general aspects the algebra of complex quantities. All the funda- mental formulae of the circular and hyperbolic functions are concisely given. The chapter on Cyclometry furnishes, presumptively, a useful generalization of the circular and hyperbolic functions. The generalized definition of a logarithm (Art. 68) and the classification of logarithmic systems,* first made pubHc, outside of the mathematical lecture-room, in a paper read before the New York Mathematical Society in October, 1 89 1, and subsequently published in the Americaji fournal of Mathematics, are here reproduced in the revised form suggested by Professor Haskell. -^ A chapter on Graphical Transformations, giving the orthomorphosis of the ex- ponential and cyclic functions, appropriately concludes this part of the subject. Many incidental problems are suggested in the form of Agenda, useful to the student for exemplification and practice. But on the other hand, many elementary * American Journal '}f Mathematics, \o\. XIV, pp. 1S7-194, and Bulletin of the New York Mathematical Society \o\. II, pp. 164-170. PREFACE. algebraic topics are not discussed, because they are not useful to the main object t)f the work, and it was especially desirable that its purpose should not be hindered by the making of a large book. A few innovations in notation and nomenclature have been unavoidably introduced. The temptation to replace the terms complex qica7itity, imaginary quantity and real q2iantity by some such terms 2iS go7iio7i, orthogon and agon has been successfully resisted. Partly in order to aid the student in obtaining a com- parative view of the subject, partly in order to indicate in some detail the sources of information and give due credit to other writers, numerous foot-note references have been introduced. I take great pleasure in acknowledging my obligations to Professor Haskell for valuable criticism and suggestion. IRVING STRINGHAM. University of California, Berkeley, July i, 1893. CONTENTS INTRODUCTION. Euclid's Doctrine of Proportion. ARTICLE. PAGE. A. Notation 3 B. Definitions and axioms 3 C. Paraphrase ft'om tiie fifth book of Euclid 7 D. Seven fundamental theorems in proportion 13 E. Agenda: Supplementary propositions 20 CHAPTER I. LAWS OF ALGEBRAIC OPERATION. I. Quantity. 1. Quantities in general 21 2. Nature of real quantities 22 II. Definitions of Algebraic Operations. 3. Algebraic addition 23 4. Zero 23 5. Algebraic multiplication 23 6. The reciprocal 24 7. Idemfactor : Real unit 25 8. Quotient 25 9. Agenda : Problems in construction 26 10. Infinity 26 ir. Indeterminate algebraic forms 27 Vlll CONTENTS. III. Law of Signs for Real Quantities. ARTICLE. PAGE. 12. In addition and subtraction 28 13. In multiplication and division 29 14. In combination with each other : + , — with X > / • • 30 IV. Associative Law for Real Quantities. 15. In addition and subtraction 31 16. In multiplication and division 32 V. Commutative Law for Real Quantities. 17. In addition and subtraction 34 18. In multiplication and division 35 19. Agenda : Theorems in proportion 36 VI. Distributive Law for Real Quantities. 20. With the sign of multiplication 36 21. With the sign of reciprocation 37 22. Agenda : Theorems in proportion, arithmetical multipli- cation and division 38 VII. Logarithms and Exponentials. 23. Definitions: Napierian definition of a logarithm . ... 39 24. Relations between base and modulus 41 25. Law of involution 42 26. Law of metathesis 43 27. The law of indices 43 28. The addition theorem . 44 29. Infinite values of a logarithm 45 30. Indeterminate exponential forms 45 CONTENTS. iX VIII. Synopsis of Laws of Algebraic Operation. ARTICLE. PAGE. 31. Law of Signs 46 32-36. Laws of association, commutation, distribution, ex- ponentiation and logaritiimication 47 37. Properties of o, i and 00 ^ 48 38. Agenda : Involution and logarithmic operation in arith- metic 48 CHAPTER IL GONIOMETRIC AND HYPERBOLIC RATIOS. IX. Goniometric Ratios. Definition of arc-ratio 49 Definitions of goniometric ratios 50 Agenda: Properties of goniometric ratios 51 Line equivalents of goniometric ratios 51 Proofthat limit (sin ^) / ^= I, when ^ J=o 52 Area of a circular sector 53 Agenda : The addition theorem for goniometric ratios . 54 X. Hyperbolic Ratios. Definitions of hyperbolic ratios 55 Agenda : Properties of hyperbolic ratios 55 Geometrical construction for hyperbolic ratios 55 Agenda : Properties of the equilateral hyperbola .... 58 The Gudermannian 58 Agenda : Gudermannian formulae 58 Proofthatlimit (sinh?/") /^^ = i. when?^ = o 59 Area of a hyperbolic sector 60 Agenda : The addition theorem for hyperbolic ratios . . 62 Approximate value of natural base 63 Agenda: Logarithmic forms of inverse hyperbolic ratios 65 X CONTENTS. CHAPTER III. THE ALGEBRA OF COMPLEX QUANTITIES. XI. Geometric Addition and Multiplication. ARTICLE. PAGE. 57. Classification of magnitudes : Definitions 66 58. Geometric addition 68 59. Commutative and associative laws for geometric addition 69 60. Geometric multiplication 70 61. Conjugate and reciprocal 71 62. Agenda: Properties of cis^ 72 63. The imaginary unit 72 64. Commutative and associative laws for geometric multipli- cation 73 65. The distributive law 74 66. Argand's diagram 75 67. Problems in complex quantities 76 XII. Exponentials and Logarithms. 68. Definitions of modulus, base, exponential, logarithm . . 78 69. Exponential of o, land logarithm of I, i5* 81 70. Classification of logarithmic systems 81 71. Special constructions 83 72. Correspondence of initial values of exponential and logarithm 84 73. The exponential formula 84 74. Demoivre's theorem 86 75. Relations between base and modulus 86 76. The law of involution 88 77. The law of metathesis 89 78. The law of indices 89 79. The addition theorem for logarithms 90 80. The logarithmic spiral 90 81. Periodicity of exponentials 91 82. Many-valuedness of logarithms 92 83. Direct and inverse processes 92 84. Agenda: Reduction of exponential and logarithmic forms 94 CONTENTS. XI XIII. What Constitutes an Algebra? ARTICLE. PAGE. 85. The cycle of operations complete 95 86. Definition of an algebra 96 XIV. Numerical Measures. 87. Scale of equal parts 97 88. For real magnitudes 98 89. For complex magnitudes 100 CHAPTER IV. CYCLOMETRY. XV. Cyclic Functions. 90. Definitions of modocyclic functions loi 91. Formulae of cyclic functions 102 92. Agenda : Examples in multiples and submultiples of the argument 106 93. Periodicity of modocyclic, hyperbolic and circular functions 107 94. Agenda: Functions of submultiples of the periods . . . 107 95. The inverse functions 108 96. Agenda: Formulae of inverse functions 109 97. Logarithmic forms of inverse cyclic functions no 98. Agenda : Examples in the reduction of inverse cyclic forms 112 CHAPTER V, GRAPHICAL TRANSFORMATIONS. XVI. Orthomorphosis Upon the Sphere. 99. Affix, correspondence, morphosis 113 100. Stereographic projection 113 loi. Transformation formulae 114 102. The polar transformation 116 103. Agenda: Properties of the stereographic transformation 117 Xll CONTENTS. XVII. Planar Orthomorphosis. ARTICLE. PAGE. 104. The Z£/-plane and the ^--plane 117 105. The logarithmic spirals of ^" non-intersecting . ... 118 106. Orthomorphosis of B"' 119 107. Isogonal relationship 121 108. Orthomorphosis of cosk (/^ + zV) 123 109. By confocal ellipses 124 110. By confocal hyperbolas 126 111. Agenda: Problems in orthomorphosis 127 CHAPTER VI. PROPERTIES OF POLYNOMIALS. xviii. Roots of Complex Quantities. 112. Definition of an «*'^ root 128 113. Evaluation of «*^ roots 128 114. Agenda: Examples in the determination of «*^ roots . 129 XIX. Polynomials and Equations. 115. Definition of a polynomial 130 u6. Roots of equations 131 117. The remainder theorem 131 118. Argand's theorem 133 119. Every algebraic equation has a root 136 120. The fundamental theorem of algebra 136 121. Agenda : Theorems concerning polynomials 138 APPENDIX. SOME AMPLIFICATIONS. Art. 23, page 40. On the notation b^ 139 Art. 24, page 42. On the proof of the theorem : If the modulus be changed from 7n to km the corre- sponding base is changed from <^ to <5'/* 139 Art. 27, pages 43, 44. Alternative proof of the law of indices 139 SIGNS AND ABBREVIATIONS. The following quantitive and operational signs are used in this work and are collected here for convenience of reference. + Plus. — Minus. (\i Difference between. ± Plus or minus. X Multiplied by. / Divided by. y/ Multiplied or divided by. = Is equal to. =: Approaches as a limit. >> Is greater than. <^ Is less than. ^ Is not less than. "^ Is not greater than. ( >>) Something greater than. (< ) Something less than. tsr Tensor of (absolute value of), vsr Versor of amp Amplitude of (argument of). In Natural logarithm of e Natural base. TT Ratio of circumference to diameter, ''log Logarithm of, to base d. log^ Logarithm of, to modulus k, •"^ [ Sine, cosine, etc., to modulus k. cos,, etc.) ' "^it j Limit of, when :r approaches a. = a) limit .*. Therefore. UNIPLANAR ALGEBRA INTRODUCTION. EUCLID'S DOCTRINE OF PROPORTION. A. — Notation. In Sections B and C of this Introduction capital letters denote magnitudes, and when the pairs of magnitudes com- pared are both of the same kind they are denoted by letters taken from the early part of the alphabet, as A, B compared with C, D ; but when they are, or may be, of different kinds, by letters taken from different parts of the alphabet, as A, B compared with P, Q or X, Y. Small italic letters m, 71, p, q denote integers. By vi . A or vi A is meant the w^^ multiple of A, and it may be read in times A ; by vin is meant the arithmetical product of the integers in and ;/, and it is assumed that vi n = nm. The combination m . 7ih. denotes the w"' multiple of the 7^^'' multiple of A, and it is assumed that m . n\ = n . in A = vi n . A. B. — Definitions and Axioms.^'' 1. " A greater magnitude is said to be a multiple of a less, when the greater contains the less an exact number of times. ' ' 2. "A less magnitude is said to be a submultiple of a greater, when the less is contained an exact number of times in the greater." * The quoted paragraphs of Section B are transcribed in part from the Sylla- bus of Plane Geometry^ published by the Association for the Improvement of Geometrical Teaching, in part from Hall and Stevens' Text Book of EucluVs Elements, Book V. 4 INTRODUCTION. The following property of multiples is assumed as axiomatic : ( i ). 7;2 A > = or < w B according as A > = or < B. (Euc. Axioms i and 3.) The converse necessarily follows : ^^ (ii). A > = or < B according as m\ > = or < vi B. (Euc. Axioms 2 and 4.) 3. ' ' The ratio of one magnitude to another of the same kind is the relation which the first bears to the second in respect of quantuplicity . ' ' "The ratio of A to B is denoted thus, A : B ; and A is called the antecedent, B the consequent of the ratio. ' ' "The quantuplicity of A with respect to B may be estimated by examining how the multiples of A are distributed among the multiples of B, when both are arranged in ascending order of magnitude and the series of multiples continued without limit." This distribution may be represented graphically thus : D A 2 A 3 A 4A 5A 6 A 7 A 8A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Multiples of A : Multiples of B : , „ „ „ „ ^ o B 2B 3B 4B 5B 6B Fig. I. 4. If, in this comparison of the multiples of two magnitudes, any multiple, as uK, of the one coincides with (is equal to) any multiple, as ;;2B, of the other, the two magnitudes bear the same ratio to one another as the two numbers vi, n, and are said to be commensurable, but * " Rule of Conversion. If the hypotheses of a group of demonstrated theorems be exhaustive (that is, form a set of alternatives of which one must be true), and if the conclusions be mutually exclusive (that is, be such that no two ot tham can be true at the same time), then the converse of every theorem of the group will necessarily be true." {Syllabus of Plane Geometry, p. 5.) INTRODUCTION. 5 Dicommensiirable if no such coincidence takes place, however far the process of comparison is carried. 5. The ratio of two magnitudes is said to be equal to a second ratio of two other magnitudes (whether of the same or of a different kind from the former), when the multiples of the antecedent of the first ratio are distributed among those of its consequent in the same order as the multiples of the antecedent of the second ratio among those of its consequent. As tests of the equality of two ratios either of the follow- ing criteria may be employed, m and n being integers : (i). The ratio of A to B is equal to that of P to O, when ;;^ A > = or < 7z B according as w P > = or < ;^ Q. (ii). If VI be any integer whatever and n another inte- ger so determined that either mh. is between ;^ B and (;^^-I)B or is equal to nV>, then the ratio of A to B is equal to that of P to Q, when rn? is between nQ and (?^+ i)Q or equal to ?^0 according as nik. is between 7^B and (;z+ i)B or equal to ;zB. (iii). It should be remarked that the rule of identity^ is applicable to this definition, and that therefore, if the ratio of A to B be equal to that of P to Q, then also w A > = or < 7z B according as w P > Z3= or < n O. 6. "When the ratio of A to B is equal to that of P to O, the four magnitudes are said to be proportionals, or to form ^proportion. The proportion is denoted thus : A : B :: P : Q, * Rule of Idf.ntity. If there be only one fact or state of things X, and only- one fact or state of things Y, then from X is Y the converse Y is X of necessity- follows. INTRODUCTION. 3\ 4A 5A 6A 7A 1' ', ' ,' ', B 2B 3B 4B 5B P 2? 3P 4P 5P 6P 7P which is read : A is to B as P is to O. A and O are called the extremes, B and P the means. ' ' The proportion A : B : : P : Q may be represented graph- ically thus : Multiples of A : Multiples of B : Multiples of P : , , , , , ^ ' I ' ,' I I ' ' Multiples of Q : 1 1 \ I L '^ o Q 2Q sQ 4Q 5Q Fig. 2. In a diagram of this kind it is obvious that in general, of the two figures thus compared, representing tw^o equal ratios, one will be an enlarged copy of the other, but in particular, if the antecedents and consequents be respec- tively equal to one another, the two figures will be con- gruent. 7. "The ratio of one magnitude to another is greater than that of a third magnitude to a fourth, when it is possi- ble to find equimultiples of the antecedents and equimulti- ples of the consequents such that while the multiple of the antecedent of the first ratio is greater than, or equal to, that of its consequent, the multiple of the antecedent of the second is not greater, or is less, than that of its consequent." That is, A : B > P : O, if integers m, n can be found such that if m A > 72 B, then w P ^ 72 Q, or if mK = n B, then 772 P < 72 Q. 8. "If A is equal to B, the ratio of A to B is called a ratio of equality. INTRODUCTION. 7 "If A is greater than B, the ratio of A to B is called a ratio of greater inequality. "If A is less than B, the ratio of A to B is called a 7'atio of less inequality. ' * 9. "Two ratios are said to be reciprocal when the ante- cedent and consequent of one are the consequent and ante- cedent of the other respectively." 10. "Three magnitudes of the same kind are said to be proportionals when the ratio of the first to the second is equal to that of the second to the third." 11. "Three or more magnitudes are said to be in con- tinued proportion when the ratio of the first to the second is equal to that of the second to the third, and the ratio of the second to the third is equal to that of the third to the fourth, and so on." C. — Paraphrase from the Fifth Book of Euclid.* Proposition i. * ^Ratios which are equal to the same ratio ai^e equal to one another. ' ' Let A : B : : P : Q and X : Y : : P : Q, then A : B : : X : Y. For, the multiples of each of the six magnitudes being ranged beside each other in pairs, as represented in Def. 6, it is obvious that in each case the multiples of the ante- cedent are distributed among those of the corresponding consequent in exactly the same order. * Enunciations and numbering quoted from Hall and Stevens' Text Book of Euclid's Elements, Book V. INTRODUCTION. Proposition 2. '■'If two ratios be equal, the antecedent of the second is greater than, equal to, or less than, its consequent, according as the antecedoit of the first is greater than, equal to, or less than, its consequent.'' Let A:B::P:0; then P > = or < O according as A > = or < B. This follows from Def 5 (iii) by taking m = n ^= i. Proposition 3. ''If two ratios be equal, their reciprocal ratios are equal.''' (Invertendo.) Let A : B :: P : O, then B : A :: O : P. This is made evident graphically by constructing dia- grams for the two ratios, as in Def 6, and reading them off as a proportion, once in direct order (downwards) for the hypothesis, and again in inverse order (upwards) for the conclusion, and applying the rule of identity. Proposition 4. "Equal magnitudes have the same ratio to the same mag- 7iitude; and the same magiiitude has the same ratio to equal magnitudes. ' ' Let A, B, C be three magnitudes of the same kind, and let A = B ; then A : C :: B : C and C : A :: C : B. For if A = B, their multiples are identical, and are therefore distributed in the same order among those of C. .-. A : C :: B : C, (Def 5.). and invertendo C : A :: C : B. (Prop. 3.) INTRODUCTION. 9 Proposition 5. ''Of two unequal magnitudes, the greater has a greater ratio to a third niag7iitude than the less has; and the same magnitude has a greater ratio to the less of two magnitudes than it has to the greater. ' ' Let A, B, C be three magnitudes of the same kind ; then if A > B, so is A : C > B : C, and if B < A, so is C : B > C : A. First : If A > B, an integer m can be found such that m A exceeds 7n B by a magnitude greater than C. Hence, the integer 7i being so chosen that mA is equal to or greater than nC and less than (?2-}-i)C, the conditions re- quire that D / ^ wB < ;^C, and therefore A : C > B : C. (Def. 7.) Second : If B < A, then taking m and 71 as in the fore- going proof the same inequalities between the multiples of A, B and C still exist, and because ?^C ^ ?;2B while 71Q is either •< 711 K, or at most = fJiA, .'