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THE PHILOSOPHY OF ARITHMETIC AS DEVELOPED FROM THE THREE FUNDAMENTAL PROCESSES or SYNTHESIS, ANALYSIS AND COMPARISON CONTAINING ALSO A HISTORY OF ARITHMETIC REVISED EDITION. EDWARD BKOOKS, PH. D., Superintendent of the Public Schools of Philadelphia. LATE PRINCIPAL OF STATE NORMAL SCHOOL. PENNSYLVANIA, AND AUTHOR OF A NORMAL SERIES OF MATHEMATICS. "The highest Science is the greatest simplicity." PHILADELPHIA : NORMAL PUBLISHING COMPANY. 1904. QA Entered according to Act of Congress, In the year 1876, by EDWARD BROOKS, In the Office of the Librarian of Congress, at Washington. COPYRIGHT, 1901, by EDWARD BROOKS LECTROTYPKI> A PRINTKD BY THE WICK F.aH*M PRINT-ISO CO., LANCASTER, PA. PREFACE. T"\ROGRESS in education is one of the most striking characteris- _L tics of this remarkable age. Never before was there so general an interest in the education of the people. The development of the intellectual resources of the nation has become an object of transcendent interest. Schools of all kinds and grades are multi- plying in every section of the country; improved methods of train- ing have been adopted ; dull routine has given way to a healthy intellectual activity ; instruction has become a science and teach- ing a profession. This advance is reflected in, and, to a certain extent, has been pioneered by, the improvements in the methods of teaching arith- metic. Fifty years ago, arithmetic was taught as a mere collection of rules to be committed to memory and applied mechanically to the solution of problems. No reasons for an operation were given, none were required ; and it was the privilege of only the favored few even to realize that there is any thought in the processes. Amidst this darkness a star arose in the East ; that star was the mental arithmetic of Warren Colburn. It caught the eyes of a few of the wise men of the schools, and led them to the adoption of methods of teaching that have lifted the mind from the slavery of dull routine to the freedom of independent thought. Through the influence of this little book, arithmetic was transformed from a dry collection of mechanical processes into a subject full of life and in- terest. The spirit of analysis, suggested and developed in it, runs to-day like a golden thread through the whole science, giving sim- plicity and beauty to all its various parts. (iii) IT PRKPACE. No one who did not in his earlier years learn arithmetic by the old mechanical methods, and who has not experienced the transition to the new analytic ones, can realize the com- pleteness of the revolution effected by this little work. But great as has been its influence, it should be remembered that it does not contain all that is essential to the science of numbers. Analysis in it.s minion, has done all that it was possible for it to accomplish, but it is not su ftiririit for the perfection of a science. There must be Hynthetic thought to build up, as well as analytic thought to separate and simplify. Comparison and generalization have an important work to perform in unfolding the relations of the various parts and in uniting them by the logical ties of thought, which should bind them together into an organic unity. What we now need for the perfection of the science of arithmetic and our methods of teaching it, is a more philosophical conception of its nature, and a logical relating of its parts which analysis leaves in a disconnected condition. It is worthy of remark that urithmetic,in respect to logical symme- try and completeness, differs widely from its sister branch geom- etry. The science of geometry came from the Greek mind almost as perfect as Minerva from the head of Jove. Beginning with definite ideas and self-evident truths, it traces its way, by the processes of deduction, to the profoundest theorem. For clearness of thought, closeness of reasoning.and exactness of truths, it is a model of excel- lence and beauty. It stands as a type of all that is best in the classi- cal culture of the thoughtful mind of Greece. Geometry is the per- fection of logic ; Euclid is as classic as Homer. The science of numbers, originating at the same time, seems to have presented less attractions or greater difficulties to the Greek mind. It is true that the great thinkers grew enthusiastic in the contemplation of numbers, and spent much time in fanciful specu- lations upon their properties, but this did comparatively little for the development of the science. The present system of arithmetic i mainly the product of the thought of the past three or four cen- tariefl. Developed by minds less logical than those of the old Greeks, and growing partly out of the necessities of business, it seems not to have acquired that scientific exactness and finish which belong to the science of geometry. That it has intrinsically as logical a basis and will admit of as logical a treatment, cannot be doubted. To endeavor to exhibit the true nature of the science, show the logical relation of its parts, and thus aid in placing it upon a logical foundation beside its sister branch, geometry, is the object of the present treatise. The work is divided into five parts, besides the Introduction. The Introduction contains a Logical Outline of Arithmetic, and a brief History of the science, including an account of the Origin of the Arabic system, the Origin of the Fundamental Operations, and an account of the Early Writers on the science. The facts pre- sented have been gathered from a variety of sources, and have been carefully compared, so far as was possible, with the originals, to secure entire accuracy in the statements. The principal author- ities followed are Leslie, Peacock, De Morgan, Fink, and Ball. As much is presented as it is supposed will be of interest to the teacher or general reader ; any who desire more detailed in- formation are referred to the writers mentioned. PART FIRST treats of the general nature of arithmetic, embracing the Nature of Number, the Nature of Arithmetical Language, and the Nature of Arithmetical Reasoning. The natu of Number is quite fully considered, especially in its relation to the idea of Time. Various definitions of Number are presented and examined, and the effort is made to ascertain that which may be regarded as the best for general use. The Nature of the Language of Arithmetic is discussed upon a broader basis than usual. The true relation of Numeration to Notation, which seems to have been overlooked by many authors, and which is frequently not understood by pupils, is explained. It is shown that Numeration is merely the oral and Notation the writ- ten language of Arithmetic. The philosophy of the Arabic system of notation, the objections to the decimal scale, and the advantages of a duodecimal system of arithmetic, are discussed. <r PRKFAC*. Considerable attention is given to the nature of Arithmetical Reasoning, a subject which seems not to have been very clearly understood by logicians and arithmeticians. The effort is made to put this matter upon a logical basis, and to ascertain and pre- sent the true nature of the logical processes by which the science of numbers is unfolded. The ground being almost entirely new, it is not to be supposed that the investigation is at all complete ; but it is hoped that what is given may induce some one to present a more thorough development of the subject. The fundamental idea of the work is that arithmetic has a triune hisit; that it is founded upon and grows out of the three logical processes, Analytic, Synthesis, and Comparison. This is a new gen- eralisation, and is believed to be correct. It has been previously maintained that all of Arithmetic is contained in the two processes, Addition and Subtraction; and that the whole science is a logical outgrowth of these two fundamental ones. In this work it is shown that Synthesis and Analysis are mechanical operations, giving rise to some of the divisions of the science, that the mechanical processes are directed by the thought process of Comparison, and that this itself gives rise to a larger part of the science. The old writers held that we can only unite and separate numbers ; in this work it is held that we can unite, $eparate, and compare numbers. Proceeding with this idea, it is shown that, regarding Addition, Subtraction, Multiplication, and Division, as the fundamental oper- ations of arithmetic, there will arise from them several other pro- cesses of a similar character, which I have called the Derivative Froceun of Synthesis and Analysis. It is then seen that for each analytical process there should be a corresponding synthetic pro- cess. There will thus arise a new process, the opposite of Factoring, to which I have given the name of Composition. This process, it will be seen, contains several interesting cases, which correlate with the different cases of Factoring. It is of especial interest in Alge- bra, as may be seen in my Elementary Algebra, Continuing this thought, it is shown that Ratio, Proportion, the PREFACE. Vli Progressions, etc., are not the outgrowth of either Synthesis or Analysis, but of the thought process Comparison. Attention is called to the nature of Ratio, a new definition is suggested, and the correctness of the prevailing method of finding the ratio of two numbers, which has been questioned, is vindicated. Suggestions are also made for improvements in some of the definitions and methods of treating Ratio, Proportion, Progressions, etc. The log- ical character of Percentage is exhibited, and the simplest and most practical method of treatment suggested. Several interesting chapters are also presented upon the Theory of Numbers. The subject of Fractions is quite fully discussed, the attempt be- ing made to exhibit their nature and their logical relation to inte- gers. The possible cases which may arise are considered, and a new case, called the Relation of Fractions, first given in one of my arithmetics, and already introduced into several other arithmetical works, is presented and explained. It is also shown that the sub- ject of Fractions admits of too methods of treatment, logically distinct in idea and form, and both treatments are presented. Especial at- tention is given to the treatment of Circulates, and the most impor- tant principles concerning them are collated. The nature of Denominate Numbers, which seems to have been imperfectly understood, is explained upon what is regarded as the correct basis. They are shown to be numerical expressions of con- tinuous quantity, in which some artificial unit is assumed as a meas- ure. This leads to the adoption of a new definition of Denominate Numbers, different from that which we usually find in our text- books. The origin of the measures in the various classes of Denominate Numbers is also stated, and many interesting facts concerning them are given. While the philosophical part of the work is that which will at- tract the most attention among thinkers, the historical part will be quite as interesting and instructive to the majority of younger readers. In the historical part; of course, no claims to original investigation are made ; but the best authorities have been con- Till tolled ; and, in many casea, their very language baa been used, their ezpremion being so clear and concise that I could not hope to im- prove it In thus combining with the philosophy of arithmetic its history, which in many case* aids in unfolding it* philosophy, I have aimed to present a work especially valuable to ttudtnlt and the younger teacher* of arithmetic. Such a work, I feel, would have been invaluable to me in my earlier years as a teacher. It is proper to remark that the work was mainly written ntx>nt twelve years ago. This might he regarded as an ad vantage; for, according to the recommendation of Horace, publication should not be hurried, but "a work thoiild be retained till the ninth year." Quin- tilian also remark* concerning his own great work on Oratory that he allowed time for reconsidering his ideas, " in order that when the ardor of invention had cooled I might judge of them on a more careful re-perusal, as a mere reader." In re-perusing the manuscript I see no reason for any change of opinion, in regard to any of the ideas presented, though I am conscious that the manner of pre- senting several subjects might, in some respects, be improved by being re-written; but I have decided to let them stand as originally conceived and expressed, thinking that they may thus gain in fresh- ness and vividness of conception what they may lack in elegance of style. Clirri-iliing ninny pleasant remembrances associated with the discussion of these ideas before my pupils in the class-room, to many of wlmm th-ir publication will prove a reminder of days gone l-y. I commit the work, with its merits and demerits, to an indulgent public, with the hope that it may be of assistance to the younger members of the profession, and contribute somewhat towards the fuller appreciation of the interesting and beautiful science of numbers. EDWARD BROOKS. Normal School. Millertrille, Pa., January 16. 1876. I revise the work after twenty-five years, giving the latest discoveries in the history of arithmetic. EDWARD BROOKS. Philadelphia, May 20, 1901. Supt. Public School*. . TABLE OF CONTENTS. INTRODUCTION. PA0K. CHAPTER I. Logical Outline of Arithmetic 9 II. Origin and Development of Arithmetic 17 III. Early Writers on Arithmetic 29 " IV. Origin of Arithmetical Processes 44 PART I. THE NATURE OP ARITHMETIC. SECTION I. The Nature of Number. CHAPTER I. Number, the Subject-matter of Arithmetic 67 " II. Definition of Number 72 " III. Classes of Numbers 76 " IV. Numerical Ideas of the Ancients 81 SECTION II. Arithmetical Language. CHAPTER I. Numeration, or the Naming of Numbers 93 II. Notation, or the Writing of Numbers 101 III. Origin of Arithmetical Symbols 108 IV. The Basis of the Scale of Numeration 113 V. Other Scales of Numeration . . . .' 121 VI. A Duodecimal Scale 126 VII. Greek Arithmetic 135 VIII. Roman Arithmetic 141 IX. Palpable Arithmetic 147 SECTION III. Arithmetical Reasoning. CHAPTER I. There is Reasoning in Arithmetic 165 II. Nature of Arithmetical Reasoning 171 III. Reasoning in the Fundamental Operations 177 IV. Arithmetical Analysis 185 V. The Equation in Arithmetic 193 VI. Induction in Arithmetic 197 PART II. SYNTHESIS AND ANALYSIS. SECTION I. Fundamental Operations. CHAPTER I. Addition 207 " II. Subtraction 213 III. Multiplication 221 IV. Division 227 SECTION II. Derivative Operations. CHAPTER I. Introduction to Derivative Operations 237 II. Composition 240 III. Factoring 244 IV. Greatest Common Divisor 249 V. Least Common Multiple 257 VI. Involution 261 VII. Evolution 267 1* fix) X CONTENTS. PART III. COMl'ARISOH. SECTION 1. Ratio and Proportion. PAaK CHAPTER I. Introduction to Comparison 291 " II. Nature of Ratio 294 " III. Nature of Proportion 805 IV. Application of Simple Proportion . .810 " V. Compound Proportion 818 ' VI. History of Proportion 826 SECTION II. The Progressions. CHAPTER I. Arithmetical Progression 841 " II. Geometrical Progression 845 SECTION III. Percentage. CHAPTER I. Nature of Percentage 855 " II. Nature of Interest 881 SECTION IV. The Theory of Numbers. CHAPTER I. Nature of the Subject 871 " II. Even and Odd Numbers 875 " III. Prime and Composite Numbers 878 " IV. Perfect, Imperfect, etc., Numbers 888 " V. Divisibility of Numbers 889 " VI. The Divisibility by Seven 897 " VTL Properties of the Number Nine 404 PART FV. FRACTIONS. SECTION I. Common Fractions. CHAPTER I. Nature of Fractions 418 " II. Classes of Common Fractions 420 " III. Treatment of Common Fractions 426 " IV. Continued Fractions 484 SECTION II. Decimal Fractions. CHAPTER I. Origin of Decimals 443 II. Treatment of Decimals 455 III. Nature of Circulates 460 IV. Treatment of Circulates 464 V. Principles of Circulates 470 ' VI. Complementary Repetends. 476 VTI. A New Circulate Form 481 PART V. DENOMINATE NUMBERS. CHAPTER I. Nature of Denominate Numbers 489 II. Measures of Extension 497 III. Measures of Weight 512 IV. Measures of Value 521 V. Measures of Time 541 VI. The Metric System 555 INTRODUCTION TO THE PHILOSOPHY OF ARITHMETIC I. LOGICAL OUTLINE OF ARITHMETIC. II. ORIGIN AND DEVELOPMENT OF ARITHMETIC. III. EARLY WRITERS ON ARITHMETIC. IV. ORIGIN OF ARITHMETICAL PROCESSES. INTRODUCTION CHAPTER I. A LOGICAL OUTLINE OF ARITHMETIC. rpHE Science of Arithmetic is one of the purest products of _L human thought. Based upon an idea among the ear- liest which spring up in the human mind, and so intimately associated with its commonest experience, it became in- terwoven with man's simplest thought and speech, and was gradually unfolded with the development of the race. The exactness of its ideas, and the simplicity and beauty of its re- lations, attracted the attention of reflective minds, and made it a familiar topic of thought ; and, receiving contributions from age to age, it continued to develop until it at last attained J,o the dignity of a science, eminent for the refinement of its principles and the certitude of its deductions. The science was aided in its growth by the rarest minds of antiquity, and enriched by the thought of the profoundqst thinkers. Over it Pythagoras mused with the deepest enthu- siasm; to it Plato gave the aid of his refined speculations; and in unfolding some of its mystic truths, Aristotle employed his peerless genius. In its processes and principles shines the thought of ancient and modern mind the subtle mind of the Hindoo, the classic mind of the Greek, the practical spirit of the Italian and English. It conies down to us adorned with (9) 10 THE PHILOSOPHY OP ARITHMETIC. the offerings of a thousand intellects, and sparkling with the gems of thought received from the profouadest minds of nearly every age. And yet, rich as have been the contributions of the past, few of the great thinkers have endeavored to unfold its logical relations as a science, and discover and trace the philosophic thread of thought that binds together its parts into a complete and systematic whole. Unlike its sister branch geometry, which came from the Greek mind so perfect in its symmetry and classic in its logic, the science of arithmetic has been treated too much as a system of fragments, without the attempt to coordinate its parts and weave them together with the thread of logic into a complete unity. To remedy this defect is the special object of a work on the Philosophy of Arithmetic, and is the task which the author of the present work has with diffidence attempted. Like all science, which is an organic unity of truths and principles, the science of arithmetic has its fundamental ideas, out of which arise subordinate ones, which themselves give rise to others contained in them, and all so related as to give symmetry and proportion to the whole. What are these fun- damental and derivative ideas, what is the law of their evolu- tion, what is the philosophical character of each individual process, and what is the logical thread of thought that binds them all together into an organic unity ? These are the ques- tions that meet us at the threshold of the effort to unfold a philosophy of arithmetic; they are the foundation upon which such a superstructure must be erected ; and we begin the answer to these questions in the first chapter, under the head of A Logical Outline of Arithmetic, which exhibits the fun- damental operations and divisions of the science. To this Logical Outline the special attention of the reader is invited, as it is not only the foundation upon which the au- thor has builded, but also the frame-work of the system. In A LOGICAL OUTLINE OF ARITHMETIC. 11 it the science is assumed to be based upon the three processes Synthesis, Analysis, and Comparison; general processes* in which each individual process must have its root, and from which it is developed. This generalization marks a new departure in the method of regarding the science, and the re- lation of its parts ; and shows the incorrectness of opinions around which has gathered the dust of centuries. Our first inquiry is, what is A Logical Outline of Arithmetic ? All numerical ideas begin with the Unit. It is the origin, the basis of arithmetic. From it, as a fundamental idea, originate all numbers and the science based upon them. Begin- ning, then, at the Unit, let us see how the science of arithmetic originates and is developed. The Unit can be multiplied or divided. This gives rise to two classes of numbers, Integers and Fractions. Integers originate in a process of synthesis, Fractions in a process of analysis. Each Integer is a synthetic product derived from a combination of units; each Fraction is an analytic product derived from the division of the unit. There are, therefore, two general classes of numbers, Integers and Fractions, treated of in the science of arithmetic. Having obtained numbers by a combination of units, we may unite two or more numbers and thus obtain a larger number by means of synthesis ; or we may reverse the operation and descend to a smaller number by means of analysis. Numbers, therefore, can be united together and taken apart; they can be synthetized and analyzed; hence Synthesis and Analysis are the two fundamental operations of arithmetic. These funda- mental operations give rise to others which are modifications or variations of them. Arithmetic, therefore, from its primary conception seems to consist of but two things, to increase and to diminish numbers, to unite and to separate them. Its pri- mary operations are Synthesis and Analysis. To determine when and how to unite, and when and how 1J 1'HILOSOPHY OP ARITUMET1C. to separii a process of reasoning cat ;>ari- son. Ti. <-in|.;in -s numbers and determines their relations. Synthe.-i.s un<l Analysis are mechanical processes , parison is the thought process. Comparison directs the original processes, modilit -s them so as to product- from them , other p: "iitained in the original ones. It is, in other u this th<> , process working upon the idea of number, that tin- original esses of Synthesis and Analysis are directed and modified, that other processes are developed from them, and that new and independent processes arise, and the science of arithi; is developed. Comparison, therefore, in arithmetic as in geom- etry, is the process by which the science is constructed, or the key with which the learner unlocks its rich storehouse of interest and beauty. Arithmetic, it is thus seen, consists fundamentally of ti things; Synthesis, Analysis and Comparison. Synlfifsis and , : >/sis are fundamental mechanical operations, suggested in the formation of numbers; Comparison is the fundamental thought process which controls these operations, unfolds their potential ideas, and also gives rise to other divisions of the science jrn>\\ in.ir immediately out of itself. In other words, the science of arithmetic has a triune basis; it has its i ^rows out of, the three processes, Synthesis I, and Cui>ii>tiri*n. Let us examine these processes and see the number, nature, and relations of the divisions growing out of the fundamental operations, and thus deter- logical character of the science of arithmetic. lit. A general synthesis is called Aildttmn. A spe- nise of -iietic process of Addition, in which the nun .^d are all ei|iial, their .-um receiving the came of iiK't. is called forming of Com/ nberg by a synthesis of factors, which may be called Composition; Multiples, formed by a synthesis of particular factor? ; and i>y a synthesis of equal factors, are A LOGICAL OUTLINE OF ARITHMETIC. 13 all included under Multiplication. Hence, since Involution, Multiples, and Composition, are special cases of Multiplication, and Multiplication is itself a special case of Addition, the pro- cess of Addition includes all the synthetic processes to which numbers can be subjected. ANALYSIS. A general analysis, the reverse of Addition, is called Subtraction. A special case of Subtraction, in which the same number or equal numbers are successively subtracted with the object of ascertaining how many times the number subtracted is contained in another, is called Division. Factor- ing is a special case of Division in which many or all of the factors of a number are required ; Evolution is a special case of factoring in which one of the several equal factors is re- quired ; and Common Divisor is a case of factoring in which some common factor of several numbers is required. The process of Division, therefore, includes the processes of Factor- ing, Common Divisor, and Evolution; and since Division is a special case of Subtraction, all of these processes are logically included under the general analytic process of Subtraction. COMPARISON. By comparison the general notion of relation is attained, out of which arise several distinct arithmetical processes. By comparing numbers, we perceive the relations of difference and quotient; and giving measures to these, we have Ratio. A comparison of equal ratios gives us Propor- tion. A comparison of several numbers differing by a common ratio gives us Arithmetical and Geometrical Progression. In comparing concrete numbers, when the unit is artificial, we perceive that they differ in regard to the value of the units, and also that we can change a number of units of one species into a number of another species of the same class ; and thus we have the process called Reduction. In comparing abstract numbers we notice certain relations and peculiarities which, investigated, give rise to the Properties or principles of num- bers. In comparing numbers, we may assume some number as a basis of reference and develop their relations in regard to 14 THE PHILOSOPHY OF ARITHMETIC. this basis; when this basis is a hundred, we have the pr- een called Percentage. we obtain a complete outline of the science of nui ami more clearly the logical relations of tin- divisions of tin- .-. \ rithmetic is conceived as based upon 1 1.< . mien ml operations, synthesis and anai lions being controlled by which develop.- esses from these and also from itself. The who!- the outgrowth of this triune basis, E thesis, Analysis, and Comparison. The rest of aritln consists of the solution of problems, either real or tin < and may be included under the bead of Applications of Arith- metic. This conception of the subject is new and important. It has re held that addition and subtraction compre- ied the entire science of arithmetic; that all other pn>- cesses are contained in them, and are an outgrowth from them. This is a fallacy, which, among other things, has led logic to the absurd conclusion that there is no reasoning in arith- metic. Assuming that there is no reasoning in the prii. esses of synthesis and analysis, and that these primary processes contain the entire science, they naturally conclude there is no reasoning in the science itself. The ana! of ti t here given dispels this error and exhibits the subject in its true light. Synthesis and Analysi- Q to lie the primary mechanical processes; Comparison, the thought .,es them with her wand of magic, and they ger- minate and bring forth other processes, having their root- tbese primary ones. Comparison also becomes the foundation rocesses distinct from those of synthesis and anal\ -ses which cannot be conceived as growing out of syn- luit which have their root in the thought process of the science in Comparison. This outline of the science grows out of the pure idea of number, iudi ; f the language of arithmetic. These A LOGICAL OUTLINE OF ARITHMETIC. 15 fundamental processes are modified by the method of notation employed to express numbers. With the Roman or Greek methods of notation, the methods of operation would not be the same as with the Arabic system. The method of " carry- ing one for every ten," of "borrowing" in subtracting, the peculiar methods of multiplying and dividing, grow out of the Arabic system of notation. A portion of the treatment of common and decimal fractions arises from the notation adopted, and the principles and processes of repetends originate in the same manner. The methods of extracting square and cube root would be different if we employed a different method of expressing numbers. It is thus seen that the fundamental divisions of arithmetic arise from the pure idea of number, that the processes in these divisions are modified by the method of notation adopted, and also that some of the principles and pro- cesses of the science grow out of this notation. It may be remarked, also, that the power of arithmetic as a calculus depends upon the beautiful and ingenious system of notation adopted to express numbers. It is believed that the above view of arithmetic must tend to simplify the subject, and that much clearer notions of the science will be attained when these philosophical relations are apprehended. A general view of the subject is presented by the following analytical outline : T 9vnthpm's / Addition ' (Composition. 8< I Multiplication. 4 ( Common Multiple. ( (.Involution. II Analvsis / Subtraction. (Factoring. 11. Analysis. -[ Division> J r Common Divisor. ( \ Evolution. 1. Ratio. 2. Proportion. -3. Progression. 4. Reduction. 5. Percentage. 6. Propertion of Numbers. Logical Outline of Arithmetic. III. Comparison. TUB PHILOSOPHY OF ARITHMETIC. D O CHAPTER II. ORIGIN AND DEVELOPMENT OP ARITHMETIC. A KNOWLEDGE of Arithmetic is coeval with the race. Every people, no matter how uncivilized, must have possessed some ideas of numbers, and employed .them in their transactions with one another. These ideas would be multiplied, and the methods of operation founded upon them gradually extended and improved as the nation advanced in civilization and intelligence. The his- tory of Arithmetic is, therefore, inseparably connected with the history of civilization and the race. The origin of its elementary processes must, of necessity, be involved in obscurity and uncer- tainty. History can speak positively only of some of the higher and more recent developments of the science. In presenting some of the principal facts concerning the history of arithmetic, we shall consider three things : the origin of our present system of arithmetic ; the early writers on the science ; and the origin of the fundamental operations. Other historical facts will be mentioned in connection with the particular subjects to which they belong. One of the most interesting inquiries is that which relates to the origin of the system of arithmetic now gen- erally adopted, which we shall consider in the present chapter. The basis of our system of arithmetic is the principle of place- value in writing numbers. All civilized nations, from the primi- tive habit of reckoning with the fingers, adopted a system of count- ing by groups of ten. Each group of ten is distinguished by a special name, and the names of the first nine numbers are used to number the groups and express the numbers between them. Thus all civilized peoples adopted the same general method of oral arithmetical language. In writing numbers, however, different 2 (17) 1" THE PHILOSOPHY OF ARITHMETIC. nations adopted widely different methods of notation. Our present simple and practical system of notation was reached by only a single nation of antiquity. The various methods of writing num- bers in use among the ancient nations and the origin of our present system will be briefly considered. The Egyptians represented numbers by written words, and also by symbols for each unit repeated as often as necessary. In one of the tombs near the pyramid of Gizeh, hieroglyphic numerals have been found in which 1 is represented by a vertical line ; 10 by a kind of horse-shoe ; 100 by a short spiral ; 10,000 by a point- ing finger ; 100,000 by a frog, and 1,000,000 by the figure of a man in the attitude of wonder. In their hieratic writing they used symbols for numbers, but they did not combine them on the prin- ciple of place-value as in the modern system of notation. There are special characters for the nine units, and also for the tens, the hundreds, etc. The following are specimens of these symbols : I II III _ 1 A A ( A j. ~1 ill 3 12 8 46 10 20 3040506070 These are combined on the additive principle, the symbol of the larger value always being placed at the left of that for the smaller value. The papyrus of Ahmes in the British Museum indicates that the Egyptians at a very early period had considerable knowl- edge of the art of arithmetic. The ancient Babylonians used the wedge-shaped characters of their cuneiform system of writing in the representation of numbers. The mark for unity, a vertical arrow head, is repeated up to ten, whose symbol is a barbed sign pointing to the left. These symbols by mere repetition served to express numbers up to one hundred, for which a new sign was employed. The characters were written sometimes one beside another, and sometimes, to save space, one over another. The symbol for the smaller number written to the right of the symbol for a hundred denoted addition ; the same sym- bol written on the left denoted multiplication, or the number of hundreds. The Babylonians thus employed the principle of place value to some extent, but having no symbol for zero they were una- ORIGIN OP THE SCIENCE. 