THE NUMBER-SYSTEM OF ALGEBRA TREATED THEORETICALLY AND HISTORICALLY BY HENRY B. FINE, Ph.D. PROFESSOR OP MATHEMATICS IN PRINCETON UNIVERSITY l3olS SECOND EDITION, WITH CORRECTIONS BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 4 4 V S jr V<\03 Mi^ 1905 COPTBIQHT, 1890, bt henry B. FmE. mdtiiematical Sciences Library 5. ^f ■ / PREFACE, The theoretical part of this little book is an elementary exposition of the nature of the number concept, of the posi- tive integer, and of the four artificial forms of number which, with the positive integer, constitute the "number- system " of algebra, viz. the negative, the fraction, the irra- tional, and the imaginary. The discussion of the artificial numbers follows, in general, the same lines as my pam- phlet : On the Forms of Number arising in Common Algebra, but it is much more exhaustive and thorough- going. The point of view is the one first suggested by Peacock and Gregory, and accepted by mathematicians gen- erally since the discovery of quaternions and the Ausdeh- nungslehre of Grassmann, that algebra is completely defined formally by the laws of combination to which its funda- mental operations are subject; that, speaking generally, these laws alone define the operations, and the operations the various artificial numbers, as their formal or symbolic results. This doctrine was fully developed for the negative, the fraction, and the imaginary by Hankel, in his Complexe Zahlensystemen, in 1867, and made complete by Cantor's beautiful theory of the irrational in 1871, but it has not as yet received adequate treatment in English. Any large degree of originality in work of this kind is naturally out of the question. I have borrowed from a iii iv PREFACE. great many sources, especially from Peacock, Grassmann, Hankel, Weierstrass, Cantor, and Thomae {Theorie der analytischen Functionen einer complexen Verdnderlichen). I may mention, however, as more or less distinctive features of my discussion, the treatment of number, counting (§ § 1-5), and the eqviation (§§ 4, 12), and the prominence given the laws of the determinateness of subtraction and division. Much care and labor have been expended on the his- torical chapters of the book. These were meant at the out- set to contain only a brief account of the origin and history of the artificial numbers. But I could not bring myself to ignore primitive counting and the development of numeral notation, and I soon found that a clear and connected account of the origin of the negative and imaginary is possible only when embodied in a sketch of the early his- tory of the equation. I have thus been led to write a resume of the history of the most important parts of ele- mentary arithmetic and algebra. Moritz Cantor's Vorlesungen iiher die Geschichte der Mathe- matik, Vol. I, has been my principal authority for the entire period which it covers, i.e. to 1200 a.d. For the little I have to say on the period 1200 to 1600, I have depended chiefly, though by no means absolutely, on Hankel : Zur Geschichte der Mathematik in Altertum und Mittelalter. The remainder of my sketch is for the most part based on the original sources. HENRY B. FINE. Princeton, April, 1891. In this second edition a number of important corrections have been made. But there has been no attempt at a complete revision of the book. HENRY B. EINE. Princeton, September, 1902. CONTENTS. I. THEORETICAL. I. The Positive Integer. PA6K The number concept 3 Numerical equality 3 Numeral symbols 4 The numerical equation 5 Counting 5 Addition and its laws 6 Multiplication and its laws 7 II. SUBTEACTION AND THE NEGATIVE INTEGER. Numerical subtraction 8 Determinateness of numerical subtraction 9 Formal rules of subtraction 9 Limitations of numerical subtraction 11 Symbolic equations 11 Principle of permanence. Symbolic subtraction 12 Zero 13 The negative 14 Recapitulation of the argument of the chapter 16 m. "Division and the Fraction. Numerical division 18 Determinateness of numerical division 18 Formal rules of division ,.' 19 Limitations of numerical division 20 Symbolic division. The fraction 21 Negative fractions 22 General test of the equality or inequality of fractions 22 Indeterminateness of division by zero •. . . . 23 V vi CONTENTS. FAGB Determinateness of symbolic division 23 The vanishing of a product 24 The system of rational numbers 25 IV. The Irrational. Inadequateness of the system of rational numbers 26 Numbers defined by " regular sequences." The irrational 27 Generalized definitions of zero, positive, negative 29 Of the four fundamental operations 29 Of equality and greater and lesser inequality 31 The number defined by a regular sequence its limiting value 31 Division by zero 33 The number-system defined by regular sequences of rationals a closed and continuous system 34 V. The Imaginary. Complex Numbers. The pure imaginary 35 Complex numbers 36 The fundamental operations on complex numbers 37 Numerical comparison of complex numbers 38 Adequateness of the system of complex numbers 39 Fundamental characteristics of the algebra of number 39 VI. Graphical Representation of Numbers. The Variable. Correspondence between the real number-system and the points of a line 41 The continuous variable 43 The complex variable 45 Correspondence between the complex number-system and the points of a plane 44 Definitions of modulus and argument of a complex number and of sine, cosine, and circular measure of an angle 45, 48 Demonstration that a -\- ib = p (cos 6-\- i sin S) = pe«9 45, 48 Construction of the points which represent the sum, difference, product, and quotient of two complex numbers 46, 47 Vll. The Fundamental Theorem of Algebra. Definitions of the algebraic equation and its roots 50 Demonstration that an algebraic equation of the nth degree has n roots 51, 53 CONTENTS. VU VIII. Numbers Defined by Infinite Series. PAGE I. Keal Series. Definitions of sum, convergence, and divergence 54 General test of convergence 55 Absolute and conditional convergence 55,57 Special tests of convergence 57 Limits of convergence 59 The fundamental operations on infinite series 61 II. Complex Series. General test of convergence 62 Absolute and conditional convergence 62 The region of convergence 63 A theorem respecting complex series 64 The fundamental operations on complex series 65 IX. The Exponential and Logarithmic Functions. undetermined coefficients, involution and evolution, the binomial theorem. Definition of function 66 Functional equation of the exponential fimction 66 Undetermined coefficients 67 The exponential function 68 The functions sine and cosine • 71 Periodicity of these functions 72 The logarithmic function 72 Indeterminateness of logarithms 75 Permanence of the laws of exponents 76 Permanence of the laws of logarithms 77 Involution and evolution 77 The binomial theorem for complex exponents 77 II. HISTORICAL. I. Primitive Numerals. Gesture symbols 81 Spoken symbols 82 Written symbols 84 VIU CONTENTS. II. Historic Systems of Notation. PAGE Egyptian and Phcenician 84 Greek 84 Koman 85 Indo- Arabic 86 III. The Fraction. Primitive fractions 90 Eoman fractions 91 Egyptian (the Booli of Ahmes) 91 Babylonian or sexagesimal 92 Greek 93 rv. Origin of the Irrational. Discovery of irrational lines. Pythagoras 94 Consequences of this discovery in Greek mathematics 96 Greek approximate values of irrationals 98 V. Origin of the Negative and the Imaginary. the equation. The equation in Egjrptian mathematics 99 In the earlier Greek mathematics 99 Hero of Alexandria 100 Diophantus of Alexandria 101 The Indian mathematics. Aryabhatta, Brahmagupta, Bhaskara 103 Its algebraic symbolism 104 Its invention of the negative 105 Its use of zero 105 Its use of irrational numbers 106 Its treatment of determinate and indeterminate equations 106 The Arabian mathematics. Alkhwarizmi, Alkarchi, Alchayyami 107 Arabian algebra Greek rather than Indian 110 Mathematics in Europe before the twelfth century Ill Gerbert Ill Entrance of the Arabian mathematics. Leonardo 112 Mathematics during the age of Scholasticism 113 The Renaissance. Solution of the cubic and biquadratic equations 114 The negative in the algebra of this period. First appearance of the imaginary 115 Algebraic symbolism. Vieta and Harriot 116 The fundamental theorem of algebra. Harriot and Girard 117 CONTENTS. ix VI. Acceptance of the Negative, the General Irrational, AND the Imaginary as Numbers. PAGE Descartes' Geometrie and the negative 118 Descartes' geometric algebra 119 The continuous variable. Newton. Euler 121 The general irrational 121 The imaginary, a recognized analytical instrument 122 Argand's geometric representation of the imaginary 122 Gauss. The complex number 123 Vll. Recognition of the Purely Symbolic Character of Algebra. quaternions, the ausdehnungslehre. The principle of permanence. Peacock 124 The fundamental laws of algebra. " Symbolical algebras." Gregory 126 Hamilton's quaternions 128 Grassmann's Ausdehnungslehre 129 The fully developed doctrine of the artificial forms of number. Hankel. Weierstrass. G. Cantor 130 Recent literature 131 PRINCIPAL FOOTNOTES. Instances of quinary and vigesimal systems of notation 82 Instances of digit numerals 83 Summary of the history of Greek mathematics 95 Old Greek demonstration that the side and diagonal of a square are incommensurable 96 Greek methods of approximation 98 Diophantine equations 102 Alchayyami's method of solving cubics by the intersections of conies 109 Jordanus Nemorarius 113 The Summa of Luca Pacioli 113 Regiomontanus 114 Algebraic symbolism 113, 116 The irrationality of e and tt. Lindemann 70 I. THEOEETIOAL. I. THE POSITIVE INTEGER, AND THE LAWS WHICH REGULATE THE ADDITION AND MULTIPLICATION OF POSITIVE INTEGERS. 1. Number. We say of certain distinct things that they form a group* when we make them collectively a single object of our attention. The number of things in a group is that property of the group which remains unchanged during every change in the group which does not destroy the separateness of the things from one another or their common separateness from all other things. Such changes may be changes in the characteristics of the things or in their arrangement within the group. Again, changes of arrangement may be changes either in the order of the things or in the manner in which they are associated with one another in smaller groups. We may therefore say : The number of things in any group of distinct things is indejyendent of the characters of these things, of the order in which they may be arranged in the group, and of the manner in which they may be associated with one another in smaller groiq^s. 2. Numerical Equality. The number of tilings in any two groups of distinct things is the same, when for each thing in the first group there is one in the second, and reciprocally, for each thing in the second group, one in the first. Thus, the number of letters in the two groups, A, B, C; . D, E, F, is the same. In the second group there is a letter * By group we mean finite group, that is, one which cannot be brought into one-to-one correspondence (§ 2) with any part of itself. 4 NUMBER-SYSTEM OF ALGEBRA. which may be assigned to each of the letters in the first : as DtoA,EtoB,FtoC; and reciprocally, a letter in the first which may be assigned to each in the second : as J. to D, Bio E, C to F. Two groups thus related are said to be in one-to-one (1-1) correspoyidence. Underlying the statement just made is the assumption that if the two groups correspond in the manner described for one order of the things in each, they will correspond if the things be taken in any other order also; thus, in the example given, that if E instead of D be assigned to A, there will again be a letter in the group D, E, F, viz. D or F, for each of the remaining letters B and C, and recipro- cally. This is an immediate consequence of § 1, foot-note. The number of things in the first group is greater than that in the second, or the number of things in the second less than that in the first, when there is one thing in the first group for each thing in the second, but not reciprocally one in the second for each in the first. 3. Numeral Symbols. As regards the number of things which it contains, therefore, a group may be represented by any other group, e.g. of the fingers or of simple marks, I's, which stands to it in the relation of correspondence described in § 2. This is the primitive method of repre- senting the number of things in a group and, like the modern method, makes it possible to compare numerically groups which are separated in time or space. The modern method of representing the number of things in a group differs from the primitive only in the substitu- tion of symbols, as 1, 2, 3, etc., or numeral words, as one, tivo, three, etc., for the various groups of marks I, II, ill, etc. These symbols are the positive integers of arith- metic. A positive integer is a symbol for the number of things in a group of distinct things. THE POSITIVE INTEGER. 5 For convenience we shall call the positive integer whicli represents the number of things in any group its numeral symbol, or when not likely to cause confusion, its number simply, — this being, in fact, the primary use of the word " number " in arithmetic. In the following discussion, for the sake of giving our statements a general form, we shall represent these numeral symbols by letters, a, b, c, etc. 4. The Equation. The iiumeral symbols of two groups being a and b ; when the number of things in the groups is the same, this relation is expressed by the equation a = b; when the first group is greater than the second, by the inequality a>b; when the first group is less than the second, by the ine- quality a or < 6 + c, { a > or < 6. 10. Formal Rules of Subtraction. All the rules of sub- traction are purely formal consequences of the fundamental 10 NUMBER-SYSTEM OF ALGEBRA. laws I-V, VII, and definition VI. They must follow, what- ever the meaning of the symbols a, b, c, -{-, — , = ; a fact which has an important bearing on the following discussion. It will be sufficient to consider the equations which fol- low. For, properly combined, they determine the result of any series of subtractions or of any complex operation made up of additions, subtractions, and multiplications. 1. a — (6 -hc^t= a — 6 — c= a — c — 6. 2. a — {b — c) = a — b + c. 3. a + b—b = a. 4. a-j-(6 — c) = a + & — c = a — c-f&. 5. a(b — c) = ab — ac. For 1. a — & — c is the form to which if first c and then b be added; or, what is the same thing (by I), first b and then c ; or, what is again the same thing (by II), b-\-G at once, — the sum pro- duced is a (by VI). a — b — c is therefore the same as a — c — b, which is as it stands the form to which if b, then c, be added the sum is a; also the same as a— (b + c), which is the form to which if & -j- c be added the sum is a. 2. a — (& — c) = a — (& — c) — c -f c, Def. VI. = a — (6 — c + c) -j- c, Eq. 1. = a — & -f c. Def. VI. 3. But a + b = a-\-b. Def. VI. .: a-}-b — b = a. Law VII. 4. a + b — c—a-]-(b—c + c) — c, Def. VI. = a-f-(& — c). Law II, Eq. 3. 6. a6 — ac = a (6 — c -f- c) — ac, Def. VI. = a{b — c) + ac — ac, LawV. = a{b—c). Eq. 3. SUBTRACTION. 11 Equation 3 is particularly interesting in that it defines addition as the inverse of subtraction. Equation 1 declares that two consecutive subtractions may change places, are commutative. Equations 1, 2, 4 together supplement law II, constituting with it a complete associative law of addi- tion and subtraction ; and equation 5 in like manner supple- ments law V. 11. Limitations of Numerical Subtraction. Judged by the equations 1-5, subtraction is the exact counterpart of addition. It conforms to the same general laws as that operation, and the two could with fairness be made to interchange their r6les of direct and inverse operation. But this equality proves to be only apparent when we attempt to interpret these equations. The requirement that subtrahend be less than minuend then becomes a serious restriction. It makes the range of subtraction much narrower than that of addition. It renders the equations 1-5 available for special classes of values of a, b, c only. If it must be insisted on, even so simple an inference as that a — (a + b)+2b is equal to b cannot be drawn, and the use of subtraction in any reckoning with symbols whose relative values are not at all times known must be pro- nounced unwarranted. One is thus naturally led to ask whether to be valid an algebraic reckoning must be interpretable numerically and, if not, to seek to free subtraction and the rules of reckoning with the results of subtraction from a restriction which we have found to be so serious. 12. Symbolic Equations. Principle of Permanence. Sym- bolic Subtraction. In pursuance of this inquiry one turns first to the equation (a — 6) + 6 = a, which serves as a definition of subtraction when b is less than a. This is an equation in the primary sense (§ 4) only when a — 6 is a number. But in the broader sense, that 12 NUMBER-SYSTEM OF ALGEBRA. An equation is any declaration of the equivalence of definite combinatioyis of symbols — equivalence in the sense that one may be substituted for the other, — (a — 6) + 6 = a may be an equation, whatever the values of a and b. And if no different meaning has been attached to a — &, and it is declared that a — b is the symbol which associated with b in the combination (a— 6) + 6 is equivalent to a, this declaration, or the equation (a — 5) + 6 = a, is a definition * of this symbol. By the assumption of the permanence of form of the numerical equation in which the definition of subtraction resulted, one is thus put immediately in possession of a symbolic definition of subtraction which is general. The numerical definition is subordinate to the symbolic definition, being the interpretation of which it admits when b is less than a. But from the standpoint of the symbolic definition, inter- pretability — the question whether a — 6 is a number or not — is irrelevant; only such properties may be attached to a — &, by itself considered, as flow immediately from the generalized equation (a — b)-\-b = a. In like manner each of the fundamental laws I-V, VII, on the assumption of the permanence of its form after it has ceased to be interpretable numerically, becomes a declaration of the equivalence of certain definite combi- nations of symbols, and the formal consequences of these laws — the equations 1-5 of § 10 — become definitions of addition, subtraction, multiplication, and their mutual * A definition in terms of symbolic, not numerical addition. The sign + can, of course, indicate numerical addition only when both the symbols which it connects are numbers. SUBTRACTION. 13 relations — definitions wMch are purely symbolic, it may be, but which are unrestricted in their application. These definitions are legitimate from a logical point of view. For they are merely the laws I- VII, and we may assume that these laws are mutually consistent since we have proved that they hold good for positive integers. Hence, if used correctly, there is no more possibility of their leading to false results than there is of the more tangible numerical definitions leading to false results. The laws of correct thinking are as applicable to mere symbols as to numbers. What the value of these symbolic definitions is, to what extent they add to the power to draw inferences concerning numbers, the elementary algebra abundantly illustrates. One of their immediate consequences is the introduction into algebra of two new symbols, zero and the negative, which contribute greatly to increase the simplicity, compre- hensiveness, and power of its operations. 13. Zero. Wlien h is set equal to a in the general equation {a — h)-\-b = a, it takes one of the forms {a — a) + a=a, {h -h) + h = h. It may be proved that a— a=:h — h. For (a — a) -\\a + 6) = (a — a) + a + 6, Law II. = a + 6, since (a — a) -f a = a. And (&- 6) 4- (a + 5) = (6 -&)+&+«, Laws I, II. since (6 —&) + & = 6. Therefore a — a = h — h. Law VII. 14 - NUMBER-SYSTEM OF ALGEBRA. a — a is therefore altogether independent of a and may properly be represented by a symbol unrelated to a. The symbol which has been chosen for it is 0, called zero. Addition is defined for this symbol by the equations 1. + a = a, definition of 0. a -\-0 = a. Law I. Subtraction (partially), by the equation 2. a-0 = a. For (a - 0) + = a. Def . VI. Multiplication (partially), by the equations 3. a X = X a = 0. For a X = a(6 — 6), definition of 0. = ab — ah, § 10, 5. = 0. definition of 0. 14. The Negative. When h is greater than a, equal say to a + d, so that b — a = d, then a — b = a — (a-\- d), = a-a-d, § 10, 1. = — d. definition of 0. For — d the briefer symbol — d has been substituted ; with propriety, certainly, in view of the lack of significance of in relation to addition and subtraction. The equation — d = — d, moreover, supplies the missing rule of sub- traction for 0. (Compare § 13, 2.) The symbol — d is called the negative, and in opposition to it, the number d is called positive. Though in its origin a sign of operation (subtraction from 0), the sign — is here to be regarded merely as part of the symbol — d. — d is as serviceable a substitute for a — b when a<,b, as is a single numeral symbol when a > 6. SUBTRACTION. 15 The rules for reckoning with the new symbol — definitions of its addition, subtraction, multiplication — are readily deduced from the laws I-V, VII, deiinition VI, and the equations 1-5 of § 10, as follows : 1. b + (-b) = -b + b = 0. For — & + 6 = (0 — &) + 6, definition of negative. = 0. Def. VI. — 6 may therefore be defined as the symbol the sum of which and b is 0. 2. a -\- {— b) = — b -\- a = a — b. For a + ( — '6) = a + (0 — 6), definition of negative. = a + - 6, § 10, 4. = a-b. § 13, 1. 3. -a + {-b) = -(a + b). For — a + (—&) = — a — &, by the reasoning in § 14, 2. = 0-(a + &), §10,1. = — (a + b). definition of negative. 4. a — (—b)=ia-\-b. For a — (—&) = a — (0 — &), definition of negative. = a - + 6, § 10, 2. = a + 6. § 13, 2. 5. (-a)-(-b) = b-a. For —a — (—b) = — a-\-b, by the reasoning in § 14, 4. = b-a. § 14, 2. Cor. (-a)-(-a) = 0. 6. a( — 6) = ( — 6)ci= — a6. For = a(& - 6), § 13, 3. = a& + a( — &). Law V. .-. a(- 6) = - a&. § 14, 1 ; Law VII. 16 NUMBEB-SYSTEM OF ALGEBRA. 7. (-a)xO = Ox(-a) = 0. Tor (- a) X = ( - a) (& - b), definition of 0. = {-a)b-{-a)b, §10,5. = 0. § U, 6, and 5, Cor. 8. (-a)(-b) = ab. For = (-a)(6-6), §14,7. = (-a)6 + (-a)(-6), Law V. = _a6 + (-a)(-6). §14,6. ... ( _ a) ( - &) = a&. § 14, 1 ; Law VII. By this method one is led, also, to definitions of equality and greater or lesser inequality of negatives. Thus 9. — a >, = or < — 6, according as 6 >, = or < a.* For as &>,=,< a, -a + a + b>, =, <-b + b-ha, § 14, 1 ; § 13, 1. or —a >,=,<— 6, Law VII or VII'. In like manner — a < < &. 15. Recapitulation. The nature of the argument which has been developed in the present chapter should be care- fully observed. From the definitions of the positive integer, addition, and subtraction, the associative and commutative laws and the determinateness of subtraction followed. The assump- tion of the permanence of the result a — b, as defined by (a — 6) -+- 6 = a, for all values of a and b, led to definitions * On the other hand, — a is said to be numerically greater than, equal to, or less than — b, according as a is itself greater than, equal to, or less than b. SUBTRACTION. 17 of the two symbols 0, — d, zero and the negative ; and from the assumption of the permanence of the laws I-V, VII were derived definitions of the addition, subtraction, and multiplication of these symbols, — the assumptions being just suificient to determine the meanings of these operation^ unambiguously. j In the case of numbers, the laws I-V, VII, and definition VI were deduced from the characteristics of numbers and the definitions of their operations ; in the case of the sym- bols 0, — d, on the other hand, the characteristics of these symbols and the definitions of their operations were deduced from the laws. With the acceptance of the negative the character of arithmetic undergoes a radical change.