. C ; B > C : A. (Def 7.) Proposition 6, ''Magnitudes which have the same ratio to the sa7ne 77iag- 7iitude are equal to one a7iother; and those to which the sa7ne 7nagnitude has the sa7ne ratio are equal to 07ie a7iother. ' ' That is, A, B, C being three magnitudes of the same kind ; if A : C :: B : C, then A = B. and if C : A :: C : B, then A = B. The proof of this proposition is made a part of the proof oi Prop. 7. lO INTRODUCTION. Proposition 7. * ' ' That magnitude which has a greater ratio than another has to the same magnitude is the greater of the two; and that magnitude to which the same has a greater ratio than it has to another magnitude is the less of the two'' That is, A, B, C being three magnitudes of the same kind; if A : C > B : C, then A > B. and if C : A > C : B, then A < B. Proof of Propositions 6 and 7. First Part : It has been proved that A : C :: B : C, if A = B, (Prop. 4.) and A : C > B : C, if A > B, (Prop. 5.) and A : C < B : C, if A < B. (Prop. 5) Hence, hy \.\\e rule of conversion, (Def 2, Ax. ii.) A > = or < B according as A : C > = or < B : C. This proves the first part of each of the two propositions. Second Part : It has been proved that C : A :: C:B, if A =- B, (Prop. 4.) and C : A< C : B, if A > B, (Prop. 5.) and C : A > C : B, if A < B. (Prop. 5.) Hence, by the 7'ule of conversion, (Def 2, Ax. ii.) A > =:- or < B according as C : A < = or > C : B. This proves the second part of each of the two propositions. (i). Corollary .• 7^ A : C > B : C, then C : A < C : B, and conversely. (ii). Corollary .• i^ A : C > P : R, then C : A < R : P, and conversely. INTRODUCTION. II Proposition 8. ''Magnitudes have the same ratio one to another which their equimultiples have. ' ' Let A, B be two magnitudes of the same kind and m any integer ; then A : B :: ;;/A : wB. For if/, q be any two integers, m ./> A > = or < w . ^ B according as / A > == or < ^B. (Def 2, Ax. ii.) But VI. p A =^p. m A and 7n.qB = q.7nB; . • . p.7nA^ = or<,q.mB according as / A > = or < ^B, whatever integers p and q represent. Hence A : B :: mA : niB. (i). Corollary.- ^ A : B :: P : Q, then w A : viB :: ?iF : ?iQy whatever integers vi and n may be. Proposition 9. ''If two ratios be equal, and any equimultiples of the antecedents and also of the co7iseque7its be taken, the 77iulti- ple of the first a7itecede7it has to that of its co7iseque7it the same ratio as the multiple of the other a7itecede7it has to that of its conseque7it.''^ Let A : B :: P : O, then ;;^A : ;^B :: 771B : ;zO, w and n being any integers. For, if/, q be any two integers, then, since by hypothesis A : B :; P: O, . • . pin.B > = or < q7i.O according as /w. A > = or < ^?^. B ; (Def 5.) that is, p . 77iB y> ~ or <^ q . nQ according as /. 7nA > ^ or < ^. 72B. .'. ?;^A : ;^B :: wP : ;^0. (Def 5.) 12 INTRODUCTION. Proposition io. ''If four magnitudes of the same kind be proportio?ials, the first is greater tJian, equal to, or less than, the third, according as the second is greater than, equal to, or less tha?i, the fourth. ' ' Let A : B :: C : D; ^n A > = or < C according as B > = = or < D. If B > D, then A : B < A : D; A : B :: C : D, .'. C :D< A : D, .-. A:D>C : D, .'. A>C. (Prop. 5.) (Hypoth.) (Prop. 7.) If B = D, then C: B :: C: D; A:B :: C:D, .-. A : B :: C : B, .-. A = C. (Prop. 4.) (Hypoth.) (Prop. I.) (Prop. 6.) If B < D, then A : B> A : D; A : B :: C : D, .'. C : D> A : D, .-. C> Aand A < C. (Prop. 5.) (Hypoth.) (Prop. 7.) Proposition it. ''If four magnitudes of the same kind be proportionals ^ the first will have to the third the same ratio as the second to the fourth. ' ' ( Alternando. ) Let A : B :: C : D, then will A : C :: B : D. INTRODUCTION. 13 For m and n being integers, A : B :: niP>^ : ;;zB and C :D :: /I C : nD, . • . mA > =^ or < 7iC according as 7;^B > But VI and 7i are any integers ; .-. A : C :: B D. (Prop. 8.) (Prop. I.) or < 7iD. (Prop. ID.) (Def. 5.) D. — Seven Fundamental Theorems in Proportion.* Proposition 12: ( Lemma). ''^'* '' If on two straight lines, AB, CD, cut by two parallel straight lines A C, B D, equimultiples of the i?itercepts re- spectively be taken; the?i the lijie joining the points of division will be parallel to AQ^ ^r B D. " On AB and CD, produced either way, let the respect- ive equimultiples BE, D F of A B, C D be taken, on the same side of B D ; then E F is parallel to B D. For, join A D, D E, B C, B F. Since the triangles A B D, C B D are on the same base BD, and their vertices A, C are in the line AC parallel to BD, they are equal in area; and whatever multiple BE is of A B, or DF of CD, the triangle DBE is that same multiple of the triangle ABD, and the triangle DBF of the triangle C B D. . ■ . area of triangle E B D =: area of triangle FBD. But these triangles E B D, FBD have the same base B D ; hence their vertices E and F must be in a straight line parallel to BD and therefore EF is parallel to BD. * Enunciations of Propositions 13-17 quoted from the Syllcifjus of Plane GeO' metry, Book IV, Section 2. **J. M. Wilson: Elementary Geometry, page 205. Fig. 3 14 INTRODUCTION*. Proposition 13. '' If two straight lines be cut by three parallel straight lines, the intercepts cni the one are to one another in the same ratio as the correspandijig iyiiercepts on the other. ' Let the three parallel straight lines A A', B B', CC be cut by t\s-o other straight lines A C, A' C in the points A. B. C and A'. B', C respectively; then AB : BC :: A' B' : B* C. For. on A C take B M = m . A B. B X = ;^ • B C. m and n being integers, M and X on the same side ot B. Also on A'C takeB'M' = ;;^. A'B', B' X' = ;^. B'C. _a a^ M' and X' being on the same side of B' as M and N of B. Then, by the foregoing lemma b 5' (Prop. 12), M M' and X X' are both parallel to ^ '^, B B' and cannot meet. Hence, whatever in- tegers in and n may represent, B' M' (or 7//. A' B' )> = or < B'X' (or n. B' C) according as B M (or m. A B ) > = or < B X (or n. B C); .-. AB : BC :: A'B' : B'C. (i). Corollary : '■' If tlie sides of a tri- ^^r- -f angle be cut by a straight line parallel to the base, the seg- yjients of one side are to one another in the same ratio as the segmetits of the other side. ' ' (ii). Corollary: '' If two straight lines be cut by four or more parallel straight lines, the ijitercepts on the 07ie are to cme ayiother in the same ratio as tlie correspondiyig iji- tercepts on the other' ' (iii). Corollary : If in any triayigle, as O A B, a straight line E F, parallel to the side A B. ait the other sides ^ O A in E and O B in F, then AB : EF :: OA : OE :: OB : OF. INTRODUCTION. 15 Proposition 14. '' A given finite straight line can be divided internally into segments having any given ratio, and also externally into segments having any given ratio except the ratio of equality ;'' and if the line be given in both length and sense, there is in each case one and only one such point of division. Let A B be the given straight Hne ; it may be divided, as at E, in a given ratio P : Q. For, on the straight line A G making any convenient angle with A B lay off A C = P, C D = Q. Then C E, drawn parallel to D B to meet AB in E, will divide AB at E in the given ratio. (Prop. 13.) Since C E and D B are par- allel, C and E lie on the same side of D and B, and hence the division will be internal if A and D are on opposite sides of C, but external if A and D are on the same side of C. If the line to be divided be esti- mated in a given sense, as from A to B, there is in each case only one point of division in the given ratio, as F, be joined to C and B G be drawn parallel to F C, then AF : FB :: AC : CG, (Prop. 13.) so that F divides AB in the ratio AC : CG, different from the given ratio. If the given ratio be a ratio of equality, the construction in the case of external division fails. Fis- 5 For if any other point? PRorosiTioN 15. ' ' A straight line which divides the sides of a triangle proportionally is parallel to the base of the triangle. ' ' i6 INTRODUCTION. Let DE divide the sides AB, AC of the triangle ABC proportionally, so that AD: DB:: AE : EC, then D E is parallel to B C. If possible, let DF be parallel to B C, F some other point than E ; then AD:DB::AF:FC. .A But by hypothesis ^ ^°P* ^^'^ AD:DB :: AE:EC .-. AF :FC:: AE : EC which is only possible when F coincides with E. Fig. 6. Proposition i6. "■' Rectangles of equal altitude are to one another in the same I'atio as their bases. Let K A, K B be two rectangles having the common alti- tude OK and their bases OA, OB extending in the same line from O to the right ; then rect. KA : rect. KB :: OA : OB. In the line O A B pro- k duced take OM-=w.OA, ON=;^.OB, m and 7i being integers, and complete the rectangles KM, Fig. 7. K N. Whatever multiples O M and O N are of O A and OB, the rectangles KM* and KN are the same respective multiples of the rectangles K A and K B ; that is, KM = 7n.KA, KN = ?z.KB, A B M N INTRODUCTION. 17 and according as OM (or ;;^.OA) > = or < ON (or w.OB) so is KM (or w.KA) > = or < KN (or?^.KB) .-. rect. KA : rect. KB :: OA : OB. (Def. 5.) (i). Corollary: '' Parallelograms or triangles of the same altitude ai'e to one another as their bases. Proposition 17. "/;^ the same circle, or in equal circles, angles at the cen- tre and sectors are to one another as the arcs on which they stand. ' ' Let there be two equal circles with centres at K and K', and on their circumferences any two arcs O A, O'B ; then angle OKA : angle O'K'B :: OA : O'B, and sector OKA : sector O'K'B :: O A : O'B. (b) On the two circumferences respectively take OM = w.OA, O'N = ?^.0'B, m and n being integers. Whatever multiples OM and O'N are of O A and O'B, the same multiples respectively are the angles or sectors O K M and O'K'N of the angles or sectors OKA and O'K'B ; that is, O KM =: W.OKA, O'K'N = ;z. O'K'B, 1 8 INTRODUCTION. and according as OM (or ;;^.0 A) > = or < O'N (or ?z.O'B) so is OKM (or ?«.OKA) > = or < O'K'N (or «.0'K'B). .-. OKA : O'K'B :; OA : O'B, (Def. 5.) wherein OKA and O'K'B represent either angles or sectors. (i). CoFtOLLARY : III any tivo given concentric circles, corresponding arcs intercepted by common radii bear always the same ratio to one ajiother. That is, if u, 21' , it", ... be arcs on one of the circles determined by a series of radii, and the same radii intercept on the other circle the corresponding arcs z\ v' , v" , . . . then , , .f „ n \v v. 2t\v :: u : V ... Proposition iS. Arcs of circles that subtend the same angle or equal angles at their centres are to one another as their radii. c'l fe A A ^ ^ Fig: 9. Let there be two arcs S ^= AD and S' = A'D', havin their angles AOD and A'OD' either equal and distinct or common, and let R, R' be their respective radii ; then S : S' :: R : R'. If the two arcs be not concentric, let them be made so, and let their hounding radii be made to coincide. Then INTRODUCTION. I9 the proposition proved for the concentric will also be true for the non-concentric arcs. Conceive the angle at O to be divided into m equal parts, in being any integer, by radii setting off the arcs S and S' into the same number of equal parts, and draw the equal chords of the submultiple arcs of S and the like equal chords of the submultiple arcs of S'. Let C and C be the respective lengths of these chords. Then, since the chords C, C cut off equal segments on the Hues OA', OB' they are parallel (Prop. 12), and C : C :: R : R'. (Prop. 13, Cor. iii.) Therefore, vi being any integer, mC : ;;^C':: R : R'. (Prop. 8.) Let 711 be the number of equal parts into which the angle at O is divided ; then in C and vi C are the lengths of the polygonal lines formed by the equal chords of S and S' respectively. If now VI be increased indefinitely, the chords decrease in length but increase in number, and the two polygonal lines which they form approach coincidence with the arcs S and S' respectively ; and by increasing in sufficiently the aggregate of all the spaces between the arcs and their chords may be made smaller than any previously assigned arbitrarily small magnitude. Under these circumstances it is assumed as axiomatic that the relation existing between the polygonal lines exists also between the arcs, which are called limits. Under this assumption it follows that S : S' :: R : R'. Q. E. D. (i). Corollary : Circumferences are to one another as their radii. (ii). Corollary : Of two arcs of circles that subtend the same angle or equal angles at their centres, that is the longer which has the longer radius. ( By Prop. 2.) 20 INTRODUCTION. E. — Agenda : Supplementary Propositions. (i). If two geometrical magnitudes A, B, have the same ratio as two integers vi, n, prove that nK = m B. (2). If A, B be two geometrical magnitudes and m, 71 two integers such that 7^ A = 7«B, prove that A : B :: w : n. Hence infer the statement in the first part of Definition 4, page 4, concerning commensurable magnitudes. (3). Given A : B :: P : Q and ;zA = ;/^B, prove that nY = mO. (4). It is a corollary of (3), that if A : B :: P : O and A be a multiple, part, or multiple of a part of B, then P is the same multiple, part, or multiple of a part of O. (5). Given A : B :: P : Q and B : C :: Q : R, prove that P > = or < R according as A > = or < C. (6). Given A : B :: P : O and B : C :: O : R, prove that A : C :: P : R. (Ex aequali.) (7). Given A : B :: P : O and B : C :: O : R and C : D :: R : S and D : E :: S : T, prove that A : E :: P : T. State and prove the general theorem of which this is a par- ticular case. (8). Given A : B :: O : R and B : C :: P : Q, prove that A : C :: P: R. CHAPTER I. LAWS OF ALGEBRAIC OPERATION. I. Quantity. I. Quantities in GeneraL Quantities, whatever their nature, may be expressed in terms of geometrical magni- tudes ; in particular they may be thought of as straight lines of definite fixed or variable length. Such mag- nitudes, in so far as they represent the quantities of ordinary algebra, are of three kinds : real, imaginary, and complex ; real if, when considered by themselves (laid off upon the real axis), they are supposed to involve only the idea of length, positive or negative, without regard to direction, imaginary when they involve not only length, but also turning or rotation through a right angle, that is, length and direction at right angles to the axis of real quantities, finally complex if they embody length and rotation through any angle, that is, length and unrestricted direction in the plane. If we think of the straight line as generated by the motion of a point, we may translate length positive or negative into motion forwards or backwards ; and it will sometimes be convenient to use the latter terminology in place of the former. It is at once evident that both reals and imaginaries are particular forms of complex quantities, reals involving motion forwards or backwards and rotation through a zero-angle, imaginaries involving motion forwards or 22 LAWS OF ALGEBRAIC OPERATION. backwards and rotation through a right angle. The three kinds of quantities will be considered in order; the distinction between them, here roughly outlined, will be made clearer by a study of their properties. 2. Nature of Real Quantities. It is evident that all real quantities may be made concretely cognizable by laying them off (in the imagination) as lengths, in the positive or negative sense, upon one straight line. In this represen- tation every straight line suffices to embody in itself all real quantities, having its own positive and negative sense, that is, its direction forwards and backwards. In particular, all the numbers of common arithmetic, both integral and fractional, are accurately represented by distances laid off from a fixed origin in the positive sense upon a straight line, and in the same way all so-called irrational numbers, though only approximately realizable as true numbers in arithmetic, are accurately represented. Hence the following proposition, which is postulated as self-evident: The laws of algebraic operatio7i that obtain with geo- metrical real magnitudes, that is, lengths laid off upon a straight line, are ipso facto trne when applied to arithmetical quantities, or Jimnbers. But the converse of this proposition is not equally self- evident. For inasmuch as so-called irrational number, that is, quantity in general, is not realizable as true number (integer or fraction) in arithmetic, the proof that the laws of algebraic operation obtain for integers and fractions constitutes not a proof, but only a presumption, that they obtain also for so-called irrational number. hi the following pages viagnitudes will be represented by straight lines of finite length. LAWS OF ALGEBRAIC OPERATION. 23 II. Definitions of Algebraic Operations. 3. Algebraic Addition. Simple addition is here defined as the putting together, end to end, different hne- segments, or Hnks, in such a way as to form a one dimen- sional continuum, that is, a continuous straight line. This kind of addition corresponds to the addition of positive numbers in arithmetic. Algebraic addition takes account of negative magni- tudes, that is, of lines taken in the negative sense (from right to left, if positive lines extend from left to right), and to add to any line-segment a negative magnitude is to cut off from its positive extremity a portion equal in length to the negative magnitude. This kind of addition includes the addition and subtraction of positive numbers in arith- metic and introduces the new rule that larger positive magnitudes may be subtracted from smaller, producing thereby negative magnitudes. We then extend the idea of negativeness also to number and produce negati\'e number, prefixing the sign — to positive number as a mark of the new quality. The result of adding together algebraically several magnitudes is called a siun. In a sum the constituent parts are terms. 4. Zero is defined as the sum of a positive and an equally large negative magnitude ; in symbols, -j- a — a = o. It is not a magnitude but indicates the absence of magnitude. 5. Algebraic Multiplication. On two straight lines making any convenient angle with one another at O, Fig. 10, lay off OA = a, OB = d, and on OA in the same 24 LAWS OF ALGEBRAIC OPERATION. direction as OA lay off 0/=j\ which shall be of fixed length in all constructions belonging to algebra and shall be called the rea/ 2init. Join J and B and draw from A a straight line parallel to JB to intersect OB in M. Then by Proposition 13 (p. 14) the intercepts on OA, OM by the parallels JB, AM are proportionals, and if OM=vi, J : a \: b : vi. The length w, thus determined, is defined as the algebraic prodtict, or simply the product, of the real magnitude a by the real magnitude b, and is denoted by ay^b, Q,x\^y a-b, or more simply still hy ab r^ In a pro- duct the constituent parts 2X^ factors. The product aV^ b may also be a factor in another pro- duct, consisting therefore of three factors, as (<3; X ^) X c. and this may in turn be a factor in a product of four factors, and so on. Fig. 10. 6. Reciprocals. J. If m be equal to j\ then J :a::b:j and b, a, are called the 7'e- ciprocals of a, b, respectively. These reciprocals are written thus: j I a=^ reciprocal oi a, j I b = reciprocal of b, etc.; or more simply, since/ remains unchanged through- out all algebraic operations, they may be conveniently * Descartes: la Geometrie, reprint of iSS6, p. 2. LAWS OF ALGEBRAIC OPERATION. 