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\ REESE LIBRARY ^ 
 
 OF THI-: 
 
 UNIVERSITY OF CALIFORNIA. 
 
 \ zAccessioiis No. (:;i^^O f. CLns No. 
 
 •li — tjr—u"\ 
 
Number and its Algebra 
 
 SYLLABUS OF LECTURES ON THE THEORY 
 OF NUMBER AND ITS ALGEBRA 
 
 INTRODUCTORY TO A 
 
 COLLEGIATE COURSE IN ALGEBRA 
 
 BY 
 
 ARTHUR LEFEVRE, C.E. 
 
 Instructor in Pure Mathematics, University of Texas 
 
 university) 
 
 BOSTON, U.S.A. 
 
 D. C. HEATH & CO., PUBLISHERS 
 
 1896 
 
Copyright, 1896, 
 By Akthue Lefevre. 
 
 C. J. Peteks & SON, Typoqbaphees, Boston. 
 
 S. J. PaEKHTI.I, & Co., PBINTEB8. 
 
DEDICATED 
 THE TEACHERS OF MATHEMATICS 
 
 IX THE 
 
 C0M:\I0X SCHOOLS. 
 
 (trNIVERSITT^ 
 
(.svs^f 
 
•0NI 
 
 G^UFORN. 
 
 CONTENTS. 
 
 CHAPTER PAGE 
 
 Introduction ' '. . . 5 
 
 I. Pbimaky Number 19 
 
 II. Counting , 21 
 
 III. Some Fundamental Theory 24 
 
 IV. Notation 25 
 
 Y. Algebra '. 31 
 
 VI. Calculation 37 
 
 VII. Primary Number. — Numerical Operations . . 42 
 
 YIII. Devices of Computation 58 
 
 IX. First Extension of the Number-Concept. . . 61 
 Eatio. — Fractions. — Surds. 
 
 X. Significance of Operations, and Special Opera- 
 tional Devices Appropriate to the First 
 
 Extension of the Number-Concept .... 65 
 
 XI. Final Extension of the Number-Concept. — 
 
 Principle of Continuity 73 
 
 xii. signlificance and efficacy of numerical op- 
 erations under the ultimate concept. — 
 Zero, Infinity. — Negative, Neomonic, Com- 
 plex Number 85 
 
 XIII. Measurement 125 
 
 XIV. Mathematics 134 
 
 XV. Some Theorems and Problems 146 
 
J 
 
INTRODUCTION. 
 
 " The scientific part of Arithmetic and Geometry would be of more 
 use for regulating the thoughts and opinions of men than all the great 
 advantage which Society receives from the practical application of 
 them : and this use cannot be spread through the Society by the prac- 
 tice ; for the Practitioners, however dextrous, have no more knowledge 
 of the Science than the very instruments with which they work. They 
 have taken up the Rules as they found them delivered down to them by 
 scientific men, without the least inquiry after the Principles from which 
 they are derived : and the more accurate the Rules, the less occasion 
 there is for inquiring after the Principles, and consequently, the more 
 difficult it is to make them turn their attention to the First Principles ; 
 and, therefore, a Nation ought to have both Scientific and Practical 
 Mathematicians." — James Williamson, Elements of Euclid with Dis- 
 sertatio7is, Oxford, 1781. 
 
 The preceding arraignment is nearly as pertinent to-day 
 in this country as it was in England more than a centui-y 
 ago. But so far as Geometry is concerned blame no longer 
 rests with the scientific mathematicians. Their investiga- 
 tions of First Principles have not only furnished us with 
 Euclid in his purity, but have developed entirely new 
 and equally consistent geometries, under postulates alter- 
 nate to Euclid's petition of the angle-sum of a rectilineal 
 triangle. Thus has been fulfilled what must at least have 
 opened up as dim vistas to Euclid's mind when he dis- 
 cerned the necessity for assuming, or petitioning as the 
 old geometers called it, his indemonstrable postulate.* 
 
 * Called variously the 5th postulate, or the 11th or 12th axiom. 
 
b INTEODUCTION". 
 
 Still further, scientific mathematicians, besides offering 
 the true Euclid in available text-books with desirable ad- 
 ditions and extensions, have corrected several errors in 
 definitions and demonstrations which constituted the sole 
 blemishes in the most perfect work ever performed by a 
 single man. There is no longer good excuse for teachers 
 choosing texts which present the postulate as a common 
 notion or axiom; to say nothing of such as baldly omit 
 the whole doctrine of ratios and proportionality. There is 
 a momentous difference between ratios and fractions, and 
 text-books which present a proportion simply as an equality 
 of fractions have set up a miserable cause of stumbling. 
 They consider "merely a special case of no importance, 
 whose only excuse for existence lies in the general case 
 omitted." * Incommensurability is the rule, commensura- 
 bility the exception. 
 
 On the other hand, when we consider Arithmetic and 
 Algebra the cap of censure fits the other head. If our 
 scientific mathematicians have furnished satisfactory text- 
 books in these subjects, I am not acquainted with them. 
 All of us who are teaching mathematics must agree with 
 good old Williamson when he complains, in the'dissertation 
 already quoted, that he found it more difficult " to make 
 a rational arithmetician than an enlightened geometer." 
 
 Let me hasten to say that the apparently controversial 
 tone of this preface springs from no polemical spirit. I 
 approach the task I have set myself with utmost modesty; 
 nay, oppressed by a sense almost of presumption in at- 
 tempting to clarify what so many have left confused. But 
 so sorely needed is a successful accomplishment of what I 
 
 * Catalogue Univ. of Texas, 1891-1892. 
 
INTRODUCTION. 7 
 
 attempt, that an honest effort needs no apology. I wish 
 also to explain that the present treatment takes its form 
 from the immediate practical aim in view ; viz., that of a 
 syllabus for a rapid review of such ground of arithmetic 
 and algebra as will best prepare for the study of what 
 goes in our curricula by the name of " higher algebra," 
 with special adaptation to the needs of that large portion 
 of my classes who are taking the course in order to qualify 
 as teachers in the public schools. 
 
 I write this Introduction, and dedicate the little work to 
 the teachers in the common schools, however, in the hope 
 of attaining a wider usefulness, in the way of awakening 
 in some Practical Mathematician a desire ""to make ra- 
 tional arithmeticians " of the youths whose studies he is 
 directing. It is proper to explain still further that, work- 
 ing away from any great library, I have been compelled 
 to prepare this matter for printing without having time to 
 procure a few published works which I would like to see 
 before committing myself to publication. 
 
 I must not be understood as advancing anything new to 
 mathematicians, though I know of no English text-book 
 which consistently expounds and maintains the theories 
 of number and algebra here presented. The work is ad- 
 dressed, not to mathematicians, but to inquiring students 
 and teachers. A sound doctrine of number and its algebra 
 seems to be left by our text-books to chance inference, or 
 deferred to stages seldom reached in undergraduate courses 
 of study. A straightforward development, comprehensible 
 by beginners, of the number concept would be of immense 
 service in mathematical instruction. 
 
 For six years I have given my classes the substance of 
 this syllabus as the best explanation I could offer of dififi- 
 
8 INTRODUCTION. 
 
 culties which could not honestly be avoided. In July, 
 1894, I read in the current issue of the Monist an article 
 by Hermann Schubert, writing in Hamburg, on Monism in 
 Arithmetic, enunciating a unifying principle which he called 
 the Principle of No Exception, referring it originally to 
 Hankel. Of course some such principle is more or less 
 clearly in the mind of every student of mathematics, but 
 having never read Hankel's own statement, I cannot say 
 whether his Prlticlple of Permanence is substantially iden- 
 tical with the developing principle I set forth, or rather 
 in line with the notion of algebra as "the science which 
 treats of the combinations of arbitrary signs and symbols, 
 by means of defined, though arbitrary, laws,"* — the view 
 of the famous Dean of Ely, and the long line of algebraists 
 of whom he is the prototype. The bare statements of 
 such a principle from radically different standpoints might 
 be confusingly similar to one not fully alive to the fun- 
 damental variance. For example, in Schubert's statement 
 of his Principle of No Exception, I recognized what I 
 conceive to be a somewhat inadequate expression of the 
 postulate I had called the Principle of Continuity (I still 
 prefer this name as pointing with direct emphasis to its 
 cardinal outgrowth — the conception of number as continu- 
 ous), whereas in the next preceding issue of the same 
 journal he is at utter variance with me in declaring that, 
 "all numbers, excepting the results of counting, are and 
 remain mere symbols, nothing but artificial inventions of 
 mathematicians." 
 
 * Peacock's Report on the Recent Progress and Present State of cer- 
 tain branches of Analysis, in the British Association Report for 1833, 
 p. 195. Cf. also Peacock's Treatise on Algebra, 1830, republished and 
 enlarged in 1842. 
 
INTRODUCTION. 9 
 
 In the article above referred to, Schubert claims that in 
 his System of Arithmetic, Potsdam, 1885, he '' was the lirst 
 to work out the idea referred to fully and logically, and in a 
 form comprehensible for beginners ; " although it had been 
 previously expressed by Grassman, Hankel, E. Schroeder, 
 and Kronecker. Such is the bibliography of this special 
 presentation of the subject, so far as I am aware, not to 
 mention Dr. Halsted's Number, Discrete and Continuous,* 
 whose title promises a treatment of this subject, but which 
 remains a fragment, dealing only with discrete number — ■ 
 what I have called Primary Number. Should I be able to 
 spur Dr. Halsted to a completion of this work I shall not 
 have written in vain. 
 
 Of course, Hankel's principle must be expounded in his 
 Theorie der complexen Zahlansysteme, Leipzig, 1867 ; but 
 I am yet ignorant of the specific publications of the other 
 authors named, except that in Zeller's jubilee work the 
 matter is referred to in an essay by Kronecker. 
 
 On the other hand, the theory here advocated must not 
 be deemed retrogressive, and referred to such writers as 
 Frend,t who, though he very philosophically maintains 
 that, since algebra has its origin and termination in arith- 
 metic, it cannot be considered independent, and fairly 
 enough regards algebra as " the science which teaches the 
 ' general properties and relations of numbers," yet ends by 
 practically throwing the greater part of the science of 
 number overboard, in rejecting all algebraic forms which 
 do not agree with his undeveloped concept of number. 
 
 My theme may be regarded as the underlying harmony 
 
 * Preface and four chapters (22 pp.) in Scientiae Baccalaureus, June, 
 1891. 
 
 t Algebra, 1796. 
 
10 INTRODUCTION. 
 
 of the great makers of analytical mathematics, — and my 
 purpose, as an attempt to present to beginners fundamental 
 theory commonly left for the speculations of the most 
 advanced. 
 
 Number is such a perfect and typical abstraction that 
 it is difiScult to see how a man who has, to use Newton's 
 phrase, " in philosophical matters a competent faculty of 
 thinking," could ever associate the terms concrete and num- 
 ber ; nevertheless this confusion muddles many popular 
 text-books. The question hardly requires or admits of 
 argument. Since it is a vicious habit rather than an illogi- 
 cal deduction which is to be combated, good-tempered ridi- 
 cule is perhaps the only fit rejoinder. In this spirit may 
 I be permitted to relate an anecdote ? Some years ago at 
 the University of Virginia the Professor of Mathematics 
 assigned several problems to be worked upon the black- 
 boards by members of the Junior Class. To one he gave a 
 problem concerning the number of oranges in a pyramidal 
 pile of stated proportions. After expounding the error or 
 propriety of the solutions of some of the other problems, 
 the turn of the orange problem came. The student stood 
 proudly beside his mechanically correct solution. " Well, 
 Mr. Blank," exclaimed the Professor, " how many apples 
 did you find ? " A look of consternation overspread the 
 youth's countenance. With a gesture of impatient annoy- 
 ance he swept the erasing brush over the figures his chalk 
 pencil had traced : '' Oh," said he, " I thought you said 
 oranges ! " In all seriousness, the text-books we have all 
 been abused by, expounding " concrete numbers," solemnly 
 cautioning against confusion of multiplicand and multi- 
 plier, divisor and quotient, and unallowable combinations of 
 the terms of a numerical proportion, are quite as ridiculous 
 
INTRODUCTION. 11 
 
 as our hero of the oranges. He displayed at least one 
 virtue, — consistency. Such questions, however, though of 
 great practical importance to the efficiency of our elemen- 
 tary schools, present no real difficulties. A little knowledge 
 of psychology and mathematics will, if attention he called 
 to the question, correct mistaken, and develop inchoate 
 concepts of Primary I^umber. A far more difficult matter 
 remains — to attain for ourselves, and to lead our pupils to 
 attain, a rational concept of number as continuous, a con- 
 cept absolutely essential to modern mathematics, and now 
 universally assumed as a fact, — implicitly so assumed, even 
 when explicitly denied. It is also necessary to pass beyond 
 the great step already made by Newton, who discerns the 
 continuity of number, but leaves it only '' triplex " : " Est- 
 que (Humerus) triplex ; integer, f ractus, et surdus : Integer 
 quem unitas metitur, Fractus quern unitatis pars submulti- 
 plex metitur, et Surdus cui unitas est incommensurabi- 
 lis " *, with the implied limits zero and infinity. Xewton 
 also recognized qualitative distinctions, positive and nega- 
 tive, but the consequent neomonic (so-called " imaginary ") 
 and complex numbers remain to be assimilated. I must 
 return, however, to notice a uniquely erroneous view of 
 primary number presented in the last issue of the Inter- 
 national Education Seriks, The T'sycliology of Number, 
 by James A. McLellan, A.M., LL.D., Principal of the On- 
 tario School of Pedagogy, Toronto, and John Dewey, Ph.D., 
 Head Professor of Philosophy in the University of Chi- 
 cago, edited like all of the series by W. T. Harris, U. S. 
 Commissioner of Education. 
 
 The astounding thesis is maintained that number is not a 
 
 * Arithmetica universalis: quoted fi-oiu Halsted's Number, Discrete 
 and Continuous. 
 
12 INTRODUCTION. 
 
 magnitude, does not possess quantity at all, and that " no 
 number can be multiplied or divided into parts." * The 
 authors vehemently assert that we might as well talk of 
 any absurdity " as to talk of multiplying a number." f 
 It is much to be regretted that a Avork of such prestige 
 should merely shift the misconception of concreteness from 
 numbers to the subjects of calculation, which we are told 
 to believe are never numbers at all. Number is most em- 
 phatically shown to be " purely abstract," | yet multipli- 
 cation is claimed to be only of concretes. It is nonsense, 
 we are told, to think of multiplying six by four ; you can 
 only multiply six inches, six oranges, by four. Of course, 
 that numbers are multiplied is a fact, a fact that psy- 
 chology may explain, but can in no wise question. After 
 repeatedly insisting upon " the absurdity of multiplying 
 pure number or dividing it into parts," § the authors admit 
 without comment, and in seeming hesitation, '•' of course, 
 in all mathematical calculations we ultimately operate with 
 pure symbols." || What are these " pure symbols " ? What 
 can they be in arithmetic but the pure numbers them- 
 selves ? It woiild be an error, shared by many algebraists, 
 to conceive algebra as lacking specific content — as operat- 
 ing with ''' pure symbols," whatever that may mean. The 
 chapter on the Psychical Nature of Nurnber is admirable, 
 and I gratefully invoke its corroboration of what will be 
 found in my syllabus on the subject; but that upon the 
 Origin of Number, though very acute in tracing the de- 
 pendence of measurement upon " adjustment of activity," 
 seems to me mistaken in finding the origin of number in 
 
 * Psychology of Number, p. 70. § Ih., p. 71, foot-uote. 
 t lb., p. 70. II lb., p. 71. 
 
 X lb., p. 69. 
 
INTRODUCTION. 13 
 
 measurement. Measurement is not the source of the con- 
 cept of number, but a stimulation to clarify and develop 
 the concept; and this is what the facts cited really show. 
 The primary concept of number, as so correctly defined 
 in the preceding chapter, is prerequisite to any attempt at 
 measurement. The savage referred to needs the concept 
 that the length of his arrow is some number of hand- 
 breadths before he can attempt to discover how many. 
 And long before this he has learned to recognize a small 
 group of objects as a "vague whole," and to "discriminate 
 the distinct individuals," i.e., in the very terms employed 
 to define number, the concept originated before any meas- 
 urement became possible. Nor, in truth, does number ori- 
 ginate in counting, as so commonly asserted; and for a 
 like reason, viz., the concept of some number must pre- 
 cede any device for naming or anywise specializing it. 
 The position that number has its origin in measurement 
 cannot seek strength from the procession toward the ab- 
 solute of Hegel's ascending categories, quality, quantity 
 (including number, as " quantum in its complete spe- 
 cialization "), measure {das Maass), essence ; * for Hegel's 
 Maass, i.e., " qualitative quantity or measure," f is a very 
 different matter from Dr. Dewey's measurement, in fact, 
 it seems very nearly the same as number according to the 
 growing insight of modern mathematics. 
 
 Having mentioned Hegel, it is proper to remark that we 
 are just now being reminded on every side — Helmholtz 
 not long ago admonished us — that students of science are 
 frequently driven by the very logic of their subjects into 
 
 * The Logic of Hegel, translation, Wallace, p. 192 ("quantum, i.e., 
 limited quantity," p. 190). 
 t lb., p. 200. 
 
 
 WV -^-T -f 
 
 •■;_ii-^ri.-<'_^-' 
 
14 INTRODUCTION. 
 
 the regions of philosophy. A vital service will be rendered 
 any serious student, should he be led to consider, either at 
 first or at second hand, the aperqus of Hegel and the power 
 of his method. Without doubt, Hegel has pointed out the 
 true way of logic, if he did not always follow that path. 
 His system is far from fully elaborated ; much is tentative, 
 doubtless much mistaken ; but the fundamental business of 
 logic (in the Hegelian sense) must remain as he appointed 
 it, a criticism of the very terms of scientific and ordinary 
 thought ; nor is a better method than his dialectic likely 
 to be discovered. Dr. Harris (who — inice — writes better 
 than he edits) has done our nation substantial service in 
 his HegeVs Logic, by condensing, elucidating, and compara- 
 tively popularizing a work of prime importance in the prog- 
 ress of human thought. If any reader has perused even 
 the preface of this book, I beg him to recall the suggestive 
 and tonic way in which Dr. Harris recounts his gradual 
 and successive attainment of various ''insights" in mat- 
 ters philosophical, and to find encouragement therefrom 
 should he stumble at the development of the concept of 
 primary number which mathematics imperatively demands. 
 D'Alembert's advice to beginners in the differential calcu- 
 lus was " allez en avant et la foi vous viendra." 
 
 It remains only to say that in this attempt to elucidate 
 a unifying principle of number, and to display the nature 
 of any algebra, I have kept in mind the capacity of ''fresh- 
 man " students, and have avoided all reference to ultimate 
 categories, psychological or ontological. 
 
 I am well aware that there are other avenues of approach 
 to the thesis here maintained, — that " various new mathe- 
 matical conceptions have been employed by Weierstrass, 
 G. Cantor, and Dedekind in establishing three independent 
 
DSfTIlODUCTION. 15 
 
 and equally cogent theories which, should prove the conti- 
 nuity of number icithout horroiving it from space,'" * to say 
 nothing of such theories (e.g., Fine's Number- System) as 
 are '•' content to get continuity from the line." f Something 
 tangible for beguaners is a great desideratum. My aim is 
 practical, and it may be claimed that even if difficulties 
 have not been surmounted, or obscurities illuminated, they 
 have at least been reduced to one clear-cut postulate. The 
 student may take stock of his knowledge, and rationally 
 prosecute his studies, even though he consider a gratuitous 
 assumption left in the rear ; " la foi viendra." 
 
 To such as may condemn the occasional analogical sug- 
 gestions, and references to general philosophy in this 
 treatise, as unbecoming the proprieties of the severest of 
 the sciences, I would beg to reply that the style of an 
 attempt to explain how mathematics came to be, and what 
 it is, of an effort to lead those who sit in darkness to form 
 the concepts with which mathematics deals, ought not to 
 be judged by the standards of the severe and self-contained 
 procedure of the full-fledged science. My subject soon 
 enters, but begins outside of mathematics ; nor is it ped- 
 antry to be philosophical in explaining the fundamental 
 concepts of any science. It would be impertinent to be 
 anything else, 
 
 I would also deprecate any charge of presumption on 
 account of several innovations in terminology. I am fully 
 aware that reformation must come, if at all, from powerful 
 leaders ; but it seemed appropriate in a work of pedagogi- 
 
 * Number, Discrete and Continuous, George Brace Halsted, Preface. 
 The italics are mine. So far as I know no one of these demonstrations 
 has appeared in English. 
 
 t /i. 
 
16 
 
 INTEODUCTIOlSr. 
 
 cal intent to point out certain misnomers, and even to 
 "practice what I preach." No confusion can arise from 
 using neomonic and protomonic for '•' imaginary " and " real " 
 etc. ; and those who deem the current terms consecrated 
 by the usage of the great geometers who have made the 
 science, may ignore the suggestions. 
 
 My practical aim must explain the apparently arbitrary 
 intrusion of detail, especially in the final chapter. In this 
 final chapter, it should be said, I have drawn freely from 
 Professor Chrystal's Text Book of Algebra, Adam and 
 Charles Black, Edinburgh, 1886. Such points only are 
 touched upon as have been shown by experience to bear 
 directly on the preparation proper to our freshman course 
 in algebra. Especially in freedom of arrangement and 
 allusion, some familiarity with the subject-matter is pre- 
 supposed; but the knowledge assumed need be neither 
 great nor accurate in order to comprehend what is presented. 
 The vague acquaintance with terms and processes possessed 
 by the ordinary high-school graduate has sometimes war- 
 ranted the projection of a particular discussion beyond the 
 parallel development of cognate topics in a way which would 
 not be admissible in teaching children. My classes stand 
 upon a vantage ground whence it is permitted to look both 
 forward and backward, and so at last to command a really 
 comprehensive view. A teacher should never forget, how- 
 ever, that at every stage there should be an index pointing 
 upward. Any period of schooling which lacks this incen- 
 tive must be a barren tract in the experience of the pupil 
 who has traversed its dull course. 
 
 I have thus here, as always, striven to avoid what may 
 be deemed the most insidious and mischievous of all mis- 
 takes in teaching and textbook-making, — such a stooping 
 
INTRODUCTION. 17 
 
 to the fancied incapacities of pupils as requires the ob- 
 scuration of pure thought, the blurring and distortion of 
 truth by substituted analogies and illustrations. Half- 
 truths are dangerous. Pupils nurtured on such philosoph- 
 ical pap too often take up the role of teachers without 
 deepened insight, and the spawn of error procreates with 
 the fecundity so characteristic of parasites. This mis- 
 take of shutting up all vistas into regions not presently 
 under exploration stultifies the learner, and necessitates 
 a weary process of unlearning at each stadium. It is in 
 the intellectual sphere the analogon of that contemptible 
 principle of school government which, in the sphere of 
 morals, appeals to timidity or vanity, and depends on 
 espionage, basely ignorant that '^ the human character is 
 susceptible of other incitements to correct conduct more 
 worthy of employ and of better effect." * 
 
 In conclusion, the difficulties, practical and theoretical, 
 of the central problem in this little work entitle it to 
 be judged with leniency. It is submitted to my classes 
 and to fellow-teachers for such uses as it may deserve. 
 
 ARTHUR LEFEVRE. 
 University of Texas, January, 1896. 
 
 * Thomas Jefferson, quoted in " The University and the Common- 
 wealth," an address by Professor Thornton of the University of Virginia, 
 delivered in the University of Texas on Commencement Day, 1894. 
 
SYLLABUS. 
 
 NUMBER AND ITS ALGEBRA. 
 
 I. Primary Number. 
 
 1. Whole, Litegral, Natural, Exact, are all terms in 
 vogue to designate the primary concept of number. The 
 former two are equivalent, and objectionable as equally 
 applicable to positive and negative numbers. They thus 
 fail of exact designation. Though they would hardly be 
 chosen de novo, they may be retained in that one of their 
 present uses which is really proper, viz., to designate Pri- 
 mary Numbers, and their negatives after the distinction of 
 positive and negative has been clearly made. Each of the 
 latter two is repugnant to any concept of number adequate 
 to the comprehension of mathematical sciences. No num- 
 ber must be conceived as either unnatural or inexact. The 
 ratio of absolutely incommensurable sects, the diagonal of 
 a square and its side, for instance, is just as exactly what 
 it is as the ratio of an inch to a foot. By the term primary 
 number no question is begged, and the very name points to 
 the development so soon found necessary. 
 
 2. Primary Number, a normal and universal creation of 
 the human mind, applies originally only to artificial wholes, 
 discrete aggregates. The group whence " twelve " is ab- 
 
 19 
 
20 NUIMBER AND ITS ALGEBRA. 
 
 stracted must be conceived as a Avhole before discriminated 
 into " twelve." Number is in nowise a sense-perception ; 
 it is purely the product of a rational process. Because the 
 adult finds a number concept in his mind when a group of 
 objects is attended to, he must by no means suppose any 
 such concept in the mind of a child, though the same objects 
 be attended to. The objects may not even be a groiqj at 
 all to the child. Adults forget how gradually any idea 
 developed in their minds. Neither is the concept one 
 necessarily in the mind of a child when a single object 
 is attended to. The mind of the child is inclined to be 
 absorbed in sense facts. The concept one is only in con- 
 trast to the concept many. It is not my purpose to inves- 
 tigate the origin and psychological processes of these 
 concepts, one and many ; nor how it comes to pass that 
 the infant mind slowly tends to group, aggregate, make 
 wholes of, distinct individual objects of sense-perception. 
 It is enough to point out that from these concepts the 
 primary concept of number springs. Various manys are 
 specialized, and so distinct numbers arise in the mind. 
 We will not enter upon the question of infant psychology 
 concerning the stages at which the manys are specialized 
 into ''two," "three," etc. It may be remarked, however, 
 that the special many " two " is recognized very early and 
 long before " three." The concepts one, many, two, come 
 almost together ; and then after a long gap further special- 
 izations are attained — another distinct gap perhaps coming 
 after fo^ir. 
 
 3. Definition-. — Primary Number is an abstraction 
 from a group of objects which represents their individual 
 existence. 
 
 4. Each number-picture of a group is wholly abstract, 
 
COUNTING. 21 
 
 in that it represents the individual existence of the ele- 
 ments of the group and nothing more. For use in pictur- 
 ing special manys a system of abstract elements is framed, 
 where no characteristic of any element is retained beyond 
 its simple separateness from all others.* This brings us to 
 Counting. 
 
 II. Counting. 
 
 5. The fundamental concept of primary numbers is 
 prior to, prerequisite for, not derived from, <' Counting." 
 The word is used in two senses, though its general syno- 
 nyms, numeration and enumeration, seem sometimes par- 
 ticularly assigned to the first meaning. In the first sense. 
 Counting is the naming of primary numbers. This nam- 
 ing, if carried to any great extent, must of necessity be 
 methodical, and of course the numbers must be conceived 
 before named. In the second sense, Counting is essentially 
 the numerical identification, by a one-to-one correspond- 
 ence, of an unfamiliar with a familiar group. In this 
 meaning. Counting consists in assigning to each individual 
 in a group one distinct individual in a familiar fixed series 
 of different things — originally the fingers, usually a fixed 
 series of different words, or different marks. 
 
 6. We must pass by many interesting facts and theories 
 concerning word- numerals (i.e., fixed series of different 
 Avords used for counting) as belonging to the domain of 
 language, only remarking that etymology confirms what 
 might have been surmised, that the fingers were the origi- 
 nal series of things which mankind made use of to apply 
 in thought to a group of objects in order to count them. 
 
 * Vide, Number, Discrete and Continuous, Halsted, cliap. i. 
 
22 DUMBER AND ITS ALGEBRA. 
 
 7. lu all systems of numeration or counting (in the first 
 of the senses defined in Section 5 ) it soon becomes neces- 
 sary, from the very limitations of human memory, to form 
 or mark off a numerical group which the reckoner can peri- 
 odically repeat. Otherwise there would be no end to the 
 number of different words required. The number-group 
 chosen by a majority of races at a pre-historic time, and 
 for the reason that we possess ten fingers, is ten. As soon 
 as any such group has been chosen, it becomes easy to 
 express by a few number-names any number within the 
 mental scope of the speakers. 
 
 For instance, in English with fifteen words (two of 
 which are disfiguringly superfluous) and two significant 
 suffixes, i.e., with seventeen words, any number whatever 
 may be expressed. 
 
 8. The student should write out a detailed explanation 
 and criticism of the English series of word-numerals, not- 
 ing the superfluous words, also such as are not internally 
 suggestive of their relation to the fundamental group, and 
 ■^hat larger groups lack the simple name that would be 
 suggested by symmetry. Let him then compare the Eng- 
 lish series with that of some other language, noting where 
 the English is better, and where worse, than the other. 
 All this totally irrespective of any system of notation, and 
 purely as a question of thought and language. 
 
 According to the best practice the last of the larger 
 groups to receive a simple name for repetition is the mil- 
 lion. Charles W. Merrifield, F.E.S., observes, " It is worth 
 while to remark that as regards billions there is a differ- 
 ence between the French and English practice ; in French 
 a hillion (or milliard) is one thousand millions, in English 
 a billion is a million millions, . . . the word is seldom used 
 
COUNTING . 23 
 
 in our language. . . . The old books use a scale of this 
 kind: A million of millions is a billion, a million of 
 billions is a trillion, and so forth; but these names are 
 never used in practice, and can hardly be said to belong 
 to the language of arithmetic or to English speech.'' * In 
 the late vulgar use of the Avord in American newspapers, 
 billion, of course, signifies one thousand millions ; for it is 
 a comment upon the vastness of such numbers that even 
 the Fifty-first Congress could not expend the thousandth 
 part of a billion dollars in the sense of one million 
 millions. 
 
 9. By the method just discussed a distinct name is 
 given to each element in the series of counters; and in 
 counting the elements, the units, the ones in any discrete 
 magnitude or manifoldness, a one-to-one application is made 
 to this series of names in a fixed order. The order being 
 learned by rote, any word-numeral, by suggesting its defi- 
 nite place in the fixed series of words, recalls all those 
 gone before ; and from this comparison the mind conveys 
 or receives an exact notion of the number of individuals 
 in the group of objects numerically characterized by any 
 such number-name. 
 
 10. These number-names are sometimes called cardinal 
 numbers, to distinguish them from a series called ordinal 
 numbers. Ordinals have little to do with arithmetic, the 
 distinction belonging to grammar : instead of saying the 
 last-counted one of five objects, we may say the fifth, etc. 
 These concepts, first, second, third, fourth, etc., are the 
 " ordinal numbers." 
 
 * Arithmetic and Mensuration, p. 4, Longmans, Green, & Co., 1882. 
 
24 NUMBER AND ITS ALGEBRA. 
 
 III. Primary Number. — Some Fundamental Theory. 
 
 11. The number of objects in one group is said to be 
 equal to the number in another when their units being 
 counted (vide § 5) come to the same finger, the same nu- 
 meral-word or mark. That is to say, two primary numbers 
 each equal to a third are equal. Also, of two such num- 
 bers one is always less than, equal to, or greater than, 
 the other, according as in a one-to-one application to the 
 counter-series the process ends with a prior, the same, 
 or a subsequent, element. Also a primary number may be 
 added to itself so as to double. (Cf.% 229.) I am not 
 concerned whether these dicta be regarded as axioms or as 
 postulates. 
 
 Primary number is thus at once classed as a magnitude. 
 
 (Indeed it may be that the method of Hegel's dialectic 
 of the mathematical catgeories would display magnitude 
 and number as essentially the same ; that is, as co-ordinate 
 transitions to the same ultimate. At least, any object — 
 as a line, a surface, a solid, a time, a temperature — is a 
 TYiagnitude or manifoldness only as number, in the final 
 concept thereof, can be abstracted.) 
 
 12. There is much of prime import to be said of meas- 
 urement {vide Chapter XIII.), but it may be remarked in 
 this connection that the concept of measurement develops 
 pari passu with that of number. To the man whose con- 
 cept of number is only what has been defined as primary 
 number, measurement is hardly to be distinguished from 
 counting. Por measurement of discrete magnitude is 
 counting ; and to the intelligence supposed there is no real 
 measurement of continuous magnitude, but any continuous 
 magnitude is "measured" by violently discreting it, and 
 
NOTATION. 25 
 
 counting the units contained, the residue being regarded as 
 merely a fractional redundancy. In short, he measures in 
 what are popularly called " round numbers." True meas- 
 urement of continuous magnitude is conceivable only under 
 the developed concept of number which includes ratios. 
 
 13. Theorem. ■ — Primary Number is independent of the 
 order of counting. 
 
 This fact is discerned immediately from the individuality 
 of the objects in the group. Since in counting the cor- 
 respondence is one-to-one, the same extent of the counter- 
 series is always necessary and sufficient to correspondence 
 with any group of objects in whatever order they be applied 
 to the counters. 
 
 The obviousness of this truth must not blind to its 
 importance ; for, as Clifford affirms, " upon this fact the 
 whole of the science of number is based." * 
 
 IV. Notation. 
 
 14. Notation is primarily the representation of primary 
 numbers by written symbols ; but in the developed science 
 of arithmetic it must include the symbolic representation 
 of ways of combining numbers, and qualitative distinc- 
 tions, which arise upon investigation. Notation in the 
 primary sense is intimately blended with numeration, for 
 it is merely the recording of the results of counting. It 
 is of vast importance, however, and a good invention for 
 the purpose could have been no easy task ; because cen- 
 turies on centuries passed after a symmetrical system of 
 numeration had been developed in thought and language 
 
 * Common Sense of the Exact Sciences, W. K. Clifford, chap. i. 
 
26 NUMBER AND ITS ALGEBRA. 
 
 before a thoroughly fitting notation was achieved. Whether 
 the beautifully simple and perfect algorithm now so famil- 
 iar to little children was perfected at a single stroke of 
 genius on the part of a nameless Hindoo, or was a grad- 
 ually consummated invention, history does not reveal. 
 
 15. Just as we passed over the etymology of numeral 
 words, we must pretermit interesting facts and surmises as 
 to how each written sign came to have its particular mean- 
 ing in the various series of signs which mankind has in 
 times past employed or still uses. Such signs for number 
 are older than any other form of writing, older even than 
 the development of language in the denary system. For 
 an entertaining monograph on this subject, consult Profes- 
 sor Eobertson Smith's article on " Numerals " in the ninth 
 edition of the Encycloprndla Britminica, from which much 
 of the following section has been taken. 
 
 16. The simplest representation of unity is a single 
 stroke. The next step would be to devise a sign to repre- 
 sent a definite group of strokes, as it would be confusing 
 to repeat single strokes too often. Soon a sign for a defi- 
 nite group of the primary groups would be required. The 
 Babylonian inscriptions well exemplify this simplest mode 
 of notation. The mark for unity, a vertical arrow-head, is 
 repeated up to ten, whose symbol is a barbed sign pointing 
 to the left. These by mere repetition serve to express 
 primary numbers up to one hundred, for which a new sign 
 was employed. 
 
 The most important principle of meaning-signified-by- 
 position appears in this system. Though the symbol of 
 the smaller number put to the right of the hundred symbol 
 represented addition, the same symbol to the left repre- 
 sented a multiplier. This principle was still more signifi- 
 
NOTATION. 27 
 
 cant in another system developed by the Babylonians. 
 Strange to say, they oftened reckoned by powers of sixty, 
 calling sixty a soss, and sixty times sixty a sar. Survivals 
 of this sexigesimal method remain in our divisions of time, 
 angles, and the circle. For example, the square of 59 is 
 found recorded (translating into our symbols) 58.1, that is, 
 58 soss and one (58 X 60 -(- 1) ; but on the same tablets the 
 cube of 30 is recorded 7.30, that is, 7 sai^ and 30 soss. We 
 thus see that because they had devised no sign for zero, it 
 could only be left to the judgment of the reader whether 
 sixty or its square was intended. 
 
 After alphabets became established in a fixed order, they 
 began to lend themselves to numerical notation. In the 
 old Greek notation, said to go back to the time of Solon, 
 and often called the Herodian system, after Herodian who 
 described it in a work written about 200 a.d., 1 stood for 
 one, n (-TrivTe) for five, A (Se'/ca) for ten, H (e/carov) for hun- 
 dred, X (xlXlol) for thousand, M (/xupt'ot) for ten thousand. 
 As an artifice of condensation a great 11 enclosing any sym- 
 bol signified five times the number represented within. 
 Another application of alphabets is more to my purpose. 
 In this system (common to Greeks, Syrians, and Hebrews 
 — in Greece displacing the Herodian), the first nine letters 
 stood for units, the second nine for tens, the third nine for 
 hundreds, and diacritic marks below the first nine trans- 
 formed them into thousands. A great M multiplied the 
 number after whose sign it was written by ten thousand. 
 The notation was subsequently improved by writing the 
 greater element always to the left, thus disjieyising with the 
 diacritic marks. The regular alphabet furnishing only 
 twenty-four letters, the necessary twenty-seven were made 
 up by calling in two old letters no longer used in phonetic 
 
28 KUMBEE, AND ITS ALGEBRA. 
 
 writing, to signify six ancV ninety, and a final symbol called 
 samjn represented nine hundred. Approaches still nearer 
 to our algorithm were devised by Greek mathematicians, 
 notably Archimedes and ApoUonius of Perga; but in all 
 the lack of the zero rendered the systems very imperfectly 
 adapted to calculation, however perspicuous as a record. 
 
 Only one more system can be glanced at before survey- 
 ing our own. This, known as the Roman, we are still 
 familiar with. It more resembles the clumsy Herodian 
 than the later Greek notation. The symbols were, 1 = 1, 
 V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000. 
 Some older forms were afterwards discarded. To the ex- 
 tent of a few subtractive forms (IV = 4, IX = 9, XL = 40, 
 XC = 90, and occasionally IIX = 8, XXC = 80) some mean- 
 ing is attached to position, but in a way rather to hinder 
 calculation. 
 
 In a mechanical contrivance, used in Europe from a very 
 early date, was attained the nearest approach to our own 
 system. The abacus (which could be ruled on waxen tab- 
 lets or roughly drawn on the ground), in a permanent form, 
 consisted of a frame in which by one means or another sets 
 of counters were kept in separate rows or columns. These 
 columns might represent various denominations of money 
 value, or weight, or units, tens, hundreds, thousands, etc. 
 In the latter case there should be only nine counters in a 
 column. From such an abacus there are but two steps to 
 our notation: first, to establish marks to represent respec- 
 tively one, two, ... or nine counters in any column ; sec- 
 ond, to conceive a sign for a vacant column. The inven- 
 tion of our nine digits and zero came slowly. The history 
 is very obscure. Our " Arabic " system is of Indian origin, 
 but appears to have been introduced into Europe by the 
 
NOTATION. 29 
 
 Arabs. It has been traced as far back as the fifth century 
 of the Christian era in India, but does not seem then to 
 have been a novelty. Hindoo writers nowhere lay claim to 
 its invention. It was probably brought to Baghdad in the 
 eighth century. In the ninth century Abu Jafar Moham- 
 med al-Ivharismi published a work on the subject, and by 
 the tenth it had spread into general use throughout the 
 Arabian world. About the twelfth century it began to be 
 received by Christian Europe. Arithmetic using this sys- 
 tem was called by the barbarous name AJgoritmus (our 
 algorithm), probably a derivation from al-Kharizmi. Leo- 
 nardo of Pisa promulgated the matter in the West, and 
 Maximus Planudes in the East. The word zero is perhaps 
 derived from the Arabic sifr, through zepliyro, used by 
 Leonardo. The algoritmus was at first used chiefly in 
 astronoDiical tables, etc. (e.g., those published about 1252 
 by Alfonso the Wise). Gradually the immense superior- 
 ity of the system above all others became apparent, and it 
 has long been used by all civilized nations. In winning its 
 way there was some confusion with prevailing notations : 
 e.g., such forms as X2 = 12 and 504 = 54 are found, where 
 the very essence of the method is lost sight of. 
 
 17. Our notation exactly conforms to our system of nu- 
 meration. The symmetr}" or regularity of the notation, how- 
 ever, is perfect. No such anomaly as is found in the word 
 " eleven " or " twelve " is tolerated. 
 
 The familiar symbols always mean one, two, ... or 
 nine ; but they signify units, tens, ten-tens, ten ten-tens, 
 etc., according to their position in the first, second, third, 
 fourth, etc., place, counting from right to left. That is to 
 say, they represent in definite positions corresponding 
 powers of ten. 
 
30 NUMBER AND ITS ALGEBRA. 
 
 Under the generalization of " powers " the notation at 
 once lends itself to the expression of fractions, the expo- 
 nents becoming negative. For example, 4072.605 means — 
 
 (10)3 -^ 4 ^ (10)2 X + (10)^ X 7 + (10)° X 2 + (10)-^ 
 X 6 + (10)-- X + (10)-^ X 5. 
 
 It is in this regular use of a base-number that the merit 
 of the system consists, and by no means in the choice of 
 the base ten. Our decimal system is a perfect instrument, 
 exciting the grateful admiration of every enlightened stu- 
 dent of science, not because it is decimal, but because the 
 digit figures by means of the zero always express, in their 
 orderly position, to left or right of a point, ascending ot 
 descending powers of one basal number. 
 
 In regard to the particular base, ten, it may be remarked 
 that, while it were idle to think of changing the confirmed 
 habits of language, it is clear that ten is an inconvenient 
 base. Twelve would be better. To see this, it is enough 
 to express decimally and duodecimally a few simple frac- 
 tions. ' , t '"' 
 
 Decimally 1/3 = 0.3333333 . . . Duodecimally 1/3 = 0.4 
 1/4 = 0.25 1/4 = 0.3 
 
 1/6 = 0.1666666 ... 1/6 = 0.2 
 
 1/8 = 0.125 1/8 = 0.16 
 
 1/9 = 0.1111111 ... 1/9 = 0.14 
 
 18. A thoughtful consideration of our notation will 
 enable the student to adapt its essential principles to any 
 base. So long as he feels hesitation in doing this, he 
 may be sure he does 7wt understand what he has deemed 
 so familiar. 
 
ALGEBRA. 31 
 
 It is obvious that a number is the same, whether ex- 
 pressed in tens, or dozens, or scores, — to take, for example, 
 two numbers, other than ten, familiar as bases to English 
 minds, but which have never been developed into symmet- 
 rical systems of counting. 
 
 Plainly the number of digit figures required is one less 
 than the base ; since 10 must represent the base, whatever 
 it may be. 
 
 19. The student should express various numbers, inte- 
 gral and fractional, on various bases, employing, say, the 
 letters of the English alphabet in order, for additional 
 digit figures when more than nine are required. He 
 should also perform additions, subtractions, multiplica- 
 tions, divisions, with numbers so expressed. 
 
 There is no other way to test or gain a thorough com- 
 prehension of the notation so glibly used. Such an exer- 
 cise will remedy many defects, and will be found to repay 
 amply the slight cost in time and labor. ( Vide § 277 et seq.) 
 
 V. Algebra. 
 
 20. An algebra is an artificial language. Its symbols 
 have laws of combination ; but these laws are the expres- 
 sion of actual properties and relations of the subject-matter, 
 not laws of the algebra in any immediate sense. There is 
 no such thing as an algebraic law ; there are only alge- 
 braic conventions. The peculiar advantage of an algebra 
 is that actual relations are given manifestations which can 
 be experimented upon, according to organized processes, to 
 give new knowledge. The first algebra was slowly formed 
 through centuries to investigate the properties of number.* 
 
 * Vide Halsted's Number, Discrete and Continuous, § 1. 
 
32 NUMBER AND ITS ALGEBRA. 
 
 21. Algebras of formal logic, of physics, of geometry, 
 have been developed to more or less usefulness. In these 
 the subject-matters immediately discoursed of are respec- 
 tively logical, physical, geometrical entities and their rela- 
 tions and combinations ; just as the algebra with which 
 we have to do discourses of number and its relations and 
 combinations. The specific entities, relations, and combi- 
 nations in any case require definition, and yield laws upon 
 their own merits. 
 
 22. A geometrical algebra must be clearly distinguished 
 from that supremely powerful and distinctively modern 
 branch of analysis (from whose establishment by Des- 
 cartes, 1637, dates the modern, the scientific era) commonly 
 called Analytical Geometry. In this discipline the alge- 
 bra is still of number, the immediate subject of discourse 
 is ever number ; but under systematic conventions the 
 algebra talks in numbers about geometry, just as it might 
 be made to talk about money or temperatures. In a true 
 and proper algebra of geometry, a and b might represent 
 sects,* and ab be defined as the definite plane surface 
 known as the rectangle of a and b. In this case there 
 could be no ratio between ab and a. Also a^ would mean 
 the actual surface, the square on a ; a^, the actual solid, 
 the cube on a ; and a^, etc., would be devoid of meaning in 
 tri-dimensional space. 
 
 23. However mechanically we may at times use the 
 symbols, it cannot be too much emphasized that in the 
 algebra of number each expression must be a rational dis- 
 course upon number to any mind, or to that mind it is 
 nonsense, or rather a blank, like a sentence in an unknown 
 
 * Definite pieces of straight lines. 
 
ALGEBRA. 33 
 
 tongue. Clifford maintains, " We may always depend upon 
 it, that algebra which cannot be translated into good Eng- 
 lish and sound common sense, is bad algebra." * 
 
 24. Although of immense utility, the algebra of number 
 must not be conceived as theoretically necessary to the 
 investigations it has so signally served. The instrument 
 has been practically prerequisite to the results that have 
 been attained on account of the limitations of mankind's 
 power of attention to complex details without symbolic 
 expression, but its essentially derivative nature must not 
 be lost sight of. Under any concept subversive of this 
 relation — the fallacy being even more baneful Avhen im- 
 plied than when explicit — the study of an algebra be- 
 comes abusive of the noblest qualities of mind ; and no 
 irrational skill in the use of the tool can compensate for 
 the intellectual debasement which is the price of content- 
 ment in its use and study upon such terms. It is as if 
 one conceived the vocabulary of a spoken language as 
 independent of the constructive thought ; back of any 
 mode of symbolic expression must lie the substantial 
 thought. 
 
 To understand our algebra of number, we must under- 
 stand number. However difficult the task, it cannot hon- 
 estly be shirked. 
 
 25. Many eminent mathematicians, to say nothing of 
 popular text-books, persist in seeking explanation of the 
 algebra of number in the facts of geometry. They seem 
 blind to the view that it is only adaptations of number 
 that they thus discover; that it is numbers, not lines, 
 surfaces, solids, that they deal Avith, even when they so 
 
 * Common Sense of the Exact Sciences, p. 21. 
 
34 NUMBER AND ITS ALGEBRA. 
 
 usefully make the algebra of number "talk geometry."* 
 The individual symbol in trigonometry or analytical geom- 
 etry, for examjjle, never means the geometrical concept. 
 An equation may under a proper system of interpretation 
 describe a line ; but no x or y in it ever means a line, 
 but the length of a line, which is a ratio, a number. 
 Would it be less sophisticated to try to discover the nature 
 and properties of number by studying temperatures, be- 
 cause, forsooth, an algebraic equation may under appro- 
 priate conditions talk temperatures as well as geometry ? 
 The confusion arising from such misconceptions is well 
 exemplified in the following quotation from an essay by 
 E. W. Hyde in the American Journal of Mathematics for 
 September, 1883, p. 3 : — 
 
 '< If, in the equation 1/1 = 1x1, 1 be taken as a unit 
 of length, then the meinbers of the equation have evi- 
 dently not the same meaning, 1/1 being merely a numeri- 
 cal quantity, while 1x1 is a uuit of area ; it being a 
 fundamental geometric conception that the product of a 
 length by a length is an area, that of a length by an area 
 a volume, while the ratio of two quantities of the same 
 order as that of a length to a length is a mere number 
 of the order zero." 
 