* It was already in a sense symbolic, expressed itself in equations and inequali- ties, and investigated the results of certain operations. But its symbols, equations, and operations were all interpretable in terms of the reality which gave rise to it, the number of things in actually existing groups of things. Its connec- tion with this reality was as immediate as that of the ele- mentary geometry with actually existing space relations. But the negative severs this connection. The negative is a symbol for the result of an operation which cannot be effected with actually existing groups of things, which is, therefore, purely symbolic. And not only do the fundamen- tal operations and the symbols on which they are performed lose reality ; the equation, the fundamental judgment in all mathematical reasoning, suffers the same loss. From being a declaration that two groups of things are in one-to-one correspondence, it becomes a mere declaration regarding two combinations of symbols, that in any reckoning one may be substituted for the other. * In this connection see § 25. 18 NUMBEB-SYSTEM OF ALGEBBA. m. DIVISION AND THE lEAOTION. .16. Numerical Division. The inverse operation to multi- plication is division. To divide a by 6 is to find a number which multiplied by h produces a. The result is called the quotient of a by h, and is written — By definition VIII. (-\ = a. Like subtraction, division cannot be always effected. Only in exceptional cases can the a-group be subdivided into h equal groups. 17. Determinateness of Numerical Division. When divis- ion can be effected at all, it can lead to but a single result ; it is determinate. For there can be but one number the product of which by 6 is a ; in other words. , If c& = dh, IX. \ c = d.* { For b groups each containing c individuals cannot be equal to b groups each containing d individuals unless c = d(§4). This is a theorem of fundamental importance. It may be called the law of determinateness of division. It declares that if a product and one of its factors be determined, the remaining factor is definitely determined also; or that if one of the factors of a product changes while the other remains unchanged, the product changes. It alone makes division in the arithmetical sense possible. The fact that * The case ft = is excluded, not being a number in the sense in which that word is here used. DIVISION AND THE FRACTION. 19 it does not hold for the symbol 0, but that rather a product remains unchanged (being always 0) when one of its factors is 0, however the other factor be changed, makes division by impossible, rendering unjustifiable the conclusions which can be drawn in the case of other divisors. The reasoning which proved law IX proves also that j As c6> or or < d, 18. Formal Rules of Division. The fundamental laws of the multiplication of numbers are III. ah = ha, IV. a {he) = ahc, V. a{h + c) =ah + ac. Of these, the definition VIII. (^ b = a, the theorem If ac = be, a = b, unless c = 0, IX. I and the corresponding laws of addition and subtraction, the rules of division are purely /ormaZ consequences, dedu- cible precisely as the rules of subtraction 1-5 of § 10 in the preceding chapter. They follow without regard to the meaning of the symbols a, b, c, =, +, — , ab, -• Thus : ^ a c _ac h d hd For ^.^.hd = -b- ^d, Laws IV, III. h d b d = ac, Def . VIII. and —■bd = ac. Def. VIII. bd 20 NUMBEB-STSTEM OF ALGEBBA. The theorem follows by law IX. ^h) _ ad c\ he a .dj For ^ • 1 = ?. I)ef. VIII. and adc^adc^ § 18, 1 ; Law IV. be d b cd _a ~&' since —cd = dc = lxcd. Def. VIII, Law IX. cd The theorem follows by law IX. o a , c ad ± be o. — ± — = • b d bd For /^- ± -V(^ = -6 • (^ ± -c^ • &, I^aws III-V; § 10, 5. \b dj. b d = ad± be, Def. VIII. and /ad ± bc \^ ^ad± be. Def. VIII. \ bd J The theorem follows by law IX. By the same method it may be inferred that as ad>, =,< be. Def. VIII, Laws III, IV, IX, IX'. 19. Limitations of Numerical Division. Symbolic Division. The Fraction. General as is thk form of the preceding equations, they are capable of numerical interpretation only when -, - are numbers, a case of comparatively rare occur- 6 d rence. The narrow limits set the quotient in the numer- ical definition render division an unimportant operation as DIVISION AND THE FB ACTION. 21 compared with, addition, multiplication, or the generalized subtraction discussed in the preceding chapter. But the way which led to an unrestricted subtraction lies open also to the removal of this restriction ; and the reasons for following it there are even more cogent here. We accept as the quatient of a divided by any number b, which is not 0, the symbol - defined by the equation b regarding this equation merely as a declaration of the equivalence of the symbols (-]b and a, of the right to sub- stitute one for the other in any reckoning. Whether - be a number or not is to this definition irrele- b a vant. When a mere symbol, r is called a fraction,, and in opposition to this a number is called an integer. We then put ourselves in immediate possession of defi- nitions of the addition, subtraction, multiplication, and division of this symbol, as Avell as of the relations of equal- ity and greater and lesser inequality — definitions which are consistent with the corresponding numerical definitions and with one another — by assuming the permanence of form of the equations 1, 2, 3 and of the test 4 of § 18 as symbolic statements, when they cease to be interpretable as numerical statements. The purely symbolic character of - and its operations 6 detracts nothing from their legitimacy, and they establish division on a footing of at least formal equality with the other three fundamental operations of arithmetic* * The doctrine of symbolic division admits of being presented in the very same form as tliat of symbolic subtraction. The equations of Chapter II immediately pass over into theorems 22 NUMBER-SYSTEM OF ALGEBRA. 20. Negative Fractions. Inasmuch as negatives conform to the laws and definitions I-IX, the equations 1, 2, 3 and the test 4 of § 18 are valid when any of the numbers a, b, c, d are replaced by negatives. In particular, it follows from the definition of quotient and its determinateness, that a a — a a — a _a -b~ 6' b 6' -b b It ought, perhaps, to be said that the determinatenesrs of division of negatives has not been formally demonstrated. The theorem, however, that if (± a) (± c) = (± 6)(± c), ± a = ± 6, follows for every selection of the signs ± from the one selection +, +, +, + by § 14, 6, 8. 21. General Test of the Equality or Inequality of Fractions. ft c Given any two fractions ± -, ± — b d ±«>,=or<±^, according as ± ad >, = or < ± 6c. Laws IX, IX'. Compare § 4, § 14, 9. respecting division when the signs of multiplication and division are substituted for those of addition and subtraction ; so, for instance, a — (b+c) = a — b — c = a — c~b . (!) (") In particular, to (a — a) + « = a corresponds -a = a. Thus a purely symbolic definition may be given 1. It plays the same role in multipli- cation as in addition. Again, it has the same exceptional character in involution — an operation related to multiplication quite as multipli- cation to addition — as in multipUcation ; for 1-" = 1", whatever the values of m and n. Similarly, to the equation (- a) + a = 0, or (0-a) + a = 0, corre- sponds (-\a = l, which answers as a definition of the unit fraction - ; and in terms of these unit fractions and integers all other fractions a may be expressed. DIVISION AND THE FRACTION. 23 22. Indeterminateness of Division by Zero. Division by does not conform to the law of determinateness ; the equa- tions 1, 2, 3 and the test 4 of § 18 are, therefore, not valid when is one of the divisors. a The symbols -, , of which some use is made in mathe- matics, are indeterminate.* 1. - is indeterminate. For - is completely defined by the equation f - j = ; but since a; x = 0, whatever the value of X, any number whatsoever will satisfy this, equation. 2. ^ is indeterminate. For, by definition, f - ) = a. \pj Were - determinate, therefore, — since then ( - ) would, a X ^^ by § 18, 1, be equal to , or to -, — the number a . • would be equal to -, or indeterminate. Division by is 7iot an admissible operation. 23. Determinateness of Symbolic Division. This excep- tion to the determinateness of division may seem to raise an objection to the legitimacy of assuming — as is done when the demonstrations 1-4 of § 18 are made to apply to symbolic quotients — that symbolic division is determinate. It must be observed, however, that -, - are indetermi- nate in the numerical sense, whereas by the determinateness of symbolic division is, of course, not meant actual numerical determinateness, but " symbolic determinateness," conform- ity to law IX, taken merely as a symbolic statement. For, as has been already frequently said, from the present stand- point the fraction - is a mere symbol, altogether without b /• X numerical meaning apart from the equation (-\b = a, with * In this connection see § 32. 24 NUMBER-SYSTEM OF ALGEBRA. which, therefore, the property of numerical determinateness has no possible connection. The same is true of the prod- uct, sum or difference of two fractions, and of the quotient of one fraction by another. As for symbolic determinateness, it needs no justification when assumed, as in the case of the fraction and the demonstrations 1-4, of symbols whose definitions do not preclude it. The inference, for instance, that because b dj [bdj ' a c ac b d bd which depends on this principle of symbolic determinate- ness, is of precisely the same character as the inference that b dJ b d ' which depends on the associative and commutative laws. Both are pure assumptions made of the undefined symbol (1 c for the sake of securing it a definition identical in form b d ^ with that of the product of two numerical quotients.* 24. The Vanishing of a Product. It has already been shown (§ 13, 3, § 14, 7, § 18, 1) that the sufficient condition for the vanishing of a product is the vanishing of one of its factors. From the determinateness of division it follows that this is also the necessary condition, that is to say : If a product vanish, one of its factors must vanish. Let xy = 0, where x, y may represent numbers or any of the symbols we have been considering. * These remarks, mutatis mutandis, apply with equal force to sub- traction. DIVISION AND THE FRACTION. 25 Since xy = 0, xy -\-xz = xz, § 13, 1. or x{y -\-z)= xz, Law V. whence, if x be not 0, y -{-z = z, Law IX. or 2/ = 0. Law VII. 25. The System of Rational Numbers. Three symbols, 0, _ d, -, have thus been found which can be reckoned with by the same rules as numbers, and in terms of which it is possible to express the result of every addition, sub- traction, multiplication or division, whether performed on numbers or on these symbols themselves ; therefore, also, the result of any complex operation which can be resolved into a finite combination of these four operations. Inasmuch as these symbols play the same r6le as numbers in relation to the fundamental operations of arithmetic, it is natural to class them with numbers. The word " number," originally applicable to the positive integer only, has come to apply to zero, the negative integer, the positive and nega- tive fraction also, this entire group of symbols being called the system of ratioyial numbers* This involves, of course, a radical change of the number concept, in consequence of which numbers become merely part of the symbolic equip- ment of certain operations, admitting, for the most part, of only such definitions as these operations lend them. * It hardly need be said that the fraction, zero, and the negative actually made their way into the number- system for quite a different reason from this ; — because they admitted of certain " real " mterpre- tations, the fraction in measurements of lines, the negative in debit where the corresponding positive meant credit or in a length measured to the left where the corresponding positive meant a length measured to the right. Such interpretations, or correspondences to existing things which lie entirely outside of pure arithmetic, are ignored in the present discussion as being irrelevant to a pure arithmetical doctrine of the artificial forms of number. 26 NUMBER-SYSTEM OF ALGEBRA. In accepting these symbols as its numbers, arithmetic ceases to be occupied exclusively or even principally , Avith the properties of numbers in the strict sense. It becomes an algebra whose immediate concern is with certain opera- tions defined, as addition by the equations a -\- b = b -{- a, a + (^ + c) = a -f 6 + c, formally only, without reference to the meaning of the symbols operated on.* IV. THE lERATIORAL. 26. The System of Rational Numbers Inadequate. The system of rational numbers, while it suffices for the four fun- damental operations of arithmetic and finite combinations of these operations, does not fully meet the needs of algebra. The great central problem of algebra is the equation, and that only is an adequate number-system for algebra which supplies the means of expressing the roots of all possible ' equations. The system of rational numbers, however, is equal to the requirements of equations of the first degree only ; it contains symbols not even for the roots of such elementary equations of higher degrees as a^ = 2, x^ = — 1. But how is the system of rational numbers to be enlarged into an algebraic system which shall be adequate and at the same time sufficiently simple ? The roots of the equation aj" +^iic"-^ +2hx''~^-i \-Pn-iX +^„ = * The word " algebra" is here used in the general sense, the sense in which quaternions and the Ausdehungslehre (see §§ 127, 128) are alge- bras. Inasmuch as elementary arithmetic, as actually constituted, accepts the fraction, there is no essential difference between it and elementary algebra with respect to the kinds of number with which it deals ; algebra merely goes further in the use of artificial numbers. The elementary algebra differs fi-om arithmetic in employing literal symbols for numbers, but chiefly in making the equation an object of investigation. THE IRRATIONAL. 2T are not the results of single elementary operations, as are the negative of subtraction and the fraction of division; for though the roots of the quadratic are results of " evolution," and the same operation often enough repeated yields the roots of the cubic and biquadratic also, it fails to yield the roots of higher equations. A system built up as the rational system was built, by accepting indiscriminately every new symbol which could show cause for recognition, would, therefore, fall in pieces of its own weight. The most general characteristics of the roots must be discovered and defined and embodied in symbols — by a method which does not depend on processes for solving equations. These symbols, of course, however character- ized otherwise, must stand in consistent relations with the system of rational numbers and their operations. An investigation shows that the forms of number neces- sary to complete the algebraic system may be reduced to two: the symbol V— 1, called the imaginary (an indicated root of the equation a;- -f- 1 = 0), and the class of symbols called irrational, to which the roots of the equation a^— 2=0 belong. 27. Numbers Defined by Regular Sequences. The Irra- tional. On applying to 2 the ordinary method for extracting the square root of a number, there is obtained the follow- ing sequence of numbers, the results of carrying the reck- oning out to 0, 1, 2, 3, 4, . . . places of decimals, viz. : 1, 1.4, 1.41, 1.414, 1.4142, ... These numbers are rational ; the first of them differs from each that follows it by less than 1, the second by less than — , the third by less than , ••• the wth by less than 10 ' 100 '' 10"-^ And is a fraction which may be made less than any assignable number whatsoever by taking n great enough. 28 NUMBER-SYSTEM OF ALGEBRA. This sequence may be regarded as a definition of the square root of 2. It is such in the sense that a term may be found in it the square of which, as well as of each fol- lowing term, diifers from 2 by less than any assignable number. Any sequence of rational numbers «•!> 0-2} 0.51 '•' tt/x? «M+1> ■•• °-IJ.+v, •" in which, as in the above sequence, the term a^^ may, by talc- ing fx great enough, be made to differ numerically from each term that folloivs it by less than any assignable number, so that, for all values of v, the difference, a^^+v — o-fi, is numerically less than S, however small 8 be taken, is called a regxdar sequence. The entire class of operations which lead to regular sequences may be called regular sequence-building. Evolution is only one of many operations belonging to this class. Any regular sequence is said to " define a number,''^ — this " number " being merely the symbolic, ideal, result of the operation which led to the sequence. It will sometimes be convenient to represent numbers thus defined by the single letters a, b, c, etc., which have heretofore represented posi- tive integers only. After some particular term all terms of the sequence a,, og, ••• may be the same, say a. The number defined by the sequence is then a itself. A place is thus provided for rational numbers in the general scheme of numbers which the definition contemplates. When not a rational, the number defined by a regular [ sequence is called irrational. The regular sequence .3, .33, •••, has a limiting value, viz., - ; which is to say that a term can be found in this sequence which itself, as well as each term which follows it, differs from - by less than any assignable number. In other words, THE IRRATIONAL. ■ 29 the difference between - and the ynth term of the sequence o may be made less than any assignable number whatsoever by taking /a great enough. It will be shown presently that the number defined by any regular sequence, aj, a2, ••• stands in this same relation to its term a^. 28. Zero, Positive, Negative. In any regular sequence aj, a2, •■• a term a^ may always be found which itself, as well as each term which follows it, is either (1) numerically less than any assignable number, or (2) greater than some definite positive rational number, or (3) less than some definite negative rational number. In the first case the number a, which the sequence de- fines, is said to be zero, in the second positive, in the third negative. 29. The Four Fundamental Operations. Of the numbers defined by the two sequences: o-ij o-iy 0-3J ••• tt/oLj o-fjL + i, ••• o-ix + v) •••, Pl> H2y Ps) • • • Pmj Hy- + h ■•• PiJ- + v^ ••• (1) Tlie sum is the number defined by the sequence : ■ "l + fil, a2 + ^2) • • • "At + filJ-J «H + 1 + ^M + Ij • • • O-M + i- + ^M + ") • • • (2) Tlie difference is the number defined by the sequence : «! — Pi, 0.0 — P2, ... a^ — p^, a.f^ + 1 — PfL + Ij • • • O-ix + v PiJ. + vy ••• (3) The jyroduct is the number defined by the sequence : tti/?!, 00^2? • • • "-ixPiJ^} tt/n + 1 P/u. + 1? • • • O-ix+vPii + V) ••• (4) TJie quotient is the number defined by the sequence : O-l 0-2 Cltt 0-ixA-X O-tx + v Pi A P^ /5ja + l Pm+v For these definitions are consistent with the correspond- ing definitions for rational numbers ; they reduce to these elementary definitions, in fact, whenever the sequences 30 NUMBER-SYSTEM OF ALGEBRA. tti, a2, ...; /8i, ^2] ••• either reduce to the forms a, a, ...; fi, (3, ... ov have rational limiting values. They conform to the fundamental laws I-IX. This is immediately obvious with respect to the commutative, asso- ciative, and distributive laws, the corresponding terms of the two sequences aj/Sj, aojSsj ••• ; fiio-i, yS^ao, ..., for instance, being identically equal, by the commutative law for ra- tionals. But again division as just defined is determinate. For division can be indeterminate only when a product may vanish without either factor vanishing (cf. § 24) ; whereas ai/3i, a2^2> ••• can define 0, or its terms after the nth fall below any assignable number whatsoever, only when the * same is true of one of the sequences ai, aj, ... ; ^i, (3^, It only remains to prove, therefore, that the sequences (1), (2), (3), (4) are qualified to define numbers (§ 27). (1) and (2) Since the sequences ai, a,, ... ; /?i, ^2: ••• ^-re, by hypothesis, such as define numbers, corresponding terms in the two, a^, /8^, may be found, such that a^a + V — tt/oi is numerically < S, and (3^ + y — jS^is numerically < 8, and, therefore, (a^ + ^,± (3^ + „) — (a^ ± ft^) < 28, for all values of v, and that however small S may be. Therefore each of the seque^ces ai-|-/8i, 02 + ^32, ••.; ai — fSi, ag — ySg, ... is regular. (3) Let a^ and ^^ be chosen as before. Then a^ + y fi^^ + y — a^^fi^., since it is identically equal to * It is worth noticing that the determinateness of division is here not an independent assumption, but a consequence of the definition of multiplication and the determinateness of the division of rationals. The same thing is true of the other fundamental laws I-V, VII. THE IRRATIONAL. 31 is numerically less than a^ + ^^ -\- ji^h, and may, therefore, be made less than any assignable number by taking 8 small enough ; and that for all values of v. Therefore the sequence aij3i, a^fi^j .-.is regular. /■<\ O-iJ. + V *^|U. O'hi. + vPiJ. Pjii. + I'^nt Pix + v PiJ. Pix. + vPfj. which is identically equal to Pfj. + y{a-IJ. + V O-tL) — a-ix + v\Pix+ V — Pix) (3 IX + vfiix By choosing a^ and ^^ as before the numerator of this fraction, 'and therefore the fraction itself, may be made less than any assignable number ; and that for all values of v. Therefore the sequence — , — , ••• is regular. ^ A (3.: 30. Equality. Greater and Lesser Inequality. Of two | j numbers, a and b, defiyied by regular sequences aj, a.,, ... ; ^i, fi^, ..., the first is greater than, equal to or less than the second, according as the number defined by uj — /?], a, — yS,, ... is greater than, equal to or less than 0. This definition is to be justified exactly as the definitions of the fundamental operations on numbers defined by regu- lar sequences were justified in § 29. From this definition, and the definition of in § 28, it immediately follows that CoR. Two numbers ivhich differ by less than any assignable number are equal. 31. The Number Defined by a Regular Sequence is its Limiting Value. The difference between a number a and the term a^ of the sequence by which it is defined may be made less than any assignable number by taking fx great enough. 32 NUMBER-SYSTEM OF ALGEBRA. For it is only a restatement of the definition of a regular sequence ai, a2, ... to say that the sequence which defines the difference a — a^ (§ 29, 2), is one whose terms after the /xth can be made less than any assignable number by choosing fi great enough, and which, therefore, becomes, as p. is indefinitely increased, a sequence which defines (§ 28). In other words, the limit oi a —a^ as fx is indefinitely increased is 0, or a = limit {a^). Hence The number defined by a regular sequence is the limit to lohich the fxth term of this sequence approaches as fi is indefi- nitely increased* The definitions (1), (2), (3), (4) of § 29 may, therefore, be stated in the form : limit (a^^,) ± limit (/3^) = limit (a^ ± jS^), limit (a^) • limit (^^) = limit (a^^^), limiUo^ = limit (^ limit {(3,) V/3, * "What the above demonstration proves is that a stands in the same relation to a^ when irrational as when rational. The principle of perma- nence (cf. § 12), therefore, justifies one in regarding a as the ideal limit in the former case since it is the actual limit in the latter (§27). a, when irrational, is limit (a^) in precisely the same sense that - is the quotient d of c by d, when c is a positive integer not containing d. It follows from the demonstration that if there be a reality corresponding to a, as in geometry we assume there is (§ 40), that reality will be the actual limit of the reality of the same kind corresponding to o;u.. ' The notion of irrational limiting values was not immediately avail- able because, prior to §§ 28, 29, 30, the meaning of difference and greater and lesser inequality had not been determined for numbers defined by sequences. THE IBBATIONAL. 