25 represented hy I a, j d, etc. Since the means, in any pro- portion between like magnitudes, may be interchanged, i^ b = I a, then also a=^ j d, and vice versa. A reciprocal, being itself a line-magnitude, may enter a product as one of its factors. 7. Idemfactor: Real Unit. If in the proportion j \a\\b \vi we write a =j\ that is, make AM in Fig. lo coincide with JB, then j \j \\ b \ w, that is, m = b. (Prop. 2.) But by the definition of a product m =j X b ; .■.jXb=b, and in particular An operator which, Hkey, as factor in a product leaves the other part of the product unchanged, is called an ide^nf actor. ^^' This particular real idemfactor is what was defined in Art. 5 as the real unit. In arithmetic it is denoted by the numerical symbol i. 8. Quotient. The product defined by the propor- tion j :c :\ I a -.m is cX I a and is called the quotient of c by a. The sign X before / may be omitted without ambiguity and this quotient be denoted by the simpler notation c \ a, in which c is called the dividejid and a the divisor. The proportion y -.awb-.j defines b = I a, <2 = / b, and a y^ b ■=^j ; hence ^X b ^=^ ayi I a = a I a =y, * Benjamin Peirce: Linear Associative Algebra (1870), p. 16, or American Journal of Mathematics, Vol. IV (1881), p. 104. 26 LAWS OF ALGEBRAIC OPERATION. and so for any magnitude whatever. Hence we may de- scribe the real unit as the quotient of any real magnitude by itself The quotient c ; a is also represented by r -f- <^, or by -. The latter notation will be frequently employed in the sequel. g. Agenda. Problems in Construction. (i). From the definitions of Arts. 5, 6 and 8 prove that the following construction for the quotient a j b is correct: On one of two straight lines, making any con- venient angle with one another, lay off OA = a, OB = b, on the other OJ=^j\ join B and yand draw AM parallel to B/to intersect 0/m M. OM\s the quotient sought. (2). Given a X a = vi, construct a. (3). Given a, b and c, construct {a y^ b) ^ c. (4). Prove that «X^i3> =or<<^ according as <3^ is > = or < I (5). Draw OX and O Kmaking any convenient angle with each other ; on O Flay off OJ=j\ OA = a, OB= b, and on OJ^take OJ^^=^j. A straight line through /par- allel to OX will be cut by J^A and J^B in two points P and Q. Show that if ^ and B are on the same side of (9, the distance between P and Q is PQ = j a ^ j b, where CO means difference between, but if A and B are on oppo- site sides of O, then PQ =^ j a-\- j b. In this construction a and b are supposed to be positive magnitudes. 10. Infinity is defined as the reciprocal of zero; in symbols /o = CO . LAWS OF ALGEBRAIC OPERATION. 27 When a magnitude decreases and becomes zero, its recip- rocal obviously increases and becomes infinite. Since zero is not a magnitude, neither is infinity as here defined. In Art. 6 it was shown that d = j a implies also a = j b] hence, fi-om the definition / o = co follows / CO = o. The construction for a product (Art. 5) shows that when one of its factors becomes o or co , the other remaininsf finite, the product itself is also o or co , so that for all finite values oi a a/ao = a y( = 0. From the definition of addition (Art. 3) it is also obvious that ±: o ± a = ± a, ±00 ih <^ = dz CO . II. Indeterminate Algebraic Forms. When a sum or product assumes one of the forms -|- co — co , o X co , 0/0, CO / CO , it is said to be indeterminate, by which is meant : the form by itself gives no information concerning its own value. (i). The form -|- 00 — co . On a straight line ABP take at random two points A, B, so that AB is any real finite magnitude whatever. Take P, Q, R, on the same line, A Q B R P a b Fig. 12. such that AP= I AQ = la, BP=I BR = \b, and let Q pass into coincidence with A. P then passes out 28 LAWS OF ALGEBRAIC OPERATION. of finite range, R passes into coincidence with B and the difference AB= a- : b, whatever its original value, assumes the form / O — /o=CO — CO. Hence, taken by itself, :o — co gives no information con- cerning its own value and is indeterminate. (ii). The forms oX'20,0/0, oo/co. In the figure of Art. 5, let tI/ and y remain fixed, while MA and/B, being always parallel to one another, turn about J/ and y until JfA coincides with, and /B becomes parallel to OM. At this instant a becomes zero, d infinite, and aV^ b assumes the form o X co ; and because the original value of ^ X ^ is anything we choose to make it, the expression o X 00 gives no information concerning its own value and is therefore an indeterminate form. Since / o = 00 and / co = o, we may replace o / o by o X 00 and co / 00 by co X o. The two forms 0/0 and CO / 00 are therefore also indeterminate. An expression, such as j a — j b, that gives rise to an indeterminate form, may nevertheless approach a deter- minate value as it nears its critical stage. To find this value is described as evaluating the indeterminate form. (See Arts. 43 and 52.) III. Law of Signs for Real Quantities. 12. In Addition and Subtraction. The sign +, by definition, indicates that the magnitude following it is to be added algebraically to what precedes, without having its character as a magnitude in any way changed ; and the sign — indicates that the magnitude immediately LAWS OF ALGEBRAIC OPERATION, 29 following it is to be reversed in sense (taken in the opposite direction) and then added algebraically to what precedes. Any symbolic representative of quantity, a letter for example, unattended by either of the signs + or—, but still thought of as part of an algebraic sum, is supposed to have the same relation and effect in such a sum as if it had before it the sign -[-. This usage necessitates the following law of signs in addition : + + = -!-, H — = — , = +> - + = -; + (+ ^) = + ^, + (- ^) = — (— a) = + a, -(+«) = thus + (+^) = + ^, +(-«) = - a, and by a, unattended by any sign, is understood + <^. 13. In Multiplication and Division. In Art. 8 it was agreed that aid shall stand for the product X (±:^) may be replaced by / (±:^), giving the corre- sponding distributive law with the sign of reciprocation : (d= ^ ± ^) / ( =b O =- (± «) / (=t + (± ^0 / ( ± 0. 38 LAWS OF ALGEBRAIC OPERATION. But while the sign X is distributive over the successive terms of a sum (Art. 20), that is: X (±^±1^ ±:...)=X (±^)+ X (lii^)-f X. . ., the sign / is not, as may be readily seen by constructing the product c X / (^ + ^), and the sum of products c y' a -f-^ X lb, and comparing the results, which will be found to differ. 22. Agenda: Theorems in Proportion, Arithmetical Multiplication and Division. (I). If A .B ::C :D y.E :F::. . . , prove that A:B: \A 4- C+^-f . .. :B -\-D-\-F-^ ... (Adde7ido.) (2). If A .B : .C :D, prove that A +^ :B : .C^ D .D (Co77ipoiie?ido.) and that Ar^B :B :: Cr^ D :D. ( Dividendo. ) (3)- If A :B :; c.nr prove that A ^ B ■ A ^ B ■: C -^ D ■ C ^. D. (4). If A.B y.P :Q, B : C :: Q : R, C :D ::B .S, and Z^ : E :: S : T, prove that A :E '.:P :T. (Ex cequali.) (5). If A '.B -..Q-.R^xiAB .C v.P -.Q, prove that A -C v.P :R. (6). Show that corresponding to ^X(X^-fX-^)=X(X^X^+X^X^) LAWS OF ALGEBRAIC OPERATION. 39 there is the analogous formula cK!a + lb)^j{lcla-\- Iclb). (7). Construct the product 3X2 and show that the result is 6. (8). Construct the quotient i 2 / 3 and show that the result is 4. (9). Construct 5 / 3 and 2X5/3- vii. Exponentials and Logarithms. 23. Definitions.'*' Suppose F, Q to be two points moving in a straight line, the former with a velocity, more strictly a speed, "^"^^ proportional to its distance from a fixed origin O, the latter with a constant speed. Let x denote O J P P P' Q Q Q Fig. 20. the variable distance of P, y that of Q from the origin, and let A. be the speed of P when x= (9/= i, /u, the constant speed of Q. As it arrives at the positions /, P\ P'' suc- cessively, P is moving at the rates : A, a:' X, :r'' A, respectively, the corresponding values of x and y are : x--=OJ=^ I, x'= I -\~JP\ x^'^i+JP'', y = o, y'=OQ\ y''=OQ'' * Napier's definition of a logarithm. Napier : Mirifici logarithmorum canonis descriptio (Lend. 1620), Defs. 1-6, pp. 1-3, and the Construction of the Wonderful Canon of Logarithms (Macdonald's translation, 1889), Def. 26, p. 19. Also Mac- Laurin: Treatise of Fluxions, vol. i, chap, vi, p. 15S, and Montucla: Histoire des Mathematiques, t. ii, pp. 16-17, 97- ** The speed of a moving point is the amount of its rate of change of position regardless of direction. Velocity takes account of change of direction as well as amount of motion. See Macgregor : Element aiy Treatise on Kinematics and Dynamics, pp. 22, 23 and 55. 40 LAWS OF ALGEBRAIC OPERATION. and Q is supposed to pass the origin at the instant when P passes y, at which point x = 0/= i. The speed of P relatively to that of Q, or vice versa, is obviously known as soon as the ratio of /x to A is given. By means of this construction the terms modulus, base, exponential, and logarithm are defined as follows : (i). /LL /A, a given value of which, say m, determines a system of corresponding distances .x', x^\ . . . and^', jv", . . . is called the modulus of the system. (ii). The modulus having been assigned, the value of X, corresponding to _y= 0J=^ i, is determined as a fixed magnitude, and is called the base of the system. Let it be denoted by b. (iii). x^ x\ x^\ . . . are called \h^ exponeiitials oi y\ y' , y'\ . . . respectively, with reference either to the modulus m or the base b, and the relation between x and y is written X ^ exp,;^ y , or x ^ b y . *^ (iv). y, y\y'\ • • • are called the logarithms of x, x\ x^\ . . . respectively, with reference to the modulus in, or the base b, and the symbolic statement of this definition is either y ^ log;;, X, or y^ ^\og x. '-!' The convention that F shall be at / when Q \<,^i O intro- duces the convenient relations log,;; I = o, and exp,;, o = i , or b^ = j, and because (by definition) y = i when x = b, log,;; b =■ I, and exp;;, ^ =b, or b' =^ b. * ^\og X is the German notation. English and American usage has hitherto favored writing the base as a subscript to log thus, log^ ; but Mr. Cathcart in his translation of Harnack's Differential u. Integral Rechming has retained the German form, which is here adopted as preferable **See Appendix, page 139. LAWS OF ALGEBRAIC OPERATION. 41 Exponentials and logarithms are said to be mvcf^se to each other. (v). The logarithms whose modulus is unity are called natural logarithms,^ and the corresponding base is called natural base, the special symbols for which are In and e respectively; thus y = ^ogi X = In .T, and x = ey = exp y represent the logarithm and its inverse, the exponential, in the natural system. 24. Relations between Base and Modulus. Let the speed of P remain equal to \x as in Art. 23 while the speed of Q is changed from /u, to k^k. The modulus, ^IX = f/i, will then be changed to k jx j \ = k 7/1, and the distance of Q from the origin, corresponding to the distance X, will become ky. Hence ky = Xogkvi ^^ = k log;;, X ; that is : To multiply the 7nodulus of a logarithm by any real quantity has the effect of multiplying the logarith7n itself by the same quantity. In particular we may write log;;/ -'^' = ^^^ In X. Corresponding to this relation between logarithms in the systems whose moduli are in and km^ the inverse, or expo- nential relation is ^^Vkm ky = exp;;, y = exp„, (| ky). * These are also sometimes called Napierian logarithms, but it is well known that the numbers of Napier's original tables are not natural logarithms. The relation between them is expressed by the formula Napierian log of ;f = lo" In (lo^ / .r). Napier's system was not defined with reference to a base or modulus. See the article on JS-apier by J. W. L. Glaisher in the Encyclopcsdia Britannica, ninth edition. 42 LAWS OF ALGEBRAIC OPERATION. Let c be the base in the system whose modulus is km; then since 7=1 when x =^b, and ky = \o%km ^^ = log^„, ^ = I , when x = c, ,'. b = exp^;« k = c^. and Hence : If the modulus be changed froDi m to kvi, the corresponding base is chajiged from b to b^^^. Again, if in the equation log^^ x = in hi x, b be substi- tuted for X, the value of m is obtained in the form m= I I In b. Hence In terms of the base of a system the logarithm is logw ^ = h^ ^ I hi b, and if to x the value e be given, m is obtained in the form m = log„, e ; whence passing to the corresponding inverse relation : exp„2 7n = b'" ^= e ; that is : The exponential of any qiiaiitity with respect to itself as a modiilns is equal to natural base. 25. The Law of Involution. By virtue of the fundamental principle of the last article — that to multiply the modulus multiplies the logarithm by the same amount — we have in general exp;.;;, z = exp„, z / k and exp,„/^sr = exp,«/^2:. Hence exp,„ hk=^ exp;„ 7, k = exp„, ^ // = exp;„/M i; * For an amplification of this proof see Appendix, page 139. LAWS OF ALGEBRAIC OPERATION. 43 or ill other terms, since changing ;// into vi / //, or ni / k^ changes b into M, or h^^, which expresses the law of involution. Obviously //, or k, or both, may be replaced by their recii)rocals in this formula and the law, more completely stated, is Evolution. If /^ be an integer, the process indicated by b^'^ is called evolution; and when k^=2^ it is usually expressed by the notation -j/ b. 26. The Law of Metathesis. Let z-r=b^^\ then by the law of involution whence, passing to the corresponding inverse relations, /ik = log„^ zk and h --= log;;, z ; . •. k log;;, Z = log;;, 5^^' , which expresses the law of interchange of exponents with coefficients in logarithms, — the law of metathesis. 27. The Law of Indices.* The law of indices, or of addition of exponents, follows very simply from the defi- nition of the exponential, thus: In the construction of Art. 23, and by virtue of Definition (iii ) of that Article O J P F P" Q Q Q Fig. 20. exp;;, /= I + JP\ exp;;, f'= I + JP^' . \ eXP;;, j/;/ eXp;;, j' = OP''/ OP' = 1 4~P'P'' OP\ For an alternative proof see Appendix, pages 139-141. 44 LAWS OF ALGEBRAIC OPERATION. But the magnitude i — F'F'^ OP^ represents the distance that P would traverse, starting at unit's distance to the left of P\ on the supposition that its speed at/'' is \x' j x'=^\^ and the corresponding distance passed over by Q isy'^—y-, hence, by definition of the exponential, I + P'P^r 0P'= exp;, (y''-y), and therefore exp,„ y'\' exprn y=exp„, (y'—y'), which is the law of indices for a quotient. In particular, if y^=o exp;;, o exp;;, ^'= exp„, (o — ^') or I / exp;;, y= exp;;, ("J'') ; (Art. 23.) and therefore, writing — y'r^y^ the foregoing equation, exp;„ 7"/exp„,7'= exp;;, {^y' — y'), becomes exp;;, y ' ' X exp„, y = exp^;, ( 7 " ^ 7) , which is the law of indices for a direct product. The com- plete statement of this law is therefore embodied in the formula exp;;, y y/ exp„, y' ^ exp,,, (7'' ±y'), or its equivalent byy/by=by'-y. 28. The Addition Theorem. Operating upon both sides of the equation last written with log;;^ we have log,. (^-^'">^^-^')=y'=^y; that is, replacing 1^', by by x'', x' 2indy'\ y' by log;;, x'\ l0g;;;:v', log;;, {X'' Yy X' )=. log,;, x' ' ± lOg,;, X\ which is the addition theorem for logarithms. LAWS OF ALGEBRAIC OPERATION. 45 29. Infinite Values of a Logarithm. If, in the construction of Art. 23, X and /a become larger than any previously assigned arbitrarily large value, while their ratio m (that is, the modulus) remains unchanged, P and Q are transported instantly to an indefinitely great distance, and OP, OQ become simultaneously larger than any assign- able magnitude. It is customary to express this fact in brief by writing b'^^OO, log;;, 00= CO ; though to suppose these values actually attained would require both A and ^u, to become actually infinite. This sup- position will be justifiable whenever we find it legitimate, under the given conditions, to assign to the indeterminate form /a ■ A. := 00 / 00 a determinate value m.-^^ In like manner, since from d^ = co we may infer ^— =^ = I / CO = o, the equations ^ — °^ = o, log;;, o = — 00 are employed as conventional renderings of the fact, that when P and Q are moving to the left, P passing from / towards O and Q negatively away from O, x, in the equa- tions X = l?^ , y = log;;, X, remains positive and approaches o, while y is negative and approaches — co . 30. Indeterminate Exponential Forms. When ^ X logw tf becomes either ±0 X ^, or ± 00 X o, it is indeterminate (Art. 11). Now log;;, ?/ is o if ?/=i, is -j- 00 if u= -^ 00, and is — co if u = o (Art. 29). Hence V X log;;, z/ will assume an indeterminate form under the following conditions : * This form of statement must be regarded as conventional. Strictly speak- ing we cannot assign a value to an indeterminate form. When the quotient x j y, in approaching the indeterminate form, remains equal to, or tends to assume, a definitive value, we substitute this value for the quotient and call it a limit. In conventional language the indeterminate form is then said to be evaluated. 46 LAWS OF ALGEBRAIC OPERATION. When z' = 00 and 71=^ i, or e^ = o and ?/ = -j- c)o> or ?^ = o and z/ --= o. But if z> X log,« ?/, or \og„i u^ is indeterminate, so is ?/^', and therefore the forms i °°, 00'', 0° are indeterminate. Whenever one of these forms presents itself, we write y = u-^ jO = log,;, .r ± log,„ J/. 48 LAWS OF ALGEBRAIC OPERATION. 37. Properties of o, i, and c»: 0= -\- a — a, ztd ^0= ±^ — o, — 0= — o, ±00 zlz ^ = =b 00, I = >■.; a I a, Y/by^ i=y/b!i, X I = I, o>^ ^ = 0, b CO = X o. Zero may be regarded as the origin of additions, unity as the origin of multiplications. 38. Agenda: Involution and Logarithmic Operation in Arithmetic. (i). Show that, if n be an integer, the index law (Art. 35) leads to the result: «« z=: ^ X <^ X ^ X • . . . to 7z factors, and that therefore 3^ = 9, 2^ = 8, 5^ = 1 25, etc. (2). vShow, by the law of involution and the index law (Art. 35), that 8^/^=2, 8i^''*=3, etc. (3). Show, by the law of metathesis (Art. 36), that 4og 32 = 5, nog 729 = 6, etc. (4). Show, by the law of metathesis and the addition theorem (Art. 36), that ^log 8 + "^log 2 = 2, "^log 2 = ^, nog(i/8)-f nog(i /27) = -3. (5). Find the logarithms: of 16 to base 2'-, of 125 to base 5 X 5' % of 128 to modulus i / In 8, of i / 81 to modulus I / In 27. CHAPTER II. GONIOMETRIC AND HYPERBOLIC RATIOS. IX. GONIOMETRIC RATIOS. 39. Definition of Arc-Ratio. In the accompanying figures JV'JVis a straight line fixed in position and direction, OP is supposed to have reached its position by turning about the fixed point O in the positive sense of rotation from the initial position OJV. Any point O C L A N JFtg-. 