19 ble to develop the modern system of notation and calculation. They used also along with the decimal system the sexagesimal system, that is one with a base of 60, and their operations with both integers and fractions show considerable mathematical facility and skill. The Chinese had a well developed number system, and seem to have come as near the present method of notation as any nation of antiquity, except the Hindoos. Of their early number symbols but little seems to be known. Later, as a result of foreign influence, there arose two new kinds of notation, whose figures are supposed to resemble the ancient symbols. Though the Chinese wrote their word-symbols in columns, yet their numbers were written from left to right, beginning with the highest order. The ordinal and cardinal numbers are usually arranged in two lines, one above another, with zeros in the form of small circles appearing as often as necessary. The following symbols will illustrate this system : II X u_ * fi o 2 4 6 10 10,000 " X /700 -*-_,_ 20,046 Their arithmetical calculations were made by means of the abacus or swan-pan, which is used at the present day among both their scholars and their merchants. The Phoenicians expressed numbers in words, and also by the use of special numerical symbols, using vertical marks for the units and horizontal marks for the tens. The Syrians somewhat later used the twenty-two letters of their alphabet to represent the numbers 1, 2, ... 9, 10, 20, ... 90, 100, ... 400 ; 500 was 400+ 100, etc. The thousands were represented by the symbols for units with a subscript comma at the right. The notation of the Hebrews followed the same plan. None of these nations had a notation that could be used in making calculations as we do with the modern system. The early Greeks seem to have used the initial letters of their 20 TIM: i-Hii.osoiMiY or AIMTIIMKTIC. number words to represent written numbers ; as (~) for 5 A for 10 (<5a), and these letters were repeated as often as neces- sary. Soon after 500 B. C. two new systems appeared among the B. < )nc used the 24 letters of the Ionic alphabet in their nat iirnl order for the numbers from 1 to 24. The other arranged these letters, together with three other symbols, in an arbitrary order, thus aI, ft = 2, . . . < = 10, =20, . . . P=WO, <> = 2( H, etc. The Greeks could perform the fundamental operations with these symbols with considerable facility, as may be seen in tin chapter on Greek arithmetic. The common method of calcu- lation, however, seems to have been with the abacus. The Greeks did not make use of the principle of place value, and they had no symbol for zero. The Romans also expressed numbers by means of letters. The characters are supposed to have been inherited from the Etruscans, and may originally have been symbolic, and subse- quently, on account of their resemblance to forms in their alpha- bet, they were replaced by letters. Mommsen says that the Roman numerals I, V, X represent the finger, the hand, and the double hand respectively. These characters were combined according to the additive principle, as in VI, VII, VIII, and also in accord- ance with the subtractive principle, as in IV, IX, XI,, XC. This Btihtractive principle is a distinctive characteristic of the Roman gystem of notation. The Romans could not use their notation for reckoning, but made their calculations with counters (calculi) or with the abacus. They seem to have had no conception of place value, or of a symbol for zero, as their system did not call for it. Our present number system, it is now known, had its origin among the Hindoos. They originated the modern position-system, Kinl introduced the zero to fill an unoccupied place. Their d books which have been in the hands of the priest- hood fur ci-nturies, contain the numerical characters. Their ear- liest symbols of the nine digits were after 3 merely abridged num- ber words, and tin- use of letters as figures is said to date from the second century B. C. The development of the system of place- ORIGI value seems to have writing numbers there is no indical value, though it appears in two other systems" prevailed in Southern India. Both of these methods tinguished by the fact that the same number can be made up in various ways. One method consisted in employing the alphabet, in groups of nine symbols, to denote the numbers from 1 to 9 re- peatedly, while certain vowels denote the zeros. A second method used type-words, and combined them according to the law of position. Thus abdhi (one of the 4 seas) = 4 ; surya (the sun with its 12 houses) = 12 ; a$vin (the two sons of the sun) = 2. The combination abdhisuryayvinas denoted the number 2124. These no doubt were stepping-stones to the present simple appli- cation of the position principle. The modern system of place value could not have been adopted before the invention of the zero, and there is no proof of its being introduced before 400 A. D. The first known use of the symbol on a document, Cantor says, dates from 738 A. D. The Arabs became acquainted with the Hindoo number-system and its figures, including zero, in the eighth century, and were instrumental in introducing the system into Europe. It was for many years thought that the present system of arithmetic origi- nated with the Arabians. The characters in general use were called Arabic characters, and the method of writing numbers was known as the Arabic system of notation. Further proof, it was thought, was found in the two words "cipher" and "zero," cipher being the Arabic as-sifr, meaning empty. This word, however, was derived no doubt from the Sanskrit name of the naught, sunya, the void. In Italy the character for naught was called " zephiro," which has, by rapid pronunciation, been changed to zero. The Arabs, however, it is now known, were not the authors of the system, but derived it from the Hindoos, and were only instrumental in introducing it into Europe. The Arabs from an early period had commercial relations with India, which brought them in contact with the Indian system of 22 THE PHILOSOPHY OP ARITHMETIC. reckoning. It is known that they were acquainted with the Hindoo number system and its figures, including the zero, as early as the eighth century. The earliest definite date, says Ball, assigned for the use in Arabia of the decimal system of notation is 773. In that year some Indian astronomical tables were brought to Bagdad in which it is almost certain the Indian numerals, including the zero, were used. The Arabs no doubt developed the system somewhat slowly, as the custom of writing out number words continued among them until the beginning of the eleventh century. In the investigation we meet with the singular fact that the Arabs employed two kinds of figures : one used chiefly in the East called " Oriental ;" another used by the Western Arabs in Africa and Spain called the Gubar or dust numerals, so called because they were first introduced among the Arabs by an Indian who used a table covered with fine dust for the purpose of ciphering. These Gubar numerals are the ancestors of our modern numerals. They are said to be modifications of the initials of the Sanskrit word-numerals. It was through Spain, however, it is generally believed, rather than directly from Arabia, that the Arabic system was introduced into Europe. The Moors as early as 747 had conquered Spain, and established there their rule. They brought with them a taste for learning, and established schools and universities, so that by the tenth and eleventh centuries they had attained to a high degree of civilization. Though the political relations of the Arabs with the caliphs of Bagdad were not entirely cordial, yet they gave ready welcome to the works of the great Arabian mathematicians. The Arabs had studied with great avidity the Greek mathematics, and their translations of Euclid, Archimedes, Ptolemy, etc., along with the works of the Arabians themselves on arithmetic and algebra, were studied at the great Moorish universities of Gren- ada, Cordova and Seville. Thus while the Christian world was enveloped in ignorance, the Arabs were cultivating the learning and literature of Greece. Though not highly gifted with creative powers of mind by ORIGIN OF THE SCIENCE. 23 which they made many valuable additions to what they thus acquired, a debt of gratitude is due them because they " preserved and fanned the holy fire." Their efforts at conquest had been crowned with brilliant success. Spain had yielded to their sway, and the Moors had become celebrated throughout Europe for the splendor of their institutions, the magnificence of their architecture, and the proficiency of their scholars. Disgusted with the trifling of their own schools, energetic and aspiring young men from England and France repaired to Spain to learn philosophy from the accomplished Moors. There they studied arithmetic, geometry and astronomy, and made themselves familiar with the Arabic method of notation and calculation. On their return they brought the characters and methods of the Arabic arithmetic with them and introduced them to the scholars of Northern Europe, and thus in time they gradually displaced the Roman system, which had been in use for many centuries. One of these tl pilgrims of science " was an obscure monk of Auvergne named Gerbert, who died in 1003. Returning to his native country he became widely celebrated for his genius and learning, and subsequently rose to the papal chair with the title of Sylvester II. His treatises on arithmetic and geometry were valuable, presenting many rules for abbreviating the operations in common use. He introduced an improvement in the use of the abacus by marking each of the nine beads in every column with a distinctive sign. These marks, called apices, are supposed to have been the same as the Gubar numerals, and thus Gerbert did much to introduce the old Hindoo numeral-forms into Western Europe. Efforts have been made to ascertain what persons were most conspicuous in the introduction of the Arabic characters into Northern Europe. There seems to have been some difficulty in obtaining access to the Moorish universities, as the Moors are said to have taken pains to conceal their learning from the Christian world. One of the earliest students from Christian Europe to acquire a knowledge of Moorish aud Arabian science was an Eng- 24 TIIK rilll.o.M.l'HY OF ARITHMETIC. Bftb monk, Adelhard <>t liath, who, disguised as a Mahommedan Indent, got into Conlo\a :iliout 1120 and obtained a copy of Euclid's Elements. This copy translated into Latin is said to have been the foundation of all the editions of the work known in Europe until 1533. Another scholar who was influential in the introduction of Moorish learning into Northern Europe was Abraham Ben E/ra. a Jewish rabbi, born at Toledo in 1097 and died at Rome in 11C7. He wrote an arithmetic in which he explains the Arabic sy of notation with nine symbols and a zero, and gives the funda- mental processes of arithmetic and the rule of three. Another eminent scholar who aided in the introduction v/a> Gerard, born in 1114 and died in 1187. He translated the Arab edition of the Almage$t of Ptolemy in 1136, which seems to have been the earliest text-book among the Arab- that contained tin- Arabic notation, and which it is thought was instrumental in the introduction of the system to the Moors in Spain. A contempo- rary of Gerard, John Ilispalensi-, a Jewish rabbi converted to Christianity, translated several Arab and Moorish works, and also wrote a treatise on algorism, which is said to contain the earliest pies of the extraction of the square root of numbers by the aid of decimal numbers. The introduction of the Arabic system throughout Europe pro- ceeded slowly. The Roman system of calculation with the abacus had been in use many centuries, and it was difficult to lead the people to make the transition from it to the new system. The struggle between these two schools of arithmeticians, the old abacistic school and the new algoristic school, was long and no doubt often bitter. It was not easy for the mathematicians and business men who had been brought up on a system of calcu- lation with the abacus to drop it and adopt the new method of ronipiitincr with abstract symbols. One of the most influential men in bringing about the general OK of the new system was Leonardo Fibonacci, born at Pisa in 1175. Educated in his youth at Bugia in Barbary, where his ORIGIN OF THE SCIENCE. 25 father had charge of the custom house, he became acquainted with the Arabic system of notation and with the great work on algebra by Al Khowarazmi. He returned to Italy about 1200, and in 1202 composed a treatise on mathematics known as Liber abaci, in which he explains the Arabic system of notation, and points out its great advantage over the Roman system. It begins thus : " The nine figures of the Hindoos are 9, 8, 7, 6, 5, 4, 3, 2, 1. With these nine figures and with this sign, 0, which in Arabic is called sifr, any number may be written." This work had a wide circulation, and practically introduced the use of the Arabic system throughout Christian Europe. It is supposed that the system was known before this time to the leading mathematicians who had read the works of Ben Ezra, Gerard and Hispalensis, and also by Christian merchants who had traded with the Mahom- edans, but the wide reputation of Leonardo gave a great impetus to its general adoption. The Arabic numerals were used at an early day by the astron- omers in composing calendars, and these calendars aided in dis- seminating a knowledge of the system. Shortly after the appear- ance of Leonardo's work, Alphonso of Castile, in 1252, published some astronomical tables founded on observations made in Arabia, which were computed by the Arabs and published in the Arabic notation. A frequent and free use of the zero in the 13th cen- tury is shown in the tables for the calculation of the tides at Lon- don and of the duration of moonlight. There is an almanac pre- served in one of the libraries of Cambridge University containing a table of eclipses for the period from 1330 to 1348. This almanac contains a brief explanation of the use of numerals and the prin- ciples of the denary notation, indicating that at that date the system was not generally understood. A little tract in the German language entitled De Algorismo, bearing the date of 1390, explains with great brevity the digital notation and the elementary rules of arithmetic. At the end of a short missal similar directions are given in verse, which from the form of the writing seems to belong to the same period. The THE PHILOSOPHY OF ARITHMETIC. characters, of uhi.-h tliuse in the margin are lac-similes, are in both manuscript* written tim right to left, the order whirl, the aus would naturally follow. Tin- ureat Italian poet, Petrarch, has the honor of leaving us one of the oldest authentic dates in the numeral characters. The .lati i- l;',7.~>, written upon a copy of St. Augustine. The college a.'.-i.uiKs in the English universities were generally kept in K.man numerals until the beginning of the sixteenth century. 1 In- Arabic characters were not used in the parish registers of England before 1600. The oldest date met with in Scotland is that of 1490, which occurs in the rent-roll of the Diocese of St. Andrews. In Caxton's Mirrour of the World, issued in 1480, there is a wood-cut of an arithmetician sitting before a table on which there are tablets with Hindoo numerals upon them. According to Fink the Roman symbols were generally used in Germany with the abacus up to the year I.'HHI. From the Kith century on, these Hindoo numerals appear more frequently in Germany on monuments and in churches, but at that time they had not become common among the people. The oldest monu- ment in Germany with Arabic figures (in Katherein near Trop- pau) is said to date from 1007, and such monuments are found in Pforzheim (1371) and in Ulm (1888). In the year 1471 there appeared in Cologne a work of Petrarch with page numbers in the Arabic figures, and in 1482 the first German arithmetic with similar numbering was published at Bamberg. It mu>t have been somewhere from the year 1400 to 1450 that the Arabic system of arithmetic began to be generally dissemi- nated throughout Europe. Men of science and astronomers had become acquainted with the system by the middle of the 13th century. Tin- trail" of Europe during the 13th and 14th centu- ries was mostly in Italian hands, and the advantages of the alg.r- istic system led to its adoption in Italy for mercantile purposes. The change, however, was not made even among merchants with- out considerable opposition ; thus an edict was issued in Florence ORIGIN OF THE SCIENCE. 27 in 1299 forbidding bankers to use the Arabic numerals, and the authorities of the University of Padua in 1348 directed that a list should be kept of books for sale with prices marked 4; non per cifras sed per literas claras." Most merchants seem to have con- tinued to keep their accounts in Roman numerals until about 1550, and monasteries and colleges until about 1650 ; though in both cases it is thought that the processes of arithmetic were performed by the Arabic system. It was not until the sixteenth century that the Hindoo position-arithmetic and its notation first found complete introduction among the civilized people of the West. The forms of several of the figures have undergone considerable change since their first introduction into Europe. In the oldest manuscripts the figures 4, 5 and 7 are most unlike the present characters. The 4 consisted of a loop with the ends pointing down thus 8; the 5 has some likeness to the figure 9, thus ^, and the 7 is simply an inverted V, thus A. In the dates used by Caxton in the year 1480, the 4 has assumed its present shape, but the 5 and 7 are still unlike the same characters of to-day. No reason is assigned for these changes ; they seem to have been gradual, and the result of chance rather than of intention. The forms of the figures at different periods may be seen in the table given on page forty-three. This explanation of the introduction of the Arabic characters and system of notation into Europe through Spain is the one now generally accepted as correct. M. Woepcke, an excellent Arabian scholar and mathematician, thinks that the Indian figures reached Europe through two different channels ; one passing through Encrypt about the third century ; another passing through Bagdad in the eighth century, and following the track of the victorious Islam. The first carried the earlier forms of the Indian figures from Alexandria to Rome, and as far as Spain ; the second carried the later forms from Bagdad to the principal countries conquered by the Kaliffs, with the exception of those where the earlier or Gubar figures had already taken firm root. The Gubar figures, he thinks, were adopted by the Neo- Pythagoreans, and introduced 28 IMI run i>.-i>mt " AKI i H.MKTIC. Italy and n province*, Gaul and Spain, as early a* nth century, w> that the Mohammedans wh-n they reached Spain in the eighth century, found these figures already estab- niii adopted them. And so, likewise, when in the ninth ami tenth centuries the new Arabic treatises on arithmetic arri\ed in Spain from the East, they naturally adopted the mi, c,t system of ciphering carried on without the ahaen>, and kept the figures to which they as well as the Spaniard had been accustomed for centuries, and thus the Gubar figure* \\< -re retained by them. The only change produced in the ciphering Europe by the Arabs was, he claims, the suppression of the abacus, and the more extended use of the cipher required by the n. w >\>reni of reckoning. In the preparation of this and the following chapter, I have re- 1 valuable assistance from Fink's History of Mathein-uio. translated from the German by Be man and Smith, and from Ball's lli>ti>ry of .Matheniatics, both valuable works to which the read* -r is referred for further information. I have al>o rec(i\ed many valuable suggestions from Dr. David Eugene Smith, I'rofe.-.-or of Mathematics in Teachers College, Columbia l'niver>ity, N The great authority on the history ot mathematics i- M-nit/ Cantor, whose works, however, have nut been tian-i.u. <i into English, CHAPTER III. EARLY WRITERS ON ARITHMETIC. ANE of the earliest known treatises on mathematics is the Ahmes papyrus of the British Museum. The manuscript was written by an Egyptian scribe named Ahmes sometime between 2000 B. C. and 1700 B. C. The title of the work is " Directions for Obtaining the Knowledge of all Dark Things." It is believed to be a copy, with emendations, of a much older treatise, so that it probably represents the knowledge of the Egyptians on arith- metic many centuries earlier than its own date. Two other mathematical papyri have recently been found belonging to a much earlier period than that of Ahmes, which without entirely agreeing with the papyrus of Ahmes, exhibit many similarities to it, especially in the method of treating fractions. So that we have some knowledge of Egyptian arithmetic as early as the twelfth dynasty, or about 2oOO B. C. The treatise of Ahmes consists of the solution of problems on arithmetic and geometry ; the answers are given, but generally not the processes by which they were obtained. It deals with both whole numbers and fractions. The treatment of fractions is peculiar in that it is limited to those having unity for the numer- ator, except in the single case of . Fractions that cannot be expressed with a unit numerator are represented by the sum of two or more fractions whose numerators are each a unit ; thus for | Ahmes writes ^. A fraction is designated by writing the denominator with a certain symbol above it to indicate its nature. Special symbols were used for , ^, and . Ahmes treats also of numerical equations, ns when he says, " heap, its seventh, its whole, it makes nineteen ;" that ie, find a number such that the (20) "' Till PHILOSOPHY OP ARITHMETIC. sum of it and one-seventh of it shall equal 19 ; the answer given is 16 + ^ -f . The word hau or " heap " signifies the unknown quantity, or x, as seen again in the following: " heap, its $, its |, its }, its whole, gives 37 ; that is, $x + $x + \x + x 37." treatise contains examples in arithmetical and geometric-ill progression, and employs the method of " false position " so popular among the Hindoos, Arabs and modern Europeans. The Greeks obtained much of their mathematical knowl- originally from the early Phoenicians and Egyptians. They culti- vated the science of numbers to some considerable extent, but failed to invent a simple and convenient method of notation by which operations with numbers could be performed with any de- gree of facility. Like many other nations of antiquity, they depended upon the abacus in performing the operations of the fundamental rules, though in the time of Archimedes and Apol- lonius they could perform these operations to some extent by means of their notation. The science of arithmetic with the Greeks was speculative rather than practical. They did not to aim at the development of skill in computation, but delighted in investigating the properties of numbers and in the discovery <! fanciful analogies among them. It is a matter of surprise that while their works on geometry have been the models of later writers on that subject, the Greeks contributed but little of value to the science and art of numbers. One of the earliest Greek writers on mathematics was Pytha- goras, an ancient geometer who is supposed to have lived from about 580 to about 500 B. C. He brought from the Kiist a pas- tor the mysterious properties of numbers, under the v which he probably concealed some of his secret and esoteric doc- trines. He regarded numbers as of divine origin the fountain of existence the model and archetype of things the essence of the universe. He divided them into classes, to each of which assigned distinct and peculiar properties. They wore Even and Prime and Composite, Plane and Solid, Triangular. Square, and Cubical. Even numbers were regarded as feminine ; odd numbers as masculine, partaking of celestial natures. EARLY WRITERS ON ARITHMETIC. 31 Euclid, born about 330 B. C., was one of the early Greek writers upon arithmetic. His treatise is contained in the 7th, 8th, 9th and 10th books of Euclid's Elements, in which he treats of the theory of numbers, including prime and composite numbers, greatest common divisor, least common multiple, continued pro- portion, geometrical progressions, etc. He develops the theory of prime numbers, shows that the number of primes is infinite, unfolds the properties of odd and even numbers, and shows how to construct a perfect number. These books of arithmetic are not included in the common editions of Euclid, but are found in an edition by the celebrated Dr. Barrow. It is supposed that Euclid was quite largely indebted to Thales and Pythagoras for his knowledge of the subject, though he undoubtedly added much to the science himself. The school at Alexandria in which he taught was highly celebrated, being attended by the Egyptian monarch Ptolemy Lagus. It was this pupil to whom Euclid, upon being asked if there was not an easier method of learning mathematics, is said to have replied, " There is no royal road to geometry ;" a statement, however, attributed to several other mathematicians of antiquity. Archimedes, born about 287 B. C., was one of the most eminent of the Greek mathematicians. He is especially celebrated for the discovery of the ratio of the cylinder to the inscribed sphere, in commemoration of which the figure of a sphere inscribed in a cylinder was engraved upon his tomb. He wrote two papers on arithmetic; the object of one, which is now lost, was to explain a convenient system of representing large numbers. The object of the second paper was to show that the method enabled one to write any number however large, in which he gave his celebrated illustration that the number of grains of sand required to fill the universe is less than 10 63 . Eratosthenes, who flourished about 250 years before the Christ- ian era, is said to have invented a method of determining prime numbers, known as Eratosthenes' sieve. He is also said to have suggested the calendar now known as the Julian Calendar, in Illl rilll.oMii'UV OK AKITIIMI which every fourth -.tain* 36G days. He determined tin- obliquity of ill-- rlijitu-, and measured a degree on the surfa .trtli \\Li.-li was subsequently found to be too long by about -. He U!M) describes an in-;. ..n.- nt fur the duplication ubi-. Nicoiiiat hns, whu is sup|H>M-d to have li\ed near the close of the first century of tin- Christian era, wrote an arithmetic \\ lndi in Latin translation remained lor a thousand years a standard authority U|>on that .-ubject. Hi- >|>< -cial aim wa- i be in\ -: Cation of the properties of numbers, and particularly of ratios. II gins with the explanation of even, odd, prime and perfect num- ; then explain:- tractions in a tedious and elnni-y manner; then .ii-i-ussvs polygonal and solid numbers, and finally treat- <>t ratios, proportion, and the pro^reion-. He pi\e> the propo.-ition that all cubical numbers are equal to the >nm ot odd numbers; as 8 = 3 :. ; >'7=7 + 0+ll; 4 = 1 + l;'i+ 17-f-19. The work was translated by Boethius, and was the rem-M/, ,1 -liook durinjr the Middle Ages. l'toh-m\ t. who died l,et\\e. n ]'2'> and \.'t\ A. 1)., was the author of numerous works on mathematics. C se on astronomy, called by the Aral's the A/nm>/rst. remaim-d a -tandard work on that subject until the time of Copernicus. In this work he treats of trigonometry, plane ami spherical, explains the obliquity of the ecliptic, uses 3-ffa as the approximate value of w, and em] 1 '1 >econds as i. Th- work exercised a strong influence in favor of sex a 21 simal iirithmetir, which uses the basis of sixty in the representation of numbers. a mathematician of Alexandria, who lived about tury, wrote a work railed Arithmetica, tfl of thirteen l>ooks>, only six of which have come down to It i< really a work on algebra, and before the discovery of the Ahmet papyrus was the oldest work extant on that su It treats of the properties of numbers, one of \\\* problems being to divide a number, as 18, which is the sum of two squares 4 and EARLY WRITERS ON ARITHMETIC. 33 9, into two other square^ which he finds to be &^-- and -fa. It presents solutions of simple and quadratic equations, uses a symbol for the unknown quantity, and shows that " a number to be sub- tracted, multiplied by a number to be subtracted, gives a number to be added." The work is purely analytic in spirit, and is thus distinguished from the works of other Greek writers like Euclid. Diophantus originated the method of investigation known as Dio- phantine Analysis. Boethius, born at Rome between 480 and 482, wrote an Arith- metic based on that of Nicomachus. The arithmetic of Boethius was the classical work of the Middle Ages, and became the model of several subsequent writers even down to the fifteenth century. It was entirely theoretical, treating of the properties of numbers, particularly their ratios, and gave no rules of calculation, and wo have no means of telling whether the arithmeticians of thitf school reckoned on their fingers, or used an abacus. In the manuscript editions of this work, current during the llth century, there is a description of the Mensa Pythagorea, also called the abacus ; and mention is made of nine figures which are ascribed to the Pythagoreans or Neo Pythagoreans. This passage is by some considered spurious, and ascribed to a continuator ol Boethius. One of the earliest Hindoo writers upon the subject of mathe- matics was Aryabhatta, who was born in Pataliputra in 476 A. D. His work, entitled Aryabltattiyam, contains a number of rules and propositions written in verse. It consists of four parts, of which three are devoted to astronomy and the elements of spherical trigonometry ; the remaining part consists of thirty-three rules in arithmetic, algebra, and plane trigonometry. The algebra shows considerable knowledge of the subject, but there is no direct evi- dence that Aryabhatta was acquainted with the modern method of arithmetic. The next Hindoo writer of note is Brahmaguptn, born in 598, and was probably living up to 6GO. His work entitled Brahma- aphtito Siddhanta (i. e., the improved system of Brahma) is 3 34 THE PHILOSOPHY OF ARITHMETIC. written in verse, and treats mainly of astronomy; though two chapters are devoted to arithmetic, algebra, and geometry. The arithmetic is entirely rhetorical ; most of the problems are worked out by the rule of three, and many of them are on the subject of interest. His algebra, which is also rhetorical, presents tin- fundamental cases of arithmetical progression, solves quadratic equations, and gives the method of solving indeterminate equa- tions of the second degree. The first known treatise among the Hindoos which contains a systematic exposition of the modern system of arithmetical nota- tion is that of Bhaskara, born 1114. This treatise was an astronomy, one chapter of which, called Lilawati, is an arithmetic written in verse, with explanatory notes in prose. After an intro- ductory preamble and colloquy of the gods, it begins with the expression of numbers by nine digits and the cipher or small 0. The characters are similar to those in present use, and the method of notation is the same. It contains the common rules of arithmetic and the extraction of the square and cube roots. The greater part of the work is taken up with the discussion of the " rule of three," which is used in solving numerous questions chiefly on interest and exchange. Another chapter of Bha-kara's work called Bjita-ganita (/. ., root computation) is a treatise in algebra. Abbreviations and initials are used for symbols ; subtraction is indicated by a dot, addition by juxtaposition merely, but no symbols are used for multiplication, equality, or inequality, these being written out at length. A product is indicated by the first syllable of the word subjoined to the factors, between which a dot is sometimes placed. In a quotient or a fraction, the divisor is written under the divi- dend without a line of separation. The two sides of an equation are written one under the other, confusion being prevented by the recital in words of all the steps which accompany the operation. Various symbols for the unknown quantity are used, but most of them nre the initials of the names of colors, and the word color is often used as synonymous with unknown quantity ; its Sanskrit EARLY WRITERS ON ARITHMETIC. 