33 For limit (a^) the more complete symbol ™^^ (a^) is also used, read "the limit which a^ approaches as /x approaches infinity " ; the phrase " approaches infinity " meaning only, " becomes greater than any assignable number." 32. Division by Zero. (1) The sequence -^, -^, ... cannot define a number when the number defined by ^i, ^2} ••• is 0, unless the number defined by aj, a.,, ... be also 0. In this case it may ; ^ may approach a definite limit as ju, increases, however small a^ and /S^u. become. But this number is not to be regarded as the mere quotient — Its value is not at all determined by the fact that the numbers defined by a,, ao ... ; (3i, ^21 ••• 3-i'6 ; for there is an indefinite number of dif- ferent sequences which define 0, and by properly choosing ai, 02 . . . ; /3i, 1^2, ... from among them, the terms of the sequence — , — , ••• may be made to take any value what- soever. '^ ^^ (2) ,The sequence ^, ^, ... is not regular when ySj, /Sg, ... defines and ai, aj, ... defines a number different from 0. No term -^ can be found which differs from the terms following it by less . than any assignable number ; but rather, by taking ix. great enough, -^ can be made greater than any assignable number whatsoever. Though not regular and though they do not define numbers, such sequences are found useful in the higher mathematics. They may be said to define infinity. Their usefulness is due to their determinate form, which makes it possible to bring them into combination with other sequences of like character or even with regular sequences. Thus the quotient of any regular sequence y„ yg ••• 34 NUMBER-SYSTEM OF ALGEBBA. by — , ^, "• is a regular sequence and defines 0; and the quotient of — , — , ... by a similar sequence 2^, ^, ... j3i ^2 Sj 82 may also be regular and serve — if a^, /Sj, y,-, 8; (i = l, 2, ...) be properly chosen — to define any number whatsoever. The term — " approaches infinity " (i.e. increases with- out limit) as /a is indefinitely increased, in a definite or determinate manner; so that the infinity which — , -^, ... defines is not indeterminate like the mere symbol - of § 22. But here again it is to be said that this determinateness is not due to the mere fact that jiy, (i^ •'• defines 0, which is all that the unqualified symbol - expresses. For there is an indefinite number of different sequences which like ^i, ^82, ... define 0, and x is a symbol for the quotient of a by any one of them. 33. The System defined by Regular Sequences of nationals, Closed and Continuous. A regular sequence of irrationals Ctj, a^, ... a„, fK-m^i, ... ttm+n> ••• (m which the differences a„^„ — a^ may he made numerically less than any assignable number by taking m great enough) defines a number, but never a number which may not also be defined by a sequence of rational numbers. For /3i, (32, ... being any sequence of rationals which defines 0, construct a sequence of rationals aj, ag, ... such that «!— tti is numerically less than (3i (§ 30), and in the same sense aj — a2<;82, ttg — ag < ^83, etc. Then limit (a„— a„) = (§§ 28, 31), or limit (a„) = limit (aj. This theorem justifies the use of regular sequences of irrationals for defining numbers, and so makes possible a simple expression of the results of some very complex THE IMAGINARY. COMPLEX NUMBERS. 35 operations. Thus a™, where m is irrational, is a number; the number, namely, which the sequence a"-i, a% ... defines, when ttj, ao, ... is any sequence of rationals defining m. But the importance of the theorem in the present discus- sion lies in its declaration that the number-system defined by regular sequences of rationals contains all numbers which result from the operations of regular sequence-building in general. It is a dosed system with respect to the four fundamental operations and this new operation, exactly as the rational numbers constitute a closed system with respect to the four fundamental operations only (cf. § 25). The system of numbers defined by regular sequences of rationals — real numbers, as they are called — therefore possesses the following two properties : (1) between every two unequal real numbers there are other real numbers; (2) a variable which runs through any regular sequence of real numbers, rational or irrational, will approach a real number as limit. We indicate all this by saying that the system of real numbers is continuous. V. THE IMAGINARY. COMPLEX NUMBERS. 34. The Pure Imaginary. The other symbol which is needed to complete the number-system of algebra, unlike the irrational but like the negative and the fraction, admits of definition by a single equation of a very simple form, viz., ar'+l = 0. It is the symbol whose square is — 1, the symbol V— 1, now commonly written i.* It is called the unit of imag- inaries. In contradistinction to i all the forms of number hitherto considered are called real. These names, "real" and "imagi- * Gauss introduced the use of i to represent -v/^. 36 NUMBER-SYSTEM OF ALGEBRA. nary," are unfortunate, for they suggest an opposition which does not exist. Judged by the only standards which are admissible in a pure doctrine of numl?ers i is imaginary in the same sense as the negative, the fraction, and the irra- tional, but in no other sense ; all are alike mere symbols devised for the sake of representing the results of opera- tions even when these results are not numbers (j^ositive integers), i got the name imaginary from the difficulty once found in discovering some extra-arithmetical reality to correspond to it. As the only property attached to i by definition is that its square is — 1, nothing stands in the way of its being "multiplied" by any real number a; the product, ia, is called a pure imaginary. An entire new system of numbers is thus created, coex- tensive with the system of real numbers, but distinct from it. Except 0, there is no number in the one which is at the same time contained in the other.* Numbers in either system may be compared with each other by the definitions of equality and greater and lesser inequality (§ 30), ia being called = ih, as a = 5 ; but a number in one system cannot be said to be either greater than, equal to or less than a number in the other system. 35. Complex Numbers. The sum a + ih is called a com- plex number. Its terms belong to two distinct systems, of which the fundamental units are 1 and i. The general complex number a -f- ih is defined by a com- plex sequence where aj, a^, ... ; ySj, ySg, ... are regular sequences. * Throughout this discussion oo is not- regarded as belonging to the number-system, but as a limit of the system, lying without it, a sym- bol for something greater than any number of the system. THE IMAGINARY. COMPLEX NUMBERS. 37 Since a = a + i0(§36, 3, Cor.) and ib = 0+ib, all real numbers, a, and pure imaginaries, ib, are contained in the system of complex numbers a + ib.- a + ib can vanish only when both a = and 6 = 0. 36. The Four Fundamental Operations on Complex Num- bers. The assumption of the permanence of the fundamen- tal laws leads immediately to the following definitions of the addition, subtraction, multiplication, and division of complex numbers. 1. (a + lb) + (a' -\- ib') = a-h a' -\-i{b + b'). For (a + ib) + (a' + ib') = a + ib + a' + ib', Law II. = a^a' + ib -f ib', Law I. = a+a'+i{b-\-b'). Laws II, V. 2. (a + ib) - (a' + ib') = a-a' -^i{b -b'). By definition of subtraction (VI) and § 36, 1. CoR. The necessary as tvell as the srifficient condition for the equality of two complex numbers a + ib, a' + ib' is that a = a' and b = b'. For if (a -f ib) - (a' + ib') =a-a'-\- i(b - b') = 0, a-a' = 0, 6 - 6' = (§ 35), or a = a', b = b'. 3. (a -I- ib) (a' -\- ib') = aa' - 66' + i{ab' + 6a'). For (a + ib) (a' + ib') = (a -f ib) a' + (a + ib) ib', Law V. = aa'+ib -a'-i-a- ib'-\- ib • ib'. Law V. = {aa' - bb') + i(ab' + 6a'). Laws I-V. CoR. If either factor of a product vanish, the product vanishes. For i X = i(b - 6) = ib - ib (§ 10, 5), = (§ 14, 1). Hence (a + ib)0 = a x + ib x = a x -\-i{b xO)= 0. Laws y, IV, § 28, § 29, 3. 38 NUMBER-SYSTEM OF ALGEBRA. . a + ih _ aa' + bb' .ba' — ab' ' a' + ib' ~ a'2+ 6'2 "^ ^'^ + b''' ' For let the quotient of a + ib by a' -\- ib' be a; + iy. By the definition of division (VIII), (x + iy) (a'+ ib') = a + ib. . • . xa' — yb' + i(xb' + ya') = a + ib. § 36, 3 .-. xa' — yb' = a, xb' + ya' = b. § 36, 2, Cor. Hence, solving for x and y between these two equations, aa' + bb' ba' — ab' X = ■ , V = . a'-' + b" ' ^ a" + b" Therefore, as in the case of real numbers, division is a determinate operation, except when the divisor is 0; it is then indeterminate. For x and y are determinate (by IX) unless a'^ + b'^ = 0, that is, unless a' = b'— 0, or a'+ ib' = 0; for a' and b' being real, a'^ and b'^ are both positive, and one cannot destroy the other.* Hence, by the reasoning in § 24, CoR. If a product of two comjolex numbers vanish, one of the factors must vanish. 37. Numerical Comparison of Complex Numbers. Two complex numbers, a + ib, a' + ib', do not, generally speak- ing, admit of direct comparison with each other, as do two real numbers or two pure imaginaries ; for a may be greater than a', while b is less than b'. They are compared numerically, however, by means of their moduli Va^ + b^, Vo '" + b''^; a + ib being said to be * What is here proven is that in the system of complex numbers formed from the fundamental units* 1 and i there is one, and but one, number which is the quotient of a + i'6 by a' + ih' ; this being a conse- quence of the determinateness of the division of real numbers and the peculiar relation (i^ = — 1) holding between the fimdamental units. For the sake of the permanence of IX we make the assumption, otherwise irrelevant, that this is the only value of the quotient whether within or without the system formed from the units 1 and i. THE IMAGINARY. COMPLEX NUMBERS. 39 numerically greater than, equal to or less than a' + ib' ac cording as Va- + b''' is greater than, equal to or less than Va'- + b". Compare § 47. 38. The Complex System Adequate. The system a + ib is an adequate number-system for algebra. For, as will be shown (Chapter VII), all roots of algebraic equations are contained in this system. But more than this, the system a + ib is a closed system with respect to all existing mathematical operations, as are the rational system with respect to all finite combinations of the four fundamental operations and the real system with respect to these operations and regular sequence-building. For the results of the four fundamental operations on complex numbers are complex numbers (§ 36, 1, 2, 3, 4). Any other operation may be resolved into either a finite combination of additions, subtractions, multiplications, divis- ions or such combinations indefinitely repeated. In either case the result, if determinate, is a complex number, as fol- lows from the definitions 1, 2, 3, 4 of § 36, and the nature of the real number-system as developed in the preceding chapter (see Chapter VIII). The most important class of these higher operations, and the class to which the rest may be reduced, consists of those operations which result in infinite series (Chapter VIII); among which are involution, evolution, and the taking of logarithms (Chapter IX), sometimes included among the fundamental operations of algebra. 39. Fundamental Characteristics of the Algebra of Num- ber. The algebra of number is completely characterized, formally considered, by the laws and definitions I-IX and the fact that its numbers are expressible linearly in terms of two fundamental units.* It is a linear, asso- ciative, distributive, commutative algebra. Moreover, the * That is, in terms of the first powers of these units. 40 NUMBER-SYSTEM OF ALGEBRA. most geaeral linear, associative, distributive, commutative algebra, whose numbers are complex numbers of the form cciei + Xoe2 + • • • + x„e„, built from n fundamental units e^, e^, '••, e„, is reducible to the algebra of the complex number a + ih. For Weierstrass * has shown that any two complex numbers a and h of the f orm a-jei + ^262 + l-^n^nj whose sum, difference, product, and quotient are numbers of this same form, and for which the laws and definitions I-IX hold good, may by suitable transformations be resolved into com- ponents Oi, a2, • • • a^ ; 61, 62? • • • ^r, such that a = cti + 02 + • • • + «» a ± & = CTj ± 61 + a2 ± 60 H + ttr ± &r> ah = a-px + «2&2H — + <^fir) h &i 62 ^r The components a;, 6^ are constructed either from one fun- damental unit fji or from two fundamental units (/„ A'^.f For components of the first kind the multiplication for- mula is (a5r.)()85r,) = («/?) fir.. * Zur Theorie der aus n Haiipteinheiten gebildeten complexen Grossen. Gottinger Nachrichten Xr. 10, 1884. Weierstrass finds that these general complex numbers difier in only- one important respect from the complex number a + ib. If the num- ber of fimdamental units be greater than 2, there always exist num- bers, different from 0, the product of which by certain other numbers is 0. "Weierstrass calls them divisors of 0, The number of exceptions to the determinateness of division is infinite instead of one. I These units are, generally speaking, not Cj, Cj, ..., e„, but linear combinations of them, as 7iei-t-72e2H h TnCn? t^e^-^- K^e.,-\ hf/iCn. Any set of n independent linear combinations of the units e^, e,, ... «» may be regarded as constituting a set of fundamental units, since all numbers of the form aiej-j-aje^H Ha„e„ may be expressed linearly in terms of them. I GRAPHICAL REPRESENTATION OF NUMBERS. 41 For components of the second kind the multiplication formula is (ag, + ^,) (a'g, + /3%) = (aa'- P(3')g, + (a^' + (3a')K And these formulas are evidently identical with the mul- tiplication formulas (al)(;81) = (a/3)l, (al + /30 (a' 1 + /3'0 = (aa'- (3(3')1 + (a;8' + /3a')i of common algebra. VI. GKAPHIOAL EEPRESENTATION OF NUMBEES. THE VARIABLE. 40. Correspondence between the Real Number- System and the Points of a Line. Let a right line be chosen, and on it a fixed point, to be called the null-point ; also a fixed unit for the measurement of lengths. Lengths may be measured on this line either from left to right or from right to left, and equal lengths measured in opposite directions, when added, annul each other ; opposite algebraic signs may, therefore, be properly attached to them. Let the sign +ibe attached to lengths measured to the right, the sign — to lengths measured to the left. The entire system, of real numbers may he represented by the points of the line, by taking to correspond to each number that point whose distance from the null-point is represented by the number. For, as we proceed to demonstrate, the distance of every point of the line from the null-point, measured in terms of the fixed unit, is a real number; and we may assume that for each real number there is such a point. 1. TJie distance of any point on the line from the null-point is a real number. Let any point on the line be taken, and suppose the seg- ment of the line lying between this point and the null-point 42 NUMBER-SYSTEM OF ALGEBRA. to contain the unit line a times, with a remainder d^, this remainder to contain the tenth part of the unit line (3 times, with a remainder dg, (^2 to contain the hundredth part of the unit line y times, with a remainder dg, etc. The sequence of rational numbers thus constructed, viz., a, a . /?, a . (Sy, . . . (adopting the decimal notation) is regular ; for the difference between its /^th term and each succeeding term is less than , a fraction which may be made less than any assignable number by taking fx. great enough ; and, by construction, this number represents the distance of the point under consideration from the null-point. By the convention made respecting the algebraic signs of lengths this number will be positive when the point lies to the right of the null-point, negative when it lies to the left. 2. Corresponding to every real number there is a x>oint on the line, whose distance and direction from the nidl-point are indicated by the number. (a) If the number is rational, we can construct the point. For every rational number can be reduced to the form of ft. • a simple fraction. And if - denote the given number, when thus expressed, to find the corresponding point we have only to lay off the ySth part of the unit segment a times along the line, from the null-point to the right, if — is positive, from the null-point to the left, if - is negative. (&) If the number is irrational, we usually cannot con- struct the point, or even prove that it exists. But let a denote the number, and «i, uo, •••, a„, ••• any regular sequence of rationals which defines it, so that «„ will approach a as limit when n is indefinitely increased. Then, by (a), there is a sequence of points on the line corresponding to this sequence of rationals. Call this sequence of points Ai, A2, •••, A^, •••• It has the property GRAPHICAL EEPRESENTATION OF NUMBERS. 43 that the length of the segment A,^An+m ^^^^ approach as limit when n is indefinitely increased. When «„ is made to run through the sequence of values «i> «2j •••? ^^16 corresponding point A^ will run through the sequence of positions Ai, Ao, •••■ And we assume that just as there is in the real system a definite number a which «„ is approaching as a limit, so also is there on the line a definite point A which A,^ approaches as limit. It is this point A which we make correspond to a. Of course there are infinitely many regular sequences of rationals «i, «2, ••• defining a, and as many sequences of cor- responding points Ai, A2, '•'. We assume that the limit point A is the same for all these sequences. 41, The Continuous Variable. The relation of one-to-one correspondence between the system of real numbers and the points of a line is of great importance both to geometry and to algebra. It enables us, on the one hand, to express geometrical relations numerically, on the other, to picture complicated numerical relations geometrically. In particu- lar, algebra is indebted to it for the very useful notion of the continuous variable. One of our most familiar intuitions is that of continuous motion. 1 \ 1 \ A P B Suppose the point P to be moving continuously from A to B along the line OAJB ; and let a, b, and x denote the lengths of the segments OA, OB, and OP respectively, being the null-point. It will then follow from our assumption that the segment AB contains a point for every number between a and b, that as P moves continuously from J. to J5, x may be regarded as increasing from the value a to the value b through all intermediate values. To indicate this we call X a continuous variable. 44 NUMBER-SYSTEM OF ALGEBRA. 42. Correspondence between the Complex Number-System and tlie Points of a Plane. The eutire system of complex numbers may be represented by the points of a plane, as follows : In the plane let two right lines X'OX and T'OYhe drawn intersecting at right angles at the point 0. Make X'OX the "axis " of real numbers, using its points to represent real numbers, after the manner described in § 40, and make Y'OY the axis of pure imaginaries, represent- ing ib by the point of OT whose distance from is 6 when b is positive, and by the corresponding point of OY' when b is negative. The point taken to represent the complex number a-\-ib is P, constructed by drawing through A and B, the points which rep- resent a and ib, parallels to F'O Fand X'OX, respectively. The correspondence between the complex numbers and the points of the plane is a one-to-one correspondence. To every point of the plane there is a complex number corre- sponding, and but one, while to each number there corre- sponds a single point of the plane.* * A reality has thus been found to correspond to the hitherto unin- terpreted symbol a + ib. But this reaUty has no connection with the reality which gave rise to arithmetic, the number of things in a group of distiuct things, and does not at all lessen the purely symbolic char- acter of a -I- ib when regarded from the standpoint of that reality, the standpoint which must be taken in a purely arithmetical study of the origin and nature of the number concept. The connection between the numbers a + ib and the points of a plane is purely artificial. The tangible geometrical pictures of the relations among complex numbers to which it leads are nevertheless a valuable aid in the study of these relations. Y B X' P X Y / 1 Fig. 1. GRAPHICAL REPRESENTATION OF NUMBERS. 45 If the point P be made to move along any curve in its plane, the corresponding number x may be regarded as changing through a continuous system of complex values, and is called a continuous complex variable. (Compare § 41.) 43. M odulus. The length of the line OP (Fig. 1), i.e. ■\/a- + b", is called the modulus of a + ib. Let it be repre- sented by p. 44. Argument. The angle XOP made by OP with the positive half of the axis of real numbers is called the angle of a + ib, or its argument. Let its numerical measure be represented by 9. The angle is always to be measured " counter-clockwise " from the positive half of the axis of real numbers to the modulus line. 45. Sine. The ratio of PA, the perpendicular from P to the axis of real numbers, to OP, i.e. -, is called the sine of 6, written sin 0. ^ Sin 6 is by this definition positive when P lies above the axis of real numbers, negative when P lies below this line. 46. Cosine. The ratio of PB, the perpendicular from P to the axis of imaginaries, to OP, i.e. -, is called the cosine of 0, written cos 0. ^ Cos 6 is positive or negative according as P lies to the right or the left of the axis of imaginaries. 47. Theorem. Tlie expression of a + ib in terms of its modulus and angle is p{cos 6 -\-i sin 6) . For by § 46 - = cos 9, .-. a = p cos 9 ; P and by § 45, - = sin 9, .: b = p sin 9. P Therefore a-\-ib = p (cos 9 + i sin 9). 46 NUMBEE-SYSTEM OF ALGEBRA. The factor cos 6 -{-i sin 6 lias tlie same sort of geometrical meaning as the algebraic signs + and — , which are indeed but particular cases of it : it indicates the direction of the point which represents the number from the null-point. It is the other factor, the modulus p, the distance from the null-point of the point which corresponds to the number, which indicates the " absolute value " of the number, and may represent it when compared numerically with other numbers (§ 37), — that one of two numbers being numeri- cally the greater whose corresponding point is the more distant from the null-point. 48. Problem I. Given the points P and P', representing a + ib and a' + ib' respectively; required' the point represent- ing a + a'-f i(6 + &'). The point required is P", the intersection of the parallel to OP through P' with the parallel to OP' through P. For completing the construction indicated by the figure, we have OD' = PE = DD", and therefore OD" = OD + OD' ; and similarly P"D" = PD + P'D'. Cor. L To get the point corresponding to a—a'+i(b—b'), produce OP' to P'", making 0P"'= OP', and complete the p„ parallelogram OP, OP'". CoR. II. The modidus of the sum or difference of two complex numbers is less than (at greatest equal to) the sum of their moduli. For OP' is less than 0P-\- PP" and, therefore, than OP + OP, unless 0, P, P' are in the same straight line, when OP" =i0P+ OP'. Similarly, PP', which is equal to the modulus of the difference of the numbers represented by P and P, is less than, at greatest equal to, OP + OP. Fig. 2. GRAPHICAL REPRESENTATION OF NUMBERS. 47 49. Problem II. Given P and P', representing a + ib and a' + ib' respectively; required the point representing (a + ib)ia' + ib'). ' Let a + ib = p(cos6 + isme), §47 and a' + ib'= p' {cos 0' + i sin 6'); then (a + ib) {a' + ib') = pp' (cos ^ + i sin 6) (cos 6' + i sin 0') + i(sin ^ cos ^' + cos ^ sin ^') ]. But cos 6 cos ^' - sin 6 sin ^' = cos (^ + e') j* and sin 6 cos 6' + cos 9 sin ^' = sin {6 + ^') .* Therefore (a+i6)(a' + «6')=pp'[cos(^+^')+^"sin(^+^')]; or, The modulus of the product of two comi-tlex numbers is the product of their moduli, its argument the sum of their arguments. The required construction is, therefore, made by drawing through a line making an angle 6-\-6' with OX, and lay- ing off on this line the length pp'. CoR. I. Similarly the product of n numbers having moduli p, p', p", ••• p<"> respectively, and arguments 0, 6', 6", ... 