21. on OP at a constant distance from O describes an arc A VQ, a Hnear magnitude. Let the ratio of this arc to the' radius OQ, both taken positively, be denoted by 6, that is, ^^ (length of arc AVQ) I (line-segment OQ). The amount of turning of OQ, that is, the angle AOQ, fixes the value of this ratio ; and since the arcs of concentric circles intercepted by common radii are proportional to those radii (Prop. 18), the ratio may be replaced by an arc CD provided only OC be taken equal to the linear unit. In 50 GOXIOMETRIC AND HYPERBOLIC RATIOS. the geometrical figures a description of the angle will be sufficient to identify the ratio itself. This magnitude 6 will be called the arc-ratio of the angle AOQ. The letter tt stands for ratio of a semi-circumference to its radius, that is, the arc-ratio of i8o°. Lines drawn parallel or perpendicular to A''iV, shall be regarded as positive when laid off from O to the right or upwards, negative when extending to the left or downwards. OP drawn outwards from O is to be considered positive in all cases. 40. Definitions of the Goniometric Ratios. LQ in the above figures being drawn perpendicular to N' N, upwards or downwards according as Q is above or below N^ N and correspondingly positive or negative, the gonio- metric ratios, called sine, cosine, tangent, cotangent, secant, cosecant, are defined as functions of the arc-ratio B by the following identities : sin 6 = LQI OQ, cos 0=OL OQ, tSLnO = LQ / OL, cot 0=OL I L Q, sec 0= OQ OL, esc 6= OQ LQ. It must be borne in mind that 6 is here not an angle expressed in degrees, but a ratio, which can therefore be represented by a linear magnitude. In elementary trigo- nometry sin 6 usually means ''sine of angle AOQ in degrees"; here it may be read "sine of magnitude ^," where = arcAVQ/ radius OAr^ If v be the number of degrees in the angle AOQ, the relation between v and 6 is 180^. (Prop. 17.) TTZ' See Lock's Elementary Trig onojne try, p. GONIOMETRIC AND HYPERBOLIC RATIOS. 51 41. Agenda. Properties of Goniometric e following : (0- sin 0^=0, cos 0=1. (2)- sin f = I, cosf = 0. (3)- sin^^ + cos^^="i. (4)- sec^^— tan^^=i. (5). csc^O — COt^^=I. If n be an integer, prove : (6). sin (0 ± [2 n + i] tt) = =b cos 0. (7). cos ((9ih [2w + i]7r) = q=sin ^. (8). sin (0± [2n -|-|]7r) = q= cos ^. (9). cos (^ ± [2 ;2 + I] tt) = dz sin 0. (10). sin, or cos of (^ zb 2 n tt) := sin, or cos of 0. (11). sin, or cos of (0 zb [2 n -f i] tt) = — (sin, or cos of 0). 42. Line-Representatives of Goniometric Ratios. If in the foregoing definitions the denominators OL, LQ be replaced by the radius OQ^ the numerators of the six goniometric ratios will be six straight lines drawn, either from the centre, or from Q, or from one of the fixed points A^ B on the circumference a quadrant's distance apart. S^P Fig. 22. If a be the radius of the circle, they may be Indicated as follows : 52 GONIOMETRIC AND HYPERBOLIC RATIOS. « sin 6=^LQ, perpendicular distance of Q from A' A. a cos B= OL, distance from centre to foot of LQ. a tan 6 = AT, distance along a tangent from A to OF. a cot = BS, distance along a tangent from B to OF, a sec = OM, intercept of tangent at Q upon OA. a CSC 0= ON, intercept of tangent at Q upon OB, These constructions are evidently only variations in the statement of the definitions of the goniometric ratios. When ti= I, the six ratios have as their geometric rep- resentatives these lines themselves. Formerly they were defined as such for all values of the radius and were therefore not ratios, but straight lines dependent for their lengths upon the arc AQ, that is upon both the angle AOQ and the radius of the circle. The older form of definition is now rare.* 43. To Prove Limit [(sin ^) /^] = i, when B^^o. Let ^^the arc-ratio of the angle FOQ in Fig. 23, draw FQF\ an arc with radius OF, draw PT'and F^l'' tangents to the arc at F and F' , join F, F\ and O, T. Then assuming that an arc is greater than the subtending chord and less than the enveloping tangents at its Fig. 23. extremities, we have 2 SF<2QF<^2FT, '''SP -^ > cos * See Todhunter's Plane Trigonometry, p. 49, and the reference there given: Peacock's Algebra, Vol. II, p. 157. See also Buckingham's Differential and Integral Calculus, 3d ed., p. 139, where the older definitions are still retained. GONIOMETRIC AND HYPERBOLIC RATIOS. 53 When ^ = o, cos 6=1] therefore by making 6 smaller than any previously assigned arbitrarily small magnitude, (sin 0) jO is made to differ from unity by a like arbitrarily small magni- tude. Under these conditions (sin 0) / is said to have i as its limit, and the fact is expressed by the formula limit ( sin 6 in which = stands for 'approaches.' 44. Area of Circular Sector. Let the sector OAQ be divided into n equal smaller sectors by radii to the points I*^, /*2, i^3, etc., which set off the arc AQ into the same number of equal parts AF^, P^P^, Pr,P^, etc., and draw P^Al perpen- dicular to OA. The area of each of the triangles OAP,, OP^P.^, OP,P^, . , .\s\a- MP, , orif (arc^(2)/C>^ = ^, it is \a'asm{AP,lOA^ = \a'-;\n{Bln), and hence the area of the entire ^'^- '^- polygon OAP, P, . . . Q \s na' . a'O s\n (0 1 n) • sm - = — • — n-i . 2 n 2 d I n Now when the number of points of division P,,P^,P^, . . . is indefinitely increased, the polygon OAP^P^ - • • Q ap- proaches coincidence with the circular sector OAP^P^ • - • Q-. that is, — • — -^r^ — ^ r= area of sector OAP^P^ - - - Q, when « = 00 ; but at the same time sin {OJji) / nzrzo and "" ^ . ^^ =zi, 54 GONIOMETRIC AND HYPERBOLIC RATIOS. and therefore also e 1 71 , when 71 = o:>. 2 Thus — • — rrr — can be made to differ, both from — 2 6 j n ' 2 and from the area of the circular sector, by quantities that are less than any previously assigned arbitrarily small magnitude. Under these circumstances it is assumed as axiomatic that the two limits which the varying quantity approaches cannot differ, and that therefore area of sector OAP^F^ ^ 2 The limits in fact could not be different unless the area of the sector were susceptible of two distinct values, which is manifestly impossible. 45. Agenda. The Addition Theorem for Goniometric Ratios. From the foregoing definitions of the goniometric ratios prove for all real arc-ratios the following formulae : (T). sin (a ± /8) =: sin a cos /8 ± cos a sin /3. (2). cos (a ± /3) = cos a cos ji + sin a sin /3. (3). tan a ± tan 6 t^"(«±«-i+tanatan^- (4)- sin 2 a = 2 sin a cos a. (5). cos 2 a = cos"" a — sin^ a. (6). I + cos 2 a = 2 COS^ a. (7). I — COS 2 a = 2 sin^ a. GONIOMETRIC AND HYPERBOLIC RATIOS. 55 X. Hyperbolic Ratios. 46. Definitions of the Hyperbolic Ratios. An important class of exponentials, which because of their rela- tion to the equilateral hyperbola are called the hyperbolic sine, cosine, tangent, cotangent, secant and cosecant, and are symbolized by the abbreviations sinh, cosh, tanh, coth, sech, csch, are defined by the following identities : sinh u^^ (e" — ^~"), cosh 2/^^ (^" -f e-"), tanh « = (^" — e~") / (e" -f e-"), coth u^ie^' + ^-") / ((?" — ^-"), sech u^2 I {e^ -{- ^~"), csch u^2 I (^" — ^~"). 47. Agenda. Properties of Hyperbol the following : (I) sinh = 0, cosh 0=1. (2) cosh^ u — sinh^ ti = 1. (3) sech" w + tanh" z^ = i. (4) coth" u — csch" u= 1. (5) sinh 2 u = 2 sinh 7i cosh u. (6) cosh 2 u = cosh" u -\- sinh" u (7) cosh 2 u -\- 1 = 2 cosh" u. (8) cosh 2 u — I = 2 sinh" u. (9) sinh ( — 7c) = — sinh?/. (10) cosh(— 21) = cosh 2t. 48. Geometrical Construction for Hyperbolic Ratios. For the representation of the hyperbolic ratios the equilateral hyperbola is employed. Its equation in Cartesian co-ordinates is y-=a^ 56 GOXIOMETRIC AND HYPERBOLIC RATIOS. Let OX, 6> F be its axes, OJ an asymptote, P any point on the curve, x and y its co-ordinates, ^(2^ the quadrant of a circle with centre at the origin and radius a, NQ a tangent to the circle from the foot of the ordinate y, PS a tangent to the hyperbola parallel to the chord FA, a, /3 the co-ordinates of (2, the arc-ratio of the angle ACQ. It is obvious from its definition that cosh u has i for its smallest and -j- oo for its largest value corresponding to 21 = o and CO respectively, and if the variations oi x j a be confined to the right hand branch of the hyperbola its range of values is likewise between i and + ^ ', hence we may assume — = cosh u, GONIOMETRIC AND HYPERBOLIC RATIOS. 57 and by virtue of the relations cosh^ it — sinh^ ic= i, and x' / a^ — y ! a-^= I, y •^ = sinh ti. a Also, since x^ — y~ z= a" and x" — NQ" = a-, therefore JVQ = NF, and we have X V — = sec 6 = cosh 7c, - = tan 6 =. sinh u. and the co-ordinates of Q being a, )8, also AH ^ y ' r\ 1 (X a f. , = - = ^ = sni u = tanh 21, - = - = cos u = sech ?^, a a X ax and finally, since OBK is similar to 6>A^i^ and a- =^y ' LO^^ BK X , LO a = — = coth u, = = csch 2C. ay 'ay Hence if a be made the denominator in each of the hyperbolic ratios, their numerators will be six straight lines, drawn from O, A, B, or P, which may be indicated thus: a sinh ii = NP, perpendicular distance of P from OX, a cosh u = ON, distance £'om centre to foot of NP, a tanh ti = AH, distance along a tangent from A to OP, a coth 21 = BK, distance along a tangent to the conjugate hyperbola from B to OP, a sech 21 ^= OM, intercept of tangent at P upon OX, a csch 2c = LO, — (intercept of tangent at P upon OY^. This construction gives pertinence to the name ratio as * Obtained bywriting the equation of the tangent iJ/Pand finding its intercept on Oy\ or thus. OL 1 OM=NP I MN, that is, OL = WM . y) I {x — OM) = a (sech ii sinh u) I (cosh ji — sech 11) = a sinh u j (cosh^ u — i)-= a csch w. 58 GONIOMETRIC AND HYPERBOLIC RATIOS. applied to the six analogues of the goniometric ratios. Compare these with the constructions of Art. 42. 49. Agenda. Properties of the Equilateral Hyperbola. Prove the following propositions concerning the equilateral hyperbola. (Fig. 25 of Art. 48.) ( I ). The tangent to the hyperbola at P passes through M, the foot of the ordinate to Q. (2). The locus of /, the intersection of the tangents NQ and MP, is A J the common tangent to the hyperbola and circle. (3). The line OIV bisects the angle and the area OA VP and intersects the hyperbola at its point of tangency with RS. (4). A straight line through P and Q passes through the left vertex of the hyperbola and is parallel to OV. (5). The angle APN= one-half the angle QON". 50. The Gudermannian. When 6 is defined as a function of 2c by the relation tan 6^sinh u (Art. 48) it is called the Guderman7iian of «* and is written gd u. Sin 6, cos and tan B are then regarded as functions of u and are written sg n, eg 21 and tg it. 51. Agenda. From the definitions of the Guder- mannian functions prove the formulae: * By Cayley, Elliptic Functions, p. 56, where the equation of definition is u = In tan (1 tt + J Q). Since tan (i ir + i ^) -= ^ + ^'" ^ and sinh-» tan ^ = In ^ "^ ^'" ^ (Art. 56). the cos Q cos Q "^ equivalence of the two definitions is obvious. The name is given in honor of Gudermann, who first studied these functions. GONIOMETRIC AND HYPERBOLIC RATIOS. 59 (l). Sg'7l-{-Cg'2C=l. y s ^ , N so; 21 -{- so; V (2). sg (u -i- v) --=^ — ^ — -^—^ . ^ ^ ^ ^ -^ I -j- Sg7t . Sg V ^ N r , N Cg7i.CgV ( 3 ) . Cg(2l + V) = —^ ^ . v.oy s V I y I -\- sgic . sgv (4). If 2 = ]/ - I , V gd0 gd 7^1 -= — 2^, or more briefly (~-g"<^) ^^ = — ^^- (Prof. Haskell.) 52. To Prove Limit [(sinh ?/) / ?^] = i , when 2i = o. In the construction of Art. 23 suppose that, during the interval of time /' — f, P moves over the distance x' — x, Q over the distance 7^' — ?^. Then speed being expressed as the ratio of distance passed over to time-interval, the speed of (2 is O J P P Q Q Fig, 26. 2i' - U and the average speed of P during the whole interval is X Let \ x^ represent the true speed of T' at a given instant within the interval considered, \x^ — 8, A .To -f S' the speeds at its beginning and end respectively ; then \x,— l<. - ^/ _ \ < A jr„ -f S'; and if the interval /' — / be made to decrease in such a way that S and o' simultaneously approach zero, the three mem- 6o GONIOMETRIC AND HYPERBOLIC RATIOS. bers of this inequality approach a common value, their limit; that is, , /' — t ~ :a.^o, u llCll L I. Hence also d"'- -b- _ hen/' =t\ ?/ - — 7t or in the language of limits (Art. 43). limit id"' 7C'=76 li^ — 7C S " A In the important case when ?/ = — 71, and 7t r= o, since u=^Q requires that ,r = i , the expression last written be- comes limit 7lr=0 f)U _ ^-« and in particular when limit j^" — , but at the same time , sinh (u / 71) . = o and ^-7— — = and therefore tc I 7i^^o ana '-j— — '- =1, U I 71 ' a? 2C sinh (u / 71) a" it , — • -, =r — , wnen 7i-=zoo. 2 U I 71 2 ' Hence by the reasoning of Art. 44, area of sector OAP, P^ . . P= . 2 54. Agenda. The Addition Theorem for Hyperbolic Ratios. From the foregoing definitions of the hyperbolic ratios deduce the following formulae : (i). sinh {ic ±v)^ sinh 7C cosh v zt cosh 2c sinh v. (2). cosh (ic ±v')=^ cosh 7C cosh v zb sinh 71 sinh v. , . . , . tanh 71 ±: tanh v (3). tanh (71 zb Z') = — -- — r : — r — . ^^'' ^ ^ I zb tanh 7C tanh v GONIOMETRIC AND HYPERBOLIC RATIOS. 63 (4). Deduce these formulae also geometrically from the constructions of Arts. 48, 53, assuming for the definitions of sinh 2t and cosh u the ratios NP / a and ON I a. [Burn- side: Messenger of Mathe?natics, vol. xx, pp. 145-148.] (5). In the figure of Art. 53 show that the trapezoids SoAP^S^, S^P^P^S^, etc., are equal in area to the corre- sponding triangles OAP^, OP^P^, etc., and consequently to each other. (6). Show that when the hyperbolic sector OAP (Art. 53) increases uniformly, the corresponding segment OS, laid off on the asymptote, increases proportionately to its own length. (7). Assuming «=i in the equilateral hyperbola of Art. 48, and that the area of any sector is ^ ?(, prove that ^^^^ [(sinh ii) 1 71']= 1. (Use the method of Art. 55.) 55. An Approximate Value of Natural Base. We may determine between what integers the numerical value of e must lie, by substituting their equivalents in X and y for the terms of the inequality : Triangle OAP > sector OAVP > triangle ORS, as represented in Fig. 28. For our present purpose it will in- volve no loss of generality and it will simplify the computation to assume OA = i , so that the equation of the hyperbola is x^ — X = I • The sectorial area OA VP, as previously found in Art. 53, is then 4?^, the area of OAP is R A Fig. 28. obviously \ y, and for that of OPS we may write ^^ OP X (ordinate of .S"). 64 GONIOMETRIC AND HYPERBOLIC RATIOS. To determine OR and this ordinate, write the equations to the tangent RS and the Hne OP^ and find the ordinate of their intersection, and the intercept of the former on the j;-axis. The results are : "n y X — ^-1/2/(^-1), the equation to the tangent RS, i and rj being the current co-ordinates of the Hne ; the equation to OP; J' the required intercept on the ^-axis ; and r}=y 2 {x— l), the ordinate of S, found by ehminating i from the equations to RS and OP. Hence the area ORS is ivOR = '^. and the inequaHty between the areas takes the form •^ J' Ify = I , then w < I , x = 1/2, and e" = X -\-y = I H- 1/2 = 2.4+; therefore d" > 2 . 4. * Had the assumption a = i not been made, this inequality would have been The equations for i^^'and OP and the expressions for OR and 'y would have been correspondingly changed, but the final results would have been the same as those given above. GONIOMETRIC AND HYPERBOLIC RATIOS. 65 1^2^^^ = I, then?^>i, ;r=| j'=^ and therefore ^ < 3- A nearer approximation to the value of e is found by other methods. To nine decimal places it is 2. 7 1828 1828. 56. Agenda. Logarithmic Forms of Inverse Hyper- bolic Ratios. It is customary to represent by sinh-^j^, cosh— ^ X, . . . , the arguments whose sinh, cosh, . . . , are J', X, etc. Let /^sinh-ij/; then y = sinh / = - (^^ —^0' whence, multiplying by ^^ and re-arranging terms, ^2^ — 2>'(?^ — 1=0, a quadratic equation in 6^, the solution of which gives or /=lnO'±lO^^+i). IfjV' be real, the upper sign must be chosen ; for \/y -f i ^j and e^ is positive for all real values of/ (Art. 23). Hence ( I }. sinh-i_>/ = In (j^/ 4- |/y^-|-i). Prove by similar methods the following formulae : (2). cosh-i ^ = In (x + ^/x' — i). (3). tanh-^^^|ln^^i^' I — 2* (4). coth-^^ = |ln^i^. (5). sech-^ -^ = In ( I / ;r + |/i / ji;^ - I). (6). csch-^j/ = In (i /_;/ -f |/i/;j/^+i). (Cf Art. 97.) CHAPTER III. THE ALGEBRA OF COMPLEX QUANTITIES. XI. Geometric Addition and Multiplication. 57. Classification of Magnitudes: Definitions. It was pointed out in Art. 2 that any one straight line suffices for the complete characterization of all so-called real quantities ; in fact the real magnitudes of algebra were defined as lengths set off upon such a line. But because, in this representation, no distinction in direction was neces- sary, all line-segments were taken to be real magnitudes, and comparisons of direction were made, by means of the principles of geometrical similarity, for the sole purpose of determining lengths. Such comparisons will still be necessary whenever the product or quotient of two real magnitudes is called for, but into the real magnitudes themselves no element of direction enters ; their sole char- acteristics are length and sense, that is, length and extension forwards or backwards. If the attempt be made to apply the various algebraic processes to all real magnitudes, negative as well as posi- tive, another kind of magnitude, not yet considered, is necessarily introduced. For example, if x be positive, no real magnitude can be made to take the place of either (— xy'^ or log,„ ( — x^\ for the square of a real quantity is always positive (Art. 14), and the definition of an exponen- tial given in Art. 23 precludes its ever assuming a negative value. In order that forms like these may be admitted into THE ALGEBRA OF COMPLEX QUANTITIES. 