 So far from just are these observations that one would 
 suppose it clear to any student of the subject that the 
 physical fact is the line, the surface, the solid, and that 
 the length, the area, the volume, are numbers, viz., the 
 ratios of the line, surface, and solid respectively to other 
 magnitudes of like kind chosen arbitrarily as units. It is 
 a theorem which we have established geometrically, that 
 
 * A felicitous i)hrase of Dr. Halsted's. 
 
ALGEBRA. 35 
 
 the ratio of the rectangle of two sects to the square on 
 any third sect equals the product of the ratios of the two 
 given sects to the third sect. That is to say, the area of 
 a rectangle equals the product of the lengths of two adja- 
 cent sides, it being distinctly understood that the unit- 
 surface is the square on the unit-line. This truth having 
 been established, consistent numerical statements may be 
 referred to such spatial entities. It is only and always in 
 some such Avay that the algebra of number '' talks ge- 
 ometry." 
 
 26. Objections to the mistake of explaining number geo- 
 metrically are often made at a fatally late stage. There 
 are writers who protest against geometric definitions of 
 the so-called imaginary numbers after having supinely 
 ignored a geometric, or some unnumerical, definition of —1. 
 Their alertness comes too late when they refuse a like 
 definition of V— 1. In this, as in many other cases, it 
 is the first principles that have been neglected. It is 
 futile to begin inquiry with V — 1. AYith beclouded 
 concepts of prior phases of number, how can it be any- 
 thing but vain to attempt to be critical at the final stage 
 of that development of number which has forced itself 
 alike upon the most practical and the most theoretical ? 
 {Cf. § 192.) 
 
 27. Concerning other extant or possible algebras than 
 that of number, I Avill only add that I have grave doubts 
 of the propriety of Professor Macfarlane's aspirations 
 towards a final and comprehensive algebra,* '' which will 
 apply directly to physical quantities, will include and unify 
 
 * "Principles of the Algebra of Physics," by A. Macfarlane, M.A., 
 D.Sc, LL.D., in Proceedings of the American Association for the Ad- 
 vancement of Science, vol. xl., 18U1, p. (i5. The italics are mine. 
 
36 NUMBER AND ITS ALGEBRA. 
 
 the several branches of analysis, and when specialized will 
 become ordinary algebra." Far from being the " special- 
 ized" form, the algebra of number appears to me to be 
 the very generalization sought by Dr. ]\Iacfarlane ; and it 
 is algebras of physics, vector algebras, etc., which are the 
 specializations. Number itself in its full development 
 appears to my mind the very ultimate common property 
 of all quantity, magnitude, manifoldness (^vlde § 229) what- 
 soever. Search for further generalization seems mistaken. 
 I set forth this opinion tentatively and in all modesty, 
 fully recognizing Dr. Macfarlane's profound learning and 
 skill in mathematics. 
 
 Negative, neomonic, and complex numbers afford the 
 qualitative distinctions under Avhich, it seems, the algebra 
 of number might be made to talk physics and geometry 
 to our full satisfaction. Should it be found inadequate to 
 the needs of the physicist, of course, a true and proper 
 algebra of physics may be fashioned. The physicists 
 must decide this question. Might not better results, how- 
 ever, be attained by seeking perfectly satisfactory means 
 for interpreting, physically or geometrically, numerical 
 statements, the algebra for which is ready to hand, than 
 by attempting to construct any real algebra of physics to 
 "apply directly to physical quantities " ? 
 
 I may invoke here the authority of no less a physicist 
 than James Clerk Maxwell. After pointing out the con- 
 tradictions which would otherwise occur in calculation, 
 he says : " We shall therefore consider all the symbols 
 as mere numerical quantities, and therefore subject to 
 all the operations of arithmetic. But in the original 
 equations and the final equations in which every term 
 has to be interpreted in a physical sense, we must con- 
 
CALCULATION. 87 
 
 vert every numerical expression into a concrete quantity 
 b}' multiplying it by the unit of that kind of quantity.'' 
 
 28. If ^ye will regard the algebra of number from the 
 standpoint of recognition of its true nature, we may take 
 up its natural use without more ado, (Vide § 156.) There 
 is nothing mysterious about the algebraic vocabulary, or 
 even recondite in the algebra, liutil we reach more ad- 
 vanced investigations concerning algebraic form. The 
 original obscurities and difficulties are in the arithmetic ; 
 that is, in the theory and import of number itself. For 
 the most j^art I shall consider what is algebraical already 
 familiar, and bend all energy to expounding the numerical 
 content, as distinguished from the algebraic form. (But 
 see § 236.) 
 
 VI. CALCUIiATIOX. 
 
 29. Calculation, or computation, is primarily counting. 
 As its methods gradually become organized, it involves a 
 thorough investigation of the laws of thought, which, upon 
 consideration by any normal mind, will be seen to govern 
 the various possible combinations of numbers and the 
 processes of these combinations. 
 
 30. In solving particular problems, whether concerning 
 numbers or the application of numbers to concrete magni- 
 tudes, it is to be borne constantly in mind that all that 
 can be tauglit in general terms is how to conceive and 
 perform numerical operations ; that, knowing this, all that 
 remains is to understand the terms of a particular problem 
 and the properties, real or conventional, of these terms, 
 under which they yield numerical relations ; and that until 
 one recognizes this fact he cannot take the first rational 
 step. 
 
38 NUMBER AND ITS ALGEBRA. 
 
 Although it is proper and necessary in teaching to make 
 constant applications of pure or theoretical arithmetic, 
 yet the way in which these applications are presented 
 in ordinary text-books is grossly misleading. There is 
 no arithmetical distinction whatever between such topics 
 as " percentage," " interest," " discount," " commission," 
 '•' brokerage," " partial payments," etc. ; yet from flaring 
 chapter-headings the distinctions appear co-ordinate with 
 those between numeration, numerical operations, and gen- 
 eral devices, such as methods for finding the greatest com- 
 mon submultiple, or the least common multiple. No new 
 arithmetical lore is required in order to calculate about 
 these mercantile transactions ; the task for the pupil is 
 merely to comprehend a few technical terms, and the 
 numerical relations subsisting, or in practice assumed to 
 subsist, among them. Nevertheless, it is a common result 
 of the misconceived method of presentation that a pupil 
 fancies he is advancing to a new development of the arith- 
 metic when he passes from ''commission" to "brokerage," 
 for instance, as if there were the faintest arithmetical dis- 
 tinction between calculating a percentage on the value of a 
 barrel of apples and the value of a block of capital stock. 
 In like manner, pupils often make pathetic attempts to 
 excogitate the conventional method of calculating a bal- 
 ance due on an account Avith partial payments, being 
 blinded by incompetent teaching to the fact that the data 
 do not afford numerical relations sufficient to a definite 
 theoretical solution. In this matter (as in many others 
 in Applied Arithmetic), arbitrary convention is neces- 
 sary to a solution ; the " rule " varies with the practice 
 of individuals, enactments of legislatures, and rulings of 
 courts of law and equity. No act of pope or parliament 
 
ARITHMETIC, PURE AND APPLIED. 89 
 
 could affect the proper decision of any truly arithmetical 
 question. Of course there is more or less numerical pro- 
 priety in the substantial justice between man and man 
 which is sought in each of the various rules for calculating 
 a balance after partial payments, but there are questions 
 involved whose decision is not afforded by inherent numer- 
 ical relations of the facts. 
 
 31. In teaching, it is supremely helpful always to empha- 
 size the difference between Pure Arithmetic and Applied 
 Arithmetic. The pupil's knowledge is surely in confusion 
 unless he sees the fundamental difference between, for ex- 
 ample, studying how to find a least common multiple (a 
 matter of insight), and a broker's commission (a matter of 
 empiric information so far as it is anything new when met 
 in a systematic course). Applied Arithmetic should be 
 presented in text-books as merely selected specimens of 
 many other practical applications which could be made, 
 and as problems, not as new arithmetical topics. The 
 necessary information should be set forth in an entirely 
 different tone from that in which arithmetical matters 
 proper are expounded. The particular applications usually 
 made are sufficiently well chosen; viz., calculations concern- 
 ing lines, surfaces, solids, times, weights, temperatures, and 
 money values, Avith special reference to the transactions of 
 mercantile and banking business. 
 
 But no candid criticism, even the most cursory, could 
 avoid complaint on account of the usual results of teaching 
 the metric system of units. Text-books are at fault here, 
 rather negatively than positively ; though some are found 
 to write ImSdmScm, when the system was devised ex- 
 pressly to avoid this — it is as if one should write 1 dollar, 
 3 dimes, 8 cents. They might be expected, however, to 
 
40 NUMBER AND ITS ALGEBRA. 
 
 put tlie matter in a]i appreciative and tonic way, instead of 
 leaving the discovery of its perfections to the chance alert- 
 ness of the pupil. Eor, because a matter is perfectly clear 
 and simple, it does not follow that it will be so esteemed. 
 The case in question demonstrates this paradox. The 
 metric system of units was invented for its perfect simpli-' 
 city ; yet, pitiful to say, it remains a bugbear to the average 
 teacher and pupil in the common schools. Nothing could 
 be more blind and irrational — it is exactly as if an Eng- 
 lishman could not be made to see that decimal money units 
 are simpler for all calculation than pounds, shillings, and 
 pence. Any student may be sure that, unless he regards 
 the metric system as perfectly clear, and vastly easier than 
 our barbarous English units, he has entirely failed to un- 
 derstand it — nor could one fail to appreciate it who really 
 understood anything of arithmetic. Its essential merit is 
 twofold : it is decimal, and therefore fits our numerical 
 notation ; its units for lines, surfaces, solids, masses, and 
 weights are all symmetrically dependent on one unit, the 
 linear. 
 
 The advantages of the second property ought to be as 
 manifest as those of the first ; but I Avill briefly illustrate. 
 If the volume of some homogeneous material is given as 
 2.76 cu. m. and its sp. gr. 3.5, the weight may be found 
 by multiplying the numbers : 2.76 X 3.5 = 9.66 tonneaux, 
 or, pointing oif to reduce to kilograms, 9660 kg. 
 
 Now, in comparison let the student calculate the weight, 
 given volume 2 cu. yd., 7 cu. ft., 6 cu. in., and sp. gr. 3.5. 
 In the first place, exact calculation is impossible in the 
 English units ; for the pound and the weight of a cu. yd. of 
 water are, of course, incommensurable. The first task is 
 to look up in some compendium of useful information the 
 
ARITHMETIC AND ALGEBRA. 41 
 
 approximate weight of a cubic yard of standard water, or 
 of a cu. ft., or of a cu. in., as may be vouchsafed. Then 
 reduce the volumetric terms accordingly, then multiply, 
 and finally (to be thoroughly English) reduce the approxi- 
 mate decimal fraction of the pound to ounces and grains. 
 One who has stupidly despised the metric system ought 
 to perform, as a penance, this calculation a V Anglaise. 
 
 32. It may be helpful to state explicitly that I always 
 use the term arithmetic in the sense of the Science of 
 Number. 
 
 There is no difference either in subject-matter or in 
 scope between arithmetic and the algebra of number. The 
 distinction made by the term algebra refers to the mode of 
 expression (vide § 20), and in a special sense to the pro- 
 fou.ndly important subject of algebraic forms of numerical 
 expression. Any arithmetical statement is of particular 
 numbers ; while, from the very nature of the conventions, 
 algebraic statements are general. It was to this end that 
 algebra was invented ; but it must never be forgotten that 
 any algebraic expression may be made particular (x'ide 
 § 23), and that the form then becomes arithmetical. 
 
 '^ Arithmetic " is too often limited (very illogically and 
 contrary to the best practice) to denote merely some primi- 
 tive developments of the science of number. Even the 
 distinction positive and negative is often expressly set 
 forth as peculiarly a matter of algebra. In our view this, 
 of course, is utterly subversive. 
 
 Arithmetic needs and uses the same symbols of opera- 
 tion and qualitative distinctions a.s the algebra of number ; 
 indeed, logically, the statement should be made the other 
 way, viz., the algebra iTses the same symbols of operation 
 and quality as arithmetic. The symbols +, — (in both 
 
42 NUMBER AND ITS ALGEBRA. 
 
 senses of each), tlie exponential notation, etc., belong 
 equally to the notation of arithmetic and to number's 
 algebra. 
 
 Newton preferred to call the algebra Universal Arith- 
 metic. 
 
 33. It would be a very good exercise for the student 
 (especially those who are taking this course in order to 
 qualify as teachers in the public schools) to critically 
 examine some text-book on arithmetic which he has heard 
 extolled.* 
 
 VII. Primary Number, — Numerical Operations. 
 
 34. There are seven distinct numerical operations. 
 Three of these are direct, and four inverses of these three. 
 The three direct arise from three different modes of com- 
 bining two numbers, and the four inverse, from inverse 
 problems, viz., given one of the two numbers in the former 
 combination and the resulting number, to find the second 
 of the two originals. Inasmuch as two of the direct oper- 
 ations are commutative (vide § 38), they give rise each to 
 only one inverse ; that is, it is the same problem, given 
 either and the result, to find the other. But the third of 
 the direct operations is not commutative ; and it therefore 
 gives rise to two inverses, it being a very different prob- 
 lem, whether the first or the second of the originals be 
 given, to find the other. 
 
 These seven operations are, by name, Addition and its 
 inverse, Subtraction ; Multiplication and its inverse, Divis- 
 
 * Upon a reperusal, this expression seems almost satirical, since it 
 would be impossible to find one which has not been extolled. I let the 
 phrase stand, however, in all its innocent irony. 
 
PRIMARY NUMBER. — NUMERICAL OPERATIONS. 43 
 
 ion ; Involution and its two inverses, Evolution and 
 Finding the Logarithm. 
 
 No numerical operations have been developed showing 
 characteristics essentially different from these modes of 
 operational combination. (For fuller discussion of this 
 point, vide § 104.) 
 
 35. It would be a sad comment on previous instruction if 
 any one is surprised to hear of the seventh of these funda- 
 mental operations ; for when we find a^ = c, the two in- 
 verse problems are equally obvious ; we may have given 
 b and c, to find a (this is familiar as evolution), or we 
 may have a and c given, to find h. Scientific mathematicians 
 (Cf. Introduction, p. 5) are to blame that no single name 
 denoting^ this operation is current. We must iise the 
 accepted phrase, finding the logarithm. Because the pro- 
 cess of this last operation is comparatively recondite, is 
 no excuse for not calling attention to the problem in the 
 very beginning of any systematic teaching of arithmetic. 
 Indeed, under any rational instruction it^ existence could 
 not be concealed. It should be said to the pupil, " Evidently 
 such a result as «* = c presents two inverse problems. At 
 this stage you will investigate only how a few very simple 
 roots may be extracted. The question how the exponent 
 or logarithm may be determined must be deferred until 
 you have acquired more knowledge and greater skill." 
 
 36. In this chapter the fundamental operations will be 
 tentatively considered for primary number. It will be- 
 come apparent that for any two primary numbers the 
 direct operations are always possible, but that the inverse 
 operations have meaning only in particular cases. Equally 
 obvious will become the urgent need and propriety ot 
 extending the concept of number, both for the theoretical 
 
44 NUMBER AND ITS ALGEBRA. 
 
 science, and its application to the measurement of concrete 
 magnitudes, 
 
 Tlte numerical symhols of the algebra evvployed in this 
 chapter are general only for irriniary number ; -j- and — 
 have only their operational meanings. This strict limita- 
 tion must be distinctly recognized. 
 
 37. To add one primary number to another is to so com- 
 bine the former with tlie latter that in the resulting num- 
 ber each unit of the components shall retain independence 
 and precisely the same functional relation to the result 
 (the sum) that it fulfilled in its original group. The con- 
 cept is so immediate to that of primary number itself (C/. 
 the specialization of various manys, % 2 et seq.), that, while 
 definition is appropriate for the purposes of scientific dis- 
 course, it hardly admits of explanation. The numbers are 
 aggregated, just as objects now thought in two groups 
 may be thought in one group. Also, the addition of any 
 two primary niimbers is always possible. (But note that 
 the definition is only for primary numbers, vide § 45.) 
 
 38. It is an immediate corollary from the absolutely 
 
 primary theorem of number {inde § 13), and the definition 
 
 of addition, that , , , , 
 
 ' a -^ h = -[- a, 
 
 that is to say, addition is a commutative operation. The 
 fact is called the Commutative law of Addition. It obvi- 
 ously extends to the sum of any number of numbers. 
 
 39. In like manner the addition of three or more pri- 
 mary numbers is associative, that is, 
 
 (,, _^ h)j^c = a^{h + r). 
 
 This fact is the Associative law of Addition. 
 
 40. An algebraic statement like the foregoing, the truth 
 
PRIMARY NUMBER. — NUMERICAL OPERATIONS. 45 
 
 of which depends on the very nature of operations, may- 
 be called a formula, as distinguished from a synthetic equa- 
 tion. In a formula any numerical symbol may be made 
 particular without restricting the generality of any other ; 
 in a synthetic equation (e.g., a -{- h = c), on the contrary, 
 to particularize any symbol more or less restricts the 
 meaning of every other. To solve a synthetic equation /or 
 any symbol, means to find a definite number which, sup- 
 posing the significance of every other symbol known, 
 substituted for the unknown symbol will satisfy the equa- 
 tions ; that is to say, make of it a formula in terms of the 
 other symbols. The name identitij is often used for for- 
 mula as here defined. When it is necessary to distinguish 
 between a formula or identity and a synthetic equation, 
 the sign = designates the former, and = the latter. 
 
 41. If the sum of tAvo primary numbers and one of 
 them be given, the other may be formed by pairing off 
 every unit of the given part with a unit of the sum, and 
 counting the unpaired units of the sum. Since addition 
 is commutative, the operation, as just defined, is the same, 
 whichever of the two parts of a sum be given. Addition 
 has therefore only one inverse, called Subtraction, and rep- 
 resented by the minus sign ( — ). 
 
 42. The problem is to solve for x the synthetic equa- 
 tion — , , 
 
 a -\- X =^ b. 
 
 Counting off a from the number represented by each mem- 
 ber of the equation, we obtain x = h — a\* that is to say, 
 
 * Of course the common notion or axiom, '"if equals be taken from 
 equals the remainders are equal" is here involved. But truly common 
 notions can be doubted by no sane man, and explicit statement of uni- 
 versal axioms is hardly required anywhere except in systematic treatises 
 on logic or epistemology. 
 
46 
 
 NUMBER AND ITS ALGEBRA. 
 
 {h — a) is the numbei- wliicli added to a gives h. This 
 number {b — a) is called the remainder or diiference result- 
 ing from the subtraction of a from b. Substituting (b — a) 
 for X gives the formula — 
 
 a -\- {b ~ a) = b ; 
 
 or, by the commutative law of addition, 
 
 b ~ a -\- a = b, 
 
 which is the formula of definition of subtraction. 
 
 43. Under the developed concept of number, any chain 
 of additions and subtractions enjoys perfect freedom of 
 commutation; but the first thing to strike the thoughtful 
 student in subtraction under the primary concept, is the 
 futility of seeking general laws, because the operation is 
 possible only in special cases. If b is less than a, b — a 
 makes no sense. 
 
 We may observe, moreover, that provided the expres- 
 sions mean anything, association may take place as fol- 
 lows : a — m — n = a — (in + n) ; for, adding m + n to 
 each member of the equation, we obtain a = a. (Cf. foot- 
 note to § 42.) 
 
 In like manner, a -i- b ~ m ~ u = a — (jii -f n — b) = 
 a — VI — (ji — b), etc. 
 
 Also a -\-b ~ m — n = a -{- (I) — vi — ?;). 
 
 This is the ground of the familiar rule about " signs " 
 and parentheses. Of course, the rule applies only under 
 great restrictions to primary numbers. 
 
 44. In practice the problem often occurs to find the 
 sum of a number of equal numbers ; e.g., how many shoes 
 are required to shoe twelve horses ? With primary num- 
 ber this is only a special case of addition. It was a true 
 
PRIMARY NUMBER. — NUMERICAL OPERATIONS. 47 
 
 instinct, however, which recognized a distinct operation. 
 But the instinct was too often disavowed in the next 
 breath by defining multiplication as " repeated addition." 
 Multiplication with primary numbers is repeated addition ; 
 ])ut this concept is incapable of development without doing 
 great violence to the word '' repeated." How can one so 
 define multiplication, and then say V2xV3 = V6? 
 Xo repeated addition can attain this result. This is antici- 
 pating ; but nevertheless we may, in the expectation of de- 
 velopment, at least be careful not to prejudice opinion. If 
 possible, let us try to say enough to define multiplication 
 for primary number without saying so much that the way 
 of development is barred. 
 
 45. The gradual extension of the meaning of terms is 
 perhaps the most powerful instrument for that ordering 
 and simplification of knowledge, that transformation of 
 chaos into cosmos, which is the vocation of science. The 
 procedure should take place with the caution befitting its 
 importance, and demands at every stadium a consummate 
 restraint of judgment in order not to say too much. The 
 severest self-criticism alone can repress the tendency of 
 tyros in every science to set delimitations which confine 
 development, and entomb thought in empiricism. 
 
 Note carefully that even addition must not be declared 
 as necessarily increasing a number. AVith primary num- 
 bers a number is increased by addition ; but to so define 
 would bar development. jSTeither in general does multipli- 
 cation increase a number, nor division decrease it, and to 
 so define would hide-bind mathematics. 
 
 46. The operation of multiplication can hardly be de- 
 fined for primary number without prejudice to the develop- 
 ment so necessary to mathematics, pure and applied. A 
 
48 NUMBER AND ITS ALGEBRA. 
 
 satisfactory definition has never been framed ; nor must it 
 be supposed tliat I consider the feat achieved in the fol- 
 lowing definition, which (or something like it) was first 
 offered, I believe, by De Morgan. The matter is one of 
 paramount importance ; for all rational views of number 
 have been developed under the principle of the persistence 
 of the laws of the operations, addition, multiplication, and 
 involution. (Their inverses would be adequately defined 
 merely as such.) Nor, until satisfactory definitions of 
 these operations in their utter generality are attained, can 
 the fundamental theory of the subject be regarded as per- 
 fect or completely established. I make an effort, not in 
 contentment with the result, but to display the difficulties. 
 
 The Multiplication of any number by another consists 
 in affecting the former (multiplicand) in precisely the 
 same way as one is affected in the other (multiplier). 
 
 Or, in Multiplication one number is so combined with 
 another that one of them shall fulfil in the result the 
 same functional relation that the number one fulfils in 
 the other. 
 
 The result is called the product. 
 
 The multiplication of any t\yo primary numbers is al- 
 ways possible. 
 
 With primary numbers the foregoing tentative definition 
 amounts to " repeated addition," nor is it claimed that it is 
 much better as a scientific achievement. The difference is 
 rather pedagogical : if you tell a pupil that " multiplica- 
 tion is repeated addition," he is disposed to think he fully 
 understands the nature of the operation ; but if you tell 
 him that in this operation the multiplicand is affected in 
 the same way as one is affected in the multiplier, although 
 he will not at first receive more information than before. 
 
PBIMAIIY NUMBER. — NUMERICAL OPERATIONS. 49 
 
 he is in a position to Avitlen liis concept of tlie " ways " in 
 which. 07ie may be affected in a nnmber. And when he 
 recognizes ratios as numbers, and that any number is its 
 own ratio to one, the composition of ratios at once falls 
 into his definition of multiplication. (Vide §§ 80, 81. 82, 
 83.) In other words, if ultimate development is not prej- 
 udiced by the definition suggested, it is only on the score 
 of its vagueness ; since in each new extension it is from 
 the principles of multiplication itself that the A\'ay in 
 which 07ie is affected, or its functional relation to the mul- 
 tiplier, can be comprehended. Nevertheless, it may be 
 considered that (when ratios have been recognized as num- 
 bers, and therefore necessarily a ratio of any two num- 
 bers, and any number as its own ratio to o7ie'), in the light 
 of the independently discovered operation, the " composi- 
 tion of ratios,'"' * the definition might be read in a fuller 
 sense than it could convey to a beginner. This may seem 
 a pitiful plight for a definition ; but I can only point out 
 that many things have to be seen to be understood, that 
 before such vision they must in their very nature be '' unto 
 the Jews a stumbing-block, and unto the Greeks foolish- 
 ness." The extended meanings of multiplication are un- 
 dreamed of to the man whose only notion of number is his 
 abstraction from a flock of sheep, or a pile of coins. 
 
 it might be better to leave the numerical operations un- 
 defined in words, and in the case of multiplication to rest 
 upon its commi;tative and associative laws (which alone 
 would not distinguish it from addition), and its law of dis- 
 tribution with addition, as at once governing and defining 
 the operation. (For these laws, vide infra.) 
 
 * Cf. Euclid, Book YI., 23; and VI., def. 5; or, better, IIalsted'.s 
 Elements of Geometry, §§ 540, 544. 
 
50 NUIMBER AND ITS ALGEBRA. 
 
 It may be objected that even addition has been defined 
 only for primary number ; but when number has been 
 seen to be a continuous magnitude, and the qualitative 
 difference, ^^osi'z^iye and negative, revealed, the "common 
 notion " of addition immediately applies. And, as sug- 
 gested in Section 37, to define addition is like defining 
 such terms as more or less. 
 
 The only point at all recondite is that, from the very 
 clean-contrary nature of positive and negative, the addi- 
 tion of a negative number to another decreases the latter. 
 Attention is called candidly to this generalization of addi- 
 tion, as well as to the application of greater and less to 
 negative numbers {Cf. §§ 116, 117, 198) ; but the propriety 
 of these concepts is left to be justified of their fruits. 
 
 47. The multiplication of primary numbers is commuta- 
 tive, i.e., — 
 
 ah = ha. 
 
 This is the Commutative Law of Multiplication. 
 
 Its truth is obvious, for three rows of four dots in each 
 row is the same group as four columns of three dots in 
 each column, thus — 
 
 Also, commutative freedom is shown to extend to the 
 factors in a sefies of successive multiplication, i.e., — 
 
 ahcde = ahedc, etc. 
 
 Multiplication with equal generality is associative ; that 
 is, any group of factors may be replaced by their product, 
 i.e, ahcde = a(hcd)e. 
 
 This is the Associative Law of Multiplication. 
 
PRIMARY NUMBER. — NUMERICAL OPERATIONS. 51 
 
 48. From the commutative nature of multiplication, it 
 follows that when a problem is discerned as requiring the 
 multiplication of one number by another, it never makes 
 the slightest difference theoretically which is taken as the 
 multiplicand. All the talk about carefully distinguishing 
 multiplier and multiplicand so prevalent in text-books is 
 sheer nonsense. If you wish to find how many oranges 
 you must provide to give 3 to each of 278 children, it is 
 utterly indifferent whether you multiply 3 by 278, or 278 
 by 3. As a matter of convenience in this particular ex- 
 ample the latter process, absurdly decried as it is, is the 
 sensible course. The problem requires the combination in 
 multiplication of the number of children and the number 
 of oranges. The product is interpreted as a number of 
 oranges. In neither case have oranges or children been 
 multiplied ; processes of horticulture or procreation would 
 be necessary in such a performance. 
 
 Concrete magnitudes can be multiplied by numbers, but 
 such processes are not purely arithmetical. For example, 
 a sect of a straight line can be really multiplied ; but the 
 process is a geometrical construction. Thus, to multiply a 
 sect by 3, lay off the given sect three times in a straight, 
 so that one of the three shall lie end-point to end-point 
 with the other two, but no other points in common. The 
 sect between the non-coincident end-points is the required 
 product. It would be an easy construction to multiply a 
 sect by ■y/2, for this would be accomplished by the fa- 
 miliar process of '^ altering " it in the ratio of the diagonal 
 of any square to its side. 
 
 Of course, it is perfectly legitimate to speak of the mul- 
 tiplication of concretes in the sense merely of an interpre- 
 tation of a numerical process. Thus, there can be no 
 
■ 62 NUMBER AND ITS ALGEBRA. 
 
 objection to saying 8 pounds multiplied by 152 make 
 1,216 pounds ; but it is utterly mistaken to protest against 
 multiplying 152 by 8 in performing the calculation. 
 
 49. Since multiplication is commutative, there is only 
 one inverse problem ; viz., given a product and one fac- 
 tor, to find the other. The operation is called Division. 
 Division requires the solution for x of the synthetic equa- 
 tion — , 
 
 ax = 0. 
 
 The formula of definition of division is — 
 
 a {l> I a) = h. 
 
 dotation ally a line laterally * presented to the number 
 symbols (— or / ), a colon ( : ), or a combination of both 
 (-f-) represents division. The first is generally to be 
 preferred. 
 
 50. With primary number division amounts to repeated 
 subtraction, but it is only safely defined as the inverse of 
 multiplication. (Vide § 45.) 
 
 51. Under the developed concept of number, if a num- 
 ber is to be combined with a series of others which operate 
 successively in multiplication and division, there is free 
 commutation and association in using the operators in the 
 manner displayed in the following : — 
 
 (1) (a X f^) -^ c = (a -i- c) X b = a X h / c = a -i- c / b; 
 
 (2) {a -~ h) -i- c = a / he = aj c -^ h ; 
 
 (3) (ft -V- h) X {c -^ d) = ac j bd = a j d -~ b / e = ac j h 
 
 -^ d, etc. ; 
 
 (4) {fi ^ b) -^ (c -^ d) ^ ad jbc = a I c-^b-d = ad / b 
 
 -f- c, etc. ; 
 
 * The " minus " sign is presented endwise to tlie number .symbols. 
 
PRIMARY KUMBER. — NUMERICAL OPERATIONS. 53 
 
 as may easily be proved from the laws of multiplication 
 and the definition of division, to be true for primary 
 number, if the opei'ations have any meaning at all. 
 
 But as in the case of subtraction (§ 43), it is vain to 
 attempt generalizations with division under the primary 
 concept of number, for division is possible only in par- 
 ticular cases. 
 
 Thus, considering that the primary numbers represented 
 are such that the statement (ct X h) -^ c makes sense, 
 (a -^ c) X l^ ii^ay, or may not, have meaning ; e.g. (3 X 4) -v- 
 6 makes sense, for there is a primary number which mul- 
 tiplied by 6 gives 12 ; but (3 -i- 6) X 4 is meaningless in 
 terms of primary number, for there is no primary number 
 which multiplied by 6 gives 3. Again (15 X 4) -;- 6 is 
 intelligible, but not (15-4-6) X 4 ; since no primary num- 
 ber multiplied by 6 gives 15. 
 
 52. If a sum of two primary numbers is to be miiltiplied 
 by a primary number, the product is the same as the sum 
 of the products of each summand by the multiplier, i.e., — 
 
 (rt -{- h) c = ac -[- be. 
 
 For 4 rows of 5 in a row is the same group as the sum 
 of two groups each of 4 rows, 2 and 3 respectively in a 
 ruw, thus : — 
 
 The principle evidently extends to the sum of any num- 
 ber of summands, and is called the Distributive Law of 
 Multiplication and Addition. 
 
 53. If the multiplier be a sum, of course redistribution 
 
 'univ.'ehsi'i 
 California 
 
54 NUMBER AND ITS ALGEBRA. 
 
 will display the final result as a sum of simple products, 
 e.g., 
 
 {a + h) {c -\- d) = a{r-\- d) -\.h {c -\- d) = ae -\- ad + he -\- hd. 
 
 54. If eacli one of a number of factors be a result of 
 mixed addition and subtraction, the Distributive Law ap- 
 plies, but with primary numbers only, under the miserable 
 restrictions inherent in inverse operations. 
 
 55. Also a series of additions and subtractions is dis- 
 tributable with a divisor. It is sufficient to give formal 
 proof in one instance : — 
 
 To prove {rt -\- V) -^ c = (a -4- r-) -f- (Ij -f- c). 
 
 Now, {(a -\- b) -^ c] c = a -\- b hy definition of division. 
 
 Again, {(a -^ c) + (i -f- <-)} = (a ^ c) c -\- (b -i- c) c by 
 distribution of multiplication and addition ; but this last 
 also = a -{- b hj definition of division, 
 .: (a -\- b) ^ c = (a -h c) -\- (b ^ c). (Cf. foot-note to § 42.) 
 
 56. If factors be sums, redistribution is possible, since 
 the original case merely recurs (vide § 53) ; but if a divisor 
 be a sum, it cannot be distributed. 
 
 (a -\-b)^ (e-\- d) = a - (^ + d) + // - (. + d), 
 but a -i- (c -)- d) does not equal (a -^ c) -f- (a -f- d), 
 as the student may easily satisfy himself. 
 
 Let this truth emphasize the principle that all such 
 questions are matters of fact, and not to be convention- 
 ally decided. 
 
 57. It frequently occurs in practice that it is required to 
 repeatedly multiply a number by itself. Given the basal 
 number and the number of times it is to occur as a factor, 
 the process is completely determined. The original num- 
 ber is called the base ; the number of times it is to occur 
 as a factor is called, according to the point of view, the 
 
PRIMARY NUMBER. — NUMERICAL OPERATIONS. 55 
 
 exponent of the base, or the logarithm of the result to the 
 specific base. The result is called the power. The expo- 
 nent is sometimes called an index. 
 
 Numbers in this relation are notationally represented 
 thus : 4^ = 64, or «* = e, where a is base ; h, the expo- 
 nent ; c, the power. The phrase logarithm of c to base 
 a is written in algebraic shorthand thus, log„ c. 
 
 58. When the exponent is two, the power is commonly 
 called the " square ; " and when the exponent is three, the 
 "cube." These names refer to true and proper geometrical 
 applications of number, but have no doubt had their share 
 in postponing general recognition of number's real nature. 
 {Cf. § 25, and §§ 230, 231.) 
 
 59. The operation of combining two numbers in the 
 sense represented notationally, as above explained, by a*, 
 is called Involution. But just as we restrained ourselves 
 from prematurely regarding multiplication as repeated ad- 
 dition, we must prejudice no subsequent questions by 
 regarding involution as repeated multiplication. It is re- 
 peated multiplication for primary numbers ; but when we 
 discern other modes of number we shall see that such is 
 by no means the essential nature of the operation. 
 
 60. It is impossible (for me) to frame a definition of 
 involution in terms of primary number which will satis- 
 factorily connote the simplest and the general meaning of 
 the operation. {Cf. §§ 45, 46.) In lieu of something more 
 satisfactory I make the following attempt : Involution is a 
 combination of two numbers such that the base shall appear 
 factorially in the result in a mode corresponding to that 
 in which unity exists additively in the exponent. While 
 this definition expresses primary involution, it is not in- 
 consistent with ultimate meanings. For example, if unity 
 
56 NUMBER AND ITS ALGEBRA. 
 
 exists three times aclditively in the exponent, the base 
 must appear three times factorially in the power ; yet when 
 niimbers are conceived in which unity fulfils a relation 
 the inverse of primary addition, we need not be surprised 
 to discover that the base appears in the power in a relation 
 the inverse of that of a direct factor. 
 
 [ «3 = aaa, and a~^ = -.-.-] . 
 \ a a aj 
 
 Again, when a number such as the ratio 1 / 3 is discerned, 
 it becomes a development, not a recantation of former opin- 
 ion, to discover that the exponent 1/3 imposes upon a base 
 an operation which shall cause it to appear in the result as 
 one of three equal factors of itself, since 1 / 3 is one of three 
 equal summands of 1. (a* = V«-) 
 
 It would be anticipating too much to carry testing any 
 further. I set forth the definition merely as the best that 
 I can offer. Perhaps the most scientific attitude in the 
 dilemma is merely to note the sense of involution for pri- 
 mary number, alertly waiting to discover what its nature 
 may be as deepening insight reveals other modes of num- 
 ber, and surmising upon general grounds that if a* means 
 repeated multiplication when i is a primary number, it 
 will not have this meaning if b is not a primary number. 
 
 61. I have dwelt upon this matter because it is an 
 exceedingly important point. The application here of the 
 Principle of Continuity {vide § 103) has led to un- 
 dreamed-of advances, not only in the mathematics, but 
 in the physical sciences. 
 
 62. Involution is evidently not commutative : «* is not 
 !)"■. A unique case is commutative ; 2* = 4^. 
 
 Neither are successive involutions associative : a/^^ is 
 not equal to ((I'^y. 
 
PRIMARY NTTMBER. — NUMERICAL OPERATIONS. 57 
 
 63. Let the student find the difference between 2(2==') 
 and (2-^) (='). 
 
 64. "Law of Indices." — For primary numbers it fol- 
 lows immediately from the definition (let the student deduce 
 the forms, however) that oVa"^ = «* + ^ + ''; (a^y = a^'^, 
 and a'^b'' = (aby. 
 
 Also iib> c, rt* ~r- a' = a^''. (See also §§ 158, 191.) 
 
 65. Because involution is not commutative there are 
 two inverse operations, requiring respectively the solution 
 for X of the synthetic equations (1) a'« = b, and (2) a-^ = //. 
 
 66. Operation (1) is called Evolutiox, or finding the 
 ath root of b. In algebraic shorthand the rtth root of b is 
 written -\^b. The radical sign is derived from the letter r 
 (radix). In actual computation (arithmetical or alge- 
 braical) after the theory of exponents has been general- 
 ized, it is far better to employ indices than radical signs. 
 
 67. Operation (2) is called Finding the Logarithm 
 {aide § 35). 
 
 68. The Formula of Definition of Evolution is (V^)« = b. 
 
 69. The Formula of Definition of finding the Logarithm 
 is a^^-a* = b. 
 
 70. As has been seen to be the case with all inverse 
 operations in terms of primary number, these inverses of 
 involution are evidently possible only in very special cases. 
 
 71. With the discovery of the seven operations, and their 
 laws, Commutative, Associative, Distributive, and the Law 
 of Indices or Exponents, the foundation of arithmetic and 
 the algebra of number is complete. I repeat (Cf. §§ 20, 23) 
 these laws could never have originated arbitrarily, or as 
 springing essentially from the algebra. As ^' algebraic 
 laws " they must be merely the expression of actual prop- 
 erties and relations of number. 
 
68 NUMBER AND ITS ALGEBRA. 
 
 VIII. Devices of Computation. 
 
 72. Various devices of computation, of more or less 
 practical utility, are familiar to all ; but it will be clear 
 upon any thoughtful consideration that they possess none 
 of the fundamental importance suggested by the promi- 
 nent role they play in ordinary text-books. AVhat is usu- 
 ally set forth as a general exhibition of addition must 
 be seen to be several partial additions and a convenient 
 association of resulting summands. The same numbers 
 would have their parts differently associated to suit dif- 
 ferent notations, e.g., XXXVII -f XXXVIII = LXXV ; or 
 37 + 38 = 75. 
 
 The average high-school graduate labors under the im- 
 pression that his fashion of " multiplying " is essential to 
 the matter, and arises from the very nature of things. In 
 " division " he learns what he sometimes regards as two 
 ways, ''Short'"' and ''Long." The names are, in truth, 
 appropriate enough, for the sole difference is that more of 
 the necessary thought is actually written down in the 
 Long than in the Short way. Yet the abbreviated form is 
 taught first, and the pupil fancies he is learning some- 
 thing new and more difficult when he learns " Long divis- 
 ion." 
 
 The rational method would be to teach first an expres- 
 sion still longer than the "Long"; then, as skill and 
 power of retaining conclusions in mind increase, conve- 
 nient abbreviations should be explained and recommended. 
 
 73. Let the student critically examine his habitual 
 ways of "adding," "subtracting," "multiplying," and "di- 
 viding " primary nunil)ers, bf)t]i in the common algorithm 
 of arithmetic, and algebraically. Let him denote every 
 
DEVICES OF COMPUTATION. 59 
 
 act of his mind in each process as an addition, subtraction, 
 multiplication, division, commutation, association, or dis- 
 tribution of numbers, under the definitions and laws set 
 forth in the preceding chapters. To take a very simple 
 example: {an^ c^) {a^ h^ c"^) ~- {aH>^ c^"-) 
 
 = (a^ a^ ¥■ V^ c^ c") -f- («•* b^ c^^) ... by association and 
 commutation. 
 
 = (a^h^c^^) -=- (a*b^c^^) ... by three partial multi- 
 plications by law of indices. 
 
 = (o^/a*) (P /h^) (c^^ I c^°) ... by association and 
 commutation. 
 
 = a^ ^^ c . . . by three divisions by law of indices. 
 74. Explain how a multiplying machine, which can 
 do no more at one time than multiply a number of ten 
 places by another of ten places, may be used to multiply 
 13693456783231 by 46381239245932. 
 . 75. The involution of primary numbers may be ac- 
 complished merely by repeated multiplication. As soon, 
 however, as one investigates logarithmic series, and the 
 construction and use of Tables of Logarithms, he learns 
 command of a more facile waj^ of performing this labo- 
 rious operation. Before learning the use of logarithms, 
 one ought to demand good wages for the toil it would cost 
 him to find 9^" ; afterwards it becomes the Avork of a few 
 minutes. 
 
 76. Evolution, as we have seen, is only occasionally pos- 
 sible under the primary concept oi number ; but even in 
 the simplest of these possible cases the device of calculation 
 familiarly used by the high-school pupil is rarely under- 
 stood, else he would be able to find (however laboriously) 
 the fifth root as well as the third. Of course evolution is 
 too laborious to be carried to any extent until Logarithmic 
 
60 NUMBER AND ITS ALGEBRA. 
 
 Tables are comprehended, Avhen it becomes easy. But if 
 one understood how his device for extracting a second or 
 third root was invented, he coukl on occasion make his own 
 rule for finding a fifth root. Let us investigate. Properly 
 distributing and associating, it is seen that — 
 
 {a J^ hf = a- -\- b {2 a -\- h). 
 
 Also {a-{-h -\- c)- = (a + i)- + c {2 (a + Z-) + c], etc. 
 
 Here is declared a rule for the evolution of a second root 
 of a number ; for a specific composition of the power is 
 displayed in a way to make decomposition easy. Likewise 
 the formulae for the evolution of a cube root are 
 
 (a + by = a^ + Z. (3 a^ _^ 3 ab + b''), 
 
 and (a-\-b -^ cf = (ci ^ b'f J^ c {^ {a -^ bf + 3 (a + b) c 
 -f- c-}, etc. 
 
 In exactly the same way the formula for the evolution of 
 a fifth root is 
 
 (ft + bf = «5 ^Jj(pa^j^l0 an> + 10 a%'' + o ah^ + b'), etc. 
 
 Suppose the fifth root of 33554432 is required. 
 
 liow the preceding formulse show that, if the root be 
 considered as the sum of three numbers, the corresponding 
 power of the sum of the first two is to be taken away, and 
 the remainder decomposed to reveal the third summand of 
 the root, and so on. Therefore we could not go wrong 
 even by choosing parts of the root at random. But a con- 
 sideration of the arithmetical notation may save much 
 trouble ; for it is plain that a fifth root of the number 
 before us has two digit figures, that is, it is to be regarded 
 as the sum of a number of tens and a number of ones. We 
 compute as follows : — 
 
FIRST EXTENSION OF THE NUMBER-CONCEPT, 61 
 
 
 
 a b 
 
 
 33554432 
 24300000 
 9254432' 
 
 30 + 2 
 
 a5_ 
 
 
 5 «" = 4050000 
 
 (Here we guess our 6 ; the calculation 
 will test accuracy.) 
 
 10 a^b = 540000 
 
 
 
 10 a%'- = 36000 
 
 
 
 5ab' = 1200 
 
 
 
 b^ = 36 
 
 
 
 4627216 
 
 9254432 
 
 (Got by multiplying 4627216 by &,as the 
 formula directs.) 
 
 77. ISTow let the student compute again, taking 20 for a 
 and 12 for b. Also let him prove 12 a cube root of 1728, 
 taking 6, then 4, then. 2, as summands of the root. 
 
 IX. First Extension of the Number-Concept. 
 Eatio. — Fractions. — Surds. 
 
 78. The first extension of the concept of number is the 
 identification of the ratio of any two magnitudes of the 
 same kind, and without qualitative distinction for the pur- 
 poses of the comparison, as a number. 
 
 79. This step was taken long ago (^Cf. Introduction, 
 p. 11), and is now universally accepted as a dictum, even 
 where not clearly discerned as a matter of insight. 
 
 80. This development of the number-concept was no 
 doubt occasioned in the history of human experience by 
 problems of practical measurement. {Cf. Introduction, 
 p. 13.) 
 
 Thought must have operated as follows : If the numeri- 
 cal relation of a yard to a foot is 3, surely there is a num- 
 ber denoting the relation of a yard to two feet, and of a 
 
62 NUMBER AND ITS ALGEBRA. 
 
 foot to a yard. That is, numbers which are fractions (vide 
 § 83) of primary number were discerned. This advance 
 still leaves number discrete, that is, increasing per saltum. 
 But again, as a second step, if there is a numerical relation 
 between two magnitudes, one of which is a fraction of the 
 other, surely there must be a numerical relation between 
 an 3^ two magnitudes of the same kind, even though neither 
 be a fraction (vide § 83) of the other. Thus, when it is 
 proved that the diagonal and side of a square are absolute- 
 ly incommensurable {Euclid, Book X, 117), the mind can- 
 not tolerate the thought that a numerical relation would 
 exist, provided the diagonal were just the least bit shorter, 
 yet, de facto, does not exist. This thought, I repeat, is in- 
 tolerable. Moreover, since the ratio of a yard to a foot 
 is an exact number, surely the ratio of a metre to a foot is 
 exactly whatever it is. It is, of course, well known that 
 the metre and foot are incommensurable 
 
 81. The connotation of all ratio (fractional and surd) as 
 number evidently makes number continuous one ivay, to 
 use a space metaphor on account of the exigencies of lan- 
 guage. Thus, under this concept, number begins with a 
 ratio smaller than any assignable fraction of 1, increases 
 continuously, passing through all the discrete stages of 
 primary number, to a ratio greater than any assignable 
 primary number. 
 
 82. To illustrate: Start with the ratio of the weight 
 of these pages to the weight of a granite bowlder. We 
 begin either with a very small fraction of 1, or a surd 
 smaller than a very small fraction of 1 (as the weights are 
 commensurable or not, probability being vastly in favor of 
 the latter case). Now, by gradual abrasion of the bowlder, 
 decrease its mass; the ratios of the weights increase con- 
 
RATIO. — FRACTIONS. — SURDS. 63 
 
 tinuously until they reach 1. Continue the abrasion, and 
 the ratios increase continuously, passing through 2, 3, 4, 
 etc. At length when the bowlder has been reduced to a 
 grain of sand, the ratio will be greater than some high 
 primary number. 
 
 83. The foregoing discourse presumes sufficient familiar- 
 ity with the subject to insure the reception of the terms 
 employed in their precise meaning ; yet it may be service- 
 able to set forth the following definitions {Cf. § 205) : — 
 
 (1) Multiple. — One magnitude is a multiple of an- 
 other when the former may be separated into equal parts, 
 each equal to the latter. (Of course '' multiple " includes 
 the limiting case where the ''part" is the whole, i.e., jiiulti- 
 plication by 1. It is merely an imperfection of language 
 which might seem to exclude this case.) 
 
 (2) SuBMULTiPLE. — lu (1) the " latter " is a submulti- 
 ple of the '' former." 
 
 (3) Fraction. — Any multiple of a submultiple is a 
 fraction. (Of course if a is a fraction of J, i is a fraction 
 of a ; also a multiple of a submultiple may reduce either 
 to submultiple or multiple.) 
 