6^"\ is the number pp'p" ... pc^ccosc^ + 6' + ^" + ... ro + isin{e-\-e' + e" + ---6^^^)\ In particular, therefore, by supposing the n numbers equal, we may infer the theorem [p (cos d + i sin 6) ]" = p" (cos n6 + i sin nO) , which is known as Demoivre's Theorem. * For the demonstration of these, the so-called addition theorems of trigonometry, see "Wells' Trigonometry, § 05, or any other text-book of trigonometry. 48 NUMBER-SYSTEM OF ALGEBRA. Cor. II. From the definition of division and the preceding demonstration it follows that 4±|-, = £^ [cos (^ - ^') + t sin (^ - ^') ] ; a' + lb' p a + ib iuii lur lilt; jjuiuu rejjrtJseiiLiiig fore, obvious. the construction for the point representing — — — is, there- a' + ib' 50. Circular Measure of Angle. Let a circle of unit radius be constructed with the vertex of any angle for centre. The length of the arc of this circle which is intercepted between the legs of the angle is called the circular measure of the angle. 51. Theorem. Any complex number may be expressed in the form pe^^ ; ivhere p is its modulus and 6 the circular meas- ure of its angle. It has already been proven that a complex number may be written in the form p^cosO + isme), where p and 6 have the meanings just given them. The theorem will be demon- strated, therefore, when it shall have been shown that e^^ = cos + i sin $. If n be any positive integer, we have, by § 36 and the binomial theorem, nj n 2 ! n' n(n-l)(n-2) (i6Y "^ 3! n^ 1-1 = i + i6 + -^(iey n A n 3! (iey + Let n be indefinitely increased ; the limit of the right side of this equation will be the same as that of the left. GRAPHICAL REPRESENTATION OF NUMBERS. 49 But the limit of the right side is \2 l + iO + jioy , jioy I 9.\ I Therefore e»* is the limit of [ 1 -| — ) as 7i approaches co I.e. e iOY ie * n To construct the point representing [1-] — On the axis of real numbers lay off OA = 1. Draw AP equal to 6 and par- allel to OB, and divide it into n equal parts. Let AAi be one of these parts. Then A^ is the point 1 H w Through Ai draw ^1^2 at right angles to OA^ and con- struct the triangle OA1A2 simi- lar to OAAi. A2 is then the point (1 -\ — ]. Fig. 3. For AOA2 = 2AOAi; and since OA2 : OA^ : : OAi : OA, and OA = 1, the length OA2 = the square of length OA^. (see § 49) In like manner construct A^ to represent ll + — ],Ai for ^!f-^-^!)" n/ Let 71 be indefinitely increased. The broken line AA1A2 ... A^ will approach as limit an arc of length 6 of the circle of radius OA and, therefore, its extremity, A^, will approach as limit the point representing cos 6 + i sin 6 (§ 47). * This use of the sjonbol e»^ will be fully justified in § 73. 60 NUMBER-SYSTEM OF ALGEBRA. Therefore the limit of fl + -yas n is indefinitely in- creased is cos 6 + i sin 0. But this same limit has already been proved to be e»^. Hence e»^ = cos ^ + i sin 6* VII. THE rUNDAMENTAL THEOEEM OP ALGEBEA. 52. The General Theorem. If where w is a positive integer, and tto, a^, ..., a„ any numbers, real or complex, independent of z, to each value of z corre- sponds a single value qf w. We proceed to demonstrate that conversely to each value of 10 corresponds a set of n values of z, i.e. that there are n numbers which, substituted for z in the polynomial ao^" + ttiZ"-^ H h a„, will give this polynomial any value, ivq, which may be assigned. It will be sufficient to prove that there are n values of z which render a^^z" + ttiZ""^ H |-«n equal to 0, inasmuch as from this it would immediately follow that the polynomial takes any other value, Wq, for n values of z ; viz., for the values which render the polynomial of the same degree, r^oz" + aiz"-^ -\ f- (a„ — Wo), equal to 0. 53. Root of an Equation. A value of z for which ttoz" + ciiZ"-^ H \-a„ is is called a root of this poly- nomial, or more commonly a root of the algebraic equation ao2;" + OiZ"-^ H h «„ = 0. * Dr. F. Franklin, American Journal of Mathematics, Vol. VII, p. 376. Also Mobius, Collected Works, Vol. IV, p. 726. FUNDAMENTAL THEOREM OF ALGEBRA. 51 54. Theorem. Every algebraic equation has a root. Given w = ao2!" + ai^""^ +••• + «„• Let 1 10 1 denote the modulus of w. We shall assume, though this can be proved, that among the values of |w| cor- responding to all possible values of z there is a least value, and that this least value corresponds to a finite value of z. Let \ivq\ denote this least value of \'w\, and ^o the value of z to which it corresponds. Then|w;o| = 0. For if not, Wq will be repre- sented in the plane of com- plex numbers by some point ' P distinct from the null- point 0. Through P draw a circle having its centre in the null- point 0. Then, by the hy- pothesis made, no value can be given z which will bring the corresponding w-point within this circle. But the w-point can he brought within this circle. For, Zq and w'o being the values of z and w which corre- spond to P, change z by adding to Zq a small increment S, and let A represent the consequent change in lo. Then A is defined by the equation (w, -f A) = a,{z, -f 8)" -f a^{z^ + 8)"-^ + a^z^ + a)"-- +- + a„_^(zo + S)+ a„. On applying the binominal theorem and arranging the terms with reference to powers of 8, the right member of this equation becomes ttoZo" + «i2;o"~^ H 1- a„-i2o + a„ + [wao2!o""' + (n - 1)01^0""' + ••• + a„-i]8 -H terms involving 8-, 8^, etc. 52 NUMBEB-STSTEM OF ALGEBBA. But lOo = tto^o" + ttiV"^ H 1- a„-iZo + ttn- . • . A = [?iao2;o"~^ + (n - 1) cii^o""^ H h a»-i] S + terms involving 8^, 8^, etc. Let p' (cos ^' + i sin $') be the complex number wao^o"-^ + (w — 1) ai2!o""^ H 1- a„_i, expressed in terms of its modulus and angle, and p (cos 6 + i sin 6) , the corresponding expression for 8. Then A = p'(gos $' + i sin $') x p (cos 6 + i sin ^) + terms involving p^, p^, etc. = pp'[cos(^ + 0') + isin {6 + ^')] + terms involving p^, p^, etc. § 49. The point which represents pp'[^cos{6 -\-0') -\-isin.(^9-\- ^')] for any particular value of p can be made to describe a circle of radius pp' about the null-point by causing to increase continuously from to 4 right angles. In the same circumstances the point representing rwo + pp' [cos (6 + $') + i sin {6 + $') ] will describe an equal circle about the point P and, there- fore, come within the circle OP. But by taking p small enough, A may be made to differ as little as we please from pp'[cos(^ -f 6') + i sm(6 + 0')^,* and, therefore, the curve traced out by P' (which represents Wq-\-^, as runs through its cycle of values), to differ as little as we please from the circle of centre P and radius pp'. Therefore by assigning proper values to p and 0, the lo- point (P') may be brought within the circle OP. * In the series Ap + Bp^ + Cp^ + etc., the ratio of all the terms fol- lowing the first to the first, i.e. Bp^ + Cp^ + etc. _ ^ B-{- Cp-\- etc. . — , — p X - > Ap A which hy taking p small enough may evidently be made as small as we please. FUNDAMENTAL THEOREM OF ALGEBRA. 53 The w;-point nearest the null-point must therefore be the null-point itself.* 55. Theorem. If a he a root o/ aoZ" -)- ajZ"-^ H +a„, this polynomial is divisible by z — a. For divide aoz" + ai2;"~^ -] + a„ by 2 — a, continuing the division until z disappears from the remainder, and call this remainder R, the quotient Q, and, for convenience, the poly- nomial /(2:). Then we have immediately f{z) = {z-a)Q + R, holding for all values of z. Let z take the value a ; then f(z) vanishes, as also the product (z — a) Q. Therefore when z = a, E = 0, and being independent of z it is hence always 0. 56. The Fundamental Theorem. The number of the roots of the polynomial a^z" -f ajZ""^ -)-••• + a„ is «. For, by § 54, it has at least one root ; call this a ; then, by § 55, it is divisible by z — a, the degree of the quotient being n — 1. Therefore we have aoZ"-fai2;"-i + ... + a„=(z-a)(a„2"-'-f&iZ"-2-|-...+6„_,). Again, by § 54, the polynomial a,^""^ + biz"-^ -\ h 6„_i has a root ; call this /S, and dividing as before, we have aoZ^H- ajz"-! + . . . -I- a„ = (2 - a) (5; - /?) (ao2"-2 -j- Ci0"-3 + . . . -f c„_2) . * In the above demonstration it is assumed that the coefBcient of 5 is not 0. If it be 0, let Ad"- denote the first term of A which is not 0. If ^ = p"(cos e" + i sin e"), we then have A = p'y[cos (re + e") + zsin (re + e")] + terms in S'+i, ••., from which the same conclusion follows as above. 54 NUMBER-SYSTEM OF ALGEBRA. Since the degree of the quotient is lowered by 1 by each repetition of this process, n — 1 repetitions reduce it to the first degree, or we have ao2!"+ai2;"-^H \.a^ = a^{z—a){z-p){z—y)'--{z-v), a product of n factors, each of the first degree. Now a product vanishes when one of its factors vanishes (§ 36, 3, Cor.), and the factor z — a vanishes when z — a, z—p when z=p, •••, z — v when z = v. Therefore aoS" + aj^"-^ + • • • + a„ vanishes for the n values, a, (3, y, ■•• v, of z. Furthermore, a product cannot vanish unless one of its factors vanishes (§ 36, 4, Cor.), and not one of the factors z—a,z — (i, ••-, z — v, vanishes unless z equals one of the numbers a, P, • • • v. The polynomial has therefore n and but n roots. The theorem that the number of roots of an algebraic equation is the same as its degree is called the fundamental theorem of algebra. VIII. Ilf UNITE SEEIES. 57. Definition. Any operation which is the limit of ad- ditions indefinitely repeated produces an infinite series. We are to determine the conditions which an infinite series must fulfil to represent a number. If the terms of a series are real numbers, it is called a real series; if complex, a complex series. I. REAL SERIES. 58. Sum. Convergence. Divergence. An infinite series represents a number or not, according as the sequence a,., S\, Sg, S3, • • Sm, Sm+1> ••• Sm+n> • • '} w tiere Si=(h, S2 = = ai + a2> ..., S< = tti + a^ + is regular or not. INFINITE SERIES. 55 If s-^ §2 • • •? ^6 ^ regular sequence, the number which it defines, or ^^^^ (^h)j is called the sum of the infinite series «i + «2 + a3+ 1- «„ H , and the series is said to be convergent. If Si, Sg, be not a regular sequence, s„ either transcends any finite value whatsoever, as n is indefinitely increased, or while remaining finite becomes altogether indeterminate. The infinite series then has no sum, and is said to be diver- gent. The series 1 + 1 + 1 + ••• and 1 — 1 + 1 — 1 + ••• are examples of these two classes of divergent series. A divergent series cannot represent a number. 59. General Test of Convergence. From these definitions and § 27 it immediately follows that : The infinite series aj + a, + ••• +a„ -{-••' is convergent when m may be so taken that the differences s„^.„ — s„ are numeri- cally less than any assignable number 8 for all values of n, where s„ and s„+„ are the sum of the first m and of the first m -\-n terms of the series respectively. If these conditions be not fulfilled, the series is divergent. The limit of the nth term of a convergent series is ; for the condition of convergence requires that by taking m great enough, s^+j — s„, i.e. a„+i, may be found less than any assignable number. But it is not to be assumed con- versely that a series is convergent, if the limit of its wth term is ; other conditions have also to be fulfilled, s„^„—s„ must be less than 8 for all values of n. Thus the limit of the nth term of the series 1-| 1 [-•••isO: but, 2 3 as will presently be shown, this is a divergent series. 60. Absolute Convergence. It is important to distin- guish between convergent series which remain convergent when all the terms are given the same algebraic signs and 56 NUMBER-SYSTEM OF ALGEBRA. convergent series which become divergent on this change of signs. Series of the first class are said to be absolutely con- vergent ; those of the second class, only conditionally con- vergent. Absolutely convergent series have the character of ordinary sums; i.e. the order of the terms may be changed without altering the sum of the series. For consider the series a^ -f- a^ + Oj + • • • supposed to be absolutely convergent and to have the sum S, when the terms are in the normal order of the indices. It is immediately obvious that no change can be made in the sum of the series by interchanging terms with finite indices ; for n may be taken greater than the index of any of the interchanged terms. Then Sn has not been affected by the change, since it is a finite sum and jt is immaterial in what order the terms of a finite sum are added ; and as for the rest of the series, no change has been made in the order of its terms. But Oj + flj + 03 -I — may be separated into a number of mfinite series, as, for instance, into the series a^ + 03 + a^ + ••• and Oj + a^ + Og + •••, and these series summed separately. Let it be separated into I such series, the sums of which — they must all be absolutely convergent, as being parts of an absolutely convergent series — are S'-^'>, S^-"), ••• ^(", respectively ; it is to be proven that S= /Sfd) + iS'(2) H- ,S'(3) + ... + ^S-tO. Let S^^\ S'''\ •••be the sums of the first m terms of the series S^^K >S'(2), •••, respectively. Then, by the hypothesis that the series a^-^ a,^-\- ••• is absolutely convergent, m may be taken so large that the sum c.(l) , c»(2) , , , o(0 shall differ from S by less than any assignable number S for all values of n ; therefore the limit of this sum is ^S*. But again, n may be so taken that S''^ shall differ from S^'^'^ by less than -, S^'\ from 8^'^ by less than -, ••• : and therefore the sum C» + C« + - + Cn f^«^ '^^ ■ + '^'^ + - + '^*' ^y !««« *^^^ (') ^ ; i.e. by less than 5. Hence the limit of this sum is -S"'!) + S^-'^ + 1- S^'\ Therefore S and /SO + S'„, as n is indefinitely increased, is C 62. Special Tests of Convergence. 1. If each of the terms of a series ai4-«2+"* ^^ numerically less than (at greatest equal to) the corresponding term, of an absolutely convergent series, or if the ratio of each term o/ tti + ttj + • • • to the corre- sponding term of an absolutely convergent series never exceed some finite number C, the series Oj -|- aj + • • • is absolutely convergent. If, on the other hand, each term o/ aj -(- a, + ••• he numeri- cally greater than (at the lowest equal to) the corresponding term of a divergent series, or if the ratio of each term of Oj + a, + ••• to the corresponding term of a divergent series be never numerically less than some finite number C, different from 0, the series a^ + % + • • • is divergent. 2. The series a^ — o, + «3 — «4 -\ , the terms of lohich are alternately positive and negative, is convergent, if after some term a^ each term be numericcdly less or, at least, not greater than the term which immediately precedes it, and the limit of a„, as n is indefinitely increased, be 0. For here Sm+n — s„ = (— l)'»[am+i — a„+2 +•••(— l)''-'am4.„]. The expression within brackets may be written in either of the forms (Om+l — am+2) + (Om+a — "771+4) + ••• (1) or a^+i — (a„+2 — a^+s) . (2) 58 NUMBEB-SYSTEM OF ALGEBRA. It is therefore positive, (1), and less than a^+i, (2); and hence ■by- taking m large enough, may be made numerically less than any assign- able number whatsoever. The series 1 1 1 is, by this theorem, con- vergent. 3. The series l-\ 1 \- --\ is divergent. Por the first 2^ terms after the first may be written 2'*' (24^1 "^2T2J "*" U^ + 1 '*' 2- + 2 "^ 2^ + 3 "^ 22 + 2^ j"*" "■ + (^2A-l + 1 "^ 2A-1 + 2 "^ "* 2^-1 + 2A-lj' •where, obviously, each of the expressions within parentheses is greater than • 2 The sum of the first 2-^ terms after the first is therefore greater than -, and may be made to exceed any finite quantity whatsoever by tak- ing K great enough. This series is commonly called the harmonic series. By a similar method of proof it may be shown that the series 1 H — - H h • • • is converajent if » > 1. Tj 1.1^2 l,l,l,l/4.,/2y and the sum of the series is, therefore, less than that of the decreasing geometric series 1 + - — f-( — ) +•••• The series 1 H — -\ 1 is divergent if » < 1, the terms 2^ 3^ * being then greater than the corresponding terms of 1 + 1 + 1 + .... 2 3 4. The series a^ + ctg -f- ctg + • • • is absolutely convergent if after some term of finite index, ai, the ratio of each term to that which immediately j^^^cedes it be numerically less than 1 and, as the index of the term is indefinitely increased, approach INFINITE SERIES. 59 a limit which is less than 1 ; but divergent, if this ratio and its limit be greater than 1. For — to consider the first hypothesis — suppose that after the term at this ratio is always less than a, where a denotes a certain positive number less than 1. Then, ^^iii < a, .-. ai^i < Uia ; ^±= ^ a, .•. a,+2 ■^ cii-j-ia ^ Uio?. • ■•••■• ^±^— < a, .-. a,_)_i < ai^^ic-i)a < ••• < ataK ai-\-(k~i) The given series is therefore < St +af[«+a2+a3+...a«^ +...]. And this is an absolutely convergent series. For a+a2+-a'=+-= ^™^* („ + a2 + ... + ««) n = CO ^ ' limit / g — a"+i \ n = co \^ l_a j — ^ — , since a is a fraction. The given series is therefore absolutely convergent, § 62, 1. The same course of reasoning would prove that the series is diver- gent when after some term a; the ratio of each term to that which precedes it is never less than some quantity, a, which is itself greater than 1. When the limit of the ratio of each term of the series to the term immediately preceding it is 1, the series is some- times convergent, sometimes divergent. The series con- sidered in § 62, 3 are illustrations of this statement. 63. Limits of Convergence. An important application of the theorem just demonstrated is in determining what are called the limits of convergence of infinite series of the form tto -(- aiX -j- a^ + a^x^ + • • •? 60 NUMBER- SYSTEM OF ALGEBRA. where x is supposed variable, but tlie coefficients ao, a^, etc., constants as in the preceding discussion. Such a series will be convergent for very small values of x, if the coefficients be all finite, as will be supposed, and generally divergent for very great values of x; and by the limits of conver- gence of the series are meant the values of x for which it ceases to be convergent and becomes divergent. By the preceding theorem the series will be convergent if the limit of the ratio of any term to that which precedes it be numerically less than 1 ; i.e. if limit n =00 ,n+r or limit f0^n±i a„ ^\ ^^^^* f — ^ V n+1. and divergent^ if 1. Thus the infiuite series a"* + mcr-'x + ^K^ — 1) (;^m-2^ _l ^ which is the expansion, by the binomial theorem, of (a + x)"* for other than positive integral values of m, is convergent for values of x numerically less than a, diver- gent for values of x niunerically greater than a. For in this case v}(m — 1) ..•(?» — 7i -f 1 )" limit n = 'n+l Umit n = 00 aX' («)! m(m — ])•••(?« — n) (n+1)! limit / , , n + 1 \ = . la X — — — 1 limit ' \ " n = 00 I 1 I '" = — a. iJ INFINITE SERIES. 61 of 2. Again, the expansion of e'', i.e. 1 + a; + — H , is con- vergent for all finite values of x. Forhere ^^^ ( ^]= ^™M <"> ' U ^™^* (« + l) = oo. H« + 1) ! The same is true for the series which is the expansion- of a'. 64. Operations on Infinite Series. 1. The sum of tioo convergent series, «! + a2 H <^""^ ^\-^\-\ > ^^ the series (ttj + bi) + (^2 + ^2) -H j '^"''^ ^'*6"* difference is the series (a, — 61) + (ttj — 62) H • The sum of the series a^ + a^+ ••• is the number defined by Sj, Sj, ..., and the sum of the series b^ + b^-i — is the number defined by t^, t.^, ..., where Sj = aj + 03 H hci and ti=b^ + b^-\ h it- The sum of the two series is therefore the number defined by Sj + t^, s.^ + t.^, •■-, § 29, (1). But if Si = (fli + 61) + ("2 + ^2) + ••• + («i + ^0. we have Si = Si + ti for all values of i. This is immediately obvious for finite values of ?, and there can be no difference between Si and Sj + ti as i approaches co, since it would be a difference having for its limit. Therefore the nimiber defined by s^ + 2-f--- are all positive. Then if Sn be the sum of the first n terms of 62 NUMBER-SYSTEM OF ALGEBRA. "i^i + ("1^2 + °2^i) + •••» ^^^ "* represent - when n is even and ■ ~ ■ when n is odd, evidently Sntn > Sn> -frntm- But n = 00 C*"^") ~ n = cx:> (sm*m). Therefore J2t (^^0 = J'^o^ («A). If the terms oi a^ + a.^+ ■■■, ^ + 1^+ ■•■ be not all of the same sign, call the sums of the first n terms of the series got by making all the signs plus, sj and tj respectively ; also SJ, the sum of the first n terms of the series which is their product. Then by the demonstration just given limit .0, ._ lunit , , , .. but *S'„ always differs from s„t,i by less than (at greatest by as much as) S'n from s'nt'n ', therefore, as before, limit . „ \ limit . . ^ 3. The quotient of the series Oq + a^x + • • • by the series liQ + h-iX -\- ••• (&o not 0) is a series of a similar form, as Cq -{- c^x -\- ' • ' , which converges when a^ + (^\X + • • • is absolutely convergent and b^x-\---' is numerically less than So- il. COMPLEX SERIES. The terms sum, convergent, divergent, have the same mean- ings in connection with complex as in connection with real series. 65. General Test of Convergence. A complex series, «i + ^2 + • • 'j ^s convergent tvhen me modulus of s„+„ — s„ may be made less than any assignable number 8 by taking m great enough, and that for all values of n; divergent, when this condition is not satisfied. See § 48, Cor. II ; § 59. 66. Of Absolute Convergence. Let tti + a2 + • • • be a complex series, and Ai-\- A2+ ■•', the series of the moduli of its terms. INFINITE SERIES. 63 ff the series Ai + A2-\ he convergent, the series ai + a2-\ — tjoill be convergent also. For the modulus of the sum of a set of complex numbers is less than (at greatest equal to) the sum of their moduli (§ 48, Cor. II). By hypothesis, Sm+n — Sm is less than any assignable number S, when /S'm = -4i + ^2+ ••• + -^"" ^^c. ; much more must the modulus of Sm+n — Sm be Icss than 5. The converse of this theorem is not necessarily true ; and a convergent series, ai + a2 + •••, is said to be absolutely or only conditionally convergent, according as the series Ai-\- A2+ ••• is convergent or divergent. 67. The Region of Convergence of a Complex Series. If the comi)lex series Oq + a^z + aiZ- + ••• is convergent idien z = Z, it is absolutely convergent for ^very value of z which is numerically less than Z, that is, it converges absolutely at every point within that circle in the plane of complex numbers which has the null-point for centre and passes through the point Z. For since the series «o + QiZ + a^Z"^ + ••• is convergent, its term anZ"^ approaches as limit when n is indefinitely increased. It is therefore possible to find a real number M which is numerically greater than every term of this series. Assign to z any value which is numerically less than Z, whose corresponding point, therefore. Lies within the circle through the point Z. For this value of z the terms of the series ao + a^z + 022^ + ••■ will be numerically less than the corresponding terms of the series '(1)%.... (1) M-\-M- + M For, since anZ''<.J^-, we have a„2«z^'>z^, etc.; i.e. that each is greater than the one following it. Since 0,5+ 0js;+ a^s^ + ••• + a„s"+ ••• converges absolutely for 2 = ^i, so also does and, therefore, Oj + OjS H — + anz"^ + •••. Hence A^ + A^z^ + ••• + ^,tn" + ••• (where At = modulus ai) is convergent, and a number M can be found greater than its sum. And since for z = z^, z^, ••• the individual terms of A,+A.,Z+... + AnZ''+- are less than the corresponding terms of A^ + A.^z^ + ••• + AnZi" + •••, this series and, therefore, modulus (a^ + a^z + ••■) remain always less than M as z runs through the sequence of values z^, ~.j, •••. Hence the values of modulus (a^z + a.^z^ + •••) which correspond to 3 = ^j, Zr, ■•■ constitute a regular sequence defining 0, each term being numerically less than the corresponding term of the regular sequence z^M, z^M, • • • which defines 0. INFINITE SERIES. 65 Cor. The same argument proves that if or z'"(a„ + a„+i 2; +...), be the sum of all terms of the series from the (m + l)th on, the series a„ + cin+iZ + ••■ can be made to differ as little as one may please from its first term a„. 