67 the category of algebraic quantity, a new kind of quantity must therefore be defined, or more properly, a new defini- tion of algebraic quantity in general must be given. Having assigned some fixed direction as that in which all real quantities are to be taken, we adopt a straight line having this direction as a line of reference, call it the real axis, and determine the directions of all other straight Hnes in the plane by the angles they make with this fixed one. Line-segments having directions other than that of the real axis are the new magnitudes that now demand consideration. They are called vectors. They have two determining ele- ments : length and the angle they make with the real axis. (i). Its length, taken positively, is called the tensor of the magnitude, and the arc- ratio of the angle it makes with the real axis is called its amplitude (or argic^nent^. Classified and defined with respect to amplitude, the magnitudes themselves are : (ii). Real, if the amplitude be o or a multiple of tt ; (iii). Lnaginary, if the amplitude be tt/ 2 or an odd multiple of TT / 2. (iv). Complex, for all other values of the amplitude. In general, therefore, vectors in the plane represent complex quantities, but in particular, when parallel to the real axis they represent real quantities ; when perpendicu- lar to it, imaginary. Any quantity is by definition uniquely determined by its tensor and amplitude, and hence : (v). Two quantities are equal if their tensors and their amplitudes are respectively equal, the geometrical rendering of which is: two magnitudes, or vectors, are equal if (and only If) they are at once parallel, of the same sense, and of equal lengths. 68 THE ALGEBRA OF COMPLEX QUANTITIES. The algebra of complex quantities, like that of real quantities, is developed from the definitions of the funda- mental algebraic operations: addition and subtraction, multiplication and division, exponentiation and the taking of logarithms. These operations applied to magnitudes represented by straight lines in the plane are called alge- braic by reason of their identity with those of the analysis of real quantities, but specifically geometric, because each individual operation has its own unique geometrical config- uration. On the other hand, the algebraic processes applied to real quantities may be described as geometric addition, multiplication, involution, etc., in a straight line. 58. Geometric Addition. Regarding lines for our present purpose as generated by a moving point, the opera- tion of addition is defined to mean that a point P, free to move in any direction, is successively transferred forwards or backwards, that is, in the positive or negative sense as marked by the signs + and — , through certain distances designated by appropriate symbols a, ^, y . . Thus the sum 4- a — ^ -|- y, in which a, ;8, y represent vectors in the plane (or in space), joined to form a zig-zag, as shown in the accompanying figure, may be read off as follows, the arrow-heads indicating direction of motion forwards: Move forwards through distance a to A, then backwards through distance y3 to B\ then forwards through distance y to C\ and the result is the same as if the motion had taken place in a direct line from (9 to C; this fact is expressed in the equation Fig. 29. J^a-[i+y=OC\ THE ALGEBRA OF COMPLEX QUANTITIES. 69 If not already contiguous, the magnitudes that form the terms of a sum, by changing the positions of such as require it without changing their direction, may be so placed that all the intermediate extremities are conterminous. Geo- metfic addition may therefore be defined as follows : The sum of two or more magnitudes, placed for the purpose of addition so as to form a continuous zig-zag, is the single magnitude that extends from the initial to the terminal extremity of the zig-zag. 59. The Associative and Cominutative Laws for geometric addition in the plane are deduced as immediate consequences of its definition. For in the first place, the ultimate effect is the same whether a transference is made from O direct to B then to C as expressed by (a -f /5) -f y, in the subjoined figure, or from O to A then direct from ^ to C as expressed by a + (/3 + 7), or from O to A to B to C as expressed by a + yS -f y; hence (a + ^)+7 = a-f-(^+T) = a-f /i+y; and in the second place, by (v) of 57, in ABCB>, a parallelogram, AD = BC = y, BC= AB = /?, and by the definition of addition AB -f BC = AC= AD + DC, whence One or more of the terms may be negative. Expressing this fact by writing zt a, zh /?, d= y in place of a, /j^, y, the two resultant equations of the last paragraph become rd=a±/?)±y==ba+ (±:^d=y)=±:azi=y5zby, ±: y ± /i = ± /? ± y. Art. 70 THE ALGEBRA OF COMPLEX QUANTITIES. which express the associative and commutative laws for addition and subtraction. It is evident that the principles of geometric addition apply equally to vectors in space, or in the plane, or to segments of one straight line. In particular algebraic addition (Art. 3) may be described as geometric addition in a straight line. 60. Geometric Multiplication. The geometric pro- duct of two magnitudes a, /?, is defined as a third magnitude y, whose tensor is the algebraic .product of the tensors of the factors and whose amplitude is the algebraic sum of their amplitudes, constructed by the rules for algebraic product and sum. (Arts. 5, 3.) If one of the factors be real and positive, the amplitude of the other reappears unchanged as the amplitude of the product, which is then constructed, by the algebraic rule, upon the straight line that represents the direction of the complex factor, and it was proved in Art. 18 that in such a construction an interchange of factors does not change the result. Hence, if a be real and positive and /3 complex, In this product, tensor o{ aX /3 = a. X tensor of fi by definition, and if tensor of /?=i, the complex quantity aX ft appears as the product of its tensor and a unit factor /?, a complex unit, which when applied as a multiplier to a real quantity a, does not change its length but turns it out of the real axis into the direction of ft. Any such complex unit is called a versoi: Let tsr stand for tensor, vsr for versor ; then every complex quantity a can be expressed in the form a ^^ tsr a X \'sr a. THE ALGEBRA OF COMPLEX QUANTITIES. 71 This versor factor is wholly determined by its amplitude, in terms of which it is frequently useful to express it. For this purpose let i be the versor whose amplitude is 7r/2, the amplitude of the complex unit: /5, OX the real axis, BiM the perpendicular to OX from the terminal extremity of ft. Then MB = sin (9, 0M= cos 6, and by the rule of Q^eometric _ ^ addition. Fig. 32. ft = 0M-\- i X MB = cos + 2 sin 6. As an abbreviation for cos -\- 2' sin 6 it is convenient to use cis 0, which may be read : sedor of B. In this symbolism, the law of geometric multiplicatioii (product of complex quantities, as above defined) is expressed in the formula, {a • cis , it becomes {a ' cis and cis ( — ) are reciprocal to each other and we may write / cis cf> = cis ( — <^). In Hke manner, since it is now evident that (^ • cis <^) X (/<2 • / cis <^) = <2 /<2 • cis o= I / (a • cis ) =^ / a ' / cis . If now in the formula expressing the law of multiplication we write / d and — i/^ for /5 and iff respectively, we have, as the law of geometric division, (a ' cis <^) / (d ' cisij/') = a / d ' cis (^ — i/^). 62. Agenda. Properties of cis <^. If 71 be an integer prove : ( I ). (a' cis 0)" = a'' ' cis n -i- [^ -{- x]) ; and by the same process, (a X ^) X y = (« X ^) X ^ • cis ([ -{- ij/ = ij/ -{- ; .-. aXf^ = /3Xo., which is the commutative law for multiplication. 74 THE ALGEBRA OF COMPLEX QUANTITIES. The letters here involved may obviously represent either direct factors or reciprocals, and the sign X may be replaced at pleasure by the sign /, without affecting the proof here given (see Art. 6i). Hence, as in Arts. 32, 33, with real magnitudes, so with complex quantities the asso- ciative and commutative laws for multiplication and division have their full expression in the formulae, 65. The Distributive Law. From the definitions of geometric addition and multipHcation (Arts. 58, 60) the law of distribution for complex quantities is an easy conse- quence. The construc- tions of the subjoined figure, in which a, ft and VI ' cis represent complex quantities will bring this law in direct evidence. The opera- tion m'cisO changes OA into OA', AB= A^ mto A E=A^\ and OB Into 0B\ that is, turns each side of the triangle OAB through the angle AOA' and changes its length in the ratio of w to I, producing the similar triangle OA' B\ in which 0~A' = VI • cis ^ X a, A^' = VI ' cis e X A Ub' = m • cis ^ X (a + /?). But, by the rule of geometric addition, Fig- 34' THE ALGEBRA OF COMPLEX QUANTITIES. 75 .' . m' ds9x{p.-\- ft)= VI • cis (9 X a + ?;? • cis ^ X /?. This demonstration i.s in no way disturbed by the intro- duction of negative and reciprocal signs. The last equation above written is, in fact, the first equation of page 37, and the subsequent equations of Arts. 20, 21 and their proofs remain intact when for the real quantities a, b, c, d, etc. , complex quantities are substituted. Hence, writing w • cis ^ = 7, (± a ± /?) >5. (±: y) = + (± a) >5^ (+ y) + (± /?) >5^ (zh y). Here, as in Arts. 20, 21, the sign X is distributive over two or more terms that follow it, but not so the sign /. 66. Argand's Diagram. It is obvious from its defi- nition as here given (Art. 57) that to every complex quantity there corresponds in the plane a unique geo- metrical figure which completely characterizes it. This figure is known as Argand's diagram,* and consists of the real axis OX with reference to which the arc-ratio 9 is esti- mated, the imaginary axis O V perpendicular to OX, the directed line OP that repre- sents the complex quantity and fis--S5- the perpendicular PA from P to OX. The axes OX and O V are supposed to be fixed in position and direction for all quantities. Any point P in the plane then determines one and only one complex quantity and one set of line-segments OA, AP, OP, different from every other set. * First constructed for this purpose by Argand : Essai snr une manUre de representer les qiiantites imaginab es dans les constructions geonietriques; Paris, 1806. Translated by Prof. A. S. Hardy.. New York, 1881. 76 THE ALGEBRA OF COMPLEX QUANTITIES. If OA = X and AF=^y, the complex quantity appears in the form x-\-iy, and if OP=a and arc-ratio of A0P=6, the relations X = a cos 0, y = a sin 0, a'' = x^ -\-y% t2in9^=y I X are directly evident from the figure and we have X ^ iy=^a(^cos6 -^ ismO), XsY{x-\-iy)==-^\/x'-\-y\ vsr {x + iy') = cos 6 -\- z sin 6, amp (.r -f- {y) = arc-ratio whose tangent isy / x. In analysis the complex quantity most frequently presents itself either expHcitly in the form x -\- iy, or impHcitly in some operation out of which this form issues. 67. Agenda. Multiplication, Division and Construc- tion of Complex Quantities. Prove the following : ( I ). (^a-\- ib) X {x -{- I'y) =^ ax — by -\- i(^ay ^ bx). (2). ia+ib) I (. + iy)^ ^^ + ^->;+ff" - "-^X (3). {a + iby + (« - iby=2 {a' + b') — I2a'b\ la -\- ib\- la — iby ^iab . . la-\- toy la — zby ^.zab U;- \^^r7^; - \«qr^/ — («= + b^y (5). TT7 + T=r7 = 3. I X- — y^ — 2ixy (x-^iyy== {x^-vyy X -\- iy x^ — 3.ry^ -f ^ iz^y — "JVO {x-^ylif — ' ix~ -{-yy (6). (7). (8). (_ ^ + /I ,/3)^= _ I _ /1 1/3. THE ALGEBRA OF COMPLEX QUANTITIES. 77 (9). (-i + ^iV3y=(-i-^iV3r=l^ (lo). \/x H- ly (II). [±: (I + / t/2]^= [± (I - / 1/2]^= - I. (12). Write down the expression for tensor in each of the above examples. (13). Prove, by the aid of Argand's diagram (Art. 66), that the tensor of the sum of two or more complex quan- tities cannot be greater than the sum of their tensors; that is, tsr(a-f ;8)5/ )8 ) = tsr a >^ tsr /8, and amp ( a >^ yS ) = amp a dz amp /?, and hence no proofs of these properties are called for. Construct the following, applying for the purpose the rules of algebraic and geometric addition and multiplication (Arts. 3, 5, 58, 60): (15). (« + /^) + (:t-H-iy). (16). {a-\-ib)-{x-\-iy). (17). (a^2b)X{x^iy), (18). (^ci-^ib)l{x + iy). (19). (^ + ^»^ (20). I / (-r + iy)- (21 ). (a • cis i>) X (b' cis tp), (22). b / (a • ciscf>). (23). (a-c[s)\ (24). (a ' cis (f>) / (b ' cis ^)^ 78 THE ALGEBRA OF COMPLEX QUANTITIES. XII. Exponentials and Logarithms.* 68. Definitions. In a circle whose radius is unity, OT is assumed to have a fixed direction, its angle with the realaxis=/(97', (Fig. 36), OR is supposed to turn about Fig. 36. O with a constant speed, Q to move with a constant speed along any line, as ES, in the plane, P along OR with a speed proportional to its distance from O. Let * The theory of logarithms and exponentials, as here formulated, was the sub- ject-matter of a paper by the author, entitled " The Classification of Logarithmic Systems,' ' read before the New York Mathematical Society in October, 1891, and subsequently published in the American Journal of Mathematics, Vol. XIV, pp. 187-194. It was further discussed by Professor Haskell and by the author in two notes in the Bulletin of the New York Mathematical Society, Vol. II, pp. 164-170. THE ALGEBRA OF COMPLEX QUANTITIES. 79 Speed of P in OR at A^X, arc-ratio of /OR ^ 0, speed of Q in BS ^/x., arc-ratio of /DS^cf>, speed of R in /RS ^ w, arc-ratio of /O 7"^ /3, OP, OQ~p, q, OM, MP^x,y, ON, NQ=u, V, 0N\ N'Q^u\ v\ OC^c, JA^a — (a possible multiple of 27r), O T^ cos (i Ar i sin /? ^ cis /?, ?;^ cis /? ^ k, i^'Z' will be called the modular line, and OF, drawn through the origin perpendicular to ET, will be called the modular 7iormal. In all logarithmic systems the relation a)/A.= tan(c^ — /5) is assumed to exist, and this, together with the equation jw. / 1,/A.^ -f w^ = m, by elimination of co, gives, as a second expression for ?;z, ni == fji / \' cos ((^ — /?). Let the values of w and /3 be assigned, and the path and speed of Q determined, by fixing the angle JDS, the position of the point Cand the value of )u,. The value of A is then completely determined through the equation m = iL I X' cos ('= exp^ {n 4- iv). THE ALGEBRA OF COMPLEX QUANTITIES. 8l or with respect to the base B, as expressed by the identity (iv). Inversely, OQ is the logarithm of OP, either with respect to the modulus /c, as expressed by the identity or with respect to the base B^ as expressed by the identity u 4- iv ^ -^log ( ^* + y')- 69. Exponential of o, i, and Logarithm of i, B. If the path of Q pass through the origin, the points E and C will coincide at O and the path of /^ will cross both the cir- cumference of the unit circle and the real axis at J. Hence y=^o and Jf = i , when 7^ = z; = o, to which correspond the convenient relations exp^o = ^°= I, and log;^ I =^log I =0. Here also, as in Art. 23, because w=^\ when z=^ B, log^ B = I, and exp^ i = B^ = B. 70. Classification of Systems. The special value zero for the modular angle /OT eliminates the imaginary term from the modulus and introduces the ordinary system of logarithms, with a real modulus. A system is called g-onic, or a-gonic, according as its modulus does or does not involve the angular element /3. The geometrical representation of agonic systems is obtained from Fig. 36 by turning the rigidly connected group of lines EN\ EQ and OF, together with the speci- fied points upon them, around the origin as a fixed centre. 82 THE ALGEBRA OF COMPLEX QUANTITIES. backwards through the angle JOT^ so that 7", S fall into the positions /, S\ D and E then coincide upon JD, OF becomes perpendicular to yZ>, ^ — ,3 remains unchanged in value but merges into <^, and the new modulus becomes ft / |/ V -f- to^, or /A / A. • cos <^, its former value with the factor cis /3 omitted ; while no change in X, /x and a> need take place. Thus the path of P remains intact, and the new Q moves with its former speed in a straight line passing through S' and through a point on OJ2X a distance to the left of O equal to OE. Hence the values of z in the two systems are identical, while w, of the original gonic system, in virtue of the back- ward rotation through the angle JOT^ is transferred to the new or agonic system, by being multiplied by cis (— /3), so that the original w and its transformed value, here denoted by w\ bear to one another the relation w = zv' cis /3. The agonic system above described obviously has for its equations of definition (Art. 68) w'=\og„,z, 2 = 5"^', in which in = jx j \- cos <^ and b is the value of 2 for which \og,^2= I. The formula connecting logarithms in the two systems therefore is ^^Sk ^ = cis /3 • \og„i2. [k = 7n cis i3. ] Finally, if in this equation 2 = B, the resulting relation between /3 and B is cis(— /3) = log„,^, or in the inverse form it is B = exp,,, ; cis (- /3) 1= ^^is(-^). THE ALGEBRA OF COMPLEX QUANTITIES. 83 71. Special Constructions. A further specializa- tion is obtained by making — 13 is then tt / 2, A. is zero, and F moves in a circum- ference concentric with the unit circle. Such constructions are possible to every logarithmic system and enable us to simplify tbe graphical representation of the relative motions ofF and Q.'-^ Expressions of the form log^;i:, for which no interpreta- tion could be found in terms of real quantities unless x were real and positive (Art. 57), will henceforth be susceptible of definite geometrical representation for all possible values of ;r. See the examples of Art. 86. * We might, in fact, propose to assign, as the path of Pin the first instance, any straight line passing through the origin, define OPas the exponential of O^). and thtn determine the modulus by the appropriate auxiliary construction. 84 THE ALGEBRA OF COMPLEX QUANTITIES. 72. Relative Positions of A and C in Fig. 36. We have by definition, c^OC, a^JA -f a possible multiple of 27r, vi^ix I \' cos (<^ — /3), w = Atan (<^ — /3). The product of the last two of these equations gives m w = /xsin (<^ — /3), But to and /x sin (<^ — /3) are the rates of change of 6 and v' respectively, and ^ = 0, v^=o are simultaneous values (Art. 68), . • . mO^=v' \ and since ^ = a, v'=- c are also simultaneous values (Art. 68) , Thus A has always a definite position depending upon the modulus and the distance from the origin at which Q crosses the modular normal. Since a is the length of arc over which R passes while Q passes from E to C, it is evident that when c / 7n lies between 2kTr and 2 (/& -f- i ) tt, say c I in = 2 kir ^ (< 2 tt) where k is an integer, the part of i^'s path that corresponds to EC encircles the origin k times before it intersects the circumference of the unit circle, and the point upon the real axis that corresponds to E is its {k -|- i)^^^ intersection with the path of F, counting from / to the left. 73. The Exponential Formula. If ^ = arc-ratio of MOP, ^=/cis^ = ^" + '^. It is required to find/ and as functions of u and v. THE ALGEBRA OF COMPLEX QUANTITIES. 85 Since the speed of yV^'' In 6^7" Is ^t cos (cf> — /3) and that of P in OJ? is Xp, and since by definition m:= fx / \ ' cos (<^ — /3), the relation between OJV\ =^ ?/ and OF, =/), two real quantities, is that of an exponential to its logarithm, with respect to the modulus m (Art. 23); that is, If ^= base corresponding to modulus m, t/= log,,, p, and p = ^"' ; and in Art. 72 it was shown that Hence 1^"+'-^= d"'cis~ • But by considering the projections of u, v upon u\ v\ in Fig. 36, we easily discover that 2t cos [i -\- vsin /3, V cos /3 — 7ism [i ^u + iv —- ^wcos^+z^sin^ff • q^^ V COS /3 — u sin ni Or since b=^e'^""^ (Art. 24), this formula may be written „ , , o , • o ^ .2/ cos /3 — 2^ sin /3 When /3 = o, it becomes )u+iv ^ ^u (cos - + / sin - ) ' and when m = i and therefore b=^e, It assumes the more special form ^ti+tv ^ ^u (cos v^i sin z'), an equation due to Euler.