 (4) Commensurable. — Two magnitudes are commen- 
 surable if either is a fraction of the other ; 
 
 (5) Incommensurable. — if neither is a fraction of the 
 other. 
 
 (6) Ratio. — That definite (exact) numerical relation 
 {Cf. § 80) of two magnitudes of the same kind, in virtue 
 of which one is either a fraction of the other, or greater 
 than one and less than another fraction of the other, 
 which differ as little as we please, is called the ratio of 
 the former to the latter. 
 
 Of course, from the very concept of ratios, and the 
 
64 NUMBER AND ITS ALGEBRA. 
 
 continuity of possible ratios, the ratio of the first of two 
 magnitudes to the second is greater than the ratio to the 
 second of any magnitude less than the first. Also two 
 ratios are equal if every numerical fraction greater than 
 either is greater than the other, and less than either is 
 less than the other. 
 
 A ratio is often spoken of as "incommensurable," of 
 course as an abbreviated expression, since it takes two 
 things to be incommensurable. You might as well say, " x 
 is equal," as to say "x is incommensurable." The abbre- 
 viation is for incommensurahle with 1. Incommensurable 
 ratios may be called surds. 
 
 Let it be clearly noted that a multiple, a submultiple, 
 or a fraction of any magnitude, is another of the same 
 kind : but that the ratio of two is a number. Thus a 
 fraction of a time is a time, of a surface a surface, of a 
 solid a solid. Bvxt the ratio of one solid to another is 
 a number, — in this case called the volume of the former 
 with respect to the latter. 
 
 Note also, any number may be regarded as its ratio to 
 1, and that all numerical fractions are ratios, but not all 
 ratios are numerical fractions. 
 
 In illustration of the definition of a ratio, and its nota- 
 tion, if of incommensurables, consider the yard and the 
 metre. Measurement {vide § 203} not excessively refined, 
 gives the number 0.9143 + for the ratio of a yard to a 
 metre. This is to be understood to mean that a yard is 
 greater than -fVVotT ^^ ^ metre and less than tVoVo- Meas- 
 urement more refined would yield a numerical fraction still 
 more closely approximating the ratio. The ratio in ques- 
 tion has been found to be greater than 0.914392, and less 
 than 0.914393. 
 
OPERATIONS UNDER FIRST EXTENSION. 65 
 
 (7) Surd. — Of the one-icay continuous Kumber, the 
 concept of "which we have now attained, those numbers 
 which are incommensurable with 1 may be called surds. 
 
 It is matter of discovery that the V^ is incommensur- 
 able or a surd. 
 
 The term surd is sometimes exclusively referred to the 
 results of such operations as V2 ; but Newton's use is 
 a philosophical one. For the V2 is found out to be 
 1.41421 -f-, that is a number, no fraction of one, but greater 
 than 1.41421, and less than 1.41422, which is surely a 
 number of precisely the same kind as the ratio of a j^ard 
 to a metre, or of a circle to its diameter (0.914392 -)- and 
 3.14159 -}- respectively). Incommensurable numbers re- 
 sulting from evolution may be distinguished as radical- 
 surds, or simply radicals. (Vide § 145.) 
 
 X. Significance of Operations, and Special Opera- 
 tional Devices, Appropriate to the First 
 Extension of the jS^umber— Concept. 
 
 84. Euclid probably never clearly unified his concepts 
 of ratio and number; but following Euclid (q.v., and cf. 
 Halsted's Elements of Geometry), it may be shown that 
 there is a combination of ratios which obeys the same 
 laws that govern the addition of primary numbers, or of 
 fractions of concrete magnitudes, an inverse operation 
 corresponding exactly to subtraction; another operation 
 ('' composition of ratios "), which obeys the same laws as 
 the multiplication of primary numbers, and an inverse 
 ("altering" a magnitude in a given ratio), corresponding 
 to division. 
 
 But, from the very definition of a submultiple of any 
 
66 NUMBER AND ITS ALGEBRA. 
 
 magnitude, the finding of a siibmultiple is identified as an 
 operation of division, since the problem is to find a magni- 
 tude which imdtijylied produces the given magnitude. Now, 
 when number has been discerned as a magnitude, these 
 reflections make it plain that a fraction of a number is the 
 number resulting from the division of that number liy 
 another, that one-half of 1 is 1 -f- 2, etc.* Also, when ratios 
 have been identified as numbers, and number thus be- 
 comes one-way continuous, the operational significance of 
 the principles, established in Chapter YII. for Addition 
 and Multiplication and their inverses, extends to all num- 
 ber (primary, fractional, and surd) thus far conceived. 
 
 Finally, inasmuch as a fractional number is the result 
 of dividing one primary number by another, it may be 
 represented most conveniently by the notation already 
 established for division. (Vide § 49.) 
 
 It would be impracticable to invent individual symbols, 
 since an unending' number of different symbols would be 
 demanded to designate even the fractional numbers lying 
 between two consecutive primary numbers ; nor could any 
 such symbol be used otherwise than as a record, since in 
 any calculation with fractions it is the generating numbers 
 which are utilized, and not the fractional number itself. 
 
 85. It seems to me that there is no way substantially 
 different from the lines of thought I have followed, where- 
 by one can really understand what he is doing in the oper- 
 ation 7/8 X 9/5 for instance. Teachers of arithmetic 
 would do well to ponder their methods at this point. 
 
 * The only explanation ( ?) of such conclusions to be found even in the 
 splendid Text Book of Algebra by Professor Chrystal, is " the statement 
 that /i X K isK oi % is merely a matter of some interpretation, arithmeti- 
 cal or other, that is given to a symbolical result demonstrably in accord- 
 ance with the laws of symbolical operation." Vol. i., p. VS. 
 
FRACTIONS. 67 
 
 86. It remains to investigate devices for performing tlie 
 seven numerical operations in this extended region of num- 
 ber, and in t^vo cases to discover the effect, the meaning, of 
 an operational combination; viz., in involution, if the expo- 
 nent be a fraction or a surd. In the first place, it is to be 
 borne in mind that it is one thing to conceive an operation, 
 and another to perform it. For example, at the conclusion 
 of these introductory lectures, it will be plain to all (if now 
 obscure) thq,t such operations as involving 10 under the 
 exponent tt, or finding the logarithm of 5 to the base 12, 
 are perfectly intelligible, even though ignorance of loga- 
 rithmic series, or of the use of a table of logarithms, should 
 leave one without devices adequate to the performance of 
 the calculations. 
 
 87. It should be observed that the terms iiumerator and 
 denoininator applied to the numbers involved in a numeri- 
 cal fraction, or even to the "terms" of a ratio of incom- 
 mensurables (e.g., V2 / 6) may be used as convenience 
 suggests ; but conceived operationally they are to be 
 thought as dividend and divisor. Tlie numerical symbols 
 ill tlie algebra of this chajjter are still to be understood as 
 representinr/ 2^^i'>^i(^fl/ ninnbers. 
 
 88. The '•' rules " for the operations of addition, multi- 
 plication, and division of fractions follow immediately from 
 the definition of a fractional number, which is merely the 
 recognition that the inverse of multiplication is always 
 possible, that the result of the division of any primary 
 number by any other is a number. 
 
 Substraction remains refractory, and meaningless unless 
 the minuend be greater than the subtrahend. 
 
 The rules are only the generalization of Sections 51 and 
 55, q.v., yet it may be serviceable to discuss them. 
 
68 NUMBEll AND ITS ALGEBRA. 
 
 89. By the distributive ]uw — 
 
 therefore the common rule. 
 
 Also a I d — h I d = by the distributive law, if 
 
 a y b ; therefore the common rule. 
 
 But how shall we perform a -{- h j c, or a j h -\- c j d, if 
 a / h, h / c, and c / d are fractions ? The operation is dis- 
 tinctly conceivable ; but the device for performing it re- 
 quires an intermediary step of multiplication, which must 
 therefore be investi.trated. Consider -^ 
 
 'o*^ 
 
 (1) a / b X c = ac / b = a -i- h / c = c -^ h / a. 
 
 (2) a/b -^ c = a I bo = ac H- b. 
 
 (3) a/b X c/d = ac/bd, etc. (Cf §51, (3)). 
 
 (4) a/b -i- c/d = ad /be, etc. (Cf §51, (4)). 
 
 (5) a / b = a / b X c / c = ac / be, also a / b = {ci J h -i- c) 
 
 ale 
 X c = -J— . 
 b I c 
 
 (6) a X b I c = ab I e — a -^ c I b, 
 
 all by the laws of division and multiplication (vide § 51). 
 Therefore the common rules : From (1), To multiply a 
 fraction, multiply the numerator or divide the denomina- 
 tor ; from (1), to multiply by a fraction, multiply by the 
 numerator and divide by the denominator ; from (2), to 
 divide a fraction multiply the denominator or divide the 
 numerator ; from (1), to divide by a fraction divide by the 
 numerator and multiply by the denominator, etc. ; from (3) 
 and (4) for cases where both terms of the operation are 
 fractions. Also from (5) it is obvious that to multiply or 
 divide both terms of a fraction by the same number neither 
 increases nor diminishes it ; and from (G), the result is 
 
FRACTIONS. 69 
 
 indifferent whether we multiply by a fraction, or divide by 
 its reciprocal. 
 
 90. It may be remarked that there is a distinction be- 
 tween dividing by a fraction and multiplying by its recipro- 
 cal, though the results are indifferent, as declared in Section 
 89 (1). The operations are not identical. The results of 
 4^, 4 X 16, 4 + 60 are the same, but the operations are by 
 no means identical, b / a is called the recijjr'ocal of a / b, 
 and may be obtained operationally from the latter by 
 dividing 1 hy a / b ; for 1 -i- a / b = b / a. Moreover " in- 
 vert " is a short-cut term which may be used among those 
 whose knowledge of first principles is assured ; but it 
 should never be used in explanation, as designating an 
 operation — one can as little turn a number upside-down 
 as inside-out. 
 
 In the United States of America the custom is almost 
 universal, never to divide by a fraction, but to choose 
 instead the equivalent operation of multiplying by its 
 reciprocal. In Europe this is not so commonly felt to be 
 more convenient. As a question of practical calculation 
 the matter is of no importance ; but it is surely lamentable 
 if pupils are led to think that they are dividing by a 
 number when they are actually multiplying by a different 
 number of such relative value that the results are equiv- 
 alent. Notationally a fractign expressly represents an 
 unperformed operation. The unexpressed result is the 
 definite number : thus, 7/6 means 7 divided by 6 ; and the 
 result is a number greater than 1 and less than 2, a defi- 
 nite value of the continuous magnitude we call Number. 
 
 A fraction in operation is to be employed as a composite 
 term consisting of a dividend and a divisor. Now, it can 
 be reasonably explained even to a very young student of 
 
70 NUMBER AND ITS ALGEBRA. 
 
 aritlimetic that to divide by a quotient is equivalent to 
 dividing by the dividend and multiplying by the divisor. 
 This having been established, he can see that the problem 
 to divide by ajh resolves itself into dividing by a and 
 multiplying by b. If the dividend is an integer, he has 
 simply to do this. If the dividend is a fraction, he must 
 first have been led to see that a fraction is rmiltiplied by 
 multiplying its numerator, or dividing its denominator ; 
 and divided by dividing its numerator, or multiplying its 
 denominator. 
 
 If these principles are discerned, he can proceed in any 
 manner he prefers. It is of no theoretical consequence 
 how he sets down on paper mental conclusions. There 
 is no obstacle to performing the division, under the princi- 
 ples stated, just as the symbols stand : 7/6-^3/5 = 
 35/18. 
 
 A very low order of convenience is subserved by mak- 
 ing a different problem of identical result :7/6 X 5/3 = 
 35/18. 
 
 This discussion may seem almost trifling ; but if one will 
 reflect that the average common-school pupil thinks he must 
 transform any such problem of division into a problem 
 of multiplication, some deficiency in the usual instruction 
 at this point will be apparent. I am convinced that our 
 schools require systematic, instruction in arithmetic of chil- 
 dren entirely too young to be capable of the reasoning and 
 insight demanded. 
 
 In such cases the best one can do is never to leave any- 
 thing totally unreasonable to the child. Even to a young 
 child very recondite matters can be a little explained — • 
 brought within a dim light of reason, if not clearly illu- 
 minated. One thing is certain, — bad history, bad gram- 
 
FRACTIONS. 71 
 
 mar, bad chemistry, or bad mathematics, is always bad 
 pedagogy as well. 
 
 If instruction in so-called arithmetic is always to have 
 reference to concrete magnitudes, as recommended * by the 
 latest "psychology of number" (Cf. Introduction), the 
 simplest method for division by a fraction would be to 
 reduce to common denominator : thus, 7/6-i-3/5 = 35/ 
 30 -^ 18 / 30 = 35 / 18. Indeed, it may well be, when 
 arithmetic has to be taught to children too young for the 
 subject, that this method is the best as a first pi'ese7itation 
 of the matter. Because the crudest notion of numerical 
 fractions, and blindness to the true significance of our nota- 
 tion of fractions, is not incompatible with some rational 
 comprehension of this process. 
 
 91. We may now return to ovir problems, a -\- h [ c and 
 a I h -{- c I d. By Section 89 (5) they may be brought under 
 the case oi a I d -\- b j d. For a -{- h j c = ac / c -|- b / c = 
 
 '^^_±A. And a/b-i- c/d = ad/bd-\- cb / db ^^I^Al^. 
 
 There is often a better way of solving the second prob- 
 lem. Evidently if b and d have a common multiple, vi, 
 less than their product, it would be advantageous, es- 
 pecially if several fractions were to be added, to reduce 
 to a common denominator by multiplying both terms of 
 each fraction by m-divided-by-the-denominator. No doubt 
 all are familiar with a device for finding the least common 
 multiple of two or more numbers. (Vide § 242.) 
 
 92. Inasmuch as an exponent of involution when a 
 primary number requires merely repeated multiplication, 
 we see — 
 
 * Psychology of Number, McLellan .and Dewey, p. IIG. 
 
72 NUMBER AND ITS ALGEBRA. 
 
 (a Jh)P = a /h-a /h-a /h • • • = av J bP. 
 
 Also, since V^ / V^ • Va / V^ r= a J b, 
 
 therefore, V* / ^ = V^/ / V^, etc. 
 
 93. If a fraction is expressed as a sum of decimal frac- 
 tions, e.g., 41.2164, evolution is apparently performed pre- 
 cisely as in Section 76. This is permissible, because — • 
 
 41.2164 = -VoVoV- and V-VoVoV" = V412164 -- VlOOOO. 
 Our notation renders it easy to perform a portion of this 
 calculation at a glance by " pointing off; " but the operation 
 must be understood as finding the •\/412164, and then di- 
 viding it by VlOOOO. 
 
 Let the student perform the calculation, not losing sight 
 of ivhat he is doing in how he does it. 
 
 Let him also fully express the operations involved in the 
 conclusion, 41.2164 = -VoVoV- ^^^^ notation is so perfect 
 that it may almost be said to work automatically, and for 
 this very reason it often blindfolds teacher and pupil. 
 
 It would richly repay the student to perform just once in 
 his life such a calculation as V41.2164 under an imperfect 
 notation. Let him do this, expressing everything in the 
 Roman characters. 
 
 94. As has been said (§ 83 (7)), it is a matter of dis- 
 covery whether or not, in any particular case, Va is a surd. 
 (Cy. § 156.) For example, if in the process displayed in 
 Section 76, it appears that no primary number is the root 
 in question, we may go on in the process of Section 93, and 
 find a fraction approximating as near as we please the surd 
 number which is the true root. Under such conditions the 
 root is a surd, and the process described interminable ; but 
 it would carry us too far afield to investigate just now 
 general criteria for deciding whether the result of given 
 
FINAL EXTENSION OF NUMBER-CONCEPT. 73 
 
 combinations of given numbers is a commensurable or 
 incommensurable number. ( Vide § 249.) Let the student 
 critically examine his familiar process in " finding V2." 
 
 95. In general an incommensurable number cannot ope- 
 rate, or be operated upon, in ultimate calculation in com- 
 binations with primary or fractional numbers. In lieu of 
 using the surd itself, we must use a fraction differing from 
 it by as little as we please ; e.g., if the ratio of a circle to 
 its diameter enter into the calculation, we employ some 
 approximate fraction, such as 3.14159. Su.rds which are 
 roots of primary numbers or of fractions may operate with 
 their exact force in special cases, and in a partial way ; e.g., 
 (a/2)3 = 2; V2 V3 = V6 ; 2 Vl2 = 4 V3 ; V2/3 = 
 
 1/3 V6, etc. ; 
 but investigations into such combinations must be post- 
 poned to the next chapter, as well as the interpretation 
 of a% if s is a fraction or a surd. 
 
 96. Finally, let it be distinctly recognized that the great 
 stumbling-block which confronts us at every turn is the 
 wretched limitation to special cases of the operation the 
 inverse of addition, that a — b is meaningless if a < b. 
 
 XI. Final Extension of the Numbek-Concept. 
 Principle of Continuity. 
 
 97. Primary number is a discrete magnitude. The first 
 extension of the number-concept (the connotation of ratios 
 as number) made number one way continuous. (^Vide 
 §81.) 
 
 The conception of number as continuous in a far more 
 general sense grew from the application of a principle, at 
 first presented as an assumption, but which is so inces- 
 
It 
 
 4 KUMBER AND ITS ALGEBRA. 
 
 santly and overwhelmingly corroborated that its rank as a 
 genuine and compulsory theory is perhaps as firmly estab- 
 lished as that of any scientific jDrinciple whatsoever. 
 
 98. As has been repeatedly shown in the foregoing 
 chapters, the combination of numbers in the inverse opera- 
 tions is meaningless under the primary concept except in 
 special cases. For example, 5 — 5, 5 — 6, 5 -f- C, V5, 
 logs 6, etc., result in no primary numbers at all. 
 
 The " first extension " gives meaning to the last three of 
 the cases just cited ; for, although in the treatment here 
 presented, a^, where s is fractional or surd, was not inter- 
 preted from the recognition of all ratio as number, the true 
 meaning might have been developed at that point, and 
 logs 6 thereby rendered intelligible.* All this, be it noted, 
 without understanding 5 — 5, or 5 — G as a number, or 
 even imagining the development yet to come after this 
 insight is attained. 
 
 99. For centuries science rested here, either not regard- 
 ing such combinations as intelligible, and their results as 
 numbers ; or only in a halting fashion, regarding the com- 
 binations as symbolic jugglery, and the results as '' imagi- 
 nary numbers." And at the present day it is only by the 
 enlightened van among men of science that this stage has 
 been passed. 
 
 Negative numbers were in this way long called " imagi- 
 nary ; " but, as they gradually forced themselves into reluc- 
 tant minds, the appellation was narrowed to denote V— 1. 
 
 100. It was only after a long struggle that negative 
 numbers gained recognition. I have not the erudition to 
 
 * It was deemed more convenient to take the final step at once; since 
 the principle which displays ratio as nuniher, and the general iirinciple 
 to which the whole treatment converges, are really one and the same. 
 
PINAL EXTENSION OF NUMBER-CONCEPT. 75 
 
 furnish exact dates, but I know that Cardan in 1545 in 
 his Ars Magna calls them ^' numeri Jicti;" and it is com- 
 monly asserted that Descartes in the seventeenth century 
 was the first to rend this portion of the veil : and I sup- 
 pose that those who half-heartedly follow in the wake of 
 science continued long afterward to regard negative num- 
 bers as " imaginary," and all operation therewith as some- 
 how a trick of algebraic signs empty of numerical meaning. 
 Certain it is that such is the attitude even to-day, not, it is 
 true, of those who follow in the wake, but of those who do 
 not follow science at all, though engaging a large share of 
 public attention as teachers thereof. For certain also it is, 
 that at the close of the seventeenth century, Newton with- 
 drew negative numbers (and therefore, as will duly appear, 
 zero, and positive and negative infinity) from the befogged 
 region of " nuvierl ficti," and revealed them as " numeri 
 veri. 
 
 The last stage of this gradual process of enlightenment, 
 in which V — 1 is still regarded as " imaginary," is yet the 
 stronghold of ignorance of fact, of prejudice, and of color- 
 blindness to philosophic evidence. 
 
 101. I would have no war of words over the appellation 
 '' imaginary." The term in this connection historically 
 has meant, and yet baldly means, " impossible," or incom- 
 prehensil/le. Of course it has no such meaning among the 
 best mathematicians of to-day; but that it is so received 
 by the unscientific, by many teachers of mathematics, and 
 by the vast majority of undergraduate students, cannot be 
 disputed. 
 
 The matter of a change in terminology is not of prime 
 importance, for terms may be disassociated in technical 
 use from their general meaning. It is a question of ex- 
 
76 NUMBER AND ITS ALGEBRA. 
 
 pediency. While sympathizing with the conservative who 
 object to all innovations as tending to confuse the vast 
 literature of the science, neomo7i{c is so much more appro- 
 priate, and '< imaginary" or "impossible" so misleading, 
 that the benefits of the change appear to outweigh the in- 
 conveniences. A reformation in terminology is not nearly 
 so confusing as changes in notation, such as have often been 
 brought about; for example, the famous propaganda of 
 "d-isni versus dot-age" (dy / dx versus y), which Dr. Pea- 
 cock began while yet an undergraduate, in league with 
 Herschel, Babbage, and Maule. The reform was finally 
 adopted at Cambridge, and Newton's notation soon became 
 entirely excluded. Nowadays mathematicians find no con- 
 fusion in using both notations. 
 
 102. The principle which has so fruitfully widened the 
 concept of number, yielding perfect self-consistency of 
 number, and ever deepening adaptation to Nature, I call 
 the Principle of Continuity, in emphasis of its most im- 
 portant outgrowth, the unlimited, twofold continuity of 
 number. 
 
 This principle may be stated as follows : — 
 
 103. Principle of Continuity. — The coynhination of 
 two numbers in any defined operation is always possible, the 
 result real, and a number ; and the precise effica-cy in any 
 operation of a number thus revealed is determined by, and 
 may be discovered from, the formula and laws of definition 
 of til e operation in question. 
 
 104. Before considering details, a glance at the results 
 which have more than justified the postulation of this 
 principle may be useful in giving the student the proper 
 perspective of the subject. 
 
 The principle at once makes negative numbers, zero, in- 
 
PRINCIPLE OF CONTINUITY. 77 
 
 finities, fractions, surds, neonionic and complex numbers, 
 all equally numbers. Also Number thus becomes unlim- 
 itedly continuous in a double sense, whereby undreamed 
 of adaptability to Nature is revealed, and all numerical 
 operations proceed untrammelled by particularity. 
 
 One who will logically apply the Fr'inciple of Continuity 
 will arrive at all classes of numbers — or divisions of 
 Number — with equal necessity and facility. Negative or 
 fractional numbers will appear as much derived, as little 
 original, or primary, as those numbers still commonly 
 called " irrational " or •' imaginary." One of these classes 
 is as foreign as any other to the primary concept of num- 
 ber ; that is, the concept of number as discrete, the concept 
 which knows only one number between, say, 5 and 7. 
 
 If the symbol i be set apart to represent the neomon 
 ( V — 1), ■we seem to have in the expression x -f- yi the 
 most general numerical form to which the laws of number 
 lead.* For it has appeared upon investigation that no 
 combination of numbers in any conceived operation can 
 result in a form essentially different. Neither has any 
 operation essentially different from the seven fundamental 
 operations developed from them. It might be surmised 
 that investigation would -reveal some fourth direct opera- 
 tion growing out of involution, as involution grew out of 
 multiplication, and multiplication out of addition ; but such 
 does not seem to be the case. No ground of distinction is 
 furnished for a new species of operation. That is to say, 
 the operation, if assumed ta be distinct, would show itself 
 not essentially so, by failing to lead to new modes of 
 Number. In other words, if the investigations referred 
 
 * For this expression, a complex number in algebraic form, is numer- 
 ically neomouic if x = 0, aud numerically whatever x is, if y = 0. 
 
78 NUMBER A^D ITS ALGEBRA. 
 
 to are trustworthy (as is no doubt the case), there can arise 
 no new opportunity to apply the Principle of Continuity, 
 so as to still further widen the meaning of ISlumber. 
 Number in its ultimate sense is therefore seen to form 
 (what primary numbers do not, nor any curtailed concept) 
 a universe complete in itself, such that starting in it we 
 are never led out of it. Cayley says, whether with sound 
 philosophy or essential contradiction of terms I will not 
 attempt to discuss, " There may very well be, and perhaps 
 are, numbers in a more general sense of the term (quater- 
 nions are not a case in point, as the ordinary laws of com- 
 bination are not adhered to) ; but, in order to have to do 
 with such numbers (if any), Ave must start with them.'' * 
 
 105. I believe that very few, even among students of 
 mathematics, are aware of the chaos of their conception 
 of number, in spite of long and familiar use. 
 
 The difficulty here, as everywhere, is the attainment of 
 true concepts, insight into the principles involved. 
 
 I balieve that the present condition is due to the fact 
 that successive generations of students have not had the 
 difficulties honestly presented to them, and have seldom 
 even considered fundamental theory. They have been 
 entrapped into .an unwarranted complacency; they have 
 juggled with symbols which are meaningless to them, and 
 for the most part without even noticing that no concept 
 
 * From note made long ago ; exact reference lost. In regard to qua- 
 ternions it may be observed, tliat though in their ordinary presentation 
 certainly not numbers, it is possible that they may yet be divested of 
 extra-numerical properties. Speaking of the anomaly according to which 
 quaternions in the common interpretation would make 'A niv- negative, 
 whereas >^ ?;i is positive and tlie wliole positive, Dr. INIacfarlane, in his 
 Algebra of Physics, remarks, " If this is a matter of convention merely, 
 then the convention in quaternions ought to conform to the established 
 convention of analysis; if it is a matter of truth, which is true ? " 
 
PRINCIPLE OF CONTINUITY. 79 
 
 rises "witli the words tliey utter, tlie symbols they write, — 
 tliat their discourse upon number is vox et i:)raeterea nihil. 
 I believe that an opposite result would be prevalent, had 
 an opposite course been pursued by teachers and authors, 
 and tliat we would now be reaping harvests instead of 
 sowing seed. 
 
 106. Some one ignorant of trigonometry, of the ana- 
 lytical treatment of geometry, of the Calculus, of the 
 varied fields of applied mathematics, and to whom the 
 boundless realms of pure mathematics loom misty and 
 fantastic — some such one, I say, may ask, "Why all this 
 striving to make number continuous, this travail to pro- 
 duce concepts of number and numerical operations, which 
 shall be perfectly general and unrestricted ? The answer 
 is, the need, intellectual and practical, is urgent, impera- 
 tive. Establish the Principle of Continuity, and Arith- 
 metic becomes a logically perfect universe, and besides, 
 all Xature becomes harmoniously numerical ; number and 
 its laws pervading it as an essential principle. Emerson's 
 noble lines, in which, with the poet's seer gift, he speaks 
 truer than he knew, then become literal fact : — 
 
 "For Nature beats in perfect tune, 
 And rounds with rliynie her every rune ; 
 Whetlier slie work in land or sea, 
 Or hide imderground her alchemy. 
 Thou canst not wave thy staff in air, 
 Or dip tliy paddle in the lake, 
 But it carves the bow of beauty there. 
 And the ripples in rhymes the oar forsake . . . 
 Not unrelated, unaffied, 
 But to each thought and thing allied 
 Is perfect Nature's every part. 
 Rooted in the mighty heart." 
 
80 NUMBER AND ITS ALGEBRA. 
 
 Besides, the assumption has been made, and its first 
 fruits are the attainments of the physical sciences during 
 the last two centuries. The progress in exact physical 
 science and the dependent arts has been due to the power 
 and freedom conferred upon analysis by this postulate ; 
 for, as I have said, it is implicit in all modern analysis, 
 even whcTi denied with the mouth of the calculator. (See 
 also §§ 110, 117.) 
 
 107. Like all profound principles, this one of the con- 
 tinuity and qualitative distinctions of number is a onatter 
 of insight, and does not admit of easy demonstration. One 
 man cannot think for another any more than he can eat for 
 him ; but if a student will fix alert and intelligent atten- 
 tion upon the inherent development of the idea, and upon 
 the manifold Avitness borne by almost every phenomenon, 
 he will at last behold the Principle, manifest in ten thou- 
 sand undreamed-of relations. 
 
 108. Tt is not practicable to give more than one example 
 of the mental attitude I desire to excite. I choose one 
 which affords a double illustration : in the first place, yield- 
 ing a geometrical instance of the Avay in which concepts in 
 every science are extended to conform to deepening insight, 
 an extension analogous to the development of the primary 
 number-concept ; and in the second place, displaying (as a 
 consequence of this attainment of an adequate geometric 
 definition) an impressive discovery of supreme law — pro- 
 vided ratios are numbers, and number positive and nega- 
 tive — in what seems, to nai've observation, utter fortuity. 
 
 109. Illustration. — (1) The primary concept of the 
 division of a sect by a point is, of course, that the point is 
 on the sect ; but investigation shows that a widening of the 
 concept is required to fit facts presented by Nature. It is 
 
PRINCIPLE OF CONTINUITY. 81 
 
 discovered that if any point, P, in the straight of a sect, 
 AB (on or out of tlie sect), shall divide it into the segments 
 FA and PB, then innumerable theorems only partially 
 true, and therefore none of their inverses true (^vide infra), 
 under the primary concept, become universally true, and 
 therefore their inverse propositions true, under the extended 
 concept. 
 
 (2) The same term, division, is necessarily retained for 
 this new relation ; for it is the very essence of the dialectic 
 to display the inherent identity of the two relations. To 
 conceive (or name) the relations in contradistinction would 
 be to miss the very truth revealed by the connotation. It 
 is everywhere discovered that the process of philosophical 
 advance is in great part the identification of old ideas, long 
 in use by the mind in its experience, with ideas which to 
 brute or naive observation appear irrelevant or distinct. 
 Reflection upon the pure thought brings out the implicit 
 identity with the category already named. 
 
 (3) In particular the case of external and internal di- 
 vision in equal ratios is discovered to be a harmony very 
 prevalent in nature. Such division of a sect is styled 
 "harmonic division." (Cy. any Geometry and any scien- 
 tific treatise on physics.) 
 
 (4) Now consider the two plane figures {A), a triangle 
 and any straight (cutting the triangle or not) ; and {B), a 
 triangle and straights joining any point (in or out of the 
 triangle) to the vertices of the triangle. Under the ex- 
 tended conception of the division of a sect by a point, the 
 straight in A divides each side of the triangle ; and of the- 
 straights in B, each divides the side of the triangle oppo- 
 site to the vertex through which it passes, in such wise 
 that the product of the three ratios of the segments of the 
 
82 NUMBER AND ITS ALGEBRA. 
 
 sides is 1 (provided that, of adjacent segments in different 
 sides, if one be tlie antecedent, then the other sliall be the 
 conseqnent of its respective ratio). Tliis is assuredly a 
 most impressive exhibition of unsuspected laivfulness in 
 a fact seemingly a very type of haphazard. But, be it 
 noted, the inverse of neither A nor B is true. Now, it is 
 an established principle that when the inverse of any prop- 
 osition is not true, it is because the subject of the direct 
 statement has been more closely limited than truth required. 
 It is clear that the inverses of A and B are flat contra- 
 dictions. 
 
 But if the ratio of sects from the same point be consid- 
 ered positive if one sect is part of the other, and negative 
 if extending oppositely, then the easily demonstrated con- 
 clusion of A is that the product of the said ratios is -f 1 5 
 and of B that the product is — 1. The inverse of each 
 now holds ; that is, if three points divide the sides of a 
 triangle so that the product of the ratios taken as stated is 
 + 1, then the points are co-straight ; and if the product of 
 the ratios is — 1, then the joins of the points with the ver- 
 tices concur. 
 
 (5) The student should fully realize what is here as- 
 serted ; and to this end let him draw a triangle and then 
 dash straights at random, cutting the triangle or not: 
 Every one of them divides the sides of the triangle in pre- 
 cisely the same way ; and ifnumher he positive and negative, 
 given three points so dividing the sides, they are co-straight. 
 Again, draw a triangle, dot at random points, in or out 
 of the triangle : Any one of these points joined to the ver- 
 tices gives straights which divide the opposite sides in 
 precisely the same way ; and if number be jjosltlve and 
 negative, given this Avay of division .of the sides by three 
 
PRINCIPLE OF CONTINUITY. 83 
 
 points, the straights joining the point to the vertices come 
 together in one point. 
 
 110. When it is considered that the preceding ilhistra- 
 tion recites merely one of ten thousand examples, number 
 is proved to be positive and negative, — not, be it under- 
 stood, as a convention, but as a necessity of thought. Men 
 who represent this qualitative distinction as arbitrary, or 
 as purely a matter of algebraic symbols, do not appreciate 
 the evidence, or do not understand what proof in such 
 premises means. Moreover, it must never be overlooked 
 that a stiir higher order of proof is afforded in the devel- 
 opment of the pure idea, regardless of any adaptations to 
 external facts. When this or that development of the 
 pure science of number is to lind application to facts of 
 other sciences is a secondary matter. {Cf. § 117). 
 
 111. Similar illustrations might be given to show the 
 adaptability of number to facts presented by nature, if the 
 other modes of number resulting from the application of 
 the Principle of Continuity are recognized. Presentation 
 of such evidence must be postponed for the most part to 
 subsequent mathematical studies ; and I shall in this con- 
 nection only ask you to observe that the Principle of Conti- 
 nuity, as enunciated in Section 103, unities all the partial 
 explanations of number which you will find advanced, or 
 implied, in various treatises ; and to reflect that the man 
 who in his own opinion discovers the entirely New is prob- 
 ably on the pathway, not of truth, but of estrangement. 
 If his system refutes, in utter antagonism, preceding sys- 
 tems, it is likely to be refuted by a successor. In all 
 philoso})hy and science, advance has been genuine only in 
 systems which have been synthetic, and unifying of pre- 
 vious efforts in a harmony of thought. No development of 
 
84 NUMBER AND ITS ALGEBllA. 
 
 thought must be regarded as a disjoined succession of dead 
 results, but as living insights in one line, each piercing 
 deeper and deeper. 
 
 112. In this light, note that the revelation in antiquity 
 of fractional and surd numbers, and the recognition of 
 number as positive and negative which has prevailed for 
 two centuries (these may be regarded as the " first " (§ 78) 
 and second extensions of the number-concept), are both 
 merely special cases of the universal principle here ad- 
 vanced. 
 
 113. To generalize is to see in a multiplicity of objects 
 similar relations to one form of mental activity that knows 
 those objects. But until one sees the need of a deeper 
 principle than that which he has hitherto employed, he 
 does not seek a way leading from what is known to him to 
 knowledge beyond. Any idea is at first bare of manifold 
 essential relations, external and internal. By reflection 
 such relations are slowly revealed. During the process 
 the idea may seem derivative from the relations {Cf. geo- 
 metric definitions of number, § 25) ; but finally this loose- 
 ness must be reduced to order, and then all its belongings 
 are seen to unfold from the idea itself, ■ — *' first the blade, 
 then the ear, after that the full corn in the ear." 
 
 114. What has just been said would do for a description 
 of the famous dialectic which Hegel describes as "the 
 self-movement of the notion (^Begrlff).'^ Indeed, it is not 
 much more than a paraphrase of its description by Dr. 
 Harris, " Seize an imperfect idea and it will show up its 
 imperfection by leading to and implying another idea as 
 a more perfect or complete form of it. Its Imperfection 
 tv'dl slioia itself as dependence on another.'''' (Italics mine.) 
 
 115. I know no other method by which the teacher can 
 
ZERO. 85 
 
 lead a student to attain for himself a concex^t of number 
 adequate to any comprehension of modern mathematical 
 analysis. Each tentative idea of number must pass over 
 into the next deeper as the result of further and further 
 insight into the subject. 
 
 It remains to apply in characteristic cases the Principle 
 of Continuity, discovering from the formula of definition of 
 any operation X^Q nature of the resulting number, as well 
 as the efficacy of any such new phase of number in any 
 combination in the defined operations. 
 
 XII. SiGXIFICAK-CE A?^D EfFICACT OF NUMERICAL OPER- 
 ATIONS Under the Ultimate Concept. 
 
 116. The very first application of the Principle of Con- 
 tinuity to the generalization of the operation Subtraction, 
 displays a number sui generis, which is of immense impor- 
 tance in analysis. The formula of definition of subtrac- 
 tion is (tnde § 42) b — a -{- a = b. Then a — a = what 
 number ? The formula declares that it is a number which, 
 added to a, makes a ; that is, it is a number which has 
 710 efficacy in addition, and therefore none in subtraction. 
 The best and only unprejudicial name for this number is 
 zero. Its symbol in arithmetic and in the algebra of 
 number is 0. 
 
 I trust that at least it has been made clear to the 
 student that it is only the very primary and crudest con- 
 cept of number which would consider zero " nothing ; " for 
 although of no efficacy in addition or subtraction, it will 
 presently be seen to exert extraordinary effect in every 
 other operation. I entreat the student not to slip at this 
 point ; for the human mind, once made sensible of its 
 
86 NUMBER AND ITS ALGEBRA. 
 
 powers, will never afterwards suffer its conception to be 
 clogged by the tyranny of material categories. Moreover, 
 it may quite commonly be found necessary to translate 
 into correct terms much discourse in mathematical trea- 
 tises, even when written by men eminent for skill and 
 learning, to say nothing of inadequate or erroneous pre- 
 sentations in works on physics and applied mathematics 
 in general. For example, you may read a Trigonometry 
 which defines the trigonometric ratios not as numbers, but 
 as sects (pieces of straight lines) ; yet you can often catch 
 the author adding one of his bits of straight lines to 2 
 or 32, and in a context where he really means the number 
 2 or 3^, etc. Occasionally you will meet denial or even 
 ridicule of all that I endeavor to lead you to see, and per- 
 haps by a man of world-wide fame. For example, in a 
 didactic treatise on Mathematics by De Morgan, published 
 in a serial Library of Useful Knowledge, London, 1836, 
 zero is conceived to be '' nothing " ; for on page 23 one 
 reads, " Above all, he must reject the definition, still some- 
 times given of the quantity — a, that it is less than nothing. 
 It is astonishing that the human intellect should ever have 
 tolerated such an absurdity as the idea of a quantity less 
 than nothing ; above all, that the notion should have out- 
 lived the belief in judicial astrology and the existence of 
 witches, either of which is ten thousand times more possi- 
 ble." The truly astonishing thing concerning the human 
 intellect is that such a man as De Morgan could have writ- 
 ten this sentence, familiar as he must have been with 
 Newton's distinction, '' Quantitates vel Aflirmativa^ sunt 
 seu majores nihilo, vel Negativse seu nihilo minores." 
 But, although deficiency is quite as quantitative as excess, 
 the whole remark is impertinent ; for zero is not " nothing." 
 
NEGATIVE NUMBER. 87 
 
 Negative numbers are unquestionably less than zero. Yet, 
 taking liim at his own word, De Morgan should have hesi- 
 tated before ridiculing as crazy the careful dictum of as 
 powerful and piercing an intellect as has ever served man's 
 will. 
 
 117. Before investigating the efficacy of zero in other 
 operations, let us look into further results of the generali- 
 zation of subtraction. 
 
 What are the properties of the resulting number in the 
 operation b — a, ii b <.a? Consider the results of the 
 following series of operations, 1 + 2; 1 + 1; 1; 1 — 1; 
 1_2; 1-3; 1-4, etc. 
 
 Here we have a series of numbers which at first decrease 
 by 1, viz., 3 ; 2 ; 1 ; 0. The subsequent numbers respec- 
 tively answer the questions, what number added to 2 
 makes 1, added to 3 makes 1, added to 4 makes 1 ? Now, 
 in these operations the sums remain the same, and the 
 given summands in each case increase by 1 ; it is clear, 
 therefore, that the required summands must decrease by 
 1. Moreover, these numbers in additive combination nul- 
 lify 1, 2, 3, etc. ; that is, make the sum in each case zero. 
 Thus, 1 + (1 - 2) = (1 + 1) - 2 == 0; 2 + (1 - 3) = (2 + 
 1) _ 3 = ; 3 + (1 - 4) = (3 + 1) - 4 = 0. Such reflec- 
 tions reveal an unending series of discrete numbers de- 
 creasing from zero, each less than the preceding by 1. 
 Their effect in nullifying 1, 2, 3, etc., in addition, renders 
 appropriate the appellations ^jostYive and negative to pri- 
 mary numbers and these now discerned. These terms are 
 established terms in logic, and are expressive of just such 
 a relation of clean-contradictory as has been discovered in 
 these modes of number. On this score, either might be 
 called positive and the other negative ; but every propriety 
 
 
88 NUMBER AND ITS ALGEBRA. 
 
 commends the course adopted — primary numbers are posi- 
 tive, and such results as we have just considered, negative. 
 That negative number finds unlimited corroboration in 
 adaptation to the facts of other sciences, has been amply 
 illustrated (§ 109) ; but its existence for pure mathematics 
 is nowise dependent upon such circumstances. Negative 
 number should never be defined or explained by such oppo- 
 sitions as right and left, up and down, forivard and hack- 
 ward, north and south, past and future, capital and debt ; 
 but always in its essential character as number. 
 
 118. Writing pos. for positive, and neg. for negative, it 
 is evident that pos. a + neg. b = pos. a — pos. b ; for pos. 1 
 
 — pos. 2 = neg, 1, therefore, by definition of subtraction, 
 pos. 2 + neg. 1 = pos. 1 ; but pos. 2 — pos. 1 = pos. 1, etc. 
 
 Also, since subtraction is the inverse of addition, 
 
 pos. a — neg. b — pos. a -f pos. h. 
 
 119. Hereby Section 42 is completely generalized, and 
 the common rule about " signs " and parentheses for addi- 
 tions and subtractions established without restriction. 
 
 120. We are arrived now at a matter of extreme impor- 
 tance, viz., the dual significance of the signs + and — . 
 One of the most salient imperfections of ordinary text- 
 books is their failure to make a clear-cut distinction be- 
 tween the essentially double meaning of +, and of — . 
 Too often the operational significance alone is defined, 
 although on the next page }'ou may find a complacent 
 statement « -f (_ o) = 0; whereas, if + means add, and 
 
 — means subtract, a -\- (— a) means, "starting with a, add 
 and then subtract a," of course, Avith the result a. And 
 under a purely operational definition such an expression as 
 a I — h is like a " sentence " made by writing words on 
 
DOUBLE MEANING OF + AND — . 89 
 
 dice and rolling them out of a box. Clifford, in liis zeal 
 against this abomination, goes too far, and gives three 
 totally distinct meanings to each of the signs.* His first 
 two for each are all that are needed or justifiable. 
 
 The names of the signs are respectively " plus " and 
 " minus ; " their meanings respectively add or positive, and 
 subtract or negative. 
 
 It is, perhaps, to be regretted that beginners are not 
 taught to use at first different symbols for these wholly 
 distinct thoughts, and afterwards led to observe that the 
 notation would be simplified if one symbol were used in 
 both meanings ; because the context always makes it clear 
 which is meant,- if the simple convention be established, 
 that, if nothing is expressed, ^^ positive " is understood, and 
 if one is omitted, it is the qualitative, and not the opera- 
 tional, symbol. Thus, in (2) (- 3), 2 / - 3, 3--, V - 1, 
 etc., the meaning subtract would not make sense, and ambi- 
 guity is impossible ; and in 2 -|- 3 — 4 the convention makes 
 it clear that the meaning is pos. 2 -\- pos. 3 — pos. 4. It is 
 true that 2 -f 3 — 4 = pos. 2 + pos. 3 -f neg. 4, and although 
 less consistent than the notational convention I recite, the 
 expression might be understood in this sense ; for the result, 
 as we have seen in Section 118, is indifferent. But see 
 clearly that the sign cannot have both meanings at one 
 time ; for 7 — 9 = pos. 7 — pos. 9 = pos. 7 -\- neg. 9 = neg. 
 2, whereas pos. 7 — neg. 9 = pos. 16. 
 
 Kote, as in accordance with the convention stated, that in 
 solving a synthetic equation for an unknoAvn number, its 
 qualitative nature is unknown, and no sign is to be under- 
 stood after the sign meaning add or subtract. 
 
 * Common Sense of the Exact Sciences, p. 34 et seq. 
 
90 NUMBER AND ITS ALGEBRA. 
 
 If for any purpose it is desirable to be quite explicit, the 
 qualitative sign may be put in parentheses with the number- 
 symbol, with the operational sign preceding. Parentheses 
 would hardly be used for the first term, or for a term stand- 
 ing alone ; e.g., + 7 - (+ 8) + (+ G) - (+ 9) = - 4, is 
 the full expression of what is meant by 7— 8 -}- G — 9 = — 4. 
 Of course, if occasion rose, write -|-7-l-(— 8) — ( — 6) — 
 (+ 9) = — 4. In short, write Avliat you mean, if you 
 express fully, but remember that abbreviations must be 
 doubly conventional. {Vide § 162.) 
 
 121. What is the product of (— a) (+ h) ? 
 Consider {-(- m — ( -{- «)} (-}- b) where vi > a. 
 
 By the distributive law, this equals + bm — (+ ba) ; but 
 by Section 118 it equals {+ m +( — a)} ( + /'); but {-{- m 
 
 + (— <0} (+ ^0 = + ^"^ +(— ") (+ f') by distributive law ; 
 therefore, since -\- bm — ( -\- ba) = -\- bm -\- (~ ba), -\- bm 
 -j-(_ ba)= -f bm+(— a) (+ b) ; therefore (— a) (+ b) = 
 
 — ba. 
 
 Hence the common rule of signs. 
 
 122. What is the product of (— a) (— b) ? 
 
 Consider {+ m — (-f a)} (— b). Distributing and ap- 
 plying Section 121 gives (4- m) (— b) — (-\- a) (— b) = 
 
 — bm — {— ba) = — bm -\- (+ ba) ; but by Section 118, 
 {+ m - (+ a)} (- b) = {+ m+ (- a)} (- b) = - bm + 
 {— a) {— b), therefore (^— a) {— b) = -[- ba = + ab. 
 
 Hence the common rule. 
 
 123. Division's definition as the inverse of multiplica- 
 tion, of course, establishes the rule of signs for division. 
 
 124. Sections 121, 122, and 123 render complete under 
 the common "■ rule of signs " the freedom of distribution 
 and commutation referred to in Sections 54, 55, and 56. 
 
 125. We are now prepared to investigate still further 
 
OPERATIONS UNDER FINAL CONCEPT. 91 
 
 the properties of zero. We have seen that it has no effi- 
 cacy in combination with other numbers in addition and in 
 subtraction. What is its efficacy in multiplication, in 
 division, in involution, in evolution, and in finding the 
 logarithm ? 
 
 126. ^Vhat is the product (a) (0) ? 
 
 Consider ba — ba = 0. By distributive and commutative 
 laws, and Section 122, ba — ba = (b — b) (+ «) = (+ a) 
 (b - b) = {b - ?y) (- o) = (- a) (b - b); whence (0) 
 
 (+ a) = (+ a) (0) = (0) (- «) = (- ") (P) = 0' o^' ^^^^^y> 
 regardless of positive or negative quality of a, 
 
 («) (<^) = (0) («) = 0. 
 
 127. Note that as an independent number zero is with- 
 out qualitive distinction of positive and negative ; for 
 a 4- = a — 0, hence -f- = — 0. 
 