69. Operations on Complex Series. The definitions of sum, difference, and product of two convergent complex series are the same as those already given for real series, viz. : 1. The sum of two convergent series, a^-\- a^-^- ••• and &i + &2 + • • ', is the series (% + 61) + {a^ + 62) + • • • > iheir difference, the series {a^ — hi) + (a^ — ^2) + •••• For if Sf = «! + a^ + ••• + ai and ti=h^-\- h.^ + ••• + hi, modulus [(s„+„ ± t^+n) - (s„ ± /„)] ^ modulus (s„+„ - s;„) + modulus (f„+„ — t„), and may, therefore, be made less than any assignable number by tak- ing m great enough. The theorem therefore follows by the reasoning of § 64, 1. 2. The jjroduct of two absolutely convergent series, «i + «2 + »3 + ••• ttwc? &i -\- b-i + l>i -\- • • ■ , is the series afii + (ajJa + «2&i) + («A + «2^2 + «3&i) •••• For, letting Si = Ai + A^+ ■■■ + Ai and T, = J5, + ^^ + ••• + -S,. ■where A,, Bt, are the moduli of a,, 6„ respectively, and representing by ffn the sum of the first n terms of the series and by 2n the sum of the first n terms of the series A^Bi + (A^B, + A,Bi)+-, we have modulus (s„«„ — (r„) A,r + ••• = A)-\-AiZ + Alt -h A^^ + 2A^t -f A.f -\ + A2" + Ajiz^-H + ■■■+ A.n^'^-V + . . . + ^r + J T_ n(n — l)---(n — k 4-1) where W;^ = ^ — ^ — - • A;! Equating the coefficients of like powers of z and t in the two members of this equation, we get Ai-hAk equal always to A^n^. In particular A^Aq = A^, therefore Aq = 1. Also AiAi = 2 A2, AoAi = 3 As, Ar^Ai = 4:Ai, ■", A^_iAi = nAn ; or, multiplying these equations together member by member, A" = A„n !, or A, = ^". A part of the equations among the coefficients are, there- fore, sufficient to determine the values of all of them in terms of the one coefficient Ai. But these values will satisfy the remaining equations ; for substituting them in the gen- eral equation An-icA/c = A,ji^, we get ^^"'^ X ^=^ X Mn-l)-(n-Tc+l) (n-ky. kl nl k\ ' which is obviously an identical equation. 70 NUMBER-SYSTEM OF ALGEBEA. The coefficient A^ or, more simply written, A, remains undetermined. ♦ It has been demonstrated, therefore, that to satisfy equa- tion (1), it is only necessary that/(2!) be the sum of an infinite series of the form where A is undetermined; a series which has a sum, i.e. is convergent, for all finite values of z and A. (§ 63, 2, § 66.) By properly determining A, f{z) may be identified with a', for any particular value of a. If a' is to be identically equal to the series (2), A must have such a value that A- AS 2r 3!^ Let e^=l+,+|l + |! + ...^ (3) where e = l + l+l + l_^...j* Then e^ = 1 + A + ~ 4-:^+ .... 21 31 Therefore 'a = e/ ; or, calling any number which satisfies the equation e" = a the logarithm of a to the base e and writing it log^a, ^=log,a. * This number e, the base of the Naperian system of logarithms, is a "transcendental" irrational, transcendental in the sense that there is no algebraic equation with integral coeflBcients of which it can be a root (see Hermite, Comptes Rendus, LXXVII). w has the same char- acter, as Lindemann proved in 1882, deducing at the same time the first actual demonstration of the impossibility of the famous old prob- lem of squaring the circle by aid of the straight edge and compasses only (see Mathematische Annalen, XX). THE EXPONENTIAL FUNCTION. 71 Whence finally, a definition of a", valid for all finite complex values of a and z, if it may be assumed that log^ a is a number, what- ever the value of a. The series (3) is commonly called the exponential series, and its sum e' the exponential function. It is much more use- ful than the more general series (2), or (4), because of its greater simplicity ; its coefficients do not involve the loga- rithm, a function not yet fully justified and, as will be shown, to a certain extent indeterminate. Inasmuch, how- ever, as e/ is a particular function of the class a", a' is some- times called the general exponential function, and series (4) the general exponential series. 74. The Functions Sine and Cosine. It was shown in § 51 that when ^ is a real number, e*^ = cos Q + i sin B. But e<.= l + »+eriodidty 2 ir. 76. The Logarithmic Function. If z = e^ and t = e'^, zt = e^e^=e^+'', §73 or log,2;i = log,2; + log,«. (7) The question again is whether a function exists capable of satisfying this equation, or, more generally, the " func- tional equation," f{zt)=f{z)+f{t), (8) for complex values of z and t. When .0=0, (7) becomes log,0 = log,0 + log,i, an equation which cannot hold for any value of t for which log^t is not zero unless log.O is numerically greater than any finite number whatever. Therefore log^O is infinite. On the other hand, when z = l, (7) becomes log,f = logel+log.«, so that logel is zero. Instead, therefore, of assuming a series with undetermined coefficients for /(z) itself, we assume one for /(I +2;), setting f{l+z) = A^z + A,z'+"-+A,X + -, and inquire whether the coefficients Ai admit of values which satisfy the functional equation (8) for complex values of z and t. THE LOGARITHMIC FUNCTION. 73 Now 1 -^ z + 1 = (1 +z)fl + ~—\ identically. .•./[l + (. + 0]=/(l+^)H-/fl+^\ or A,{z + + A.Xz -\-ty+...+AXz + ty + -" =:AiZ + A2Z^-\ \-A^Z"-] + A,(l+z)~H+A,(l-{.z)-H'+..-+A,(l+z)-V-i-.- Equating the coefficients of the first power of ^ (§ 72) in the two members of this equation, Ai + 2A2Z-\-3AsZ^-\ h (n -\- 1) A^+iZ" -\ = A,{1 -z -^-z' -z' -\- •>• + (-l)"z'^+ ...) ; whence, equating the coefficients of like powers of z, A, = Ai, 2^2 = -^!, ..., nA^ = {-iy-'Ai,"; or A=-4 .-, A=(-ir^4-. Z n As in the case of the exponential function, a part of the equations among the coefficients are sufficient to determine them all in terms of the one coefficient Ai. But as in that case (by assuming the truth of the binomial theorem for negative integral values of the exponent) it can be readily shown that these values will satisfy the remaining equa- tions also. The series 2-- +- +(- 1)"-'-+ ••• 2 3 ^ ^ n converges for all values of z whose moduli are less than 1 (§62,3). For such values, therefore, the function ^(^^-|+-+(-l)»-^^V-) (9) satisfies the functional equation /[(i + z)(i + 0]=/(H-^)+/(i + 0- 74 NUMBEB-ST8TEM OF ALGEBRA. And since z = l —(1 — z) and t=l—(l — t), the function - Afl-z+^^-^-\-"- + ^^—^+.'.\ satisfies this equation when written in the simpler form f(zt) =f(z) +/(0, for values of 1 — z and 1 — t whose moduli are both less than 1. 1. Log^ b. To identify the general function /(I + z) with the particular function \oge(l-{-z) it is only necessary to give the undetermined coefficient A the value 1. For since log,(l+z) belongs to the class of functions which satisfy the equation (8), Therefore eiog,(i+^)_/(^-f+") l0g,(l+2!) = ^^2-|+-\ But eioge(i+-) = 1 + 2. Hence or, equating the coefficients of the first power of z, A = 1. The coefficients of the higher powers of z in the right number are then identically 0. It has thus been demonstrated that log, 6 is a number (real or complex), if when b is written in the form l-{-z, the absolute value of z is less than 1. To prove that it is a number for other than such values of b, let b = pe^^ (§51), where p, as being the modulus of b, is positive. Then log,b = log,p-\-ie, and it only remains to prove that log^p is a number. Let p be written in the form e" — (e" — p), where e" is the first integral power of e greater than p. THE LOGARITHMIC FUNCTION. 75 Then since e" - (e" - p) = e^fl — ^" ~ P\ log^p = log.e" + log/l - ?^P^ and log/l ^^^ 1 is a number since ^ ~^ is less than 1. 2. io^'a 6. It having now been fully demonstrated that a^ is a number satisfying the equation a^a^ = a^^ for all finite values of a, Z, T; let a^=:z, a^z=t, and call Z the logarithm of z to the base a, or log„2, and in like manner T, logj. Then, since zt = a%'^ = a^% loga(20 = log«z + log„f, or logo 2 belongs, like log<,z, to the class of functions which satisfy the functional equation (8). Pursuing the method followed in the case of log^ft, it will be found that loga(l + z) is equal to the series A(z (-•••) when A = . This number is called the modulus of logea the system of logarithms of which a is base. 77. Indeterminateness of log a. Since any complex num- ber a may be thrown into the form pe*'^, logea = log,p + tU (10) This, however, is only one of an infinite series of possible values of log^a. For, since e»^ = e*(^±2n7r) (§ 75)^ loge a = log, pe*(« ± 2«;r) = log^ p ^ ^ (^ -J- 2 riTr) , where n may be any positive integer. Log^a is, therefore, to a certain extent indeterminate ; a fact which must be care- fully regarded in using and studying this function.* The * For instance loge (zt) is not equal to loge z + loge t for arbitrarily chosen values of these logarithms, but to log^z + log<,< ± i2nw, where n is some positive integer. 76 NUMBER-SYSTEM OF ALGEBRA. value given it in (10), for which w=0, is called its principal value. When a is a positive real number, 6 = Q, so that the prin- cipal value of log^a is real; on the other hand, when a is a negative real number, ^ = tt, or the principal value of log, a is the logarithm of the positive number corresponding to a, plus iV. 78. Permanence of the Remaining Laws of Exponents. Besides the law cv'a^ = a^+' which led to its definition, the function a^ is subject to the laws : 1. {a'y = a''. 2. {aby = a'h'* 1. (a^)' = a^. For a" = (ei°se<.) - = 1 + (log.a) 2 + G^g^^)^^^ + • • • § 73, (4) = l+2log,a+ '^ °' ' +••♦ = e^'^e^". § 73, (3) .-. {e^^^'^y = 6"^°^^", and log^a"^ = z logett. From these results it follows that (a^)' = eioge(«^)' __ gtlogeO* _. gezlogg* = a^. 2. (aby = a'b\ For (aby^e^"^^^""^' gzlogeo5 — - gZlogea+ zloggb . gzlogeo , gzlogeft = o* . &^ § 76, (7) § 73, (1) * — = a"-', which is sometimes included among the fundamental laws to which a" is subject, follows immediately from a'^af = a«+< by the lafinition of division. INVOLUTION AND EVOLUTION. 77 79. Permanence of the Remaining Law of Logarithms. In like manner, the function log„2; is subject not only to the log„ (zt) = log,, z + log„ t, but also to the law log^z^ = tlog^z. For z = a'°ea^, and hence z' = (a'"""^)' _ tt«log„^_ §78,1 80. Evolution. Consider three complex numbers ^, z, Z, connected by the equation ^^ = z. This equation gives rise to three problems, each of which is the inverse of the other two. For Z and t, may be given and z sought ; or ^ and z may be given and Z sought ; or, finally, z and Z may be given and t, sought. The exponential function is the general solution of the first problem (involution), and the logarithmic function of the second. For the third (evolution) the symbol -y^ has been devised. This symbol does not represent a new function ; for it is defined by the equation {^^z)'^=z, an equation which is 1 satisfied by the exponential function z'^. Like the logarithmic function, ^z is indeterminate, though not always to the same extent. When Z is a positive integer, t,^ = z is an algebraic equation, and by § 56 has Z roots for any one of which -M <^H-ij • • • f*o ^-re all numbers less than 10 ; and then repre- sented by the mere sequence of numbers «„tt„_i-"ao — it being left to the x>osition of any number a^ in this sequence to indicate the power of 10 with whijch it is to be associ- ated. For a system of this sort to be comj)lete — to be capable of representing all numbers unambiguously — a sym- bol (0), which will indicate the absence of any particular power of 10 from the sum a„10" -|- a„_ilO"~^ + • • • + «ilO -f cio, is indispensable. Thus without 0, 101 and 11 must both be Avritten 11. But this symbol at hand, any number may be expressed unambiguously in terms of it and symbols for 1,2, ...9. The positional idea is very old. The ancient Babylonians commonly employed a decimal notation similar to that of the Egyptians ; but their astronomers had besides this a very remarkable notation, a sexagesimal positional system. HISTORIC SYSTEMS OF NOTATION. 87 In 1854 a brick tablet was found near Senkereh on the Euphrates, certainly older than 1600 b.c, on one face of which is imjjressed a table of the squares, on the other, a table of the cubes of the numbers from 1 to 60. The squares of 1, 2, ••• 7 are written in the ordinary decimal notation, but 8^, or 64, the first number in the table greater than 60, is written 1, 4 (1 x 60 + 4) ; similarly 9^, and so on to 59-, which is written 58, 1 (58 x 60 + 1) ; while 60^ is written 1. The same notation is followed in the table of cubes, and on other tablets which have since been found. This is a posi- tional system, and it only lacks a symbol for of being a perfect positional system. The inventors of the 0-symbol and the moderu complete decimal positional system of notation were the Indians, a race of the finest arithmetical gifts. The earlier Indian notation is decimal but not positional. It has characters for 10, 100, etc., as well as for 1, 2, ••• 9, and, on the other hand, no 0. Most of the Indian characters have been traced back to an old alphabet* in use in Northern India 200 b.c. The original of each numeral symbol 4, 5, 6, 7, 8 (?), 9, is the initial letter in this alphabet of the corresponding numeral word (see table on page 89,t column 1). The characters first occur as numeral signs in certain inscriptions which are assigned to the 1st and 2d centuries a.d. (column 2 of table). Later they took the forms given in column 3 of the table. When was invented and the positional notation replaced the old notation cannot be exactly determined. It was * Dr. Isaac Taylor, in his book " The Alpliabet," names this alpha- bet the Indo-Bactrian. Its earliest and most important monument is the version of the edicts of King Asoka at Kapvu--di-giri. In this inscription, it may be added, numerals are denoted by strokes, as I, II, III, Mil, illll. t Columns 1-5, 7, 8 of the table on page 89 are taken from Tay- lor's Alphabet, II, p. 266 ; column 6, from Cantor's Geschichte der Mathematik. 88 NUMBEB-STSTEM OF ALGEBRA. certainly later than 400 a.d., and there is no evidence that it was earlier than 500 a.d. The earliest known instance of a date written in the new notation is 738 a.d. By the time that came in, the other characters had developed into the so-called Devanagari numerals (table, column 4), the classical numerals of the Indians. • The perfected Indian system probably passed over to the Arabians in 773 a.d., along with certain astronomical writings. However that may be, it was expounded in the early part of the 9th century by Alkhwarizml, and from that time on spread gradually throughout the Arabian world, the numerals tak- ing different forms in the East and in the West. Europe in turn derived the system from the Arabians in the 12th century, the " Gobar " numerals (table, column 5) of the Arabians of Spain being the pattern forms of the Euro- pean numerals (table, column 7). The arithmetic founded on the new system was at first called algorithm (after Alkhwarizml), to distinguish it from the arithmetic of the aibacus which it came to replace. A word must be said with reference to this arithmetic on the abacus. In the primitive abacus, or reckoning table, unit counters were used, and a number represented by the appropriate number of these counters in the appropriate columns of the instrument; e.g. 321 by 3 counters in the column of lOO's, 2 in the column of lO's, and 1 in the column of units. The Romans employed such an abacus in all but the most elementary reckonings, it was in use in Greece, and is in use to-day in China. Before the introduction of algorithm, however, reckon- ing on the abacus had been improved by the use in its columns of separate characters (called apices) for each of the numbers 1, 2, • • • 9, instead of the primitive unit counters. . This improved abacus reckoning was probably invented by Gerbert (Pope Sylvester II.), and certainly used by him at Rheiras about 970-980, and became generally known in the following century. HISTORIC SYSTEMS OF NOTATION. 89 TABLE SHOWING THE EVOLUTION OF THE ARABIC CIPHERS. z H OC h INDIAN ARABIC C U R O PE A N A.D. COBAR APICES LETTERS OF INDO-BACT ALPHABE Sec. I. Sec. V. Sec.X. Sec.X. SecX. Sec.Xli. SCG.XIV. C^\ Cj ) 1 \- 1 — e^-^ \ ^ TT I X B. C. Sec. II. 1 >.v ^ ^ f ^ > } V ¥ ^ 1 r 7^ A ^ b h Y \ 1 S ^ ? 5^ 4 c, S \ b ^ (/ 1 7 7 J ./^ 1 7 l 6> ( 9 8 ^e g i ? r 9 !) ^ ? o « .^ 90 NUMBER-SYSTEM OF ALGEBRA. Now these apices are not Koman numerals, but symbols wbicli do not differ greatly from the Gobar numerals and are clearly, like them, of Indian origin. In the absence of positive evidence a great controversy has sprung up among historians of mathematics over the immediate origin of the apices. The only earlier mention of them occurs in a pas- sage of the geometry of Boetius, vs^hich, if genuine, was written about 500 a.b. Basing his argument on this pas- sage, the historian Cantor urges that the earlier Indian numerals found their way to Alexandria before her inter- course with the East was broken off, that is, before the end of the 4th century, and were transformed by Boetius into the apices. On the other hand, the passage in Boetius is quite generally believed to be spurious, and it is maintained that Gerbert got his apices directly or indirectly from the Arabians of Spain, not taking the 0, either because he did not learn of it, or because, being an abacist, he did not appreciate its value. At all events, it is certain that the Indo- Arabic numerals, 1, 2, ••• 9 (not 0), appeared in Christian Europe more than a century before the complete positional system and algorithm. The Indians are the inventors not only of the positional decimal system itself, but of most of the processes involved in elementary reckoning with the system. Addition and subtraction they performed quite as they are performed nowadays ; multiplication they effected in many ways, ours among them, but division cumbrously. III. THE FKAOTION. 89. Primitive Fractions. Of the artificial forms of num- ber — as we may call the fraction, the irrational, the nega- tive, and the imaginary in contradistinction to the positive THE FRACTION. 91 integer — all but the fraction are creations of the mathe- maticians. They were devised to meet purely mathematical rather than practical needs. The fraction, on the other hand, is already present in the oldest numerical records — those of Egypt and Babylonia — was reckoned with by the Romans, who were no mathematicians, and by Greek mer- chants long before Greek mathematicians would tolerate it in arithmetic. The primitive fraction was a concrete thing, merely an aliquot part of some larger thing. When a unit of measure was found too large for certain uses, it was subdivided, and one of these subdivisions, generally with a name of its own, made a new unit. Thus there arose fractional units of measure, and in like manner fractional coins. In time the relation of the sub-unit to the corresponding principal unit came to be abstracted with greater or less completeness from the particular kind of things to which the units belonged, and was recognized when existing be- tween things of other kinds. The relation was generalized, and a pure numerical expression found for it. 90. Roman Fractions. Sometimes, however, the relation was never completely enough separated from the sub-units in which it was first recognized to be generalized. The Romans, for instance, never got beyond expressing all their fractions in terms of the uncia, sicilicus, etc., names origin- ally of subdivisions of the old unit coin, the as. 91. Egyptian Fractions. Races of better mathematical endowments than the Romans, however, had sufficient appreciation of the fractional relation to generalize it and give it an arithmetical symbolism. The ancient Egyptians had a very complete symbolism of this sort. They represented any fraction whose numer- ator is 1 by the denominator simply, written as an integer with a dot over it, and resolved all other fractions into 92 NUMBER-SYSTEM OF ALGEBRA. sums of such unit fractions. The oldest mathematical treatise known, — a papyrus * roll entitled " Directions for Attaining to the Knowledge of All Dark Things/' written by a scribe named Ahmes in the reign of Ra-a-us (therefore before 1700 b.c), after the model, as he says, of a more ancient work, — opens with a table which expresses in this manner the quotient of 2 by each odd number from 5 to 99. Thus the quotient of 2 by 5 is written 3 15, by which is 11 ... meant -H ; and the quotient of 2 by 13, 8 52 104. o Only -, among the fractions having numerators which o differ from 1, gets recognition as a distinct fraction and receives a symbol of its own. 92. Babylonian or Sexagesimal Fractions. The fractional notation of the Babylonian astronomers is of great interest intrinsically and historically. Like their notation of in- tegers it is a sexagesimal positional notation. The denomi- nator is always 60 or some power of 60 indicated by the position of the numerator, which alone is written. The 22 30 ^, , .^ 60 602 this notation be written 22 30. Thus the ability to represent fractions by a single integer or a sequence of integers, which the Egyptians secured by the use of fractions having a com- mon numerator, 1, the Babylonians found in fractions having common denominators and the principle of position. The Egyptian system is superior in that it gives an exact expres- sion of every quotient, which the Babylonian can in general do only approximately. As regards practical usefulness, however, the Babylonian is beyond comparison the better system. Supply the 0-symbol and substitute 10 for 60, and * The Rhind papyrus of the British Museum ; translated by A. Eisenlohr, Leipzig, 1877. fraction ^, for instance, which is equal to ^ + -^^, would in THE FRACTION. 93 this notation becomes that of the modern decimal fraction, in whose distinctive merits it thus shares. As in their origin, so also in their subsequent history, the sexagesimal fractions are intimately associated with astronomy. The astronomers of Greece, India, and Arabia all employ them in reckonings of any complexity, in those involving the lengths of lines as well as in those involving the measures of angles. So the Greek astronomer, Ptolemy (150 A.D.), in the Almagest (/xeydXr} avvra^ts) measures chords as well as arcs in degrees, minutes, and seconds — the de- gree of chord being the 60th part of the radius as the degree of arc is the 60tli part of the arc subtended by a chord equal to the radius. The sexagesimal fraction held its own as the fraction par excellence for scientific computation until the 16th century, when it was displaced by the decimal fraction in all uses except the measurement of angles. 93. Greek Fractions. Fractions occur in Greek writings — both mathematical and non-mathematical — much earlier than Ptolemy, but not in arithmetic* The Greeks drew as sharp a distinction between pure arithmetic, aptOix-qTiKri, and the art of reckoning, Xoyto-TtK?;, as between pure and metrical geom- etry. The fraction was relegated to XoyiorriK-^. There is no place in a pure science for artificial concepts, no place, there- fore, for the fraction in apiOiLrjTLKrj ; such was the Greek posi- tion. Thus, while the metrical geometers — as Archimedes (250 B.C.), in his "Measure of the Circle" {kvkXov ix€Tpr)ai<:), and Hero (120 b.c.) — employ fractions, neither of the trea- tises on Greek arithmetic before Diophantus (300 a.d.) which * The usual method of expressing fractions was to write the numer- ator with an accent, and after it the denominator twice with a double 17 accent : e.g. i(' ku" ku" = ^. Before sexagesimal fractions came into vogue actual reckonings with fractions were effected by unit fractions, of which only the denominators (doubly accented) were written. 