* * Init oductio ni Ai/a/ysi/i hifinitorum, cd. Nov., 1797, Lib. I, p. 104. 86 THE ALGEBRA OF COMPLEX QUANTITIES. 74. Demoivre's Theorem. When zc = o, Euler's formula becomes giv __ cis 1)^ and by involution ^inV -_ ^(.jg y^n __ (xjg ^^^ or (cos V ^ i sin z^)" = cos 7iv -f- ^ sin wz^ for all real values of n. This equation is known as De- moivre's theorem.^ 75. Relations between Base and Modulus. Let the lines EN\ EQ and OF be regarded for the moment as rigidly connected with one another and be turned con- jointly in the plane about the fixed point O through an arbitrary angle, whose arc-ratio may here be denoted by y. ON^ in the new position thus given it, then forms with 0/ an angle whose arc-ratio is y3 -j- y, the modulus k, = m cis /3, by virtue of this change, becomes m cis ( /3 -f y) = ffi cis /3 • cis y = a; • cis y, and since OQ, in common with the other lines with which it is connected, is turned about O through the angle of arc- ratio y, w is hereby transformed into zt^cisy; while the locus of F is in no way disturbed by any of these changes. Hence ze; cis y = cis y ' log^ z = log^ cisy-S". In a second transformation, let the motion of F still remain undisturbed, while the speed of Q is changed from fx to nfj. (?i = a real quantity). By this change the modulus Kcisy becomes ?ZKcisy, the distance of Q from the origin becomes ?z^ instead of ^, and zc^ cis y is transformed into 71^ cis y. Hence, writing -pi cis y = v, we have Demoivre : Miscellanea Analytica (Lond., 1730), p i. THE ALGEBRA OF COMPLEX QUANTITIES. 87 VW=V \0g^2 = \0g.,^Z, in which v is any complex quantity, and we may reiterate for gonic systems of logarithms the first proposition of Art. 24 : (i). To viidtiply the fnodtdiis of a logaritnni by any quantity has the effect of multiplying the logarithm itself by the same quantity. Corresponding to this equation connecting logarithms in two systems whose moduli are k and vk, the inverse, or exponential relation is expy;^ vw = exp^ w -^Pk(-'<^)- Let C be the base in the system whose modulus is vk ; then the following equations co-exist : 2e;=log^2', ^^ = exp^ 2X^ = j5^, vw = log^K z, 2 = exp;,^ vza = C^, in which are involved, as simultaneous values of w and 2, zv-= i^ when z^= B, w= ijv, when z=C. (Art. 68 (ii).) These pairs of values, substituted successively in the fourth and second of the previous group of equations, give, as the relations connecting B, C and v^ B=C\ C=B"\ Hence we may reiterate for gonic systems of logarithms the second proposition of Art. 24 : (ii). If the 7nodulus be changed from k to vk, the corre- sponding base is cha7igcd from B to B^' '^. 88 THE ALGEBRA OF COMPLEX QUANTITIES. The third proposition of Art. 24 is a corollary of this second; for if the modulus be changed from k to i, the base is thereby changed from B to B^ \ that is, exp^ K = B*^ = ^ ; or (iii). The exponential of any quantity with respect to itself as a modulus is equal to 7iatural base. Finally, if B be substituted for 2 in the equation \og^z=K\n2, which is a special case of the formula of proposition (i), the resulting relations between k and B are K=il\nB, B = e^'^, and, in terms of its base and of natural logarithms, the logarithm to modulus k is log^ 2=-.\x\.2 In B. 76. The La\v of Involution. By virtue of propo- sition (i) of last Article we have in general exp^K w = exp^ cv ! /, and exp^ iW = exp^ tw. Hence exp^ /ze; = exp^ / ^ ze; == exp^ .^ t = exp^ j^^i. Otherwise expressed, since changing k into k / 1, or k / w, changes B into B^, or B'^, the statement contained in this set of equations is that B^'^ =(B^y'' = (B'^y. Obviously t or za, or both, may be replaced by their re- ciprocals in this formula, and the law of involution, more completely stated, is (B^^-)>^'^=(B>-y'^)^'. (Cf Art. 25.) THE ALGEBRA OF COMPLEX QUANTITIES. 89 77. The Law of Metathesis. Let 2=^J3^; then by the law of involution and to these there correspond the inverse relations ze; = log^2-, whence the law of metathesis, t\o%^^z = \Qg^zK (Cf. Art. 26.) Also, by changing the modulus (Art. 75) we may write / loge„ 5- = ze/ log/ -2". (Cf. Art. 36.) 78. The Law of Indices. Let w and / be any two complex quantities, w^ii -\- iv, t=r -]- is, in which ?^, v, r and J are real. By the exponential formula (Art, 73), „ , ff , • tf • ^ cos /3 — u sin /3 VI r>t 7 ^^^.# a.. <.;„/? • ^cos /3 — rsiny3 VI in which vi and b correspond to one another as modulus and base respectively in an agonic system of logarithms, and are both real. Hence, by the laws of geometric multi- phcation and division (Arts. 60, 61), „ ^, „, J , A- \ p , , 4- X • o . (z'=t5)cos/3 — (?<;z!=r)sin,3 '^ m But, by the laws of geometric addition and subtraction (Art. 58), 90 THE ALGEBRA OF COMPLEX QUANTITIES. {u ±r)^i{v±s') = {2C-\- iv) zh (r + is) Thus the law of Indices obtains for complex quantities. (Cf. Art. 27.) 79. The Addition Theorem. Operating upon both sides of the equation last written with log^, we have log^(^->^^0=^±A which, if B"'^ = z and B^ = z\ and therefore ^-^log^z^^ and 2-'= log^ /, becomes which is the addition theorem for complex quantities. (Cf Art. 28.) 80. The Logarithmic Spiral. The locus of /*, in Fig. 36 of Art. 68, is obviously a spiral. Its polar equation may be obtained by considering the rates of change o( p and 0, which are respectively Xp and so that for all integral values of /^, cis {9 4- 2/(:7r) = cis 9. In consequence of this property, cis^ is said to h^periodiCy having t\\Q period 2 7r. The exponential B^ has similarly a period. Solving for 7C and V the two linear eqations, ?/= 21 cos /3 4- ^ sin /3, v'^=vcosiS — ?^sin/3, (Art. 73.) we easily find whence 7c^ cos /3 — v^ sin /3 z/ sin ^ -}- ^' cos /3 ; It -J- iv = (?^7r= i, and therefore Hence -5^ has the period 27 kit. In particular c'" has the period 2irr, 92 THE ALGEBRA OF COMPLEX QUANTITIES. It is obvious that the only series of values of v^ that will render q\?>v' lm=^\ is 2k7mr, where k is an integer (Art. 63). Hence the only value of w, that will render Bw+w,^j^-a ig ic^=:z2kiKTr, The function B"^ therefore has only one period and is said to be singly periodic. 82. Many-Valuedness of Logarithms. As a con- sequence of the periodicity of B'^ = z^ the logarithm (for all integral values of k) has the form log^ Z=^W ^ 2ikKTZ, that is, log^-s, for a given value of z, has an indefinitely great number of values, differing from each other, in suc- cessive pairs, by 2ZK7r. The logarithm is therefore said to be many-valued ; specifically its many-valuedness is infinite. We shall discover this property in other functions. In the natural agonic system, 111 ^7= W + ^.ikTT. 83. Direct and Inverse Processes. In the present section and in Sec. VII we have had occasion to speak of logarithms and exponentials as inverse to one another. More explicitly, the exponential is called the direct function, the logarithm its inverse. We express the operation of inversion in general terms by letting f (or some other letter) stand for any one of the direct functional symbols used, such as B, or ^, thus expressing 5- as a direct function of w in the form z=f(w), and then writing for the purpose of expressing the fact that w is the corre- sponding inverse function of z. Thus when B''^ takes the place ofy(tC'), log^ B takes the place of /"' (5-). THE ALGEBRA OF COMPLEX QUANTITIES. 93 In Other terms, inversion is described as that process which annuls the effect of the direct process. If2=/(w), the effect of the operation / is annulled by the operation /-% thus: In accordance with this definition the following processes are inverse to each other : (i). Addition and subtraction : (x -|- a) — a = X, (x — a) -\~ a = x. (ii). Multiplication and division: (x >( a) I a =^ X, (.r / a) >< a -~ x, (iii). Exponentiation and logarithmic operation : (iv). Involution and evolution : {^x°yia = x, (x^'^Y=x. In the use of the notation of inversion here described, its application to symbolic operation, in the form//""' (^w) = ic, must be carefully distinguished from its application to products and quotients by which from ah=^c is derived a=^b-^c. Whenever the direct function is periodic its inverse is obviously many-valued; for, if / be a period oi f(w'), so that then y*""' (2') = ze; -f- 7lp, for all integral values of ?^. (Cf Art. 82.) 94 THE ALGEBRA OF COMPLEX QUANTITIES. 84. Agenda. Reduction of Exponential and Loga- rithmic Forms. Prove the following : ^^u /2-in(a=+3=)-z.tan-^ b/a ^ig [iv\n {a' -^ b') + u tan"^^/^] . (^il^yc-Tiv ^ ^«]n3-z/-/2cls (z;ln^ -i- ^ -mi). log„i+/n {'^-\-ij') = \ m In (.r" +_>/=) — 71 tan-^j'/x logm+in iy = VI \\\y — \mr -\-i(^n \\\y -^ \ vni)? lo&j ( — x) = zlnx — TT. log_i ( — x) = — Inx — ZTT. (2) (3) (4) (5) (6) (7) (8) (9) logz^' ■^^ l0gz(-2) = (10). Given a -\~ zb as the base of a system of loga- rithms, find the modulus and reduce it to the form 7^. -f zv. (11). Express ^+^'^log (.r -|- ?» in the form u -{- zv in terms of a, b, x andj^'. (12). If 7/= 7^ cos /3 -j- z; sin /3, z'' = z; cos /3 — 7^ sin -3, 2v=^zc-\- iv and k =-. m (cos /3 + i sin /3), prove : 7;2 ^/ - (e"'/'^ — ^— «'/'^) = sinh — cos — + 7 cosh — sin — 2 ^ ■' VI VI ■ VI VI (13). Vxovci f{w')=^aw'' ^ 2bw ^ c=^z deduce /-^ {2•) = {-b±,^ b^-ac-\- az)la. THE ALGEBRA OF COMPLEX QUANTITIES. 95 XIII. What Constitutes an Algebra? 85. The Cycle of Operations Complete. It is obvious from the manner in which in the preceding pages the algebraic processes as appUed to complex quantities have been defined, that out of their operation in any possible combination no new forms of quantity can arise. But in particular the examples of Art. 67 illustrate this fact in an expHcit way in the case of geometric addition and multipli- cation ; and tlie exponential equation „ , . , p , • o . ^ cos ,3 — 7c sin /3 J^ic + tv — — 1)11 COS ii+vswiji • Qjg ! L_ , m (Art. 73.) and the logarithmic equation ^og^(x -{- (v) -= Khi (x -j- ly) (Art. 75.) = Kln(TA^-+7-^^) = ^ In (a- + j.=) + k In cis (tan-^^J , or log^ (.r + /» = 2 In (^^ + r) + ^''< tan-^l » exhibit with equal clearness that only complex quantities can result from the application of exponential and loga- rithmic operations. Hence the processes of addition and subtraction, multiplication and division, involution and evolution, exponentiation and the taking of logarithms complete the cycle of operations necessary to algebra. Other operations indeed will be introduced and applied to complex quantities, such as those defined in Art. 90, but though expressed in abbreviated form as single operations, they are combinations of those already described. 96 THE ALGEBRA OF COMPLEX QUANTITIES. 86. Definition of an Algebra. When a series of elements operating upon each other in accordance with fixed laws produce only other elements belonging to the same series, they are said to constitute a group. Thus all positive integers, subject only to the processes of addition and multiplication, produce only positive integers, and hence form a group. Such a group is an algebra. The effect of introducing into the arithmetic of positive integers the further processes of subtraction and division is to break the integrity of the old group and form a new one whose elements include, not only positive integers, but all rational numbers, both positive and negative, integral and fractional. A final step through evolution, or the extract- ing of roots, bringing with it the logarithmic operation, leads to imaginary and complex numbers — that is, numbers composed of both a real and an imaginary part. If now, as is legitimate, we regard all reals and imag- inariesas special forms of complex quantities — reals having zero imaginary parts, imaginaries having zero real parts — then, as pointed out in the preceding article, the algebraic processes of addition, subtraction, multiplication, division, evolution, involution, exponentiation, and the taking of logarithms (logarithmication), applied to complex quantities in any of their several forms, produce only other complex quantities. And hence : The aggregate of all complex quantities — includijig all reals and imagi7iaries^ both 7-ational and irrational — ope7^at- ing upoyi each other in all possible ways by the rules of algebra^ form a closed group. Such a groiip is again an algebra. If, in an algebra the elements that constitute the sub- jects of its operations form a closed group when subjected to a complete cycle of such operations, such an algebra THE ALGEBRA OF COMPLEX QUANTITIES. 97 may be said to be logically complete. It is incomplete if the cycle of its operations be not a closed one. An algebra, for example, that admits evolution and the logarithmic pro- cess, but precludes the imaginary and the complex quantity is logically only the fraction of an algebra. The inability of the earlier algebraists to recognize this fact made it also impossible for them to carry out the algebraic processes of evolution and the taking of logarithms to any except real and positive numbers. With this fundamental characteristic of the algebra of complex quantities the reader will find it interesting to compare the defining principles of Peirce's linear associa- tive algebras, outlined in his memoir on that subject,* and Cayley's observations on multiple algebra and on the defi- nitions of algebraic operations, contained in his British Association address at Southport in 1883, and subsequently amplified in a paper published in the Quarterly Jouryial of Mathematics for 1 887. * * XIV. Numerical Measures. 87. Scale of Equal Parts. Every magnitude, real, ivtiaginaiy or complex, as represented by a straight li7ie, can be measicred by means of an arbitrary scale of equal parts, called units, and can be expressed in terms of the assumed 2cnit by a rational 7mmber, real, imaginary or co^nplex, with an error that is less than any assignable mwtber. The method may be exemplified as follows : * Benjamin Peirce : Linear Associative Algebra (Washington, 1870), or Ameri- can Journal of Mathematics^ Vol. IV (1881), p. 97. **Cayley: Presidential Address in Report of the British Association for the Advancement of Science, for the year 18S3, and on Multiple Algebra in the Quarterly Jo7irnal of Mathenialics {iS'&-]), Vol. XXII, p. 270. 98 THE ALGEBRA OF COMPLEX QUANTITIES. 88. For Real Magnitudes. Suppose the scale of equal parts to be constructed, of which the arbitrary unit shall be OA^AB = BC= = MN=j\ and suppose the outer extremity, P, of the given magnitude OP to fall between the vi^^^ and (?;? -{- i)^^ points of division B A' O A B C M P N — 2 — I 123 VI m + i / '^- 37- of the scale. Then 7?i is the integral number of full unit- lengths contained in OP, and OM, or w Xj\ differs from OP by a magnitude that is at least less than yJ/A^ that is, less than i xy, so that OP=jXm-\-jX«i), where (< i) means 'something less than i.' Divide the segment MJV into r equal parts and suppose P to fall between the /i^^ and (/i -\- i)^^ points of division. Then k is the integral number of r^^ parts of the unity by which OP exceeds OAf and the excess of OP over OM-^j X C-^^/^) is less than the r^^^ part ofy ; that is op=yx(« + p)+yx(o+y ■-J X (w + tn) +y X\ = r] e, or <^ = tan"^ ( rj /e). Since c and 1; are separately less than an arbitrarily small real number, so is -\/e- -f- '>?'' * cis <^ less than an arbitrarily small complex number. Hence the rational complex number m + 2?i is the measure of x -}- zy within any re- quired range of approximation. CHAPTER IV. CYCLOMETRY. XV. Cyclic Functions. go. Definitions.^ The general cyclic functions, of which the circular and hyperbolic ratios of Chapter II are special forms, are defined by the following identities, in which w and w^ stand for complex quantities and K^7n * cis /3 ^ modulus. («). sin^ w^'^ {e'"l'^ — \ (7). cos^(za zh'W^) >■ = cos^ Z£^ ' cos^ 7£/ it K~^ sin^ zv ' sin^ zc^. From (6) and (7) by making zv^ = zv\ (8). sin^ 2z^'= 2 sin^ze/ • cos^ze', (9). cos^ 2zv^= cos^ z£/ + *^~~ sin^K 7^. From (3) and (9) by addition and subtraction: (10). cos^ 2te^ — I = 2/<~=sin^z^^, (11). cos^ 2ZV -|- I = 2 cos^ zv. From (6) and (7) by division : tan^7£/ ±: \.-AXi^zv^ (12). tan^ (ze' di ze'') = f \ 4. / , /\ doXf^zv^K =tan^7^/ (13). QO\.,^{w :^zv ) — ^ ^ I ±L cot,^ w ' tan,^ zju I04 CYCLOMETRV. From (12) and (13) by making w^=^w\ r \ ^ 2tan^7/y (14). tan^2Z£; = — ■ — _,/" ^ — > (15). cot;^ 2r£/ = ^ (cot^zc + '^~*tan^zt/). From the two forms of (6) by addition and subtraction: (16). sin^ (7*7 -f ze'') -|- sin^ (z£/ — ze;') = 2 si (17). sm.f^{iv -\- w'') — sin^ (?£; — w^) From the two forms of (7) by addition and subtraction : (18). COS^ (t(7 -f ^'') + COS^ (Z£/ — Z£'') = 2 COS^ 7l' ' COS;( 7C>^, (19). cos^ (r^ + ^<^') ~ cos^ (z^; — zt'') = 2K~''sin^ 7i' ' sin^ za\ In (16), (17), (18), (19) write 2 for z<7 + ze/', 2^ for w — zv^; they then take the respective forms : ( 20). sin^ 2 -{- sin^ 2^ = 2 sin^ K^ + ^'0 ' cos^ i (-^ - ^'). (21). sin^^ — sin^^''' = 2cos^4 (-3'-}-^') -sin^-K^- — ^'), (22). COS^ -S' -f COS^ 2-' = 2 COS^ -2 (^ + ^0 • COS;, K-^ — ^0> (23). COS;, 2- — COS;, 2-' = 2K~= siU;,^ (2 -\- 2^) ' siU;,^ (2 — 2^). CYCLOMETRY. 105 From (12) and (13) by transposition of terms and reduction: (24). tan^ 7C' ± tan^ 7^' = sin^ (za zh w^) ' sec^ za ' sec^ za^\ (25). cot^ Z£' dz cot^ a'' = sin^ (w^±: w) ' csc^ w ' csc^ w\ From (i) and (2) by involution: (26). (cos^ ze/'dr /^'"'sin^ze/')^^ = cos^ Tvza^ ± K~' sin^ ww^, a generalized form of Demoivre's theorem. (Cf Art. 74.) All of the foregoing formulae pass into the corresponding circular and hyperbolic special forms, by the respective substitutions k = z and k= i. Making the proper substitutions from (g-), (/z), (z), (7)' {^)y (0 i^ (6) and (7), after assigning to k the values / and i successively, we obtain : (27). sin(zv ±tw^) =:s'mw ' cosh w^±: i cos w ' sinh za% (28). cos(76' zb ze£^') = cos 7u ' cosh ze^'nF z'sin za ' sinh za^, (29). sinh (e£/ zb /?£/') = sinh zv ' cos zv^ ±z i cosh zv ' sin zv^, ( 30) . cosh ( 7C^ ziz zzf'' ) = cosh TV ' cos w' ziz 2* sinh zu ' sin ze/'. By definitions (a) and (/5) and the definitions of cir- cular and hyperboHc sines and cosines (Art. 90) : (31). sin^e£/ = K sinh z£J!k = zk sin za/'tK, (32). cos^ za = cosh zcj/k = cos zc/zk. I06 CYCLOMETRY. 92. Agenda. Prove the following formulae (i). sm^^7c>=: (2). cos^^u>= ±z^ — (3). (4). sin^ 7u 4- sin;^ 7^' tan^ ^ (7v -f- ?£/') sin^ 7£' — sin^ ?<:'' tan^ ^ (w — w' ) COS;^ w + cos^ w' K= cot^ J (?£/ -4- 7<."''') (5). — ''- j ^ — j = t2in^^(7£j dzTjf'y (6). ^^ "^ = K-COt^i (7C' -^ 7C'). COS,. 