35 equivalent also signifies a letter, and letters are sometimes used either from the alphabet or from the initial syllables of subjects of the problem. In one or two cases symbols are used for the given as well as the unknown quantities. The work contains also a treatise on trigonometry. The first Arabic arithmetic known to us is that of Al Kho- warazmi, written about the year 830. It begins with the words, " Spoken has Algoritmi. Let us give deserved praise to God, our leader and defender." Here the name of the author has passed into Algoritmi, from which comes our modern word algorism, meaning the art of computing in any particular way. The work treats of the fundamental rules by the Hindoo method, though the forms of operation are not so simple as those now used. Al Kho- warazmi also wrote a work on algebra in which the term " algebra," al-gebr, first occurs. This work holds an important place in the history of mathematics, as not only subsequent Arabian, but nearly all the early mediaeval works on algebra were based on it. It was from the writings of Al Khowarazmi that the Italians first obtained their ideas of algebra and of the modern method of arith- metic. This arithmetic was long known as algorism, or the art of Al Khowarazmi, in distinction from the arithmetic of BoethiuB, and this name was retained until the eighteenth century. The work had great influence in introducing the Arabic method of arithmetic to the scholars and mathematicians of Europe. Some of the early European writers on arithmetic were men- tioned in the previous chapter on the origin and development of arithmetic. These are Gerbert of the 10th century, and Leonardo, Ben Ezra and Gerard of the 12th century. From the 12th to the 15th century there seem to have been few writers of note on the subject of mathematics, the most noted being Jordan us of Ger- many in the 13th century. One of the most distinguished mathe- maticians of the 15th century was Regiomontanus, who composed a work on trigonometry in 1464. This work contains the earliest known instances of the use of letters to denote known as well as unknown quantities. 36 nil nm.osoi'iiY oi AKI i HMI.I K . In 1482 there appeared at Bamberg a small treatise on arith- metic which WHS attributed to Ulridi Wagner of Nuremburg. It was printed on parchment, and only fragments of a single copy of it are now extant. In 1483 the same Bamberg publishers brought out a second arithmetic, printed <>n paper, and covering seventy- seven pages. The work is anonymous, but I'lricli Wagi Mippo.-rd tn be its author. This Bamberger arithmetic of 1483, says Unger, bears no resemblance to previous Latin treatises, but aims especially at facility of computation in mercantile allairs. The method of solution, as in all the early books on arithmetic. was that of" the rule of three," known also as the " merchant.-' rule " or the " golden rule." An arithmetic by John Widmann was published in Leip/ig in 1489, which is noted as being the earliest book in which the symbols -f- and have been found, though they had pre\ i appeared in a Vienna manuscript. They were not used, hov. as symbols of operation, but apparently merely as marks signify- ing excess or deficiency. It is supposed by some that they were originally warehouse marks to indicate more or less than the normal weights of boxes or chests containing goods. In Widmann 's book we find equation of payments treated according to the methods still in use. Problems of proportional parts and alliga- tion were solved by the use of as many proportions as there were groups to be separated. The work is obscure and deficient in rules for operations, and abounds in fanciful names of topics which Stifel in later years pronounced to be simply laughable. LII i, or Lucas di Borgo, an Italian monk, published }\\< great work entitled Xmnma de Arithmftica, Gcometria, Pi lif<i in Venice in 1 41) 4. The work consists of two p:irt. the fir.-t dealing with arithmetic and algebra, the second with geometry. This is one of the earliest printed treatises on arithmetic and algebra, and the earliest work presenting a >y>t in- atic exposition of nl^oristie arithmetic. It treats of the four fundamental rules, and present* methods of extracting the square root. In its practical application it deals largely with questions EARLY WRITERS ON ARITHMETIC. 37 relating to mercantile transactions, including bills of exchange, working out numerous examples in these subjects. It also con- tains the first known treatise on double entry book-keeping. In this work the term " million " and also " nulla " or cero (zero) occurs for the first time in print. The work had a wide influence in the general introduction of the new arithmetic throughout Europe. Philip Calandri published a work on arithmetic at Florence in 1491. It begins' with a picture of Pythagoras teaching, headed " Pictagoras Arithmetrice introductor." His notion of division is curious. When he divides by 8, he calls the divisor 7, demand- ing, as it were, that quotient which, with seven more like itself, will make the dividend. He describes the rules for fractions, and gives some geometrical and other applications. Jacob Kobel, in 1514, published, at Augsburg, a work on arithmetic in which the Arabic numerals are explained, but not used. The computation was by counters and Roman numerals. In the frontispiece is a cut representing the mistress settling accounts with her maid-servant by an abacus with counters. Cuthbert Tonstall, in 1522, published an arithmetic in Latin which had great influence on the development of the science in England. He gives the multiplication table in the form of a square, and also addition, subtraction, and division tables. For | of of J? he writes f ^ ^ ; and be gives a clear explanation of the multiplication of fractions. De Morgan says this book is " decidedly the most classical which ever was written on the sub- ject in Latin, both in purity of style and goodness of mutter." Jerome Cardan published, at Milan, in 1539. a work entitled Practica Arithmetica. It shows, as might have been expected from an Italian of that age, more power of computation than the French and German writers. It contains a chapter on the mystic properties of numbers, one use of which is in foretelling future events. These are mostly the numbers mentioned in the Old and New Testaments, but not altogether. In another treat- ise, Cardan expresses his opinion that it was Leonardo of Pisa who first introduced the Arabic numbers into Europe. ;;.s mi. niii.o.-oriiY OF ARITHMETIC. iert Recorde publi.-ln-d hi.- r< Irbratcd work on arithmetic, called ' The Grounde of Art. -," about 1540. It was originally ated to Kdwanl VI. '!'!. work was subsequently revi.- 1 and enlarged by John Dee, and published in 1 "ring tin* original dedication, which had b n omitted in the edition pre- pared during the reign of Mary. This work contains u number of the subjivts of modern text-books, including the rule of three, alligation, fellow.-hip, false position, and the method of testing operations by " casting out the 9's." He uses + and with tin- explanation, " + whyche betokeneth too muche, as this line, , pluine without a crosse line, betokeneth too little." It was sub- sequent lv revised by Mellis, who added a third part on practice and other things, and also by Hartwell. The last edition known in by Edward Hattoii, 10'J'J, which contains an additional book called " Decimals made easie." It is said to contain a large number of the principles and problems of modern text-books. Recorde introduced the sign of equality ( = ) in a work >: published in 1 ;">'><>. The work was called by the odd title, " The Whetstone of Witte," in which he gives his reason for the symbol in the following quaint language : " And to avoid the tedious repetition of these words, I will settle, as I doe often in worke use, a pair of parallel or Gemowe lines of one length, thus, =, because noe 2 thynges can be more equalle." Michael Stifel published, at Nuremberg, in 1">44, his celebrated work entitled Arittnnetica Integra. The first two books are on the properties of numbers, on surds and inconimi-n>urahles, learnedly treated, and with a full knowledge of what Km lid had done on the subject. The third book is on algebra, and did much for the introduction of algebra into Germany. Stifel acknowledges his obligations to Adam Riese, and professes to have tak'-n his examples from Christopher RudolflT. Stifel was the first to use the symbols + and to denote the operations of addition and sulv- traction. He introduced also the symbol of evolution, f / ', or- initial of radix or root, though Cantor says that Rudolff had previously used it. EARLY WRITERS ON ARITHMETIC. 39 Nicholas Tartaglia, an eminent Italian mathematician, pub- lished a work on arithmetic, vols. 1 and 2 of which appeared in 1556, and vol. 3 in 1560. The works are verbose, but give a clear account of the various arithmetical methods then in use, and present a large number of notes on the history of arithmetic. The work on arithmetic contains an immense number of questions on every kind of problem which would be likely to occur in mercan- tile arithmetic, and attempts are made to frame algebraic formulas applicable to particular problems. 'It contains also a large collec- tion of arithmetical puzzles and questions of an amusing character, among which is found the question, " What would 10 be if 4 were 6 ?" and the problem of the three jealous husbands and their wives who were to cross a river with a single boat that would carry only two persons. The treatise on numbers was really an algebra, in which are found some interesting investigations. Tartaglia is believed to be the author of a method of solving cubic equations which Cardan obtained from him under a promise of secrecy, and afterward published under his own name in violation of his promise. Simon Stevinus published, at Leyden, in 1585, a work which was edited by Albert Girard in 1634. This work is character- ized by originality, accompanied by a great want of the respect for authority which prevailed in his time. For example, great names had made the point in geometry to correspond with the unit in arithmetic. Stevinus tells them that 0, and not 1, is the representative of the point. " And those who cannot see this," he adds, " may the Author of nature have pity upon their un- fortunate eyes; for the fault is not in the thing, but in the sight which we are not able to give them." A portion of this work contains " Les Tables d' Interest " and " La Disme," the latter of which exerted a great influence on the introduction of decimal fractions. John Mellis, in 1588, at London, published, " A briefe instruc- tion and manner how to keepe bookes of Accompts after the order of Debitor and Creditor," etc. This is the earliest English work on book-keeping by double entry. At the end of the book- 40 THE PHILOSOPHY OP ARITHMETIC. keeping it * short tmitinc on arithmetic. Mellis says : Truly, 1 am but the renuer and reviver of un uuncient old copie. printed here In London the 14 of August, 1543. Then collected, pub- liiihed, made and set forth by one Hugh Oldcastle, Scholemaster, who, as appcareth by his treatise then taught Arithmetike and this booke, in Saint Olluves parish in Marke Lane." In 1596, a work entitled, " The Pathway to Knowledge," was published in London, which was a translation from the Dutch, by W. P. The translator gives the following verses, of which he is supposed to be the author: Thirtic dales bath September, Aprill, June, and November, Februarie, eight and twentie alone ; all the rest thirtie and one. Mr. Davies, in his Key to Hutton's Course, quotes the follow- ing from a manuscript of the date of 1570, or near it : Multiplication is mie relation, And Division is quite as bad, The Golden Rule is mie stumbling stule. And Practice drives me mad. Cataldi, successively Professor of Mathematics at Florence, Perujiia and Bologna, published a work on the square root of numbers at Kolomna, in 1613. The rule for the square root is exhibited in the modern form, and he shows himself a most in- trepid calculator. The greatest novelty of the work is the intro- duction of continurd fractions, then, it seems, for the first time presented to the world. He reduces the square roots of even numlHTs to continued fractions, and then uses these fractions in approximation, but without the aid of the modern rule which tit-rives each approximation from the preceding two. Richard Witt, in 1613, published a work containing "Arith- metical questions " on annuities, rents, etc., ** briefly resolved by means of certain Breviats." These Breviats are tables, and this work is said to be the first English book containing tables of com- pound interest. Decimal fractions are really used. The tables being constructed for ten million pounds, seven figures have to be EARLY WRITERS ON ARITHMETIC. 41 cut off; and the reduction to shillings and pence, with a temporary decimal separation, is introduced when wanted. The decimal separator used is a vertical line, and the tables are expressly stated to consist of numerators, with 100... for a denominator. John Napier, born 1550, died 1617, wrote a treatise on arith- metic which was published at Edinburgh in 1617, after the author's death. It contains a description of Napier's rods with applications. It is remarkable because it expressly attributes the use of decimal fractions to Stevinus. It also states that Napier invented the decimal point. De Morgan says this is not correct, since 1993.273 is written 19932'7"3'". Napier is illustrious as the inventor of logarithms. Robert Fludd, in 1617 and 1619, published a work on mathe- matics at Oppenheim. It contains two dedications, the first, signed Ego, homo, to his creator ; the second, on the opposite side of the leaf, to James I. of England, signed Robert Fludd. The first volume contains a treatise on arithmetic and algebra. The arithmetic is rich in the description of numbers, the Boethian divisions of ratios, the musical system, and all that has any con- nection with numerical mysteries of the sixteenth century. The algebra contains only four rules, referring for equations, etc., to Stifel and Recorde. The signs of addition and subtraction are P and M with strokes drawn through them. The second volume is strong upon the hidden theological force of numbers. Albert Girard published a treatise on algebra at Amsterdam in 1629, which contains a slight treatise on arithmetic. The arithmetic contains no examples in division by more than one figure. On one occasion the decimal point is used, though this was not the first time it had been employed. Girard introduced the parenthesis in place of the vinculum, which had been used by Recorde. Wm. Oughtred's Claris Mathematica, a work on arithmetic and algebra of great celebrity, was first published in 1631. It retains the old or scratch method of division which. Dr. Peacock observes, lasted nearly to the end of the seventeenth century. He 42 THE PHILOeol'IIY OP AKITIIMtmc. not use the decimal point, but writes 12.3456 thus : 1213456. The symbol for multiplication, X, St. Andrew's cross, was intro- duced by Oughtred. lie seems to have first employed the symbol : : to denote the equality of ratio*. He wrote a treatise on trigonometry in 1657, in which abbreviations for tine, corine, etc., were employed. Nicholas Hunt published, in 1633, " The Hand-Maid to Arith- metick refined." The book is full on weights and measures, and commercial matters generally. It does not treat of decimal frac- tions, however. The author calls " dec i mull Arithmeticke, * a division of a pound into 10 primes of two shillings each ; each shilling into six primes of two pence each. It expresses the rule* in verse, of which the following is an example : Adde tbou upright, reserving every tonne. And write the digits downe all with tbj pen. Subtract the lesser from the greet noting the rest, Or ten to borrow you are ever prest To pay what borrowed was think it no paine. But honesty redounding to your gaiue Peter Herigone, in 1634, published at Paris" a work entitled 44 Cursus Mathematici tomus secundus." It introduces the deci- mal fractions of Stevinus, having a chapter " des nombres de la dixme." The mark of the decimal is made by marking the place in which the last figure comes. Thus when 137 livres 16 sous ift to be taken for 23 years 7 months, the product of 1378' and 23583'" is found to be 32497374"", or 3249 liv., 14 sous, 8 denier*. William Webster published, in 1634, tables for simple and compound interest. This work treats decimal arithmetic as a thing known. No decimal point is recognized, only a partition lin<- to be used on occasion. It contains the first head-rule for turning a decimal fraction of a pound into shillings, pence and farthings. Many other interesting details will be found in the works of De Morgan, Unger, Fink, Ball, Gow and Cantor. EARLY WRITERS ON ARITHMETIC. 43 1C N * O 1 < ^ tf NOTE. This page of symbols is taken from Cajori's "History of Elementary Mathematics " by permission of the author and publisher. CHA1TKU IV. ORIGIN OF AK1TIIMKT1CA1. rKOCESSES. /~\NE of the most interesting |>oints connected with the hi.-tor\ ^ of arithmetic, would be a full and complete account of the genesis of the different divisions and processes of the sci- This, ho u impossible. The origin of the elementary or fundamental processes dates back before the invention of printinjr, and can never be determined. Some of the principal facts, how- ever, upon this point, in addition to those already given, will be stated. ARITHMETICAL LANGUAGE. The notation of the nine dL'ii.- and zero, upon which the science of arithmetic is based and developed, originated, as we have already shown, among the Hindoos. This notation was adopted by the Arabians, and be- came general among Arabic writers on astronomy, as well as arithmetic and algebra, about the middle of the 10th century. From the Arabs, who, in the llth century, held jxissession of the southern provinces of Spain, the knowledge was communicated to the Spaniards and other nations of Europe. The Italians, from an early period, adopted tin- method of dis- tributing the digits of a number into groups or period- of six, and consequently proceeding by millions. This is the method of numeration given by I'acioli, 1494. The method of reckoning by three places, as used in this country and on the Continent, seem- to have originated with the Spanish. In a work on arithmetic by Juan de Ortega, 1.030, we find the following method of numera- tion ; 10, dc/.ena ; HIM. centena ; 1000, millar ; 10000, dezena de millar; 100000, centena de millar; 1000000, cuento. The term ii//in, however, had not yet been introduced, and it has not been fully ascertained at what time this introduction took place. (44) ORIGIN OF ARITHMETICAL PROCESSES. 45 Cantor says that the term millione occurs the first time in print in the summa do arithmetica of Paciola. Bishop Tonstall, 1522, speaks of the term million as in common use, but rejects it as bar- barous, being used only by the vulgar. Stevinus divided numbers into periods of three places, called each period membres, aad distinguished them as le premier membre, le seconds membre, etc. Instead of million he says mille mille ; for a thousand million he uses mille mille mille; and for a million- million he uses mille mille mille mille. It would appear from the practice of Stevinus, and from the observation of his contempor- ary, Clavius, that the term million was not at this time in general use amongst mathematicians. Albert Girard divides numbers into periods of six places, which he terms premiere masse, seconde masse, troisieme masse, etc., the first of which only is divided into periods of three places each ; but he does not use the word million. Ducange of Rymer mentions the word million in 1514, and in 1540 it occurs once in the arithmetic of Christopher Rudolff. The term was introduced into Recorde's arithmetic, 1540, and subse- quently appeared in all succeeding authors. It appears to have been admitted into German works much later than into the French and English. The terms billion, trillion, etc., so far as known, appeared first in a manuscript work on arithmetic by Nicolas Chuquet, a gifted French physician of Lyons, and appear in 1520 in a printed work of La Roche. FUNDAMENTAL OPERATIONS The fundamental operations of arithmetic were, without doubt, invented by the Hindoos at a very early period. The work from which our knowledge of Hindoo arithmetic has been mainly derived, is the Lilawati of Bhaskara, who lived about the middle of the 12th century. The work is named after the author's daughter, Lilawati, who, it appeared, was destined to pass her life unmarried and re- main without children. The father, however, having ascer- tained a lucky hour for contracting her in marriage, left an hour- cup on a vessel of water, intending that when the cup should subside, the marriage should take place. It happened, 46 TIIK PIIII.060PHY OF ARITHMETIC. however, that the girl, from a curiosity natural to children, looked into the cup to see the water coming in at the hole, when, by chance, a pearl separated from her bridal dress, fell into the cup, nnd rolling down to the hole, stopped the influx of water. When the operation of the cup had thus been de- layed, the father was in consternation ; and, examining, he found that a small pearl had stopped the flow of the water, 'and the long expected hour was passed. Thus disappointed, the father said to his unfortunate daughter, " I will write a book of your name, which shall remain to the latest times, for a good name is a second life, nnd the groundwork of eternal existence." This work frequently quotes Brahmagupta, an author who is known to have lived in the early part of the 7ih century, and portions of whose works, containing treatises on arithmetic and mensuration, are still extant. Brahmagupta also refers to an earlier author, Arabhatta, who wrote an algebra and arithmetic as early as the 6th century, and who is considered one of the old- est writers among the Hindoos. In tracing the history of the operations of arithmetic, we must therefore begin with the Lilu- wati of Bhaskara. The fundamental operations of arithmetic, as given in the Lil-nnitl, are eight in number ; namely, addition, subtraction, multiplication, division, square, square root, cube, cube root. To the first of these the Arabs added two, namely, duplation and mediation or halving, considering them as operations distinct from multiplication and division, in consequence of the readiness with which they were performed ; and they appear as such in many of the arithmetical books in the 16th century. Addition. The rule given in the Lilawati for addition is as follows : ' The sum of the figures, according to their places, is to be taken in the direct or inverse order," which is interpreted to mean, * from the first on the right towards the left, or from the last on the left towards the right." In other words, they commenced indifferently with the figures in the highest or low- est places, a practice which would not lead to much incon ORIGIN OF ARITHMETICAL PROCESSES. 47 venience in their mode of working. Thus, to add 2, 5, 32, 193, 18, 10, 100, they proceed as follows: Sum of the units, 2, 5, 2, 3, 8, 0, 0, 20 Sum of the tens, 3, 9, 1, 1, 0, 14 Sum of the hundreds, 1, 0, 0, 1, 2 Sum of the sums, 360 Subtraction. The process of subtraction was also com- menced either at the right or the left, but much more commonly at the latter ; and it is remarkable that this method of begin- ning to subtract at the highest place, which is subject to considerable inconvenience, should have been so general. It is found in Arabic writers, in Maximus Planudes, a Byzantine writer of about the middle of the 13th century, and in many European writers as late as the end of the 16th century. In Planudes, numbers to be added or subtracted are placed one underneath another, as in modern works on arithmetic ; and the sum or difference is written above these numbers. When a term in the subtrahend is greater than the correspond- ing one in the minuend, a unit is written beneath them, as in the example in the margin. In performing the operation, 3 is increased 18769 rem. by the unit in the next place to the right, and 54612 rain, also 5, 8, 4, and the terms thus increased are !??, subtracted from the terms above, increased by 10, to find the remainder. In other cases, the numbers are arranged, as 06779 rem. in the margin, the digits 3, 0, 0, 2 in the minuend ^9( being replaced by 2, 9, 9, 1, and then 5 is 2 3245 Tub subtracted from 4, 4 from 1, 2 from 9, 3 from 9, and 2 from 2, in order to get the remainder. It is obvious, that when such a preparation is made, it is indifferent where we commence the operation. Bishop Tonstall attributes the invention of the modern practice of subtraction to an English arithmetician of the name of Garth. This method he has illustrated with great detail, 48 THE PHILOSOPHY OF ARITHMETIC. and added, for the assistance of the learner, a subtraction table, giving the successive remainders of the nine digits when sub- traeted from the series of natural numbers from 11 to 19 inclu- tbe only cases which can occur in practice. In speaking of the methods of preceding writers, 2 91010 he has presented tin- example in the margin, in 3 ]. whi-h it will be seen that the numbers from . which the subtraction is actually made, are placed above the terms of the minuend. In the arithmetic of Ramus, which was published in 1584, though written at an earlier period, we find the operation performed from left to right, and this method is followed by some other writers. Thus, in subtracting 345 from 7 432 the terms to be subtracted and the remainder are written as in the margin. When 3 is subtracted from 4, the remainder should be 1 ; but it is replaced by zero, since the next term in the subtrahend is greater than the corres- ponding term of the minuend ; in the second term the remainder, which should be 9, is reduced to 8, since 5, the next term of the subtrahend, is greater than 2, the term above it, but the last remainder 7, is not changed. Orontius Fineus, the predecessor of Ramus in the professor- ship of Mathematics at Paris, in his De Arithrtielica Practica, 1555, subtracts according to the method now used ; and it is difficult to account for the adoption by Ramus of so inconven- ient a method as he employed, when the method of Fineus must have been familiar to him, unless we attribute it to that love of singularity which led him to aspire to the honor of founding a school of his own. Multiplication. The author of Lilawati has noticed six different methods of multiplying numbers, and two others are mentioned by his commentators. These may be illustrated by their application to the following example: "Beautiful and dear Lilawati, whose eyes are like a fawn's, tell me what are the numbers resulting from one hundred and thirty-five taken ORIGIN OF ARITHMETICAL PROCESSES. 49 into twelve ? If thou be skilled in multiplication, by whole or by parts, whether by division or separation of digits, tell me, auspicious woman, what is the quotient of the product divided by the same multiplier ?" Here the multiplicand is 135, and the multi- 135 plier 12; and the first method, which consists of 12 12 12 multiplying the terms of the multiplicand sue- 12 60 oessively by the multiplier, is indicated in the 3 6 margin. 16 20 The second method, which consists in sub- dividing the multiplier into parts, as 8 and 4, 135 8 1080 and severally multiplying the multiplicand by 135 4 540 them, is also indicated in the margin. 1620 The third method, which con- sists in separating the multiplier 12, into its two factors, 3 and 4, and 1354 205403 120 multiplying successively by these factors, the last product being the result, is also represented in the margin. The fourth method consists in taking the digits as parts, viz., 1 and 2, the multiplicand 135 135 being multiplied by them severally, and the products being added together according to the places of the figures, as is represented in the 1620 margin. The fifth method consists in multiplying the multiplicand by the multiplier less 2, namely, 135 10 1350 10, and adding the result to twice the multipli- 135 2 270 eand, as may be seen in the margin. 1620 The sixth method consists in multiplying the multiplicand by the multiplier increased by 8, namely, 20, and subtracting 8 times the multiplicand, as represented in the margin. 50 THE PHILOSOPHY OF ARITHMETIC. 135 12 ~To 11 5 The other two methods are given in the Commentary of Ganesa. The first of these, which is represented in the margin, appears to have been very popular in the East, and was adopted by the Arabs, who termed it shabacah, or net-work, from the reticulated appearance of the figure which it formed, and also by the Per- * sians under a slight alteration of form. It is found likewise in the works of the early Italian writers on algebra, and the same principle may be recognized in the process of multipli- cation by Napier's rods. The second of these two methods of multiplica- tion, as represented in the margin, is described in full by Ganesa. He, however, considers this method difficult, and not to be learned by dull scholars with- l out oral instruction. 1620 The number and variety of these methods would seem to show that the operation of multiplication was regarded as difficult, and it is remarkable that the method now used is not found amongst them. We find no notice of the multiplication table among either them or the Arabs. At all events, it did not form a part of their elementary system of instruction, a circumstance which would account for some of the expedients which they appear to have made use of, for the purpose of relieving the memory from the labor of forming the products of the higher digits with each other. The Arabs adopted most of the Hindoo methods of multi- plication, and added some others of their own ; among which are some peculiar contrivances for the multiplication of small numbers. They may also be considered as the authors of the method of quarter squares, or of finding the product of two num- bers by subtracting the square of half their difference from the square of half their sum. The Arabs were most probably the in- ventors of the method of proof by casting out 9's, which is as yet unknown to the Hindoos ; they called it tarazit, or the balance. ORIGIN OF ARITHMETICAL PROCESSES. 51 The work of Planudes was chiefly collected from the Arabic writers, as appears from his being acquainted with the method of casting out 9's. In multiplication he has g,Q chiefly followed the method of multiplying crosswise or 35 Kara TOV xiaapav, from the figure x, which is employed to x connect the digits to be multiplied together. Thus, in 24 multiplying 24 into 35, we should write the factors as in the margin ; and then multiply 4 into 5 (/wwaArf^ write down and retain 2 for the next place ; multiply 4 into 3, and 3 into 5, the sum is 22, which added to 2, makes 24 (<*<*<*<%), write down 4 and retain 2 ; lastly, multiply 2 into 3, add 2, which makes 8 (mzTwraJef) ( and the product is 840. He also gives another method which he acknowledges to be very difficult to per- form with ink upon paper, but very commodious on a board strewed with sand, where the digits may be readily effaced and replaced by others. Thus, taking the same example, we multiply 2 into 3, write 6 above the 3 ; * , , multiply 2 into 5, the result is 10; add 1 to 6, and 35 replace it by 7, or write 7 above it; multiply 4 into 3, 24 the product is 12; write 2 above 5, and add 1 to 7, which is replaced by 8, or 8 written above it ; lastly, multiply 4 into 5, the result is 20; add 2 to 2, place 4 above it and after it the cipher ; the last figures, or those which remain without accents, will express the product required. Division. The extreme brevity with which the rules of division are stated in the Lilawati renders it difficult to describe the Hindoo method of dividing numbers. We are directed to abridge the dividend and divisor by an equal number, whenever that is- practicable; that is, to divide them both by any common measure; thus, instead of dividing I' 1 *-* 1 by 12, we may divide 540 by 4, or 405 by 3. We find, how over, in one of the commentators on this work, a description of the process of long division, which, if exhibited in a schomr. would exactly agree with the modern rule ITALIAN METHODS. The Italians, who cultivated arithmetic 52 THE PHILOSOPHY OF ARITHMETIC. with so much zeal and success, from a very early period adopted from their Oriental masters many of their processes for the multiplication and division of numbers ; adding, how- ever, many of their own, and particularly those which are practiced at the present time. In the Summa de Arithmetical, of Lucas di Borgo, we find eight different methods of multi- plication, some of which are designated by quaint and fanciful names. We shall mention them in their order. 