94 NUMBER-SYSTEM OF ALGEBRA. have come down to us — the 7th, 8th, 9th books of Euclid's "Elements" (300 B.C.), and the *' Introduction to Arithmetic " (eiVaycoy^ apid ix-qTiKTj) of Nicomachus (100 A.D.) — recognizes the fraction. They do, it is true, recognize the fractional relation. Euclid, for instance, expressly declares that any number is either a multiple, a part, or parts (ju-cpr/), i.e. multiple of a part, of every other number (Euc. VII, 4), and he demonstrates such theorems as these : If A he the same parts of B that C is of D, then the sum or difference of A and C is the same parts of the sum or difference ofB and D that A is of B (VII, 6 and 8). If A he the same jyarts of B that C is of D, then, alternately, A is the same parts of C that B is of D (VII, 10). But the relation is expressed by two integers, that which indicates the part and that which indicates the multiple. It is a ratio, and Euclid has no more thought of expressing it except by two numbers than he has of expressing the ratio of two geometric magnitudes except by two magni- tudes. There is no conception of a single number, the fraction proper, the quotient of one of these integers by the other. In the apiOfi-qTiKa. of Diophantus, on the other hand, the last and transcendently the greatest achievement of the Greeks in the science of number, the fraction is granted the position in elementary arithmetic which it has held ever since. IV. OEiaiN or THE lEEATIONAL. 94, The Discovery of Irrational Lines. The Greeks attrib- uted the discovery of the Irrational to the mathematician and philosopher Pythagoras* (525 b.c). * This is t^exElicit declaration of the most reliable document extant on the history of geometry before Euclid, a chronicle of the ancient OBIGIN OF THE IRRATIONAL. 95 If, as is altogether probable,* the most famous theorem of Pythagoras — that the square on the hypothenuse of a right triangle is equal to the sum of the squares on the geometers which Proclus (a.d. 450) gives in his commentary on Euclid, deriving it from a history written by Eudemus about 330 b.c. This chronicle credits the Egyptians with the discovery of geometry and Thales (600 b.c.) with having first mtroduced this study into Greece. Thales and Pythagoras are the founders of the Greek mathematics. But while Thales should doubtless be credited with the first conception of an abstract deductive geometry in contradistinction to the practical empirical geometry of Egypt, the glory of realizing this conception belongs chiefly to Pythagoras and his disciples in the Greek cities of Italy (Magna Grsecia) ; for they established the principal theorems respecting rectilineal figures. To the Pythagoreans the discovery of many of the elementary properties of numbers is due, as well as the geometric form which characterized the Greek theory of numbers throughout its history. In the middle of the fifth century before Christ,Athens became the principal centre of mathematical activity. There Hippocrates of Chios (430 B.C.) made his contributions to the geometry of the circle, Plato (380 B.C.) to geometric method, Thesetetus (380 b.c.) to the doctrme of incommensurable magnitudes, and Eudoxus (300 b.c.) to the theory of proportion. There also was begun the study of the conies. About 300 B.C. the mathematical centre of the Greeks shifted to Alexandria, where it remained. The third century before Christ is the most brilliant period in Greek mathematics. At its beginning — in Alexandria — Euclid lived and taught and wrote his Elements, collecting, systematizing, and per- fecting the work of his predecessors. Later (about 250) Archimedes of Syracuse flourished, the greatest mathematician of antiquity and founder of the science of mechanics ; and later still (about 230) Apol- lonius of Perga, "the great geometer, '^ whose Conies mai'ks the cul- mination of Greek geometry. Of the later Greek mathematicians, besides Hero and Diophantus, of whom an account is given in the text, and the great summarizer of the ancient mathematics, Pappus (300 a.d.), only the famous astrono- mers Hipparchus (130 b.c) and Ptolemy (150 a.d.) call for mention here. To them belongs the invention of trigonometiy and the first trigonometric tables, tables of chords. The dates in this summary are from Gow's Hist, of Greek Math. * Compare Cantor, Geschichte der Mathematik, p. 153. 96 NUMBER-SYSTEM OF ALGEBRA. Other two sides — was suggested to him by the fact that 32 4- 42 = 52^ in connection with the fact that the triangle whose sides are 3, 4, 5, is right-angled, — for both almost certainly fell within the knowledge of the Egyptians, — he would naturally have sought, after he had succeeded in demonstrating the geometric theorem generally, for number triplets corresponding to the sides of any right triangle as do 3, 4, 5 to the sides of the particular triangle. The search of course proved fruitless, fruitless even in the case which is geometrically the simplest, that of the isosceles right triangle. To discover that it was necessarily fruitless ; in the face of preconceived ideas and the appar- ent testimony of the senses, to conceive that lines may exist which have no common unit of measure, however small that unit be taken ; to demonstrate that the hypothenuse and side of the isosceles right triangle actually are such a pair of lines, was the great achievement of Pythagoras.* 95. Consequences of this Discovery in Greek Mathematics. One must know the antecedents and follow the consequences of this discovery to realize its great significance. It was * His demonstration may easily have been the following, which was old enough in Aristotle's time (340 b.c.) to be made the subject of a popular reference, and which is to be found at the end of the 10th book in all old editions of Euclid's Elements : If there be any line which the side and diagonal of a square both contain an exact number of times, let their lengths in terms of this line be a and b respectively ; then lfl — 2 a^. The numbers a and b may have a common factor, 7 ; so that a=ay and b = 0y, where a and are prime to each other. The equation b^ = 2a^ then reduces, on the removal of the factor 7^ common to both its members, to ^2 _ 2 a^. From this equation it follows that ^, and therefore 0, is an even number, and hence tha|; a which is prime to is odd. — ■^=7''^ But set $ = 2 p', where &' is integral, in the equation 0^ = 2a.^ ; it becomes 4 j3'2 = 2 a'^, ^r 2 0''^ = a^, whence o^, and therefore a, is even. o has thus been proven to be both x>dd and even, and is therefore not a njimber. ORIGIN OF THE IRRATIONAL. 97 the first recognition of the fundamental difference between the geometric magnitudes and number, which Aristotle for- mulated brilliantly 200 years later in his famous distinc- tion between the continuous and the discrete, and as such was potent in bringing about that complete banishment of numerical reckoning from geometry which is so character- istic of this department of Greek mathematics in its best, its creative period. No one before Pythagoras had questioned the possibility of expressing all size relations among lines and surfaces in terms of number, — rational number of course. Indeed, except that it recorded a few facts regarding congruence of figures gathered by observation, the Egyptian geometry was nothing else than a meagre collection of formulas for com- puting areas. The earliest geometry was metrical. But to the severely logical Greek no alternative seemed possible, when once it was known that lines exist whose lengths — whatever unit be chosen for measuring them — cannot both be integers, than to have done with number and measurement in geometry altogether. Congruence be- came not only the final but the sole test of equality. For the study of size relations among unequal magnitudes a pure geometric theory of proportion was created, in which proportion, not ratio, was the primary idea, the method of exhaustions making the theory available for figures bounded by curved lines and surfaces. The outcome was the system of geometry which Euclid expounds in his Elements and of which Apollonius makes splendid use in his Conies, a system absolutely free from extraneous concepts or methods, yet, within its limits, of great power. It need hardly be added that it never occurred to the Greeks to meet the difficulty which Pythagoras' discovery had brought to light by inventing an irrational number, itself incommensurable with rational numbers. Eor arti- ficial concepts such as that they had neither talent nor liking. 98 NUMBER-SYSTEM OF ALGEBRA. On the other hand, they did develop the theory of irra- tional magnitudes as a department of their geometry, the irrational line, surface, or solid being one incommensurable with some chosen (rational) line, surface, solid. Such a theory forms the content of the most elaborate book of Euclid's Elements, the 10th. 96. Approximate Values of Irrationals. In the practical or metrical geometry which grew uj) after the pure geometry had reached its culmination, and which attained in the works of Hero the Surveyor almost the proportions of our modern elementary mensuration,* cqyproximate values of irrational numbers played a very important rdle. !Ni or do such approx- imations appear for the first time in Hero. In Archimedes' " Measure of the Circle " a number of excellent approxima- tions occur, among them the famous approximation — for TT, the ratio of the circumference of a circle to its diam- eter. The approximation - for V2 is reputed to be as old as Plato. It is not certain how these approximations were effected, f They involve the use of some method for extracting square roots. The earliest explicit statement of the method in common use to-day for extracting square roots of numbers (whether exactly or approximately) occurs in the com- mentary of Theon of Alexandria (380 a.d.) on Ptolemy's * The formula Vs (s — a)(s — b) (s — c) for the area of a triangle in terms of its sides is due to Hero. t Many attempts have been made to discover the methods of approximation used by Archimedes and Hero from an examination of their results, but with little success. The formula Va'^ ±h — a± — 2a will account for some of the simpler approximations, but no single method or set of methods have been foimd which will account for the more difficult. See Giinther: Die quadratischen Irrationalitaten der Alten und deren Entwicklungsmethoden. Leipzig, 1882. Also in Handbuch der klassischen Altertums-Wissenschaft, liter. Halbband. GREEK ALGEBRA. 99 Almagest. Theon, who like Ptolemy employs sexagesimal fractions, thus finds the length of the side of a square con- tainiu? 4500° to be 67° 1' 55". 97. The Later History of the Irrational is deferred to the chapters which follow (§§ 106, 108, 112, 121, 129). It will be found that the Indians permitted the simplest forms of irrational numbers, surds, in their algebra, and that they were followed in this by the Arabians and the mathema- ticians of the Kenaissance, but that the general irrational did not make its way into algebra until after Descartes. V. OEIGIN or THE NEGATIVE AND THE IMAGINARY. THE EQUATION. 98. The Equation in Egyptian Mathematics. While the irrational originated in geometry, the negative and the imaginary are of purely algebraic origin. They sprang directly from the algebraic equation. The authentic history of the equation, like that of geome- try and arithmetic, begins in the book of the old Egyptian scribe Ahmes. For Ahmes, quite after the present method, solves numerical problems which admit of statement in an equation of the first degree involving one unknown quantity.* 99. In the Earlier Greek Mathematics. The equation was slow in arousing the interest of Greek mathematicians. They were absorbed in geometry, in a geometry whose methods were essentially non-algebraic. To be sure, there are occasional signs of a concealed algebra under the closely drawn geometric cloak. Euclid * His symbol for the unknown quantity is the word hau, meaning heap. 100 NUMBER- ST STEM OF ALGEBRA. solves three geometric problems -which, stated algebraically, are but the three forms of the quadratic ; x' + ax = 6^, Qi? = ax-\- b-, x^-{-b- = ax* And the Conies of Apollonius, so astonishing if regarded as a product of the pure geo- metric method used in its demonstrations, when stated in the language of algebra, as recently it has been stated by Zeuthen,t almost convicts its author of the use of algebra as his instrument of investigation. 100. Hero. But in the writings of Hero of Alexandria (120 B.C.) the equation first comes clearly into the light again. Hero was a man of practical genius whose aim was to make the rich pure geometry of his predecessors available for the surveyor. With him the rigor of the old geometric method is relaxed ; proportions, even equations, among the measures of magnitudes are permitted where the earlier geometers allow only proportions among the magnitudes themselves ; the theorems of geometry are stated metrically, in formulas ; and more than all this, the equation becomes a recognized geometric instrument. Hero gives for the diameter of a circle in terms of s, the sum of diameter, circumference, and area, the formula : t , Viols -I- 841 -29 He could have reached this formula only by solving a quadratic equation, and that not geometrically, — the nature of the oddly constituted quantity s precludes that suppo- sition,— but by a purely algebraic reckoning like the following : The area of a circle in terms of its diameter being — -, * Elements, VI, 29, 28 ; Data, 84, 85. t Die Lehre von den Kegelschnitten im Altertum. Copenhagen, 1886. X See Cantor ; Geschichte der Mathematik, p. 341. GREEK ALGEBRA. 101 the length of its circumference Trd, and tt according to 22 Archimedes' approximation — ^, we have the equation : , , Trd^ , , 11 ,2 , 29 , s = cl-\ ^ttcL or :r~dr-\ d = s. 4 14 7 Clearing of fractions, multiplying by 11, and completing the square, 121 d^ + 638 d + 841 = 154 s + 841, whence 11 d + 29 = Vl54 s + 841, V154S4-841-29 or d = 11 Except that he lacked an algebraic symbolism, therefore, Hero was an algebraist, an algebraist of power enough to solve an affected quadratic equation. 101. Diophantus (300 a.d. ?). The last of the Greek mathematicians, Diophantus of Alexandria, was a great algebraist. The period between him and Hero was not rich in cre- ative mathematicians, but it must have witnessed a grad- ual development of algebraic ideas and of an algebraic symbolism. At all events, in the d/at^/tr/riKa of Diophantus the alge- braic equation has been supplied with a symbol for the unknown quantity, its powers and the powers of its recip- rocal to the 6th, and a symbol for equality. Addition is represented by mere juxtaposition, but there is a special symbol, //i, for subtraction. On the other hand, there are no general symbols for known quantities, — symbols to serve the purpose which the first letters of the alphabet are made to serve in elementary algebra nowadays, — there- fore no literal coefficients and no general formulas. With the symbolism had grown up many of the formal rules of algebraic reckoning also. Diophantus prefaces the 102 NUMBER-SYSTEM OF ALGEBRA. apiOfirjTiKd. with rules for the addition, subtraction, and mul- tiplication of polynomials. He states expressly that the product of two subtractive terms is additive. The apiOjj.rjTiKo. itself is a collection of problems concern- ing numbers, some of which are solved by determinate algebraic equations, some by indeterminate. Determinate equations are solved which have given posi- tive integers as coefficients, and are of any of the forms ax*" = hx^, ax'' + 6cc = c, aa.*^ -f- c = 6a;, el's? = hx-\- c\ also a single cubic equation, ar* + x = 4.^'^ + 4. In reducing equa- tions to these forms, equal quantities in opposite members are cancelled and subtractive terms in either member are rendered additive by transposition to the other member. The indeterminate equations are of the form ?/" = (^^ -}- 6x + c, Diophantus regarding any pair of positive rational numbers (integers or fractions) as a solution which, substi- tuted for y and x, satisfy the equation.* These equations are handled with marvellous dexterity in the dpiOixrjTiKo.. No effort is made to develop general comprehensive methods, but each exercise is solved by some clever device suggested by its individual peculiarities. Moreover, the discussion is never exhaustive, one solution sufficing when the possible number is infinite. Yet until some trace of indeterminate equations earlier than the apiOfx-qriKa. is discovered, Diophan- tus must rank as the originator of this department of mathematics. The determinate quadratic is solved by the method which we have already seen used by Hero. The equation is first multiplied throughout by a number which renders the co- efficient of a^ a perfect square, the " square is completed," the square root of both members of the equation taken, and * The designation "Diophantine equations," commonly applied to indeterminate equations of the first degree when investigated for inte- gral solutions, is a striking misnomer. Diophantus nowhere considers such equations, and, on the other hand, allows fractional solutions of indeterminate equations of the second degree. INDIAN ALGEBRA. 103 the value of x reckoned out from the result. Thus from acc2 _|_ c = 5x is derived first the equation ara? -\- ac = abx, then a?^ — abx + ( - ) = ( - ) —ac, then *^^ ~ o ~ v ( r ) — <^*c, V(IT- and finally, x = ^^ The solution is regarded as possible only when the num- ber under the radical is a perfect square (it must, of course, be positive), and only one root — that belonging to the positive value of the radical — is ever recognized. , Thus the number system of Diophantus contained only the positive integer and fraction ; the irrational is excluded ; and as for the negative, there is no evidence that a Greek mathematician ever conceived of such a thing, — certainly not Diophantus with his three classes and one root of affected quadratics. The position of Diophantus is the more interesting in that in the apid^i-qTiKa the Greek science of number culminates. 102. The Indian Mathematics. The pre-eminence in math- ematics passed from the Greeks to the Indians. Three mathematicians of India stand out above the rest: Arya- bhatta (born 476 a.d.), Brahmagupta (born 598 a.d.), Blids- kara (born 1114 a.d.). While all are in the first instance astronomers, their treatises also contain full expositions of the mathematics auxiliary to astronomy, their reckoning, algebra, geometry, and trigonometry.* * The mathematical chapters of Brahmagupta and Bhaskara have been translated into English by Colebrooke : "Algebra, Arithmetic, and Mensuration, from the Sanscrit of Brahmagupta and Bhaskara," 1817 ; those of Aryabhatta into French by L. Rodet (Journal Asiatique, 1879). 104 NUMBER-SYSTEM OF ALGEBRA. An examination of the writings of these mathematicians and of the remaining mathematical literature of India leaves little room for doubt that the Indian geometry was taken bodily from Hero, and the algebra — whatever there may have been of it before Aryabhatta — at least powerfully affected by Diophantus. Nor is there occasion for surprise in this. Aryabhatta lived two centuries after Diophantus and six after Hero, and during those centuries the East had frequent communication with the West through various channels. In particular, from Trajan's reign till later than 300 A.D. an active commerce was kept up between India and the east coast of Egypt by way of the Indian Ocean. Greek geometry and Greek algebra met very different fates in India. The Indians lacked the endowments of the geometer. So far from enriching the science with new discoveries, they seem with difficulty to have kept alive even a proper understanding of Hero's metrical formulas. But algebra flourished among them wonderfully. Here the fine talent for reckoning which could create a perfect nu- meral notation, supported by a talent equally fine for sym- bolical reasoning, found a great opportunity and made great achievements. With Diophantus algebra is no more than an art by which disconnected numerical problems are solved ; in India it rises to the dignity of a science, with general methods and concepts of its own. 103. Its Algebraic Symbolism. First of all, the Indians devised a complete, and in most respects adequate, sym- bolism. Addition was represented, as by Diophantus, by mere juxtaposition ; subtraction, exactly as addition, except that a dot was written over the coefficient of the subtra- hend. The syllable hha written after the factors indicated a product ; the divisor written under the dividend, a quo- tient ; a syllable, ka, written before a number, its (irrational) square root; one member of an equation placed over the other, their equality. The equation was also provided with INDIAN ALGEBRA. 105 symbols for any number of unknown quantities and their powers. 104. Its Invention of the Negative. The most note- worthy feature of this symbolism is its representation of subtraction. To remove the subtractive symbol from be- tween minuend and subtrahend (where Diophantus had placed his symbol //>), to attach it wholly to the subtrahend and then connect this modified subtrahend with the minuend additively, is, formally considered, to transform the sub- traction of a positive quantity into the addition of the •corresponding negative. It suggests what other evidence makes certain, that algebra oices to India the immensely useful concept of the absolute negative. Thus one of, these dotted numbers is allowed to stand by itself as a member of an equation. Bhdskara recognizes the double sign of the square root, as well as the impossi- bility of the square root of a negative number (which is very interesting, as being the first dictum regarding the imag- inary), and no longer ignores either root of the quadratic. More than this, recourse is had to the same expedients for interpreting the negative, for attaching a concrete phys- ical idea to it, as are in common use to-day. The primary meaning of the very name given the negative was debt, as that given- the positive was means. The opposition between the two was also pictured by lines described in opposite directions. 105. Its Use of Zero. But the contributions of the Ind- ians to the fund of algebraic concepts did not stop with the absolute negative. They made a number of 0, and though some of their reckonings with it are childish, Bhdskara, at least, had sufficient understanding of the nature of the " quotient " ~ (infinity) to say "it suffers no change, however much it is increased or diminished.'' He associates it with Deity. 106 NUMBER-SYSTEM OF ALGEBRA. 106. Its Use of Irrational Numbers. Again, the Indians were the hrst to reckon with irrational square roots as with numbers; Bhdskara extracting square roots of binomial surds and rationalizing irrational denominators of fractions even when these are polynomial. Of course they were as little able rigorously to justify such a procedure as the Greeks ; less able, in fact, since they had no equivalent of the method of exhaustions. But it probably never occurred to them that justification was necessary; they seem to have been unconscious of the gulf fixed between the discrete and continuous. And here, as in the case of and the negative, with the confidence of apt and successful reckoners, they were ready to pass immediately from numerical to purely symbolical reasoning, ready to trust their processes even where formal demonstration of the right to apply them ceased to be attainable. Their skill was too great, their instinct too true, to allow them to go far wrong. 107. Determinate and Indeterminate Equations in Indian Algebra. As regards equations — the only changes which the Indian algebraists made in the treatment of determinate equations were such as grew out of the use of the negative. This brought the triple classification of the quadratic to an end and secured recognition for both roots of the quadratic. Brahmagupta solves the quadratic by the rule of Hero and Diophantus, of which he gives an explicit and gen- eral statement, ^ridhara, a mathematician of some dis- tinction belonging to the period between Brahmagupta and Bh^skara, made the improvement of this method which consists in first multiplying the equation throughout by four times the coefficient of the square of the unknown quantity and so preventing the occurrence of fractions under the ra,dical sign.* Bhdskara also solves a few cubic and biquadratic equa- tions by special devices. * This method still goes under the name " Hindoo method." ARABIAN ALGEBRA. 