 128. It may be a profitable comparison to call atten- 
 tion expressly to a unique property of 1 in multiplication 
 and division ; thus, — 
 
 (r?) (-(- 1) = rt, and a / -{- 1 = a, that is to say, x (+ 1) = 
 
 - (+ !)• 
 
 Also (a) (— 1) = — a, and o / — 1 = — a, i.e., X (— 1) 
 
 = - (- !)• 
 
 129. X = 0, for (a _ a) (Z» - ?;) = X = ab — ah 
 
 _ ah + aJ> = 0. 
 
 130. r.ut what is the result /O ? 
 
 This case is of extreme importance. Failure to compre- 
 hend it when it comes into systematic use in the Calculus 
 has put a veil of irrational mystery over that whole dis- 
 cipline. Thousands of students, although they have met 
 and slightly used this indeterminate form before, yet inas- 
 much as they have regarded it a matter of special con- 
 
92 NUMBER AND ITS ALGEBRA. 
 
 vention that 0/0 should represent any number, are 
 dumfounded to find a discipline where a number is, they 
 say, made zero in one member of an equation, and some- 
 thing else in the other. After a pathetic struggle to see 
 reason in their procedure, they commonly give over, and 
 accept the outrageous extravagance that a concatenation of 
 deductions to be valid need not have meaning in every 
 link ; that a compulsory conclusion of an argument does 
 not require intelligibility of its several steps; or that 
 results may be thoroughly made out true for reasons no- 
 wise understood. 
 
 131. "When the ratio 0/0 is first presented for consider- 
 ation, one may be disposed to jump to the decision that 
 0/0 = or 0/0 = 1; but it is clear, from the definition 
 of division, that in the synthetic equation 0/0 = y, any 
 number (0, 3/5, 1, V2, 2, 3, etc.) substituted for y will 
 make a formula (§ 40), an identity. That is to say, / = 
 anv number. 
 
 V 
 
 Indeed, this statement is merely another way of saying, 
 "any number multiplied by zero gives zero," which is com- 
 monly accepted without objection. And both of these 
 statements are only particular applications of the postulate 
 expressed in the Principle of Continuity. 
 
 The ratio 0/0, then, may be any number; but in par- 
 ticular instances it is often a number which may be deter- 
 mined by independent considerations. 
 
 132. If two numbers (or any other two magnitudes of 
 the same kind) vary, their ratio varies ; but the ratio at 
 any assigned limits of the variables is the same as at values 
 of the variables only infinitesimally (vide § 222) removed 
 from such limits. In fact, the original definition of equality 
 of ratios contains this doctrine. • ( Vide § 83 (6).) 
 
OPEKATIONS UNDEll FINAL CONCEPT. 93 
 
 Consider the functions of x, x^ — 1, and x — 1. 
 
 3.2 _ 1 
 What is the ratio wlien x = 1? 
 
 X — 1 
 
 The ratio may be evahiated Avithout hesitation, as x 
 assumes various values, until a; = 1 is reached, when both 
 functions vanish, and the ratio assumes the indeterminate 
 form, 0/0. But when x differed only infinitesimally from 
 
 1, beyond objection, = x -\- 1, which differs only 
 
 X — 1 
 
 infinitesimally from 2. Therefore, when x = 1, the ratio 
 
 - = ~ — = a; + 1 = 2, absolutely, 
 x-1 ^ ' -^ 
 
 There is no trickery here. The Calculus, with its 
 astonishingly powerful algorithm, applies such numerical 
 interpretations to concrete magnitudes ; nor would it, in 
 my opinion, be out of place in this connection to give illus- 
 trations of the wonderful propriety, and accordance with 
 independent facts, of this method, but out of deference to 
 established custom — usus tyranmis — I leave such corrobo- 
 rations to future studies, with the simple assurance of 
 their cogency. I shall only set forth one more very simple 
 illustration of an. evaluation of a ratio 0/0. Consider 
 the following two functions of y, 2 y -|- 3 y- + 4 y^, and 
 3 y -\- ^ y"^ -\- 21 y^. Their ratio would be easily evaluated 
 for particular finite values of y ; but suppose the variable 
 y becomes zero, what then is the ratio of the functions ? 
 
 If y = Q, li^L+ll! + li^ = 0/0. 
 
 And if each term be divided by y we have 
 
 2 + 3.v-f-4.v^ _ 2 /3, when y = 0. 
 
94 NUMBER AND ITS ALGEBRA. 
 
 Now, it might well be objected that in dividing by y, if 
 y = 0, the most we could say would be 
 
 2(y/y) + 3yO////)+4rO////) 
 
 3(y/z/) + 9y(y/y) + 27y^(y/y)' 
 
 and that this remains as obscure as the original if / 
 is any number. The explanation is, that the ratio of one 
 and the same variable to itself is constantly 1 ; a J a = 1 
 always. Therefore, even when y == 0, y / y = 1, and if so, 
 
 ^(y/y)+3y(y/y)+4y^(y/y) _ 2 + 3y + 4y^ _^ 
 
 It would take us too far afield to go further into the doc- 
 trine of limits of variable magnitudes and infinitesimals, 
 and the appropriate application of number. The whole 
 question of the use of this indeterminate form 0/0 may 
 not improperly be postponed by the student, who for the 
 present might content himself with the discernment that, 
 whether it be possible to evaluate / or not in particular 
 problems, 0/0 may be any number. 
 
 133. What is the result of the operation / ct ? and 
 what of ffl / ? 
 
 The first asks the question, what number multiplied by 
 a gives zero ; and from the formula of definition and Section 
 126, the answer is evidently zero. 
 
 Also 0/ + a = = 0/-a. 
 
 The second asks the question, what number multiplied 
 by zero gives a ? 
 
 Erom Section 126 it is evident that no number 3-et dis- 
 cerned answers this question. 
 
 But a consideration of the continuously increasing ratios 
 (vide §§ 81, 82) of the same number to a decreasing series 
 
INFINITY. 95 
 
 of numbers, reveals that, if the ratio -)- a / is a number, 
 it is one greater than any primary number, and of peculiar 
 "efficacy in operational combination. This number, whose 
 reality is requisite for untrammelled numerical analysis, 
 is called 2^ositive infiiiity, and notationally expressed as 
 
 -|- CO - 
 
 Similar generalization under the Principle of Continuity 
 makes — a / negative infinity, written -co . 
 
 134. The discovery of many properties of infinity, posi- 
 tive and negative, must be left to future studies ; as well 
 as the principles of evaluation of ratios of infinities dif- 
 ferently derived, analogous to evaluations of ratios 0/0. 
 {Vide § 132.) 
 
 It Avill be found that in the application of ISTumber to cer- 
 tain magnitudes (e.g., straight lines in Euclidean Geome- 
 try) that for them it appears the points at infinity coincide. 
 Other " one-dimensional " (vide § 232) magnitudes show 
 a double absolute : for example, ProhahiUty ranges from 
 absolute certainty /or, to absolute certainty against. 
 
 Without going too deeply into philosophical questions, 
 it may be remarked that Hegel, in discussing the mathe- 
 matical infinite, " points out that the mathematical infinite 
 . . . uses the idea of the true infinite, and therefore stands 
 higher than the so-called metaphysical infinite. The latter 
 opposes the infinite to the finite as the mere negative of 
 the latter, and thereby makes two finites, the former the 
 void of the latter ; whereas the mathematical infinite ex- 
 presses self-relation as its true form."* Much might be 
 said also of how important to philosophy is the mathe- 
 matical concept of continuity. Indeed, many of Hegel's 
 
 * HegeVs Logic, Harris, !>. 278. 
 
96 NUMBER AND ITS ALGEBRA. 
 
 conceptions are true only as glimmerings of wliat mathe- 
 maticians had before made clear, or have since illuminated. 
 
 135. I present in tabular form * the possible meanings' 
 of the ratio x j ij, as x and y independently vary from 
 to ao . The student can readily verify the statements, and 
 extend them to cover distinctions of positive and negative 
 in X and y : — 
 
 ( 1 ) X I y'\'& finite if x is finite and y finite. 
 
 ( 2 ) may be finite if a; = and y = 0, 
 
 ( 3 ) or if a; ^= CO and v/ = cc . 
 
 (4)a:'/?/ = ifa; = and y not 0, 
 
 ( 5 ) or if .T not co and y ^ c/j . 
 
 ( 6 ) may =0 if a; = and y = 0, 
 
 ( 7 ) or if a- = oo and y = cc . 
 
 (8) x / y = ao ifa-=co and y not cci , 
 
 ( 9 ) or if a; not and y = 0. 
 
 (10) may = oo if a; = and y = 0, 
 
 (11) or if a; = CO and ?/ = oo . 
 
 136. Of course oo + oo = co . But oo — oo is indeter- 
 minate ; since any number (0, finite, or infinite) substi- 
 tuted for X satisfies the synthetic equation oo + a; = oo . 
 
 137. Of course oo X co = oo . But OXco = coXO is 
 indeterminate ; since the multiplications of which Section 
 135 (5), (7) are the inverses, show X oo = any number. 
 
 138. Various considerations dependent iq^on the con- 
 tinuity of number confirm the interpretation that a;" = 1, 
 if X is finite. But it may suffice to consider that if x, y, 
 and z are finite, a;^ -=- a;^ = x^-' ; and it y = z, 1 = x'J -i- 
 
 xy = xy-y = x". 
 
 * A similar table occurs in Chrystal's Text Book of Algebra, Part I., 
 p. 317. 
 
OPERATIONS UNDER FINAL CONCEPT. 97 
 
 139. Evidently (1) 0-^ = if a: is finite. 
 
 (2) a;+* =00 if ic is finite and > 1. 
 
 (3) = if rr < 1 and > 0. 
 
 (4) x"^ = if 33 is finite and > 1. 
 
 (5) = CO if ic < 1 and > 0. 
 
 (6) 0+- =0. (8) a; +=» = 00 . 
 
 (7) 0-= = 00 . (9) ^ -=" = 0. 
 
 As the student may convince himself. (§ 143 is anticipated.) 
 
 140. But the results (1) 0°, (2) ^ °, (3) 1+^, (4) l-« are 
 indeterminate ; as may be seen most readily by considering 
 that x^ = m2'^°^ra-^, where m is finite and greater than -|- 1. 
 x^' is accordingly determinate when y log„j x is determinate, 
 and indeterminate when ylog^a; is indeterminate. The 
 cases when ?/log,„« is indeterminate are, by Section 137 : — 
 
 (1) When y = 0, log,„ x = -co"; i.e., when y = 0, cc = 0. 
 
 (2) When y = 0, log„j,x = -[- co ; i.e., when y = 0, .r = oo . 
 
 (3) When ?/ = -j- oo , log,„a3 = ; i.e., when y = -j-c/^ ,x ^ 1. 
 
 (4) When y = — cc , log,„ic = ; i.e., when y = — ct) ,x = 1. 
 
 141. Every indeterminate form may be reduced to 0/0, 
 and in this sense it may be said that the one fundamental 
 case of indetermination is 0/0. For example : — 
 
 00-^ = 1/0-1/0 = 1^ = 0/0; 
 ^/^ = ^ = 0/0. 
 
 142. Let the student tabulate from the foregoing sec- 
 tions all the indeterminate operations. 
 
 He must be content to postpone investigation into the 
 evaluation of these indeterminate results as they arise 
 
98 NUMBER AND ITS ALGEBRA. 
 
 from particular functions of variables, regarding Section 
 132 as a simple example of the general principle. 
 
 143. It remains to investigate the efficacy, as exponents 
 of evolution, of fractional, surd, and negative numbers. 
 
 What is the meaning of the operation a^, if a is positive 
 and finite, and x a fraction ? 
 
 The conclusion is corroborated by the continuity of num- 
 ber, by countless correspondences, and by perfect consist- 
 ency with all other laws ; but regarding the Law of Indices 
 as the essential definition of the operation, the meaning is 
 immediately revealed. Thus : let a'" ' ", where 7n and n are 
 positive integers, equal z. Then, since «"«/« is subject to 
 the Law of Indices, z^ = zzz . . . n factors = a"'^"a'"^" 
 
 . . , n factors = «"-/"+«/» ■ ■ ■ n terms ^ „m_ rj^^r^^ -^ ^^ 
 
 say, z is a. number whose nth power is a'"; or z is an «th 
 root of a"'; i.e., a""'" = V«'". 
 
 In particular, if m = 1, ct} ' ^ = V«. 
 
 As we saw in Section 94, the operation ■>/« (where a is 
 positive) is always possible, in the sense that, if the result 
 be a surd number, it can be determined to any degree of 
 approximation. (But see § 153, et seq.) 
 
 144. It will appear in the studies to which these lec- 
 tures are introductory that there are n nth. roots of a, 
 where n. is a primary number ; but the student may observe 
 now, that when n is even there are at least two roots of 
 a, one the negative of the other; e.g., 4}/^= J^ 2. But 
 note that the law of indices has regard only to the corre- 
 sponding roots of numbers, simply because V« Va does 
 not equal a, if one positive and one negative root be taken. 
 (ride%^ 191 and 146.) 
 
 145. It is necessary to say at this point that we must 
 either use the terms " rational/' " irrational," '' real," and 
 
TERMINOLOGY. 99 
 
 " imaginary," or invent equivalents. ( Vide Introduction, 
 p. 16, and § 101.) The terms are unquestionably abusive, 
 and perhaps the time is ripe for a protest. No number is 
 irrational, and all numbers are real. Therefore, if merely 
 as an experiment, I shall be consistent in calling numbers 
 commensurable (tvith 1, understood) where the text-books 
 say "rational ; " either incommensurable or surd, where they 
 say "irrational;" radical-surd, where they say ''surd" 
 (^Cf. § 83) ; protomonic, where they say "real" ; neoinonic, 
 where they say " imaginary " (unless they say " imaginary " 
 when they mean complex) ; and when functions or opera- 
 tions are spoken of as "rational" or "irrational," in sub- 
 stituting stlrpal and radical respectively. These words, 
 except 2}'>'otomonic and sthpal, are in good usage either 
 exactly or approximately in the senses defined. Proto- 
 monic and sthyal I coin ; reluctantly, but unavoidably. I 
 hope they justify themselves as antitheses of neomonic and 
 radical. Of course, surd is not much better etymologically 
 than " irrational ;" but the metaphor is dead, and. conse- 
 quently harmless. Concerning commensurable, see Section 
 205. (See also § 156.) 
 
 146. Before passing to other cases of the exponential 
 function a^, it is proper to call attention to certain para- 
 doxes which may arise in the interpretation of such 
 functions. (Cy*. § 191.) For example, a* ^-= a'^. But as a 
 fractional index, a^ ''^ means Va* = i a'^ which at first sight 
 might seem to assert that a" = -^ a~. Likewise, one might 
 be led to say, since (a'")" = a""> = (a")"', (-i^'-y = (■i-y^, 
 and so (-t- 2)^ = J- 4, that is, + 4 = ^ 4. (Cf. § 144.) 
 
 Such difficulties will arise in a"'^", etc., when mfn is 
 not in its lowest terms, a*'^ = a^ is not a radical function 
 at all; though it is quite true that the second roots of 
 
100 NUMBER AND ITS ALGEBRA. 
 
 a* are + a^ and — a^. The law of indices is not a matter 
 of arbitrary or meaningless symbols, but of facts. If 
 algebraic expressions are not regarded as logical state- 
 ments, and full account taken of the nature of the 
 derivation of one equation from another, apparent con- 
 tradictions will often arise. (^Cf. § 319 et. seq.) 
 
 147. What is the effect of a negative exponent of invo- 
 lution ? 
 
 Consider «-"» = a"'" X a™- j a"^. 
 
 By law of indices, 
 
 a-'" X a''' I a''' = a- '" + '"' -r- a'" = a" / a'" = 1 /a™, 
 by Section 138 ; therefore, 
 
 «-"» = 1 /a"". 
 
 That is to say, a~'" is the reciprocal of a"^. 
 
 148. The continuity of number at once extends all that 
 has been shown to be true of integral and fractional ex- 
 ponents to surd exponents. 
 
 Thus in the function a^, whether x be commensurable or 
 
 surd, we can always find two fractions, m / n and , 
 
 between which x lies, and which differ by as little as we 
 please. As stated in Section 95, in calculation we use 
 a^jn instead of a^, where m/nis a fraction closely approxi- 
 mating the surd x. 
 
 149. When a is positive and > 1, and regarding only 
 protomonic positive roots, a^ is a continuous function of x, 
 passing through all values from to -(- co , as x varies from 
 — oo to -f CO . Thus, — 
 
 a^ is 0, < 1, 1 /«, 1, > 1, a, + 00 
 
 when X is — oo , — , — 1, 0, -\-, -]^ 1, -]- cc . 
 
LOGARITHMS. 101 
 
 When a is positive and < 1, the vahies of «^ are + co , > 1 
 1 / a, 1, < 1, a, 0, corresponding to the same values of x. 
 
 150. As has been explained, b = a^ and x = log„ b, are 
 merely different ways of writing the same functional rela- 
 tion. Thus all laws and properties of logarithms are de- 
 rivable from the principles of involution, in brief, from the 
 law of indices. Until the uses of logarithms and the con- 
 struction of logarithmic tables are investigated, it is enough 
 to say that for the same base the following are the leading 
 properties of logarithms, — as the student may easily dis- 
 cover from the law of indices : — 
 
 (1) The log. of a product of positive numbers is the 
 sum of the logs, of the factors. 
 
 (2) The log. of the quotient (ratio) of two positive 
 numbers is the log. of dividend minus log. of divisor. 
 
 (3) The log. of any power of a positive number is the 
 log. of the number multiplied by the exponent. (Power is 
 used in the general sense ; for the statement is true for all 
 exponents, and therefore inclusive of the commonly sepa- 
 rated rule for roots.) 
 
 (4) (log,, b) (logs a) = 1, and log„m = -51^ . 
 
 The base of " common " logarithms for piirposes of final 
 calculation is 10 ; but the base discovered to be primarily 
 appropriate to mathematical investigations is an incommen- 
 surable number, called e. 
 
 e = l-ul + — 4- — H \- to CO terms * 
 
 = 2- 7182818284 +. 
 
 * 1 X 2 may be abbreviated 2 ! -p^^^^ „ factorial two," 
 
 1X2X3 may be abbreviated 3! .. factorial three," etc. 
 
 1X2x3X4 may be abbreviated 4 ! 
 
102 NUMBER AND ITS ALGEBRA. 
 
 The base 10 gives logarithms vastly more convenient in 
 calculation ; the base e, in analysis. 
 
 Formulae (4) yield the simple process for deducing from 
 a given table of logarithms to any base, the logarithms to 
 any other base. Thus, to deduce log„ m from logj m, mul- 
 tiply by . 
 
 logs a ^ 
 
 The constant multiplier, ^j , is called the modulus of 
 
 the system whose base is a with respect to the system 
 whose base is b. 
 
 The modulus of the system whose base is 10 with respect 
 
 to the system whose base is e, is :; — = 0-4342944819 +. 
 
 ^ loff.lO 
 
 ■'a e 
 
 The modulus of the system, base e, with respect to the 
 
 common system, is = 2-7182818284 -\-. 
 
 -I logio e 
 
 Of course, = logio ^ ; that is to say, the recipro- 
 
 log.lO 
 cal moduli of two systems are reciprocals in the numerical 
 
 sense. 
 
 151. For interesting historical sketches, the student is 
 referred to the articles, " Logarithm's " and " John iSTapier of 
 Merchiston," by J. W. L. Glaisher, in the Encydopcedia Bri- 
 tannica, ninth edition. A perusal of these monographs will 
 lead him to appreciate the brilliancy of Napier's invention, 
 and the merit of Briggs and Vlacq, as well as the claims of 
 Byrgius, a Swiss contemporary of Napier, as an indepen- 
 dent but crude inventor. He should bear in mind that this 
 achievement came prior to the exponential notation, or any 
 clear idea among mathematicians of exponential functions. 
 An attempt to prove — to say nothing of discovering — the 
 laws of logarithms, after divesting one's self of knowledge 
 of the generalization of involution and all moderji advan- 
 
INADEQUACY OF PROTOMONIC NUMBER. 103 
 
 tages from the correspondences of two series of numbers, 
 one in " arithmetic " and the other in " geometric " progres- 
 sion, would alford a very high estimate of ISTapier's genius 
 and acumen. 
 
 152. The student can easily discern tlie laws of the 
 function a^ if a is negative, and x zero or integral. If x 
 is zero or a multiple of 2, the power is positive ; if x is 
 odd, the power is negative. 
 
 153. Also, if a is negative, and x fractional, with odd 
 denominator, the power is protomonic (vide § 145). In 
 other words, if a is negative and n odd, there is always 
 a protomonic nth. root of a. For consider the function y", 
 where n is an odd positive integer. The function passes 
 through all values from -co to 0, as ?/ passes from — co 
 to 0. Therefore, there must be some protomonic negative 
 number, y, for any negative nvipiber, a, such that y^ = a. 
 That is to say, there is always a protomonic odd root of 
 a negative protomonic number. 
 
 154. But if in the operation a-^, a is negative and x a 
 fraction in its lowest terms, with even denominator, there 
 is no result whatever, nor is the operation intelligible with- 
 in protomonic number. For the function y", where n is 
 an even integer, is always positive. Therefore there is no 
 protomonic number, y, such that ?/" is negative when n is 
 even. That is to say, there is no protomonic even root of 
 
 4 
 
 a negative protomonic number.* 
 
 155. Evidently, then, unless the Principle of Continuity 
 
 * Also in terms of protomonic number there is no logarithm of a 
 negative number to a positive base. At this stage we cannot investigate 
 such functions ; but log (+ a) ( — b) has been shown to be indeterminately 
 any member of an infinite series of complex numbers. Thus in no case 
 are we led out of complex number as the ultimate generalization. 
 (Cf. § 202.) 
 
104 NUMBER AND ITS ALGEBRA. 
 
 shall widen our concept of number, the generality of 
 numerical operations abruptly fails at this point. But the 
 Principle of Continuity does apply here as everywhere 
 else ; and the power of analysis is enhanced, and the appli- 
 cability of Number to the relations of concrete magnitudes 
 perfected beyond the dreams of mathematical science prior 
 to this development. 
 
 Before taking this step, however, we must investigate a 
 few fundamental properties of radical-surds. 
 
 156. In all algebraic expression of number the student 
 must avoid confusion on account of any possible value of a 
 function for particular numbers in place of the algebraic 
 symbols. Thus {j^^'^)"" is a radical function of 2> in alge- 
 braic form ; although, of course, in cases where m = 2n, 
 and n an integer (p^ ' "^y = j)^, the nature of which again 
 depends on the character of p. Or, the -y/x is algebrai- 
 cally a radical-surd ; although if a? = 4, it is commensura- 
 ble, and so forth. 
 
 It is not necessary to be constantly " providing " obvious 
 conditions. Intelligent attention will always secure com- 
 prehension of the algebraic statements in the sense in- 
 tended, whenever explicit provision is omitted. 
 
 157. A radical-surd number, or multiple, or fraction 
 thereof, is called a simple, monomial radical-surd. The 
 suD^ or difference of two such, or of one such and a com- 
 mensurate number, is called a simple binomial radical-surd. 
 
 It will be seen that every stirpal function of a radical- 
 surd can be expressed as a simple radical-surd. 
 
 Two radical-surds are called similar when they can be 
 expressed as multiples or fractions of the same radical- 
 surd : e.g., V3/4 and VlS are similar; for V3/4 = 
 1/2 V3, and Vl2 = 2 V3. 
 
EADICAL-SURDS. 105 
 
 E-adical-surcIs with the same base and same root-index 
 are called equiradical ; e.g., a^'^, a"^, a"'^. 
 
 Radical-surds with the same root-index are called of the 
 same order — quadratic, cubic, biquadratic, quintic, . . . 
 n-tic; e.g., V3, 5^'", x"'^ are quadratic surds; V3, 5^^^, 
 x"'^ are quintic surds. 
 
 158. From the Law of Indices (a'» a" = a"' + ") it is 
 easily proved for protomonic numbers (but see § 191) that 
 
 a"' a"'' = (aa.y^, or (('"■ [>'"■ = (ab)"K Thus, if m is integral 
 f^m ^m _ ^^^(j . . . m factors) X (phb . . . m factors) by 
 definition, 
 = {ab.ah.ab . . . m factors) by laws of association 
 
 and commutation, 
 = (a^)™ by definition. 
 
 And if m is fractional, say ?/i. = 1 / n where w is a posi- 
 tive integer, («i'"a"''a^"' . . . n factors) X (^^ '" ^^ '"^' ^" 
 . . . n factors) = (a^'^'h^"') (a^'^b^'") . . . n factors; 
 but the left-hand member equals ab; therefore (a^'"i^'") 
 (a^^"6^'") . . . n factors = ab, therefore a^'^'b^'" (aby'\ 
 if positive roots of a, b, and ab are alone considered (vide 
 §§ 144, 146). _ 
 
 159. A special case, -\/a"b = a^b, is important in re- 
 ducing radical-surds to similarity. 
 
 160. Note also ^a = VaP ; for aV» = «"/"? = V«p. 
 
 pn + q 
 
 161. Also, ^(1^^ + " = aP^ai;iov a » =aPaih. 
 
 162. Similar radical-surds are " added " or " subtracted " 
 by distributing the radical-surd factor with the coefficients. 
 {Vide § 73.) If possible, first reduce by the principle of 
 Sections 159, 1 60, e.g. , 1/3 V 32 - V l8 + 3 -v'64 = 1/3 
 V(16) (2) - V(9H2) + 3 V(4) (2) = 4 /3 V2 - 3 V2 
 -\-Q-j2 =(4/3 - 3 -f 6) V2 = 13/3 V2. 
 
106 NUMBER AND ITS ALGEBRA. 
 
 Staxements involving radicals are usually intended to 
 concern only positive roots ; but in abstract operation such 
 statements are necessarily various, including the roots in 
 every combination. The whole truth about the result in 
 the example is 
 
 {±4/3 - (± 3) + (± 6)} V2 = ± 13/3 V2, or 
 ±5/3 V2, or -i-23/3 V2, or ±31/3 V2. 
 
 The V2 is also both positive and negative; but since 
 each commensurable factor has already occurred with both 
 signs, no new value would be obtained from the double 
 value of the V2. But if all this is to be signified, it would 
 be better to be explicit, and write 1/3 (zl=V32) — (-j-VlS) 
 + 3 (±-^64). (Vide § 120.) 
 
 163. Section 158 affords the rule for the multiplication 
 or division of similar radical-surds, or of radical-surds of 
 the same order. 
 
 If radical-surds are not of the same order they may be 
 made so by Section 160. 
 
 The Law of Indices immediately furnishes the rule for 
 the involution or evolution of radical-surds. 
 
 164. The student should exercise himself in these opera- 
 tions. 
 
 165. Two simple binomial quadratic surds are called con- 
 jugate when one is the sum and the other the difference of 
 the same two terms :. e.g., a -\- -y/h and a — -y/h, or V« + 
 V^ and -yja — -y/b. 
 
 166. Theorem. — The product of conjugate binomial 
 quadratic surds is a stirpal function of their bases (a com- 
 mensurate number if the bases are commensurate numbers), 
 namely, the difference of the squares of the terras. 
 
 Proof: (Va -f V^*) (Va — ^h) = a + V« V^* - -\/h 
 ■yja — h = a — b. 
 
CLASSIFICATION OF ANALYTICAL FUNCTION. 107 
 
 167. It is usuall}^ preferable in the division of one radi- 
 cal-surd by another, or of a commensurable number or 
 non-radical surd by a radical-surd, to stirpalize * the de- 
 nominator. 
 
 This is accomplished when the divisor is a monomial 
 radical-surd, as Va"*, by multiplying both dividend and 
 divisor by -v/«"~"'. For example, — 
 
 3 _ 3 V2 _ ^ _c c Va^ _ 
 
 4V2 ~4V2V2~^/^^^' bVa'~ hVa'Va^~ 
 
 When the divisor is a binomial quadratic surd, multiply 
 both dividend and divisor by the conjugate quadratic surd ; 
 when a trinomial, make it a binomial by association, and 
 apply the principle twice. 
 
 168. Let the student find a stirpalizing multiplier for 
 
 This is the most general case of a monomial. 
 
 169. A stirpal integral tervi with respect to any num- 
 bers, means the product of positive integral powers of those 
 numbers. 
 
 A stirpal integral function of any numbers is a series 
 (one or more) of stirpal integral terms combined in addi- 
 tion or subtraction. 
 
 Where no ambiguity is to be feared we may say merely 
 "integral function." xj a -\- y fh + ;s/c — 1 is an integral 
 function of .r, y, z ; but is not an integral function of a, b, c. 
 
 In integral functions the degree of any term is the sum 
 
 * The common term is " rationalize; " but having eschewed this, we 
 must say stirpalize. 
 
108 NUMBER AND ITS ALGEBRA. 
 
 of the exponents of the numbers considered (commonly 
 called variables) ; and the degree of the function is the 
 highest of the degrees of its terms. 
 
 An integral function of the 1st degree is often called a 
 linear function. 
 
 The term degree applies only to integral functions. 
 
 Thus, - -I ^ + 1 is of no degree at all : the term does not 
 
 X x^ 
 
 apply. 
 
 Functions in Avhich the variables are affected by positive, 
 but not integral, ex ponent s are called radical functions. 
 For example, a + -yjh + x, or a + (Z> + x f ''\ is a radical 
 function of x (also of I) ; and Vx + \j]), or {x — if '^f '«, 
 is a radical function of x and y. 
 
 Functions in which the variable occurs with negative 
 index are called fractional functions, and distinguished as 
 stirpal or radical fractional functions, according as the nega- 
 tive index is integral or not. Thus, ^-, or «a;-^, is a stirpal 
 
 X 
 
 fractional function of a;; and ~, or ax-^'-, is a radical 
 fractional function of x. "^^ 
 
 Integral, radical, and fractional functions are classed, not 
 very felicitously, as "algebraical" functions, in distinction 
 from others equally algebraical, called " transcendental." 
 I shall have no occasion to use these objectionable terms, 
 since the other functions are all particularly named upon 
 their own merits. 
 
 Functions in which the number considered occurs as an 
 exponent are called exponential functions; e.g., «^, a-^"" 
 are exponential functions of x. 
 
 The foregoing classes of functions are those organically 
 involved in numerical operations. Others, less essentially 
 
FUNCTIONS OF EADICAL-SUEDS. 109 
 
 connected with orgauic laws, are named from their several 
 points of view ; e.g., log cc, logarithmic function ; sin a;, 
 cos X, tan x, etc., trigonometric functions, etc. 
 
 Numerical functions (Cf. §§230, 234) of every variety 
 are termed analytical functions (Cf. §§ 145 and 156.) 
 
 170. Theorem.- — ^ Every integral function of quadratic 
 surds (V«, V^, Vc . . .) can be expressed as a sum of a 
 non-radical term and multiples or fractions of the radicals 
 and their products — 
 
 (V'«5 V^, V c . . . 's/iiO, \ac, -yhc . . . -y/abc . . .). 
 
 Proof: Consider any integral function of one quad- 
 ratic surd, say </> (Va). Terms of even degree are non- 
 radical, and terms of odd degree can all be reduced to the 
 form na"^ V«. Collecting the even and odd degree terms, 
 we have <^ ( Vc') = Z; -)- A Va, where k and h are stirpal. 
 
 If we have <^ (V«, V^), proceeding as before, we get 
 ^ ( V«, V^) = K -\- H a/ a, where K and If are stirpal so 
 far as V^ is concerned, and each. an integral function of 
 V^. These can be reduced, and will yield only terms such 
 that ^ (Va, V^) = k -{- h V« + '»i' 'Vb + n ^/ab. 
 
 171. CoKOLLAKY. — It follows that <^ (— V«) ■= k — h 
 Va ; and therefore if </> Va be any integral function of Va, 
 then, </) ( — V«) is a stirpalizing factor of ^ V«. {Cf.% 166.) 
 
 Also if in </> (Va, V^, Vc, . . . ) we change the sign of 
 any one, say, V^, then ^ (Va, V^, Vc, . . . ) X <^ (Va 
 — Vi, Vc, . . . ) is stirpal so far as V^ is concerned. 
 
 172. Extension of the theorem to all stirpal functions, 
 integral or not, of quadratic surds — and of the corollary 
 
 to the entire stirpalization of ^ {-yja, -\/b, Vc, . . .) is 
 
 left as an exercise to the student. 
 
 173. As a very simple example of the utility of these prin- 
 ciples, suppose one had to calculate to five decimal places, 
 
110 NUMBER AND ITS ALGEBRA. 
 
 1 . Time and labor would be saved by redu- 
 
 1 -)- V2 -|- V3 cing to the equivalent integral function 
 of the radicals, 1/2 + 1/4 V2 - 1/4 V6, before cal- 
 culating. 
 
 174. Theorem. — '^ If p, q, A, B, be all commensurable, 
 and V/? and Vg" incommensurable, then we cannot have 
 Vi> = A + B Vq. 
 
 " For, squaring, we should have, as a consequence, j) = A^ 
 
 -\- B"^ q -\- 2 AB V«7 ; whence, Vv = —^ ~ , which 
 
 -r i -r I, , 1 2AB 
 
 asserts, contrary to our hypothesis, that -y/q is commen- 
 surable." 
 
 The proof of this theorem, Avhich is copied verbatim 
 from Chrystal's Text Book of Ahjebra, Vol. I, p. 200, estab- 
 lishes what may seem at first sight a contradiction of the 
 doctrine of the Continuity of Number. Especially so, under 
 the somewhat ambiguous title of the section in Professor 
 Chrystal's work (perhaps the best yet written in English), 
 the ^^Independence of Surd Numhers.^^ Eadical-surds are 
 definite parts of the continuous magnitude, Number ; nor 
 does the theorem contradict this ; nor are radical-surds 
 <' independent " in any other sense than that there are no 
 commensurable numbers such that V^^ = A -{- B ^ q. 
 
 175. Since, by Section 170, any integral function of a 
 quadratic surd can be expressed as in the form, A -\- B -y/q, 
 it follows from Section 174 that one quadratic surd cannot 
 be expressed as an integral function of a dissimilar surd. 
 
 176. It is an obvious corollary of Section 174 that if A; + 
 h -yj a -\- m V^ + n "y/ab = 0, where neither a nor b is zero, 
 then k = 0, A = 0, m = 0, and n = 0. 
 
 177. One case, whose utility is experienced very early 
 in algebraic studies deserves special mention. If a -\- Va; 
 
NEOMONTC NUMBER. Ill 
 
 = b -\- -y/i/, then a = h and x = y, provided a, b, x, and y 
 are all commensurable, and -\/x and V*/ surds. 
 
 178. Let the student prove that the product or quotient 
 of two similar quadratic surds is commensurable ; and 
 inversely. 
 
 The like is not true for radical-surds of higher orders ; 
 but let him show that the product of two similar, or of two 
 equiradical, surds is either commensurable or an equirad- 
 ical surd. 
 
 179. We are now prepared to take up the consideration 
 of the problem presented in a}''^ where a is negative, and 
 n an even, positive integer. 
 
 As Ave saw in Section 154, the operation is unintelligi- 
 ble under the concept of Number thus far attained. But 
 if the Principle of Continuity is valid, the result must be 
 a number ; and if not any number hitherto conceived, we 
 must investigate the characteristics of this unknown num- 
 ber, X in the synthetic equation (— 1) Y^ = x. 
 
 180. Whether fortunately or unfortunately, this prob- 
 lem confronts pupil and teacher at a very elementary 
 stage of numerical analysis. In every high school the 
 solution of quadratic equations is attempted; and these, 
 even in the simplest form, are in general solvable only in 
 terms of neomonic and complex numbers. The question, 
 therefore, cannot be postponed ; and it behooves every 
 teacher to clear up his ideas on this subject. 
 
 181. Mathematicians of to-day have left the point of 
 view of the sixteenth century, from which numbers were 
 characterized as " rational " and " irrational," " real " and 
 "imaginary ; " they use V— 1 as naturally as — 1. Neo- 
 monic one, and negative one, bear a similar relation to 
 Primary Number. 
 
112 NUMBER AND ITS ALGEBRA. 
 
 The conception of neomonic number is not essentially 
 more difficult than that of negative number. He who can 
 conceive the one, can conceive the other. The V— 1 is no 
 more an impossible and meaningless operation in terms of 
 protomonic number, than 1 — 2 is impossible and unintel- 
 ligible in terms of primary number. Terms are often bab- 
 bled in unconscious vacuity of thought. Many speak quite 
 familiarly of negative number, who nevertheless regard 
 neomonic number as some irrational and meaningless trick 
 of handwriting. As suggested in Chapter XII, I lament 
 imperfect concejits of Number on the imrt of us all, but let 
 no man pigeon-hole in his mind contradictory opinions. It 
 seems to me something to put neomonic numbers on the 
 same footing as negative numbers, or even numerical frac- 
 tions. 
 
 When this point of view is attained, I think we stand 
 in the dawn ; or rather that the sun has risen upon Arith- 
 metic, even as it has risen upon Geometry. Perhaps we 
 shall not have long to wait for still fuller and more satis- 
 fying interpretations of number than have been expounded 
 hitherto ; because not one man, but hundreds, have reached 
 some such standpoint as that from which I have endeav- 
 ored to present the subject. During two thousand years 
 after Euclid saw that he must assume the " parallel postu- 
 late " it was universally regarded either as an axiom, or as 
 a theorem capable of demonstration. But finally the true 
 insight was gained (regained) by many minds about the 
 same time ; and then the Non-Euclidean Geometry, and 
 daylight became, indeed, "inevitable."* 
 
 * The MoJiist, July, 1894, ' Xon-Eudidean Geometry Inevitable, by- 
 George Bruce Halsted. Of course tVie majority of text-books still pre- 
 sent Geometry at this crucial place from the mediaoval standpoint ; but 
 
THE NEOMON. 113 
 
 182. If the V — 1 is a number, we have by definition 
 
 V-1 V-l= -1, 
 also Va ^a = a, where a is any positive protomonic 
 
 number ; 
 therefore (Va V— 1) (V« V— 1) = — a, multiplying 
 
 member by member ; 
 therefore Va V— 1 = V— «, taking square root of each 
 
 member. 
 
 Consequently it appears that the square root of any 
 negative number is the product of the square root of the 
 corresponding positive number and V — 1- Considering 
 also all multiples and fractions of V— 1, and the nega- 
 tives of each, we discern a continuous Number whose unit 
 is V— 1> and which has, therefore, been called Neomonic 
 Number. The Number whose unit is 1 may be called Pro- 
 tomonic in contradistinction. 
 
 Writing i for V— Ij this continuous series may be rep- 
 resented — 
 
 — cx> i . . — 2 i . . — -y/2 i . . — i . . — ^ i . . (i) . , 
 -f ^ i . . . -\- i . . + V2 i . . + 2 t . . + CO i. 
 
 The protomonic series may be represented — 
 
 — CO 
 
 . - 2 . . . - V2 . . . - 1 . . - 1/2 . . ... 
 
 -f 1/2 . . + 1 . . + V2 . . . + 2 . . . + 00 . 
 
 183. No neomonic number can equal any protomonic 
 number except i = 0. For it is deducible from various 
 
 this is probably as much due to the mercantile rule of using up a stock- 
 on-liand before advancing to something better, as to ignorance of recent 
 developments. No doubt hundreds of teachers put the " axiom " in its 
 right place in their expositions of the text ; and so, as it were by a note, 
 bring their text-books " up to date." 
 
114 NUMBER AND ITS ALGEBRA. 
 
 premises that 1 = 0. Thus, ii xl = 0, then (xi) (xi) = 0, 
 that is, — x^ = 0; therefore x = 0, and therefore * = 0. 
 
 184. Most laws of operation with neomonic numbers 
 are evident from familiar princixDles. Thus: — 
 
 ai 4- bi = (a -\- h) i . . . hence the sum of two neomonic 
 numbers is neomonic. 
 
 ai — hi = (a — b) i . . . hence the difference of two neo- 
 monic numbers is neomonic. 
 
 ai X b = abl . . . hence the product of a neomonic and a 
 protomonic number is neomonic. 
 
 ai X bi = — ab . . . hence the product of two neomonic 
 numbers is protomonic. 
 
 ai -ir b r= (a 1 0)1 i _ , hence ratios of protomonic and 
 h -r- ai = (^— b I a) I ) neomonic numbers are neomonic. 
 ai ^ bl = a I b . . . hence ratios of neomonic numbers are 
 
 protomonic. 
 ^2 = - 1 ; P = - 1 V- 1 = - * ; ^^ = *2 r _ + 1 ; and 
 
 where n is a positive integer, 
 
 (aiy = {ai . ai . . . n factors) = {aaa . . . n factors) 
 (iii . . . n factors) = a"*", 
 
 that is, the positive integral power of a neomonic number 
 is protomonic or neomonic according as the same power of 
 i is protomonic or neomonic. Moreover, the integral powers 
 of * are seen to recur in a period or cycle of four different 
 values. Negative exponents result as always in the recip- 
 rocal of the same number Avith like positive exponent. 
 
 185. Discussion of radical functions of i, and the inter- 
 pretation of neomonic exponents, is postponed to more 
 
COMPLEX NUMBER. 115 
 
 advanced studies ; but we are not led to any new applica- 
 tion of the principle of Continuity, and therefore to no new 
 mode of jSTumber, beyond the result of combining proto- 
 monic and neomonic numbers in addition and subtraction. 
 
 186. The extension of the number-concept reaches its 
 own essential terminus in the operation a + hi, where a 
 and h are protomonic. 
 
 In a -{-hi we have the most general expression of num- 
 ber ; for it is protomonic, neomonic, or complex, according 
 as ^ = 0, a = 0, or neither equals 0. 
 
 187. The result of the operation a -\- hi, is called a com- 
 plex number ; and is seen to be really a new mode of Num- 
 ber by considering the series of complex numbers formed 
 in a + hi, as a and h pass independently through all pro- 
 tomonic values. 
 
 188. It is highly important to note this two-fold, two- 
 dimensional (vide § 229, et seq.), character of complex 
 number, and its consequent contrast with protomonic and 
 neomonic number. There is only one way of varying x 
 continuously (without repetition of intermediate values) 
 from — 2 to + 3, if it remains protomonic. Likewise, only 
 one way for continuous passage of x from — 2 i to -\- 3 i, 
 if it is to be always neomonic. But in utter contrast, there 
 is an infinite variety of ways for x to pass continuously 
 from — 2 + 3 i to + 2 + 3 /, remaining always a complex 
 number. (Vide § 197.) 
 
 189. If a = and b = 0, a -\- hi = 0; and inversely. 
 
 190. Complex number contains all protomonic and all 
 neomonic number as special cases, and is therefore Number 
 in its final generalization. 
 
 191. The student should everywhere carefully avoid con- 
 fusion in dealing with the alternate square roots of any 
 
116 NUJIBEE, AND ITS ALGEBRA. 
 
 number ; but especially is this the case with neoinonic 
 numbers. Having been accustomed to write {vide §§ 64, 
 158) Va V6 = V"^, he may fall into the error of writing 
 V— a V^^ = V(— «) (— h) = -y/ah. I call this an error 
 because we must be consistent in algebraic conventions ; and 
 in such contexts the positive root is understood by -y/ab* 
 
 It is not a true statement that Vf'- V^ = Vo^, if the 
 square roots are to be taken at random. One cannot make 
 various assertions in the same sentence. Therefore, in 
 Va V^ = Vab, we evidently mean only the positive 
 square roots to be considered. If negative roots are to be 
 taken into account, we must say what we mean. Thus 
 (writing -j- V* for j^ositive square root of a, and — V« for 
 negative square root of a) (— V«) (— V^) = + 'Vab; or 
 (— V") (+ "v^) = — Vcib, etc. 
 
 Now, if in accordance with the algebraic convention 
 plainly exhibited above, we consider only positive square 
 
 * In a translation just published of Durege's Theory of Functions of 
 a Complex Variable, by Professors Fischer and Schwatt of the Univer- 
 sity of Pennsylvania, Philadelpliia, 189(3, it is stated on Page 10 of the 
 Introduction: "Euler himself taught, as n ow g enerally accepted, that, 
 if a and b denote two positive quantities, V— « V^^ = y/ab ; i.e., that 
 the product of two imaginary quantities is equal to a real quantity." 
 
 The omission of the minus sign before Vab may be a typographical 
 error; for the authors, like all others, use \/— a ■\/^^ = — Vab. 
 
 In the translators' Introduction it is very appropriately remarked: — 
 
 "To follow the gradual development of the theory of imaginary quantities 
 is especially interesting, for the reason that we clearly perceive with what diffi- 
 culties is attended the introduction of ideas, either not at aU known before, or 
 at least not sufficiently current. The times at Avhich negative, fractional, and 
 irrational quantities were introduce.l into mathematics are so far removed from 
 ns, that we can form no adequate conception of the difficulties which the intro- 
 duction of those quantities may have encountered. Moreover, the knowledge 
 of the nature of imagi)iiiry quantities has helped us to a better understanding 
 of negative, fractional, and irrational quantities, a common bond closely unit- 
 ing them all." 
 
 Of course I would have one read numbers in the place of " quantities." 
 
EFFICACY OF NUMERICAL OPERATIONS. 117 
 
 roots of neomonic numbers, V— « V— ^ does not equal 
 ■\/ab, but — V«6 ; — 
 
 for V— a V— b =^ai^bi =i^^ab =( — 1) V«^ = — Vf<6. 
 One need find no difficulty in reconciling with the Prin- 
 ciple of Continuity the statements that, regarding only 
 positive roots, Va V^ = -s/ab, while V— « V— * is not 
 equal to V(— «) (— b). The law of indices must be 
 applied with due regard to other laws. The essential 
 statement of the law of indices is «^ av = a^ + y. This 
 includes all particular cases as a, x, and y assume differ- 
 ent characters. But it has been necessary with every 
 phase of number to understand in this statement that only 
 corresponding roots are considered when x and y are frac- 
 tional with even denominators. (Cy. §§ 144, 146.) For 
 example, V'^ X 1^^^ would not equal 11/2 + 1/2^ ^^ \^ if 
 one positive and one negative root were taken. Now, 
 this fundamental statement of the law of indices hokls 
 
 for all number. It is the very definition of V— 1, that 
 (_ 1)W2 (^_ iy/2 ^ ^_ ;Ly/2 + i/'2^ (-_ ly = _ 1. 
 
 It was easily proved for protomonic number that, regard- 
 ing only corresponding roots when a; is a fraction with 
 even denominator, a^ a^ = («ff)^, and a'-' b^ = (abY ; but 
 when a and aa differ in quality, the very conditions of 
 the original statement are abolished (it is as if one posi- 
 tive and one negative root of a^ had been taken), and 
 different conclusions might be anticipated under the same 
 laws. 
 
 In fine, all this is not an anomaly of V— 1 in operation, 
 but merely an alternative statement of its existence. The 
 difficulty lies in the origin of neomonic number, not in its 
 operation. 
 
 On the other hand, a^/b'^ = (ct / by, established for pro- 
 
118 NUMBER AND ITS ALGEBEA. 
 
 tomonic number, does hold if a^ and h'^ are neomonic, — ■ 
 simply because, in this case, no qualitative difference arises 
 in the direct performance of the operations indicated by 
 the two members of the equation, if, in accordance with 
 the meaning of the formula, only positive roots are 
 regarded. 
 
 For example, (- ^Y'-i (- 9)^ ^2= (4/9)^ '^ ; for (- 4)^ '^ 
 = 2 i, and (- 9)7^ = 3 * ; therefore, (- 4)77 (_ 9)^ '''- 
 = 2ilZi = 2j2>. Also the positive square root of 4/9 
 is 2 / 3. 
 
 Note, also, that for a like reason V« V— h = V— ab; 
 for -y/a -y/ — 6 = -y/a -\b i = '\ab i, and V — cih = ~vab i. 
 