1. Multiplicatio : bericuocoli e schacherii. The second of these names is derived from the resemblance of the written process to the squares of a chess-board ; the first from its resemblance to the check- 456 ers on a species of sweetmeat or cake, 3 7 made chiefly from the paste of bacochi or apricots, which were commonly used at j 3 1 1 648 festivals. The process is exhibited in the margin. This method is presented by 92 Tartaglia and later Italian writers with- 172368 out the squares; and it thus became the method which is now universally used, and which was adopted from the beginning of the 16th century by all writers on arith- metic, nearly to the exclusion of every other method. 2. Castelluccio ; by the little castle. This method, as indicated in the margin, uses the |876 upper number as the multiplier, and begins with the higher terms. This method was much prac- ticed by the Florentines, by whom it was some- ' 4175230 times called alV indietro, from the operation 40734 beginning with the highest places, more Arabum, 67048164 according to the statement of Pacioli. 3. Columna, o per tavoletta ; by the column, or by the tablets. These were tables of multiplication, arranged in columns, the first containing the squares of the digits, the second the pro- ducts of 2 into all digits above 2 ; the third, of 3 into all digits above 3 ; and so on, extending in some cases as far as the pro- ORIGIN OF ARITHMETICAL PROCESSES. 53 ducts of all numbers less than 100 into each other. Pacioli says that these tablets were learned by the Florentines, and their familiarity with them was considered by him as a princi- pal cause of their superior dexterity in arithmetical operations. This method is used in multiplying any number, however large, into another which is within the limits of the table. Thus, to multiply 4685 by 13, the terms of the multiplicand are multiplied successively by 13, and the results formed in the ordinary manner. 4. Crocetta sive casella ; by cross multi- plication. This method is said to require more mental exertion than any other, par- ticularly when many figures are to be combined together. Pacioli expresses his admiration of this method, and then takes 20 7 9 3 6 the opportunity of enlarging on the great difficulty of attaining excellence, whether in morals or in science, without labor. 5. Quadrilatero ; by the square. This is a method which has been characterized as elegant, and as not requiring the operator to attend to the places of the figures when performing the multiplications. The method is represented in the margin, and will be readily understood. 6. Gelosia sive graticola ; latticed multiplication. called," says Pacioli, " because the dispo- sition of the operation resembles the form of a lattice, a term by which we designate the blinds or gratings which are placed in the windows of houses inhabited by ladies so that they may not easily be seen, as well as by other nuns, in which the lofty city of Venice greatly .abounds." The method will "9 7 4 be readily understood by the example given in the margin, which multiplies 987 by 987. It is the same as one previously 5 4 5 4 3 3 1 G i 9 2 1 7 9 2 7 1 5 2 9 on. " It 4 is > 54 THE P1IILOSOPHY OF ARITHMETIC. noticed, which was in common use among the Hindoos, Ara- bians, and Persians. 7. Ripiego ; multiplication by the unfolding or resolution of the multiplier into its component factors. Thus, to multiply 157 by 42, resolve 42 into its ripieghi or factors, 6 and 7, and multiply successively by them. 8. Scapezzo ; multiplication by cutting up, "* 5, 6 or separating the multiplier into a number of '. '. . oarts, which compose it by addition. Thus, 81624 30 1 1 A 1 9 A I Q A I f* A to multiply 2093 by 17, we separate 17 into ' g.jy 10 and 7, multiply by each, and take the sum _ iii. bU yU IoU of the products. In some cases both multi- plicand and multiplier were separated into parts. Thus, the multiplication of 15 by 12 was performed as in the margin. In another Italian arithmetic, published in 1567, by Pietro Cataneo Sienese, we find the same distinctions preserved, and the same names, or nearly so, attached to them ; the method of cross multiplication is expressly attributed to Leonardo ,. of Pisa, who derived it, in common with Maximus >^ Planudes, from the Hindoos, through the Arabians. 4 7 It is not impossible tliat St. Andrew's cross, which ~~^~ is the sign of multiplication, was derived from the custom of uniting the numbers to be multiplied together by lines which crossed each other, as in the example given in the margin. Both Lucas di Borgo and Tartaglia mention other methods of multiplication which were made use of in their time. An extraordinary passion seems to have prevailed in that age for 'the invention of new forms of multiplication, and every pro- fessional practitioner of arithmetic considered it as an important triumph of his art if he could produce a figure more elegant and more refined in its composition and arrangement than those which were used by others. They are, all of them, how- ever, characterized by Pacioli as inconvenient, at least com- pared with those which he had given ; and Tartaglia treats them as trifling and superfluous, such as any one may invent who is acquainted with the 2d proposition in the 2d Book of Euclid. ORIGIN OF ARITHMETICAL PROCESSES. 55 The Hindoos, as has been stated, had no proper knowledge of the multiplication table, and the Arabs do not appear to have made use of the table of Pythagoras as the basis of their arithmetical education ; the credit of introducing it, therefore, is due to the early Italian writers on the science, who probably found it in the writings of Boethius, and adopted it thence. Even after the Italian arithmeticians were familiar with this table, many writers of other countries considered it important to relieve the memory from the labor of retaining it for the products of all digits exceeding 5, by giving rules for their formation. The principal rule for this purpose, called regula ignavi, or the sluggard's rule, was adapted from the Arabians, and is found in Orontius Fineus, liecorde, Laurenberg, and most other writers between the middle of the 16th and Itth centuries. The rule is as follows: Sub- 73 82 91 tract each digit from 10, and write down the XXX difference ; multiply these differences together, "^ *j __ and add as many tens to their product as the ^ 56 72 first digit exceeds the second difference, or the second digit the first difference. The Arabians made use of this and other similar rules which applied to numbers of two places of figures, a practice which may be accounted for by their very general use of sexagesimals, and the consequent importance of being able to form the products which are found in a sexagesimal table. Many other expedients were proposed to relieve the mem- ory, in the process of multiplication, from the labor 514*) of carrying the tens. An interesting one is pre- 43 sented by Laurenberg, an author who endeavored lou to elevate the character of the common study of 1532 arithmetic by collecting all his examples from clas- sical authors, and by making them illustrative of the geography, chronology, weights and measures 110fl of antiquity. It will be understood from the example given, without explanation. 66 THE PHILOSOPHY OF ARITHMETIC. Division. Neither Planudes nor the early Arabic writers seem to have presented any methods of dividing that merit the special notice of the writers on the history of arithmetic. Lucas di Borgo gives four distinct methods which we proceed to explain. These methods had particular names, as in mul- tiplication. 1. Partire a regolo, sometimes called also partire per testa or division by the head, was used when the divisor was a single digit, or a number of two places, such as 12, 13, etc., included in the librettine or Italian tables 6 of multiplication. The method will be readily 3478 understood from the example given. Di Borgo says: 579|- " This method of division is called by the vulgar, the rule, from the similitude of the figure to the carpenter's rule which is made use of in the making of dining-tables, boxes, and other articles, which rules are long and narrow." 2. Per ripiego ; which consists in resolving the divisor into its simple factors, or ripieghi. It will be readily understood from the example g 35721 given, and be recognized as a common method of 3969 modern arithmetics. 3. A danda ; which the author says is thus called for rea- sons which will be readily seen in the opera- tion itself, which represents the division of D ^ s 7 r ' ^g^ 1 *' 230265 by 357, giving a quotient of 645. 230 265 The process is the same as our common 2142 method of long division, only the numbers ~1606~ are not so conveniently written. It was 1428 called a danda, or by giving, because after 1785 every subtraction we give or add one or 1785 more figures on the right hand. The author, however, prefers the next method. 4. Galea vel galera vel batello ; so called from the process resembling a galley, " the vessel of all others most foared on ORIGIN OF ARITHMETICAL PROCESSES. 57 the sea by those who have good knowledge of it ; the most secure and swiftest ; the most 975 rapid and lightest of the boats that pass on T$9Ai the water. " The method may be illustrated 9/535399(9 by dividing 97535399 by 9876. We first 98,70 write the dividend, and underneath it the divisor, and commence with the second figure of the dividend, since the divisor is not contained in the first four terms of the divi- dend. Multiplying the divisor by the first term of the quotient, 9 times 9 are 81, which sub- 86 tracted from 97 leaves 16, which is written $%& above 97 ; then cancel 97 and 9 in the divi- 97-35399/98 sor ; 9 times 8 are 72, which taken from 165, 98766 leaves 93 ; write 9 above 16 and 3 above 5 987 in the dividend, and cancel 165, and 8 in divisor ; 9 times 7 are 63, which subtracted from 933 leaves 870; cancel 933 in remainder, and 7 in divisor; 9 times 6 are 54, which subtracted from 705 leaves 651 ; cancelling 705, and 6 in the divisor, we have as a remainder 8651399. For multiplying by the second quotient figure, we arrange the divisor as in the margin, and proceed as 8(31022 before. The complete operation is repre- 9/55$5 sented by the last work in the margin, and ^$5^5/3 is so apparent that it needs no further expla- 9/75^5^99(9876 mm> nauon. 98777 Tartaglia states that it was the custom in 988 Venice for masters to propose to their pupils 9 as the last proof of their proficiency in this process of division, examples which would produce the com- plete form of the galley, with its masts and pendant. The last addition to the work was supplied by the scheme for the proof of the accuracy of the operation by casting out the 9's. Dr. Peacock gives an example showing the numbers 58 THE PHILOSOPHY OF ARITHMETIC. thus arranged, which is very curious, but too long for insertion here. The same process is illustrated by an example ^ ^ , from the numerous calculations by Regiomon- 3134 tanus, in his tract on the quadrature of the 154750 [4 circle, written as early as 1464, though not 276548 published until 1532. The question proposed is to divide 18190735 by 415. The divisor is 4HH placed under the dividend and repeated at 444 every step backward, and all the figures erased 43333 in succession. The quotient, 43833 is placed down the side and along the bottom, the remainder 40 being the only digits left on the board. It is amusing to observe the enthusiastic admiration of Di Borgo for this method of division. When describing the pre- ceding method he seems impatient, and looks forward with pleasure to the description of the method a la galea, as pos- sessing a certain charm and solace, remarking that it is a noble thing to see in any species and scheme of numbers, a galley perfectly exhibited, so as to be able to observe its mast, its sail, its yards and its oars, launched in the spacious ocean of arithmetic. This method, we are surprised to learn, appears to have been preferred by nearly every writer on arithmetic as late as the end of the 17th century. It was adopted by the Spaniards, French, Germans, and English; and it is the only method which they have thought necessary to notice. It is found almost universally in the works of Tonstall, Recorde, Stifel, Ramus, Stevinus, and Wallis ; and it was only at the beginning of the 18th century that this method of division, called by the English arithmeticians the scratch method of division, from the scratches used in cancelling the figures, was superseded by the method now in common use, which was specifically called Italian division, from the country whence it was derived. ORIGIN OF ARITHMETICAL PROCESSES. 59 Recorde noticed the Italian method of division, which, he says, " I first learned of, and is practiced by my ancient and espe- cial loving friend, Master Henry Bridges, wherein not any one figure is cancelled or defaced. He illustrates the method by an ^ example which we subjoin ; though, as before stated, he preferred the scratch method of dividing. POWEUS AND ROOTS. The author of the Lilawati has given rules for the formation of squares and cubes, as well as for the extraction of the corresponding roots. The rule for the formation of the square, which is gj very ingenious, is as follows: Place the square of 28 the last digit over the number, and the rest of the 126 digits doubled and multiplied by the last are to be 1? placed above them respectively ; then repeating the _J^ number with the omission of the last digit, perform 88209 the same operation. This is illustrated in squaring the num- ber 297. In performing the converse operation, every uneven place is marked by a vertical line, and the intermediate digits by a horizontal one ; but if the place be even, it is joined with the contiguous odd digit. It may be illus- ' ' trated by extracting the square root of 88209, enough of the work being indicated to show the 48209 nature of the method. We subtract from the last i uneven place, 8, the square 4, and there remains 12209 48209, represented as in the margin. Double the ' root 2, making 4, and divide 48, the number de- noted l>y the next two terms, by the result, obtaining '.) (10 would be too large), and subtracting 9 times 4 or '.',('}, \v<- have 12209. From the uneven place, with the resi- due, 122, subtract the square of 9, or 81; the remainder is 4109 Double 9, giving 18, and unite the result with 4, giving 58, and divide 410 by it, and we have 7, and the remainder, 60 THE PHILOSOPHY OF ARITHMETIC. 8 49, to which the square of the quotient 7, or 49, answers with- out a residue. The double of the quotient, 14, is put in a line with the preceding double number, 58, making 594, the half of which is the root sought, 297. This account of the Hindoo method of extracting square root, is taken from the commentators on the Lilawati, and does not differ essentially from the method now used ; and the same may be said of the method of extracting the cube root, the principal difference from the present method being found in their peculiar methods of multiplying and dividing. The method of extracting the square root used by the Ara- bians resembled their method of division ; and it is prob- able that they are both founded on the Greek methods of performing these operations with sexagesimals. The example given will show the form of operation. Vertical lines being drawn and the numbers distinguished into periods of two figures, the nearest root of 10 is 3, which is placed both below and above, and its square, 9, subtracted ; the 3 is now doubled, and 6 being writ- ten in the next column, is contained twice in 17, or the remainder with the first figure of the next period ; the 2 is therefore set down both above and below, and being multiplied into 6 gives 12, which is subtracted from 17, leaving 5 ; the square of 2, or 4, is now subtracted from 55, and 518, the re- mainder, with the succeeding figure, is divided by 64, or the double of 32, giving 8 for the quotient ; then 8 times 64 are 512, which, subtracted from 618 leaves 6-, and 64 is exhausted by taking from it the square of 8. It is said that this mode was adopted from the Arabs by the Hindoos. 1 7 5 8 4 9 1 7 1 2 5 5 4 5 1 8 5 1 2 6 4 6 4 4 8 6 6 3 2 ORIGIN OF ARITHMETICAL PROCESSES. 61 ^2416 604 4(304 2416 00 m The earlier mathematicians of Europe employed a similar method of extracting the square root, though perhaps not quite so systematic and regular. In proof of the rule which they followed, they constantly refer to the 4th proposition of the 2d book of Euclid. I will give several examples illustrating their methods The first is from the arithmetic of Pelletier, the first edition of which was published in 1550. It represents his method of extract- ing the square root of 92416, and is so sim- ple it needs no explanation. It will be seen that the dots marking the periods into which the number is separated are placed under the number, instead of above it as is now the custom. The second example is from the work of Lucas di Borgo, and is in the form of the process which was most commonly adopted. The example, as will be seen, is the extrac- tion of the square root of 99980001. The scheme will require no explanation, but will be readily understood by those who are fam- iliar with the galley form of division. We present another illustration taken from the tract, already mentioned, of Regiomontanus. The question is to find the square root of the number 5261216896. Now the nearest square to 52 is 49, leaving 3 to be set above the 2, while 7, the root, is placed in the vertical line ; then double of 1, or 14, being set under the 36, is contained twice, and 2 is accordingly placed under the 7 ; but twice 1 is 2, which taken from 3 leaves 1, and twice 4 are 8, which taken from 6, or 16, leaves 8, and extinguishes the 1 before it ; and twice 2 are 4, which taken from 1, or 11, leaves 7, and converts the pre- ceding 8 into 7. In this way the process advances till the 123 2465 1757174 38796595 5261216896 14406 430 145 14 1 72534 02 THE PHILOSOPHY OF ARITHMETIC. figures become successively effaced. The root, 72534, is placed both at the right hand side and also immediately below the work. The divisors do not appear to be right, but we do not feel sufficiently acquainted with the subject to change them, and do not possess the original work by which we can verify them. The method of extracting cube root used by the Arabians and Persians, and by them communicated to the Hindoos, resembles likewise their method 5 of performing division. We will illus- trate it by extracting the cube root of 91125. Having drawn the vertical lines as indicated, the several digits of the num- ber are inscribed between them, and dots set over the first, fourth, seventh, etc., reckoning from the right. The nearest cube to 91 is 64, which is set down and subtracted, leaving 27. To obtain the next term of the root, 3 times 18, which is 3 times the square of the root found, is written below, and being contained 5 times in 271, the divisor is completed by adding 3 times the product of 4 and 5, or 60, and then the square of 5, or 25, mak- ing f in all 5425, each term of which is multiplied by 5, and the products sub- tracted in succession. The ancient mode of extracting the cube root practiced in 9 1 1 a 5 6 4 2 7 2 5 2 2 1 1 '2 2 5 4 8 6 8 2 5 1 5 4 2 5 Europe was similar to the process just explained, but not so regular and formal. The annexed example is taken from the Ars Supputandi of the famous Cuthbert Tonstall, Bishop of Durham, the earliest treatise on arithmetic published in England, and a work of no common merit. The number 250523582464, 4' 7'6' 3'" 4'0'" 2'5'0'5'2'3'5'8'2'4'6'4' 6 3'4'l / 8 / 7'8'9 / 8 7 9 7 0'4 ' 1'0'2'1'5'9' 1'2'2'S'l' 0' 4'9'fi' 8'2'4'6' 7' ORIGIN OF ARITHMETICAL PROCESSES. 63 whose root is to be extracted, is placed above two parallel lines, between which the root 6304 is inserted ; the successive divisors and the corresponding remainders being written alter- nately below and above, and the figures erased as fast, as che operation advances, the operation of erasure being here denoted by accents. M. Stifel, who usually sought to generalize the methods of his predecessors, has considered the process of extracting the square root in connection with those of higher powers. By observing the formation of the powers themselves, he discovered certain schemes, or pictures as he calls them, for extracting the square, cube, biquadrate, etc., roots. If we indicate the terms of a binomial root by a and b, his scheme for the square root would consist of a-20-6 and 6 2 written under the b to denote addition. The meaning of the scheme is $ that in extracting the square root, the first 070r)2p / T(2601 term, a, must be multiplied by 20 to get 2 - 20-6. the divisor from which we determine the 36-276 26 - 20 0-0 second term, o; after which the sum of 2-60-20 1 the product of a, 20, and b, and b 1 must 1-5201 be subtracted from the first remainder. His method is illustrated by the extraction of the square root of 6765201, as here given. The history of the origin of these arithmetical processes is derived from Prof. Leslie and Dr. Peacock, much of it having been copied word for word from the originals. The origin of methods in Fractions, Decimals, Rule of Three, Continued Fractions, etc., will be given in connection with those subjects ; and such other historical information as it is thought will be of interest to the reader will be presented in its appropriate place. Occasionally the same fact is repeated, in order to give a completeness to tho particular subject discussed. PART I. THE NATURE OF ARITHMETIC. SECTION I. THE NATURE OF NUMBER. SECTION II. ARITHMETICAL LANGUAGE. SECTION III. ARITHMETICAL REASONING. SECTION 1. THE NATURE OF NUMBER. I. SUIUECT MATTER OP ARITHMETIC II. DEFINITION OF NUMI;ER III. CLASSES OF NUMBERS. IV. NUMERICAL IDEAS OF THE ANCIENT* CHAPTER I. NUMBER, THE SUBJECT MATTER OF ARITHMETIC. \TUMBER was primarily a thought in the mind of Deity. -L i He put forth His creative hand, and number became a fact of the universe. It was projected everywhere, in all things, and through all things. The flower numbered its petals, the crystal counted its faces, the insect its eyes, the evening its stars, aud ihc moon, time's golden horologe, marked the months and the seasons. Man was created to apprehend the numerical idea. Finding it embodied in the material world, he exclaimed, with the enthu- siasm of Pythagoras, " Number is the essence of the universe, the archetype of creation." He meditated upon it with enthu- siasm, followed its combinations, traced its relations, unfolded its mystic laws, and created with it a science the beautiful science of Arithmetic. Let us consider the origin and nature of the idea out of which man has created this science of exact relations and interesting principles. Origin. The conception of number begins with the contem- plation of material objects. Objects are found in combinations or collections, and the inquiry, how many of such a collection, gives rise to the idea of number. The young mind looks out upon nature, communes with its material forms, sees unity and plurality, the one and the many, all around it, and awakens to the numerical idea. Strange law of spiritual development! the material thing calls into being the immaterial thought. The unity and plurality, as it dwelt in the God-mind and was ( 07 ) 68 THE PHILOSOPHY OF ARITHMETIC. embodied in the material world, passes over to the mind of man, and appears as an idea of the immaterial spirit. The idea of definite numbers is developed by a mental act called counting. We ascertain the how-many of a collection, by counting tb*> objects in the collection. The act of counting, (one, two, three, etc.), is the foundation of all our knowledge of number. In counting, we pass in succession from one object to another. Succession implies time, and is only possi- ble in time. The idea of number, therefore, has its origin in the fact of time, and is possible only in this great fact. A brief consideration of this relation will not be uninteresting. Time is one of the two great infinitudes of nature. Space and Time are the conditions of all existence. Time enables us to ask the question, when ; Space, the question, where. Space is the condition of matter regarded as extended, and is thus the condition of extension. Extension has three dimen- sions, length, breadth, and thickness. The science of extension is geometry. Space is thus seen to be the basis or condition of the science of geometry. Time is the condition of events, as Space is of objects. Every event exists in Time, as every object must exist in Space. Time has somewhat the same relation to the world of mind, that Space has to the world of matter. Matter extends in Space, as mind protends in Time. This intimate relation of Number and Time leads me to present a few thoughts concern- ing the nature of Time, and the development of the idea of Number from it. Time is not a mere abstraction. It is not a quality per- ceived in an object and drawn away from it by the power of abstract thought, and conceived as an abstract notion. Neither is it a general idea, or a concept. We do not first get partic- ular notions of Time, and then, by putting these together, form a general idea of it. No summation of particular times can give the grand, unlimited idea of Time that the mind possesses. Indeed, we do not consider particular times as examples of NUMBER, THE SUBJECT MATTER OF ARITHMETIC. 69 Time in general ; but we conceive all particular times to be parts of a single endless Time. This continually flowing and endless time is what offers itself to us when we contem- plate any series of occurrences. All actual and possible times exist as parts of this original and genera. Time. There- fore, since all particular times are considered as derivable from time in general, it is manifest that the notion of time in general, cannot be derived from the notions of particular times. Time is a grand intuition. It is an idea which is formed in the mind when the proper occasion of sensible experience is presented. Sensible experience is not the cause, but the occa- sion upon which the mind conceives or originates this idea. It is the product of the higher intuitive power known as the Reason. But Time is not only an idea, it is a great reality. It has a real objective existence, independent of the mind which conceives it. Were there no minds to conceive it, time would still exist as the condition of events. Were all events blotted out of existence, time would remain an endless on-going. Time is infinite. No mind can conceive its beginning ; no mind can conceive its end. All limited times merely divide, but do not terminate the extent of absolute time. In it every event begins and ends, while it never begins and never ends. It is, in its very nature, like Him who inhabiteth eternity, with- out beginning and without end. Time gives rise to succession, as space does to extension. Out of succession grows the idea of Number, and the science of Number is Arithmetic. Arithmetic, therefore, has somewhat the same relation to time, that geometry has to space. In view of this fact, some philosophers have called geometry the science of space, and arithmetic the science of time. This view of ju-ithmetic, however, has not been adopted by all writers, since t here are other ideas growing out of time than that of number. VVhewell, in writing of the Pure Sciences, speaks of the three great ideas Space, Time, and Number ; thus distinguishing between Number and Time. Several efforts have been made 70 THE PHILOSOPHY OF ARITHMETIC. to construct a science of Time; the most remarkable is that of Sir William Rowan Hamilton, which resulted in the invention of the wonderful Calculus of Quaternions. Time is considered as having but one dimension. In this respect it differs from Space, which has three dimensions, length, breadth, and thickness. Time may be regarded as analogous to a line, but it has no analogy to a surface or a vol- ume. Time exists as a series of instants which are before and after one another ; and they have no other relation than this of before and after. This analogy between Time and a line- is so close, that the same terms are applied to both ideas, and it is difficult to say to which they originally belonged. Time and lines are called long and short; we speak of the beginning and the end of a line, of & point of time, and of the limits of a portion of duration. There being nothing in Time which corresponds to more than one dimension of extension, there is nothing which bears any analogy with figure. Time resembles a line extending indefinitely both ways ; all partial times are portions of this line ; and no mode of conceiving time suggests to us a line making an angle with the original line, or any other combina- tion which might give rise to figures of any kind. The anal- ogy between time and space, which in many circumstances is so clear, here disappears altogether. Spaces of two and of three dimensions, surfaces and volumes, have nothing to which we can compare them in the conceptions arising out of time. The conception which peculiarly belongs to thrre, as figure does to space, is that of the recurrence of times similarly marked. This may be called rhythm, using the word in a general sense. The forms of such recurrence are noticed in the versification of poetry and the melodies of music. All kinds of versification, and the still more varied forms of recur- rence of notes of different lengths, which are hoard in all the varied strains of melodies, are only examples of such modifica- tions or configurations, as we may call them, of time. They HUXBEB, THE SUBJECT MATTER 07 ABITHMETIC. 71 involve relations of various portions of time, as figures involve relations of various portions of space. But jet the analogy between rhythm and figure is by no means very elose ; for in rhythm we hare relations of quantity alone in parts of time, whereas in figure we hare relations not only of quantity, but of a kind altogether different namely, of position. On the other band, a repetition of similar elements, which does not necessarily occur in figures, is quite essential in order to impress upon us that measured progress of time of which we here speak. And thus the ideas of time and space bare each their peculiar and exclusive relations; position and figure belonging only to space,- while repetition and rhythm are ap- propriate only to time. One of the simplest forms of recurrence is alternation, as we have alternate accented and unaccented syllables For example: " Come one', cotne all', this rock' shall fly V Or without any subordination, as when we reckon numbers, and call them in succession, odd, even, odd, even, etc. But the simplest of all forms of recurrence is that which has no variety, in which a series of units, each considered as exactly similar to the rest, succeed one another; as one, one, one, and so on. In this case, however, we are led to consider each unit with reference to all that have preceded ; and thus the series one, one, one, and so forth, becomes one, two, three, four, five, and so on; a series with which all are familiar, and which may be continued without limit. We thus collect from that repetition of which time admits, the conception of Number. This view of the origin of the idea of Number is now accepted by a large number of thinkers, bat there are those wbo bold other theories. Toe most objectionable view is tbat number is a cense perception, the absurdity of which is seen in the fact that number IMS no color or form or any attribute of a percept. Number is not a percept ; it is an intuition. CHAPTER II. DEFINITION OF NUMBER. THE idea of number is so elementary that it is difficult to define it scientifically. Various definitions have been pre- sented by different writers upon the subject, though no one has hitherto given one which is, in all respects, satisfactory. The two most celebrated definitions are those of Newton and Euclid, both of which will be briefly considered. Newton defined number as " the abstract ratio of one quan- tity to another quantity of the same species." This definition is philosophical and accurate. It shows number to be a pure abstraction derived from a comparison of things. In discrete quantity, it regards one of the individual things as the unit of comparison ; while in continuous quantity the unit is assumed to be some definite portion of the quantity considered. This definition was no doubt primarily intended to apply to extended quantity, in which there is no natural unit, but in which some definite portion of the quantity is assumed as a unit of measure, and the quantity estimated by comparing it with this unit as a standard. Such comparison gives rise to three kinds of numbers; integral, fractional, and surd numbers. When the quantity measured contains the unit a definite number of times, the number is integral ; when it is only a definite part of the measure, the number is fractional ; when there is no common measure between the unit and th quan- tity measured, the number is a surd or radical. The definition of Newton, though admirable in many respects, is not suitable for popular use. It is too abstract and (72) DEFINITION OF NUMBER. 