107 The theory of indeterminate equations, on the other hand, made great progress in India. The achievements of the Indian mathematicians in this beautiful but difficult depart- ment of the science are as brilliant as those of the Greeks in geometry. They created the doctrine of the indetermi- nate equation of the first degree, ax + by = c, which they treated for integral solutions by the method of continued fractions in use to-day. They worked also with equations of the second degree of the forms aa:^-\- b = cy^, xy = ax -'t-hy + c, originating general and comprehensive methods where Dio- phantus had been content with clever devices. 108. The Arabian Mathematics. The Arabians were the instructors of modern Europe in the ancient mathematics. The service which they rendered in the case of the numeral notation and reckoning of India they rendered also in the case of the geometry, algebra, and astronomy of the Greeks and Indians. Their own contributions to mathematics are unimportant. Their receptiveness for mathematical ideas was extraordinary, but they had little originality. The history of Arabian mathematics begins with the reign of Almanstir (754-775),* the second of the Abbasid caliphs. It is related (by Ibn-al-Adami, about 900) that in this reign, in the year 773, an Indian brought to Bagdad certain astronomical writings of his country, which contained a method called " Sindhind," for computing the motions of the stars, — probably portions of the Siddhdnta of Brahma- gupta, — and that Alfazdri was commissioned by the caliph to translate them into Arabic, f Inasmuch as the Indian * It was Almansur who transferred the throne of the caliphs from Damascus to Bagdad which immediately became not only the capital city of Islam, but its commercial and intellectual centre. t This translation remained the guide of the Arabian astronomers until the reign of Almamiin (813-833), for whom Alkhwarizmi pre- pared his famous astronomical tables (820). Even these were based chiefly on the "Sindhind," though some of the determinations were made by methods of the Persians and Ptolemy. 108 NUMBER-SYSTEM OF ALGEBRA. astronomers put full expositions of their reckoning, algebra, and geometry into their treatises, Alfazdri's translation laid open to his countrymen a rich treasure of mathematical ideas and methods. It is impossible to set a date to the entrance of Greek ideas. They must have made themselves felt at Damascus, the residence of the later Omayyad caliphs, for that city had numerous inhabitants of Greek origin and culture. But the first translations of Greek mathematical writings were made in the reign of H^rfln Arraschid (786-809), when Euclid's Elements and Ptolemy's Almagest were put into Arabic. Later on, translations were made of Archimedes, Apollo- nius. Hero, and last of all, of Diophantus (by Abft'l Wafd, 940-998). The earliest mathematical author of the Arabians is Alkhwarizmi, who flourished in the first quarter of the 9th century. Besides astronomical tables, he wrote a treatise on algebra and one on reckoning (elementary arithmetic). The latter has already been mentioned. It is an exposition of the positional reckoning of India, the reckoning which mediaeval Europe named after him Algorithm. The treatise on algebra bears a title in which the word Algebra appears for the first time: viz., Aldjebr walmu- Mbala. Aldjebr (i.e. reduction) signifies the making of all terms of an equation positive by transferring negative terms to the opposite member of the equation ; almuJcabala (i.e. opposition), the cancelling of equal terms in opposite mem- bers of an equation. Alkhwarizmi's classification of equations of the 1st and 2d degrees is that to which these processes would naturally lead, viz. : ax^ = bx, bo^=c, bx = c, a^ + bx = c, a^ + c=bx, a^ = 6a; + c. These equations he solves separately, following up the solution in each case with a geometric demonstration of its ARABIAN ALGEBRA. 109 correctness. He recognizes both roots of the quadratic when they are positive. In this respect he is Indian ; in all others — the avoidance of negatives, the use of geometric demonstration — he is Greek. Besides Alkhwarizmi, the most famous algebraists of the Arabians were Alkarchl and Alchayyaml, both of whom lived in the 11th century. Alkarchi gave the solution of equations of the forms : ax^p + hx^ = c, ao(?p + c = hx^, hx'' -\-c = ay?^. He also reckoned with irrationals, the equations V8 + V18=V50, a/sI - ^/2 = a/16, being pretty just illustrations of his success in this field. Alchayydmi was the first mathematician to make a system- atic investigation of the cubic equation. He classified the various forms which this equation takes when all its terms are positive, and solved each form geometrically — by the intersections of conies.* A pure algebraic solution of the cubic he believed impossible. Like Alkhwarizmi, Alkarchi and Alchayydmi were Eastern Arabians. But early in the 8th century the Arabians con- quered a great part of Spain. An Arabian realm was established there which became independent of the Bagdad caliphate in 747, and endured for 300 years. The inter- course of these Western Arabians with the East was not * Thus suppose the equation x^-\-hx = a, given. For h substitute the quantity f; and for o, f^r. Then x^ = p\r — x). Now this equation is the result of ehminating y from between the two equations, x^ = p.y, y^r=x{r — x); the first of which is the equa- tion of a parabola, the second, of a circle. Let these two curves be constructed ; they will intersect in one real point distinct from the origin, and the abscissa of this point is a root of a:3 + 6z = a. See Hankel, Geschichte der Mathematik, p. 279. This method is of greater interest in the history of geometry than in that of algebra. It involves an anticipation of some of the most important ideas of Descartes' Geometrie (see p. 118). 110 NUMBER-SYSTEM OF ALGEBRA. frequent enougli to exercise a controlling influence on their aesthetic or scientific development. Their mathematical productions are of a later date than those of the East and almost exclusively arithmetico-algebraic. They con- structed a formal algebraic notation which went over into the Latin translations of their writings and rendered the path of the Europeans to a knowledge of the doctrine of equations easier than it would have been, had the Arabians of the East been their only instructors. The best known of their mathematicians are Ibn AflaJi (end of 11th century), Ibn Albannd (end of 13th century), Alkasddi (15th century). 109. Arabian Algebra Greek rather than Indian. Thus, of the three greater departments of the Arabian mathematics, the Indian influence gained the mastery in reckoning only. The Arabian geometry is Greek through and through. While the algebra contains both elements, the Greek pre- dominates. Indeed, except that both roots of the quadratic are recognized, the doctrine of the determinate equation is altogether Greek. It avoids the negative almost as care- fully as Diophantus does ; and in its use of the geometric method of demonstration it is actuated by a spirit less modern still — the spirit in Avhich Euclid may have con- ceived of algebra when he solved his geometric quadratics. The theory of indeterminate equations seldom goes beyond Diophantus ; where it does, it is Indian. The Arabian trigonometry is based on Ptolemy's, but is its superior in two important particulars. It employs the sine where Ptolemy employs the chord (being in this re- spect Indian), and has an algebraic instead of a geometric form. Some of the methods of approximation used in reckoning out trigonometric tables show great cleverness. Indeed, the Arabians make some amends for their ill-advised return to geometric algebra by this excellent achievement in algebraic geometry. The preference of the Arabians for Greek algebra was EARLY EUROPEAN ALGEBRA. Ill especially unfortunate in respect to the negative, which was in consequence forced, to repeat in Europe the fight for recognition which it had already won in India. 110. Mathematics in Europe before the Twelfth Century. The Arabian mathematics found entrance to Christian Eu- rope in the 12th century. During this century^and the first half of the next a good part of its literature was translated into Latin. Till then the plight of mathematics in Europe had been miserable enough. She had no better rei^resentatives than the Romans, the most deficient in the sense for mathematics of all cultured peoples, ancient or modern ; no better lit- erature than the collection of writings on surveying known as the Codex Arcerianus, and the childish arithmetic and geometry of Boetius. Prior to the lOtli century, however, Xorthern Europe had not sufficiently emerged from barbarism to call even this paltry mathematics into requisition. What learning there was was confined to the cloisters. Reckoning {computus) was needed for the Church calendar and was taught in the cloister schools established by Alcuin (735-804) under the patronage of Charlemagne. Reckoning was commonly done on the fingers. Not even was the multiplication table gen- erally learned. Reference would be made to a written copy of it, as nowadays reference is made to a table of loga- rithms. The Church did not need geometry, and geometry in any proper sense did not exist. 111. Gerbert. But in the 10th century there lived a man of true scientific interests and gifts, Gerbert,* Bishop of Rheims, Archbishop of Ravenna, and finally Pope Sylvester II. In him are the first signs of a new life for mathematics. His achievements, it is true, do not extend beyond the revival of Roman mathematics, the authorship of a geom- * See § 88. 112 NUMBER-SYSTEM OF ALGEBRA. etry based on the Codex Arcerianus, and a method for effect- ing division on the abacus with apices. Yet these acliieve- meuts are enough to place him far above his contemporaries. His influence gave a strong impulse to mathematical studies where interest in them had long been dead. He is the fore- runner of the intellectual activity ushered in by the trans- lations from the Arabic, for he brought to life the feeling of the need for mathematics which these translations were made to satisfy. 112. Entrance of the Arabian Mathematics. Leonardo. It was the elementary branch of the Arabian mathematics which took root quickest in Christendom — reckoning with nine digits and 0. Leonardo of Pisa — Fibonacci, as he was also called — did great service in the diffusion of the new learning through his Liber Abaci (1202 and 1228), a remarkable presentation of the arithmetic and algebra of the Arabians, which remained for centuries the fund from which reckoners and algebraists drew and is indeed the foundation of the modern science. The four fundamental operations on integers and frac- tions are taught after the Arabian method ; the extraction of the square root and the doctrine of irrationals are presented in their pure algebraic form ; quadratic equations are solved and applied to quite complicated problems ; negatives are accepted when they admit of interpretation as debt. The last fact illustrates excellently the character of the Liber Abaci. It is not a mere translation, but an inde- pendent and masterly treatise in one department of thf» new mathematics. Besides the Liber Abaci, Leonardo wrote the Practicu Geometriae, which contains much that is best of Euclid, Archimedes, Hero, and the elements of trigonometry ; also the Liber Quadratorum, a collection of original algebraic problems most skilfully handled. EARLY EUROPEAN ALGEBRA. 113 113. Mathematics during the Age of Scholasticism. Leo- nardo was a great luatiieinatician,* but line as his work was, it bore no fruit until the end of the 15th century. In him there had been a brilliant response to the Arabian impulse. But the awakening was only momentary ; it quickly yielded to the heavy lethargy of the " dark " ages. The age of scholasticism, the age of devotion to the forms of thought, logic and dialectics, is the age of greatest dul- lness and confusion in mathematical thinking. f Algebra owes the entire period but a single contribution ; the concept of the fractional power. Its author was Nicole Oresme (died 1382), who also gave a symbol for it and the rules by which reckoning with it is governed. * Besides Leonardo there flourished in the first quarter of the 13th century an able German matliematician, Jordanus Ncinorarius. He was the author of a treatise entitled De numeris daiis, in which known quantities are for the first time represented by letters, and of one De tranguUs which is a rich though rather systemless collection of theorems and problems principally of Greek and Arabian origin. See Giinther : Geschichte des mathemathischen Untenlchts im deutschen JMittelalter, p. 156. t Compare Hankel, Geschichte der Mathematik, pp. .349-352. To the unfruitfulness of these centmles the Summa of Luca Pacioli bears witness. This book, which has the distinction of being the earliest book on algebra printed, appeared in 1494, and embodies the arith- metic, algebra, and geometry of the time just preceding the Renais- sance. It contains not an idea or method not already presented by Leonardo. Even in respect to algebraic symbolism it surpasses the Liber Abaci only to the extent of using abbreviations for a few frfe- quently recurring words, as p. for "plus," and R. for "res" (the unknown quantity) . And this is not to be regarded as original with Pacioli for the Arabians of Leonardo's time made a similar use of abbreviations. In a translation made by Gerhard of Cremona (12th century) from an unknown Arabic original the letters r (radix), c (census) , d (dragma) are used to represent the unknown quantity, its square, and the absolute term respectively. The Summa of Pacioli has great merits, notwithstanding its lack of originality. It satisfied the mathematical needs of the time. It is very comprehensive, containing full and excellent instruction in the 114 NUMBER-SYSTEM OF ALGEBRA. 114. The Renaissance. Solution of the Cubic and Bi- quadratic Equations. The lirst achieveiuent in algebra by the mathematicians of the Eenaissance was the algebraic solution of the cubic equation : a fine beginning of a new era in the history of the science. The cubic cc^ + mx = n was solved by Ferro of Bologna in 1505, and a second time and independently, in 1535, by Ferro's countryman, Tartaglla, who by help of a transfor- mation made his method apply to x^ ± moi? —±n also. But Cardan of Milan was the first to publish the solution, in his Ars Magna,* 1545. The Ars Magna records another brilliant discovery : the solution — after a general method — of the biquadratic ic* + 6 a^ + 36 = 60 cc by Ferrari, a pupil of Cardan. Thus in Italy, within fifty years of the new birth of algebra, after a pause of sixteen centuries at the quad- ratic, the limits of possible attainment in the algebraic solution of equations were reached; for the algebraic solution of the general equation of a degree higher than 4 is impossible, as was first demonstrated by Abel.f The general solution of higher equations proving an obstinate problem, nothing was left the searchers for the art of reckoning after the methods of Leonardo, for the merchant- man, and a great variety of matter of a purely theoretical interest also — representing the elementary theory of numbers, algebra, geometry, and the application of algebra to geometry. Compare Cantor, Geschichte der Mathematik, II, p. 308. It should be added that the 15th century produced a mathemati- cian who deserves a distinguished place in the general history of mathematics on account of his contributions to trigonometry, the astronomer Regiomontanus (1436-1476). Like Jerdanus, he was a German. * The proper title of this work is : " Artis magnae sive de regulis Algebraicis liber unus." It has stolen the title of Cardan's "Ars magna Arithmeticae," published at Basel, 1570. t Mdmoire sur les Equations Algebriques : Christiania, 1826. Also in Crelle's Journal, I, p. 65. EARLY EUROPEAN ALGEBRA. 115 roots of equations but to devise a metliod of working them out approximately. In this the French mathematician Vieta (1540-1603) was successful, his method being essen- tially the same as that now known as Newton's. 115. The Negative in the Algebra of this Period. First Appearance of the Imaginary. But the general equation presented other problems than the discovery of rules for obtaining its roots ; the nature of these roots and the relations between them and the coefficients of the equation invited inquiry. We witness another phase of the struggle of the negative for recognition. The imaginary is now ready to make com- mon cause with it. Already in the Ars Magna Cardan distinguishes between numeri veri — the positive integer, fraction, and irrational, — and numeri jicti, or falsi — the negative and the square root of the negative. Like Leonardo, he tolerates negative roots of equations when they admit of interpretation as " debitum," not otherwise. While he has no thought of accepting im- aginary roots, he shows that if 5 + V— 15 be substituted for a; in a; (10 — x) = 40, that equation is satisfied ; which, of course, is all that is meant nowadays when 5 + V— 15 is called a root. His declaration that 5 ± V— 15 are " vere sophistica " does not detract from the significance of this, the earliest recorded instance of reckoning with the imaginary. It ought perhaps to be added that Cardan is not always so successful in these reckonings ; for in another place he sets \{-i 47 \64 8 Following Cardan, Bombelli* reckoned with imaginaries to good purpose, explaining by their aid the irreducible case in Cardan's solution of the cubic. * L' Algebra, 1579. He also formally states rules for reckoning with ± V— 1 and a + 6 V — 1. 116 NUMBER-SYSTEM OF ALGEBRA. On the other hand, neither Vieta nor his distinguished follower, the Englishman Harriot (1560-1621), accept even negative roots ; though Harriot does not hesitate to perform algebraic reckonings on negatives, and even allows a nega- tive to constitute one member of an equation. 116. Algebraic Symbolism. Vieta and Harriot. Vieta and Harriot, however, did distinguished service in perfect- ing the symbolism of algebra ; Vieta, by the systematic use of letters to represent known quantities, — algebra first became "literal" or ''universal arithmetic" in his hands,* — Harriot, by ridding algebraic statements of every non- symbolic element, of everything but the letters which rep- resent quantities known as well as unknown, symbols of operation, and symbols of relation. Harriot's Artis Analy- ticae Praxis (1631) has quite the appearance of a modern algebra, t * There are isolated instances of this use of letters much earlier than Vieta in the De numeris datis of Jordanus Nemorarius, and in the Algorithmus demonstratus of the same author. But the credit of making it the general practice of algebraists belongs to Vieta. t One has only to reflect how much of the power of algebra is due to its admirable symbolism to appreciate the importance of the Artis Analyticae Praxis, in which this symbolism is finally established. But one addition of consequence has since been made to it, integral and fractional exponents introduced by Descartes (1637) and Wallis (1659). Harriot substituted small letters for the capitals used by Vieta, but followed Vieta in representing known quantities by consonants and imknown by vowels. The present convention of representing known quantities by the earlier letters of the alphabet, miknown by the later, is due to Descartes. Vieta' s notation is unwieldy and ill adapted to purposes of alge- braic reckoning. Instead of restricting itself, as Harriot's docs, to the use of brief and easily apprehended conventional symbols, it also employs words subject to the rules of syntax. Thus for A^ — 3 B^A = Z (or aaa — 3bha = z, as Harriot would have written it), Vieta writes A culms — B quad 3 in A aequatur Z solido. In this respect Vieta is inferior not only to Harriot, but to several of his predecessors and EARLY EUROPEAN ALGEBRA. 117 117. Fundamental Theorem of Algebra. Harriot and Girard. Harriot has been credited with the discovery of the " fundamental theorem " of algebra — the theorem that the number of roots of an algebraic equation is the same as its degree. The Artis Analyticae Praxis contains no mention of this theorem — indeed, by ignoring negative and imagi- nary roots, leaves no place for it; yet Harriot develops systematically a method which, if carried far enough, leads to the discovery of this theorem as well as to the relations holding between the roots of an equation and its coefficients. By multiplying together binomial factors which involve the unknown quantity, and setting their product equal to 0, he builds "canonical" equations, and shows that the roots of these equations — the only roots, he says — are the positive values of the unknown quantity which render these binomial factors 0. Thus he builds aa — ha — ca= —be, in which a is the unknown quantity, out of the factors a — b, a -\-c, and proves that 6 is a root of this equation and the only root, the negative root c being totally ignored. While no attempt is made to show that if the terms of a " common " equation be collected in one member, this can notably to his contemporary, the Dutch mathematician Stevmus (1548-1620), Miio would, for instance, have written x^ + Sr — 8 as 1© + 30 — 80. The geometric aiiiliations of Vieta's notation are obvious. It suggests the Greek arithmetic. It is surprising that algebraic symbolism should owe so little to the great Italian algebraists of the 16th century. Like Pacioli (see note, p. 113) they were content with a few abbreviations for words, a " syncopated" notation, as it has been called, and an incomplete one at that. The current symbols of operation and relation are chiefly of English and German origin, havuig been invented or adopted as follows : viz. = , by Becorde in 1556 ; -y/, by Rudolf in 1525 ; the vinculum, by Vieta in 1591 ; brackets, by Bombelli, 1572 ; -^ , by Rahn in 1659 ; x , >, <, by Harriot in 1631. The signs + and — occur in a 15th century manu- script discovered by Gerhardt at Vienna. The notations a — h and - for the fraction were adopted from the Arabians. 118 NUMBER-SYSTEM OF ALGEBBA. be separated into binomial factors, the case of canonical equations raised a strong presumption for the soundness of this view of the structure of an equation. The first statement of the fundamental tlieorem and of the relations between coefficients and roots occurs in a remarkably clever and modern little book, the Inven- tion Nouvelle en VAlyebre, of Albert Girard, published in Amsterdam in 1629, two years earlier, therefore, than the Artis Analyticae Praxis. Girard stands in, no fear of imag- inary roots, but rather insists on the wisdom of recognizing them. They never occur, he says, except when real roots are lacking, and then in number just sufficient to fill out the entire number of roots to equality with the degree of the -equation. Girard also anticipated Descartes in the geometrical in- terpretation of negatives. But the Invention Nouvelle does not seem to have attracted much notice, and the genius and authority of Descartes were needed to give the interpreta- tion general currency. VI. AOOEPTANOE OF THE NEaATIVE, THE GENEEAL lEEATIONAL, AND THE IMAGINAEY AS NUMBEES. 118. Descartes' Geometrie and the Negative. The Geome- trie of Descartes appeared in 1637. This famous little trea- tise enriched geometry with a general and at the same time simple and natural method of investigation : the method of representing a geometric curve by an equation, which, as Descartes puts it, expresses generally the relation of its points to those of some chosen line of reference.* To form such equations Descartes represents line segments by letters, — the known by a, b, c, etc., the unknown by x and y. He * Seo Geometrie, Livre II. In Cousin's edition of Descartes' works, Vol. V, p. 337. THE NEGATIVE. 119 supposes a perpendicular, y, to be dropped from any point of the curve to the line of reference, and then the equation to be found from the known properties of the curve which connects y with x, the distance of y from a fixed point of the line of reference. This is the equation of the cu.rve in that it is satisfied by the x and y of each and every curve-point.