 The safe practice is to express every neomonic number 
 in its essentially proper form, as based upon a new unit. 
 
 Rules of thumb would conduct one to true results in all 
 operations except multiplication ; but for many reasons, 
 always express -y/— a as -\/at. If you do this, correct 
 calculation will be easy under the very definition of the 
 neomon, i^ = — 1. 
 
 192. As a natural consequence of the view that Algebra 
 is some mysterious conglomeration of " pure symbols " 
 (C/l Introduction, pp. 8, 12) without content, existing for 
 itself, void of numerical meaning, it was long discussed, 
 as if it were a matter to be settled by parliament, whether 
 V— (f^ V — b should equal V— «^, or — V«^. Only one 
 hundred years ago English mathematicians were divided 
 on this question. One party argued that the product must 
 be V — ab; because the product of one "impossible quan- 
 tity " by another, could not possibly equal a " real 
 quantity " — as if a priori deduction of Avhat is, or is 
 not, possible with imjjossihle quantities was not ab initio 
 an impossible discussion within the realm of Reason. 
 
COMPLEX NUMBERS. 119 
 
 May not tlie foregoing discussion (as well as every other 
 investigation we have pursued) serve to emphasize the car- 
 dinal thesis of these lectures ; namely, that the essential 
 nature of any algebra is as defined in Section 20 ; that it is 
 Arithmetic, as the science of Number, which everywhere 
 underlies, shapes, and organizes our Algebra ; that it is 
 real numerical laws and operations that the algebra conven- 
 tionally expresses; that, although Number is certainly a 
 creation of the human intellect, it is not, therefore, the 
 creature of our choice or whim; that, once formed, the 
 Idea unfolds itself ; that every numerical problem is a 
 question of Truth ; that the explanation is to be discov- 
 ered; and that the verdict is nowise subject to conven- 
 tional decision or parliamentary settlement. 
 
 193. It might be very helpful to illustrate the proper- 
 ties of complex number by the graphic representation 
 known as Argand's diagram, which constitutes the foun- 
 dation of a beautiful application to geometry; but we 
 shall here confine ourselves to purely analytical inves- 
 tigations. 
 
 We have seen (§ 188) the two-dimensional nature of 
 complex number, and the infinite variety of ways in 
 which it may vary continuously from a -\- hi to c -f di, 
 because the protomonic and neomonic parts may vary 
 independently. 
 
 In order that x -\- yl shall become zero, x and y must 
 vanish simultaneously. Por, li x -\- yi = 0, ic = — yi, and 
 hence cc = and y = 0, — else would a protomonic number 
 equal a neomonic, Avhich is impossible except both be zero. 
 (Vide § 183.) 
 
 On the other hand, if either x or y becomes infinite, x -{- 
 yi=cc. (FicZe§198). 
 
120 NUMBER AND ITS ALGEBRA. 
 
 194. li a -\- bi = G -\- di, then a = c and b = d. 
 
 For, subtracting c -f- di from each member of the given 
 equation, a — c-\-(b — d)i = 0; therefore, by Section 193, 
 d — c = and b — d ^= ; that is, a = c and b = d. 
 
 Of course, since x + (— y) = x — (+ y) ox x — y {% 120), 
 the preceding formula inckides all combinations as a, b, c, 
 and d are positive or negative ; e.g., if a -f bi = c — di, 
 a = c, b = — d. 
 
 195. Two complex numbers which differ only in that 
 one is the result of the addition, the other of subtraction, 
 of the neomonic part, are called conjugate ; e.g., — 1 / 2 + 2 i 
 and — 1/2 — 2i, 0T2i and — 2 i, or generally x -\- yi and 
 X — yi. 
 
 Obviously the sum of conjugate complex numbers is pro- 
 tomonic, but so also is their product : — 
 
 (x -j- yi) (x — yi) = x^ — y"^ i"^ = x"^ -[- y^. 
 
 196. Let the student prove the inverse proposition. 
 
 197. x^ 4- ?/^ is called the norm of the complex number 
 X -|- yi, ox X — yi] and, as seen in Section 195, the product 
 of conjugate complex numbers is the norm of each. 
 
 But note that also norm (— a? — yi) = (— xy -\- y'^ = 
 x^ + if'j although — X — yi is not conjugate with x + yi, 
 nor is their product the norm of either; for (— x —yi) 
 (x -\- yi) = y"^ — x"^ — 2 xyi. 
 
 198. The positive square root of the norm of a complex 
 number is called its modulus : mod (x + yi) = + Vic^ + y'^. 
 
 This modulus has extremely important properties. 
 
 The attentive student may have already discerned diffi- 
 culty in applying comparisons of greater or less to complex 
 numbers ; for example, which is the greater, 3 + 4 i or 
 2 + 5i? 
 
MODULUS OF COMPLEX NUMBER. 121 
 
 The quantity {vide § 229) of a complex number is discov- 
 ered to depend upon its modulus. Complex numbers with 
 equal moduli are quantitatively equal, though not identical 
 numbers. Any magnitude of two dimensions must exhibit 
 this mode of equivalence without congruence. Argand's 
 diagram would give a good illustration of this relation : 
 the points representing (or terminating the radii which 
 represent) complex numbers of equal moduli would all lie 
 on a circle ; points corresponding to complex numbers of 
 less moduli would lie within the circle, and of greater 
 moduli without. 
 
 This property of the modulus is exhibited analytically in 
 the fact that, since mod {x + yi) = + V^^ + y^ which is 
 positive regardless of the quality of x or y, if either x ox y 
 increases, the modulus increases, and if either x ox y de- 
 creases, the modulus decreases. And this change is continu- 
 ous, the modulus vanishing with the number, and inversely. 
 
 If two numbers are equal, their moduli are equal ; for 
 we have seen (§ 194), \i a -\- hi = c -\- di, a = c and b = d. 
 But the inverse is not true ; for if cv' -{- b'^ r= d^ -\- d% it 
 does not follow that a = c and b = d. ^ 
 
 Note that if 7/ = in x + yi, that is, if the complex 
 number be wholly protomonic, the modulus becomes + Vx^ 
 = _|_ X, — and this whether x in the complex number be 
 positive or negative. Thus, the mod (+ 3) = + V(+ 3)"'^ 
 = + 3 ; and mod (- 3) = + V(- 3)^ = + 3. 
 
 For this reason, many European continental writers use 
 the term modulus of x ('^mod x") where a:; is a protomonic 
 number, instead of the term " numerical value of a;," em- 
 ployed by English writers. For example, we constantly 
 speak of + 3 and — 3 as "numerically equal," whereas, 
 if equal — being numbers — they could only be numerically 
 
122 NUMBER AND ITS ALGEBEA. 
 
 equal ; and they are not equal, for their difference, instead 
 of being zero, is 6. 
 
 It would therefore serve accuracy and propriety to fol- 
 low the practice of the writers referred to. 
 
 199. Evidently the sum of any number of complex num- 
 bers is a complex number. 
 
 Likewise the product of any number of complex num- 
 bers is a complex number. 
 
 Also the ratio of two complex numbers is a complex 
 number. For — 
 
 a -\- bi _ (a -\- hi) (c — di) _ (ac -\- h d) — (ad — cb) i 
 c + di ~ c2 + cZ- ~ ~c2 + d^ 
 
 _ f ac-\- bd \ _ 
 
 -\c^^d^' 
 
 which is a complex number. 
 
 Since stirpal functions can involve only the operations 
 of addition, multiplication, and their inverses, we have 
 thus established the theorem : Every stirpal function of 
 one or more complex numbers is a complex number. 
 
 200. Several other fundamental theorems concerning 
 stirpal functions of complex numbers, and moduli of com- 
 plex numbers, are deferred to the final chapter. 
 
 In regard to radical functions of complex numbers, we 
 can consider here only the particular case of the square 
 root : — 
 
 Assume that the square root of a complex number is a 
 complex number, and let — 
 
 V« + bi = A + Bi, 
 
 where a, b, A, and B are protomonic. 
 
 Squaring each member : a -\- bi = A"^ — B^ -\- 2 ABi ; 
 
EFFICACY OF NUMERICAL OPERATIONS. 123 
 
 therefore, by Section 194, a = A^ — B^ (1) 
 
 and b = 2AB. (2) 
 
 Adding the squares of (1) and (2) 
 
 a2 -|_ J2 = (^2 _^ ^2y^ (3>) 
 
 Taking square roots of (3), and remembering that 
 j2 _|_ ^2 jg essentially positive : 
 
 Adding (1) and (4) : + V«M^ -\-a = 2A^ 
 
 therefore 
 
 = .v/ 
 
 + Va"-^ + b'^ + 
 
 a 
 
 2 
 Subtracting (1) from (4) : + Va^ + b'^ — a = 2 B% 
 
 therefore ^ = ± y/+ Va2 + 6^ _^ 
 
 Since -J- Va'^ + ^^ > «> these values of A and i> are pro- 
 tomonic. Since b = 2 AB, like signs in the values of A 
 and B must be taken if b is positive, and unlike, if b is 
 negative, therefore if b is positive, 
 
 v;r+l5=±{v/+^ffl±i!+.-y/+Vi!±^!^| r. 
 
 and if b is negative, 
 
 Let the student verify by multiplication. 
 
 If the protomonic part of the complex number vanish, 
 we have here the formula for the square root of a neomonic 
 number. 
 
 Particularly if a = 0, and b = -\-l, formula I becomes, — 
 / -. 1 + i 
 
124 NUMBER AND ITS ALGEBRA. 
 
 If a = 0, and & = — 1, we have from formula II, — 
 
 1 -i 
 
 V-^= ± 
 
 V2 
 
 By means of these results the student can readily find 
 four 4th roots of -|- 1> and of — 1. 
 
 201. Since radical functions of any number involve only 
 fractional exponents besides stirpal operations upon the 
 number, if we show that the nth. roots, when w is a primary 
 number, of a complex number are complex numbers, we 
 establish the theorem : All radical functions of complex 
 numbers are complex numbers. 
 
 The investigation must be postponed to future studies ; 
 for more powerful instruments of analysis (e.g., Demoivre's 
 theorem) are required than are at the command of the 
 students to whom these lectures are primarily addressed. 
 But the theorem has been established. 
 
 202. Command of the proper means of analysis (e.g., 
 logarithmic series) would enable the student to prove that 
 exponential functions (^vide § 169) of complex numbers 
 lead to no new mode of number. 
 
 Thus, finally, it has appeared that the ultimate gener- 
 alizatioii, (a -\- hiY+y\ is still a complex number; and that 
 therefore the Universe of Number closes, returns upon 
 itself, is comjjlete. 
 
MEASUREMENT. 125 
 
 XITI. Measurement. 
 
 203. The measurement of any magnitude (concrete or 
 abstract) is the process of finding its ratio to another mag- 
 nitude of the same kind, arbitrarily chosen as a unit. 
 
 204. The measure of a magnitude is this ratio — a 
 number. 
 
 Under the conventions of English speech, the measure 
 of any magnitude is expressed by a phrase made up of this 
 number and the name of the chosen unit. 
 
 205. The noun, measure, is commonly used in the sense 
 of Section 204, in the sense of submultiple {vide § 83), and 
 in the sense of unit. There may be no very good ground 
 of choice in these terms, but consistency in the same dis- 
 course is desirable. It may be better not to say, '* the 
 greatest common measure " of two or more magnitudes, 
 since any magnitude of the same kind woiild be a common 
 measure, in another meaning of the word ; for example, 
 the yard may be the common measure of all lines, and so 
 may the metre. It may be better to say, instead, the 
 greatest common submultiple. On the other hand, commen- 
 surable and incommensurable point the same way as common 
 measure. 
 
 The use of measure in the sense of unit is superfluous 
 in the presence of the clearer term, unit, and appears to 
 foster a confusion of concepts with commensurability ; 
 whereas it is very seldom that a unit is commensurable 
 with the magnitude measured. Attention is merely called 
 to this confusion in our language, and consequently in our 
 thought. Under the necessity of some choice, I have se- 
 lected cornvfiensurahle, submultiple., and unit, and have simply 
 avoided measure as the inconsistent synonym both of sub- 
 multiple and unit. 
 
126 NUMBER AND ITS ALGEBRA. 
 
 206. The unit of any kind of magnitude may be any 
 magnitude of the same kind. 
 
 Convenience, or lack of concerted choice, ofteia establishes 
 in common usage many units for magnitudes of the same 
 kind. 
 
 207. Magnitudes are of the same kind when, of any two, 
 one is necessarily greater than, equal to, or less than the 
 other. 
 
 Magnitudes between which there is no such comparison 
 are of different kinds ; and between such there is no ratio, 
 nor could one be added to the other. 
 
 208. The ratio of any two magnitudes is independent of 
 any unit, or units, of measurement. Their absolute values 
 can in no way depend upon the arbitrary standard, or stan- 
 dards, by which they may happen to be estimated. For 
 example, the ratio of the time of rotation of Mars to the 
 time of rotation of Venus is that exact numerical relation 
 of the former to the latter, in virtue of which the former is 
 a fraction of the latter ; or greater than one, and less than 
 another fraction of the latter, which differ as little as we 
 please. (Cf.^ 83.) Evidently this ratio can nowise de- 
 pend upon other comparisons of these times with any other 
 periods of time whatsoever. 
 
 209. But the ratio of any two magnitudes equals the 
 ratio of their respective measures in comparison with the 
 same unit. For example, the ratio of the two periods of 
 planetary rotation just mentioned equals the ratio of their 
 respective ratios to any third period of time, say, the time 
 of the earth's rotation, ■ — tlie period we name a day. 
 
 210. There is such a thing as direct operation with con- 
 crete magnitudes ; but it is only through their measures, 
 that is, their ratios to some unit, that magnitudes other 
 
MEASUREMENT BY PROPORTIONALITY. 127 
 
 than Number can become subjects of genuine calculation, 
 the proper subjects of which are numbers, and numbers 
 alone. {Cf. §§ 27 and 48.) For example, the sum of two 
 sects is a sect, which may be found directly by placing the 
 given sects end to end in a straight, with none but these 
 end-points in common. The sect between the non-coinci- 
 dent end-points is the sum. But in calculation we add the 
 lengths (i.e., the numbers which are the ratios of the sects 
 to any unit-sect) of the sects, and obtain the length of their 
 sum (i.e., the number which is the ratio of the sum-sect to 
 the same unit). 
 
 211. As stated in Section 203, the measurement of a 
 magnitude consists in finding its ratio to another magni- 
 tude of the same kind, chosen as a basis of comparison. 
 Howsoever this ratio may be found, the magnitude is 
 measxwed. 
 
 In physical science magnitudes are commonly measured, 
 not directly, but indirectly; that is, the direct comparison 
 is not between the magnitude which is to be measured and 
 a chosen unit, but between two magnitudes of a different 
 kind which are proportional to the magnitude which is to 
 be measured and its unit. It is highly important that this 
 fact be recognized by all students of physical science. It 
 also emphasizes very clearly the absurdity of omitting a 
 sound exposition of the doctrine of proportionality from 
 elementary instruction in mathematics. 
 
 The doctrine of proportionality is not especially difficult 
 or recondite ; but, even if it were, its thorough exposition 
 cannot be postponed, because comprehension thereof is pre- 
 requisite for understanding ordinary measurements in the 
 most elementary physical science, and the commonplace 
 problems of daily life. For example, temperatures are 
 
128 NUMBER AND ITS ALGEBRA. 
 
 never measured directly, but always by raeaus of tlieir 
 assumed proportionality to the volumes of some body at 
 the temperatures in question. Again, masses are usually 
 measured by their proportionality to the corresponding 
 weights in the same place, etc. 
 
 212. Arcs of a circle are so conveniently measured by 
 means of their proportionality to the angles they subtend 
 when the vertices of the angles are at the centre of the 
 circle, that they are seldom measured directly. It must 
 be carefully noted, also, that only angles less than a peri- 
 gon are so proportional, and therefore so measurable. The 
 indirectness of such measurement of arcs is not sufficiently 
 emphasized in many text-books. The most faithful Eng- 
 lish translator of Euclid long ago warned teachers of the 
 dangers lurking around this question. In the first of his 
 introductory Dissertations, he gives good advice for leading 
 a pupil to attain an exact and adequate concept of an 
 angle, and especially deprecates any association of angles 
 and arcs, averring himself at this stage " afraid to meddle 
 with circular arches, lest we should conjure up a prejudice 
 which we might want art afterwards to layP 
 
 In more than one instance — old Roger Ascham's sage 
 counsel anent teaching Latin comes to mind — modern 
 tyros in pedagogics would have done better to consult wise 
 predecessors than to follow every fad of educational milli- 
 ners as they vie with each other in designing latest fash- 
 ions. In many of our high schools it would be difficult to 
 find a pupil who knows exactly what an angle is ; and not 
 impossible to find some who would speak of an arc as 
 equal to, or half of, an angle. 
 
 213. Definition. — One series of magnitudes of the 
 same kind is proportional to another series of magnitudes 
 
CRITERIA OF PROPORTIONALITY. 129 
 
 of the same or of a different kind, corresponding one-to-one 
 to the first series, when the ratio of any two of the one 
 series equals the ratio of the corresponding two of the 
 other series. 
 
 Tliis is the direct meaning of the statement tliat one 
 series of magnitudes is proportional to another series. 
 Criteria sufB.cient to prove this relation Avill be discussed 
 presently. 
 
 214. Too much prominence is commonly given to the 
 case where each series consists of two magnitudes. Of 
 course two magnitudes are proportional to two others, 
 when a ratio of the one pair equals the corresponding ratio 
 of the other pair. 
 
 215. Criteria sufficient to prove the relation of propor- 
 tionality are often, and on high authority, set forth as defi- 
 nitions of proportionality. Of course there is no error in 
 this ; but it appears to create confusion. I believe it is 
 the principal explanation of the not uncommon opinion 
 that the true doctrine of proportionality is unteachable to 
 high-school pupils. 
 
 216. Euclid's definition of equality of ratios affords the 
 usual criterion of proportionality : Two series of magni- 
 tudes will be proportional provided that, if any equimulti- 
 ples of a corresponding pair of magnitudes one in each 
 series be taken, and any equimultiples whatsoever of any 
 other corresponding pair be taken, then the multiple of the 
 first magnitude in one series is greater than, equal to, or 
 less than the multiple of the second of the same series, 
 according as the multiple of the first taken of the other 
 series is greater than, equal to, or less than the multiple 
 of the second of that series. 
 
 217. That these requirements are capable of being ap- 
 
130 NUMBEE, AND ITS ALGEBRA. 
 
 plied as a test, may be shown by the following case : Rec- 
 tangles of equal bases are proportional to their altitudes 
 upon those bases. 
 
 Here a series of surfaces (vide § 229), corresponding one- 
 to-one to a series of lines, is declared proportional to the 
 latter. It is so, for the ratio of any two of the surfaces 
 equals the ratio of the corresponding two of the lines ; be- 
 cause, if upon sects equal to the given base two rectangles 
 be constructed whose altitudes are any multiples of the 
 altitudes of any two of the given rectangles, the rectangles 
 so formed are respectively the same multiples of the ori- 
 ginal rectangles. Thus equimultiples of a corresponding 
 pair, one in each series, and any equimultiples of a second 
 corresponding pair, have been taken. Also, if the altitude 
 of one of these trial rectangles be greater than the altitude 
 of the other, the first rectangle is greater than the second ; 
 and if equal, equal ; and if less, less. Therefore, the ratio 
 of any two of the series of rectangles equals the ratio of 
 the corresponding two of the altitudes ; that is to say, the 
 rectangles are proportional to the altitudes. 
 
 Note that all this is regardless of commensurability of 
 the altitudes or of the rectangles. 
 
 218. Euclid's criterion has been objected to because it 
 is required that the conditions be satisfied for any, that is 
 all, multiples ; and it is impossible to try all primary num- 
 bers. This objection is not valid, though there may be 
 cases which require a searching test in order to avoid error. 
 For example : Consider the numbers, 4 and 3, and 5 and 4. 
 
 ]\Iultiplying the two antecedents each by 6, and the two 
 consequents each by 9, we get 24, 27 ; 30, 36 — where 
 24 < 27, and 30 < 36. Making multiples in like manner 
 with 6 and 7, we get 24, 21 ; 30, 28 — where 24 > 21, and 
 
CRITEIIIA OF PROPORTIONALITY. 131 
 
 SO > 28. ISTevertheless, 4 and 3 are not proportional to 
 5 and 4. Thus we see that the criterion may be satisfied 
 for certain multiples, and yet not satisfied ; for it demands 
 that the excess or defect be on the same side for all mul- 
 tiples under the stated conditions. In the example cited, 
 if Ave use 10 and 13 for multipliers, Ave get 40, 39 ; 50, 
 52 — Avhere 40 > 39, but 50 < 52. 
 
 Of course, Avhere the question concerns the jDroportion- 
 ality of four given numbers there is no occasion to apply 
 any general criterion ; but the relation may be tested im- 
 mediately by a comparison of the ratios. Thus, if 4, 3 ; 5, 
 4 are proportional 4/3 equals 5/4; but 4/3 does not 
 equal 5/4, because 4/3 - 5/4 = 16/12 - 15/12 = 1/12. 
 
 219. Alternative criteria, especially adapted to test pro- 
 portionality in many cases Avhich arise in geometry and 
 physics, are presented and their adequacy established, on 
 page 93 of Halsted's Synthetic Geometry (John Wiley and 
 Sons, NcAV York) : — 
 
 Tavo series of magnitudes Avhicli correspond one-to-one, 
 are proportional (that is to say, the ratio of any tAvo of the 
 first series equals* the ratio of the corresponding two of 
 the second series) provided (1) If any tAvo of the one 
 series are equal, so are the corresponding two of the other 
 series ; and (2) To the sum of any tAvo of the One series 
 corresponds the sum of the corresponding tAvo of the other 
 series. 
 
 For example : — The intercepts made by a system of 
 parallel straights upon one transversal are proportional to 
 the intercepts made upon any other transversal : for if any 
 two intercepts on one transversal are equal, so also are 
 
 * Fide Section 83 (6). 
 
132 NUMBER AND ITS ALGEBRA. 
 
 the corresponding two on another transversal ; and to the 
 sum of two on one transversal corresponds the sum of 
 the corresponding two on the other transversal. 
 
 Again, consider arcs of the same or equal circles and 
 their chords. Arcs are not proportional to their chords, 
 because, although if two of the arcs are equal their chords 
 are equal, yet to the sum of two arcs does not correspond 
 the sum of their chords. 
 
 220. For the continuous magnitude, Space, the scientific 
 fundamental unit is the metre, which is the sect between 
 two marks on a metal bar preserved at Paris. The sect 
 is to be taken when the bar is at the temperature of melt- 
 ing ice. This temperature has the advantage of being 
 readily fixed ; but a point so far from ordinary working 
 temperatures requires the correction of all observations of 
 objects not iced, and coefficients of expansion need to be 
 accurately known for all substances employed. The origi- 
 nal (1799) French standard metre is a platinum bar 
 end-standard about 1 inch wide and i inch thick. End- 
 standards are objectionable because they can be observed 
 only by contact, and attrition at the ends is inevitable. 
 The new standard of the International Metric Commission 
 is a line-standard of platino-iridium, about 40 inches long 
 and 0.8 inch square, grooved on four sides so that its 
 section is between an X and H form. This gives rigidity 
 and a surface in the axis of the bar to bear the lines of the 
 standard. 
 
 This standard is preserved at the International IMetric 
 Bureau at Paris, where the most refined methods of com- 
 parison are provided for, and which is supported and di- 
 rected by seventeen nations. 
 
 The legal theory of the Metric System of Units is : — 
 
PHYSICAL UNITS. 133 
 
 (1) The standard metre, with decimal fractious and mul- 
 tiples thereof. (2) The litre (declared to be a cube of 0.1 
 metre edge), with decimal fractions and multiples. (3) 
 The kilogram (defined as the weight in vacuum of a litre 
 of water at 4°C.), with decimal fractions and multiples. 
 
 No standard litre exists, all liquid measures being fixed 
 by weight. 
 
 When established in 1799 the metre was supposed to be 
 one ten-millionth of the terrestrial quadrant through Paris. 
 It differs from this fanciful value by about ^oVo- 
 
 The merits of the metric system of units were briefly 
 discussed in Section 31. 
 
 221. The fundamental units for the measurement of 
 physical magnitudes, chosen by the Units Committee of 
 the British Association, and unquestionably the most sci- 
 entific ever agreed upon, are the centimetre, gram, and 
 second. The system is known as the C.G.S. sj^stem. For 
 details of its application to all branches of physical sci- 
 ence (e.g., to electricity) the student is referred to Pro- 
 fessor Everett's Units and Physical Constants, Macmillan 
 and Co. 
 
134 ^*u:MBER a>'d its algebra. 
 
 XIV. ^Mathematics 
 
 222. rormal thouglit, consciously recognized as such, is 
 the means of all exact knowledge ; and a correct under- 
 standing of the main formal sciences, Logic and !Mathe- 
 maticSj is the proper and only safe fovmdation for a scientific 
 education. 
 
 The origin and nature of the truths of the formal sci- 
 ences are not so recondite as they are often made to appear. 
 The validity of Eeason is the sole postulate. Mathematical 
 truths are discovered as the results of rational operations 
 upon certain elementary concepts determined by the defini- 
 tions with which the science begins. The operations are 
 not capricious, nor is their nature arbitrary. They are not 
 empty words, but realities — not '• material "' realities, but 
 all the more real. For example, numbers, as we have seen, 
 are not concrete things ; and as soon as we forget that they 
 are the products of rational processes, we at once fall into 
 error and confusion. Such confusion is most prominent 
 in concepts of zero and infinity. A vague concreting of 
 infinity is often observable, even among those who do not 
 make a like mistake with any other number. Because In- 
 finity as a concrete is inconceivable, the number infinity 
 is commonly spoken of as inconceivable, and a prevalent 
 opinion regards finite numbers as the only ones we can 
 reason about. Charles S. Pierce, eminent as logician and 
 mathematician (and mastery of both sciences is requisite 
 to authority in either), says, "I long ago showed that finite 
 collections are distinguished from infinite ones only by one 
 circumstance and its consequences ; namely, that to them 
 {the finite) is applicable a peculiar and unusual mode of 
 reasoning called by its discoverer, DeMorgan, the 'syllo- 
 
IKFlNItY. 135 
 
 gism of transposed quantity.' . . . DelVIorgan, as an actu- 
 ary, might have argued that if an insurance company pays 
 to its insured on an average more than they have ever paid 
 it, including interest, it must lose money. But every mod- 
 ern actuary would see a fallacy in that, since the business 
 is continually on the increase. But should war, or other 
 cataclysm, cause the class of insured to be a finite one, the 
 conclusion would turn out painfully correct after all. . . . 
 If a person does not know how to reason logically, and 
 I must say that a great many fairly good mathematicians 
 — yea, distinguished ones — fall under this category, but 
 simply uses a rule of thumb in blindly drawing inferences 
 like other inferences that have turned out well, he will, 
 of course, be continually falling into error about infinite 
 numbers. The truth is, such people do not reason at all. 
 But for the few who do reason, reasoning about infinite 
 numbers is easier than about finite numbers." * 
 
 In regard to infinitesimals (the word is simply the Latin 
 ordinal form of infinity), and contending opinions concern- 
 ing the methods of the Infinitesimal Calculus, it may be 
 remarked that, under the true doctrine of continuity and 
 limits, infinitesimals are presupposed, and that there can 
 be no reason except expediency to shun them in the differ- 
 ential calculus. And since they are indispensable for the 
 integral calculus, Mr. Pierce is probably right in his view 
 of the proper procedure of the Avhole discipline, when he 
 says, in the paper quoted above, '■'■ as a mathematician, I 
 prefer the method of infinitesimals to that of limits, as 
 far easier and less infested with snares." At all events^ 
 any avoidance of infinitesimals as absurdities, or as offer- 
 
 * "Law of Mind," Monht, July, 1892. 
 
136 NUMBER AND ITS ALGEBRA. 
 
 ing obstacles to sound and lucid reasoning, is unneces- 
 sary. 
 
 223. Mathematics has often been characterized as the 
 most conservative of all sciences. This is true in the 
 sense of the immediate dependence of new upon old re- 
 sults. All the marvellous new advancements presuppose 
 the old as indispensable steps in the ladder. It is on this 
 account that " there is no royal road " to mathematics. 
 This inaccessibility of special fields of mathematics, ex- 
 cept by the regular way of logically antecedent acquire- 
 ments, renders the study discouraging or hateful to weak 
 or indolent minds. In reality similar demands are made 
 by every science ; but elsewbere they are not so imperious, 
 so uncompromising. It is possible for one who has not 
 mastered fundamental knowledge in the sciences of phi- 
 lology, history, biology, physics, chemistry, etc., to nurse 
 the delusion of proficiency and comprehension of advanced 
 problems ; but mathematics is inviolable against such vain 
 assaults. Instant and conscious is the curb upon her vota- 
 ries of inadequate knowledge. 
 
 The modern tendency to dangerously narrow specializa- 
 tion within the bounds of one science is, also, more surely 
 checked in mathematics than elsewhere. The attempt 
 was made in mathematics as in the other sciences ; but it 
 has been restrained. Professor Felix Klein remarked at 
 the opening of the Mathematical and Astronomical Con- 
 gress at Chicago, in 1893 : " When we contemplate the de- 
 velopment of mathematics in this nineteenth centur}^, we 
 find something similar to what has taken place in other 
 sciences. The famous investigators of the preceding pe- 
 riod were all great enough, to embrace all branches of 
 mathematics. . . . With the succeeding generation, how- 
 
MATHEMATICS. 137 
 
 ever, tlie tendency to specialization manifests itself. . . . 
 Such conditions are unquestionably to be regretted. . . . 
 I wish on the present occasion to state and to emphasize 
 that in the last two decades a marked improvement from 
 within has asserted itself in our science with constantly 
 increasing success. The matter has been found simpler 
 than was at first believed. It appears indeed that the dif- 
 ferent branches of mathematics have actually developed — 
 not in opposite, but in parallel directions, that it is possible 
 to combine their results into certain general conceptions. 
 ... A distinction between the present and the earlier 
 period lies evidently in this : that what was formerly 
 begun by a single master mind, we now must seek to ac- 
 complish by united efforts and co-operation." 
 
 224. Another trait of mathematics which renders it at- 
 tractive to some minds and repellant to others, is its self- 
 sufficiency, its isolation, its independence of other sciences. 
 But it must never be forgotten that mathematics is ever at 
 the service of other sciences ; and it is for them to so 
 formulate their problems as to make them susceptible of 
 mathematical treatment. Indeed, in some instances, the 
 difficulties which balk a thorough investigation of certain 
 physical phenomena consist in the mathematical problems 
 encountered in the solution of numerical equations, the 
 summation of numerical series, etc., which the skill of 
 experimenters has succeeded in deducing from the phenom- 
 ena in question. Thus, on the one hand, the physicist or 
 economist is unceasingly occupied in attempting to express 
 the relations of the entities with which he deals, as some 
 numerical function known to be within the reach of mathe- 
 matical reduction, — for so compendious is the language of 
 Algebra, that theoretically most quantitative and many 
 
138 NUMBER AND ITS ALGEBRA. 
 
 qualitative relations are somehow expressible as numerical 
 relations. On the other hand, the algebraist is constantly 
 striving to bring more and more algebraic forms within the 
 powers of his analysis. By their joint labors the confines 
 of knowledge are steadily widened. 
 
 225. It may be helpful to offer a definition of INIathe- 
 matics, not in the sense of final delimitation, but in order 
 to afford a clear notion of what is meant by subjects or 
 relations capahle of viathematlcal treatment. I cannot do 
 better than quote Professor George Chrystal in his article 
 on " Mathematics," Encydopcedia Britannica, ninth edi- 
 tion, who makes the f olloAving definition : " Any concept * 
 which is definitely and completely determined by means 
 of a finite number of specifications, say by assigning a 
 finite number of elements, is a mathematical concept. 
 Mathematics has for its function to develop the conse- 
 quences involved in the definition of a group of mathe- 
 matical concepts. Interdependence and mutual logical 
 consistency among the members of the group are postu- 
 lated, otherwise the group would either have to be treated 
 as several distinct groups, or would lie beyond the sphere 
 of mathematics." 
 
 226. Examples of concepts completely determined by a 
 finite number of specifications are familiar. On the other 
 hand, horse, tree, gold, becmty, love, are examples of non- 
 mathematical concepts. Of course, Kumber may be ab- 
 stracted from these, or any other separate objects of 
 thought or sense-perception, and Number is a mathemati- 
 cal concept ; but the concept of a number of trees is not at 
 all the concept of the trees. Again, the form of an irregu- 
 
 * I have taken the liberty of changing the word " conception " to 
 concept three times in this passage. 
 
ELEMENTAL CONCEPTS. 139 
 
 lar piece of wood canuot be determined by a finite number 
 of specifications, and its form therefore cannot be mathe- 
 matically treated (its weight of course could). But if from 
 this irregular piece of wood a sphere be turned, its form 
 is specified by stating that it is a sphere, and giving the 
 length of its radius. This illustrates at once the bounda- 
 ries of mathematics, and the relation of mathematics to 
 the arts. 
 
 227. ]Mensuration is an important function of mathemat- 
 ics ; but it occupies too prominent a place in some notions 
 of the subject-matter of the science. I have already ((7f. 
 Introduction, p. 13, and § 12) referred to the mistake of 
 assigning the origin of Number to measurement. Nor is 
 the prevalent notion that mathematics is the '^ science of 
 quantity " correct. Projective geometry in the purity of 
 its recent development is displayed as a mathematical 
 treatment and method well-nigh void of quantitative rela- 
 tions, and dealing for the most part with qualitative re- 
 lations of spacial manifoldnesses.* 
 
 228. When we reach elementary concepts we always 
 find that they cannot be defined except in cognate terms. 
 Such elemental concepts are quality, one, many, space, time, 
 and the interrelated concepts, whole, part, more, less, equal, 
 quantity. All that can be done in the way of defining such 
 concepts, is to exhibit the phenomena from which they 
 have been abstracted, and the processes of abstraction ; 
 and then, for purposes of exact expression, make the def- 
 inition in cognate terms. A good, short definition of 
 quantity according to a standard dictionary (the Century) 
 is : " The intrinsic mode by virtue of which a thing is more 
 
 * See a work on projective geometry by Dr. Halsted, just now in 
 press. 
 
140 NU5IBER AND ITS ALGEBRA. 
 
 or less than another." {Mode = system of relationship.) 
 It is plain that some things exist in this mode; that is, 
 possess quantity, which are not magnitudes, or manifold- 
 nesses, in the mathematical sense. For example, beauty is 
 quantitative, is more or less ; but in no proper sense can it 
 be added to itself so as to double. It is a loose figure of 
 speech to say, the beauty of one thing is tAvice that of 
 another, as is at once apparent, should we go on to say 
 that it was eleven times that of some other. 
 
 229. ]\Iany words have been used to denote the charac- 
 teristic relation of a mathematical concept to its elements. 
 Magnitude and quantity are the familiar terms. In the 
 preceding discourse I have employed the former term ; but 
 for reasons both of intrinsic propriety, and less ambiguity 
 owing to irregular usage, the word manifoldness, which 
 has lately come into use,* is perhaps the most fitting 
 term ; though manifoldness is also used to denote a group 
 of correlated magnitudes differing in kind. 
 
 Quantity and magnitude are each used in two respec- 
 tively synonymous senses. Either magnitude or quantity 
 may be found defined for mathematical purposes as '' any- 
 thing which may be added to itself so as to double ; " and 
 yet the same writer may be found speaking of the magni- 
 tude of some such magnitude, or the quantity of some 
 such QUANTITY. The ancient, and still universally cur- 
 rent, categorical sense of quantity seems to me to render 
 it the more appropriate terra for the sense of the italicized 
 words in the phrases cited ; and therefore, by exclusion, 
 
 * Cf. the article on " :Matliematics" by Prof. Chrj'stal, above referred 
 to; also the article on "Measurement" by Sir Robert Ball, Royal As- 
 tronomer for Ireland. 
 
MANIFOLDNESS. 141 
 
 magnitude should be confined to the sense of the capi- 
 talized words. 
 
 Manifoldness in one sense is entirely synonymous with 
 magnitude in the use I have made of the latter. But be- 
 sides a single totality, manifoldness often means a single 
 system of different totalities, and the difference may be in 
 kind. Thus, a line is a manifoldness, so is a surface, so is 
 an angle ; yet the system of lines, along with the angles 
 and surface determined, which we call a triangle, is also 
 termed a manifoldness. 
 
 Now, a triangle, in the sense of the whole figure, is not a 
 magnitude ; its surface is a magnitude, its sides are magni- 
 tudes, and its angles are magnitudes. The word triangle 
 often plainly means exclusively the surface of the triangle, 
 and the abbreviation is legitimate where there is no danger 
 of confusion ((7/*. § 217) ; but if triangle means the entire 
 definite system of surface, lines, and angles, then, clearly, 
 a triangle is not a magnitude, but a system of different 
 magnitudes. But a triangle, in this sense of the whole 
 figure, the system of magnitudes, three sects, three angles, 
 and one surface, is called a manifoldness. Such manifold- 
 nesses have been termed discrete ; but this is a totally dif- 
 ferent sense of discrete, from its meaning in any statement 
 that a single magnitude is discrete, e.g., Primary Number 
 is a discrete magnitude. Discrete is the antithesis or an- 
 tonym of co7itimious. Most magnitudes are continuous ; 
 number, time, and space are the great continuums, with 
 which mathematics has most to do. If mayiifoldness is to 
 be used in this double sense, it is necessary to distinguish 
 the meanings by some adjectives ; and discrete is not a 
 good term for the latter sense. Homogeneous and dis- 
 parate would not be abusive terms. I shall use them. 
 
142 NUMBER AND ITS ALGEBRA. 
 
 230. Xumber is tlie very web of mathematics, the mani- 
 foldness iipon which are woven investigations concerning 
 all other manifoldnesses whatsoever. All other manifold- 
 nesses are even fundamentally determined (as will pres- 
 ently appear) by means of Xumber ; but Number determines 
 itself. 
 
 Geometry cannot even apparently proceed without arith- 
 metic. Euclid makes the formal connection in his fifth 
 book ; but there is a more primary and essential connec- 
 tion. We have considered the error of seeking geometric 
 definitions of number, particularly negative, neomonic, and 
 complex number. But the tables are entirely turned Avhen 
 we consider that geometric or any other manifoldnesses are 
 defined in some very fundamental properties by means of 
 number. 
 
 231. Most text-books on stereometry set forth that all 
 solids have three dimensions, length, breadth, and thick- 
 ness. But what does this exactly mean ? What is the 
 length, breadth, and thickness of a pyramid, a rough stone, 
 a bunch of grapes ? Xo solids, except cubes or right paral- 
 lelopij)eds, clearly determine three principal directions in 
 which length, breadth, and thickness may be discerned. 
 
 The dimensions are clearly and sharply defined only by 
 considering the number of specifications necessary and suf- 
 ficient to fully determine any element. Thus, solid space 
 regarded as point aggregates is tri-dimensional, because, 
 given three concurrent straights or planes, as ground of 
 reference, three numbers are necessary and sufficient to 
 determine any one point-element, distinguishing it from all 
 others. 
 
 Note also that the space of our experience is four-dimen- 
 sional if regarded as an assemblage of geodesic lines, 
 
MANIFOLDNESSES. l-i3 
 
 because in that case four numbers are required to deter- 
 mine one element. 
 
 232. Manifoldnesses, homogeneous or disparate, are one- 
 dimensional, two-dimensional, etc., (or one-fold, two-fold, 
 etc.), according as in the totality or system considered, one 
 number, or two numbers, etc., are necessary and sufficient 
 to determine and distinguish any particular element in the 
 homogeneous totality, or in the system. 
 
 The distinction between homogeneous and disparate mani- 
 foldnesses must not be confounded with that between 
 continuous and discrete manifoldnesses. A homogeneous 
 manifoldness is either continuous or discrete ; a dispa- 
 rate manifoldness is a system of homogeneous (continuous 
 or discrete) manifoldnesses. Disparate denotes a system 
 of manifoldnesses differing in kind ; that is, such as could 
 not be compared with one another {vide § 207), e.g., the 
 surface, lines, and angles of a triangle. As already said, 
 most homogeneous manifoldnesses are continuous. Pri- 
 mary Number is the conspicuous discrete magnitude with 
 which we have to do. 
 
 According to different standpoints, the same manifold- 
 ness may be of various dimensions. 
 
 233. Examples. — A straight line regardless of position, 
 time, temperature, probability, the totality of all spheres dis- 
 tinguished, not in respect of position, but solely in regard 
 to size or quantity, are one-fold manifoldnesses. All such 
 are homogeneous, for of course no one-fold manifoldness 
 could be disparate. 
 
 The assemblage of points on a plane, the sphere as sur- 
 face {Cf. latitude and longitude), are two-fold manifold- 
 nesses. 
 
 Space as an assemblage of points is a tri-dimensional 
 
144 NUMBER AND ITS ALGEBRA. 
 
 manifoldness. A triangle considered without reference to 
 position' (because it may be completely determined in vari- 
 ous ways by assigning three elements) is a triple disparate 
 manifoldness. 
 
 The totality of all spheres each to be completely deter- 
 mined is a four-fold manifoldness. 
 
 Since a plane quadrilateral is completely determined 
 when five elements are known, it is a quintuple or five- 
 fold disparate manifoldness. 
 
 A plane w-gon in like manner is a (2 ?i — 3)-fold dis- 
 parate manifoldness. 
 
 234. There are two general methods in the mathemati- 
 cal investigation of manifoldnesses. They are called the 
 synthetic, or synoptic method, and the analytic method. 
 The analytic method is mainly numerical; the synthetic 
 deals directly with the magnitudes considered, and only 
 unavoidable numerical relations are involved. Of course 
 there is no sharp line of demarcation, and the two methods 
 yield identical results. 
 
 In geometry metrical relations are in general more readily 
 investigated by the analytic ; descriptive properties by the 
 synoptic method. 
 
 235. The synthetic method is peculiarly fitted to pure 
 geometry, but this is not its only field. Ever since Rie- 
 mann's epoch-making dissertation, Ueher die Hypothesen 
 u-elclte (lev Geometrie zu Grunde Ueyen, 1854, synoptic 
 methods have been applicable to w-fold manifoldnesses ; 
 and the applications to Statistics and Physics are familiar. 
 
 236. In mathematics all analytic methods employ an 
 algebra {vide § 20 et seq.) ; but it is the Algebra of Number 
 which is the most highly developed and powerful instru- 
 ment of such methods of research. It is to the study of 
 
ALGEBRAIC FORM. 145 
 
 this organized and compendious instrument of numerical 
 expression that these lectures are introductory. Plainly 
 the first step to the understanding of the algebra of num- 
 ber is to understand the nature and laws of number. It 
 is hoped that these lectures have been a fairly adequate 
 guide and stimulus to this step. After mastering what 
 may be called the vocabulary of the language (proficiency 
 in this matter has been assumed), the next step is to 
 grasp the idea of algebraic /b;v/i. In the study of Algebra 
 this should be the main standpoint. It is only by follow- 
 ing out the problems which arise in a systematic study 
 of algebraic form tljat the modern developments of pure 
 algebra, or its applications to geometry, can be rightly 
 comprehended. 
 
 237. In conclusion, I may say, in reference both to this 
 little work, and to any text-book which may engage j^our 
 attention, that if a mathematical treatise is worth reading 
 at all, it is worth re-reading, and reading backwards and 
 forwards, and in special topics. As Professor Chrystal 
 says in the preface to his Text Book of Algebra, ''When 
 you come on a hard or dreary passage, pass it over ; and 
 come back to it after you have seen its importance or 
 found the need for it further on." 
 
 "UNIVE 
 
 C.4-.. '"' 
 
146 NUMBER AND ITS ALGEBRA. 
 
 XV. Some Theorems axd Problems. 
 
 238. Every primary number is a multiple (§ 83) of one 
 and of itself : if it has no other submultiple, it is called a 
 prime number ; if it has another submultiple, it is called 
 comjiosite. 
 
 If one primary number is a submultiple of each of two 
 or more others, it is called a common suhmidtij)le. 
 
 Primary numbers (prime or composite) with no common 
 submultiple other than unity, are said to be prime to each 
 other. 
 
 239. Theorem. — Every composite primary number can 
 be resolved into factors which are positive integral powers 
 of prime numbers. 
 
 Every primary number less than a composite number 
 either is, or is not, a submultiple of the latter : let a be 
 the least primary number (> 1), that is a submultiple of 
 the composite number, A. 
 
 Then A = ax. If x be also a multiple of a, x = oij, and 
 A = a^y. Einall}' A = a"^i(, Avhere u is either 1, or prime 
 to a, and either prime, or a multiple of some prime > a 
 and < A, say, b. 
 
 In like manner ti=h"v, where v<Cn- and v^cPw, 
 where to < v, and so on. 
 
 Clearly the process must end with 1 ; therefore 
 
 A = aI"h'"cP . . . , 
 
 where a, b, c . . . are prime numbers. 
 
 It will be seen below that this resolution can be effected 
 in only one way ; also, that positive integral powers of 
 prime numbers are prime to each other. 
 
 240. Understanding the numerical symbols as represent- 
 
HIGHEST COMMON SUBMULTIPLE. 147 
 
 ing integers (positive or negative), and extending* the 
 meaning of the term multiple to include the relation of 
 one number to another if the ratio of the former to the 
 latter be integral, then : — 
 
 If ^ is a multiple of a, any multiple of A is a multiple of 
 a. Tliis is obvious. 
 
 Also, if A and B have a common submultiple, m, then 
 Ax -J- By is a multiple of m. 
 
 For, say, A = pm. and B = qm ; 
 
 then Ax -|- By = xpm -\- yqvi, 
 
 and therefore, distributing the right-hand member, 
 Ax -J- By = m (xp + yj). 
 
 From these two theorems is deduced a means of finding 
 the highest f common submultiple (h. c. s.) of two or more 
 integers. 
 
 For, if A = pB -\- c, the h. c. s. of A and B is the h. c. s. 
 of B and c. To prove this it is necessary and sufficient to 
 show — 
 
 (1) Every submultiple of B and c is a submultiple of A 
 and B. 
 
 (2) Every submultiple of A and i? is a submultiple of B 
 and c. 
 
 (1) As just shown, every submultiple of B and c is 
 a submultiple of p) B -\- c, that is of A; therefore, every 
 submultiple of B and c is a submultiple of A and B. 
 
 (2) Since A = pB ^ c, c = A — pB. Therefore, again, 
 every submultiple of A and i? is a submultiple of ^ — ^; B, 
 
 * A violent extension {Cf. definition, § 83) ; but custom is a tyrant, 
 and brevity tempting. Some such term as co-multiple would adequately 
 distingui-sh this relation, e.g., of 12 to — 3. 
 
 t The term highest is employed in order to avoid contradictory uses of 
 " greatest" in regard to negative numbers. {Cf. § 242.) 
 
148 
 
 NUMBER AND ITS ALGEBRA. 
 
 that is of c, and consequently every submultiple of A and B 
 is a submultiple of B and c. 
 