73 difficult to be understood by young pupils ; and cannot, there- fore, be recommended for our elementary text-books. It may be said, also, that it does not express clearly the process of thought by which we attain the idea of number. It is more appropriate as applied to continuous than to discrete quantity, while the idea of number begins with discrete rather than con- tinuous quantity. In this latter respect it may possibly be improved by changing the form of expression, while retaining its spirit: thus, A number is the relation of a collection to the single thing. This is simpler than the original form, and is in many respects a very satisfactory definition. Euclid defined number to be "an assemblage or collection of units or things of the same species." This definition, slightly modified, has been generally adopted by mathematicians. In its original form it excluded the number one, since one thing is not an assemblage or collection, and hence it has been changed to read A number is a unit or a collection of units. This is the definition which is now found in a large number of text- books. This definition, 'however, is not strictly correct. A number is not precisely the same as a collection of units, and a collec- tion of units is not necessarily a number. In other words, there is a difference between a collection of things and a num- ber of things. This may be more clearly seen by the use of the corresponding verbs. To collect and to number -are two different things. We may collect without numbering, and we may number without collecting ; I may collect & number of things, and I may number a collection of things. If a basket of apples were strewn over the floor and I were told to collect them, I might do so without numbering them ; or, if told to number them, I might do so without collecting them. In the latter case I would have a number of apples without having a collection of apples, except the mental collection, from which it appears that a number is not precisely the same as a collection. Number is more definite than collection. A collection is an 74 THE PHILOSOPHY OF ARITHMETIC. indefinite thing, numerically considered ; number is that which makes it definite. Number and collection are not, therefore, identical. Number is rather the how many of the collection. It is thus seen that Euclid's definition, as modified and now introduced into most of our text-books, is not without scien- tific objections. It must be admitted, however, that there is no other one word which so nearly expresses the idea of the word number as collection; and, for ordinary purposes, they may be used interchangeably. Thus we may say, in analysis, we pass from the collection to the single thing ; from a number to one. It is, therefore, regarded as the best definition for the ordinary text-book, that has hitherto been presented. From this discussion it will appear, as above stated, that it is difficult to present a good definition of Number. This diffi- culty is due to the fact that Number is a simple term express- ing a simple idea, for which we have no other word of precisely the same signification. Simple terms are always difficult to define, from the very fact that they define themselves. Indeed, perhaps there is nothing in the way of a definition of number clearer than the identity "A Number is a Number." The following, though liable to a verbal objection, seems to me to come as near the truth as anything that has yet been pre- sented: A Number is the how-many of a collection of units; or, A Number is how many times a single thing is reckoned, or is contained in a collection The first excludes the number one, unless, as some writers propose, we give a special signification to collection. The second provides for the number one, but is not, in other respects, eo satisfactory as the first. These definitions express pre- cisely the idea of a number, but the use of the expression hoiv many as a noun, is not elegant in the English language. The simplest and most satisfactory definition for a text-book is, "A Number is a unit or a collection of units." The definitions of a number, as given in some of our text- books, are very objectionable. One author says : " Numbers DEFINITION OF NUMBER. 76 are repetitions of units." This may answer as a popular state- ment, but is very far from meeting the requirements of a sci- entific definition. Another author says: "A number is a definite expression of quantity." So is a triangle or a circle, each of which should be a number if this definition is correct Another says: "A number is an expression that tells how many." The two errors are, first, that a number is not an expression; and, second, that a number does not tell anything. The following definitions have also been given by different writers: "Number is a term signifying one or more units;" "A number is an expression of one or more things of a kind;" " A number is an expression of quantity by a unit, or by its repe- tition, or by its parts;" "Number consists of a repetition of units;" "A number is either a unit or composed of an assem- blage of units;" "A number is a term expressing a particular sameness of repetition." Other definitions, equally incorrect, may be found by leafing over text-books upon the subject A very simple definition, and especially suitable for a primary text-book is, "A number is one or more units." It may bo remarked that authors seem to be adopting the definition of Euclid, with the modification presented above, so that the standard definition in our text-books is becoming, " A number is a unit or a collection of units." To give a perfect definition of Number is exceedingly diffi- cult, if not impossible. Stevinus defines it as "that by which the quantity of anything is expressed," but mathematicians have not adopted it. Euler's definition, " number is nothing else than the ratio of one quantity to another quantity taken as a unit," has been highly commended. "Number is a do fi- nite expression of quantity," has its advocates. "Number is quantity conceived as made up of parts, and answers to the question, How man}'?" has the authority of a very c-nrcful writer. The world, however, still waits for a simple and uc- carate definition, which may be generally adopted. CHAPTER III. CLASSES OF NUMBERS. N" UMBERS have been variously classified with respect to different properties, or by regarding them from different points of view. The fundamental classes to which attention is here called, are Integers, Fractions, and Denominate Num- bers. These three classes are practically and philosophically distinguished, and constitute the basis of three principal divisions of the science of arithmetic. Logically, the distinc- tion is not without exception, for a Fraction may be denomi- nate, and a Denominate Number may be integral; but the division is regarded as philosophical, since they are not only different in character, but require distinct methods of treat- ment, and give rise to distinct rules and processes. The philosophical character and relation of these three classes of numbers, will appear from the following considerations : Integers. The Unit is the basis or beginning of numbers. A number is a synthesis of units; it is the how -many of a collection of units. These units, as they exist in nature, are whole things, undivided ; hence the first numbers of which a knowledge is acquired, are whole numbers, that is, collections of entire or undivided units. Such units, being entire, are called integral units, and the numbers composed of them are Called integral numbers, or Integers. An Integer is, therefore, a collection of integral units, or, as popularly defined, it is a whole number. It is a product of pure synthesis. Fractions. The Unit, as the basis of arithmetic, may be multiplied or divided. A synthesis of units, as we have seen, (76) CLASSES OF NUMBERS. 77 gives rise to Integers ; a division of the unit gives rise to Fractions. Dividing the unit into a number of equal parts, we see that these parts bear a definite relation to the unit divided, and by taking one or more of these parts, we have a Fraction. It is thus seen that the conception of a fraction implies three things: first, a division of the unit; second, a comparison of the part to the unit; and third, a collection of the fractional parts. In other words it is the product of three operations, division, comparison, and collection ; or, like the logical nature of the science of arithmetic itself, a fraction is a triune product, consisting of analysis, comparison, and synthesis. Denominate Numbers. The unit of a simple integral num- ber exists in nature. A Denominate Number is a collection of units not found in nature ; it is a collection of artificial units adopted to measure quantity of magnitude. The philosophical character of a denominate number is indicated in the following statement: Nature, regarded as how many and how much, gives rise to two distinct forms of quantity ; quantity of multitude, and quantity of magnitude. Quantity of multitude is primarily expressed by numbers, since it exists in the form of individuals, or units ; quantity of magnitude does not admit, primarily, of being expressed in numerical form. To estimate quantity of magnitude, we must fix upon some definite part of the quantity considered as a unit of measure, by which we can give it a numerical form of expression. A Denominate Number may, therefore, be defined as a numerical expression of quantity of magnitude. Or, since the unit is a measure by which the quantity is estimated, we may define it to be a number whose unit is a measure. Again, since tho unit is not natural but artificial, we may de- fine it to be a number whose unit is artificial. Either of these definitions suffices to distinguish it from the other two classrj- of numbers. It differs from them in respect of the nature of the quantity to which it refers, and also in its origin and com- position. In the simple integral numbers, the units, as found 78 THE PHILOSOPHY OP ARITHMETIC. in nature, are collected ; in the denominate number, the unit is assumed, the quantity compared with the unit, and the result expressed numerically. The same kind of quantity may be measured by different units, bearing a definite relation to each other, which gives rise to a scale of units. Taking our scales as they now exist, we have a series of units definitely related to each other, forming a Compound Number, which does not appear in the other classes of numbers. This, how- ever, is rather incidental than essential, as it partially vanishes when we apply the decimal scale to quantity of magnitude, as in the metric system of weights and measures. It is thus seen that there are three distinct classes of num- bers; and, since they require different methods of treatment, they will be considered independently. The remainder of this chapter will be devoted to the discussion of some of the pecu- liarities of integral numbers. Classes of Integers. Simple Integral Numbers, being learned before Fractions and Denominate Numbers, are the first class to which the term number was applied; they have consequently appropriated to themselves the almost exclusive use of the word number. Thus, it is the general custom to speak of Numbers, Fractions, and Denominate Numbers, appar- ently forgetful that they are all numbers. This custom being o common, the word Integer being somewhat inconvenient, and some of the properties which belong to integral numbers applying also to the other two classes, I will also use the word number in place of integral number in considering this part of the subject. Numbers are of two general classes, Concrete and Abstract. A Concrete Number is a number in which the kind of unit is named. An Abstract Number is a number in which the kind of unit is not named. A concrete number may also be defined as a number associated with something which it numbers. This is seen in the etymology of the term, con and cresco, a growing together. An abstract number may also be defined CLASSES OF NUMBERS. 79 as a number not associated with anything numbered. This is indicated by the etymology of the term, ab and traho, a drawing from. It is not true, therefore, as has been asserted, that " all numbers are concrete." Number is never concrete, in the popular sense of material. When I think of four apples, the apples are concrete, but the four is purely numerical and in no sense material. It would be much nearer the truth to say that all numbers are abstract; for the number itself is always a pure abstraction. The distinction between an abstract and a concrete number is not a difference in the numbers them- selves, but a distinction founded upon the fact of their being associated or not associated with something numbered. This distinction is clearly seen in the origin of the idea of number. The idea of number is awakened by the contem- plation of material objects. The mind takes the thought of the how-many, abstracts it from the material things with which it was at first associated, lifts it up into the region of the ideal, and conceives it as pure number. Though the idea was pri- marily awakened by the objects of the material world as the occasion, yet so distinct is number from matter, that if all material things were destroyed, we could still have a science of number as complete as that which now exists. There is still another method of conceiving the distinction between concrete and abstract numbers. All numbers are composed of units. The unit gives character and value to the number of which it is the basis. A number is clearly appre- hended only as we have a clear apprehension of the unit: thus, 6 pounds or 6 tons are only clear and definite ideas to us as we have clear and definite ideas of the units, pound and Ion. Hence, also, the nature of numbers depends upon the nature of the units which compose them. Fundamentally, units are of two classes, concrete and abstract. A concrete unit is some object in nature or art, as, an apple, a book ; or some definite quantity agreed upon to measure quantity of magnitude; as, a yard, a pound, etc. An abstract unit is 80 THE PHILOSOPHY OF ARITHMETIC. merely one without any reference to any particular thing. The concrete unit is not a number, it is only one of the things num- bered ; the abstract unit is the number one. A collection of abstract units gives us an Abstract Number; a collection of concrete units gives us what is called a Concrete Number. An Abstract Number is thus merely a number of abstract units ; a Concrete Number is a number of concrete units. The number itself and the things numbered, considered together, constitute what is called the Concrete Number. This is the usual method of conceiving the distinction between an abstract and a concrete number ; but it is not as simple as the one pre- viously presented. From either method of conceiving the difference between these two classes of numbers, it will be seen that the Concrete Number is dual in its nature, consisting of two classes of units. Thus, in the concrete number, four apples, the concrete unit is one apple; while the basis of the number four itself is the abstract unit, one. Both of these classes of units must be clearly apprehended in order to have a clear and adequate idea of any concrete number. CHAPTER IV. NUMERICAL IDEAS OF THE ANCIENTS. AMONG the ancients, much time was spent in discussing the properties of numbers. The science, with them, was mainly speculative, abounding in fanciful analogies. Pythag- oras, the greatest mathematician of his age, was deeply imbued with this passion for the mysterious properties of numbers. He regarded number as of Divine origin, the foun- dation of existence, the model and archetype of things, the essence of the universe. Plato ascribed the invention of numbers to Theuth, as may be seen in the following passage in the Phsedrus: " I have heard, then, that at Naucratis, in Egypt, there was one of the ancient gods of that country, to whom was consecrated the bird which they call Ibis ; but the name of the deity himself was Theuth. He was the first to invent numbers, and arith- metic, and geometry, and astronomy, and moreover draughts and dice, and especially letters." In the Timseus, he presents the conception of the relation of numbers to time, with great beauty of expression. " Hence, God ventured to form a cer- tain movable image of eternity; and thus, while he was disposing the parts of the universe, he, out of that eternity which rests in unity, formed an eternal image on the principle of numbers, and to this we give the appellation of Time." Aristotle, in speaking of the Pythagoreans, says, "They supposed the elements of numbers to be the elements of all entities, and the whole heaven to be an harmony and number.' 1 6 (81 ) 82 THE PHILOSOPHY OF ARITHMETIC. And again he says, "Plato affirmed the existence of numbers independent of sensibles; whereas, the Pythagoreans say that numbers constitute the things themselves, and they do not set down mathematical entities as intermediate between these." The views of Pythagoras are so curious and interesting that they may be stated somewhat in detail. He regarded Numbers as of Divine origin, as above stated, and divided them into various classes, to each of which were assigned distinct proper- ties. Even numbers he regarded as feminine, and allied to the earth ; odd numbers were supposed to be endued with masculine virtues, and partook of the celestial nature. One, or the monad, was held as the most eminently sacred, as the parent of scientific numbers. Two, or the duad, was viewed as the associate of the monad, and the mother of the elements, and the recipient of all things material ; and three, or the triad, was regarded as perfect, being the first of the mas culine numbers, comprehending the beginning, middle, and end, and hence fitted to regulate by its combinations the repetition of prayers and libations. It was the source of love and sym- phony, the fountain of energy and intelligence, the director of music, geometry, and astronomy. As the monad represented the Divinity, or Creative Power, so the duad was the image of matter ; and the triad, resulting from their mutual con- junction, became the emblem of ideal forms. Four, or the tetrad, was the number which Pythagoras affected to venerate the most. It is a square, and contains within itself all the musical proportions, and exhibits by sum- mation (1 + 2+3-f 4) all the digits as far as ten, the root of the universal scale of numeration. It marks the seasons, the elements, and the successive ages of man ; and also represents the cardinal virtues, and the opposite vices. It marked the ancient fourfold division of science into arithmetic, geometry, astronomy, and music, which was termed tetractys, or quater- nion. Hence, Dr. Barrow explains the oath familiar to the NUMERICAL IDEAS OF THE ANCIENTS. 83 disciples of Pythagoras: " I swear by him who communicated the Tetractys." Five, or the pentad, being composed of the first male and female numbers, was styled the number of the world. Repeated in any manner by an odd multiple, it always reappeared ; and it marked the animal senses and the zones of the globe. Six, or the hexad, composed of the sum of its several fac- tors (1 + 2+3), was reckoned perfect and analogical. It wa. likewise valued as indicating the faces of the cube, and as entering into the composition of other important numbers. It was deemed harmonious, kind, and nuptial. The third power of 6, or 216, was conceived to indicate the number of years that constitute the period of metempsychosis. Seven, or the heptad, formed from the junction of the triad and tetrad, has been celebrated in every age. Being unpro- ductive, it was dedicated to the virgin Minerva, though pos- sessed of a masculine character. It marked the series of the lunar phases, the number of the planets, and seemed to modify and pervade all nature. It was called the from of Amalthea, and reckoned the guardian and director of the universe. Eight, or the octad, being the first cube that occurred, was dedicated to Cybele, the mother of the gods, whose image, in the remotest times, was only a cubical block of stone. From its even composition, it was termed Justice, and made to signify the highest or inerratic sphere. Nine, or the ennead, was esteemed as the square of the triad. It denotes the number of the Muses; and, being the last of the series of digits, and terminating the tones of music, it was inscribed to Mars. Sometimes it received the appellation of Horizon, because, like the spreading ocean, it seemed to flow around the other numbers within the decad ; for the same reason, it was also called Terpsichore, enlivening the productive principles in the circle of the dance. Ten, or the decad, from its important office in numeration, was, perhaps, most celebrated. Having completed the cycle, 84 THE PHILOSOPHY OF ARITHMETIC. and begun a new series of numbers, it was aptly called apo- catastasic, or periodic, and therefore dedicated to the double- faced Janus, the god of the year. It had likewise the epithet of Atlas, the unwearied supporter of the world. The cube of the triad,or the number twenty-seven, expressing the time of the moon's periodic revolution, was supposed to signify the power of the lunar circle. The quaternion of celestial numbers, one, three, five, and seven, joined to that of the terrestrial numbers, two, four, six, and eight, compose the number thirty-six, the square of the first perfect number, six, and the symbol of the universe, distinguished by wonderful properties. In pursuit of these mystical relations and analogies, every number became, as it were, possessed of a property; and all numbers possessed some relative analogy with each other to which a name could be given. Numbers also became the sym- bols of intellectual and moral qualities. Thus, perfect numbers compared with those which are deficient or superabundant, are considered as the images of the virtues, regarded as equally remote from excess and defect, aud constituting a mean point between them: thus, true courage is a mean between audacity and cowardice, and liberality between profusion and avarice. In other respects, also, this analogy is remarkable, as perfect numbers, like virtues, are few in number, and generated in a constant order; while superabundant and deficient numbers are like vices, infinite in number, disposable in no regular series, and generated according to no certain and invariable law. The tracing of these analogies, accompanied, as they usually were, with moral illustrations of uncommon elegance and beauty, may be considered as furnishing a pleasing, if not a useful exercise of the understanding; but such analogies were often taken for proofs, and assumed as the bases of the most absurd and inconsistent theories. Thus Pythagoras considered "number as the ruler of forms and ideas, and the cause of NUMERICAL IDEAS OF THE ANCIENTS. 85 gods and dafemons;" and again that "to the most ancient and all-powerful creating Deity, number was the canon, the efficient reason, th6 intellect also, and the most undeviating of the composition and generation of all things." Philolaus declared "that number was the governing and self-begotten bond of the eternal permanency of mundane natures." Another said, "that number was the judicial instrument of the Maker of the universe, and the first paradigm of mundane fabrication." It appears to have been a favorite practice with the Greeks of the latter ages to form words in which the sum of the num- bers expressed by their component letters, should be equal to some remarkable number ; of this kind were the words aSpaaat and afipaaata, the letters in which express numbers, which added together, are equal to 365 and 366, the number of days in the common and bissextile years respectively; and it was also remarked that the word vedas possessed the same property as the first of these words. Words in which the sums of the numbers expressed by the letters were equal, were called m>6fiaTa h6^a; and we have an example in the Greek anthol- ogy, where a poet, wishing to express his dislike to a fellow of the name of Aa^oyopaf, says, that having heard that his name was equivalent in numeral value to Ao<//of, a pestilence, he pro- ceeded to weigh them in a balance, when the latter was found to be the lighter. Observations like these, however trifling, are not without their portion of curiosity ; but the same indulgence cannot be shown to the absurdities of those Pythagorean philosophers, who, among other extraordinary powers which they attributed to numbers, maintained that, of two combatants, the one would conquer, the characters of whose name expressed the larger sum. It was upon this principle that they explained the rela- tive prowess and fate of the heroes in Homer, tta-pa^if, unri, and A,Y"LA'c, the sums of the numbers in whose names are 871, 1225, and 1276 respectively. This very singular superstition continued in force as late tu> 86 THE PHILOSOPHY OF ARITHMETIC. the sixteenth century, and was transferred from the Greek to the Roman numeral letters, I, U or V, X, L, C, D, and M, which correspond to the numbers 1, 5, 10, 50, 100, 500, and 1000; thus the numeral power of the name of Maurice (Mau- ritius) of Saxony, was considered as an index of his success against Charles V. It was the fashion, also, to select or form memorial sentences or verses to commemorate remarkable ilates. Thus the year of the Reformation (1517) was found tc I*} expressed by the numeral letters of this verse of the Tt Deum, Tibi cherubin et seraphin incessabili voce proclamant, in which there is one M, four C's, two L's, two IPs or V's, and seven I's. The Chinese, also, are distinguished for their arithmetical fancies. They regarded even numbers as terrestrial, and par- taking of the feminine principle Yang; while odd numbers were regarded as of celestial extraction, and endued with the masculine principle Y. Even numbers were represented by small black circles; odd numbers by small white ones, vari- ously disposed and connected by straight lines. Thirty, the sum of the five even numbers, 2, 4, 6, 8, and 10, was called the number of the Earth ; twenty-five, the sum of the odd numbers, 1, 3, 5, 7 t 9, and also the square of five, was called the number of Heaven. The nine digits were grouped /*\ O-O O O p ^t in two ways called Lo-chou and \^/ Ho-tou. The former expres- sion signifies the Book of the o \ / o River Lo, or what the Great Yu 1 \/ o saw delineated on the back of J / \ o the mysterious tortoise which \ Q Q rose out of that river. It may be represented as follows : Nine * / *'^ was the head, one the tail, ^ ^ i( ^+ ^ ' O ^^ ^^ three and seven its left and \ s* *\ * right shoulders, four and two NUMERICAL IDEAS OF THE ANCIENTS. 87 its fore feet, eight and six its hind feet. The number five, which represented the heart, being the square root of twenty five, was also the emblem of Heaven. It will be noticed that this group of numbers is the common magic square of nine digits, each row of which amounts to fifteen. The Ho-tou was what the Emperor Fou-hi observed on the body of the horse-dragon which he saw spring out of the river Ho. It consists of the iirst nine numbers arranged in the form of across. The central number was ten, which, it is remarked by the commentators, terminates all the operations on numbers. Other facts equally curious will be found in the literature of other nations, a full collection of which would make an interesting volume. For the facts here presented, and the manner in which they are stated, I am indebted to Leslie. This passion for discovering the mystical properties of num- bers descended from the ancients to the moderns, and numer- ous works have been written for the purpose of explaining them. Petrus Bungus, in 1618, wrote a work on the mysteries of numbers, extending to seven hundred quarto pages. He illustrates all the properties of numbers, whether mathemat- ical, metaphysical, or theological ; and not content with col- lecting all the observations of the Pythagoreans concerning them, he has referred to every passage in the Bible in which numbers are mentioned, incorporating, in a certain sense, the whole system of Christian and Pagan theology. He holds that the number 11, which transgresses the decad, denotes the wicked who transgress the Decalogue, whilst 12, the numbe' of the apostles, is the proper symbol of the good and the just 88 THE PHILOSOPHY OF ARITHMETIC. The number, however, upon which, above all others, he haa dilated with peculiar industry and satisfaction, is 666, the num- ber of the beast in Revelation, the symbol of Antichrist ; and he seems particularly anxious to reduce the name of Martin Luther to a form which may express this formidable number. It may also be remarked that Luther interpreted this number to apply to the duration of Popery, and also that his friend and disciple, Stifel, the most acute and original of the early math- ematicians of Germany, appears to have been seduced by these absurd speculations. The numbers 3 and 7 were the subject of particular specula- tion with the writers of that age ; and every department of nature, science, literature, and art, was ransacked for the pur- pose of discovering ternary and septenary combinations. The excellent old monk, Pacioli, the author of an early printed treatise on arithmetic, has enlarged upon the first of these numbers in a manner which is rather amusing, from the quaint and incongruous mixture of the objects which he has selected for illustration. " There are three principal sins," says he, "avarice, luxury, and pride ; three sorts of satisfaction for sin, fasting, almsgiving, and prayer; three persons offended by sin, God, the sinner himself, and his neighbor; three witnesses in heaven, the Father, the Word, and the Holy Spirit; three degrees of penitence, contrition, confession, and satisfaction, which Dante has represented as the three steps of the ladder that leads to Purgatory, the first marble, the second black and rugged stone, the third red porphyry. There are three Furies in the infernal regions; three Fates, Atropos, Lachesis, and Clotho; three theological virtues, faith, hope, and charity; three enemies of the soul, the world, the flesh, and the devil ; three vows of the Minorite Friars, poverty, obedience and chastity ; three ways of committing sin, with the heart, the mouth, and the act; three principal things in Paradise, glory, riches, and justice; three things which are especially displeas- ing to God, an avaricious rich man, a proud poor man, and a NUMERICAL IDEAS OF THE ANCIENTS. 