* To meet the difficulty that the mere length of the perpen- dicular {y) from a curve-point will not indicate to which side of the line of reference the point lies, Descartes makes the convention that perpendiculars on opposite sides of this line (and similarly intercepts {x) on opposite sides of the point of reference) shall have opposite algebraic signs. This convention gave the negative a new position in mathematics. Not only was a " real " interpretation here found for it, the lack of which had made its position so dif- ficult hitherto, but it was made indispensable, placed on a footing of equality with the positive. The acceptance of the negative in algebra kept pace with the spread of Descar- tes' analytical method in geometry. 119. Descartes' Geometric Algebra. But the Geometrie has another and perhaps more important claim on the atten- tion of the historian of algebra. The entire method of the book rests on the assumption — made only tacitly, to be sure, and without knowledge of its significance — that two algebras are formally identical whose fundamental opera- tions are formally the same ; i.e. subject to the same laws of combination. For the algebra of the Geometrie is not, as is commonly said, mere numerical algebra, but what may for want of a better name be called the algebra of line segments. Its symbolism is the same as that of numerical algebra; but * Descartes fails to recognize a number of the conventions of our modem Cartesian geometry. He makes no formal choice of two ax^'s of reference, calls abscissas >/ and ordinates .r, and as frequently regards as positive ordinates below the axis of abscissas as ordinates above it. 120 NUMBER-SYSTEM OF ALGEBRA. symbols which there represent numbers here represent line segments. Not only is this the case with the letters a, b, x, y, etc., which are mere names {noms) of line segments, not their numerical measures, but with the algebraic combina- tions of these letters, a + h and a — h are respectively the sum and difference of the line segments a and 6; 06, the fourth proportional to an assumed unit line, a, and h ; -, the b fourth proportional to b, a, and the unit line ; and Va, -Va, etc., the first, second, etc., mean proportionals to the unit line and a* Descartes' justification of this use of the symbols of numerical algebra" is that the geometric constructions of which he makes a + b, a — b, etc., represent the results are " the same " as numerical addition, subtraction, multiplica- tion, division, and evolution, respectively. Moreover, since all geometric constructions which determine line segments may be resolved into combinations of these constructions as the operations of numerical algebra into the fundamental operations, the correspondence which holds between these fundamental constructions and operations holds equally between the more complex constructions and operations. The entire system of the geometric constructions under consideration may therefore be regarded as formally iden- tical with the system of algebraic operations, and be represented by the same symbolism. In what sense his fundamental constructions are "the same" as the fundamental operations of arithmetic, Des- cartes does not explain. The true reason of their formal identity is that both are controlled by the commutative, associative, and distributive laws. Thus in the case of the former as of the latter, ab = ba, and a{bc) = abc; for the fourth proportional to the unit line, a, and b is the same as the fourth proportional to the unit line, b, and a ; and the fourth proportional to the unit line, a, and be is the same as * Geomgtrie, Livre I. Ibid. pp. .31.3-314. THE GENERAL IRRATIONAL. 121 tlie fourth proportional to the unit line, ab, and c. But this reason was not within the reach of Descartes, in whose day the fundamental laws of numerical algebra had not yet been discovered. 120. The Continuous Variable. Newton. Euler. It is customary to credit the Geometrie with having introduced the continuous variable into mathematics, but without suffi- cient reason. Descartes prepared the way for this con- cept, but he makes no use of it in the Geometrie. The x and y which enter in the equation of a curve he regards not as variables but as indeterminate quantities, a pair of whose values correspond to each curve-point.* The real author of this concept is Newton (1642-1727), of whose great inven- tion, the method of fluxions, continuous variation, "flow," is the fundamental idea. But Newton's calculus, like Descartes' algebra, is geo- metric rather than purely numerical, and his followers in England, as also, to a less extent, the followers of his great rival, Leibnitz, on the continent, in employing the calculus, for the most part conceive of variables as lines, not num- bers. The geometric form again threatened to become para- mount in mathematics, and geometry to enchain the new " analysis " as it had formerly enchained the Greek arith- metic. It is the great service of Euler (1707-1783) to have broken these fetters once for all, to have accepted the con- tinuously variable number in its purity, and therewith to have created the pure analysis. For the relations of con- tinuously variable numbers constitute the field of the pure analysis ; its central concept, the function, being but a device for representing their interdependence. 121. The General Irrational. While its concern with variables puts analysis in a certain opposition to elementary algebra, concerned as this is with constants, its establish- * Ggometrie, Livre II. Ibid. pp. 337-338. 122 NUMBER-SYSTEM OF ALGEBRA. ment of the continuously variable number in mathematics brought about a rich addition to the number-system of alge- bra — the general irrational. Hitherto the only irrational numbers had been ''surds," impossible roots of rational numbers ; henceforth their domain is as wide as that of all possible lines incommensurable with any assumed unit line. 122. The Imaginary, a Recognized Analytical Instrument. Out of the excellent results of the use of the negative grew a spirit of toleration for the imaginary. Increased atten- tion was paid to its properties. Leibnitz noticed the real sum of conjugate imaginaries (1676-7) ; Demoivre dis- covered (1730) the famous theorem (cos 6 -\-i sin Oy = cos n6 + i sin n$ ; and Euler (1748) the equation cos 6 -f i sin = e'^, which plays so great a rdle in the modern theory of functions. Euler also, practising the method of expressing complex numbers in terms of modulus and angle, formed their prod- ucts, quotients, powers, roots, and logarithms, and by many brilliant discoveries multiplied proofs of the power of the imaginary as an analytical instrument. 123. Argand's Geometric Representation of the Imaginary. But the imaginary was never regarded as anything better than an algebraic fiction — to be avoided, where possible, by the mathematician who prized purity of method — until a method was discovered for representing it geometrically. A Norwegian, Wessel,* published such a method in 1797, and a Frenchman, Argand, the same method independently in 1806. * See W. W. Beman in Proceedings of the American Association for the Advancement of Science, 1897. THE COMPLEX NUMBER. 123 As + 1 and — 1 may be represented by unit lines drawn in opposite directions from any point, 0, and as i (i.e.V — 1) is a mean proportional to + 1 and — 1, it occurred to Argand to represent this symbol by the line whose direction with respect to the line + 1 is the same as the direction of the line — 1 with respect to it ; viz., the unit perpendicular through to the 1 — line. Let only the direction of the 1 — line be fixed, the position of the point in the plane is altogether indifferent. Between the segments of a given line, whether taken in the same or opposite directions, the equation holds : AB+BC=Aa It means nothing more, however, when the directions of AB and BC are opposite, than that the result of carrying a moving point from A first to B, and thence back to C, is the same as carrying it from A direct to C. But in this sense the equation holds equally when A, B, C are not in the same right line. Given, therefore, a complex number, a + ib ; choose any point A in the plane ; from it draw a line AB, of length a, in the direction of the 1 — line, and from B a line BC, of length b, in the direction of the i — line. The line AC, thus fixed in length and direction, but situated anywhere in the plane, is Argand's picture of a + ib. Argand's skill in the use of his new device was equal to the discovery of the demonstration given in § 54, that every algebraic equation has a root. 124. Gauss. The Complex Number. The method of rep- resenting complex numbers in common use to-day, that described in § 42, is due to Gauss. He was already in pos- session of it in 1811, though he published no account of it until 1831. To Gauss belongs the conception of i as an independent unit co-ordinate with 1, and of a + ib as a complex number, 124 NUMBER-SYSTEM OF .ALGEBRA. a sum of multiples of the units 1 and i; his also is the name " complex number " and the concept of complex num- bers in general, whereby a + ib secures a footing in the theory of numbers as well as in algebra. He too, and not Argand, must be credited with really breaking down the opposition of mathematicians to the imaginary. Argand's Essai was little noticed when it ap- peared, and soon forgotten ; but there was no withstanding the great authority of Gauss, and his precise and masterly presentation of this doctrine.* VII. EEOOGNITION OF THE PUEELY SYMBOLIC OHAEAOTEE OF ALGEBEA. QUATERNIONS. AUSDEHNUNGSLEHRE. 125. The Principle of Permanence. Thus, one after another, the fraction, irrational, negative, and imaginary, gained entrance to the number-system of algebra. Not one of them was accepted until its correspondence to some actually existing thing had been shown, the fraction and irrational, which originated in relations among actually ex- isting things, naturally making good their position earlier than the negative and imaginary, which grew immediately out of the equation, and for which a " real " interpretation had to be sought. Inasmuch as this correspondence of the artificial numbers to things extra-arithmetical, though most interesting and the reason of the practical usefulness of these numbers, has not the least bearing on the nature of their position in pure arithmetic or algebra ; after all of them had been accepted as numbers, the necessity remained of justifying this * See Gauss, Complete "Works, II, p. 174. SYMBOLIC CHARACTER OF ALGEBRA. 125 acceptance by purely algebraic considerations. This Avas first accomplished, though incompletely, by the English mathematician. Peacock* Peacock begins with a valuable distinction between arilh- metical and symbolical algebra. Letters are employed in the former, but only to represent positive integers and frac- tions, subtraction being limited, as in ordinary arithmetic, to the case where subtrahend is less than minuend. In the latter, on the other hand, the symbols are left altogether general, untrammelled at the outset with any particular meanings whatsoever. It is then assumed that the rules of operation applying to the symbols of arithmetical algebra apply without altera- tion in symbolical algebra ; the meanings of the operations themselves and their results being derived from these rides of ojieration. This assumption Peacock names the Principle of Perma- nence of Equivalent Forms, and illustrates its use as follows : t In arithmetical algebra, when a^b, c> d, it may readily be demonstrated that (a — 6) (c — d) = ac — ad — bc-\- bd. By the principle of permanence, it follows that (0-b)(0-d) = OxO-Oxd-bxO-\-bd, or (-b)(-d) = bd. Or again. In arithmetical algebra a'"a" = a'"+", when m and n are positive integers. Applying the principle of permanence, p p p (a?)' = ai • av '-'to q factors = a? p whence a« = i/a". p p -+ -H — to q terms * Arithmetical and Symbolical Algebra, 1830 and 1845 ; especially the later edition. Also British Association Reports, 1833. t Algebra, edition of 1845, §§ 631, 569, 639. 126 NUMBER-SYSTEM OF ALGEBRA. Here the meanings of the product {—b)(-d) and of the symbol a? are both derived from certain rules of operation in arithmetical algebra. Peacock notices that the symbol = also has a wider mean- ing in symbolical than in arithmetical algebra ; for in the former = means that "the expression which exists on one side of it is the result of an operation which is indicated on the other side of it and not performed." * He also points out that the terms "real" and "imagi- nary" or "impossible" are relative, depending solely on the meanings attaching to the symbols in any particular application of algebra. For a quantity is real when it can be shown to correspond to any real or possible existence ; otherwise it is imaginary. f The solution of the problem : to divide a group of 5 men into 3 equal groups, is imaginary though a positive fraction, while in Argand's geometry the so-called imaginary is real. The principle of permanence is a fine statement of the assumption on which the reckoning with artificial numbers depends, and the statement of the nature of this dependence is excellent. Regarded as an attempt at a complete presen- tation of the doctrine of artificial numbers, however. Pea- cock's Algebra is at fault in classing the positive fraction with the positive integer and not with the negative and imaginary, where it belongs, in ignoring the most difficult of all artificial numbers, the irrational, in not defining- arti- ficial numbers as symbolic results of operations, but princi- pally in not subjecting the operations themselves to a final analysis. 126. The Fundamental Laws of Algebra. "Symbolical Algebras." Of the fundamental laws to which this analysis leads, two, the commutative and distributive, had been noticed years before Peacock by the inventors of symbolic * Algebra, Appendix, § 631. t Ibid. § 557. SYMBOLIC CHARACTER OF ALGEBRA. 127 methods in the differential and integral calculus as being common to number and the operation of diiierentiation. In fact, one of these mathematicians, Servois,* introduced the names coinnintative and distributive. Moreover, Peacock's contemporary, Gregory, in a paper "On the Real Nature of Symbolical Algebra," which ap- peared in the interim between the two editions of Peacock's Algebra,! had restated these two laws, and had made their significance very clear. To Gregor}'- the formal identity of complex operations with the differential operator and the operations of numer- ical algebra suggested the comprehensive notion of algebra embodied in his fine definition : " symbolical algebra is the science which treats of the combination of operations de- fined not by their nature, that is, by what they are or what they do, but by the laws of combination to which they are subject." This definition recognizes the possibility of an entire class of algebras, each characterized i)rimarily not by its subject- matter, but by its operations and the formal laics to which they are subject; and in which the algebra of the complex num- ber a -\- ib and the system of operations with the differential operator are included, the two (so far as their laws are identical) as one and the same particular case. So long, however, as no " algebras " existed whose laws differed from those of the algebra of number, this definition had only a speculative value, and the general acceptance of * Gergonne's Annales, 1813. One must go back to Euclid for the earliest known recognition of any of these laws. Euclid demonstrated, of integers (Elements, VII, 16), that ah= ba. t In 18.38. See The Mathematical Writings of D. F. Gregory, p. 2. Among other writings of this period, which promoted a correct under- standing of the artificial numbers, should be mentioned Gregory's interesting paper, "On a Difficulty in the Theory of Algebra," Writ- ings, p. 235, and De Morgan's papers "On the Foundation of Algebra " (1839, 1841 ; Cambridge PhOosophical Transactions, VII). 128 NUMBEE-SYSTEM OF ALGEBRA. the dictum that the laws regulating its operations consti- tuted the essential character of algebra might have been long delayed had not Gregory's paper been quickly followed by the discovery of two "algebras," the quaternions of Hamilton and the AusdehnungsleJire of Grassmann, in which one of the laws of the algebra of number, the commutative law for multiplication, had lost its validity. 127. Quaternions. According to his own account of the discovery,* Hamilton came upon quaternions in a search for a second imaginary unit to correspond to the perpendicu- lar which may be drawn in space to the lines 1 and i. In pursuance of this idea he formed the expressions, a-{-ib-\-jc, x-\-iy-\-jz, in which a, h, c, x, y, z were sup- posed to be real numbers, and j the new imaginary unit sought, and set their product (a + ib -\-jc){x-\- iy -\-jz) — ax — by — cz-\-i {ay + hx) +j {az + ex) + ij {bz + cy) . The question then was, what interpretation to give (/. It would not do to set it equal to a'+ib'+jc\ for then the theorem that the modulus of a product is equal to the product of the moduli of its factors, which it seemed indis- pensable to maintain, would lose its validity ; unless, in- deed, a' = b' = c' = 0, and therefore ij = 0, a very unnatural supposition, inasmuch as 1 i is different from 0. No course was left for destroying the ij term, therefore, but to make its coefficient, bz -f cy, vanish, which was tanta- mount to supposing, since b, c, y, z'are perfectly general, that ji = — ij. Accepting this hypothesis, denial of the commutative law as it was, Hamilton was driven to the conclusion that the system upon which he had fallen contained at least three imaginary units, the third being the product ij. He called * Philosophical Magazine, II, Vol. 25, 1844. SYMBOLIC CRARACTEE OF ALGEBRA. 129 this Jc, took as general complex numbers of the system, a + ib -\-Jc + kd, X + iy -^-jz -j- Jew, quaternions, built their products, and assuming r =j- = k- = -l jk = -Jcj= i ki = — iJc=j, found that the modulus law was fulfilled. A geometrical interpretation was found for the 'imag- inary triplet" ib ^jc -\-kd, by making its coefficients, b, c, d, the rectangular co-ordinates of a point in space ; the line drawn to this point from the origin picturing the triplet by its length and direction. Such directed lines Hamilton named vectors. To interpret geometrically the multiplication of i into j, it was then only necessary to conceive of the j axis as rigidly connected with the i axis, and turned by it through a right angle in the jk plane, into coincidence with the k axis. The geometrical meanings of other operations fol- lowed readily. In a second paper, published in the same volume of the Philosophical Magazine, Hamilton compares in detail the laws of operation in quaternions and the algebra of numbv'^r, for the first time explicitly stating and naming the asso- ciative law. 128. Grassmann's Ausdehnungslehre. In the Ausdeh- nungslehre, as Grassmann first presented it, the elementary magnitiides are vectors. The fact that the equation AB + BC= AC always holds among the segments of a line, when account is taken of their directions as well as their lengths, suggested the probable usefulness of directed lengths in general, and led Grassmann, like Argand, to make trial of this definition of 130 NUMBER-SYSTEM OF ALGEBRA. addition for the general case of three points, A, B, C, not in the same right line. But the outcome was not great until he added to this his definition of the product of two vectors. He took as the product ab, of two vectors, a and b, the parallelogram gen- erated by a when its initial point is carried along b from initial to final extremity. This definition makes a product vanish not only when one of the vector factors vanishes, but also when the two are parallel. It clearly conforms to the distributive law. On the other hand, since (a -f b) (a -{-b) = aa + ab + &a + bb, and (a + b) {a -\-b) = aa = bb = 0, a6-f-6a = 0, or ba = — ab, the commutative law for multiplication has lost its validity, and, as in quaternions, an interchange of factors brings about a change in the sign of the product. The opening chapter of Grassmann's first treatise on the AusdehnungsleJire (1844) presents with admirable clear- ness and from the general standpoint of what he calls " Formenlehre " (the doctrine of forms), the fundamental laws to which operations are subject as well in the Aus- dehnungslehre as in common algebra. 129. The Doctrine of. the Artificial Numbers fully Devel- oped. The discovery of quaternions and tlie AusdeJuiungs- lehre made the algebra of number in reality what Gregory's definition had made it in theory, no longer the sole algebra, but merely one of a class of algebras. A higher standpoint was created, from which the laws of this algebra could be seen in proper perspective. Which of these laws Avere distinctive, and what was the significance of each, came out clearly enough when numerical algebra could be com- pared with other algebras whose characteristic laAvs were not the same as its characteristic laws. SYMBOLIC CHARACTER OF ALGEBRA. 131 The doctrine of the artificial numbers regarded from, this point of view — as symbolic results of the operations which the fundamental laws of algebra define — was fully- presented for the negative, fraction, and imaginary, by Ilankel, in his ComxAexe Zahlensystemen (1867). Hankel re-announced Peacock's principle of permanence. The doctrine of the irrational now accepted by mathematicians is due to Weierstrass and G. Cantor and Dedekind.* A number of interesting contributions to the literature of the subject have been made recently ; among them a paper f by Kronecker in which methods are proposed for avoiding the artificial numbers by the use of congruences and "inde- terminates," and papers t by Weierstrass, Dedekind, Holder, Study, Scheffer, and Schur, all relating to the theory of general complex numbers built from n fundamental units (see page 40). * See Cantor in Mathematische Annalen, V, p. 123, XXI, p. 567. The first paper was written in 1871. In tlie second, Cantor compares liis tlieory with that of "Weierstrass, and also with the theory proposed by Dedekind in liis Stetigkeit unci irrationale Zahlen (1872). Tlie theory of the irrational, set forth in Chapter IV of the first part of this book, is Cantor's. t Journal fiir die reine und angewandte Mathematik, Vol. 101, p. ;5.J7. t Gottinger Xachrichten for 1884, p. 395 ; 1885, p. 141 ; 1889, p. .')4, , p. 237. Leipziger Berichte for 1889, p. 177, p. 290, p. 400. Mathe- mathische Annalen, XXXIII, p. 49. SuppLEMENTARr NoTE, 1902. An elaborate and profound analysis of the number-concept from the ordinal point of view is made by Dedekind in his Was sind und was sollen die Zahlen? (1887). This essay, together with that on irrational numbers cited above, has been translated by W. W. Beman, and published by the Open Court Company, Chicago, 1901. The same point of view is taken by Kronecker in the memoir above mentioned, and by Helmholtz in his Zahlen und Messen (Zeller- Jubelaum, 1887). G. Cantor discusses the general notion of cardinal number, and 132 NUMBER-SYSTEM OF ALGEBRA. extends it to infinite groups and assemblages in his now famous Memoirs on the theory of infinite assemblages. See particularly Mathematische Annalen,' XL VI, p. 489. Very recently much attention has been given to the question : What is the simplest system of consistent and independent laws — or "axioms," as they are called — by which the fundamental operations of the ordinary algebra may be defined ? A very complete resume of the literature may be found in a paper by O. Holder inLeipziger Berichte, 1901. See also E. V. Huntington in Transactions of the American Mathematical Society, Vol. Ill, p. 264. 4 4 7 3 9 UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. JAN 5 - 1961 OCT 2 Rtt*^ DEC 13 « DEC B*^''' MAR 1 1970 j JUN 1 5 1984 RECEIVED MAY 23 1984 17 1987 im PSTU INTERLIBRARY FEB^'O 1979 mm WE£KS FROM DATE Of RECEiPJ JIOH-RE!l£ttABL£ MAR ? 1979 Form L9-32m-8,'58(5876s4)444 8 RECU F_v'^^"P'^'^P^ LO/NS UNIVi'^^^^ITY of CALIFORWI^ AT LOS ANGELES LIBRARY UC SOUTHERN REGIONAL LIBRARY FACILITY III II lllllill M II lllllill lllllll AA 000 507 337 4 Library 2-4-; I^S'^.vu M72