 Thus, to find the h. c. s. of A and B, where yl > J5, we 
 have, by successive divisions, — 
 
 A = 2>B -\- c . . . Avhere c <i B, 
 
 c = rd -\- e 
 
 where d <i c, 
 where e < rf. 
 
 where m < /, 
 
 Avhere n < ?», and may be 1, 
 where, if ?i = 1, «• = vh. 
 of ^ and B ; for the h. c. s. of A 
 
 Iz = vl -\- m 
 
 I = vm -\- n 
 
 and, finally, vi = loi 
 
 Whence, 7i is the h. c. s 
 and B = h. c. s. of B and e = h. c. s. of c and d = . . . = 
 h. c. s. of m and n. But m = wn, and therefore 7i is the 
 h. c. s., since n can have no submultiple higher than itself. 
 
 If n = 1, A and B have no common submultiple but 
 unit}', and are prime to each other. 
 
 For example, find the h. c. s. of A = 2911 and B = 1763. 
 
 The calculation may be compared with the foregoing as 
 
 follows : — 
 
 ^ = 1763)2911 =.4(1 =i> 
 1763 
 
 = 1148)1763(1 = q 
 
 (^ = 615) 1148(1 = r 
 615 
 
 e = 533 ) 615 ( 1 = 5 
 533 
 /* = 82 ) 533 (6 = t 
 492 
 
 whence, y = 41 is the h. c. s. 
 
 ^ = 41)82(2 = ?^ 
 82 
 
 
 
RELATIVE PRIMENESS. 149 
 
 It must be discerned that the essence of this process is 
 merely that the quotients be integral, and the moduli (^vide 
 § 198) of the dividends be in decreasing order, for qualita- 
 tive distinctions are ignored ; -[- 4, for instance, being in- 
 differently the h. c. s. of 8 and 12. 
 
 In accordance with these considerations the process may 
 be abbreviated in various ways. If convenient, remainders 
 may be negative, and any submultiple of a divisor evi- 
 dently prime to the dividend, or submultiple of dividend 
 prime to divisor, may be cast out. The above calculation 
 might have been abbreviated thus : — 
 
 *&' 
 
 Since neither 
 3 nor 5 is a sub- 
 multiple of 1763, 
 15 may be cast 
 out of 615. 
 
 1763)2911(2 
 3526 
 
 - 15) -615 
 
 41)1763(43 
 164 
 
 123 
 123 
 
 Every common submultiple of A, B, C . . . is a common 
 submultiple of A and B, and therefore of vi, the li. c. s. of 
 A and B. Consequently, to find the h. c. s. oi A, B, C . . . , 
 find the h. c. s. of m and C, and so on. 
 
 241. It follows from the preceding discussion, that, if a 
 and b be prime to each other, any common submultiple of 
 aN and b must be a submultiple of N. 
 
 Also, if a be a submultiple of bN and prime to b, it is a 
 submultiple of N. 
 
 Also, if a be prime to I, m, n . . ., it is prime to their 
 product, Imn ; and consequently if a, b, c . . . , be each 
 prime to all of I, vi, n . . . , the product, abc . . . , is prime 
 to the product, Imii .... 
 
150 NUMBER AND ITS ALGEBRA. 
 
 In particular, if a be prime to h, «" is prime to Z/™. This 
 is true, of course, when a and h are prime numbers ; that 
 is to say, positive integral powers of prime numbers are 
 prime to each other. 
 
 Moreover, an integer can be resolved into factors which 
 are powers of prime numbers in only one way. {Vide 
 § 239.) 
 
 For, if two resolutions be possible, let alcd = hnnrs. 
 
 Then ahcd is a multiple of Z; but since / is a positive 
 integral power of a prime number, it is prime to each of a, 
 h, c, d, except one which is a not less power of the same 
 prime number; and there must be such a one, or I could 
 not be a submultiple of cd>nl ; — say a is this one. Again 
 hnnrs is a multiple of a, and it follows as before that I 
 must be a multiple of a. But if a is a multiple of I, 
 and / of a, a = I. Likewise three more of vmis must 
 respectively equal h, c, and d, and tlierefore the unpaired 
 factor must be 1. 
 
 242. The lowest common multiple* of two integers equals 
 their product divided by their highest common submultiple. 
 
 For if A = sx and B = sy, where s is the h. c. s. of A and 
 B, then AB = s^-xy. But s, x, and y are prime to each 
 other, and therefore sxy is the 1. c. m. of A and B, — and 
 
 AB 
 
 sxy = 
 
 * In respect to primary numbers, the term least common multiple 
 means exactly what it says; hut in reference to both positive and nega^ 
 tive integers a variance in the meaning of the term "least" is to he 
 noted, such as was remarked in the foot-note of Section 240 concerning 
 "greatest." Indifferent alternatives— one positive, the other negative 
 — are always considered in the highest common submultiple, and the 
 lowest common multiple of two integers. 
 
LOWEST COMMON MULTIPLE. 151 
 
 Therefore, to find the lowest common multiple of two 
 integers, we have the rules : — • 
 
 Divide their product by their h. c. s. ; or 
 
 Divide either by their h. c. s., and multiply the other by 
 the quotient ; or 
 
 Divide each by their h. c. s., and take the product of the 
 quotients and the h. c. s. 
 
 Any one of these three rules may in a special case be the 
 most convenient. 
 
 The 1. c. m. of more than two integers is the 1. c. m. of 
 the 1. c. m. of the first two and the third, and so on. 
 
 243. Plainly (symbols meaning integers) a = xh -\- r in 
 an infinite variety of ways ; for x may be fixed arbitrarily 
 and r found, so that r = a — xh. But important special 
 cases arise if (/, h, and x are positive, and r restricted : — 
 
 (1) When r < b. 
 
 (2) When, though r is negative, mod r < mod h. (Vide 
 § 198.) 
 
 In both cases a = xb -\- r in only one yvciy. 
 
 (1) Tf xb be the greatest multiple of b, not > a, then 
 r = a — xb, where r < b. Nor could there be a second 
 resolution under the same conditions, else xb -f- *' would 
 equal x' b -}- r ', and therefore r — r' = (x' — x) b, and there- 
 fore )• — r' would be a luultiple of b, — an impossibility, 
 since r and r', being each less than b, r — r' is less than b. 
 
 (2) If xb be the least multiple of b not < a, then 
 a — xb = r, where r is negative, but mod /■ < mod b ; and 
 the resolution is unique as before. 
 
 In these cases r is called the least positive remainder 
 and " least " * negative remainder of a with respect to b. 
 
 * Cf. foot-notes to Sections 240 and 242. 
 
152 NUMBER AND ITS ALGEBRA. 
 
 Least remainder^ unqualified, is to be understood in the 
 former sense. 
 
 Obviously a is prime to h if the least remainder of a 
 with respect to h does not vanish, and not prime if it does 
 vanish. 
 
 244. Let the student prove, if the least remainders of x 
 and y with respect to z be equal, a; — ?/ is a multiple of z, 
 and inversely. 
 
 245. When the ratio a; /y is not integral, a- /y is said to 
 be essentially fractional, or briefly, fractional. 
 
 li a jh = c I cl when ay h and c > d, prove that the frac- 
 tions, reduced to form n -{- r/h, where r < h, must have 
 their integral and fractional parts equal separately. 
 
 246. Pi^ove : 11 A [ B = a / h and a / h is at its lowest 
 terms (i.e., a prime to h), then A = na and B = nh. 
 
 247. Prove that, using only positive remainders in the 
 process of finding the h. c. s. of two positive integers, A and 
 B, every remainder equals i {Ax — By), where x and y 
 are positive integers, and the upper sign goes with the 1st, 
 3d, etc., and the lower with the 2d, 4th, etc., remainders. 
 
 Also, if a and h be prime to each other, positive integers 
 can always be found such that xa — yh = J^l. 
 
 It is obvious that these numbers, when determined, will 
 be prime to each other, for by Section 240, 1 is a multiple 
 of every common submultiple of x and y. 
 
 248. Prove : 
 
 (1) If X prime to y, {x + y)" and {x — ?/)" have h. c. s. 
 
 not > 2". 
 
 (2) If X prime to y, x" + ?/" and a;" — y" are prime, or 
 
 have h. c. s. = 2. 
 
 (3) If X prime to y, a; + y and x^ -\- y"^ — oty are prime, 
 
 or have h. c, s. = 2 or 3. 
 
RADICAL SURDS. 153 
 
 (4) The difference of the squares of two odd integers is 
 
 a multiple of 8. 
 
 (5) The difference of the squares of two consecutive 
 
 integers equals their sum. 
 
 (6) The product of three consecutive even integers is a 
 
 multiple of 48. 
 
 (7) The sum of the squares of three consecutive odd 
 
 numbers and 1 is a multiple of 12, but never 
 of 24. 
 
 (8) The product of the cubes of three consecutive inte- 
 
 gers is a multiple of their sum. 
 
 249. At several points in preceding chapters, it has been 
 taken for granted that the operation of evolution upon 
 many integers results in essentially surd or incommensu- 
 rable number ; that is to say, that no fraction can possibly 
 be the required root — • although fractions approximating 
 the surd as nearly as desired can be obtained. Fractional 
 number is still discrete, fractions are continuous through 
 surds. (Vide U 94, Sl-82.) 
 
 To demonstrate these propositions, it is enough merely 
 to consider that no power of an essentially fractional num- 
 ber can be an integer. For, ii x / y is a fraction in its 
 lowest terms, x is prime to i/, and therefore, by Section 241, 
 any power of x is prime to any power of i/, and consequently 
 any power of a-/ ^ is still essentially fractional. 
 
 For example : Obviously no integer is the square root 
 of 7, but some number greater than 2 and less than 3. But 
 this number is no fraction, for, as just shown, no power 
 Avhatsoever of any essentially fractional number can be an 
 integer. Thus, it is proved that the familiar process of ap- 
 proximate calculation of roots of such integers is absolutely 
 interminable. (Moreover, the endless decimal fraction 
 
154 NUMBER AND ITS ALGEBRA. 
 
 obtainable can never form a repeating period of figures — 
 (vide § 284). 
 
 In tliis Avay it is plain that no integers except the series, 
 1, 4, 9, 16 . . . , (the squares of 1, 2, 3, 4 . . . , and called 
 " square numbers ") can have any but incommensurable 
 square roots ; that the cube roots of all integers but 1, 8, 
 27, 64 . . . (1^, 2% 3^, 4^ . . .) are incommensurable, and 
 so on. 
 
 250. For proof of the proposition : The number of prime 
 integers is infinite (see Euclid, IX, 20). 
 
 251. Attentive perusal of the following sections Avill 
 bring out a general distinction (correct apprehension of 
 which is highly important) between the applications of a 
 confusingly similar terminology to individual numbers and 
 to analytical functions of such numbers, — the distinction 
 between algebraic form, and particular numerical values. 
 
 For example, note the distinction between " exactly di- 
 visible " applied to algebraic forms, and stibmu/fiple applied 
 to numbers. It is not even true that the highest common 
 submultiple of two niimbers which are obtained from the 
 substitution of particular numbers for the numerical sym- 
 bols in two analytical functions, is the same number that 
 would be obtained by substituting the same values in the 
 highest common factor of the two algebraic forms ; nor 
 would it be possible to make a definition of the algebraical 
 highest common factor, so that this should be true. 
 
 The investigations immediately following apply only to 
 
 integral functions. 
 
 A 
 
 252. If A and 1> be integral functions of x, and -- = Q, 
 
ALGEBRAIC DIVISION. 155 
 
 () is a stirpal but not necessarily an integral function of x. 
 ( Vide § 169.) 
 
 When Q is an integral function of the variables, A is 
 said to be exactly* divlsihle by D. 
 
 When ^ (x) cannot be transformed into an integral func- 
 tion, it is said to be essentially fractional, or fractional. 
 
 An essentially integral function cannot be identically 
 
 {vide § 40) equal to an essentially fractional function. 
 
 A 
 In — = (), if all the functions are integral, the degree of 
 
 Q is the degree of A minus the degree of D. 
 
 If the degree of yl ? 9 less than the degree of D, Q is 
 essentially fractional. 
 
 253. If ^ = PD + P (all integral functions) P is ex- 
 actly divisible by P or not, according as A is exactly 
 divisible l)y P or not. 
 
 For, since A = pp _]_ p 
 
 A_ 
 P 
 
 PP 
 
 + 
 P 
 
 R 
 
 = P 
 
 + 
 
 R 
 P 
 
 A 
 
 
 P -- 
 
 P 
 
 , 
 
 
 P 
 
 
 
 P 
 
 
 
 A ■ ■ R . . 
 
 therefore, as — • is integral or not, — is integral or not. 
 ' i> ° 'Z> " 
 
 254. Fundamental theorem in algebraic division : — 
 
 ~ = Pm-n + -^ , where m > n, 
 
 and where R vanishes or is an integral function of degree 
 < n. 
 
 * There are not the same ohjections to this phrase, as against terming 
 one numher exact and another inexact. Of- Sections 1 and 80. 
 
156 NUMBER AND ITS ALGEBRA. 
 
 (The subscripts represent the respective degrees of the 
 functions.) 
 
 Arranging A,„ and X*,, according to descending powers of 
 the variable, we would get by dividing the first term of A^ 
 by the first term of i>„ , 
 
 ■^m — 1^ -^ -'^H ~r -"'TO — 1 (at utmost). 
 
 Dividing by D„ gives 
 
 /f 7? 
 
 B ~ '"-"^ J) • 
 
 Moreover, this result can occur in only one way ; for, if 
 
 — = PH — —^=F'A — ^, where the functions satisfy the 
 
 D 1) D . ■ 
 
 foregoing conditions, then Avould 
 
 -D r,, R' -K /i 1 i- ^- T>/ , -K from each\ 
 P—P'= by subtracting P -| ■ ; 
 
 D D \ D member j 
 
 -pt r> 
 
 and therefore P~P'^= — ., which is impossible; since 
 
 7?' 7? 
 
 P—P' is an integral function, and cannot be inte- 
 
 D 
 
 gral, since the degrees of P' and R are less than the degree 
 
 of D. 
 
 1 P 
 
 255. If — = P-\ — - , the degrees and character of the 
 
 functions being as stated in the preceding section, P is 
 called the integral quotient and R the remainder (^par 
 excellence). 
 
 Plainly, the necessary and sufficient condition for " ex- 
 act divisibility " is that the remainder vanish. 
 
 256. Example of the " long rule " for division of inte- 
 gral functions : — • 
 
 Divide \ x^ -f ^\ j-ir + ^\ /f ^'J i ^ + i 2/- 
 
ALGEBRAIC DIVISION. 157 
 
 The work may conveniently be arranged thus : — 
 
 1 
 
 4 
 
 1 
 4 
 
 
 -X^lJ 
 
 + 
 + 
 
 1 i/3 
 
 1.2 U 
 
 -I 
 
 xhj — 
 
 
 - tV y 
 
 
 
 
 \^y 
 
 1 ,,3 
 
 T^ y 
 
 ^ a;2 — 1 a-y + 1 y2 
 
 + 0, = 7? ; 
 
 therefore the latter function is an exact divisor of the 
 former. 
 
 257. The special case of the division of the general inte- 
 gral function of the ?«th degree by a binomial divisor of the 
 1st degree, of form x — a, is of extreme importance. 
 
 If the student will closely examine his results in the 
 operation {ax'' -\- hx''-~'^-\- ex"-- -\- . . . Ix + A} ^ (x — a), 
 he Avill discover the following general laws : — 
 
 The degrees of the terms of the integral quotient regu- 
 larly descend. 
 
 The first coefficient of the integral quotient is the first 
 coefficient of the dividend. 
 
 Each subsequent coefficient is the next preceding multi- 
 plied by a, -{- the corresponding (in orxler, not degree, of 
 term) coefficient in the dividend. 
 
 The remainder, if it does not vanish, may be obtained 
 precisely as if it were a subsequent coefficient. 
 
 Care must be taken to supply by zeros any lacking terms 
 in a particular case. 
 
 Example. — Divide 3 a;^ + 5 x^ — 9 x -\- 11 by x — 2. 
 
 -|-3-|-0-|- 5 — 9-|-ll... (Coefficients of dividend). 
 
 I Ci _i -| <> _|_ O < _[_ KA (Each preceding number in third line 
 
 ~^ ~r '^ r ~r ... multiplied by 2). 
 
 -1-3 + 6+17+25+61 
 
158 NUMBER AND ITS ALGEBRA. 
 
 Therefore integral quotient = S x^ -\- 6 x^ -}- 17 x -\- 25, 
 and remainder = -f Gl. 
 
 258. The general process which disj)layed the foregoing 
 theorem proves * also the following : — 
 
 Remainder Theorem. — If any integral function of x 
 be divided by x — a, the remainder is the same function of 
 a as the dividend is of x. That is to say, the remainder 
 may be obtained by substituting a for x in the dividend. 
 
 Thus, in the example above, 
 
 61 = 3 (2y + 5 (2)2 _ 9 (2) + 11. 
 
 If the divisor were x -\- 2, we need only consider x -\- 2 
 = X — (— 2), where a is — 2. 
 
 For instance, {3x* -\- ox^ — 9 x -\- 11} -^ (.y + 2) 
 
 gives +3+0+5-9 + 11 
 
 _ 6 + 12 - 34 + 86 
 _|_ 3 _ 6 + 17 - 43 + 97 
 
 Therefore integral quotient = 3 x^ — 6 ./■- + 17 ic — 43, 
 and remainder = + 97. 
 
 And, in accordance with the remainder theorem, 
 
 97 = 3 (- 2)-' + 5 (- 2)2 - 9 (- 2) + 11. 
 
 259. The remainder theorem is clearly proved in the 
 process of dividing the general function of x of the ?ith 
 degree by x — a; but on account of its fundamental im- 
 portance in the theory of equations, I transcribe an inde- 
 pendent proof : — 
 
 Let <f>,^ (x) be an integral function of x of the nth 
 degree ; then 
 
 * Proved iudependently in Section 359. 
 
KEMAINDER THEOllEM. 159 
 
 ^" ^' ' = Qn-\-\ where R does not involve x ; 
 
 X — a X — a 
 
 therefore, (/>„ (.r) = <?« _ i (a? — «) + ^• 
 
 But this equation, being an identity, holds Avhen cc = a, 
 when, (/)„ («) = + -^j 
 
 which, remembering the meaning of <^„ (o), is merely an 
 algebraic statement of the ''remainder theorem." 
 
 The full meaning of this statement must not be missed ; 
 for it at once declares the remainder when </> (x) -^ (x — a), 
 and the value of ^ (x) when x — a: — the statement is 
 <^ {(I) = E. 
 
 Thus in the preceding examples 
 
 3 ^^ + 5 U-- — 9 a; + 11 = Gl when x = 1, and = 97 when x 
 _ _ 2. 
 
 This method of calculating the value of an integral 
 function of x for a particular value of the variable gen- 
 erally saves work in comparison with direct substitution. 
 
 260. Prove : If an integral function of x, ^ (x), be di- 
 vided by aa; -f J, / / \ 
 
 261. Note that if <^ (x) vanishes for any value of a-, 
 say V, then upon division by a; — r, 7? = 0, and inversely. 
 
 262. If «!, ^2, «3 . . . a^ be r different values of x, for 
 Avhich an integral function of x of the ?ith degree vanishes 
 where n > r, then 
 
 (|) (.7-) = {x — «,) {x — Qa) ... {x. — a;)f„ _ r (x), 
 where/,, _^ (x) is an integral function of x of the (n — r)th. 
 degree. And when 71 = r, 
 
 ^ (x) = (x — «i) (x — a^) . . . (x — a„) /, (;x), 
 where /, (x) must be a constant. (But see § 268.) 
 
160 NUMBER AND ITS ALGEBRA. 
 
 263. An integral function of any number of variables 
 is called, homogeneous when the degree of every term is 
 the same ; e.g., ax -\- hi/, or ax^ -\- hxy -(- y^. 
 
 264. Prove : I-f each variable in a homogeneous function 
 of the nth degree be multiplied by iii, the result is the 
 same as if the function were multiplied by ?>i". 
 
 Also : The product of two homogeneous functions of the 
 mt\\ and nth. degrees respectively, is a homogeneous func- 
 tion of the (jn -\- ?^)th degree. 
 
 Let this last theorem always be applied to test the accu- 
 racy of distribution of a jjroduct of homogeneous functions. 
 
 265. An integral function is called symmetrical with 
 respect to its variables when their interchange leaves the 
 function unaltered. Several approximations to symmetry 
 have received special names ; e.g., if a function be not 
 altered except in sign by interchange of variables, it is 
 called alternating. Functions are often both homogeneous 
 and symmetrical. 
 
 266. From the definition, it follows that the sum, differ- 
 ence, product, or quotient of two symmetrical functions is 
 a symmetrical function, — a useful rule in testing and 
 abbreviating algebraic work. 
 
 Since symmetry concerns only coefficients, general forms 
 are easily written down. 
 
 Write down the general integral symmetrical function of 
 X, y, z of third degree. 
 
 267. Since the coefficients are independent of the varia- 
 bles, if two integral functions are equal as an identity (vide 
 § 40), and the coefficients of one are determined by any 
 means, then these coefficients are determined once for all. 
 
 This theorem has been called (not very happily) the 
 Theorem of Undetermined Coefficients. 
 
UNDETERMINED COEFFICIENTS. 161 
 
 It is most useful even in its elementary applications to 
 integral functions, and becomes an indispensable instru- 
 ment in dealing Avith infinite series.* 
 
 For example : Required the product 
 
 (^ + 2/ + -) (^'' + y' + -' - ^y - ^- - y^) ; 
 
 we can write down by symmetry 
 
 (x + y + z) (x^ -\. ,f ^ z^ - xy - xz - yz) = A {x^ + 7/ 
 
 + ^3) + ^ (xhj + x-'z + xy^ + xz'' + y-'z + zhj) 
 
 -\- Cxyz. 
 Since this identity must hold for all values of x, y, z, 
 
 taking 
 
 X = 1, y = 0, z = 0, gives 1 = A. 
 
 Putting x = 1, y = 1, and ;v = 0, and using the discov- 
 ered permanent value of A, we have 
 
 * Since the whole matter of infinite series is postponed to subsequent 
 studies, this subject cannot be entered upon further than to caution tlie 
 student that in sucli an algebraic statement as 
 
 =^ \ -\- X -\- X- -\- x^ + x^ . . . it is never to be understood that 
 
 1 — X 1 — X 
 
 equals, or even approximates, the infinite series unless the series be con- 
 vergent; i.e., unless the sum continually approximates a definite limit. 
 Evidently if a; > 1, it would be absurd to take the above statement into 
 consideration for a moment. In fine, such statements are understood as 
 plainly concerning only such values of the variable as make the series 
 convergent. Compare various obvious ellipses common in all expression 
 of thought. 
 
 Let this be the student's reply to the cavilling he may sometimes hear 
 upon this matter. 
 
 Of course if the remainder is added at any point, the expression is an 
 
 identity, always true ; e.g., = 1 + x + x'^ -\ — ; thus, if x = 10, 
 
 1 — X 1 — X 
 
 we have = 1-1-10-1- 100 -{- 1000 + more and more untrue, the more 
 
 numerous the terms; but if the remainder be added at any stage, we 
 have a true equation: 
 
 _L^ 1 + 10 + 100+ T<^^ ^ + ffl + ^o _lM = _ 1. 
 
162 NUMBER AND ITS ALGEBRA. 
 
 1 ^1 + 1; (^i + 1 - 1; = 1 (1 + 1; + i>Hi + 1; ; 
 
 or 2 = 2 + 2i?5 
 
 therefore i? = 0. 
 
 Using these determined values of A and B and x = 1, 
 9/ = 1, z = 1, we get, 
 
 (1 + 1 + 1)0 = 1(1 + 1 + 1)+ C; 
 therefore C = — 3. 
 
 Therefore the required product is a;^ + y^ + z^ — 3 xj/z. 
 
 268. Returning now to Section 262, it is plain from Sec- 
 tion 267 that/o (x) must equal the coefficient of x" in <^ (x). 
 
 The " if " in Section 263 must be carefuPy noted. It 
 has not been shown that n integral, 1st degree functions 
 can be found, such that 
 
 4>n {x) = A- {x — «i) {x — «o) (.r — 03) . . . (x. — a,). 
 
 This question is also deferred to subsequent studies in 
 Theory of Equations, when it will be proved that every 
 equation has a root, and that every equation of the ni\\ 
 degree has n roots (all of Avhich need not be different). 
 
 By a root of the equation ^ (x) = is meant a value of 
 the variable which causes the function to vanish ; that is, 
 satisfies the equation ^ (x) = 0. AVe have seen (§ 261), 
 that when an integral function of x is exactly divisible by 
 x — a, a is, 2^, root of the equation, and inversely. 
 
 The general formal proof that " every eqiiation has a 
 root " must be postponed ; yet Ave might almost assume 
 the fact as implicit in the Principle of Continuity (§ 103). 
 Assuming this, we can prove that every integral equation 
 of the «th degree has n roots, and no more. 
 
 Let a be one root ; then, 
 
 ,/; (.r) = (x- — «)/„_i(.r); 
 
ROOTS OF INTEGRAL EQUATIONS. 163 
 
 but/„_i(.r) must have a root, and so on for u roots (some 
 of which might be repeated), and a constant factor, fg (x) 
 (vide § 262). Moreover, ^„ (ic) cannot have more than 7i dif- 
 ferent roots, because if any integral function of ?/th degree 
 vanish, for more than ii values of the variable, it must 
 vanish identically; that is, for all values of x (i.e., every 
 coefficient in form cr" -j- ^*"~' + ex""- -\- . . . -\- dx -{- k 
 must be zero). For, let 
 
 <|)„ (a:) = a (x — i\) (x — r.) {x — r^) . . . (x — ?'„) ... (1). 
 
 Now, if possible, let ;• be another value of the variable 
 for which the function vanishes. Since (1) holds for all 
 values of x, then 
 
 ^ (.'') = f* ('■ — ''i) ('' — ^'2) ('■ — ''3) ..•('• — r„) = ; 
 
 and since each '' r" by hypothesis is different, a must be 
 zero. 
 
 But a is the coefficient of the .r" term in ^„ (x). In this 
 way, step by step, each coefficient in (f),, (x) is shown to 
 vanish if more than n values of the variable satisfy the 
 equation (|)„ (./;) = 0. 
 
 For example, x'^ — (x -\- 1) (x — 1) — 1 is of the 2d degree, 
 yet plainly it vanishes for 0, 1, 2, — and therefore for all 
 values of x. 
 
 269. The preceding section affords an independent proof 
 of the theorem of undetermined coefficients, Avhich may be 
 re-stated as follows : — 
 
 Any function of x is transformable into an integral func- 
 tion in only one way. For, if possible, suppose the two 
 following different integral functions, derived from the 
 same function, as identities, and therefore equal for all 
 values of x : 
 
164 NUMBER AND ITS ALGEBRA. 
 
 No generality is lost in regarding them as of the same de- 
 gree ; for, if not, it would simply mean that the coefficients 
 concerned were zero. Subtracting the right-hand member 
 froin each, we get 
 
 {a — ai) cc" + {h — h^) a;"-' + (c — c^) a;"-^ -|- . . . 
 
 for more than n values of x. 
 
 Therefore, a — a^ = 0, b — h^ = 0, . . . k — k^ = ; 
 that is to say, a = a^, b = l/^, . . . k = k^. 
 
 270. Professor Chrystal remarks at this point in his 
 Text Book of Algebra, " the danger with the theory we 
 have just been expounding is not so much that the student 
 may refuse his assent to the demonstration given, as that 
 he may fail to apprehend fully the scope and generality of 
 the conclusions." Their utility cannot fail to be more and 
 more highly appreciated by the attentive student. 
 
 271. (1) Determine the value of Z; such that 2 a'^ — 8 
 x"^ -{- 1 X -\- k shall be exactly divisible by x -{-2. By 
 Section 259, the remainder to division by x — (— -) is 
 2 (_ 2)3 _ 8 (- 2)2 + 7 (- 2) + /.• = - 62 + k. 
 
 If the function is to be exactly divisible by a- -J- 2, this 
 remainder must vanish, or — G2 -|- ^^ must be zero ; i.e., 
 k = 62. 
 
 (2) In like manner the question of exact divisibility 
 may be readily tested : 
 
 /y»n j,n 
 
 AVhen ■— , 72 = y" — ?/" = ; the division is always 
 
 x — y 
 
 exact. 
 
GIVEN ROOTS, TO FORM EQUATION. 165 
 
 When - ~^^^i , i? = (_ y)" — ?/" — 0, if n be even, = 
 ^' + y 
 — 2 y if n be odd. 
 
 When — J^^ , ^ = _j/» -}-?/" = 2 ^" ; the division is 
 
 ■^ - y 
 
 never exact. 
 
 When ^^li.^", R = (— yY -]- y'' = 0, if ?i be odd, = 
 2 ?/" if n be even. 
 
 (3) If A -~ D gives remainder R, and B -^ D remainder 
 i2', show that AB -=- 2) and RR' -^ i>* give identical re- 
 mainders. 
 
 (4) Observe that, in the proposition that an equation of 
 the ?ith degree has n roots and no more, we prove that 
 any finite number has n nth roots and no more, — ■ all of 
 which need not be different. 
 
 To find these roots of any number, a requires the solu- 
 tion of the equation »•" = a, or ic" — a = ; that is to say, 
 the factorization of a-" — a in the form, 
 
 {x — ri) (a- — 7\) {x — I's) . . . (x — ?•„). 
 
 (5) We are also enabled to make an integral equation of 
 given roots. Thus, to form an equation whose roots are 0, 
 + 1, — V2, — 1, we have simply to write, 
 
 Cx (x — l)(x-\- V2) (x ~\-l) = 0, 
 
 where C is any constant we please; e.g., thts equation, 
 taking C = 1, is 
 
 a-" + V2 a;3 _ a;2 _ ^2 a- = : 
 or, taking C = V2, 
 
 V2 a;-* + 2 a;=5 ^ V2 a;- - 2 x = 0. 
 
166 NUMBER AND ITS ALGEBRA. 
 
 272. Having thoroughly explained the meaning of " ex- 
 act" divisibility as api^lied to the division of one integral 
 function by another, the sense in which one function 
 is termed the highest common factor of two others is 
 apparent : — 
 
 The integral function of x of highest degree which 
 ''exactly divides" each of two or more integral functions 
 of X, is their highest common factor (h. c. f.). (But see 
 § 251.) 
 
 If the given functions are easily resolvable into factors 
 which are integral functions of the first degree, the h. c. f. 
 is readily taken by inspection; since it is simply the 
 product of such of these first degree factors as are com- 
 mon, each raised to the lowest power in which it occurs 
 in either of the given functions. 
 
 Otherwise we may proceed very much as in Section 240, 
 since if A = BQ-]- R, the h. c. f. of A and B is the h. c. f. 
 of B and R : proved by considering Section 253. 
 
 Consequently, to find the h. c, f. of two integral func- 
 tions of X, A and B, where the degree of B is less than 
 that of A, we may 
 
 divide A hy B so that A = BQ^-^- Ry 
 
 and divide B by ^i so that B = R^Q^ -\- Rr,, 
 and divide R^ by R^ so that R^ = R^Q^ -\- R., etc., until 
 
 Rn-i/Rn gives R„_i = R„ ^„+i + R, 
 where R vanishes, or is of zero degree, that is, a constant. 
 In the latter case, there is no h. c. f. ; in the former R^ is 
 the h.c.f. For by Section 253, A and B, B and ^i, 
 ^1 and R2, . . . R„_i and R„, are of descending degree, 
 and all have the same h. c. f., and no factor of higher 
 degree than R^ can exactly divide R„. In case R is a 
 constant, R„_i and R„ have no common exact divisor other 
 
HIGHEST COMMON FACTOR. 167 
 
 than R ; that is to say, there is no common clivisoi- in the 
 sense intended, although any constant will ''exactly di- 
 vide " any integral function in the sense of giving an 
 integral quotient ; i.e., remainder zero. ( Vide § § 255, 
 256.) 
 
 It follows from the nature of this process of finding the 
 h. c. f. that at any stage either divisor or dividend may 
 be multiplied, or divided by any integral function of the 
 variables (of course including any constant), provided it is 
 certain that the factor so introduced or removed has no 
 factor in common with the other functions. Any function 
 which is obviously a common factor of both dividend and 
 divisor at any stage may be removed from each, provided 
 we multiply the h. c. f. afterwards resulting by the re- 
 moved common factor. In dealing with factors which are 
 constants, regard "factor" in the sense of common suh- 
 multlple of the coefficients. Finally, it must be observed 
 that the recurring operations are, on account of such modi- 
 fications as have been ascribed, not divisions in the ordi- 
 nary sense ; for the " division " may, if convenient, be 
 arrested at any stage (while the remainder is yet of higher 
 degree than the divisor), to remove common, or introduce 
 independent, factors. 
 
 273. (1) Find h. c. f. of 9 .^^ - 30 .r* + 4.j .7-^+ 24 x and 
 15 x^ - 30 X* — 90 a;3 + 60 x^ -f 195 a; + 90. (Problem 
 worked out on page 168.) 
 
 Of the originally removed factors, 3 x and 15, 3 is com- 
 mon ; therefore, cc^ — 3a:;^-|-3a;-|-l must be multiplied by 
 3 to obtain the h. c. f., 3 a-^ + 9 a;^ -f 9 a; + 3. 
 
168 
 
 NUMBER AND ITS ALGEBRA. 
 
 o 
 CO 
 
 
 »o 
 
 5^ 
 
 00 
 tH 
 
 + 
 
 o 
 
 I 
 
 O 
 
 + 
 
 Kl 
 
 cr. 
 
 tH 
 
 + 
 
 ?^ 
 O 
 CO 
 
 + 
 
 O lO 
 C5 
 
 + 
 
 cr, 
 
 + 
 
 + 
 
 
 « 
 
 CO 
 
 CC 
 
 
 T-f 
 
 
 + 
 
 
 ^ 
 
 f^ 
 
 a 
 
 o 
 
 CO 
 
 
 + 
 
 1 
 
 CI 
 
 N 
 
 ^i 
 
 ^i 
 
 (M 
 
 C^l 
 
 T-t 
 
 T-l 
 
 + 
 
 1 
 
 » 
 
 CO 
 
 ^ 
 
 w 
 
 CO 
 
 CO 
 
 1 
 
 1 
 
 1 
 
 1 
 
 5S 
 
 « 
 
 CO 
 
 TJH 
 
 1 
 
 + 
 
 o 
 
 US 
 
 H 
 
 H 
 
 CO 
 
 CO 
 
 CC 
 + 
 
 CI 
 
 + 
 
 eo 
 
 O 
 
 o 
 
 o 
 
 CO 
 
 O 
 
 CO 
 
 5S 
 O 
 
 CO 
 
 CO 
 
 1—1 
 
 T-l 
 
 + + 
 
 H 
 
 H 
 
 ^ 
 
 "^ 
 
 o 
 
 LO 
 
 + + 
 
 cq 
 
 C) 
 
 H 
 
 f^ 
 
 '^ 
 
 -tth 
 
 lO 
 
 O 
 
 + + 
 
 M 
 
 CO 
 
 H 
 
 f^ 
 
 CC 
 
 CC 
 
 T-i 
 
 T-i 
 
 
 CO CO 
 
 <3! 
 
 lO 
 
 O' -^ 
 
 
 1—1 
 
 tH 
 
 
 O lO 
 
 
 + + 
 
 1 
 
 1 
 
 1 1 
 
 H 
 
 H H 
 
 f^ 
 
 « 
 
 ?^ H 
 
 ^ 
 
 O C5 
 
 '^ 
 
 c^ 
 
 O (M 
 
 oi 
 
 T-t CO 
 
 C^l 
 
 T-t 
 
 c^i CO 
 
 + 
 
 + + 
 
 1 
 
 1 
 
 1 1 
 
 CI 
 
 (N 
 
 CI 
 
 « 
 
 1 • 1 
 
 f^ 
 
 c^ 
 
 H 
 
 f^ 
 
 ^ CI 
 
 >o 
 
 C^ 
 
 <M 
 
 CO 
 
 '^ ^ 
 
 "d^ 
 
 
 1 
 
 1 
 
 CO t- 
 
 
 t 
 
 CO 
 
 CO 
 
 1 1 
 
 
 i^ H 
 
 H 
 
 H 
 
 CC TO 
 
 
 O CC 
 
 GO 
 
 ^ 
 
 S H 
 
 
 tH -H 
 
 
 
 O' CC 
 
 + 
 
 
 + 
 
 1^ 
 
 + 
 
 -* CO 
 + + 
 
 o 
 
 CO 
 
 CO 
 
 CO 
 
 ?^ ~ 
 
 CO 
 
 
 
 o 
 
 o o 
 
 1 
 
 1 1 
 
 C^ tH 
 
 CO CO 
 
 
 U3 U5 
 
 * 
 
 
 C5 
 
 CO CO 
 
 
 
 
 O' -^ -* 
 
 + 
 
 tH 
 + 
 
 ?^ 
 
 + 
 
 TO 
 
 CO 
 
 + 
 
 CI 
 
 CO 
 
 + 
 
 CO 
 
 CO 
 
 o +^ 
 
 CO 
 
 CO 
 
 w 
 
 ^ 
 
 
 >i 
 
 a:> 
 
 
 ^ 
 
 
 
 bf 
 
 o 
 
 
 
 
 
 'S 
 
 O 
 
 
 
 
 
 
 
 
 •^ 
 
 a 
 
 
 4^ 
 
 o 
 
 
 
 
 
 P! 
 
 5 
 
 
 r^ 
 
 ® 
 
 
 •^ 
 
 
 
 S-l 
 
 
 
 O 
 
 £ 
 
 
 r-^ 
 
 
 CO 
 
 
 
 r^^ 
 
 e8 
 
 -f^ 
 
 ^ 
 
 aj 
 
 
 
 +3 
 
 «t-i 
 
 5i 
 
 CO 
 
 o 
 
 r^* 
 
 c 
 
 
 •?o 
 
 
 
 ;:> 
 
 c 
 
 S 
 
 
 o 
 
 
 "S 
 
 V 
 
 tji 
 
 J 
 
 
 ^ 
 
 +-» 
 
 
 
 
 ^ 
 
 <— " 
 
 o 
 
 <a 
 
 '^^ 
 
 +^ 
 
 
 c 
 
 ^ 
 
 <D 
 
 
 OJ 
 
 ai 
 
 rb 
 
 >3 
 
 (H 
 
 
 e3 
 
 ffl 
 
 "U 
 
 o 
 
 
 (1 
 
 a 
 
 ^ 
 
 g 
 
 on 
 
 
 
 a 
 
 r^ 
 
 1 
 
 
 S 
 
 1 
 
 a 
 _a 
 
 
 f-l 
 
 
 
 
 o 
 
 
 > 
 
 © 
 
 "o 
 
 -S 
 
 S 
 
 o 
 
 V 
 
 aa 
 
 a> 
 
 s 
 
 (!) 
 
 
 ci 
 
 -^ 
 
 +3 
 
 M 
 
 
 _C 
 
 2 
 
 a 
 
 o 
 
 +2 
 
 a 
 
 T— 1 
 
 
 ^ 
 
 >S 
 
 I — 1 
 
 o 
 
 r^ 
 
 +J 
 
 -CS 
 
 _>5 
 
 i 
 
 -to 
 
 
 
 O) 
 
 "Si 
 
 M 
 
 m 
 
 
 
 
 -*J 
 
 
 o 
 
 
 
 9 
 g 
 
 s 
 
 c3 
 
 
 O 
 
 a 
 
 .2 
 '•5 
 
 * 
 
 -73 
 C 
 e8 
 
 o 
 
ALGEBRAIC PKIMENESS. 
 
 169 
 
 (2) What is the necessary relation among their coeffi- 
 cients in order that aar- + hx + c and cx^ + te + a may 
 have an exact common divisor of the first degree ? 
 
 ax 
 
 /^_.^)' 
 
 ax"^ -\- hx -|- c 
 
 ax 
 
 2 I «(« + g) 
 
 X 
 
 ^^_a{a+c)\ 
 
 cx'^-\-l>x-{-a 
 
 cx^-\ x-\ — 
 
 a a 
 
 (dividing by 
 coef . of X gives) 
 
 , fl. + C 
 
 V' 
 
 c a 
 
 
 c — a — c-\ ^ — ! — -^Jt. 
 
 Now, if the functions have an exact common divisor of 
 the first degree, It must vanish ; therefore the condition 
 
 c — a — c A ^^ ■ — ^— = U. 
 
 ^ b'' 
 
 Whence - ab"" + a {a ^ cf = ; 
 
 or, dividing by «, {a -\- cy = b- ; 
 
 or ■ a -{- c = ^ b. 
 
 274. From Sections 253 and 272 it is plain that the 
 h. c. f . of three integral functions is the h. c. f. of the h. c. f. 
 of two and the third, and so on. 
 
 275. Integral functions which have no common exact 
 divisor are said to be algebraically prime. Many condi- 
 tions of algebraic primeness, more or less analogous to 
 
170 NUMBER AND ITS ALGEBRA. 
 
 those estaBlished concerning absolute numerical primeness, 
 might be investigated. 
 
 276. The precise meaning of algebraic lowest common mul- 
 tijjle of two integral functions will now easily be understood 
 as the integral function of lowest degree exactly divisible 
 by both. 
 
 Let A = HQ and B = HQ', where the symbols repre- 
 sent integral functions, and H the h. c. f. of A and 7>. Let 
 M be any common multiple of A and B ; then — 
 
 M = AE, where E is an integral function of x. 
 
 Therefore M = HQE. 
 
 But Jf is an algebraic multiple of i? = HQ'\ 
 
 ^1 r M HQE QE , QE , , 
 
 therefore ■ = — - — = -^ — , where -i — must be an m- 
 
 HQ' HQ' Q' Q' 
 
 tegral function. But since Q and Q' are by hypothesis 
 algebraically prime, E jQ' = X or E = (i'X, where X 
 is integraL Consequently 
 
 Jf = HQE = HQQ'X. 
 
 But this last algebraic statement (translated) declares 
 
 that any common multiple of A and B is the product of 
 
 H, Q, and^', as defined, and some other integral function, 
 
 X. Hence, 31 is of the lowest possible degree when X is 
 
 of zero degree ; that is, a constant. And since constants 
 
 are not altogether ignored in the desired result, M is the 
 
 ''lowest common multiple" when X=l; that is to say, — 
 
 A P 
 since HQQ' = ■ — -, the 1. c. m. of two integral functions is 
 
 the quotient of their product divided by their h. c. f. 
 
 Alternative rules are similar to those for single numbers 
 {vide § 242). The algebraic 1. c. m. has neither the practi- 
 cal nor the theoretical importance of the algebraic h. c. f. 
 
SCALE NOTATION. 171 
 
 277. The fundamental theorem in the expression of 
 numbers, in a notation such as our common system, is tlie 
 following : — 
 
 Any primary number may be expressed finitely, and in 
 only one way in the form ■ — • 
 
 ^o + ^1 (''i) + ^2 OVo) + ^3 (rir^rs) -j- . . . c„ ()\r.r3 . . . r„), 
 
 where Vi, Vo, r^, . . . r„ is a series of primary numbers, 
 unrestricted except that there are as many as may be 
 required, and Co< rj, c^-C r.2, c.2< r^, etc. 
 
 For if / be any primary number, dividing I liy r^ gives 
 
 (1) / = Co + Qi)\ where CoKn; 
 and dividing Q^ by r^ gives 
 
 (2) ' Qi = r^ -{- Q.,r-2 where <?i< r^; 
 and dividing Q.^ by rg gives 
 
 (3) Q. = C.2 + (^3/-3 where Co< rg; 
 and so on until Qn< r„^i is reached. 
 
 (1) and (2) give 
 
 ^ =- <'o + (''i + Q'^ro) i\ = ^0 + ^I'-i + Q-ir^ro ; 
 and svibstituting for Q.2 from (3) 
 
 1= Co-\- c^r^ + c, (ri7\) + Qsnr.Vs ; 
 
 and so on until Q„_i = <"„_i + (?„'"«> where, writing 6'„ for 
 Q„, we have 
 
 I = Co -\- c^ (ri) + C2 (i\r.2) + (-3 (t\r^rs) + . . . r„ 
 (7\r^r3 . . . ?•„). 
 
 Moreover, this expression is unique for the same series 
 of r's, because, if not, let 
 
 ^0 + <-i'-i + ^2 QV2) + . . . = Co' + r^'r^ + e/ (}\ro) + . . . 
 
172 NUMBER AND ITS ALGEBRA. 
 
 dividing each member by ?*i, gives 
 
 -0 
 
 + q + f2»-2 +...= — + Cl' + C2''-2 + 
 
 y-l 
 
 But since Cq <ri, and Cq' < i\, by Section 245 the essen- 
 tially integral and essentially fractional parts of these 
 numbers must be equal separately; that is to say, — 
 
 £2 = £2. , or Co = Co , and so forth. 
 
 Example. — Express the number represented in our 
 notation by 200 on the scale, 7, 3, 5, 2, etc. 
 
 7 1 200 
 3 1 28 ... 4 
 5\j)_. . . 1 
 1 ... 4 
 
 since the quotient 1 is less than i\ = 2, the process ter- 
 minates, and 
 
 200 = 4 + 1 X 7 + 4 (7 X 3) + 1 (7 X 3 X 5). 
 
 Express 100 in the scale 3, 4, 5, 6, 7, etc. 
 
 278. A corresponding theorem for the expression of 
 numbers essentially fractional (§ 245) where numerator < 
 denominator (''proper" fractions), may be proved; that is 
 to say : — 
 
 — = r ~ "T" r • • • ' r»' ' 
 
 D ri i\i\ r^r^n n^'2^3 • • • »« ^ 
 
 where Jj' = {riTor^ . . . r„) D, 
 
 where d^ < i\ , d^ < i\ , etc., and where / may vanish. The 
 general proof, and demonstration that / = 0, when r, r^ is 
 ... r„ is a multiple of B, is left as an exercise for the 
 student. 
 
RADIX NOTATION. 173 
 
 Example. — In this way express 7/ 10 in scale 5, 7, 9, 
 11, 13, 15, 17, . . . 
 
 5 
 
 
 
 
 035 
 
 
 
 
 3 
 
 . . . 5 
 
 7 
 
 10)35 
 
 3 
 
 . . . 5 
 
 9 
 
 
 
 
 10)45 
 4 
 
 . . . 5 
 11 
 
 10)55 
 
 5 
 
 . . 5 
 
 evidently this cannot terminate, because j\, Vo, r^ . . . r„ 
 can never be a multiple of 10. 
 
 Therefore, 7/ 10 = | + ^-i. + ^^ + ^rj:^^, • 
 
 279. Since the scale of r's may be arbitrary, it may be 
 chosen so that all the fZ's shall be one. Decompose 7/10 
 as a sum of fractions with unit numerators. 
 
 280. When the scale of notation is constant ; that is, 
 when all the r's are equal is the important special case of 
 the foregoing Theorem. 
 
 In this case 
 
 7= c^r'^ + c^r + Coj" + c^r^ + • • • cj 
 
 1 
 
 and — = d^r-' + cLr-'- + d,r-^ + • • • dj--" + -^ 
 
 Avhere / may be zero. 
 
 Here, of course, is recognized our system of notation, 
 where r = 10. 
 
174 NUMBER AND ITS ALGEBRA. 
 
 It is our custom to omit the 7''s, whose powers are under- 
 stood from the order of the c's and c^'s, the proper place 
 being displayed by never failing to express the zero when 
 any c or d has this value. Also we omit the signs of addi- 
 tion, and write the integral series from right to left, so 
 that it may be regularly continued by the fractional series, 
 a mere point amply serving to separate the two. In this 
 way the powers of the radix or base decrease by ones from 
 left to right, thus : 
 
 Cn ''" + ^„-i ''""' + • • • ^2 '•'' + ^1 ^^ + -^0 '>'" + ^1 ^~^ + ^2 r-^ 
 -\- d^r-^, etc. ; or, omitting -f's and r's and pointing off the 
 
 281. From the condition that the c's and d's must all be 
 less than r, it is obvious that in any such notation r — 1 
 figures are required to uniquely designate the jDOssible 
 values of c's and c^'s. 
 