7<:' — COS,. 7a (7). Sin^27£/ — K - tan^ Z£/ (8 ). sin^^ 37£' = 3 sin^ 7C' • cos^ 7v -f k-= sin]^ 7£'. (9). cos^ 37<.v = 3 K-- cos^ 7CI ' s\n\ 7a -f- cos]^ 7£'. / \ 4. 3 tan^ 7£/ + K~= tan!. 7a (10). tan,.-^ 7£' = ^^ '^ '^ . N secI- 7a (11). sec^ 2w r \ ^ 1 sin,^ 7a (12). tan^-|7(' = COS^ 7£' -j- I (13). By making k = / and /c=i successively in the formulae of Art. 91 deduce the corresponding formulae of the circular and hyperbolic functions. CYCLOMETRY. 107 93. Periodicity of Modocyclic Functions. In Art. 81 it was shown that all exponentials are periodic having the period 2/K7r. The forms e"^"^, e^""" and t'^ have therefore the periods 2ZK7r, 27r, and 2zV respectively; hence also the modocyclic, circular and hyperbolic func- tions, whose definitions render them explicitly in terms of gwiK^ ^w and e" (Art. 90) have the same respective periods, so that, \i f stand for any one of the symbols sin^, cos^, tan^, etc., g for sin, cos, etc., and h for sinh, cosh, etc., the law is, n being any integer : For modocyclic functions f{u< -\- 2niK-K) z=f(w)^ for circular functions g(iw -\- 2inr)=g{7v), for hyperbolic functions h (Z£/ -f 21llTr) = h (?£/). 94. Agenda. Functions ofSubmultiples of the Periods. (i). The Period iK-rr. Show that tan^a/ and cot^7£/ have also the shorter period iK-rr. (2). Range of Vahies. Compute the following table of the range of values of the modocyclic functions: w = i-^-- to t-Kr: 2 7K- to 3'"'^^ 2 •^"^" to 2/K7Z 2 sinK ^(^ + zKO to +iK + 7K to ±7/C0 ±/kO to — /K — 7k: to TtKO COSk zh' + 1 to ±0 ±0 to —I ~i to TO + to +1 tan/c w TikO to +//C00 ±7 /coo to +tKO + //C0 to +7KC0 i/K GO to +Z/CO cot AC 7XJ + i , . / 3 z'ktt \ . , (11). cos^l^^^ ± x^l= -h z/KSin^ Z£/, (12). Write similar formulae for all modocyclic functions 3 /ktt/ 2 ± w, and 2 /ktt zb a^. 95. The Inverse Functions. The modocyclic, circular and hyperbolic functions, being defined in terms of exponentials, are direct; and corresponding to them, through inversion (Art. 83) we have the inverse modo- cyclic, circular, and hyperbolic functions. CYCLOMETRV. lOQ Applying the operation of inversion to the three equations 2=:^ f {w -\- 27iiKTr') =y(z£/), 2 =g {W -f- 27^7^) =g (Z£/), z=- h {w -f- 271177') = h {w), of Art. 93 we have the corresponding inverse forms : y~' (2') = ?£/ -|- 2;^ZK7^, g-^ (2-) =^W ^ 271-7?, /2~' (^z) = W -{- 27lZ7r, in which n is any integer whatever. Thus, Hke logarithms, the various forms of inverse cyclic functions are many- valued to an infinite extent. We may however define the interval over which the values of w shall range in such a way as to make it one- valued within the interval considered. Care must be taken to do this in writing the formulae of inverse functions. 96. Agenda. Formulae of Inverse Functions. By inverting the corresponding direct formulae prove the follow- ing, and assign in each case the interval for which the formula is true. (i). sin^'^f = cos^' y I -f- *< ''■^^• (2). sin7 X = tan-' . . _^ -^ • "|/ I -f- K ^ X^ (3). sin7 a- ± sin-> = sin";:' {x ^^l -\- K~^y- ±y \^ i -\- k~^ x^). (4). s'm-'x±:sm-'j' = cos-' (i/(i + K-^'^0(i+K-=x) zh K-\xy). no CYCLOMETRV (5). cos-^r zb COS-' J cos^^ {.vy ± V(x' — I) (y — i)), _ _ -^ —J' (6). tan^' X ± tan^> = tan^^ --_^^ (7). cot^^^dtcot^'j/ = cot^ ^, I dz AC ""^ ,-ry (8). 2 sin^';f = sin";^' (2:r|/i -j- k~-x~), (9). 2sin~'jr= cos'J^' (i -f 2/<~-ji."). (10). 2 cos^'.r = cos"^' (2jr- — i). 2 i* (11). 2 tan";:' X = tan"' — —-^r^ — -, ' (12). 2 tan^' X = cos^' ^ _ ^_/^^, - (13). 2 tan-^ X = sin-' ^ _ ^,3 y . ' (14). Deduce the corresponding formulae for inverse circular and hyperbolic functions by assigning to k the values 4" ^' ^i^d — I successively. 97. Logarithmic Forms of Inverse Cyclic Functions. The inverse cyclic functions are obviously logarithmic. We may obtain them as such in the following manner. Let 21, v, x, y be four quantities, in general com- plex, such that u = log^ z>, X = cos^ 2i, y = sin^ 21. Then ^u/K __ ^, __ QQ^^ 2c -L K~' sin^ 26 = X ■-]- K~^jy, CYCLOMETRY. in x= it y I -h K~j\ K j= ±y X — I. (Art. 91, 3.) . • . sin^'j' = AC In (j'/k ± Vy/K^ + i), and cos";^' X = K In (x dz i/.r" — i ). The logarithmic equivalent of tan~' 2 may be deduced from the definition tan^w^K {e^/>^ — ^-"^/'^) / {f'/^ -f- ^-«'/'<) ==^, by solving this equation as a quadratic In ^'^'/'^. Thus, tan^ w = ~^,/^:^~ = 2, whence K 2 w = tm\J 2- = - hi -' Since cot^w= ijx, the corresponding formula for cot^^'^ is K K? -|- I cot^ 2 := - In — '^ 2 KZ — I The forms for sec;^' x and csc^'j/ are also obtained from those for cos';^':^ and sin~'_y by changing x and^ into ijx and ily respectively. They are sec^'^ = K In (i/.r ±: ^' ijx^ — i}- csc~'j/ --^ K In (i/kj/ ±: \/ iJK^y' + i)- 112 CYCLOMETRY. The Student can easily verify these results by making the transformations independently. The hyperbolic forms, obtained from the foregoing by putting K = I are important, and are of frequent application in the integral calculus. They are given in Art. 56. As there explained, for real values of 21 the positive sign before the radicals must then be chosen. The forms for the inverse circular functions, got by putting k = 7, are less frequently useful. 98. Agenda. Prove the following, n being an integer: ( I ). sin-' i K (^ - 2-') = cos-' ^ (.3- + ^-) = log^ 2 -\- 2711 KTT. (2). tan-'K(2-=— l)/(<^^-[- l) = \og^2-^2?l2K7r. (4). cos^' (-^" 4" i)^) ^ ^'^ (cos /3 cosh"' p — sin /3 cos"' o-) -|- im (sin /3 cosh"' p -\- cos /3 cos"' o-), where p= zt i/(^ + lA=— ^0. «■= dz \/\s — ]/j' — x=), s = 1 (^x^ -^ y -\- i) , and Krr=;;/cis/3. (5). sin"' (x -\- ry) = vi (cos /3 cosh"' P— sin /3 sin"' 2) -\- im (sin /3 cosh"' P+ cos /3 sin"' 2), where, as before, k = m cis /3, and P=iii|/'(5H-i/6^^-^0, ^=±.x/(^S-VS-'-X^), .Y = (— ^ sin /3 4- jj/ cos /3) / w, Y= (^ cos /3 + j/ sin /3) / ?«. (6). Reduce tan"' {x + ?» to the form it + iv. CHAPTER V. GRAPHICAL TRANSFORMATIONS. XVI. Orthomorphosis Upon the Sphere. gg. Affix, Correspondence, Morphosis. Every complex quantity, defined geometrically by a vector drawn from the origin with proper length and direction, de- termines uniquely a point in its plane, namely the extremity of the vector ; and conversely, to every point in the plane corresponds one and only one complex quantity. It is convenient therefore to assign, as the geometrical repre- sentative, or affix, "^ of a given complex quantity, a point in a plane, to note its different states by a series of affixes, and to represent a continuous change in it by a line, its path, in general not straight. This relation between point and complex quantity is described as a one-to-one correspondejice, and the spreading out upon the plane of the points or paths of a varying complex quantity is its morphosis in the plane, or its planar morphosis. 100. Stereographic Projection. By means of a stereographic projection we may establish a one-to-one correspondence between the plane and a sphere, so that a point upon the plane determines uniquely a point on the sphere, and vice versa; and the spreading out of all the C. Jordan : Cottrs cT Analyse de T ^cole Polytechnique , Vol. I, p. io6. Art. ii6. 114 GRAPHICAL TRANSFORMATIONS. points or lines on the sphere that correspond to the affixes or paths of a complex quantity in the plane, will be its morphosis, more specifically its orthomorphosis^^^ upon the sphere. We accomplish the transformation in the following manner. We place the sphere with its center at the origin of complex quantities in the plane. Regarding the plane as fixed in a horizontal position, all projecting lines are drawn from the upper extremity of the vertical diameter as a center of projection (Fig. 39 of Art. loi). Then if from this center of projection a straight line be drawn to any point in the plane, it will cut the spherical surface in one other point. The two points thus uniquely determined, one in the plane, the other on the sphere, are said to correspond to each other, or to be correspoyiding points. Thus a complex quantity may have an affix in the plane and a corresponding affix on the sphere. Points in the plane inside the great circle in which it cuts the sphere correspond to points on the lower hemisphere; points in the plane outside this circle have their corre- spondents on the upper hemisphere. loi. Transformation Formulae. ^^^ In order to con- nect algebraically the two forms of the complex quantity, in the plane and on the sphere, assume as the origin of co-ordinates in both systems the center of the sphere, let i and y) be the horizontal, t the vertical co-ordinates of points on the sphere, x and y the co-ordinates of corre- *An orthomorphic transformation of the plane into the spherical surface. This knid of transtormation is called orthomorphosis by Cayley : Jotirnal fur die teine utid atigewandte Mathematik, Bd. 107 (1891), p. 262, and Quarterly Journal 0/ Mathematics, Vol. XXV (1S91), p. 203. See also the first footnote to Art. 108. ** Cf . Klehi : Vorlesungen iiber das Ikosaeder, p. 32. GRAPHICAL TRANSFORMATIONS. 115 spending points in the plane. It will involve no loss of generality to assume the radius of the sphere to be i. Its equation then is Fig' 39- and OC being the vertical axis, P and O corresponding points on the plane and sphere respectively, we have, by similar triangles, whence and therefore X _0P _}>__ I ^ V I _ ^ ^ I — ^ zy Il6 GRAPHICAL TRANSFORMATIONS. From these and the equation of the sphere we readily obtain .T'+y+I 2 and thence the values of ^, i and rj in terms of x and j/, namely : . 2X v 102. The Polar Transformation. If it be desired to present the formulae of transformation in terms of tensor and amplitude, we may write x = r cos 6, y=^r sin Q, I = cos , and accordingly sm cos <^ _ . :r + z v = r CIS c^ = ^ — r cis c', •^ I — sm o ' cos<^ I — sin ^ Thus the expression cos<^ (i — sin <^) "cis^, in which <^ and are independent of each other, suffices to represent all possible complex quantities. By easy substitutions, I, rj, t, are found, in terms of r and 0, to be . 2rcos^ 2rsin^ r^ — i ^ "" r^ + i" ' '^ "" r^+ I ' ^ "" r^-f- i ' GRAPHICAL TRANSFORMATIONS. 1 17 103. Agenda. Properties of the Stereographic Pro- jection. (i). Prove analytically that a circle, or a straight line in the plane, corresponds to a circle on the sphere. (2). Prove geometrically that any two lines in the plane cross each other at the same angle as the corresponding lines in the sphere. (Cf. Art. 107.) (3). Show that to the centre of projection correspond all points at infinity in the plane, and that it is therefore consistent to say : there is in the complex plane but one point at infinity. (4). Show that meridians on the sphere through the centre of projection, and parallel horizontal circles on the sphere correspond respectively to straight lines through the origin and concentric circles in the plane. XVII. Planar Orthomorphosis. 104. W-plane and Z-plane. In the graphical rep- resentation of an equation connecting two complex varying quantities w and z, it conduces to clearness of delineation and exposition to separate the figures representing the variations of lu and z, and to speak of the a/-plane and the ^--plane as though they were distinct from one another. This language and procedure help us to see more clearly that the plane with the ze^-markings upon it has a distinctive character and presents in general an appearance different from that which it has when its markings represent the variations of the functions of w, and to distinguish more easily the two groups of markings from one another. It is the purpose of the present section to describe the planar orthomorphosis of some of the functions that have been defined in the foregoing pages, that is, to cause the Il8 GRAPHICAL TRANSFORMATIONS. point Q, the affix oiw, to traverse the ze'-plane in a specified manner, and to mark out the paths that P, the affix of a function of w, will follow, in consequence of the assumed variations of w. 105. The Logarithmic Spirals of B^ Non-inter- secting. The function ^"', is singly periodic; that is, there is only one quantity, the period siktt, multiples of which, when substituted for w, will render B'^ = \, (Art. 81). If now Q^ and Q, the affixes oiw^ and w, move in the ze^-plane upon parallel straight lines, the variable quantities Wq and w may be assumed to have the relation w = w^ -f- «, where <« is a constant quantity (fixed in length and direc- tion);* and the paths of >5^'o and B'"" will either not inter- sect at all, or will coincide throughout their whole extent. For, in order that the two paths may have a point in common there must be a pair of values w^, zi'^-j-a, for which and a must be a multiple of 2ZKTr (Art. 81). But if a be a multiple of p/ktt, then for all values of za,^ and the two ze'-curves have all their points common. Hence, since in the construction of Fig. 36, Art. 68, a vector representing 21 kit must lie in the direction 01^, we conclude : Tf the paths of w^ and w in the w-plane be parallel straight liiies, the paths of B''' and B'"^ in the z-plane will *This is merely a way of saying, that if a link, or rod, while remaining parallel to a fixed direction, move with one of its extremities upon a fixed straight line, its other extremity generates a second straight line parallel to the first. GRAPHICAL TRANSFORMATIONS. 119 be coincident, or distinct and not intersecting, according as the intercept viade by the two w-lincs on the modnlar 7ior- vial is or is not a vmltiple of the period ^/ktt. 106. Orthomorphosis of B^. The fixed elements in the z<7-plane are the real axis, the modular line and the modular normal, — in Fig. 36, the lines O/, ET and OF; in the ^--plane they are the real axis and the unit circle. By the operation of exponentiation, indicated by ^^, a straight line in the ze'-plane is transformed, metamorphosed, into a logarithmic spiral (Art. 80). Hence if the variable elements of the ze'-plane be assumed to be straight lines, in the ^'-plane they will be logarithmic spirals. Assigning as the path of w^ a straight line OS^ passing through the origin (Fig. 40), write W=W^'\- aiK, in which a is a real quantity. The path of w, for a given value of a, will then be a line EC, parallel to OS^, to which will correspond in the ^--plane, a logarithmic spiral E'C^ (Fig 41). In particular to the path of w^ corres- ponds the spiral z,^ that passes through the intersection of the real axis with the unit circle. w-plane. Fig. 40. z-plane. Fig. 41. I20 GRAPHICAL TRANSFORMATIONS. To the straight lines in the zu-plane, obtained by giving different values to a, there correspond in the ^--plane, so long as a is less than 2- and not less than o, distinct non- intersecting logarithmic spirals (Art. 105). And since OC=c= ma (Art. 72), when a varies from o to 2-, c varies from o to 2 m tz, or as represented in Fig. 40, from o to (9 C and when ^C moves from the position OS^ to the position £iSj , the corresponding logarithmic spiral makes a complete revolution and sweeps over the entire -S'-plane. To every point in the ^--plane corresponds one and only one point in the band between the parallels OS^, BiS,, whose width is 2W7rcos (<^ — /3), and also one and only one point within every band, in the zc-plane, having* this width and parallel to OS^. The construction therefore shows graphically how, to every value of 2, there cor- respond an infinite number of values of w, namely all the values w -}- 2k2KTr, wherein k is any integer. The suc- cessive affixes of ze^, w -\~ qikit^ w — iiktv, w -f- 4Z'<7r, etc., are obviously situated at the division-points of equidistant intervals, each equal to 2imr, along a straight line through the affix of w parallel to OF. Whenever a z£/-line crosses OF, the corresponding ^■-spiral crosses the unit circle (Art. 68). Hence, to points in the z£^-plane below or above the modular normal OF, correspond respectively points in the ^--plane within or or wfthout the unit circle. Thus the shaded and unshaded portions of Figs. 40, 41 correspond respectively. The points where any spiral in the ^'-plane crosses the real axis are those for which B'"^ is real. But the necessary and sufficient condition for the vanishing of the imaginary term of B'^" is . V cos /3 — u sin /3 , „ ^ sm ^-— ^ = 0, (Art. 73.) GRAPHICAL TRANSFORMATIONS. 121 for which purpose it is necessary and sufficient that V cos /3 — 7c sin /3 = /t;;^7^, or It cos ( ^ + /^) + ^^ sin^^ 4- /3j == ^WTT, wherein k is any integer ; and the locus of the equation last written is a straight line parallel to 7" and distant from the origin a multiple of mir. Hence, whenever a z£^-path crosses such a line the corresponding ^■-spiral crosses the real axis. (Cf. Arts. 72, 80.) As particular constructions we have : When the w-Ymes are parallel to the modular line O T, the ^--spirals degenerate into straight lines passing through the origin; and when the ze^-lines are parallel to the modular normal OF, the ^--spirals become circles concentric with the unit circle. (Cf. Art. 71.) 107. Isogonal Relationship. Any two spiral paths of B'^ cross one another at the same angle as the correspond- ing straight paths of w. Let B^, p^y v^, 7C^ and B, p, v\ u^ be two sets of cor- responding values of 0, p, v^ and it' (Fig. 36). It was shown in Art. 72 that mO = v^ ; hence and Ml:zlol _ A . ^JJUlo . P-Po ^'^ P—Po But (Fig. 36) v'— v^ = (u'— u.^) tan (<^ — /3), whence, smc& p = b"' and p,^=zb"o (Art. 73), A(^-^o) -P: . -'<=i^a tan (d. _ ,3). 122 GRAPHICAL TRANSFORMATIONS. and in Art. 52 it was shown that limit \ b''' — b"o \ p^ . limit \p, (^ - ^ J /=/oi P-P. = tan(<^-/3). In Fig. 42 let 7^^/* represent a segment of the spiral of B'", P^T 2. tangent line at P^ and let P^,p=OPr_,, OP, 6^ = arc- ratio of XOP,^, z-plane. = arc-ratio of XOP, Then / -P,= HP and p^ {B - 0^) = P^//; whence limit \pj^ P=Po that is, =1 tan (arc-ratio of KP^ 7"), arc-ratio of KP-^ T= — /3. Thus the angle between the radius vector p and the spiral is constant and equal to that between the path of zc and the modular line (Art. 68). It follows that the angle between any two spirals is the same as that between the two corresponding paths of w. O. E. D. It follows also, that to a small triangle formed by any three ze^-lines in the ze^-plane corresponds a small curvilinear triangle in the ^--plane, whose angles are respectively equal to those of the former and whose sides become more and more nearly proportional to those of its correspondent as GRAPHICAL TRANSFORMATIONS. I23 the two triangles become smaller. The two figures there- fore approach the condition of similarity to one another as a limit, when they themselves approach the vanishing point. It is because of this property that the transformation of of w into B^ is called orthomorphic.