89 luxurious old man ; three things which are in no esteem, the strength of a porter, the advice of a poor man, and the beauty of a beautiful woman. And all things, in short, are founded in three, that is, in number, in weight, and in meas- ure." In these fanciful speculations, the number seven has received an equal, if not a greater distinction than the number three. In the year 1502, there was printed at Leipsic a work in honor of the number seven, especially composed for the use of the students of the university, which consisted of seven parts, each part consisting of seven divisions. In 1624, William Ingpen, Gent., of London, published a work entitled " The Secrets of Numbers, according to Theological, Arithmetical, Geometrical, and Harmonical Computation. Drawn for the better part, out of those ancients, as well as Neoteriques. Pleasing to read, profitable to understand, opening themselves to the capacities of both learned and unlearned, being no other than a key to lead men to any doctrinal knowledge whatso- ever." Di Borgo seems to have been influenced by the same principle in determining the number of the divisions of arith- metic; for he says: "The ancient philosophers assign nine parts of algorism, but we will reduce them to seven, in reverence of the seven gifts of the Holy Spirit; namely, numeration, addition, subtraction, multiplication, division, progressions, and extraction of roots." Some of these fancies are not entirely extinct at the present day. In England, seven constitutes the term of apprenticeship, the period for academical degrees, and as in our own country, the product of these two magic numbers three and seven con- stitutes the legal age of majority ; and the frequent use of the number seven in the Bible has given it associations which have caused it to be regarded as a sacred number. SECTION II. iRITHMETICAL LANGUAGE I. NUMERATION. II. NOTATION. III. ORIGIN OF SYMBOLS. IV. BASIS OF THE SCALE. V. OTHER SCALES OF NUMERATION VI. A DUODECIMAL SCALE. VII. GREEK ARITHMETIC. VIII. ROMAN ARITHMETIC. IX. PALPABLE ARITHMETIC. CHAPTER I. NUMERATION, OR THE NAMING OF NUMBERS. T)EGINNING at the Unit, we obtain, by a process of syn- -L) thesis, arithmetical objects which we call Numbers. These objects we distinguish by names, and thus obtain the language of arithmetic. This language is both oral and written. The oral language of arithmetic is called Numera- tion ; the written language of arithmetic is called Notation. Numeration treats of the method of naming numbers; Nota- tion treats of the method of writing numbers. As oral language always precedes written language, it is seen that Numeration precedes Notation, and that the practice of arith- meticians in reversing this order is illogical. Numeration is the method of naming numbers. It also includes the reading of numbers when expressed by characters. The oral language of arithmetic is based upon a principle peculiarly simple and beautiful. Instead of giving independ- ent names to the different numbers, which would require more words even to count a million than one could acquire in a life- time, we name a few of the first numbers, and then form groups or collections, name these groups or collections, and then use the first simple names to number the groups. The method is really that of classification, which performs for arithmetic somewhat the same service of simplification that it does in natural science. This ingenious, though simple and natural method of breaking numbers up into classes or groups, seems to have been adopted by all nations. With the civilized world and with most uncivilized tribes, these groups generally con- (93) 94 THE PHILOSOPHY OF ARITHMETIC. sist of ten single things, suggested, undoubtedly, by the practice among primitive races, of reckoning by counting the fingers of the two hands. Method of Naming. The fundamental principle of naming numbers, then, is that of grouping by tens. We regard ten single things as forming a single collection or group; ten of these groups forming a larger group, and so on; ten groups of any one value forming a new group of ten times the value, each group being regarded and used as a single thing. In this way, by giving names to the first nine numbers, and names to the groups, and employing the first nine to number the groups, we are enabled to express the largest numbers in a concise and convenient form. The value of this method of naming may be seen from the consideration that, without it, the memory would be overwhelmed by the multiplicity of disconnected words, and we should require a lifetime to learn the names of numbers, even up to a few hundred thousands. It also enables us to form a clear and distinct conception of large numbers, whose composition we discover in the words by which they are expressed, or in the symbols by which they are represented. It serves, also, as a basis for the ingenious and useful method of writing numbers, without which arithmetic would be almost useless to us. Naming numbers in this way, a single thing is called one ; one and one more are two ; two and one more are three ; and in the same manner we obtain four, five, six, seven, eight, and nine, and then adding one more and collecting them into a group, we have ten. Now, regarding the collection ten as a single thing, and proceeding according to the principle stated, we have one and ten, two and ten, three and ten, etc., up to ten and ten, which we call two tens. Continuing in the same manner, we have two tens and one, two tens and two, etc , up to three tens, and so on until we obtain ten of these groups of tens. These ten groups of tens we now bind together by a thread of thought, forming a new group which we call a hun- NUMERATION, OB THE NAMING OP NUMBERS. 95 dred. Proceeding from the hundred in the same way, we unite ten of these into a larger group which we name thousand, etc. This is the actual method by which numbers were originally named ; but unfortunately, perhaps, for the learner and for sci- ence, some of these names have been so much modified and abbreviated by the changes incident to use, that, with several of the smaller numbers at least, the principle has been so far disguised as not to be generally perceived. If, however, the ordinary language of arithmetic be carefully examined, it will be seen that the principle has been preserved, even if disguised so as not always to be immediately apparent. Instead of one and ten we have substituted the word eleven, derived from an expression formerly supposed to mean one left after ten, but now believed to be a contraction of the Saxon endlefen, or Gothic ainlif (ain, one, and lif, ten); and instead of two and ten, we use the expression twelve, formerly supposed to have been derived from an expression meaning two left after ten, but now regarded as arising from the Saxon twelif, or Gothic tvalif (tva, two, and lif, ten.) With the numbers following twelve, the principle can be more readily seen, though by constant use the original expres- sions have been abbreviated and simplified. The stream of speech, "running day by day," has worn away a part of the primary form, and left us the words as they now exist. Thus, supposing the original expression to be three and ten, (orig- inally the Anglo-Saxon thri and tyri) if we drop the conjunction and, we shall have three-ten ; changing the ten to teen we have three-teen; then changing the three to thir, and omitting the hyphen, we have the present form thirteen. In a similar manner the expression four and ten becomes fourteen; five and ten, fifteen ; six and ten, sixteen, etc. By the same prin- ciples of abbreviation and euphonic change, we might have obtained twenty, thirty, etc. Supposing the original form to be two tens, or twain tens (in the Saxon twentig, from twegen, 96 THE PHILOSOPHY OF ARITHMETIC. two, and tig, ten), then changing the twain to twen, and the tens to ty, we shall have the common form, twenty. In three tens, changing the three to thir and the tens to ty, we have thirty. In the same way we obtain forty, fifty, sixty, etc., and from these by omitting the and in the expression two tens and one, two tens and two, etc., we have twenty-one, twenty- two, thirty-three, forty-seven, etc. To illustrate the law of the formation of these names, we have used the present English forms rather than those in which the transformations actually occurred. It will be remembered that these names were derived from the Anglo-Saxon, and the changes which we have illustrated took place in that language before the names were adopted in the English tongue. The word thirteen was actually derived from the Anglo-Saxon threo-tyne, which was composed of thri, three, and tyne, ten; fourteen from feowertyne, composed offeower, four, and tyne, ten, etc. We get the word twenty from the Anglo-Saxon twentig, which is composed of the Anglo-Saxon twegtn, two, and tig, ten ; thirty from thritig, which is composed of thri, three, and tig, ten, etc. The law of the composition of these original words is no doubt the same as that illustrated by the use of the English words given above. In a similar manner we name the numbers from one hundred to the next group, consisting of ten hundreds, to which we assign a new name, calling it thousand. After reaching the thousand, a change occurs in the method of grouping. Previ- ously, ten of the old groups made one of the next higher group, but after the third group, or thousands, it requires a thousand of an old group to form a new group, which receives a new name. A thousand thousands forms the next group after thousands, which we call million from the Latin mille, a thousand. In the same manner, one thousand millions gives a new group which we call billion, one thousand billions a new group which we call trillion, etc. This change in the law by which a new group is formed from NUMERATION, OR THE NAMING OF NUMBERS. 97 an old one, is not an accident; it is intentional. It is due to science, rather than to chance. The method of counting ten in a group was commenced in an age anterior to science, and proceeded no further than hundreds and thousands, since the wants of the people did not require larger numbers; but when arithmetic began to be cultivated as a science, it was seen to be a matter of convenience to increase the size of the groups receiving a new name, and then the law became changed. The reason that the law of naming numbers does not appear in the names of the smaller numbers, is, that they became changed from the original form on account of their frequent use. The same fact appears in grammar in the irregularity of the verbs expressing ordinary actions, as run, go, eat, drink, etc., which became thus irregular in the formation of their tenses from the constant and careless use of the common peo- ple, before the language was fixed by the rules of science or the art of printing. Utility. The utility of the method of naming numbers by collecting them into groups or bunches, is generally imperfectly appreciated. The method which naturally would be first sug- gested to the mind, is to give each number an independent name, just as we distinguish rivers, cities, states, etc. This would, of course, require a vocabulary of names as extensive as the series of natural numbers, a vocabulary which, even for the ordinary purposes of life, could be learned only by years of labor. By the method of groups, the vocabulary is so sim- ple that it can be acquired and employed with the greatest ease. It may be remarked, that this method of grouping, though suggested by the accidental circumstance of counting the fingers, is in accordance with that universal operation of the mind by which it binds up its knowledge into bunches or packages. It is, in fact, based upon the principle of generaliza- tion and classification. Origin of Names. The origin or primary moaning of the names applied to the first ten numbers, is not known. It hn? 98 THE PHILOSOPHY OF ARITHMETIC. been supposed that the names of the simple numbers were originally derived from some concrete objects, and there are a few facts which seem to indicate the correctness of this suppo- sition. Thus, the Persian name for five is pendje, while pentcha means the expanded hand, and the corresponding terms in the Sanskrit are said to have a similar meaning. The term linia, which with slight modifications is used for five through- out the Indian Archipelago, means hand in the language of the Otaheite and other islands. Among the Jaloffs, an African tribe, the word for five, juorum, likewise signifies hand. Among the Greenlanders the term for twenty is innuk, or man; that is, after completing the counting of fingers and toes, they say innuk or man; and there are also examples of the identity of the term for man and twenty among some of the tribes of South America. Among the Indians of Bogota, New Grenada, the term quicha, meaning a foot, is used to number the second decade, while twenty is named gueta, which signifies a house. Nearly all the South American tribes use the word for hand to express five, and in many cases the word for man is used to express twenty. A tribe in Paraguay denote four by an expression which means the fingers of the Emu, a bird common in Par- aguay, possessing four claws on each foot, three before, and one turned back ; and their word for five is the name of a beautiful skin with five different colors. The same number is, however, more commonly expressed by hanam begem, the fingers of one hand; ten is expressed by the fingers of both hands; and for twenty they say hanam rihegem cat gracha- haka nnomichera hegem, the fingers of both hands and feet. Among the Caribbeans, the fingers are termed the children of the hand, and the toes children of the feet ; and the phrase for ten, chou oucabo raim, means all the children of the hands Humboldt has given from the researches of Duquesnej the etymological signification of some of the numerals of the Indians of New Grenada. Thus, ata, one, signifies water; NUMERATION, OR THE NAMING OF NUMBERS. 99 bosa, two, an enclosure; mica, three, changeable; muyhica^ four, a cloud threatening a tempest ; hisa, five, repose ; ta, six, harvest; cahupqua, seven, deaf; suhuzza, eight, a tail; and ubchica, ten, resplendent moon. No meaning has been discovered for aca, the numeral for nine. It would seem im- possible, amidst such various meanings, to discover any prin- ciple which may seem to have pointed out the use of these terms as numerals. In the Mexican numeral symbols there is an intelligible con- nection between the sign and the thing signified, though the association seems to be entirely arbitrary. Thus, the symbol for one is a frog; for two, a nose with extended nostrils, part of the lunar disk, figured as a face; for three, two eyes open, another part of the lunar disk; for four, two eyes closed; for Jive, two figures united, the nuptials of the sun and moon, conjunction ; for six, a stake with a cord, alluding to the sacrifice of Guesa tied to a pillar ; for seven, two ears ; for eight, no meaning assigned ; for nine, two frogs coupled ; for ten, an ear ; for twenty, a frog extended. The following theory, advanced by Prof. Goldstiicker, in a paper read before the Philological Society in 1870, in which he gives good linguistic evidence in support of the origin of the Sanskrit numerals, and consequently of our own, is at least plausible, and will be interesting: One, he says, is "he," the third personal pronoun; two, "diversity;" three, "that which goes beyond ;" Jour, "and three," that is, "one and three;" five, "coming after;" six, "four," that is, "and four," or "two and four;" seven, "following;" eight, "two fours," or "twice four ;" nine, " that which comes after" (ch. nava, new); ten, "two and eight." Thus, only one and two have distinct orig- inal meanings. After giving these, our ancestors' powers needed a rest; then they made three, and added to it one for four; then took another rest, repeated the notion of three in five, and the notion of four in six ; then rested once more, and again repeated the notion of three and five iu seven ; took 100 THE PHILOSOPHY OF ARITHMETIC. another rest, and got a new idea of two fours for eight ; but for nine repeated for the fourth time the " coming after" notion of three, five, and seven ; while for ten they repeated for the third time the addition-notion of four and six. The Professor insists strongly on this seeming poverty and helplessness of the early Indo-European mind. He does not put forward the above meanings of the numerals as new, though he believes that his history of most of the forms of their names is so. The anomalous form of the Sanskrit shash, six the hardest of them first set him at work on the numerals, and the Zend form kshvas led him to the true explanation of this, and thence to that of the other numerals. In closing this chapter, we remark that the names of the periods above duodecillions have not been fully settled by usage. Prof. Henkle, who has examined the subject with considerable care, finds a law which he maintains should hold in the forma- tion of the names of the higher periods. The terms quintillions, sextillions, and nonillions are formed, not from the cardinals, quinque, sex, and novem, but from the ordinals, quintus, sextus, and nonus. From this he infers that analogy plainly demands that the names beyond duodecillions should be formed from the Latin ordinal numerals. For the names thus formed, see appendix. CHAPTER II. NOTATION, OR THB WRITING OF NUMBERS. A RITHMETICAL language is the expression of arithmet- -lA- ical ideas. These ideas may be expressed in sound to the ear, or in visible form to the eye ; arithmetical language is, therefore, both oral and written. The oral language is called Numeration; the written language, Notation. Numeration is the method of naming numbers; Notation is the method of writing numbers. From this consideration it would seem that the written language of arithmetic must bear an intimate rela- tion to the oral language, which we find to be the case. The general method of writing numbers, now adopted by all civil- ized nations, is the Hindoo, usually called the Arabic method. This method is based upon, and arises naturally out of, the method of naming numbers by groups. The fundamental principle of the Arabic system is the ingenious and refined idea of place value. Recognizing the method of naming numbers by groups, it assumes to represent these groups by the simple device of place. It fixes upon a few characters to represent a few of the first numbers, uml then employs these same characters to number the groups, the group numbered being indicated by the place of the char- acter. This leads to the distinction of the intrinsic and local value of the numerical characters. Each character has a defi- nite value when it stands alone, and a relative value when used in connection with other characters. The number of the arithmetical characters is determined by the number of units in the group. The grouping being by ( 101 ) 102 THE PHILOSOPHY OF ARITHMETIC. tens, the number of characters needed is only nine, one less than the number of units in the group. These characters are called digits, from the Latin digitus, a finger, the name com- memorating the ancient custom of reckoning by counting the fingers. In the combination of these characters to express numbers, it will often be required to indicate the absence of some group ; hence arises the necessity of a character which expresses no value, a character which denotes merely the absence of value. This character is known as naught, or zero. We thus have the following ten characters : 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, with which we are able to express all possible numbers. Utility. The Arabic system, based upon the refined idea of place value, is one of the happiest results of human intelli- gence, and deserves our highest admiration. Remarkable as is its simplicity, it constitutes, regarded in its philosophical char- acter or its practical value, one of the greatest achievements of the human mind. In the hands of a skillful analyst, it be- comes a most powerful instrument in wresting from nature her hidden truths and occult laws. Without it, many of the arts would never have been dreamed of, and astronomy would have been still in its cradle. With it, man becomes armed with prophetic power, predicting eclipses, pointing out new planets which the eye of the telescope had not seen, assigning orbits to the erratic wanderers of space, and even estimating the ages that have passed since the universe thrilled with the sublime utterance, "Let there be light !" Familiarity with it from child- hood detracts from our appreciation of its philosophical beauty and its great practical importance. Deprived of it for a short time, and compelled to work with the inconvenient methods of other systems, we should be able to form a truer idea of the advantages which this invention has conferred on mankind. Relation to Numeration. Though the methods of notation and numeration are intimately related, there is also an essential distinction between them. Though similar, they are by no means identical in principle. Their similarity is seen in the fact that NOTATION, OR THE WRITING OP NUMBERS. 103 the method of notation could not be applied without the method of numbering by groups ; their distinction is seen in the fact that we could have the present method of numeration without the Arabic system of notation. The notation seems to be an immediate outgrowth from the numeration, yet not a necessary one ; for many nations who had the same method of naming numbers, employed other methods of writing them. Their true relation also appears in considering their common relation to the decimal scale. The decimal principle belongs both to our method of naming and of writing numbers. This coincidence is not accidental, but essential to the harmony of oral and written expression. The necessity of this would be very apparent if we should attempt to change the base of the scale of notation without changing the base of the method of naming numbers. With our present base we say one and ten, two and ten, etc., or at least their equivalents; and our written expressions are read in the same manner. Should we adopt any other scale of notation, retaining our present base in naming numbers, the reading of numbers in this new scale would be so awkward and inconvenient as to be almost impossible. Hence it follows, that for a scale of nota- tion to be advantageously employed, the methods of naming and writing numbers should possess the same basis. Thus, if the scale of notation be quinary, instead of naming numbers five, six, seven, etc., we should say Jive, one and five, two and five, etc.; if the scale were senary, we should say six, one and six, two and six, etc. Relation to the Base. It will also be seen that the princi- ple of the methods of naming and writing numbers is entirely distinct from the number used as the base. The intimate asso- ciation of the Arabic system with the base, has sometimes led to the idea that the base is a part of the system itself. This error should be carefully avoided. The Arabic method assumes that we name numbers by groups, and that each group contains ten; but it is in principle entirely independent 104 THE PHILOSOPHY OF ARITHMETIC. of the number constituting a group. The number in the group determines the base of the scale, and consequently the number of characters to be used, but does not afi'ect the principle of the method, which is simply that of place value. Should we change the base of numbering, it would change the names of the numbers after twelve, and the base of the Arabic scale; but it would in no wise affect the principle of cither the method of numeration or of notation. Number of Characters. The number of characters in the Arabic system of notation depends upon the number of units in the groups of numeration. Thus, we must have as many simple characters as will express the different numbers from one until we reach within a unit of the group. We shall have no character for the group, since, according to the device of place value, it is to be indicated by changing the place of the symbol which represents one, it being one of the first group. The number of significant characters must, therefore, be always one less than the number denoting the base of the system. In the decimal scale the number of digits is nine ; in an octary scale it would be seven ; in a quinary scale, four, etc. Origin. The origin of this system of notation is now uni- versally accredited to the Hindoos. When, by whom, and how it was invented, we do not know. It is not improbable that it began with the representation of the spoken words by marks, or abstract characters. They may at first have given inde- pendent characters to the numbers as far as represented. It then probably occurred to them that, since they gave independ- ent names to a few numbers and then numbered by groups, they could simplify their system of notation by making it cor- respond to their system of numeration. Then first dawned upon the mind the idea of a few characters to represent the first simple numbers, and the use of these same characters to number the groups. They now stood on the threshold of one of the greatest discoveries of all time. Here arose the ques- NUTATION, OR THE WRITING OF NUMBERS. 106 tion How are these groups to be distinguished? How shall we determine when a character denotes a number of units or tens, or hundreds, etc.? How many methods occurred to them before the method of place, who can tell ? This might ha've been done by slightly varying the character, by attaching some mark to it, by annexing the initial of the group, etc.; either of which would .have been comparatively complicated and incon- vonicnt. At last, to the mind of some great thinker, occurred the simple idea of place value, and the problem was solved. "Who was the man ?" is a question answered only by its own echo, for his name sleeps in the silence of the past. Were it known, mankind would feel like rearing a monument to his memory, as high and enduring as the Pyramids of Egypt; but now it can only raise its altar to the Unknown Genius. Origin of Character*. The origin of the characters, like that of the system, is shrouded in mystery ; but little light upon the subject comes down the historic path. Many of the early writers gave some ingenious speculations concerning their origin. Gatterer imagined that he had discovered iu Egyptian manuscripts written in the enchoriac character, tliiit the digits were denoted by nine letters; and Wachter supposed them to have a natural origin in the different com- binations of the fingers: thus, unity is expressed by the outstretched finger ; two by two fingers, which may have been represented by two marks that, by long use, passed into the present form, and so on for all the other symbols. In the absence of facts, three theories have been presented, which are at least interesting on account of their ingenuity, and are certainly somewhat plausible. One of these theories is that they are formed by the combination of straight lines, as tin 1 primary representation of numbers; another is that they are formed by the combination and modification of angles; and still ;i noilicr and more recent theory is that they are the initial letters of the Hindoo numerals. These three theories may be distinguished as the theories of lines, angles, and initial letters. 106 THE PHILOSOPHY OF ARITHMETIC. The first theory is based on the primary use of straight lines to represent numbers. By this method, one straight line, |, would represent one,- two straight lines which may have been connected thus, L, two; three lines, thus, ^, or with a connecting curve, thus, -jj, three ; four lines arranged thus, Q or thus, 4 four; five lines arranged thus, Jjj, five; six lines arranged thus, |j, six; seven lines, thus, 9, seven ; eight lines thus, g, or thus Y, eight; nine lines, thus, ^, nine. The zero is supposed to have been originally a circle, suggested from counting around the fingers and thumbs held in a circular position. The second theory is based upon the use of angles to repre- sent numbers. The ancient mathematicians were noted for their astronomical observations and calculations, and being thus familiar with the use of angles, it is not unreasonable to suppose that they would employ the angle in their representa- tion of numbers. Thus, they might very naturally have used one angle, / |, for one ; two angles, ", for two ; three angles ], for three; four angles, A for four; five angles, g, for Jive, six angles, , for six; seven angles, 9, for seven; eight angles, 0, for eight ; nine angles, ^, for nine. These char- acters being frequently made, would eventually assume the rounded form which they now possess. By this theory, the character for zero is easily and naturally accounted for Jf angles were used to represent numbers, nothing would be rep- resented by a character having no angles, which is the closed curve. The latest and most plausible theory for the origin of Arabic characters is, that they were originally the initial letters of the Sanskrit numerals. This theory is presented by Prin- seps, a profound Sanskrit scholar, and is indorsed by Max Miiller. Such a use of initial letters was entirely feasible in the Sanskrit language, as each numeral began with a dif- ferent letter. The plausibility of the theory further appears <rom the fact that it follows the general law of representing NOTATION, OR THE WRITING OF NUMBERS. 107 numbers by letters, as in the Roman, Greek, and Hebrew systems. This theory does not account for the origin of the zero, the luost important character of them all, in fact, the key to the system of modern arithmetic. No other system of notation except the sexagesimal system, had it. Max Miiller says : " It would be highly important to find out at what time the naught first occurs in Indian inscriptions. That inscription would deserve to be preserved among the most valuable monu- ments of antiquity, for from it would date in reality the beginning of true mathematical science impossible without the naught nay, the beginning of all the exact sciences to which we owe the invention of telescopes, steam engines, and electric telegraphs." Dr. Peacock supposes that it was derived from the Greek o, introduced by Ptolemy to denote the vacant places in the sexagesimal arithmetic; the Hindoos, he says, having used a dot for this purpose. It seems to have been difficult at first to comprehend the pre- cise force of the cipher, which, insignificant in itself, serves only to determine the rank and value of the other figures. When they were first introduced into Europe, it was deemed necessary to prefix to any work in which they were used, a short treatise on their nature and application. These notices are often met with attached to old vellum almanacs, or inserted in the blank leaves of missals, and frequently intermixed with famous prophe- cies, most direful prodigies, and infallible remedies for scalds and burns. A sort of mystery, which has imprinted its trace on our language, seemed to hang over the practice of using the cipher; and we still speak of deciphering and writing in cipher, in allusion to some dark or concealed art. Indeed, in the early history of arithmetic in Europe, either on account of its association with the infidel Mohammedans from whom it ivas derived, or of the popular prejudice against learning which prevailed at that time, the system was regarded as belonging to black art and the devil ; and it was, no doubt, this popular prejudice that delayed its general introduction into Christian Europe. CHAPTER III. ORIGIN OF ARITHMETICAL SYMBOLS. rpHE symbols of arithmetic may be divided into three general JL classes : Symbols of Number, Symbols of Operation, and Symbols of Relation. What is the origin of these symbols ; who invented them, or first employed them? This question, a very interesting one, I shall endeavor to answer in the present chapter. I. SYMBOLS OF NUMBER. The Symbols of Number em- ployed by different nations, are the Arabic figures and the letters of the alphabet. Nearly all civilized nations seem to have made use of the letters of the alphabet to represent num- bers. The Greeks divided their letters into several classes, to represent the different groups of the arithmetical scUe. The Roman system employed the seven letters, I, V, X, L, C, D, and M, to represent numbers. The Arabs at first used the Greek method, and afterward exchanged it for that of the Hin- doos. There are three theories given for the origin of the Arabic symbols of notation, known respectively as the theory of lines, of angles, and of initial letters. These three theories are explained in the chapter on Notation. It may also be remarked that some of the Arabian authors who treat of astrological signs, allege that the Indian or Arabic numerals were derived from the quartering of the circle, and Leslie says that the resemblance of these natural marks to the derivative ones appears very striking. The Roman symbols are supposed to have originated in the use of simple straight lines or strokes, (108) ORIGIN OF ARITHMETICAL SYMBOLS. 