 It is also plain that all the rules of the decimal algo- 
 rithm apply to any other base, say 12, except that the 
 " carriages " would go by 12's instead of lO's. Of course 
 for radix 12, tAvo new digit figures would be required ; and 
 for radix 2, symbols for 1 and only could be used. Thus 
 teM on the binary scale would be 1010 ; that is, 
 1 X 2^ + X 2-2 + 1 X 2^ + X 2°. 
 
 282. Example. — Express 102305 (radix ten) on base 
 twelve 121102305 
 
 ■19 ^KOK K 
 
 12 710 ... 5 
 
 12 59 ... 2 
 
 (using a and h as digits for ten and eleven). 
 
 Therefore 102305 (r = ten) is 4 ^^ 255 (r = twelve). 
 
RADIX NOTATION. 
 
 175 
 
 Inversely, to express 4 h 255 (radix twelve) on decimal 
 base. 
 
 Consider the expression means (using our common nota- 
 tion for calculation), 
 
 4 (12)* + 4 (12/ + 2 (12/ + 5 (12; + 5 ; 
 or, performing the indicated operations. 
 
 5 = 
 
 5 
 
 60 = 
 
 5(12) 
 
 288 = 
 
 2 (12)2 
 
 19008 = 
 
 11 (12)3 
 
 82944 = 
 
 4 (12)* 
 
 102305 
 
 If the student will refer to Sections 257-259, he will 
 notice that the remainder theorem yields an easier way 
 for this calculation. (1) The problem is merely the evalu- 
 ation of the given function of twelve. We may therefore 
 write : — 
 
 4, 11, 
 
 5, 
 
 48, 708, 8520, 102300, 
 4, 59, 710, 8525, 102305. 
 
 It would be good practice to Avork^ out as follows 
 the duodecimal algorithm, on its own merits, "carrying 
 twelve — but from our fixed habit of thought this is much 
 more difficult : — 
 
 m 
 
 «|4/>255 
 a 5 b 00 . 
 
 . . 5 
 
 a 713 . 
 
 . . 
 
 a 86 . 
 
 a a . 
 
 1 . 
 
 . 3 
 
 . -J 
 . 
 
 that is, 4 h 255 (radix twelve) is 102305 (radix ten). 
 
176 NUMBER AND ITS ALGEBRA. 
 
 283. Fractions expressed in sucli notations as are under 
 discussion are called radix fractions, decimal if the base is 
 ten, duodecimal if the base is twelve, etc. 
 
 A fraction ^^ expressed as a radix fraction cannot termi- 
 nate unless iVr" is a multiple of D ; for 
 
 ±=.d,d,d, ' ^^ 
 
 JJ ^ ^ " ' r"i> 
 
 Multiplying each member by ?•" reduces the radix fraction 
 part of the right-hand member to an integer, giving 
 
 ^-_ = d^d„ds . , . d,^-\- j- , 
 
 where tZj do . . . is the integral part of the quotient ; and 
 there must be a fractional part unless lir" is a multiple 
 of D. Also if N I D is in "lowest terms," i.e., if N be 
 prime to D, it is plain that a radix fraction cannot termi- 
 nate unless ?'" is a multiple of I). Nor can ?•" be a multiple 
 of D unless it be resolvable into powers of primes which 
 are jDrime factors of ;•. For example, to express N j D (in 
 its lowest terms) as a decimal fraction, we must have 
 D = 2'' 5", where either x or y may be zero. 
 
 N . 
 284. If, when the proper fraction — in its lowest terms 
 
 is expressed as a radix fraction, the latter does not termi- 
 nate, its digit figures must repeat in a cycle of not more 
 than _D — 1 figures. For, evidently only D — 1 different 
 remainders can occur, and Avhen one recurs, the figures of 
 the quotient must repeat. Such radix fractions are called 
 repeating, recurring, or circulating. 
 
 The repeating period may begin at once, or may begin 
 after figures which do not repeat, — commonly distin- 
 guished as ^^wre and mixed circulates. The repeating 
 
RADIX FRACTIONS. 177 
 
 period is sometimes called perfect when it consists of the 
 full complement {D — 1) of figures. The repeating period 
 is denoted by dotting its first and last figures. 
 
 This subject could be better discussed in connection 
 with infinite series, and "geometrical" progression; but 
 repeating decimals occur so frequently in practice that 
 their reduction to simple fractions cannot be left in the 
 dark. 
 
 Consider — 
 
 3[lj 1 
 
 0-333333 + 3 X ]^Q6 • • • 1/S =0-3 
 
 7L1: 1 
 
 0-142857142857 + ^-TTTTTT • • • 1/7 =0-142857 
 
 <xioi- 
 
 1111- 
 
 '•'''^'+1T^^ ...1/11 = 0-09 
 
 24 [Ij 
 
 0-04133 + ^—- ... 1/24 = 0-0413. 
 
 ^ 24 X 10^ ' 
 
 The remainders, inexpressible as a radix fraction, may 
 
 be introduced at any point. I express them to avoid 
 
 discussion of infinitesimals; and if regarded as implicit 
 
 in the notation of the repeating decimals, the reasoning 
 
 in this section is exact in terms of thought familiar to 
 
 beginners. 
 
 Now 1 = 0-111 + 
 
 ^^ =0-010101 + 
 
 ^1^ = 0-001001 + 
 
 j__ = 0-00010001 + 
 
 The law is plain, and furnishes a way to transform 
 repeating decimals into simple fractions. 
 
178 NUMBER AND ITS ALGEBRA. 
 
 For example : Express 0-324 as a common fraction in its 
 lowest terms. Evidently 0-324 = 324 x 0-OOi. But 0-6oi 
 = ^iT ; therefore 0-324 = §|f = ^Yt = if- 
 
 Hence the rule: To express any pure circulating deci- 
 mal as a common fraction, write the repeating period for 
 numerator, and for denominator as many nines as there are 
 decimal places in the repeating period. ' 
 
 To find the rule for mixed circulates, consider : — 
 
 F = 0-3i8 
 
 1000 F = 318 'ISIS 4- ft ^^'^^^^6 these remainder frac- 
 
 -tf\ p< S-1 S1 S -U V ( tions, /, are absolutely the 
 
 ^^ — "^ '"■ 1 same. 
 
 therefore 990i^=315 
 
 and F=U^ = ^^. 
 
 Again consider : * — 
 
 i^=0-03G93is 
 10,000,000 F = 369318-18 +/ ) ^^'^ remainder frac- 
 
 100,000 i^= 3693-18 + /•( ['^'"f t'' ^^""^ 
 
 ,, „ — — — —^ I lutelv the same. 
 
 therefore 9900000 F = 365625 ^ 
 
 and F= r^f^-Aesji- — jis 
 
 !? 9 00000 — 35 J" 
 
 Hence the rule .- To express a mixed circulating deci- 
 mal as a common fraction, subtract the non-repeating part 
 from the whole circulate for the numerator, and for the 
 denominator write as many nines as there are decimal 
 
 * The same result may he obtained thus : 
 
 0-03693i8 = 0-0.3693 + Q-OOOOOiS 
 
 ^3fi9 3._ _| 1 V n-i«! 
 
 — TOOOOO ' 10000(7 A U lO 
 3fi93 I 1 s/18 
 
 — T05p(J + JT^h^V — TT fff f §7 = ^V 
 
REPEATING RADIX FRACTIONS. 179 
 
 places in the repeating period, followed by as many zeros 
 as there are places in the non-repeating part. 
 
 285. Inasmuch as we have seen that any integer is ex- 
 pressible in only one way in any radix scale, it is clear 
 that a common fraction in any scale of notation is expres- 
 sible as a common fraction in any other scale. Con- 
 sequently any terminating or repeating radix fraction in 
 any scale transforms into a common fraction, and therefore 
 into a terminating or repeating fraction in any other scale. 
 
 Note carefully that a terminating radix fraction in one 
 scale need not transform into a terminating radix fraction 
 in another scale, but into a terminating or repeating radix 
 fraction. 
 
 286. To transform a fraction from one scale to a radix 
 fraction in another, simply multiply by the new base, and 
 the fractional part of the product again by the new base, 
 and so on. The integral parts of these products in due 
 order are the figures of the transformation. For example, 
 to express | as a duodecimal fraction : — 
 
 3 X 12 = 4i 
 i X 12 = 6, 
 
 therefore | = 0.46 (radix twelve). 
 
 Or again, to express 0.13 as a radix fraction in the seven 
 scale : — q -l^g 
 
 7 
 
 0.91 
 
 7 
 
 6.37 
 
 7 
 
 2.59 
 
 7 
 
 4.13 
 
180 NUMBER AND ITS ALGEBRA. 
 
 here 13 recurs, and the fraction repeats this period, so, — 
 
 0.13 (radix ten) = 0-6024 (radix 7). 
 
 To prove the propriety of this process, consider a proper 
 fraction, F, and let — • 
 
 r I- r 
 
 in some new scale of notation whose base is r. 
 
 Then rF =7-, + ^^^+ . . . 
 
 r /•- 
 
 say vF = x^ -\- F, where F must be a proper fraction ; 
 therefore x^ is the integral part of rF. 
 
 Again rF = :r„ -[- ^ -f "li. _)_ 
 
 And in like manner Xn is the integral part of rF ; and 
 so on. 
 
 287. If / be any integer, and s the sum of its digits, and 
 r the radix of the scale of notation, then the remainder of 
 
 = the remainder of — ^^ — . 
 
 r - 1 r -1 
 
 For, let / = Co + ("i ?' 4- ^2 '"^ + ^ 3 '"^ + • • • <'n ^■''• 
 Subtracting the sum of the digits from each member of 
 the equation gives 
 
 / _ s = q (;• - 1; + r^ (/-'^ - 1) + ^3 (>■''- 1) + . . . c,^ (/■" - 1). 
 
 Since, by Section 271, each term of the right-hand mem- 
 ber is a multiple of (r — 1), if we divide each member by 
 (?' — 1) (or any submultiple) we get 
 
 = some integer. 
 
 r-1 r-1 " 
 
 Therefore, by Section 245. the essentially fractional parts 
 of I J r — 1 and s J r —1 must be equal. 
 
EEMAINUEUS TO NINE. 181 
 
 288. From this theorem follows the special corollary 
 that in our decimal notation any integer and the sum of 
 its digits give the same remainders to 9 or 3. 
 
 This is the reason of the familiar rule for "casting out 
 9's," in order to test the accuracy of calculations. 
 
 If P = MN = 9 a; + ^> = (9 y + m) (9 s + n), where jh 
 111, and n are the respective remainders to 9 of P, M, and 
 N, it follows that 2^ ^^tl- "^'^ gi'^e the same remainders to 9, 
 
 since ^ x -\- p = (9 y + ?») (9 z -\- n) 
 
 = 9-1/." + 9 (/?y + mz) -J- 17171 
 
 = 9 (9 l/Z -(- 711/ -\- mz) -\- 77171. 
 
 In practice, find 7?, 171, and n, not hy dividing P, M, and 
 N, by 9, but, in accordance with the theorem, by dividing 
 their digit-sums by 9 ; '' cast out " the nines. It is plain 
 also that the remainder to 9 (or 3) of ^ + i> -|- C equals 
 the like remainder to a -\- h -\- c, where a, h, and c are the 
 respective remainders to A, B, and C. 
 
 Therefore to test addition : — • 
 
 (1) 
 
 8277 
 
 remainder to 9 
 
 . . 6 
 
 3485 
 
 remainder to 9 . 
 
 
 
 • > ^ 
 
 7146 
 
 remainder to 9 . 
 
 . . 
 
 8036 
 26944 
 
 remainder to 9 . 
 
 . . 8 
 16 
 
 The sums, 26944, and 16, each, give the same remainder, 
 7 ; consequently the addition * is probably correct, — only 
 probably, because this check could not take note of an 
 error of any multiple of 9, or compensating errors, or 
 transposition of figures. 
 
 * Strictly, partial additions, and associations to suit our notation. 
 (Fide §§72, 73.) 
 
182 NUMBER AND ITS ALGEBRA. 
 
 (2) To test subtraction : — 
 
 87235 remainder to 9 ... 7 
 14505 remainder to 9 ... 3 
 72670 X 
 
 The difference, 72670, and 4 give same remainder, 4, to 9. 
 
 (3) To test multiplication : — 
 
 349751 remainder to 9 . . . 2 
 
 28637 remainder to 9 . . . 8 
 
 10015819387 l6 
 
 The product and 16 give the same remainder, 7. 
 
 (4) To test division, let the student prove that if 
 
 The remainder to nine of P = remainder to nine of 
 {qd 4- ?•), where q, d, and r are the respective remainders 
 to nine of Q, D, and B. Thus to test the division, 
 
 27220662 ..o , 398 
 
 47923 47923 
 
 remainder to 47923 (divisor) =^ 7 
 
 remainder to 568 (quotient) = 1 
 
 remainder to 398 = 2 
 
 remainder to 7 X 1 + 2 =0 
 
 remainder to dividend = 
 
 289. Problems : — 
 
 (1) Expressed in a certain scale seventy-nine becomes 
 142, what is the radix ? 
 
 (2) In what scale of notation does 301 express the 
 second power of an integer ? 
 
 (3) Dediice a test of multiplication by "casting out 
 elevens." 
 
 (4) Prove that any integer of four digits in the scale of 
 
ALGEBRAIC SQTTARE ROOT. 183 
 
 ten is a multiple of 7, if its first and last digits be equal, 
 and the hundreds digit twice the tens digit. 
 
 (5) In ten scale a number of 6 digits whose 1st and 4th, 
 2d and 5th, 3d and 6th digits are respectively the same, is 
 a multiple of 7, 11, and 13. 
 
 290. The common process of finding algebraic square 
 roots, cube roots, etc., is familiar to all, most text-books 
 making far too much of it. The method has little interest, 
 theoretical or practical.* Even the analogous numerical 
 calculations are better dispensed with, if a table of log- 
 arithms is at hand ; and the method for the algebraic 
 problem is rendered superfluous by the simpler method 
 of "undetermined coefficients." AVe consider only cases 
 where the function is a perfect square, because further 
 discussion would take us into the question of infinite 
 series. 
 
 Example. — Eequired the algebraic square root of — 
 
 ^ 12 3 9 
 
 If a " perfect square," the root must be of the form, ax^ 
 -\-l>x -\- c, the square of which is (rx* + 2 abx^ -\-(2 ac -f Z»^) 
 a;2 _|_ 2 l,e.v + cl The corresponding coefficients must be 
 equal ; therefore, o = 1. 2 ab = 1 .-. h = 1 / 2. 2 he = 
 — 1/3. '.6=— 1/3; therefore the required square root is 
 
 x^-\--- 1/3. 
 ^2 ' 
 
 * Professor Chrystal remarks: "The metlioil was probably obtained 
 by analogy from the arithmetical process. It was first given by Recorde 
 in The Whetstone of Witte (black letter, 1557) the earliest English work 
 on algebra." It would be serviceable to the student to compare the 
 difference between the numerical and the algebraic ijroblems. 
 
184 NUMBER AND ITS ALGEBRA. 
 
 To find c we might have taken either of the last three 
 coefficients. 
 
 A similar method would yield the cube root of a function 
 which is a " perfect cube," etc. 
 
 291. Without going too far into the subject, it is proper 
 to add here several fundamental theorems concerning com- 
 plex numbers, postponed from Chapter XII. 
 
 If <f> (x 4" yi) be an integral function of a complex num- 
 ber, we saw in Chapter XII. that it is reducible to a com- 
 plex number, say A -j- BL Now, if all the coefficients of 
 ^ (x -{- yi) are protomonic, A and B are protomonic, and A 
 can contain only even, and B only odd, powers of y ; there- 
 fore, if X -\- yi be changed to x — yi, A will remain unal- 
 tered, and B changed to — B. That is to say, if (|) {x -{- 
 yi) = A -\- Bi, ^ (x — yi) = A — Bi. 
 
 The theorem is readily extended to include all stirpal 
 functions, integral or fractional, of a complex number, and 
 generalized for such functions of more than one complex 
 number. 
 
 292. As a corollary, if all the coefficients of the stirpal 
 function ^ (ii) be protomonic, and if ^ (u) = 0, when ?* = 
 a + bi, then ^ (u) = 0, when u = a — bi; for if ^ -|- -^'<- = 
 0, ^ = and ^ = (§ 193). 
 
 State the corollary for ^ (u, v, iv . . .). 
 
 293. Since <|) (x -f yi) = A -{- Bi and ^ (x — yi) = A 
 — Bi, when all the coefficients in the functions are proto- 
 monic ; and since 
 
 norm 4, (x + y i) =^ norm {A + Bi) = A"-\-B-= (A + Bi) 
 (A — Bi) ; therefore 
 norm <^ (x -\- yi) = norm ^(x — yi) = ^ (.« -(- yi) (f) (x — yi) ; 
 and therefore : — 
 
COMPLEX NUMBERS. 185 
 
 mod <f)(x -\- yi) = mod <^ (re — y?") = -f- V<^ (x + yi) (j> (x —yl) ; 
 
 and in general 
 
 mod ^ [x + yi, u -\- v'l, . . .^ = mod <^{x — yl, ?< — vi, . . . } 
 
 = + V {(/) (a- + 2/^', n + '(;/, . ■. . ) (fi(x — yl, w 
 
 -t-.-. . . )}. 
 
 294. If the function be the product of several complex 
 numbers, this theorem gives 
 
 mod{(r+ si) (t -\- ui){y -\- u-lj) = V{(^"+ si) (t + nl) (v -\- n'l) 
 (r —si){t —in)(v — 2rl)} = + V{{r-~\-s-)(t'-\-u^)(v^-\-tv^)} 
 = V/'^ + s' ^/t'^ + u- Vv- -\- w'^ — mod l^r -\- si) mod 
 (t -\- III) mod (y + wi) ; 
 
 that is to say, the modulus of the product of any number 
 of complex numbers equals the product of their moduli. 
 
 It might plausibly be taken for granted (since we have 
 seen that it x -\- yi = 0, x = 0, and ?/ = 0) ; but it is better 
 to prove distinctly that the product of two complex num- 
 bers cannot be zero, unless one of the complex numbers is 
 zero : 
 
 If yz = 0, where y and z are complex numbers, mod (yz) 
 = 0. But mod (yz) = mod y mod z ; therefore, mod y mod 
 z = 0. 
 
 But mod y and mod z are protomonic. Therefore, either 
 mod y = 0, or mod z = ; and consequently, by Section 
 198, either y = 0, or z = 0. 
 
 295. Again, as a special case of the general theorem in 
 Section 293, if the particular function be the quotient of 
 two complex numbers, we have 
 
 mod S ^ + "\ \^+J\ ^+"\ . ^ - '"' 
 I V -{- tvl ) \ I V -\- ivl V — u-i 
 
 Y I v-'-{- 'W-' \ Vw' + W mod (v + wl) 
 
186 NUMBER AND ITS ALGEBRA. 
 
 that is to say, the modulus of the quotient of two complex 
 numbers is the quotient of their moduli. 
 
 296. The modulus of the sum of complex numbers may 
 equal the sum of their moduli, cannot be greater, and 
 is in general less. For consider two complex numbers, 
 t -f- «^ and v -f- iH. 
 
 By Section 293, mod (t -\- ul -[- v -{- vi) = 
 
 -\- s/ {{t -\- ui -}- V -\- wi) (t — ui -\-v — u-i)} = 
 + ^ {(* + ^)^ + (^ + '^y^}} therefore we desire to prove, 
 + ^{(t-\-vy-\-{u-^wy} not > + V(^^+<) + V(z;^+w;2), 
 or, since only positive roots are concerned, that (t + i.')^ + 
 (u + tvf not > t^-\- u' + V- + IV- + 2 V{b-' + u') (v'^ + w^). 
 Subtracting t' + n^ -{- v" -\- n-^ from both members of this 
 inequality gives, 2 fv -\- 2 mv not > 2 V(^^ + u^) (v'^ + w"^), 
 and dividing by 2, tv -\- uw not > -\J {^^ + m^) (v'^ -f it--). 
 
 The right-hand member is essentially positive, and there- 
 fore not less than the left, if the latter is negative (as 
 might be on accomit of the original quality of t, n, v, 
 or ir) ; and the theorem is consequently proved for that 
 case. 
 
 If the left-hand member is not negative, by squaring 
 both sides* we get 
 
 t-v- + 2 tuviv -f iihv'^ not > f-r" + nhr" + fic^ + v^u^, 
 or 2 tvwu not > t'-^u-" -\- v-i(% 
 
 or not > t-ir- + r'~n- — 2 tvwu, 
 
 or not > (fir — ?';/)-. 
 
 But this is true, since the right-hand member is essen- 
 tially positive. 
 
 297. Argand's diagram beautifully apjilies to geometri- 
 cal relations these properties of complex numbers, thus 
 analytically displayed. 
 
THEOKY OF NUMBERS. 187 
 
 • 298. It would be interesting and instructive to follow 
 a great many very curious and useful investigations of 
 various properties of primary or discrete (to say nothing 
 of complex, or continuous) number^of which no mention 
 ever has been made. But to do so Avould carry us into 
 ideas and notations equally strange, and would be deemed 
 a transgression of appropriate bounds for such an elemen- 
 tary treatise as is this little work. For instance, Gauss 
 makes the notion of congruence fundamental in his Disqui- 
 sitiones Avithmeticae, Congruence meaning the relation of 
 I and J, if 7 = ayx + r, and J =^ h^-\- r, where [x. is termed 
 the modulus of / and J, and / and J are called congruent 
 with respect to modulus /a. Some astonishing facts are 
 directly deducible from this simple mode of classification. 
 It is not from the difficulties of the more elementary por- 
 tion of the Theory of Numbers * that the field lies fallow 
 for our undergraduate courses in mathematics, and I be- 
 lieve the interest of students would be less disposed to 
 flag if the firmer grasp of thought were commanded Avliich. 
 such studies would infallibly encourage. 
 
 299. If the equation ^ (x, y, z) = ifr (x, y, z) is satisfied 
 for all values of the variables, it is called an identical 
 equation, or an identity, or a formula. ( Vide § 40.) 
 
 In this case the equation is formally true, under the 
 very laws of numerical operation, regardless of particular 
 values of the variables. 
 
 If, on the other hand, an equation is satisfied only for 
 special values of the variables, it is called a synthetic, or 
 conditional, equation. From this point of view, the con- 
 
 * For bibliography of tlie interesting and important subject which 
 bears tliis name, see Numbers, Theory of, Cayley, Ency. Brit., 9tli eil. 
 
188 NUMBER AND ITS ALGEBRA. 
 
 stants are commonly spoken of as known, and the varitl- 
 bles as unknown "quantities," — numbers, in the algebra 
 of number. 
 
 Synthetic equations are classified and named with refer- 
 ence to their unknown numbers, precisely as functions are 
 characterized in regard to their variables. {Vide § 169.) 
 
 Synthetic equations involving only stirpal and radi- 
 cal functions (exponential, etc., equations are deferred to 
 future studies) can always be made to depend upon an 
 equation of the form 
 
 <\> (a-, y, z, . . . ) = Q, 
 where (/> is an integral function. 
 
 This form, therefore, is of prime importance in the 
 theory of equations. 
 
 300. Synthetic equations concerning the same variables 
 may occur in sets, or systems. In this case they are 
 called simultaneous, and the problem is to find the sets of 
 values of the variables which render every equation of the 
 
 . system an identity. 
 
 Such a set of values is said to satisfy the system, and 
 is called a solution of the system. 
 
 Such solutions are to be distinguished in many ways 
 from the solutions of one integral equation in one variable, 
 where a solution is called a root. 
 
 301. It is important to distinguish between two differ- 
 ent kinds of solution : — (1) Numerical solution, exact or 
 approximate, which can often be obtained where formal 
 algebraical solution would be out of the question ; and (2) 
 What may be called formal solution, that is, a solution in 
 which the variables are expressed as definite analytical 
 functions of the constants. Such solutions of equations of 
 degree higher than the fourth cannot, in general, be found. 
 
THEORY OF EQUATIONS. 189 
 
 302. The final test of any solution is the satisfaction 
 of the equation, upon substitution therein of the values 
 obtained for the unknown numbers. No matter how the 
 solution has been obtained, if it does not stand this test, 
 it is no solution ; and no matter how obtained, if it does 
 stand this test, it is a solution. It is often a good way to 
 guess a solution, and make the test. 
 
 303. FUK^DAMEKTAL PkOPOSITIOX IX THE TlIEORY OF 
 
 Equations. — If in the equation <j),^ (x) = 0, (/>„ (x) be an 
 integral function of x of the ?ith degree (the coefficients, in 
 general complex, in particular, protomonic, numbers) where 
 the coefficient of the a;" term is not zero, then ^„ (x) is the 
 product of n factors, each of the first degree. 
 
 With one provision we proved this proposition in Section 
 268, and it has also been shown that these factors can 
 always be in the form 
 
 C (x — r{) (x — ?-2) (a- — Ts) . . . (x — ?■„), 
 
 where C is the coefficient of a:" in (^,^ (x), and Vi, ?-2, r^, 
 
 ...?•„ are the roots of the equation. Consequently, the 
 
 problem of solving an integral equation with one unknown 
 
 number, is identical with the problem of resolving the 
 
 general function of one variable, of like degree, into factors 
 
 of form 
 
 C (x — ?-j) (x — r^) (x — vs) . . . {x — ?■„). 
 
 304. It is worth while to call attention to the fact that 
 
 x^ -\- X -{- 1 = (x -{- 1 -\- Va-) (x -\-l — Va-), 
 
 often given by beginners when required to factor x'^ ~{- x -\- 1, 
 although a true identity, is no factorization in the sense 
 intended, because the factors are not integral functions. 
 
 305. ISTothing need be said of the solution of integral 
 
190 ISrUMBEK AND ITS ALGEBRA. 
 
 equations of the first degree : properly associating the 
 
 terms, and reducing by the distril^utive law to the form 
 
 iV" 
 Cx = N. gives X = — . 
 
 306. Eecurring to Section 271 (4), we know that 
 
 (x — a) (x — h){x — c)... {x — n) = 
 
 is an equation whose roots are a, h, c, . . . n. 
 
 Performing the multiplications, we have the form : 
 
 a.» + r^.r«-i + r„.r"-^+ . . . +r-„_j a- + c„ = 0, 
 where, c^ = — {a, J^ h -{- c -\- . . . -\- n) 
 Co = ab -f- "<' -{- he -\- . . . -\- vin 
 Cz= — {'ihe 4- uhd -\- acd -[-.,. -)- liiui' 
 r„ = -j- ahrd . . . n. 
 (Plus or minus, as 7i is even or odd). 
 
 Hence, if an integral equation of the ?ith degree is in 
 the above general form : 
 
 The coefficient of the second term is minus the sum of 
 the roots. 
 
 The coefficient of the third term is the sum of their 
 products, taken two at a time. 
 
 The coefficient of the fourth term is minus the sum of 
 their products, taken' three at a time, etc. 
 
 The last term (the constant) is plus or minus the product 
 of all the roots, according as w is even or odd. 
 
 307. It follows : In every equation of the nth. degree 
 in the general form, 
 
 If the second term is wanting, the sum of the roots is 
 
 I the last term is wanting, at least one root is zero, 
 [f all the roots are integral, they are submultiples of 
 tiie last term, which must be integral. But the inverse 
 
THEORY' OF EQUATIONS. 191 
 
 does not follow ; siuce tlie last term may be integral, yet 
 roots be fractional. But if the last term is not integral, 
 some of the roots are not integral. 
 
 If all but one of the roots are known, the remaining one 
 may be found by adding the sum of the known roots to 
 the coefficient of the second term, and changing the quali- 
 tative sign of the result. Or, by dividing the last term by 
 l^lus or minus the product of the known roots, according 
 as 71 is even or odd. 
 
 If m roots are known, the equation may be depressed 
 to another of the {ii — m)th degree, by dividing by the 
 product of lit factors of the form, 
 
 (x — 7\) (x — Vo) . . . (x — ''m)j ^^^^ therefore : — 
 
 If all but two roots are known, the coefficient of the 
 depressed equation is the sum of the known roots and the 
 coefficient of the second term of the given equation. And 
 the last term of the depressed equation is the last term of 
 the given equation, divided by plus or minus the product 
 of the known roots, according as n is even or odd. 
 
 308. From the process of multiplication required in Sec- 
 tion 306, it is evident that if all the r's are positive, the 
 quality of the terms is alternately + ^^^^ — • Hence, if 
 the roots of an equation are all positive, the signs of its 
 terms (supplying missing terms by zeros) are alternately 
 -J- and — , and inversely. 
 
 Again, if all the r 's be negative, there is no change in 
 the signs of the terms. 
 
 It would not be difficult to deduce here Descarte's Rule 
 of Signs : An integral equation cannot have more positive 
 roots than it has changes of signs, nor more negative roots 
 than it has continuations of the same sign. 
 
102 NUMBER AND ITS ALGEBRA. 
 
 309. Prove : Any integral equation may be transformed 
 into another whose roots are the negatives of the original 
 roots, by changing the signs of alternate terms, beginning 
 with the second. 
 
 310. To transform an integral equation into another, 
 whose roots are the roots of the original equation multi- 
 plied by a given number, k : — 
 
 In tlie general form substitute y / /- for x, obtaining, — 
 
 Multiplying by k"- gives 
 //" + ^iAr-' + ^./.y--+ . . .r„_i7.-«-V+^„A- = (2) 
 The roots of (2) are the values of ?/ that satisfy it; but 
 1/ = kx ; therefore, noting the coefficients in (2), to effect 
 the desired transformation, multiply the second term by k, 
 the third by k'-^, and so on. 
 
 311. Equations may be transformed in many other use- 
 ful ways ; for example, so that the roots shall be the ori- 
 ginal roots ^ some constant. This mode of transformation 
 is most serviceable in the special case of making the exact 
 increment which will cause the second term to vanish, — a 
 device for preparing cubic and biquadratic equations for 
 solution. For a simple illustration see Section 316. 
 
 312. Seeing that we have the unique resolution : ■ — - 
 
 ^u (^-) = = c(x — ?-j) (x - ;•.,) . . . (x - ?•„), 
 it follows from Section 292 that if ^„{x) has all its coeffi- 
 cients protomonic, and vanishes when x = a-{- hi, it must 
 vanish when x = a — hi. 
 
 This is to say, that in any integral equation whose 
 coefficients are protomonic, roots which are complex num- 
 bers must occur in conjugate pairs. 
 
QUADRATIC EQUATIONS, 193 
 
 In like manner {vide § 170) surd roots can enter equa- 
 tions with commensurable coefficients only in conjugate 
 pairs. 
 
 Thus, all such equations, if of an odd degree, must have, 
 in the former case at least one protomonic, and in the 
 latter at least one commensurable, root. 
 
 313. The general equation of the second degree in one 
 variable is ax^ + hx -(- ^ = 0. The general theory of solu- 
 tion is already in our hands, and in this case the formal 
 solution (vide § 301) is always obtainable. Various methods 
 may be followed. 
 
 The general equation, 
 
 ax^ -f- ^■^-" + c = Oj 
 may be reduced without altering the roots (§ 303) to 
 
 a a 
 
 or X- -\ — x = . 
 
 a a 
 
 From consideration of the formula (x -\- yy z= x- -\-2 xy 
 -f- ij'\ it is plain that the left-hand member may be made a 
 
 / h \2 
 
 '' complete square " in x by adding f 7y~~ ) to each member, 
 which gives — 
 
 X'^ -\ X -\ = 1 = . 
 
 a 4 a^ a 4 a^ 4 a^ 
 
 Taking the square root of each member, * 
 
 * The double sign before the left-hand member would be superfluous, 
 since nothing more would be said than is expressed as the statement 
 stands; e.g. : — 
 
 ± (a -\- h) — ^ {c -\- d) says no more than 
 
 a -f ?> = ± {c-^ d), as one may readily satisfy himself. 
 See also Section 325. _„.»—__ 
 
194 NUMBER, AND ITS ALGEBRA. 
 
 X + ^-^ = ± .— V^/' - 4 ae, 
 la la 
 
 or 3. ^ Z-IL i "^^''' — 4 «c 
 
 2 a 
 
 We have here a formal solution of the general quadratic 
 equation. 
 
 Also the quadratic function, ax^ -\- hx -[- c, has been fac- 
 tored. For, by the principles clearly exhibited in Section 
 303, — 
 
 ax--\-bx-\-c=^n{ x ■ — ■ " -' 
 
 2 a J 
 
 314. In solving a particular quadratic in one variable, 
 we may give this process of *' completing the square " its 
 particular application ; or we may employ the formal solu- 
 tion as a rule; that is, after reducing the given equation 
 to the form ax" -)- /y^ -|- c = 0, simply write down the partic- 
 ular values in 
 
 X 
 
 — — ^ jz V^' - 4 ac 
 
 2 a 
 
 Of course, if the given equation in form ax- -{- hx -\- c = 
 0, affords a function readily factorable by inspection, it 
 would be absurd to feign an investigation for what is 
 already known. For instance, one with any skill in the 
 algebra cannot fail to see that in a-^ + 5 a; -f G = 0, we 
 have (x + 3) (a; -f 2) = ; which is to say, that x = — 3, 
 and X = — 2. 
 
 The device of reducing the given equation to the 
 form, 4 a'^x- -\- 4 ahx + 4 ac = 0, before "completing the 
 square" (known as the Hindoo Method), is hardly Avorth 
 mentioning, since it merely avoids fractions which offer no 
 obstacle to calculation. It is doubtless a relic of the times 
 when fractional number was regarded with suspicion. 
 
HOOTS OF QUADRATIC EQUATIONS. 195 
 
 315. If i\\e formal solution of ax- -\- hx -\- c ^ 0, 
 
 — h A- -Vb- — 4 ac 
 X = ^ , 
 
 2a 
 
 be coDsidered, it will be seen that, when the coefficients are 
 protomonic, the roots are : • — • 
 
 (1) Protomonic and unequal, if Ir — 4 ac is positive. 
 
 (2) Protomonic and equal, if //- — 4 ac = 0. 
 
 (3) Commensurable, if V^" — 4 ac is commensurable. 
 
 (4) Conjugate surds, if V^- — 4 ac is incommensurable. 
 
 (5) Conjugate complex numbers, if ^- — 4 ac is negative. 
 
 (6) Equal, if //- = 4 ac. 
 
 (7) Equal moduli, but one positive, other negative, if 
 
 as — - is + or — ] . 
 
 protomonic or neomonic, 
 
 (8) One zero, other = — h/ a, if c = 0. 
 
 (9) Both zero, it b = and c = 0. 
 
 It may be profitable to find, from a different standpoint, 
 more or less the same criteria : — • 
 
 From Section 306, the equation, ax' -j- bx -\- c ^ 0, gives 
 the following relations of roots and coefficients, — 
 
 r^ -(-?•„ = , and /^ ?'o == - . 
 
 a 'a 
 
 Consequently 1\ and r^ are 
 
 . ." . b ■ . c 
 
 positive if - is negative and - positive ; 
 
 a a 
 
 negative if - is positive and - positive ; 
 a a 
 
196 
 
 NUMBEE AND ITS ALGEBKA. 
 
 of opposite quality if - is negative. 
 
 a 
 
 Tliese statements presuppose (1) above, ^- — 4 ac > 0. 
 
 ri = — To if 
 
 0. 
 
 a 
 
 ri = or ?-o = if - = 0. 
 
 a 
 
 ri = and r. = if - = and - =- 0. 
 
 ' a a 
 
 If ax^ -|- ix + c = be still regarded as a quadratic 
 when a = 0, then one root is co . If ^ also is zero, both 
 roots become infinite. 
 
 These criteria may be tabulated : — 
 
 KouTs. 
 
 Ckitekiox. 
 
 ]{OOT.S. 
 
 Ckiteiuon. 
 
 Protomonic . . . 
 Commensurable . 
 
 Surd 
 
 Complex .... 
 
 Equal 
 
 Equal moduli, but 
 one +, other— . 
 
 62 — 4 ac> 0. 
 
 Positive . . 
 Negative . . 
 
 One+, One — 
 
 One, . . . 
 Both, . . 
 One, CO . . . 
 Both, 00 . . 
 
 ^+,and^-. 
 a a 
 
 ^+,and^+. 
 a a 
 
 c __ 
 
 a 
 
 c = 0. 
 
 6 = and c = 0. 
 
 a = 0. 
 
 rt = and 6 = 0. 
 
 i/b- — iac, commen- 
 surable. 
 
 V'62 — -lac, surd. 
 0- — 4ac <0. 
 62 — 4 ac = 0. 
 
 6 = 0. 
 
 316. Another method of solving a quadratic equation 
 is important from its bearing on the solution of cubic 
 equations. 
 
 The general equation, ax^ -\- bx -\- c = . . . (1), may be 
 reduced by a change in the variable to the form ai/^ -|- (Z = 
 . . . (2), from the immediate solution of Afhich the origi- 
 nal variable is recovered. To discover Avhat change must 
 be made in the original to serve this purpose (^vide § 311), 
 let X — 1/ -\- e. 
 
INDETERMINATE SYSTEMS. 197 
 
 Ji X = 7/ -{- e, (1) is equivalent to 
 
 « (1/ + ey-i-b(y + e) + c = 0; 
 or of?/2 ^ (2 ae -\- b) y -\- ae"- -\- he -\- c = 0. (3) 
 
 To make the second term vanish, 2 ae -\- b must be zero, 
 
 b 
 
 or e = 
 
 
 2 a 
 
 With this 
 
 vah;e of e, (3) becomes 
 
 
 <, b- — 4: ae ^ 
 4 a 
 
 whence 
 
 ±■^/b^ -4= ac 
 ^ 2a 
 
 But 
 
 x = y -\- e = y - --; 
 
 2 a 
 
 therefore 
 
 — Z* ± V/'^ — 4 ac , r 
 X = , as before. 
 
 2a 
 
 317. If an equation contaifis two {a fortiori, more than 
 two) unknown numbers, it is obviously indeterminate. An 
 extraneous condition (e.g., that the variables shall be in- 
 tegers) sometimes affords a basis for a determinate solution, 
 
 A system of simultaneous equations is in general deter- 
 minate when the number of equations equals the number 
 of the variables. 
 
 If the number of equations is less than the number of 
 variables, the solution is in general indeterminate. 
 
 If the number of equations is greater than the number 
 of variables, there is in general no solution, the system 
 being inconsistent, contradictory. 
 
 These are ultimate logical principles ; special limitations 
 of the statements are needed rather than proof. 
 
 It must suffice here to point out that a system may be 
 apparently determinate, yet indeterminate by reason of one 
 
198 NUMBER AND ITS ALGEBRA. 
 
 being analytically derivable from the others. Also it may 
 happen that a system of analytically independent equations 
 may have more equations than variables, yet not be con- 
 tradictory. 
 
 Let the student frame examples of such conditions. 
 
 318. A determinate system of integral equations involv- 
 ing the variables, x, y, z, . . . , cannot have more than, and 
 in general has exactly abc . . . solutions, where a, b,c, . . . , 
 are the degrees of the system in the respective variables. 
 
 Proof of this proposition must await future studies ; but 
 it is useful to know the theorem, and the question presents 
 itself at once, and should not be ignored by the teacher. 
 
 319. Two systems of equations, each of which may con- 
 sist of only one, are termed equivalent when every solution 
 of each is a solution of the other. 
 
 From any system we may, in an infinite variety of ways, 
 deduce another system ; but the derived system is not gen- 
 erally equivalent to the original. 
 
 This matter is of fundamental importance, even at the 
 most elementary stages. It is commonly (with several 
 notable exceptions) left in the dark by our text-books, 
 though " there are few parts of algebra more important 
 than the logic of the derivation of equations, and few, un- 
 happily, that are treated in more slovenly fashion in elemen- 
 tary teaching. Xo mere blind adherence to set rules will 
 avail in this matter ; while a little attention to a few simple 
 principles will readily remove all difficulty." * 
 
 320. If A and B are two functions, which do not become 
 infinite for any finite values of the variables (such cases 
 must be considered separately), the only values of the vari- 
 
 * Text Book of Algebra, Chrystal, vol. i., p. 285. 
 
EQUIVALENCE OP EQUATIONS. 199 
 
 ables which make ^ x ^ = are such as make yl = 0, or 
 ^ = 0, according to laws already fully demonstrated. 
 
 321. Axiomatically, ii A = I>, (1) 
 then A-^C= B ^ C. (2) 
 
 Also, (1) and (2) are equivalent, for neither can be true 
 xmless the other is true. 
 
 Note the corollaries whereby we " transpose a term with 
 changed signs," or " change all signs," or reduce any equa- 
 tion to the form () = 0, without destroying equivalence. 
 
 322. On the other hand, although, if 
 
 A = B, (1) 
 
 then AC=BC, ' (2) 
 
 the derivation being perfectly legitimate, and the resulting 
 equation true, yet (2) is not equivalent to (1), unless C is 
 a constant not zero ; for, by Section 321 (2) is equivalent 
 
 ^° AC-BC=0 
 
 that is to C(A- B) =0 (3) 
 
 Now, if C is a constant not zero, (3) is equivalent to (1) by 
 Section 320 ; but not otherwise, for if C is a function of 
 the variables, (3) is satisfied by all values of the variables 
 that satisfy the equation, C = . . . (4), which in general 
 will not satisfy (1). Therefore (2) is not equivalent to 
 (1), but to (1) and (4). 
 
 In this way it is plain that multiplying both members of 
 an integral equation by an integral function introduces 
 roots, and dividing the members of such an equation by 
 an integral function loses roots. 
 
 Also, from any integral equation another equivalent 
 equation can always be derived in which the coefficient of 
 any term shall be as desired, say -f- 1 for the highest term ; 
 for this is obtainable by multiplying by a constant. 
 
200 NUMBEK AND ITS ALGEBRA. 
 
 323. Fractional equations must never be confounded, 
 in the matter of degree and number of roots, with, integral 
 equations. The very term degree does not apply to frac- 
 tional equations. Fractional functions of x may sometimes 
 
 be integral functions of some function of .r f e.g., - j; but 
 
 in general no sucli relations as obtain between degree and 
 number of roots in integral equations subsist for fractional 
 equations. The latter must be solved under the logic of 
 the equivalence of derived integral equations. 
 
 From any fractional equation an integral equation may 
 be deduced, which may or may not be equivalent. If 
 E ^= F, where E and F are fractional functions, and 
 Jf = 1. c. m. of the denominators in E and F, then EM = 
 FM is integral. 
 
 Here extraneous solutions of M = may be introduced, 
 but not necessarily or generally. E and F contain frac- 
 tions whose denominators are factors in 31, and in general 
 roots of M =0 would make E ov F infinite, and conse- 
 quently M (E — F) not necessarily zero. 
 
 See examples below for clear understanding of this 
 point. 
 
 324. If both members of an equation be raised to the 
 same power, in general the resulting equation is not equiva- 
 lent. Thus A = B; then A^ = B% or A^ - B^ = 0. But 
 the last is equivalent to {A -\- B) {A — B) = 0; hence the 
 solutions of A -\- B = would in general be introduced. 
 
 It may be noted that in squaring A = B the result is 
 the same as if the members of the equivalent equation, 
 A- B = 0, were multiplied by A + B. {Vide § 322.) 
 
 325. Neither the equation between the positive, nor that 
 between the negative, square roots of the members of the 
 
EQUIVALENCE OF EQUATIONS. 201 
 
 equation A = B, is an equivalent equation ; but the two 
 equations (generally written together with double signs) 
 between the positive root of one, and both roots of the 
 other, constitute an equivalent system. (Vide § 313.) 
 
 + VX= + V^ (1) 
 
 nd + VA = -VB (2) 
 
 is a system equivalent to ^ = Z?. 
 
 For A = B is equivalent to A — B = 0, which is equiva- 
 lent to {-y/A -\- -y/B) (VA — -y/B) = 0, which is equivalent 
 to the system (1) and (2). 
 
 326. li A = B he Q, radical equation, repeated involu- 
 tions Avill deduce an integral equation which may or may 
 not be equivalent. Extraneous solutions may be intro- 
 duced ; and, if like roots in the original equation alone be 
 regarded, often no solution of the derived equation will 
 satisfy the original. 
 
 327. Two equations which are not equivalent are called 
 indejjendent. Two or more independent equations involv- 
 ing a corresponding number of variables may be capable of 
 coincident solution ; if so, they are termed simultaneous, 
 that is, consistent, or involving variables which, though un- 
 known, are the same. Contradictory statements, no matter 
 how artfully veiled the contradiction, can lead only to non- 
 sense in algebra, as elsewhere. 
 
 Compare again Sections 300, 317, 318. 
 
 The devices of elimination, whereby an equation in one 
 variable is deduced from a system of simultaneous equa- 
 tions in several variables, are familiar ; but the logic of 
 such derivations, and the paramount question of the equiv- 
 alence of the derived and original systems may have been 
 overlooked. 
 
202 NUMBER AND ITS ALGEBRA. 
 
 The present discussion must be concluded with two 
 propositions specially concerning the equivalence of simul- 
 taneous systems. The subject will have been by no means 
 exhausted ; but my purpose of stimulating alert and intelli- 
 gent observation in the important matter of solving alge- 
 braic equations will probably be fulfilled. The student's 
 skill and knowledge will steadily increase, if strict atten- 
 tion be always paid to the question of equivalence. 
 
 328. The system, 
 
 P ^^ n S^^ ! I i« equivalent to ^ = (1) | 
 
 for any solution of I makes A zero, and B zero, and there- 
 fore satisfies II; and any solution of II makes A zero, and 
 therefore reduces II (2) to q B = 0, ov B = 0. 
 
 Conseqviently any solution of either satisfies both. 
 
 It may be suggestive to state this proposition again in 
 the form 
 
 A = B) A — P 
 
 ^ ? is equivalent to 
 C = D\ AA-C = B-\-D 
 
 On the other hand, — 
 
 ^~ ^}I is not equivalent to ^ = ^ | ,j 
 C = B\ ^ AC = Bd] 
 
 For, though all the solutions of I are solutions of II, II 
 has in addition all the solutions of C = 0, and D = 0. 
 Let the student satisfy himself of the truth of this propo- 
 sition. It explains many ''answers" which may have 
 been incomprehensible to him. 
 
 The following examples may serve to impress what has 
 been said concerning the equivalence of derived equations 
 with their originals, although at every point the student 
 should have found specific illustrations. 
 
EQUIVALENCE OF EQUATIONS. 203 
 
 329. (1) Solve 
 
 ^ ^^ =1. (1) 
 
 X — 3 X -\- (J 
 
 Multiplying each member by (.r — 3) (.r -f G) gives 
 a-2 -3x - 18 = 0, 
 or, (x _ 6) (x + 3) = 0, (2) 
 
 whence x = 6 and x ^= — 3. 
 
 Both of these are solutions of (1). i^o roots of (x — 3) 
 (x -j- 6) = were introduced, because x ^ 3 or x = — 6 
 would make the left-hand member of (1) infinite, and 
 therefore M {E - F) not zero. {Cf. § 323.) 
 
 (2) Solve 1 — = — 6. (1) 
 
 Transposing, and adding the fractions, gives 
 
 1 - -^ J = - 6, 
 
 X — 1 
 
 or 1 — (x + 1) = — G, 
 
 or a- = G . . . equivalent to (1). 
 