^^ It is a property that can be proved to be generally true of functions of a complex variable. ^-'^ 108. Orthomorphosis of cos^(u -h iv.) In Art. 8 1 it was shown that u -\- iv = (?/-}- ry) cis /3, and it is obvious from the definitions of Art. 90 that . w . w cos^ w = cosh - = cosh — -• — ^ • '^ K ni CIS p ' whence cos^ in -\- 2V)=^ cosh — — — • Here u^ and v^ have the significations attached to them in Art. 68, Fig. 36. For brevity write ?// VI = r, v^l m = s. By formula (30) of Art. 91 we have cosh (r -[- ?V) = cosh r cos s ^ i sinh r sin s. Let ji: ^ cosh r cos i^, j/ ^ sinh r sin j-; then X- , v' -j— . — — J cosh^ r sinh" r ' and X' y- cos ■ s sin" s *"Orthomorphic. . . . Preserving the true or original shape of the infini- tesimal parts, though it may be expanding or contracting them unequally." ( Cenijay Dictionary. ) **Byerly: Integral Calculus, 2d ed., Art. 211, p. 275. 124 GRAPHICAL TRANSFORMATIONS. • In these equations x and y are the Cartesian co-ordinates of the varying affixes of cos^^ (?^ -f- iv'). As equations of loci they make the following correlations apparent : When r remains constant and j- varies, the affix of 21 + iv describes a straight line parallel to the modular normal while the affix of cos^ (?^ + z<^0 moves upon an ellipse; when s remains constant and r varies, the affix of u -f- iv describes a straight line parallel to the modular line while the affix of cos^ (?/; -f z^) moves upon a hyperbola. 109. By Confocal Ellipses. The former of the last two equations represents, for r= constant, an ellipse whose principal semi-diameters are cosh r and sinh r. Since coshV — sinh-r= i, any two or more of the ellipses ob- tained by giving different values to r are confocal,^ having their foci on the real axis at unit's distance on either side of the origin. If r be regarded as a parameter and be allowed to vary from o to +00, the ellipse represented by the equation .T7cosh-r-T-_>/"/sinh-r= I , starting from the segment of straight line joining the foci as its initial form,^-^ expands and sweeps over the entire plane. Since the equation is the same whether r be positive or negative, the variation of r from o to — 00 gives a like result. It is only necessary, therefore, in order to make x andjv pass once through all real values, to retain the positive values of r. In any one of the ellipses the variations oi x and y are governed by the variations of cos s and sin j ; real values of x lie between — cosh r and -f- cosh r, real values of y between — sinh r and + sinh r. To the values * Charles Smith: Conic Sections, Art. 221, p. 244. GRAPHICAL TRANSFORMATION'S. 125 O, -TT, TT, -TT, 2 7r, correspond respectively X = cosh ;•, o, — cosh ;', o, cosh ;-, y = o, sinh ?', o, — sinh ;-, o. By comparing these limits of variation we easily dis- cover, r remaining constant and positive, that when s passes through each successive quadrant of a unit circle, the point {x, y) describes, in the same order, the successive quad- rants of an ellipse. If r be negative, the same ellipse reappears, but its periphery is described in the reverse direction. Thus the affix of cosh (r -f- 'is') describes the entire ellipse when s passes from o to 27r; or obviously also, when .c passes from any value s^ to Sq~\- 27r. If r and s be now replaced by u^ j 7ii and v^ j vi and cosh(r+?V) by cos^{u -\- iv), the results of this ortho- morphosis may be collated as follows : When u^ varies from o to -f 00 and v^ from some fixed value v^ to v^-^ 2imr, the affix of cos^iji -\- iv) ranges over the entire ^--plane; that is, to the band A A in the a/-plane (Fig. 43), of width 2imr, extending from the TV-plane. z-plaii(\ Pis. 43- Fig. 44- 1^6 GRAPHICAL TRANSFORMATIONS. modular normal OF upwards to infinity parallel to the modular line, corresponds the entire s-plane (Fig. 44). In like manner to the band BB, below the modular normal, and also to every other rectangular band con- gruent to AA having its base in the modular normal, corresponds the entire 2'-plane. no. By Confocal Hyperbolas. By an analysis similar to that of Art. 100 we derive from the equation cos-i" sin-^ a series of confocal hyperbolas, having cos s and sin s as their principal semi-axes and, since cos* «? -f sin- ^ = i , having their foci at the points -f i and — i, and forming therefore with the ellipses of Art. 109 a series of confocal conies. -'^ Any hyperbola of the series may be generated by giving to s the four values 2it — s, s, tt -f ^, it — s \x\. succession and in each case allowing r to vary from o to + oc ; . the values 2 7r — s and s produce one branch of the hyper- bola, IT -^ s and IT — s the other. If then, in addition to the variations of ?', s pass continuously from s^ to s-^ -\- ^tt, the series of hyperbolas will pass over the whole of the z-plane and the affix of cosh (r -f is) will range over the band A A, Fig. 43. This co-ordination of the two figures is thus the same as that established in Art. 109. But the co-ordination may take place in other ways. It is possible, for example, to generate one entire branch of * As is well known, these confocal conies cut one another orthogonally. fSee Charles Smith's Conic Sections. Art. 224, p. 246.) In other words, the transforma- tion of u-\-iv into cos,^ («+2z/) is orthomorphic ; and in virtue of this ortho- morphism, when the path oi ti-\-iv is any straight line, that of cos^ (z<+zz/) is a spiral that cuts the ellipses, and also the h>-perbolas, at a constant angle. GRAPHICAL TRANSFORMATIONS. I 27 any hyperbola of the series by keeping s fixed and varying r fi-om — CO to -]- ^ ) -^ passing from cos ^ to -f- 00 , ^ from — 00 through o to -f- co. The other branch of the same hyperbola is then obtained by changing «? into tt -\- s and causing ?^ to vary again from — cc to -\- cc . If then, in addition to the variation of r, s be given all values between s^ and s,^ -{- i-n-, the morphosis is completed. To the 2'-plane in this case are co-ordinated two parallel bands, a a and b b o{ Fig. 43, having a width equal to \in'K and extending to infinity in both directions, and separated from one another by a third band of like dimensions. III. Agenda. Problems in Orthomorphosis. De- scribe the graphical transformation of it -\- iv into each of the following functions : (i). sin^ {2C + hi). (2). tan^ {ic -f iv). (3). sec^(/^ + ?V). (4). {it -{-ivy. (5). Prove that the graphical transformation of tt^ iv into cos^ {it -f zV), or into any of the other cyclic functions is an orthomorphosis. (6). Perform ypon ^("+'^)/'<^, sm^{u-\- iv), cos^(?^-f- /z') and tan^ {u -|- iv), the transformation of Art. loi and trace, upon the surface of the sphere, the path of each of these functions when the affix of 2t -j- iv moves upon a straight line in the plane. CHAPTER VI. PROPERTIES OF POLYNOMIALS. XVIII. Roots of Complex Quantities. 112. Definition of an n^^ Root. The n^'^^ root of a given quantity is defined to be such another quantity as, when muhiplied by itself ?^ — i times (used n times as a factor), will produce the given quantity. An 7i^^ root of zc is denoted by w^^'^ Throughout the discussion concerning roots, whether of quantities or equations, n is supposed to be an integer, and unless statement be made to the contrary, a positi\'e integer. 113. Evaluation of n*^ Roots. Every complex qua^itity has n 7i^^^ roofs of the form , . 2kTr-^6 11 ' in which ?'^/" is a tensor, {ikir ^- 6) j 11 an amplitude, and k has one of the values o, i, 2, . . . , 7^ — i. Let the complex quantity, whose roots are to be in- vestigated, be denoted by r cis 0. Since r is a real positive magnitude, r^^'" has one real positiv^e value (Art. 23). How to find this value we do not here enquire. By Demoivre's theorem (Art. 74), for all integral values ofX^ \ rv« cis ^-^ I = r cis ( 2X^7r + ^) = r cis 0, PROPERTIES OF POLYNOMIALS. 1 29 and the complex quantities , . e , . 217 ^- e , . 2(11 — 1)17-^6 ^i/n CIS - , r'/" CIS - - -- > , r'/'' CIS -^ J—1-. n n n are all different. Hence each of them is a distinct 7^^^' root of rcis^, and there are n of them. Thus rcis^ has n n^^ roots, which was to be proved. No additional values are derived from r^/« cis {2kir -f (f)ln by giving to k any values other than those contained in the series o, i, 2, . . . . 7^— i, and r^/" has only one real positive value. Hence A complex qiiantity has only ii /^^'^ 7'oots. 114. Agenda. Examples in the Determination of z?*^ PvOOts. Prove the followmg:-'^ (2). The cube roots of — i are — i and \ ±i\ y^2>' (3). The fourth roots of — i are iti — 7- and ± — ^— • ^^^ 1/2 1/2 (4). The cube roots of i -\- i are -1+^ (l/3_±„0±i(j/3ziO, 2^/3 ' 24/3 and - ( 1/ 3_-_ i) ±i_(j/ 3_±0 . 24/3 (5). The sixth roots of 4- i are ± I, ^ ±: ii 1/3 and -\± i\ 1/3. *Cf. ChrystaL Algebta. Vol. 1, pp 242, 248. 130 PROPERTIES OF POLYNOMIALS. (6). Find the fifth roots of -f i and the sixth roots of— I. (7). If w be one of the complex cube roots of -f i, then ( I -l- (o-y = - = o, and (i — oj -f- u)-y= (^1 -\- w — o)-)> = — 8. Show that I -r <^"is one of the twelfth roots of -f i. (8). The twentieth roots of + i are the successive powers, from the first to the twentieth inclusive, of ih' (10 + 2^/5) + ^'(1/5 -I) J. (9). Representing the complex number of example (8) by 0-, show that 0-' — o-" 4- o-^ — o-= + I = o, o-" + I = o, and that or is therefore a tenth root of — i. Show that the even powers of cr are the tenth roots of -f- i- XIX. Polynomials and Equations. 115. Definition of Polynomial. An algebraic ex- pression of the form a^^a,z-f- a,2^ -\- . . . . -j- a„ 2", in which «q, «,, . . . an are any quantities not involving 2, and in which the exponents of 2 are all integers, is called a rational, integral polynomial iii z. In what follows it will be sufficient to speak of it more briefly as a polynomial, the qualifying adjectives being understood. The highest ex- ponent oi z contained in it is its degree. PROPERTIES OF POLYNOMIALS. 131 116. Roots of Equations. The investigaaon of Art. 113 solves the problem of finding what are called the roots of the equation ^71 -- r^^ in which ze^ is a known complex quantity, and shows that such an equation, which would commonly present itself in the binominal form az'' 4- <^ = o, \bla^= — w'X has exactly n roots. If additional terms containing powers oi z lower than the 71^^ be introduced into this equation, the problem of its solution becomes at once difficult, or impossible. In fact, the so-called algebraic solution of a general algebraic equa- tion of a degree higher than the fourth, that is, a solution involving only radicals and having a finite number of terms, is known to be impossible.^ A discussion of the methods that may be employed in solving equations is beyond the intended scope of the present work, but the so-called fundamental theorem of algebra (Art. 120), accompanied by those propositions that are prerequisite to its demonstration, find a fitting place here. 117. The Remainder Theorem. If a polynojnial of the 71^^ degree in z be divided by z — y, the remainder, after n successive divisimis, is the result of substituting a for z in the polynomial. Let /"(£-) denote the polynomial. * Proved to be so by Abel; Journal fur die reine und angewandte Mathematik (1826), Bd. I, pp. 65-84, and CEuvres covipietes de N. H. Abel, nouv. ed., Vol. I, pp. 66-94. 132 PROPERTIES OF POLYNOMIALS. and let the division by 2 — y be performed. It is obvious that the remainder after the first division will be of a degree lower by i than the dividend, that each succeeding re- mainder will be of a degree lower by i than its predecessor, and that therefore the 71^^ remainder will not involve z. If the final quotient be denoted by O and the final remainder by R, then z — y ^ ' 3" — y in which R does not involve z\ whence, by multiplying by fiz) = Q(z-y) + R. This equation has the properties of an identity,''^ and in it z may therefore have any value whatever. Accord- ingly, let y be substituted for z, and let the result of this substitution in the polynomial be denoted byy(y); then /(y)=-^(7-y) + ^, in which R remains unchanged from its former value, and Q, being now a polynomial in y, is finite. Hence and ^-/(y). Q. E. D. If the remainder obtained in dividing /" (5') hy z — y vanish, then /(y) ^o and y is said to be a root of the equation/C^-) = o. Hence: *This may be shown by actually evolving it in specific instances. Thus, if the process here described be applied to the quadratic a^-^-a^z-Va,^'^, the result is and the principle of this procedure is obviously general, and independent of the degree of the polynomial. It should be observed that the identity does not depend upon the process of division; we might, in fact, produce it by the processes of addition, subtraction and rearrangement of terms. The di\ ision process is used as a convenience, not by necessity. PROPERTIES OF POLYNOMIALS. 133 (i). If fi^^^ be exactly divisible by z — y, y is a I'oot of the equation f {^z') = o. Conversely, the remainder will vanish ify(y)^o. Hence : (ii). jfy be a root of the equation f(^z) =^ o, thenf(^z) is exactly divisible by z — y. 118. Argand's Theorem. If for a given vahie of z the polynomial of the n^^^ degree, a^-\-a,z-^ a.z'' -\- . . . . -j- a„ z", have a value w ^ different from zero, its coefficients a^, a^, a^, ... an being given quantities, real, iinagiiiary, or com- plex, there exists a second value of z, of the form x -f- y', for which the polynomial has another value w such that tsr w < tsr w^. The following demonstration of this theorem is a modifi- cation of Argand's original proof.* Let Zq be the given value of 5- and let w^ be the resulting value of the polynomial, different from zero, so that 1^^ = a^-\- a^z -^ a^z" -{- . . . . + ^,, z". Add to Zq an arbitrary complex increment z, whose tensor and amplitude are disposable at pleasure, and let w be the resulting value of the polynomial, so that 7£/ = ^, + ^,(0^ + s)+«,(0^ + 0)=+ . . . +^»G-o + ^)"- If the several powers of the binominal z^ -j- z, in this equation be expanded by actual multiplication, or by the *Argand: Essai sur nne manicre de reprcsenter les quantites imaginaires dans les constructions geometriques (Paris, 1806;, Art. 31. The demonstration was reproduced by Cauchy, in \.he Journal de V Ecole Royale Pidytechnique (1S20), Vol. XI, pp. 411-416, in his Analyse algebrique (1821), ch. X, and again in his Exercises d^ analyse et de physique mathnnatique. Vol. IV, pp. 167-170. 134 PROPERTIES OF POLYNOMIALS. binominal theorem, and its terms be then arranged accord- ing to the ascending powers of z, it may be written in the form 7^' = ^o + ^i^o + ^3^!) + ^3^o-r . • . ^a^zl in which b^, <^,., , . . bn_^ involve z^ but not z. By hypothesis a„ is not zero, but any or all of the coefficients b^, b^, . . . b„_^ may possibly vanish. Let b„i be the first that does not vanish, b,i standing for the same thing as a,i, so that 7U = 7C'^-^ b„,Z"^ -f {b„,+ , -}- K + 22^ + b„Z"-"^-^) Z"^+\ and let b„i = a cis a, bm-ri^bm+2Z-\- . . . +-^.,2"-'«-^=^cis/3, and s = r cis 6, the quantities a, a, b, /3, r and B being real. Then w = 7i'^ -[- ar"' cis (a -f ;;/ 0) -f ^/-"'+^ cis ) /3 -r (^'^ — i ) ^ J • Since the length of any side of a closed polygon cannot be greater than the sum of the lengths of all the other sides, or in other words, since the tensor of a sum cannot be greater that the sum of the tensors of the several terms (Art. 58), . • . b< tsr/^„,+i -I- r tsr b„,^^ -f . . . -f ^«-;;^-I tsr ^„. Hence, by diminishing r sufficiently b may be made to differ from tsr^,„+i by an arbitrarily small quantity, and a maximum limit to the variation of r may be assigned such that b shall not exceed a fixed finite value. It follows that r may be taken so small that br<^a, or br'"^'' • • • > y« of 2-, and for no others. 121. Agenda. Prove the following theorems con- cerning polynomials : ( I ). Every polynomial in x -f iy can be reduced to the ioxmX-YiY. (2). If/(jir4-t;0 be a polynomial in x -^ iy having all its coefficients real, and if f{x-\-7y) = X-^iY, then f{^x-iy')=^X—iY. (3). If all the coefficients of the polynomial y(-3') be real, and if f{a^ib)-=o, then f(^a — id) = 0. (4). In an algebraic equation having real coefficients imaginary roots occur in pairs. APPENDIX SOME AMPLIFICATIONS. Art. 23, page 40. The notation b^ does not here presuppose any knowledge of the fact, easily proved as indicated in Art. 38, that by has the usual arithmetical meaning when b andj/ are numbers. Art. 24, page 42. The following systematic arrangement of the steps of the proof of the first proposition of page 42 will aid the student to a clearer apprehension of it. First System. Second System. base = b, base = c, modulus = m, modulus = km, y = \og„i X, ky = \ogkm X, by = X, c^y = X, x^=b when_y =1, x=^c when j/ = ijk, .-. b^'k=,c and c^=^b. Art. 27, pages 43-44. The following alternative proof of the law of indices, though longer than that given on pages 43-44, has the merit of greater explicitness : 140 APPENDIX. Let the straight line SP, while remaining transversal to the two intersecting straight lines OP, OS, move parallel to a fixed direction with a speed proportional to its perpendicular distance from O. Since OP and OS are proportional to this dis- tance, P and 6^ move also with speeds respectively proportional to to their distances from O. Let Q move along a straight line through O with a constant speed /x, and let the following sets of values corre- spond respectively, as designated by the accents : OP, 0P\ OP''=x, x\ x'\ OS, OS', OS'' = s, s\ s'\ 00 OQ' 0Q''=y, y, y\ Finally, let OS' and OJ be each equal to the linear unit and let speed of P at /= A.. If t represent the time it takes SP to pass to the position S''P'' , then designating ratio of distance traversed to time- interval as average speed (abbreviated to av. sp.), we have av. sp. of 6*= ( /' — s) / f, av. sp. ofP=lx''-x)/t, and (av. sp. of S) I (av. sp. of P) = (s''~ s) / (x''- x). But s''/s = x'\/x, that is, (/'- s)/s=(x''- x)/x, or (s''— s) I (x''— x)^=sj X, av. sp. of 6' av. sp. of P APPENDIX. 141 This proportion remains, however small the interval be, and the limit of average speed, as the interval approaches zero, is the actual speed at the beginning of the interval. It follows that speed of S s speed of S speed of P x A .r whence speed of S= \s, and speed of S at unit's distance from O is A.. But by the definition of an exponential (Art. 23), since the positions 0\ Q" correspond respectivelv both to P\ P^' and t^ S\ S'\ x'= exp„,y, x''=exp,ny', and, since 0S^= i, s''=exp,„(y'—y), where m = fx / A. But again, since s^= i, . • . exp„,y' j exp„,y'= exp,,, ( j'' — J'')- END OF THE UNIPLANAR ALGEBRA. ■?^ •^k>^' '^B w^^ %SAJ^'