109 variously combined, for which were subsequently substituted the letters of the alphabet. This theory is explained at length in the chapter on Roman Notation. SYMBOLS OF OPERATION. The Symbols of Operation are the signs of addition, subtraction, multiplication, division, involution, evolution, and aggregation. The origin of most of these symbols has been definitely determined. The Symbols of Addition and Subtraction were first introduced as symbols of operation by Michael Stifel, a German mathematic- ian, in a work entitled Arithmetica Integra, published at Nurem- burg in 1544. These signs had appeared previously in a work of Johann Widmann called the Mercantile Arithmetic, published at Leipzig in 1489. They are, however, not used by him as symbols of operation, but merely as marks of excess or deficiency. The next oldest book extant in which these signs are found is that of Christopher Rudolff, published in 1524, though he does not use them as symbols of operation. Stifel was a pupil of Rudolff, and it is supposed that he obtained the symbols from him, for, as he himself admits, he took a large part of his work from that of Rudolff. Stifel introduces the symbols as if he had originated them or their new use, for he says, " thus, we place this sign," etc., and " we say that the addition is thus completed," etc. To Stifel, therefore, belongs the credit of first using these symbols as signs of operation. Why these particular signs were adopted has been a matter of conjecture. Prof. Rigaud supposed that + was a corruption of P, the initial for plus, and Dr. Davis thought that it was a cor- ruption of et or Sf. Stifel, however, does not call the signs phis and minus, but signum additorum and signum subtractorum, which renders these suppositions improbable. Dr. Ritchie suggested that perhaps + was two marks joined together in addition, and that was taken to indicate subtraction, since it is what is left after one of the marks is removed. De Morgan thought that the minus sign was first used, and that + was derived from it by putting a small cross-bar for distinction. " The sign -f ," he 110 THE PHILOSOPHY OF ARITHMETIC. says, " in the hands of Stifel's printer has the vertical bar much shorter than the other, and when it is introduced into the wood- outs of the engraver, the disproportion is greater still." The Hindoos, from whom our knowledge of algebra was originally derived, used a dot for subtraction, and the absence of the dot for addition, and De Morgan suggests that the Hindoo dot may have been elongated into a bar to signify subtraction, and that an addi- tional line to the symbol of subtraction gave the sign of addition. M. Libri attributes the invention of + and to Leonardo da Vinci, the celebrated Italian artist and philosopher, but it is probable that Da Vinci used the symbol + for the figure 4. The most recent explanation of these signs is that they were originally warehouse marks. In Widmann's arithmetic they occur almost exclusively in practical mercantile questions. Goods were sold in chests which when full were expected to hold a certain established weight. Any excess or deficiency was indicated by -f- or , and these signs may have been marked with chalk on the chests as they came from the warehouses. Usually the weight of the chest, it may be supposed, would be deficient, and this was marked with the sign ; when a cask or chest was above the standard weight the line may have been crossed with a vertical line giving the symbol +. It may be remarked that these symbols were not immediately adopted by mathematicians. In a work on algebra, published in 1619, the signs of addition and subtraction are P and M with strokes drawn through them. The Symbol of Multiplication (X), St. Andrew's cross, was in- troduced by William Oughtred, an eminent English mathematic- ian and divine, born at Eton in 1573. The work in which this symbol first appeared was entitled Clavis Mathematica, " Key of Mathematics," and published in 1631. Oughtred was a fine thinker, and was honored by the title " prince of mathematicians." What led to this particular form for the symbol is unknown. Two other signs for multiplication were proposed, the (.) by Descartes and the curve (^) by Leibnitz, but though having the authority of great names they failed of adoption. ORIGIN OF ARITHMETICAL SYMBOLS. Ill The Symbol of Division (-*-) was introduced in 1630 by Dr. John Pell, Professor of Philosophy and Mathematics at Breda. This symbol was used by some old English writers to denote the ratio or relation of quantities. I have also noticed it used thus in some old German mathematical works. The Arabs used a dash, writing one number under the other, in the form of a fraction. Dr. Pell was highly regarded as a mathematician. It was to him that Newton first explained his invention of fluxions. The System of Exponents, to represent the powers of a num- ber, has been generally ascribed to Descartes, 1596-1650, the illustrious metaphysician and inventor of Analytical Geometry. Exponents were, however, employed by De la Roche as early as 1520, but Descartes' extensive use of them led lo their general adoption. The earliest writer on algebra denoted the powers of a number by an abbreviation of the name of the power. Harriot, a mathematician of the 17th century, repeated the quantity to indi- cate the power ; thus, for a* he wrote aaaa. The Radical Sign (p/) was introduced by Stifel, the introducer of -f and . This symbol is a modification of the letter r, the initial of radix, root. The root of a quantity was formerly de- noted by writing the letter r before it, and this letter was grad- ually changed to the form i/. The Vinculum or Bar, placed over quantities to connect them together, thus, 4X3 + 5, was first used by Vieta in 1591, the in- troducer, in algebra, of the system of representing known quanti- ties by symbols. The Parenthesis and Brackets were first used by Albert Girard, a Dutch writer on algebra, in 1629. III. SYMBOLS OF RELATION. Symbols of Relation are the signs of equality, ratio, equal ratios, inequality and deduction. The origin of a few of these has been ascertained. The Symbol of Equality (=) was introduced by Robert Re- corde, an English physician and mathematician of the sixteenth century. It first appeared in 1556 in his work on algebra, called by the odd title The Whetstone of Witte. This sign was also employed by Albert Girard. The French 112 THE PHILOSOPHY OF ARITHMETIC. and German mathematicians used the symbol oo to denote equality, even long after Recorde. This symbol is said to be a modifica- tion of the diphthong , the initial of the Latin phrase <equale est. It is also stated that the symbol = was often used as an abbrevia- tion for cst in mediaeval manuscripts. The Symbol of Ratio (:) is supposed to be a modification of the sign of division. The sign of division was frequently employed by the old English and German mathematicians to indicate the relation of quantities. Who first omitted the dash and employed the present form of the symbol of ratio, I have not been able to ascertain. It occurs in a work by Clairaut, published in 1760. The Symbol of Equal Ratios (: :) may be a modification of the sign of equality (=) or a duplication of the symbol of ratio (:), but this is not certain. It seems to have been introduced by Oughtred, in a work published in 1631, and was brought into common use by "Wallis in 1686. The Symbols of Inequality (> and <) are evidently modifica- tions of the sign of equality. If parallel lines denote equality, oblique lines would naturally be used to denote inequality, the lines converging toward the less quantity. They are said to have been introduced by Harriot in 1651. I have now presented, in a connected and systematic manner, about all that is known concerning the origin of the ordinary arithmetical symbols. All of them belong to the period of mod- ern history and are the products of the revival of learning. One each of the signs of operation was furnished by France, England and the Netherlands, and three by Germany. Of the other sym- bols named, all were introduced by Englishmen, with the excep- tion of the vinculum, which is due to Vieta, a Frenchman, and the parenthesis and brackets, which were invented by the Dutch mathematician Girard. CHAPTER IV. THE BASIS OF THE SCALE OF NUMERATION. rpHE Basis of our scale of numeration and notation is dec JL imal. This basis is not essential, but accidental. Man- kind commenced reckoning by counting the fingers of the left hand, including the thumb, and thus at first probably reckoned by fives. As the art of numbering advanced, they adopted a group, derived from the fingers of both hands, and thus ten became the basis of numbering. The decimal base was con- sequently determined by the number of fingers on each hand. Had there been three fingers and a thumb, the scale would have been octary; had there been five fingers and a thumb, the scale would have been duodecimal, which would have been a great advantage to arithmetic, whatever it might have been to the hand itself. The universal use among civilized nations of the decimal scale of numeration seems to imply some peculiar excellence in it. It appears as if nature had pointed directly to it, on account of some essential fitness of the number ten, as the numerical basis. Indeed, this opinion has been quite general, and the habit acquired from the use of the system has served to confirm the belief. Many persons get the base of numera- tion and the mode of notation so mingled together that they see in the Arabic system nothing save the decimal basis of numeration, and attribute to it all those high qualities which belong to the mode only. It is this which has led some per- sons to regard the decimal basis as the perfection of simplicity and utility. 8 (113) 114 THE PHILOSOPHY OF ARITHMETIC. A little reflection, however, will prove that such an assump- tion is groundless. Although the decimal scale has been adopted by every civilized nation, yet, as has been shown, the selection was accidental, and the base entirely arbitrary. The selection occurred before attention was given to a general sys- tem, in short, without reflection, and its supposed perfection is a mere delusion. Any other number might have been taken as the root of the numerical scale ; and, were a new basis to be selected by mathematicians familiar with the properties of numbers, there are several considerations that would lead them to adopt some other basis than the decimal. Some of the objections to the decimal basis will be stated, and a few consid- erations presented in favor of some other number as the basis of the language of arithmetic. First, the decimal scale is unnatural. It has been super- ficially urged that the decimal scale is the most natural one that could have been selected. On the contrary, there is no- thing natural about it, except the fingers, and a little reflection would have shown that these are grouped by fours instead of fives. In fact, a group by tens is seldom seen, either in nature or in art. What things exist by tens, associate by tens, or separate into tenths? Nature groups in pairs, in threes, in fours, in fives, and in sixes; but seldom, if ever, in tens. Man doubles and triples and quadruples his units ; he divides them into halves and thirds and quarters ; but where does he estimate by tens or tenths ? It is thus seen that the grouping by tens is an unnatural method, suggested neither by nature nor the practical requirements of art. Second, the decimal scale is unscientific. The confused idea of the relation of the base of the scale to the mode of notation, has led some to suppose that the decimal scale is one of the triumphs of science. The truth is, as has already been shown, that not only was it not established upon scientific principles, but it is really a violation of those principles. The decimal scale originated by chance, by a mere accident. Men THE BASIS OF THE SCALE OF NUMERATION. 115 had ten fingers, including the thumbs, and found it convenient to reckon by counting their fingers ; and thus acquired the habit of counting by tens. Had science, instead of chance, presided at its birth, we should have a basis that would have given a new beauty and a greater simplicity to our already admirable system of arithmetical language. Third, the decimal scale is also inconvenient. It has been held not only that the decimal basis is scientific, but that it is the most convenient one that could have been selected. It needs but little reflection to see the incorrectness of this assumption. One essential for the basis of a scale is the property of its being divisible into a number of simple parts, so that it may be a multiple of several of the smaller numbers. The number ten will admit of only two such divisions, the half and the fifth. The third, fourth, and sixth are not exact parts of the denary base, in consequence of which it is incon- venient to express these fractions in the scale. Were the basis twelve instead of ten, we could obtain the half, third, fourth, and sixth, and these fractions could be expressed by the scale in a single place ; whereas the fourth now requires two places (.25), and the third and sixth cannot be expressed exactly in a decimal scale, except as a circulate. Essentials of a Base. It will be interesting to notice some of the essentials of a base, and to observe what number com- plies most fully with these requirements. The first essential of a good base is that it will admit of being divided into the simple fractional parts ; the second is that the number be neither too large nor too small. The advantage of the capability of being divided into simple fractional parts is that such fractions may be readily expressed in the terms of the scale as we now express decimal fractions. In the decimal scale only one-half and one-fifth can be expressed in one place of decimals, since they are the only exact parts of ten. With a scale whose basis is a multiple of two, three, four and six, each correspondiug frac- tion could be expressed in terms of the scale in a single plucv. 116 THE PHILOSOPHY OF ARITHMETIC. In respect to the size of the base, if the number is too small, it will require too many names and places to express large numbers. If the number is very large, it will group together too many units to be apprehended and easily used in numerical operations. Other Scales. There are several other bases which have been recommended as preferable to the decimal ; the most important of which are the Binary, the Octary, and the Duodecimal. The Binary scale was proposed and strongly advocated by Leibnitz. He maintained that it was the most natural method of counting, and that it presented great practical and scientific advantages. He even constructed an arithmetic upon this basis, called Binary Arithmetic. The obvious objection to this base is, that it would require too many names and too many places in writing large numbers. The Octary system has also been strongly advocated. A very able article in an American journal says that the binary base is the only proper base for gradation, and the octary is the true commercial base of numeration and nota- tion. It is probable, however, taking all things into consideration, that the duodecimal scale would be the most suitable. The number twelve is neither too large nor too small for conveni- ence. Its susceptibility of division into halves, thirds, fourths, and sixths, is an especial recommendation to it. So great are these advantages, that, if the base were to be changed, the duodecimal base would, without doubt, be selected. The advantage of the duodecimal scale is especially apparent in the expression of fractions in a form similar to our decimal fractions. In the decimal scale, and \ are the only simple fractions that can be expressed by the scale in a single place ; i cannot be expressed at all as a simple decimal ; requires two places, and , like , gives an interminate decimal. With a duodecimal scale we could express ^, , , and in a single place; while and would require only two places. Thus, in the duodecimal scale, we should have =.6; =.4; THE BASIS OF THE SCALE OF NUMERATION. 117 i=-3; =.2; =.16, and =.14. This is a very great sim- plification; and since all combinations of 2 and 3 could be readily expressed, and since these constitute such a large pro- portion of numbers, it is evident that the simplification of the subject, by means of a duodecimal scale, would be very con- siderable. I will arrange the expressions of these fractions in the deci- mal and duodecimal scales, side by side, that the advantage of the latter may be more clearly seen. DECIMAL SCALE. DUODECIMAL SCALE. i=.5 = .166+ =.333+ |=. 142857 {=25 =.125 =.2 =.111+ 1=6 =.2 =.4 =.16 =.2497 It will be seen that in the decimal scale all the simple frac- tions used in practice, except , give circulates or require two or three figures to express them; while in the duodecimal scale all the fractions ordinarily used in business transactions are expressed in a single place, and even and require only two places. The fractions and cannot be exactly expressed in the scale, but these fractions are seldom used in business. It will be interesting to notice that and both give perfect rep- etends in the duodecimal scale, and that they possess the same properties as perfect repetends in the decimal scale. There seems to have been a natural tendency towards a duo- decimal scale. Thus, a large number of things are reckoned by the dozen, and this scale is even extended to the gross and the great-gross; that is, to the second and the third powers of the base. Again, in our naming of numbers, the terms eleven and twelve seem to postpone the forming of a group until we reach a dozen. A similar fact appears in extending the multi- plication table to include twelve times, since, with the deci- mal scale, it could conveniently stop with nine or ten times The division of the year into twelve months, the circle into twelve signs, the foot into twelve inches, the pound into twelve 118 THE PHILOSOPHY OF ARITHMETIC. ounces, etc., are each a further indication of the same ten- dency. Change of Base. The objections to the decimal base have led scientific men to advocate a change in our scale of numer- ation and notation. Such a change would, without doubt, be a great advantage, both to science and to art ; yet the practi- cal difficulties attending such a change are so great that it seems to be almost impossible. A change in the base would require a complete change in the oral language of arithmetic. The decimal scale is so interwoven with the speech of nations, that such a change could be effected only after years of labor. For a while, it would be necessary to have two methods of arithmetic taught and in use, as in Europe at the time of the transition from the Roman to the Arabic system of nota- tion. The learned would soon adopt the new method, but the common people would cling with such tenacity to the old, that even a century might intervene before the new method would become generally established. Will this change ever be made ? is a question which is sometimes asked. I do not know ; but I am strongly in favor of it, and believe it possible. The diffusion of popular educa- tion will prepare the way for it, by removing the difficulties of its adoption. These difficulties, though great, are not insur- mountable. Changes of notation have taken place in several different nations, and some nations have changed two or three times. The Greeks changed theirs, first for the alphabetic, and afterwards, with the rest of the civilized world, for the Arabic system. The Arabs themselves first adopted the Greek, and afterwards changed it for the Hindoo method. The peo- ple of Europe changed from the Roman to the Arabic system even as late as the fourteenth century, though it took one or two centuries to effect the transition. What was done thus early in the history of science, could, with the increased intel- ligence of our people, be much more readily accomplished at the present day. A writer in one of our American periodicals THE BASIS OF THE SCALE OF NUMERATION. 119 says: "The probability is that it will be done. The question is one of time rather than of fact, and there is plenty of time. The diffusion of education will ultimately cause it to be de- manded." It is a curious fact, and one worthy of remembrance, that Charles XII. of Sweden, a short time before his death, while lying in the trenches before the Norwegian fortress of Freder- ickshall, seriously deliberated on a scheme of introducing the duodecimal system of numeration into his dominions. CHAPTER V. OTHER SCALES OF NUMERATION. AS we have seen, any number might have been taken as the basis of the scale of numeration, the number ten, the basis of our present scale, being selected from the circumstance of there being ten fingers on the two hands. Some other scales have actually existed, and it will be interesting to notice, in various languages, traces of an earlier and simpler mode of reckoning. In order to a clearer notion of the subject, it may be premised that a scale whose basis is two is called Binary ; three, Ternary ; four, Quaternary ; five, Quinary ; six, Senary; seven, Septenary ; eight, Octary ; nine, Nonary ; ten, Denary; twelve, Duodenary, etc. The earliest method of numeration was that of combining units in pairs. It is still familiar among sportsmen, who reckon by braces or couples. Some feeble traces of the Binary system are found in the early monuments of China. Fouhi, the founder and first emperor of that vast monarchy, is vener- ated in the East as a promoter of geometry and the inventor of a science, the knowledge of which has been lost. The em- blem of this occult science appears to consist of eight separate clusters of three parallel lines or trigrams, drawn one above the other after the Chinese manner of writing, and represented either as entire or broken in the middle. These varied tri- grams were called Koua or suspended symbols, from the cus- tom of hanging them up in the public places. In the formation of such clusters, we may perceive the application of the binary (120) OTHER SCALES OF NUMERATION. 121 scale as far as three ranks, or the number eight. The entire lines are supposed to signify one, two, or four, according to their order, while the broken lines are valueless, and serve merely to indicate the rank of the others. If this be true, it furnishes an example of a species of arithmetic with the device of place, possessing an antiquity of more than 3000 years. The Binary scale, though never fully adopted by any nation as a method of counting, has been recommended by one of the most celebrated modern philosophers, Leibnitz, as presenting many advantages, from its enabling us to perform all the operations in symbolic arithmetic by mere addition and sub- traction. Such a system would, of course, require but two symbols, unity and zero, by means of which all numbers could be expressed. Thus, two would be expressed by 10, three by 11, four by 100, five by 101, six by 110, seven by 111, eight by 1000, etc. This system was studiously circulated by its author by means of scientific journals and his extensive cor- respondence; and was communicated by him to Bouvet, a Jesuit missionary at Pekin, at that time engaged in the study of Chinese ambiguities, and who imagined that he had discov- ered in it a key to the explanation of the Cova, or lineations previously referred to. This system was also recommended by the theological idea associated with it, of which it was claimed to be the represent- ative. As unity was considered the symbol of Deity, the forma- tion of all numbers out of zero and unity was considered, in that age of metaphysical dreaming, as an apt image of the crea- tion of the world, by God, from chaos. It was with reference to this view of the binary arithmetic, that a medal was struck, bear- ing on its obverse, as an inscription, the Pythagorean distich, Numero Deu impari gaudet, and on its reverse the appropri- ate verse descriptive of the system which it celebrated, Omnibus ex nihilo ducendis sufficit Unum. The good Jesuit, who seemed to have caught the spirit of Chinese belief, regarded 122 THE PHILOSOPHY 01 ARITHMETIC. the Cova, which were supposed to conceal great mysteries, as the symbols of binary arithmetic, as a most mysterious testi- mony to the unity of the Deity, and as containing within them the germ of all the sciences. To count by threes was another step, and this has been pre- served by sportsmen under the term leash, meaning the strings by which three dogs, and no more, can be held at once in the hand. The numbering by fours has had a more extensive application ; it was evidently suggested by the custom of tak- ing, in the rapid counting of objects, a pair in each hand, and thus reckoning by fours. English fishermen, who generally count in this way, call every double pair (of herring, for instance), a throw or cast ; and the term warp, which origin- ally meant to throw, is employed to denote four, in various articles of trade. It is alleged that the Guaranis and Sulos, two of the lowest races of savages inhabiting the forests of South America, count only by fours ; at least they express the number five by four and one, six by four and two, seven by four and three, etc. It has been inferred, also, from a passage in Aristotle, that a certain tribe of Thracians were accustomed to use the quaternary scale of numeration. The Quinary system, which reckons by fives, or pentads, has its foundation in the practice of counting the fingers of one hand. It appears, from the statements of travellers, to have been adopted by various savage nations. Thus, certain tribes of South America were found to reckon by fives, which they called hands. In counting six, seven, and eight, they added to the word hand the names one, two, three, etc. Mungo Park found that the same system was practiced by the Yolofs and Foulahs of Africa, who designate ten by two hands, fifteen by three hands, etc. The quinary system seems also to have been formerly used in Persia; the word pende, which denotes five, having the same derivation as pentcha, which signifies a hand. It is even partially used in England among whole- sale traders. In reckoning articles delivered at the warehouse, OTHER SCALES OF NUMERATION. 123 the person who takes charge of the tale, having traced a long horizontal line, continues to draw, alternately above and below it, a warp, or four vertical strokes, each set of which he crosses by an oblique score, and calls out tally as often as the number Jive is completed. This custom is a very general one in assemblies where votes are counted, and in similar circum- stances elsewhere. The Senary method, so far as we can learn, was never used by any tribe or nation; at least never arose spontaneously. It is said to have been adopted at one time in China by the order of a capricious tyrant, who, having conceived an astro- logical fancy for the number six, commanded its several combi- nations to be used in all concerns of business or learning throughout his vast empire. The Septenary scale has not, so far as we can learn, been used anywhere. The number seven has been regarded as a kind of magic number, but nothing in nature suggested the method of counting by sevens. The division of the year into periods consisting of seven days each, a custom among nearly all nations, has given the number seven a wide distinction, and its frequent use in the Bible has caused it to be regarded as a sacred number, the basis of a celestial system of reckoning. The Octary scale, also, though it would possess many advant- ages, and has been recommended by scientific writers, has never made its appearance in any language. A Nonary scale has also never been used, and would be the most inconvenient of the smaller scales except the septenary. The Denary scale is the system which has prevailed among all civilized nations, and has been incorporated into the very structure of their language. This universal method manifests the existence of some common principle of numbering, which was the practice, so familiar in the earlier periods of society, of reckoning by counting the fingers on both hands. The origin of the terms used in the more polished ancient languages is not easily traced, but in the roughness of savage dialects 124 THE PHILOSOPHY OF ARITHMETIC. these names vary less from the primitive words. The Muysca Indians were accustomed to reckon as far as ten, which they called quihicha or a foot, referring, no doubt, to the number of toes on their bare feet; and beyond this number they used terms equivalent to foot one, foot two, etc., for eleven, twelve, etc. Another South American tribe called ten, tunca, and merely repeated the word to signify a hundred, or a thousand, thus : tunca-tunca, tunca-tunca-tunca. The Peruvian language was actually richer in the names of numerals than the Greek or Latin. The Romans went no higher than mille, a thou- sand, and the Greeks than fmpia, or ten thousand. But the Peruvians had the expressions, hue, one ; chunca, ten ; pachac, a hundred; huaranca, a thousand; and hunu, a million. It appears from an early document, that the Indian tribes of New England used the Denary scale, and had distinct words to ex- press the numbers as far as a thousand. The Laplanders join the cardinal to the ordinal numbers ; thus, for eleven they say auft nubbe lokkai, that is, one to the second ten. The origin of the numerals in our own dialect will be found treated at greater length in another place. The mode of reckoning by twelves or dozens, may be sup- posed to have had its origin in the observation of the celestial phenomena, there being twelve months or lunations commonly reckoned in a solar year. The Romans likewise adopted the same number to mark the subdivisions of their unit of measure or of weight. The scale appears also in our subdivisions of weights and measures, as twelve ounces to a pound, twelve inches to a foot ; and is still very generally employed in wholesale business, extending to the second and even to the third term of the progression. Thus, twelve dozen, or 144, make the long hundred of the northern nations, or the gross of traders; and twelve times this again, or 1728, make the double or great gross. The scale of numeration by twenties has its foundation in nature, like the quinary and denary. In a rude state of society, before the discovery of other methods of numeration, OTHER SCALES OF NUMERATION. 125 men might avail themselves for this purpose, not merely of the fingers on the hands, but also of the toes on the naked feet ; and such a practice would naturally lead to the formation of a vicenary scale of numeration. The languages of many tribes indicate this method, and many savage tribes do thus actually reckon. It is said of the inhabitants of the peninsula of Kam- schatka, that "it is very amusing to see them attempt to reckon above ten ; for having reckoned the fingers of both hands, they clasp them together, which signifies ten; then they begin at their toes and count twenty, after which they are quite confused and cry matcha, where shall I take more?" Among the Caribbees who constituted the native population of Barbadoes and other islands of the Caribbean sea, the numeration beyond five was carried on by means of the fingers and toes, and their numer- ical language became generally descriptive of their practical method of counting. The Abipones, an equestrian people of Paraguay, to express five show the fingers of one hand; to express ten, the fingers of both hands; "for twenty, their expression is pleasant, being allowed to show all the fingers of their hands and the toes of their feet." Traces of reckoning by scores or twenties, are found in our own and other European idioms. The expression threescore and ten is familiar. The term score itself, which originally meant a notch or incision made on a tally to signify the suc- cessive completion of such a number, seems to indicate that such a mode of counting was most familiarly used by our ances- tors. The vicenary scale seems to have prevailed very exten- sively among the Scandinavian nations, as is shown by the vestiges of it both among them and the languages partly derived from them. The French language has no term for the numbers in the second series of the denary scale above soix- ante or sixty. Eighty is expressed by quatre-vingts, four twenties, and ninety by quatre-vingts-dix, four twenties and ten. The people of Biscay and Armorica are said still to reckon by the powers of twenty, and, according to Humboldt, the same mode of numeration was employed by the Mexicans. CHAPTER VI. A DUODECIMAL SCALE. AS already explained, any number may be made the basis of a system of numeration and notation. The decimal basis is a mere accident, and in some respects an unfortunate one, both for science and art. The duodecimal basis would have been greatly superior, giving greater simplicity to the science, and facilitating its various applications. In this chapter it will be explained how arithmetic might have been developed upon a duodecimal basis. In order to make the matter clear, I call attention to two or three principles of numeration and notation. First, the bases of numeration and notation should be the same ; that is, if we write numbers in a duodecimal system, we should also name numbers by a duodecimal system. Second, in naming num- bers by any system, we give independent names up to the base, and then reckon by groups, using the simple names to number the groups. Bearing these principles in mind, we are ready to understand Numeration, Notation, and the Fundamental Rules in Duodecimal Arithmetic. NUMERATION. In naming numbers by the duodecimal system, we would first name the simple numbers from one to eleven, and then, adding one more unit, form a group, and name this group twelve. We would then, as in the decimal system, use these first names to number the groups. Naming numbers in this way, we would have the simple names, one, two, three, etc., up to tivelve. Continuing from twelve, we would have one and twelve, two and twelve, three and twelve, etc., up to twelve and twelve, which we would call two twelves. Passing on from this (126) A DUODECIMAL SCALE. 