 But a beginner might multiply by a; — 1, deriving 
 
 ic - 1 - a--^ = - 1 - G a; + 6 (2) 
 
 whence x = \ and a* = G, 
 
 where 1 is no solution of the original, and therefore (2) is 
 not equivalent to (1). 
 
 Multiplying by any integral function, not nesessary to 
 clear of fractions, will derive an equation not equivalent. 
 Accordingly, every device for identical simplification should 
 be employed before multiplying by the lowest common 
 multiple. 
 
 (3) Solve i_^ ^'' + ^-6 ^ ^^-3a- + 2 ^ ^ 
 
204 NUMBER AND ITS ALGEBRA. 
 
 Multiplying each member by (x — 2) (x + 2), and redu- 
 cing identicall}-, gives 
 
 3 «2 _ 4 a; _ 4 = 0, (2) 
 
 whence x = ^ Jz VI6 + 48 ^^_Al = 2o.- 2/3. 
 
 6 6 ' 
 
 Equation (2) is not equivalent to (1), the root, 2, of 
 (x — 2) (x 4- 2) = having been introduced, because the 
 fraction in the left-hand member of (1) is not in its lowest 
 terms. If (1) be reduced before clearing of fractions we 
 obtain 
 
 14-0^ + 3^ ^ ~ o"^*^ ? 
 X -^ 2 
 
 whence, multiplying hj x -j- 2, 
 
 x^ -\- 6 X -\- 8 = x^ - 3 X -\- 2, (3) 
 
 or X = - 2/3, 
 
 where (3) is equivalent to (1). 
 
 (4) Solve V4 - .T = ic — 4. (1) 
 
 Squaring 4 — x = x- — 8 x -\- 16, 
 
 or a"2 — 7 a- 4- 12 = 0, 
 
 or (x - 3) (x - 4) = 0, (2) 
 
 whence x = S and x = 4. 
 
 Of these solutions of (2), 4 is a solution of (1) if the 
 l^ositive square root be taken, and 3 is not a solution ; 
 whereas, if the negative root be taken, 3 is a solution and 
 4 is not. Thus (2) is equivalent to 
 
 -\- V4 — X = X — 4: and — V4 — x = x — 4. 
 
 (5) Solve V3rK-|- 1 = V9a; + 4 - V2 x-l (1) 
 
 Squaring twice, and reducing identically, gives 
 
 a;2 _ I a; _ 5 == (2) 
 
 whence x = 5 and x = — ^ 
 
EQUIVALENCE OF EQUATIONS. 205 
 
 Using only positive roots of the radicals, 5 is a solution of 
 (1) ; but — ^ substituted in (1) gives 
 
 or ^ V2 i = h V2 i — V2 i, (3) 
 
 an absurdity if the statement be restricted to positive 
 roots ; but if the negative root of the left-hand member be 
 taken with the positive roots of the terms in the right- 
 hand member, (3) is an identity. 
 Therefore (2) is equivalent to 
 
 -f VSu; + 1 = + VOu; -f 4 - (+ V2 X — 1), 
 
 and — V3 a; + 1 = + V9 ic + 4 — (+ V2 a; — 1). 
 
 (6) Solve 2 - V2 X + 8 -f 2 v./; + 5 = (1) 
 
 Squaring twice^ we deduce 
 
 a-2 = 16 (2) 
 
 whence a; = -JL 4. 
 
 In this case, using the positive roots of the radicals in 
 (1), neither -|- 4 nor — 4 is a solution. 
 
 So far as I am acquainted with them, treatises upon 
 algebra, if they notice such cases, merely declare that the 
 original equation is impossible and has no solution. Pro- 
 fessor Chrystal states the theorem : — 
 
 " From every algebraical equation we can derive a 
 rational integral equation, ivhlch to'ill he satisfied Inj antj 
 solution of the given equation ; but it does not follow that 
 every solution, or even that any solution, of the derived 
 equation will satisfy the original one." 
 
 The italics are mine, and would mark logical contradic- 
 
206 NUMBER AND ITS ALGEBRA. 
 
 tions if there is " any solution " of the original. Professor 
 Clirystal's example is : — 
 
 V* + 1 4- V^' — 1 = 1. 
 
 The derived equation yields the single solution, x ~ & . 
 The only remark is, " it happens here that a; = | is not a 
 solution." 
 
 Note, I is a solution of + V*' + 1 + (— V.e — 1) = 1. 
 
 Professor Taylor, in his Academic Algebra, Boston, 1893, 
 which deserves rare praise for emphasizing from the begin- 
 ning the question of equivalence of equations, uses example 
 (6) above, concluding "2 — V2 x + 8 + 2 Vx + 5 = is 
 an impossible equation, for it has no solution." 
 
 Now, I must not be understood as disputing these state- 
 ments ; they are true, taking the numerical statements to 
 be restricted to positive roots. But it seems to nie that 
 the student stands in need of further explanation : he 
 should be directed to observe, that though one may write 
 down what he pleases, as an isolated statement, no restric- 
 tions can be put upon the operational effect of such nu- 
 merical relations. The square root of 4, as an inexorable 
 fact, is -|- 2 or — 2. In general operation, radical surds 
 necessarily include all their roots. If one says, V-*^, he has 
 expressed six distinct subjects of affirmation, nor can the 
 logical consequences of these alternatives be avoided in 
 numerical analysis. 
 
 The conclusion of the particular problem under consid- 
 eration is, that no finite number satisfies the equation, 
 taking positive square roots ; but by reason of the perfect 
 generality and freedom of numerical operations, if there is 
 a number such that either of the square roots concerned 
 fulfils the conditions, it must be yielded as a solution of 
 
EQUIVALENCE OF EQUATIONS. 207 
 
 the equation. We had occasion to notice in Section 323, 
 and in example (1) above, that indeterminate infinite solu- 
 tions do not obliterate or interfere with finite solutions, if 
 there be any such. 
 
 And in general, the complete analysis of any radical 
 equation would seem to require the investigation of all the 
 alternative equations arising from the indifferent roots of 
 radical surds. Some of these niay be impossible, in the 
 sense of having no finite solution ; but if a finite number 
 will satisfy any one in the system, it will certainly discover 
 itself in the attempted solution of an}' other, — and simply 
 because the choice of particular roots is arbitrary, and an 
 equation cannot be made to yield nonsense, or contradic- 
 tion, so long as there is possible consistency of its terms. 
 
 (7) Solve the simultaneous system 
 
 x-2 + 2y2 = 9 (2) f 
 
 Solving (1) for x x = 5 — 2>j (3) 
 
 Substituting in (2) from (3) (5 - 2//)^ -(- 2i/ = 9. 
 
 or 3y- - lOy + 8 = 
 
 or (3y_4)(y-2) = • (4) 
 
 System A is equivalent to system B (calling (3) and (4) 
 system B). But system B is equivalent to the double 
 system 
 
 ^ = ^--^U, and ^ = ^-^^1^ 
 3y_4 = 0) 7/_2 = 0[ 
 
 r' A 
 
 The solution of c is a; = —,?/ = —. 
 
 3 -^ 3 
 
 The solution of fZ is x = 1, ?/ = 2. 
 
 Hence these are the two solutions of A. (Vide § 318.) 
 
208 NUMBER AND ITS ALGEBKA. 
 
 (8) Solve the simultaneous system 
 
 x^-2xy= (1) I 
 
 4x2 ^ 9^2 _ 225 (2) I ^ 
 
 Factor (1) x{x- 2y) = 0. 
 
 Hence A is equivalent to the double system 
 
 4^2 ^ 92 = 225 ") , , 4*2 -L 9y2 _ 225 ) 
 
 ' -^ V ^, and ' "^ - c. 
 
 a; = ) ic-2^ = i 
 
 The solutions of •'^ are obviously a- = 0, ?/ = 5 ; and a* = 
 0, ^ = — 5. Substituting ;r = 2// in the first equation of 
 
 ' ^^^^^ r = 0, or y = ± 3. 
 
 Substituting in a: — 2// = we have for the solutions of 
 c, X ^ Q>, 1/ = 3 ; and x = — 6, y = ~ 3. 
 
 Hence the four (vide § 318) solutions of A are x = 0, 
 7/ = 5 ; X = 0, 7/ = ~ 5; X ^6, 7/ = 3; X = — 6,7/ = —3. 
 In the solution of simultaneous systems, attention must 
 always be given to the correct association of values of 
 the variables. 
 
 330. When a simultaneous system has its equations of 
 the second degree, its solution demands in general the 
 solution of a biquadratic equation in one variable. Inas- 
 much as the studies to which these lectures are intro- 
 ductory may be regarded as beginning about at this point, 
 I bring these discussions to a close, without treating of 
 the solution of simultaneous quadratic systems, or of cubic 
 equations, or of biquadratic equations, to say nothing of 
 equations of liigher degree, except in so far as the general 
 fundamental theory may suffice in particular instances. 
 
 Such matters are to be studied in detail ; but it may 
 be remarked in closing tliat, if a simultaneous quadratic 
 system has only one of its equations of the second degree, 
 or if the equations are homogeneous or symmetrical (vide 
 
HIGHER EQUATIONS. 200 
 
 § 263), means are offered for the deduction of equivalent 
 equations in one variable of the second degree, and the 
 system may in these cases be solved by the methods for 
 quadratics. Indeed, it is often the case that, on account of 
 symmetry, this is true for a system of simultaneous equa- 
 tions of degree higher than the second. Again, any equa- 
 tion of form, aa;-" + hx^ -|- c = 0, may be solved as a 
 quadratic in cc", and the two unaffected equations, a;" = 
 -1- k, which result may then be solved by factoring the 
 functions a'" -|- k and a,-" — 1:, if n be integral, or by invo- 
 lution of the members of x"- = -J- k if n be fractional with 
 numerator 1, or by both devices if n be fractional with 
 numerator > 1. For it must never be overlooked that the 
 solution of an integral equation in one variable, in form 
 yi = 0, is identical with the problem of factoring the func- 
 tion A into the form c (.« — )\) (x — r^) . . . (x — ?•„). 
 Example. — -Find the six sixth roots of -f 1, and of — 1. 
 
 (1) Let x^ = 1- 
 
 then ic" - 1 = = (x^ + 1) (x^ - 1), 
 
 or (x + 1) (x- - X + 1) (.c - 1) (a- + x + 1) = 0. 
 
 This equation is satisfied when any factor = 0. Taking 
 the factors in order, and equating to zero, gives the follow- 
 ing six roots : — 
 
 any one of Avhich, of course, taken six times as a factor, 
 makes -j- 1. 
 
 (2) Let a-« = _ 1 ; 
 
 then a-« + 1 == = (x^ + i) (x^ - i), 
 
 or (x — i) {x- -f IX — 1) {x + t) (a;^ — /^ — 1) = ; 
 
 whence, as before, 
 
210 NUMBER AND ITS ALGEBRA. 
 
 %Aj V • *Aj ' — - ^" • t// ~^~ €■ a «jC ■ ■ ■ 
 
 ' • rvt ^.^ ^^ ^ « ^y, _^ 1 * 
 
 2 ' ' 2 
 
 any one of which, taken six times as a factor, makes — 1. 
 
 331. In the application of JSTumber to concrete problems, 
 the logic of the connection of the numerical statements 
 with the particular concrete conditions must be thoroughly 
 comprehended. It should constitute one of the most im- 
 portant parts of mathematical studies and training. It 
 ought to be no matter for surprise that numerical results 
 are often obtained, totally meaningless in regard to the 
 particular problem. On the contrary, such results should 
 be generally expected, alertly watched for, in order to reject 
 them from the problem in question. 
 
 Number is a twofold continuous magnitude, and there- 
 fore its thoroughgoing application is possible only to two- 
 fold continuous magnitudes. {Cf. § 188). In reference 
 to time, a one-dimensional continuum, all protomonic num- 
 ber (positive and negative, fractional and surd) has intel- 
 ligible application ; but neomonic and complex number 
 could have no application to temporal relations. To space, 
 all number, protomonic, neomonic, and complex, may have 
 due application. Sj)ace, in fact, being a threefold con- 
 tinuum, in a manner transcends Kumber, in the sense of 
 permitting an infinite reapplication of number. We have 
 seen, however, that, given three planes of reference, it is 
 possible to uniquely determine any point in solid space by 
 means of three protomonic numbers, and that it is this 
 circumstance which constitutes the ultimate meaning of 
 the statement that space is tri-dimensional. 
 
 On the other hand, if a problem require a number of 
 men, it is limited in its very terms to primary number; 
 since \ men, or V3 men, would be as inapplicable as 2 -|- 7 t 
 
CONCRETE PROBLEMS. 211 
 
 men, unless, indeed, implicit reference to some continuous 
 magnitude afforded ground for the application of such 
 results ; e.g., if a problem concerns the number of men in 
 a regiment, applicable results are exclusively in primary 
 number, and if such are not found, there is contradiction 
 in the problem as given ; whereas, if a problem concerns 
 the number of men required to dig a ditch, any positive 
 protomonic number might be interpretable. 
 
 Not only must the student expect to find solutions of 
 his equations which have no bearing on a particular prob- 
 lem, but it may be that no solution of a correct algebraic 
 translation of the numerical conditions of a problem is 
 applicable. The interpretation of such results is that 
 the problem is self-contradictory, the required conditions 
 impossible. 
 
 The clear logical principle is, that, if the problem have 
 any solution, it must be yielded among the solutions of 
 any system of algebraic equations which correctly state 
 the numerical conditions of the prolilem, no matter how 
 many inapplicable solutions may also be yielded. If no 
 numerical solution is applicable, the problem is impossible, 
 that is to say, its conditions constitute an absolute contra- 
 diction of any such outcome as was contemplated. 
 
 In many minor ways, also, it is impossible to restrict 
 the perfect generality of numerical operations, and the 
 numerical symbols of the algebra. For example, an un- 
 known number may be added to another; but whether the 
 addition increases or decreases a given number, it is rash 
 to say before the quality of the unknown is discovered. 
 
 Thus it is ill-considered to demand that 15 be divided 
 '' into two such parts that the greater shall exceed 3 times 
 the less by as much as half the less exceeds three." For 
 
212 NUMBER AND ITS ALGEBRA. 
 
 (representing the greater by x, and the less by 15 — x) the 
 numerical conditions are plainly intended to be 
 
 a; _ 3 (15 - a-) = i (15 - »•) - 3, 
 whence x = 11, and 15 — a; = 4. 
 
 But on turning to the requirement it is seen that 11 falls 
 short of, not " exceeds " 3 times 4 by as much as half of 4 
 falls short 3. In line, one cannot choose the issue of abso- 
 lute facts according to his whim, and the problem as given 
 is presumptuous ; all that could have been safely required 
 were numbers which would give equal differences for the 
 intended subtractions. 
 
 The indeterminate result - has already been referred to ; 
 
 it may mean that any number answers the requirement, or 
 it may be susceptible of evaluation. 
 
 332. Very often all that is required may be discovered 
 from equations Avithout solving them, by transformations 
 into various equivalent forms. Consequently the principles 
 governing the equivalence of derived and original systems, 
 and the study of functions, as distinguished from equations, 
 have, besides their theoretical importance, a practical use- 
 fulness quite apart from their bearing upon solution. In- 
 deed, the whole subject of the solution of equations has 
 widened into that of the variation of functions. For a long 
 time equations have been losing, and functions gaining, 
 prominence, both in analytical importance and practical 
 utility. Nowadays, instead of seeking merely the values 
 of the variables which cause the function to vanish, that is, 
 solving the equation ^ (x) = 0, all values of the variable, 
 as it varies continuously, and the corresponding values of the 
 function, are considered. The function is calculated for 
 enough specific values of the variable to give a clear idea 
 
VARIATION AND GRAPHS OF FUNCTIONS. 213 
 
 of its variation. Especial attention must be given to such 
 values of the variable as cause the function to pass through 
 critical values, - — zero among others. 
 
 Independently of the analytical treatment of geometry 
 (where the purpose of geometrical investigation is so 
 powerfully served by the numerical analysis), this modern 
 way of regarding analytical functions receives reciprocal 
 assistance — if not theoretically, at least as affording the 
 bodily eye a clear representation — by drawing what is 
 called the graph of the function. 
 
 The graph of a function of one variable is plotted by 
 laying off, to any scale, sects proportional to {vide § 213) 
 arbitrarily chosen values of the variable, in a straight line, 
 to the riglit or left of a point, according as the chosen 
 value is positive or negative ; and at the points so deter- 
 mined, laying off perpendicularly (one way for -|-, the other 
 for — ) sects projjortional to the corresponding values of 
 the function, plotting the end points of tiiese sects. By 
 sketching a curve through such points, a representation of 
 the corresponding variations of function and variable is 
 afforded. The curve so obtained will generally give warn- 
 ing of critical values of the function, at which stages closely 
 consecutive values of the variable must be taken to insure 
 a correct graph of the function. 
 
 It is iisual to write j/ = 4> (-^O' ^^^*^ ^^^^ ^^^® values of y 
 corresponding to selected values of x. 
 
 For example, let the student plot the graphs of the 
 following functions, also tabulating the chosen values of x 
 with the corresponding ij 's. 
 
 (l)y = l-^ (2) y=(yi^. (3)y = r4i;- 
 
214 NUMBER AND ITS ALGEBRA. 
 
 At first one may be disposed to examine far more values 
 than necessary. Always plot first the y's corresponding 
 to X 's which allow evaluation by inspection, — often these 
 will suffice. 
 
 A systematic study of the variations of functions would 
 be surprisingly interesting, even to students who have 
 hitherto found their mathematics dull. The subject could 
 be introduced profitably, even at very elementary stages of 
 algebraic studies, and, while stimulating interest and sus- 
 taining attention, would give a better preparation, both for 
 continued study of pure mathematics, and for the manifold 
 I^ractical uses of mathematics in other sciences, than do the 
 methods at present in vogue. 
 
 333. The general theor^^ of Inequalities, and of Maxima 
 and Minima values of functions also, deserves a more 
 thorough and independent treatment than it commonly 
 receives in our elementary text-books. The fundamental 
 ■principles are of so simple and instructive a character, and 
 form so valuable an introduction to the methods of analysis 
 employed in more advanced studies, that our usual elemen- 
 tary courses need in this matter thoroughgoing reformation. 
 The theory of inequalities is the best introduction to that 
 of infinite series, and the latter is indispensable in the 
 study of logarithms and many other subjects which are at 
 once entered upon in the first-year courses of our colleges 
 and universities. 
 
 For the most part, the logic of inequalities, and the deri- 
 vation of equivalent inequalities, runs parallel to the 
 analogous theory for equations, except where restrictions 
 intervene in regard to inequalities, owing to the fact that 
 the members of an inequality cannot, like the members of 
 anequation, be interchanged. 
 
INEQUALITIES. 215 
 
 The student may be reminded (vide § 198), in this con- 
 nection, that there is no comparison in the ordinary sense 
 of greater and less between complex numbers, because such 
 numbers are in terms of heterogeneous units. Of course 
 this general statement includes particular cases where one 
 of the numbers is either protomonic or neomonic, and the 
 other complex, or where one is protomonic and the other 
 neomonic. With complex numbers, as we have seen, the 
 comparison must be between their moduli. 
 
 A fruitful source of error with beginners (on account of 
 the prevailing inadequacy of number concepts) is neglect 
 of the fact that any negative number is less than zero 
 (—00 < 0), and that « > ?/, or x < y, according as x — ?/ 
 is positive or negative. 
 
 The freedom of transposition of terms with changed 
 signs, in an inequality, is quite as immediate a corollary of 
 axiomatic judgments, and the significance of the symbols, 
 as the like freedom in equations. For it is the same 
 axiom that, if equals be added to unequals, the results are 
 correspondingly unequal, as that, ''if equals be added to 
 equals, the results are equal." (Vide § 42, foot-note.) 
 
 Examples. 
 
 (I.) Prove: x" -\- y- ':> 2 xy, if x and y are protomonic 
 numbers, (x — y)' is positive whether x > ?/ or a- < y ; but 
 (x — yY — cf- — 2 xy + y-, therefore x"^ — 2 xy -\- y- is posi- 
 tive, and therefore x- + y"^ > 2 xy. 
 
 In order to emphasize the extreme importance of limit- 
 ing values, I have allowed a fallacy to pass unchallenged 
 in this argument. It is not true that x"^ -]- y^ > 2 xy. For, 
 although (x — ?/)- is positive, it ma}^, if x = y, be zero, 
 when x"^ -\- y'^ = 2 xy ; consequently the true statement is 
 
 x- -\- y" not < 2 xy. 
 
216 NUMBER AND ITS ALGEBRA. 
 
 (II.) Prove: The sum of a positive fraction and its 
 reciprocal is not less than 2. 
 
 Consider ? + 1^ not < 2. (1) 
 
 y ^ 
 
 Multiplying each member by ocij gives 
 
 a-2 + f- not < 2 :nj ' (2) 
 
 But (2) has just been proved ; therefore its equivalent 
 inequality, (1), is true. 
 
 (III.) Prove : Half the sum of two iwsitive numbers is 
 not less than the square root of their product. 
 
 OC 1 7/ 1 
 
 Consider — ~^—-^ not < (xi/)^ (1) 
 
 '7*" I ' '/*?/ I 7/ 
 
 Squaring gives — ' " ' -^ ' "^ not < xy. (2) 
 
 But, by Ex. I, a:^ -|- y- not < 2xij; 
 
 therefore x^ -\- 2 xy -\- y- not < 4 a-^/ ; 
 
 therefore (2), and therefore its equivalent inequality (1), is 
 
 true. 
 
 This proposition is readily generalized by the reasoning 
 called " mathematical induction," * by showing that if it 
 is true for any number of numbers, it is true for one more : 
 — but it is true for two, therefore for three, and so on. 
 Thus we prove for n numbers t : — 
 
 a-{-h -\-c-\- . 
 
 n 
 
 not < {(the . . .) '/'\ 
 
 * Not true and proper iuduction, but absolutely cogent deduction, 
 involving no assumption except the validity of reason, the postulate of 
 all thought. 
 
 t Tlie left-hand member is called the "arithmetic mean," and the 
 right, the "geometric mean," of the n numbers. 
 
MAXIMA AND MINIMA. 217 
 
 A maximzim of a function does not mean its greatest pos- 
 sible, nor a minimum its least piossihle, value. A maximum 
 value of a function is a value toward which it increases, 
 and from which it decreases as the variable continuously 
 varies, whether by increasing or decreasing. And a mini- 
 mum value of a function is a value before which the func- 
 tion decreases, and after Avhich it increases as the variable 
 varies continuously, whether by increasing or decreasing. 
 Maxima and minima for a function may repeat, definitely 
 or indefinitely ; or there may be only one maximum or one 
 minimum for a function, in which case the maximum is 
 the greatest possible, or the minimum the least possible 
 value. 
 
 The general connection between inequalities and the 
 theory of maxima and minima values of functions is ex- 
 emplified in the principle, that if <^ (.y, y, z, . . .) and 
 xj; (x, y, z, . . .) be two functions of the same variables 
 such that 
 
 <^ {x, y,z, . . .)= N, (1) 
 
 and xp (x, y,z,.. .) not > <^ (.r, y,z, . . .); (2) 
 
 and if any values oi x, y, z, . . . , say, a, h. c, . . . , can be 
 found which satisfy (1) and at the same time make (2) 
 an equation, then i/^ («, h, c, . . .) is a maximum value of 
 
 Also, if xp (x, y,z, . . .) = N, (3) 
 
 and </, (.r, y, z, . . .) not < ^ {x, y, z, . . .) ; (4) 
 
 and if any values oi x, y, z, . . . , say, a, b, c, . . . , can 
 be found which satisfy (3) and simultaneously make (4) 
 an equation, then ^ (a, h, c, . . .) is a minimum value of 
 
 Example. — Find the maximum volume of a rectangular 
 
218 NUMBER AND ITS ALGEBRA. 
 
 parallelopiped of given surface, and minimum surface for 
 given volume. 
 
 Let X, y, and z be the lengths of three adjacent edges ; 
 then the geometrical data of the problem are, the area of 
 the surface is 2 (xij -f- ^^ -\-y^i ^-nd the volume of the solid 
 is xyz. ( Vide § 25.) Writing u = xy, v = xz, w = yz, the 
 area becomes 2 (w + v -)- iv), and the volume, ^ uviv. 
 
 Hence the analytical problem is to find the maximum 
 (or maxima) of the function -y/uviv, given the function 
 2 (^u -\- V -\- w) = a constant. Since the meaning of the 
 problem excludes negative number (ynVZe § 331), the prob- 
 lem as assigned is equivalent to finding the maxima of 
 
 ■y/uvw given, — ^ — = k ; for (considering only posi- 
 tive protomonic numbers, all that apply to the problem) 
 ■yjuvw is maximum for the same values of the variables 
 that ■y/uvtv is maximum ; and given 2 (^u -\- v -{- ?/') = a 
 
 constant, we have —^ — — — = k. But this transforma- 
 ' 3 
 
 tion was adopted because we know (Ex. Ill, above) that 
 
 ' - not < {iivn'Y 
 
 o 
 
 which is to say mat 
 
 (iivw)''^ not > -— t — IL — J 
 o 
 
 Consequently we have 
 
 "+;+" = k, (1) 
 
 and (uvwy not > !L±_L+J1'. (2) 
 
 o 
 
 It only remains to find values of u, r, ?<• Avhich satisfy 
 (1) and make (2) an equation. But (2) cannot be an equa- 
 
MAXIMA AND MINIMA. 219 
 
 tion unless if = v = v\ This, therefore, is the condition, 
 and (uviv)^^ is uniquely maximum when m = i; = ?<; ; and 
 (remembering the meaning of xi, v, and w), if u = v = w, 
 
 then X = 2/ = z =i — j , where K = the given area. 
 
 The reciprocity, implicit in the theorem immediately pre- 
 ceding this example, gives the same condition (x = >/ = z) 
 for the solution of the second part of this problem ; but the 
 beginner may have failed to note the reciprocal relations 
 of the conditions for maxima and minima of two functions 
 displayed in the general investigation. 
 
 In like manner, then, the second part of the problem 
 
 gives 1 
 
 (uvwy = I, (1) 
 
 11 -A- V -\- w 1 
 
 and not < (uvw')^ ; 
 
 o 
 
 w + V + ?y . . . . 
 
 whence, is uniquely minimum when ic = v = tv , 
 
 and therefore, as before, the area is minimum when x = y 
 = .v = (i)"% where L is the given volume. 
 
APPENDIX. 
 
 PEDAGOGICAL NOTE. 
 
 The primary concept of number is the same in all men, and the 
 conception conld not be obstructed, even if teachers set tliemselves 
 to thwart it. As an original question, therefore, there is little peda- 
 gogical import in discussion of methods of stimulating the infant 
 mind to definite specialization of various manys. ( Vide § 2.) 
 
 It would be enough to jjoint out to the inexperienced teacher that 
 when the time for definite and systematic specialization of manys 
 comes, a child can learn the general system as a Avhole better than 
 he can learn it piecemeal; that the so-called arithmetic of the first 
 two or three grades in our schools is properly a matter of language, 
 a matter of naming, in the manner of the child's linguistic environ- 
 ment, universal concepts already attained by the young innocents 
 when committed to the mercies of the primary school. There is no 
 more sense in attempting to explain icJiut "twelve" is, than in 
 making a like effort in regard to "time" or "space," or such 
 concepts as "more," " less," "greater," "equal." The child really 
 knows these things as well as his teacher. Even if a child lived 
 eight years in an English-speaking society Avithout learning the 
 English name for the special many, " twelve," or even without 
 having definitely recognized it, the substance of the thought, as 
 distinguished from the symbolism of a particular language, would 
 nevertheless be familiar to him, and nearly as well known as it can 
 be until one gives profound study to epistomology. 
 
 The simple and easily taught subjects of counting, and the ele- 
 mentary phases of numerical operations, have been confused by the 
 inane verbosity of pedagogical writers. In his admirable Philos- 
 ophy of Education* (one of the best books ever written on the 
 
 * Translated in the International Echication Serks. 
 221 
 
^ 
 
 222 APPENDIX. 
 
 subject) Piosonkranz justly remarks, " Treatises written upon it 
 [education] abound more in sballowness than any other literature. 
 Shortsightedness and arrogance find in it a most congenial atmos- 
 phere, and uncritical methods and declamatory bombast flourish as 
 nowhere else." 
 
 It is enough to point out one example of injurious methods of 
 dealing with imaginary difficulties. Ignorance of psychology and 
 lack of common-sense have led many superintendents, even where the 
 minimum school age is eight years, to i:>rohibit all mention of num- 
 bers greater than ten in the "first grade," and greater than twenty 
 in the "second." This makes both the teaching and the learning a 
 sham, and the nemesis of all dishonesty dogs it. It is benumbing to 
 honest, depraving to vain or deceitful, pupils. 1 know a city whose 
 school superintendent has instituted such methods with fatuous 
 braggadocio, where a visitor, after witnessing an hour's counterfeit 
 teaching, — What is one and two? one and three? two and three ? 
 If you had five apples, and gave one to Mary and one to John, how 
 many would you have left ? and so forth, with occasional introduc- 
 tion of such prodigious numbers as nine or ten, — followed the class 
 to the playground. It was the season of huUn-gnll. Each urchin 
 knew well the score of the treasures in his bulging pockets. "Iluliy- 
 gull, hand-full, how many ? " challenged one 3'oung plunger. 
 "Twenty-two," guessed his opponent. One second for the count 
 and the subtraction, and back came the triumphant cry, " Give me 
 seven to make it twenty-tMo! " 
 
 On the other hand, there seem to be peculiar difficulties, even for 
 adults, in attaining the concept of nmnber absolutely essential to 
 comprehension of arithmetic, — the discernment of number as a 
 continuous magnitude with fractional parts and qualitative distinc- 
 tions termed positive and negative. Here pedag6gical devices are 
 sorely needed. It is not enough to warn against mistaken interfer- 
 ence; the teacher's skill will be taxed to the utmost to stimulate the 
 minds he is guiding to develop concepts of a high order of abstrac- 
 tion, and such as, left to himself, the pupil would never form at all. 
 
 As "object-lessons" to young children — the aim being to clear 
 up normal and universal concepts of quantity — presentation of 
 yard-sticks and foot-rules, gallon and quart measures, etc., may be 
 ii useful practice, and it does teach about fractions ; but it does not 
 
APPENDIX. 223 
 
 at all immediately suggest fractions of numbers. A fraction of a 
 line is a line, of a solid is a solid ; and these can be and universally 
 are discerned under the primary concept of number, and without 
 discernment of numerical fractions. Every savage knows that a 
 quart is a fraction of a gallon. The "object-lessons" mentioned 
 really constitute an elementary discipline in geometry (if every 
 primary school exercise must be labelled with the name of some 
 science). Lines and solids are spacial entities, and contempla- 
 tion of their relations is primarily a geometrical exercise. I say 
 X>riniarili/, because any two magnitudes of the same kind have an 
 absolute numerical relation ; but to see that a quart is one-fourth of 
 a gallon (only another way of saying that four quarts equal a 
 gallon) is not at all to see the number called one-fourth in the 
 systematic terminology of arithmetic. Every child sees the former, 
 an obvious geometric fact, — too many of his teachers have never 
 discerned the latter. {Vide §§ TS-90.) The primary concept of 
 number is universal and normal to the human mind, just as the 
 concei^t of space is common and original to all men. Systematic 
 development of the latter gives geometry, of the former gives arith- 
 metic. The developments of the one are quite as much matters of 
 fact as the developments of the other. 
 
 Ontological definition of number is as little to be required of 
 arithmetic as like definitions of space of geometry, or of matter and 
 force of physics. Each science simply takes its respective common 
 notions, which it develops according to inherent characteristics. 
 The developed science always casts light back upon primary notions 
 (Cy. the effect of Xon-Euclidean geometries upon native ideas of 
 space, or the exigencies of dilemmas in physics upon naive con- 
 cepts of matter) ; but no such questions are to be raised for young 
 students beginning to study arithmetic, geometry, or physics. The 
 most important maxim for wise teaching in any science is never to 
 set delimitations which confine development and entomb thought in 
 empiricism, — never to clip the growing tree at the top. 
 
 Now every man (and every dog) knows that one side of a triangle 
 is less than the sum of the other two sides ; but no one would sup- 
 pose that this circumstance entitled every man to opinions concern- 
 ing the conclusions of geometry ; yet similar presumptions are rife 
 among teachers of arithmetic. Men possessing (in common with 
 
224 APPENDIX. 
 
 their most savage brethren) only the primary concept with which 
 arithmetic begins, often misrepresent as matter of convention or 
 symbolic jugglery the arithmetical conclusions that number is a 
 continuous magnitude, with fractional parts, and qualitative dis- 
 tinctions — • as much matters of fact as any conclusions of geometry. 
 
 The developments of the number concept are undreamed of to the 
 man whose only thought thereof is his abstraction from a flock of 
 sheep or pile of coins. As soon as man's energetic and organizing 
 thought develops this concept, the insight is infallibly attained that 
 number is a continuous magnitude, not concrete, not material, but 
 none the less real. 
 
 The concept which appears to me most like the first development 
 of primaiy number, which includes all ratio (including fractions), 
 and the qualitative distinctions, positive and negative, is Time. 
 Even children recognize time as a continuous magnitude, — as more 
 or less; that of two times one must be definitely greater than, equal 
 to, or less than the other,- — and the qualitative distinctions of i^ast 
 and future. The analogy of jjresejii and zero is also perfect. 
 
 I suggest that teachers, called upon as they always are to teach 
 arithmetic to children somewhat too 3'oung for the reasoning and 
 insight required, would do well, in attempting to stimulate the con- 
 ceptual energies of their pupils, to use definite times rather than 
 lines, surfaces, solids, etc., in illustrating numerical relations sub- 
 sisting between any two magnitudes of the same kind. Altliough 
 no better success can be assured in this way (for any fraction or part 
 of a time is a time and not a number); yet from the very fact that 
 times cannot be seen or handled, the abstracting functions of the 
 mind are brought into play, and there is better ground of hope that 
 the desired conception will take place than if objects of sense- 
 perception had been presented. It may be well to remark in this 
 connection that in all illustration great care is demanded lest the 
 analog hide instead of revealing. Rosenkranz, in his valuable Phi- 
 loHophy of Education, already referred to, wisely cautions : " Our age 
 inclines at present to the superstition that man is able, by means of 
 simple sense-perception, to attain a knowledge of the essence of 
 things, and thereby dispense with the trouble of thinking. It is 
 vain to try to get behind things, or to comprehend them, except by 
 thinking," 
 
APPENDIX. 225 
 
 I am not aware that the suggestion has been made hitherto ; but, 
 in the Hght of tlie above warning against abuse, 1 am convinced that 
 teachers of arithmetic would do well to contemplate the similarity 
 of the concepts Time and Number, as the latter is conceived, not in 
 the savage stadium of thought, but in its first scientific develop- 
 ment. 
 
 There are many subsidiary advantages also in choosing the uni- 
 versally conceived magnitude, time, for such illustrations. The 
 mind is unconsciously but directly led from the tyranny of material 
 categories of thought; and the human mind, once made sensible of 
 its powers, will never again suffer its conceiJtions to be shackled in 
 this native slavery of the race. 
 
 Rightly employed, arithmetic might be used with more efficacy in 
 the intellectual emancipation, which is one of the chief ends of edu- 
 cation, than any subject in the curricula of common schools. There 
 is no other field where one pure idea is developed in such unbroken 
 consistency, and such freedom from involvement in complex rela- 
 tions with foreign elements. 
 
 In conclusion, no matter whether the pupil at a given stage be in 
 a position to see the end of his studies or not, it is evident that the 
 teacher, with no notion of the end, will be a faulty guide, since he 
 leads he knows not whither. 
 
 
INDEX. 
 
 The numbers refer to sections. 
 
 Addition, Si, 37, 45-, 73, 88, 162, 184, 
 199. 
 
 Algebra, 20-, 32, 71, 15(3, 192, 224, 
 236, 251. 
 
 Algebraic Form : vide Form. 
 
 Analytics, 27, 169, 234. 
 
 Angles, 211, 212. 
 
 Arithmetic, 14, 30-32, 71, 10(5, 192. 
 
 Arithmetic, Pure and Applied, 31. 
 
 Association, Laws of, 39, 43, 46, 47, 
 51, 62, 71, 73. 
 
 Axioms, 42, 333. 
 
 Base of Notation : cf. Radix, Scales, 
 7, 17. 
 
 Billion, 8. 
 
 Calculation, 29, 95, 210. 
 
 Calculus, 130, 132, 222. 
 
 Cardinal Numbers, 10. 
 
 Circulating Decimals: vide Re- 
 peating. 
 
 Coefficients, Theorem of Undeter- 
 mined, 267-268. 
 
 Commensurable, 83, 145, 174, 205. 
 
 Commutative Laws, 34, 38, 43, 46, 
 47, 51, 62, 65, 73. 
 
 Complex Number, 27, 145, 180, 
 186-, 193-, 291. 
 
 Comijosition of Ratios, 84. 
 
 Computation, Devices of, 72, 93, 95. 
 
 Concept, Number: vide Number 
 Concept. 
 
 Concepts, Elemental Mathemati- 
 cal, 222, 228-. 
 
 Concrete Problems, 27, 30, 48, 210, 
 
 331. 
 Congruence, 298. *■ 
 Continuity of Number, 80-82, 97, 
 
 188, 198. 
 Continuity, Principle of: vide 
 
 Principle. 
 Counting, 5, 9, 12, 14. 
 Cube : Cube Root, 58, 76, 249, 290. 
 Decimal Fractions: cf. Radix, 17. 
 Decimal Notation : cf. Notation, 17. 
 Definitions : cf. respective heads : — 
 
 Addition, 37. 
 
 Algebra, 20. 
 
 Arithmetic, 32. 
 
 Calculation, 29. 
 
 Commensurable, 83. 
 
 Counting, 5. 
 
 Division, 49. 
 
 Evolution, GG, 68. 
 
 Finding Logarithm, G9. 
 
 Fraction, S3. 
 
 Incomtnensurablc, 83. 
 
 Inrolution, (50. 
 
 Mathematics, 225. 
 
 Multiple, 83. 
 
 Multiplication, 40. 
 
 Notation, 14. 
 
 Primary Number, 2, 3. 
 
 Proportional, 213. 
 
 Patio, 83. 
 
 Submultiple, 83. 
 
 Subtraction, 41, 42. 
 
 Surd, 83. 
 
 Etc. 
 
 22*7 
 
228 
 
 INDEX. 
 
 Degree, 169, 323. 
 Denominator, 87. 
 Dialectic, 11, 4.5, W, 80-, 110, 113- 
 
 114, 181, 202, 230. 
 Dimensions, 188, 231. 
 Discrete, 2, 80-81, 97, 229. 
 Distributive Law, 52-, 73. 
 Division, 49, 73, 84, 89-, 90, 98, 123, 
 
 103, 184, 199. 
 Division, Algebraic, 254-. 
 Duodecimals, 17, 282-283. 
 Enumeration, 5. 
 Equation, Synthetic, 40. 
 Equations, — 
 
 Classification of, 299. 
 Equivalence of : vide Equiva- 
 lence. 
 Higher: cf. Equations, Theory 
 
 of, 330. 
 Indeterminate, 317. 
 Quadratic, 313-. 
 Roots of : vide Hoots. 
 Simultaneous, 300, 317, 327, 330. 
 Solutions of: vide Solutions. 
 Systems of: vide Equivalence. 
 Theory of, 2G8-, 30.3-. 
 Transformation of, 309-, .332. 
 Equivalence of Equations, 319-. 
 Evolution, 34, 68, 76, 92, 98, 143, 
 
 163. 
 Exponential Notation, 57, 146, 150, 
 
 156. 
 Exponents, Law of, 64, 146, 158, 
 
 191. 
 Factors, Algebraic : cf. Hiyhest and 
 Theory of Equations, 251, 304. 
 Finding Logarithm, 34, 69. 
 Form, Algebraic, 28, 156, 2.36, 251, 
 
 301. 
 Formula, 40, 299. 
 
 Formulae of Definition, 42, 49, 68, 
 69. 
 
 Fractions, 78-, 80-85, 89-, 143, 278, 
 285. 
 
 Functions, — 
 
 Classification of, 169, 263, 265. 
 Variation of, 332. 
 Graphs of, 332. 
 
 Fundamental Theory: vide The- 
 ory. 
 
 Geometry, 22, 25, 181, 193, 227, 230, 
 2;!5, 297. 
 
 Graph of Function, 332. 
 
 Greater, 11, 46, 116, 198, 240, 242, 
 252, 333. 
 
 Greatest Common Measure: vide 
 Submultiple. 
 
 Highest Common Factor, Alge- 
 braic, 251, 272. 
 
 Homogeneous Functions, 263. 
 
 Homogeneous Manifoldness, 229. 
 
 Identity: vide Formula. 
 
 "Imaginary:" cf. Neomonic, 99, 
 101, 145, 181, 191. 
 
 Incommensurable, 83, 95, 145, 174, 
 205. 
 
 Indeterminate Equations: vide 
 Equations. 
 
 Indeterminate Forms, 130-, 135-. 
 
 Indices, Law of, 64, 146, 158, 191. 
 
 Inequalities, 3.33. 
 
 Infinitesimals, 222. 
 
 Infinity, 133-, 222. 
 
 Integers, 1, 241. 
 
 Involution, 34, 59, 60, 75, 92, 163, 
 184. 
 
 " Irrational " : (/. Surd and Incom- 
 mensurable, 104, 145, 181. 
 
 Less, 11, 46, 116, 198, 240, 242, 252, 
 333. 
 
 Logarithm, Finding the : vide 
 Finding the Logarithm. 
 
 Logarithms, 57, 150-, 333. 
 
 Lo"-ic 21 222 
 
INDEX. 
 
 229 
 
 Lowest Conimou Multiple : vide 
 
 Multiple. 
 Lowest Common Multiple, Alge- 
 braic, 251, 276. 
 Magnitude, 11, 207, 210, 229. 
 Manifoldness, 229-. 
 Many, 2, 228. 
 Manys, Specialized, 2, 37. 
 Mathematics, 105, 222-, 234. 
 Maxima, 333. 
 Measure, 204, 205, 209. 
 Measurement, 12, 80, 203, 211, 227. 
 INIensuration, 203-, 211, 220, 227, 
 
 231. 
 Metre, 220. 
 
 Metric System, 31, 220. 
 Minima, 333. 
 
 Minus, Double Meaning of-, 120. 
 Modulus of Complex Numbers, 
 198, 293, 333. 
 Congruence, 298. 
 Logarithms, 150. 
 Multiple, 83. 
 
 Multiple, Lowest Common, 242. 
 Multiplication, 34, 44-48, 73, 89, 
 
 121, 163, 184, 199. 
 Negative Number, 27, 99, 104, 110, 
 
 117, 143, 181, 333. 
 Neomon, 26, 104, 181, 182, 191. 
 Neomonic Number, 27, 104, 145i 
 
 181- 191. 
 Nine, Remainders to, 287-. 
 Nines, Casting out : vide Nine, 
 
 Eemainders to. 
 Norm of Complex Numbers, 197, 
 
 293. 
 Notation, 14, 17, 57, 90, 120, 150, 
 
 156, 277-, 280-, 284-. 
 Notation, History of, 16. 
 Number Concept, 2-, 27, 34,71, 78-, 
 
 80, 84-, 96-115, 155, 179-, 186, 
 
 192, 202, 226, 230. 
 
 Number, — 
 Development of : vide Number 
 
 Concept. 
 Origin of : vide Origin. 
 Primary, Fractional, Surd, Posi- 
 tive and Negative, Neomomic, 
 
 Complex : vide Corresponding 
 
 heads. 
 Numbers, Theory of : vide Theortj 
 
 of Numbers. 
 Numerals, 6, 8, 15. 
 Numeration, 5, 7, 17. 
 Numerator, 87. 
 One, 2, 128, 181. 
 Operations, 30, 34, 45-, 60. 73, 86, 
 
 104, 116, 130-, 135-184, 199, 
 
 210. 
 Ordinals, 10. 
 
 Origin of Number, 2, 5, 2.'50. 
 Physics, 21, 27, 211, 221, 224. 
 Plus, Double Meaning of +, 120. 
 Primary Number, 1-4, 11, 54, 60. 
 
 70, 96, 181. 
 Prime Numbers, 238-. 
 Primeness, Algebraic, 251, 275-. 
 Principle of Continuity, 61, 97-, 
 
 103-, 107, 155. 
 Projective Geometry, 227. 
 Proportionality, 211-219. 
 Protomonic Number, 101, 145, 154. 
 Quadratics : vide Equations. 
 Quality, 14, 27, 32, 78, 110, 117, 120, 
 
 224, 228. 
 Quantity, 198, 224, 228-. 
 Radix Fractions, 17, 284-. 
 Radix Notations, 17, 280-. 
 Radicals, Radical-Surds, 83, 94, 
 
 145, 157-, 170-, 249. 
 Ratio, 25, 78-, 83, 84, 132, 203, 208. 
 Reciprocal, 90. 
 Remainder, Least, 243, 255. 
 Remainder Theorem, 258-. 
 
230 
 
 INDEX. 
 
 Remainder to Nine, 287-. 
 Repeating Decimals, 249, 284-. 
 Roots of Equations, 268, 271, 300, 
 
 306-, 315, 323. 
 Rules, 43, 88, 93, 119, 121, 124, 242. 
 Scales, Notational, 277-. 
 Series, 267, 333. 
 Solutions of Equations, 300-302, 
 
 331. 
 Square : Square Root, 58, 76, 93-, 
 
 191, 200, 249, 290. 
 Stirpal, 145. 
 
 Submultiple, 83, 84, 205, 251. 
 Submultiple, Highest Common, 
 
 205, 240, 251. 
 Subtraction, 34, 41-, 45, 89, 98, 117, 
 
 162, 199. 
 Surds, 80, 83, 94, 143, 145, 174, 249. 
 Symbols, 14, 17, 20, 23, 32, 192. 
 Symmetrical Functions, 265. 
 
 Synoptic Mathematical Methods, 
 
 234. 
 Synthetic Mathematical Methods : 
 
 vide Synoptic. 
 Synthetic Equation, 40. 
 Systems : vide Equivalence of 
 
 Equations, and Manifoldness. 
 Terminology, 83, 101, 145, 198, 205. 
 Theory, Fundamental, 2, 11, 26, 30, 
 
 71, 97-, 107, 132, 207, 222, 225, 
 
 228, 230, 331. 
 Theory of Equations, 268-, 303-. 
 Theory of Numbers, 298. 
 Undetermined Coefficients, 267- 
 
 268. 
 Unit, 203-206, 220. 
 Unity, 2, 16, 128, 181, 203-. 
 Variation of Functions, 332. 
 Variables, 299. 
 Zero, 16, 116, 125-, 182, 189, 222. 
 
 
ERRATA. 
 
 Page 52. 
 
 2d. line from bottom : instead oi b — d read h j d. 
 
 Page 83. 
 
 Top line : instead of " point to the vertices " read points 
 to the ojjj^osite vertices. 
 
 Page 105. 
 
 13th line from bottom : sign of equality is omitted in 
 latter portion of the line. 
 
 Page 115. 
 
 7th line from bottom : read § 198. 
 
 Page 165. 
 
 3d line : read x — y in denominator. 
 
 Page 174. 
 
 11th line: read c?3r~^ 
 
 Page ISO. 
 
 7th line: read^ +^. 
 r v 
 
 Page 203. 
 
 11th line : instead of 1 — 6 in second denominates-, read 
 1-x. 
 
 
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