12 / we would have two twelves and one, two twelves and two, etc., to three twelves, and so on until we reach twelve twelves, when we would form a new group containing twelve twelves, and give this new group a new name, as gross, and then employ the first simple names again to number the gross. In this way we would continue grouping by twelves, and giving a new name to each group, as in the decimal scale by tens, as far as is necessary. These names, in the duodecimal system, would naturally become abbreviated by use, as the corresponding names in the decimal system. Thus, as in the decimal system ten was changed to teen, we may suppose twelve to be changed to teel, and omitting the " and" as in the common system, we would count one-teel, two-teel, thir-teel, four-teel, Jif-teel, six-teel, etc., up to eleven-teel. Two-twelves might be changed into two-tel, or twen-tel, corresponding to two-ty or twenty, and we would continue to count twentel-one, twentel-two, etc. Three-twelves might be contracted into three-tel or thirtel, corresponding to three-ty or thirty of the decimal system; four-twelves to fourtel, five twelves to fiftel, etc., up to a gross. Proceeding in the same manner, a collection of twelve gross would need a new name, and thus on to the higher groups of the scale. In this manner, the names of numbers according to a duo- decimal system could be easily applied. Were we actually forming such a system, the simplest method would be to intro- duce only a few new names for the smaller groups, and then take the names of the higher groups of the decimal system, with perhaps a slight modification in their orthography and pronunciation, to name the higher groups of the new scale. Thus, million, billion, etc., could be used to name the new groups without any confusion, as they do not indicate any definite number of units to the mind, but merely so many col- lections of smaller collections. Indeed, even the word thou- sand, with a modification of its orthography, say thousun, might be used to represent a collection of twelve groups, 128 THE PHILOSOPHY OF ARITHMETIC. each containing a gross, without any confusion of ideas, Their etymological formation would not be an objection of any particular force, as no one in using them thinks of their pri- mary signification. These terms are not suggested as the best, but as the simplest in making the transition from the old to the new system. It will also be noticed that our departure in the decimal scale from the principle of the sys- tem, by using the terms eleven and twelve, would facilitate the adoption of a duodecimal system. To illustrate the subject more fully, let us adopt the names suggested, and apply them to the scale. Naming numbers according to the method explained, we would have the names as indicated in the following series : one oneteel twentel-one one gross and one two twoteel twentel-two one gross and two three thirteel twentel-eight two gross and five four fourteel twentel-eleven six gross and seven five flfteel thlrtel-one ten gross and eight six sixteel fortel-two eleven gross and nine seven seventeel flftel-six one thousun eight eighteel sixtel-eight one thousun and five nine nineteel seventel-nine one thousun four gross ten tenteel tentel-ten and seven eleven eleventeel eleventel-eleven two thousun seven gross twelve twentel one gross and fortel-one NOTATION. The writing of numbers by the duodecimal system would be an immediate outgrowth of the method of naming numbers in this system. As in the decimal system of notation, it would be necessary to employ a number of char- acters one less than the number of units in the base, besides the character for nothing. Since the group contains twelve units, the number of significant characters would be eleven two more than in the decimal system. For these characters we should use the nine digits of the decimal system, and then introduce new characters for the numbers ten and eleven. To illustrate, we will represent ten by the character * and eleven by n. These characters, with the zero, would be combined to rep- resent numbers in the duodecimal scale in the same manner as the nine digits represent numbers in the decimal scale. Thus, A DUODECIMAL SCALE. 129 twelve would be represented by 10, signifying one of the groups containing twelve; 11 would represent one and twelve, or oneteel; 12 would represent two and twelve, or twoteel; 13 would represent thirteel; 14, fourleel; 15, fifteel, etc. Con- tinuing thus, 20 would represent two twelves, or twentel; 21, twentel-one; 23, twentel-three, etc. The notation of numbers up to a thousun may be indicated as follows: one, 1 twelve, 10 thirtel, 30 two, 2 oneteel, 11 thirtel-two, 32 three, 3 twoteel, 12 thirtel-five, 35 etc., etc. twentel, 20 thirtel-ten, 3* nine, 9 twentel-one, 21 thirtel-eleven, 3n ten, $ twentel-ten, 2* one gross, 100 eleven, n twentel-eleven, 2n one thousun, 1000 Extending the series as explained above, we should have the following notation table : TABLE. g s * i I & I 3 s > ^ ^ s S s ^ o H P. p j> '- H o 00 00 o E Twelves 00 a p V- O O Twelves Millyuns. *o 00 ao S Twelves Thousuns 00 00 o 1-1 Twelves. *3 85n46H857*365 From the explanation given it is clearly seen that a system of duodecimal arithmetic might be easily developed, and read- ily learned and reduced to practice. Employing the names which I have indicated, or others similar to them, the change from the decimal ' to the duodecimal system would be much less difficult than has usually been supposed. It would be necessary to learn the method of naming and writing num- bers, which we have seen is very simple, and a new addition 9 ISO THE PHILOSOPHY OF ARITHMETIC. and multiplication table, from which we could readily derive the elementary differences and quotients. The rest of the sci- ence would be readily acquired, as all of its methods and principles would remain unchanged. Indeed, so readily could the change be made, that in view of the great advantages of the system, one is almost ready to believe that the time will come when scientific men will turn their attention seriously to the matter and endeavor to effect the change. FUNDAMENTAL OPERATIONS. In order to show how read- ily the transition could be made, I will present the method of operation in the fundamental rules. We would proceed first to form an addition table containing the elementary sums, which, as in the decimal system, we would commit to memory. From this we could readily derive the elementary differences used in subtraction. Such a table is given on page 131. By means of this table we can readily find the sum or dif- ference of numbers expressed in the duodecimal system. To illustrate, required the sum of 487n, 5438, OPERATION 63n7, 4>856. The solution of this would be as 487n follows: Adding the column of units, 6 units 5*38 and 7 units are 11 units, and 8 units are 19 63n7 units, and n units are 28 units, or 2 twelves and 8 units ; writing the units, and carrying 2 to the column of twelves, we have 2 twelves and 5 twelves are 7 twelves, and n twelves are 16 twelves, and 3 twelves are 19 twelves, and 7 twelves are 24 twelves, or 2 gross and 4 twelves; writing the twelves, and carrying 2 to the third column, we have 2 gross and 8 gross are * gross, and 3 gross are 11 gross, and $ gross are In gross, and 8 gross are 27 gross, or 2 thousuns and 7 gross; 2 thousuns and $ thousuns are 10 thousuns, and 6 thousuns are 16 thousuns, and 5 thousuns are In thousuns, and 4 thousuns are 23 thousuns; hence the amount is 23748. To illustrate subtraction let it be required to find the differ- A DUODECIMAL SCALE. 131 ADDITION AND MULTIPLICATION TABLES IN THE DUODECIMAL 8CALB. M to x|x . .. ^ e X X X X X X X X 00 -1 OS I 01 unit X X 00, GO 00 00 xix x!x g II ii I *|i- * Cj f f i i o td ta ic + + CC QO 8 f Tlflti? -I - I - I + '+ II 9 o HH H I I O w 132 THE PHILOSOPHY OF ARITHMETIC. ence between 6428 and 2564. We would solve this as follows- Subtracting 4 units from 8 units we have 4 units remaining; we cannot take 6 twelves OPERATION. from 2 twelves, so we add 10 twelves and have 12 twelves; 6 twelves from 12 twelves leaves 8 twelves; carrying 1 to 5 we have 6 gross; 3*84 we cannot take 6 gross from 4 gross ; adding 10 as before we have 6 gross from 14 gross leaves 4> gross; adding 1 thousun to 2 thousuns, we have 3 thousuns from 6 thousuns leaves 3 thousuns; hence the remainder is 3*84. In order to multiply and divide, we first form a multiplica- tion table similar to that now used in the decimal system, and commit it to memory. This table need not extend beyond " twelve times," as in our present system there is no need of extending beyond "ten times." From this table of elementary products, we can readily derive the table of elementary quo- tients as we do in the decimal system. Such a table will be found on page 131. It will be interesting to notice several peculiarities of this *,able, similar to those of the decimal system. As the column of "five times" ends alternately in 5 and 0, making it so easily learned by children, so the column of "six times" in the duodecimal table will end alternately in 6 and 0. In our present table the sum of the two terms of each product in the column of "nine times" equals nine, so in the duodecimal table, the sum of the two terms of each product in the column of "eleven times" equals eleven. We also notice that each product in the column of " twelve times" ends in 0, as does each product in the column of "ten times" of our present table. By means of the multiplication table we can readily find the product or quotient of numbers expressed in the duodeci- mal scale. To illustrate multiplication, let it be required to find the product of 54$8 by 3n7. We would solve this as follows: Using the first term of the multiplier, 7 times 8 are A DUODECIMAL SCALE. 188 48, 7 times 4> are 5$, and 4 are 62, 7 OPERATION. times 4 are 24 and 6 are 2*, 7 times 5 54*8 3n7 are 2n and 2 are 31, making the first partial product 3U28; multiplying by n we have n times 8 are 74, n times $ 14280 are 92 and 7 are 99, n times 4 are 38 and 9 are 45, n times 5 are 47 and 4 are 4n ; 3 times 8 are 20, 3 times $ are 26 and 2 are 28, 3 times 4 aw 10 and 2 are 12, 3 times 5 are 13 and 1 are 14. Adding up the partial products, we have as the complete product, 1953768. To illustrate division, let it be required to find the quotient of 1953768 divided by 3n7. We would OPERATION. solve this as follows : We find that the 3n7)1953768(54*8 divisor is contained in the first four 179n terms of the dividend 5 times, and mul- 1747 tiplying 3n7 by 5 we have 179n ; sub- 13*4 tracting this from the dividend we 3636 have a remainder, 174; bringing down 337$ the next figure of the dividend and 2788 proceeding as before, we have for the quotient 54*8. The method of finding the square or cube root of a number expressed in the duodecimal scale is similar to that used in the decimal scale, as may be shown by an OPERATION. example. Thus, find the square root of ir53'01(347 115301. The greatest square in n is 9 ; subtracting and bringing down a period, and dividing by 2 times 3 or 6, we find the *"O J ftQ t 7V-)nnl second term of the root to be 4; complet- 8n01 ing the divisor and multiplying 64 by 4, we have 214; subtracting and bringing down, we have 3n01, and dividing by 2 times 34, or 68, we have 7 for the last figure of the root; completing the divisor and multiplying it by 7, we have 3n01, which leaves no remainder. The above tables and calculations seem awkward to one 134 THE PHILOSOPHY OF ARITHMETIC. who is familiar with the decimal system ; but it should be remembered that a beginner would learn the addition and mul- tiplication tables and the calculations based on them, just as readily as he now learns them in the decimal system. The practical value of such a system, in addition to what has already been said, may be seen in the calculation of interest, the rules for which would be greatly simplified on account of the relation of the number of months in a year (12) to the base, and also of the relation of the rate to the same, which would be some S% or 9% ; that is, 8 or 9 per gross. I hope to be able in a few years to publish a small work in which the whole science of arithmetic shall be developed on the duodeci- mal basis. CHAPTER VII. GREEK ARITHMETIC. C\ REEK Arithmetic, like that of all other nations of anti- vT quity, began in the representation of numbers by strokes or straight lines. This system, in the progress of thought and civilization, was finally discarded, and the letters of the alpha- bet taken as the symbols of numbers. After adopting the letters of their alphabet, the Greeks seem to have had no less than three distinct methods of notation. They used the letters in their natural order, to signify the smaller ordinal numbers. In this way the books of Homer's Iliad and Odyssey are usually marked. They employed also the first letters of the words for numerals as abbreviated symbols, mak- ing use of an ingenious device for augmenting the powers of these symbols; thus, a letter enclosed by a line on each side and another drawn over the top, as Fl, was made to signify five thousand times its original value. A more complete method consisted in the distribution of the twenty-four letters of their alphabet into three classes, corre- sponding to units, tens, and hundreds, adding another character to each class to complete the symbols for all of the nine digits. This latter method was the one in common use, and that which was made the basis of their arithmetic. The units from one to nine inclusive, were denoted by the letters a, /3, y, 6, e, r, C, 7, ; the tens by t, K, X, //, v, f , o, JT, h I and the hundreds by P , a, T, v t $, x, V*) w > ) Thousands were represented by the first series with the iota, or dash subscribed, thus: .#,? <? etc. With these characters they could readily express any number under (135) 136 THE PHILOSOPHY OF ARITHMETIC. 10,000, or a myriad. Thus, 991 was expressed by 2) '/a; 1382, by fap; 6420, by p; 4001, by |. It will be noticed that neither the order nor the number of characters was considered in expressing numbers. The value of the expression was the same in whatever order the letters were placed ; though as regularity tended towards simplicity, they generally wrote the characters according to value, from left to right. Myriads, or ten thousands, were denoted by the letter M, a letter representing the number of myriads indicated being written above it. Thus, denoted 10,000; M, 20,000; M, 30,000, etc. Thus, also, j denoted 370,000 ; 1P 43720000 ; and in general, the letter M placed beneath any number had the same effect as our annexing four ciphers. This is the notation employed by Eutocius in his commenta- ries on Archimedes, but it is evidently inconvenient in calcula- tion. Diophantus and Pappus expressed the myriad more simply by the two letters M v placed after the number, and afterwards by merely writing a point after it. This enabled them to express 100,000,000, which was the greatest extent of the ordinary Greek arithmetic. This system had been extended by Archimedes and Apollo- nius, for the purpose of astronomical and other scientific calculations. Archimedes, in order to express the number of grains of sand that might be contained in a sphere that had for its diameter the distance of the fixed stars from the earth, found it necessary to represent a number which, with our nota tion, would require sixty-four places of figures; and in order to do this, he assumed the square myriad, or 100,000,000, as a new unit, and the numbers formed with these new units he called numbers of the second order ; and thus he was enabled to express any number which in our notation requires sixteen figures. Assuming again 100,000, OOO 2 as a new unit, he could represent any number that requires in our scale twenty-four GREEK ARITHMETIC. 137 figures, and so on ; so that by means of his numbers of the eighth order, he could express the number in question, which requires sixty-four figures in our scale. By this system all numbers were separated into periods or orders of eight figures. This was afterwards considerably improved by Apollonius, who, instead of periods of eight places, which were called by Archimedes octates, reduced num- bers to periods of four places ; the first of which, on the left, were units, the second period myriads, the third double myri- ads or numbers of the second order, and so on indefinitely. In this manner Apollonius was able to write any number that can be expressed by our system of numeration ; as for example, if he had wished to represent the circumference of a circle whose diameter was a myriad of the ninth order, he would have written it thus: y.nvie. Bci-e. y<pir6. j^A/J. yuft?. $XP/~ %uA(3. {2)v. (fond- 3.1415 9265 3589 7932 3846 2643 3832 7950 2824 The learned astronomer Ptolemy modified this system in its descending range by applying it to the sexagesimal subdivisions of the lines inscribed in a circle. He likewise advanced an important step, by employing a small or accentuated o to supply the place of any number wanting in the order of progression. The Greek method of expressing fractions was also peculiar. An accent set on the right of a number, made of that number the denominator of a fraction whose numerator was a unit, thus, /=$, <i'=J, f<J'=3r, pa'=Tir> e ^ c - When the numerator is not unity, the denominator is placed as we set our exponents. Thus, t'e^ represented 15 64 , or ^|, and (, l>Ka represented 7 121 , or Y^y. The fraction \ had a particular character, as C, <, C', or K. The notation of the Greeks was not adapted to the descending scale, and consequently they had no decimals. The notation of the Greeks, though much inferior to that of the present day, was formed upon a regular and scientific basis, and could be employed with considerable convenience as an instrument of calculation. We will present two or three 138 THE PHILOSOPHY OF ARITHMETIC. examples taken from Barlow's Theory of Numbers, from which some of the previous facts are gathered. Addition. The following example in addition is from Eutocius, Theorem 4, of the Measure of the Circle. 847 3921 60 8400 Tff ft 908 2321 The method, it will be seen, is similar to compound addition, but is simpler on account of the constant ratio of ten between any character and the succeeding one. Subtraction. The following example in subtraction is from Eutocius, Theorem 3, on the Measure of the Circle. O.yxte 93636 /8.V 9 23409 CT~^? 70221 The method is simple, proceeding from right to left as in our subtraction, which seems so obviously advantageous and simple that one can hardly conceive why the Greeks should ever proceed in the contrary way, although there are many instances which make it evident that they did, both in addition and subtraction, work from left to right. Multiplication. In multiplication they most commonly pro- ceeded in their operations from left to right, as we do in mul- tiplication in algebra, and their successive products were placed without much apparent order; but as each of their characters retained always its own proper value, in whatever order they stood, the only inconvenience of this was, that it rendered the addition of them a little more troublesome. The following example is from Eutocius. As it is dim- pvy 153 ' l/'ft'S cult to remember the value I L _ of all the Greek characters, a '"^ pv ^//'l/'/g/q//^ we will indicate the opera- '* Q Z rf l rf 5 f $ tion by writing 1, 2, 3, ^ yve 2 3 ///4// go etc., for the series of units ' ' GREEK ARITHMETIC. 139 1', 2', 3', etc., for the series of tens ; I", 2", etc., for the hundreds, etc., and denote the myriads by writing m as an exponent. Division. The division of the Greeks was still more intri- cate than their multiplication, for which reason it seems they generally preferred the sexagesimal division, and no example is left at length by any of those writers except in the latter form ; but these are sufficient to throw some light on the pro- cess they followed in the division of common numbers, and Delambre has accordingly supposed the following example : ay 332 W 3'"3"2'9 (1" / 8"2'3 P.B 182 s 1"'8"2'3 150 145 8 3 4 2 9 4 3 1 6 9 4 2 6 9 ' 5 5 4 4 6 G 9 9 This example will be found, on a slight inspection, to resem- ble our compound division, or that sort of division that we must necessarily employ, if we were to divide feet, inches and parts by similar denominations, which, together with the number of different characters that they made use of, must have rendered this rule extremely laborious ; and that for the extraction of the square root was, of course, equally difficult, though the principle was the same as ours, except in the difference of the notation. It appears, however, that they fre- quently, instead of making use of the rule, found the root by successive trials, and then squared it in order to prove the truth of their assumption. This beautiful system was vastly superior in simplicity and practical utility to that transmitted to and retained by the Romans, and by them bequeathed to the nations of modern Europe. It was, at least when it had reached its highest development through the genius of Archimedes and Apollo 140 THE PHILOSOPHY OF ARITHMETIC. nius, quite well fitted for an instrument of calculation; and though somewhat cumbrous in its structure, was capable of performing operations of very considerable difficulty and mag- nitude. It will be seen, however, that though much more refined and pliant than that of the Romans, the notation of the Greeks is very much inferior to the common or Hindoo method ; and one cannot help wondering that so ingenious and philosophical a people failed to conceive the simple idea of place value, and construct a system of notation upon it. This seems all the more astonishing when we remember that Archimedes invented a system of octates, or system of eights, which was subse- quently improved by Apollonius, by making the periods con- sist of only four places, and dividing all numbers into orders of myriads. In this form, as Barlow remarks, it seems most astonishing that he did not perceive the advantage of making the periods to consist of a less number of characters; for, hav- ing given a local character to his periods of four, it was only necessary to have done the same for the single digits, in order to have arrived at the system in present use. And this is the more singular, as the use of the cipher was not unknown to the Greeks, being always employed in their sexagesimal operations where it was necessary ; and consequently the step between this improved form of their notation and that of the present system was extremely small, although the advantages of the latter when compared with the former are incalculably great. It seems to have been the lot of the metaphysical mind of the Hindoos to make this "brilliant invention of the decimal scale," one of the greatest improvements in the whole circle of the sciences, and to which we are indebted for all the remarkable advances in modern analysis. CHAPTER YIII. ROMAN ARITHMETIC. 1'HE arithmetic of the Romans was quite inferior to that of the Greeks, a necessary consequence of the inferiority of the method of notation adopted. The method of notation, though usually ascribed to the Romans, was probably invented by the Greeks, and communicated by them to the Romans, who in turn transmitted it to their successors in modern Europe. It no doubt originated in the use of simple strokes, variously combined, to represent numbers. Subsequently it was found convenient to represent numbers by the letters of the alphabet, and the numerical strokes were finally displaced by such alpha- betic characters as most nearly resembled them. The origin of the Roman characters is not certainly known; but the theory, as given by Leslie, and by many regarded as correct, is interesting and plausible. It is certain that the first numerical characters consisted simply of strokes or straight lines. This was the method primarily used by nearly every nation of antiquity, and was the beginning of a philosophical and universal system alike intelligible to all nations. Such characters are still preserved in the Roman notation with very little change, and were probably adopted before the importation of the alphabet itself, by the Grecian colonies that settled Italy and founded the Latin commonwealth. Assuming, then, a perpendicular line | to signify one, two such lines 1 1 to signify two, three lines 1 1 1 to signify three, and so on up to ten, and we have the first series of the numerical scale. They might then (141) 142 THE PHILOSOPHY OF ARITHMETIC. be supposed to throw a dash across the last stroke or unit, to mark the completion of the series; and thus, a cross, X, would come to signify ten. The continued repetition of this mark would denote twenty, thirty, etc., until they reached a hundred, or ten tens, which completes the second series, and might be denoted by adding another dash to the mark for ten, or by merely connecting three strokes, thus Q. The repetition of this symbol would, in like manner, indicate the successive hundreds, the tenth of which would be marked by the addition of another stroke, so that four combined strokes, M> would express a thousand. Such were probably the symbols originally employed in the Roman notation ; in process of time it would be perceived that the inconvenience in writing, arising from so many repetitions of the same character, might be avoided by adopting symbols for the intermediate numbers; and it was seen that these might be furnished by the division of the symbols already in use. Thus, having parted in the middle the two strokes, X, either the under half, /\, or the upper half, V, was employed to signify Jive, or the half of ten. Next, for fifty, the half of a hundred, the symbol Q was divided into two equal parts, (~~ and |_, either of which represented fifty. Again, the symbol for thousand having come to assume a rounded shape, thus ft, or thus CD, the half of this, either CI, or ID, was taken to represent the half of one thousand or five hundred. The symbol Q, to represent a hundred, would, in process of time, being frequently made, have its corners rounded and attain the form C- Lastly, noticing that these characters closely resemble some of the letters of the alphabet, it was agreed to employ those letters as the symbols of the numbers mentioned. The notation of numbers by combined strokes, was evi- dently founded in nature, and may be regarded as the begin- ning of a philosophical language of arithmetic. That this was the foundation of the Roman system is confirmed from the analogous practice of other nations. It is quite clear that the ROMAN ARITHMETIC. 143 Egyptians and Chinese must have followed the same method. The inscriptions on the ancient obelisks present a few numerals which are easily distinguished. The substitution of capital letters for the combined strokes which they chanced most to resemble, though it gave uniformity to the system of notation, prevented any farther improvements of the system. The only simplification which the Romans appear to have introduced, was to diminish the repetition of letters by reckoning in some cases backwards, as in IY, which was originally represented by four strokes, and IX, which was probably at first written vim. Their method of representing large numbers was a little diiferent from that now used, as may be seen by the following examples : DorD MorCD 133 CC133 1333 CCC1333 500 1000 5000 10,000 50,000 100,000. In illustration, it is interesting to notice that Cicero in his fifth oration against Verres expresses 3600 by CI3 CI3 CI3 13C. The Romans often contracted or modified the forms of their numerals, especially in carving inscriptions upon stones, in which case the abbreviated letters were called lapidary char- acters. The marks for any number could also be augmented in power one thousand times, either by enclosing them with two hooks or C's, or by drawing a line over them. Thus, CXO, or X denoted 10,000, and CLVIM given by Pliny, means 156,000,000. Sometimes a letter was placed over another to indicate their product ; thus, ^ would express 500,000. The multiplier was also sometimes written like an exponent, thus IIP was used to express three hundred. In expressing very large numbers, points were sometimes interposed: thus, Pliny writes XVI. XX.DCCCXXIX for 1,620,829. It may be remarked that if this practice had become more general it would probably have effected a material improvement of the system. 144 THE PHILOSOPHY OF ARITHMETIC. In the latter ages of the Roman Empire, the small letters of the alphabet seem to have been used in imitation of the numeral system of the Greeks. The letters a, b, c, d, e, f, g, h, and i represented the nine digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 ; the next series k, 1, m, n, o, p, q, r, and s expressed 10, 20, 30, 40, 50, 60, 70, 80, and 90 ; and the remaining letters t, u, x, y, and z denoted 100, 200, 300, 400, and 500. To represent the rest of the hundreds it was necessary to employ capitals or other characters, and 600, 700, 800, and 900 were repre- sented by I, V, hi and hu. But this mode of notation never obtained any degree of currency, being mostly confined to those foreign adventurers from Greece, Egypt or Chaldea, who, pretending to skill in judicial astrology, were enabled to prey on the credulity of the wealthy Romans. In modern Europe the Roman numerals were supplied by Saxon characters. Thus, in the accounts of the Scottish Ex- chequer for the year 1331, the sum of 6896 5s. 5d. stated as paid to the King of England is thus marked: o c xx vj. viij. iiij. xvj. Ij. v. s. v. d. The Roman system, as now used, employs seven characters, of which I represents one, V five, X ten, L fifty, C one hun- dred, D five hundred, M one thousand. To express other numbers these characters are combined according to the fol- lowing principles : 1. Every time a letter is repeated its value is repeated. 2. When a letter is placed after one of greater value, the sum of their values is the number expressed. 3. When a letter is placed before one of a greater value, the difference of their values is the number expressed. 4. When a letter stands between two letters of a greater value, it is combined with the one following it. 5. A letter is placed before one of its own order only, or the unit of the next higher order. 6. A dash over a letter increases its value a thousand fold. HUMAN ARITHMETIC. 1-ifi In accordance with the fifth principle it would be incorrect to write VC for ninety-Jive, or 1C for ninety-nine. It is also to be noticed that the letter V is never used before a letter of greater value, since the only case in which it could be thus used according to the fifth principle is before X, giving VX for five, which is more concisely expressed by Y itself. In expressing numbers by the Roman method we always write the different orders of units successively, beginning with the higher orders. Thus, in expressing four hundred and ninety-nine, we would not write ID, though this, by principle second, would be the difference of one and five hundred, but we first write CO CO for four hundred, then XC for ninety, and then IX for nine, giving CCCCXCIX. It may be interesting to notice, however, that though the Roman method was not employed in numerical calculations, it might have been so employed by slightly modifying the usual mode of notation. Thus, by not using the third principle, but writing IIII for IV, and YIIII for IX, or by using some mark to show that the letters written according to that prin- ciple are taken together, as XXIV, we can perform the four fundamental operations without much inconvenience. To illus- trate, we give a problem in multiplication, with its explanation. Explanation. VIII multiplied by VII equals LVI, X multiplied by VII equals LXX, L multiplied XXXVI by VII equals CCCL ; III multi- plied by X equals XXX, X multi- plied by X equals C, L multiplied DCLXXX by X equals DCL ; multiplying by DCLXXX X a second and third time, and MM?) XVI taking the sum of the four partial products, we have MMDXVI, or two thousand five hundred and sixteen. This result may be obtained by multiplying by VII and XXX; or by II, V, X, and XX, etc. The multipli- cand also may be variously separated in the multiplication. 10 146 THE PHILOSOPHY OF ARITHMETIC. It is clear, however, that this operation would be very com- plicated with large numbers, so much so, indeed, as to be unfitted for general use, and it is believed that it was not used in performing numerical calculations. These calculations were performed by means of counters, or other palpable em- blems. The instrument generally used was called the Abacus. Leslie says that "the system of characters among the Romans was so complex and unmanageable as to reduce them to the necessity in all cases of employing the Abacus." The Abacus appears to have continued in use among the people of Europe until quite a recent period. The counters or pebbles were, from a corruption of the word algorithm, called in England augrim, or awgrym, stones. Thus, in Chaucer's description of the chamber of Clerk Nicholas, he says : " His almageste and bokes grete and smale, His astrelabre, longing for his art, His augrim stones layen faire apart On shelves couched at his beddes head." Indeed, the modern method of arithmetic was not known in England until about the middle of the sixteenth century ; and the common people, imitating the clerks of former times, were still accustomed to reckon by the help of the awgrym stones. Thus, in Shakespeare's comedy of the Winter's Tale, written at the beginning of the seventeenth century, a clown, staggered at a very simple multiplication, exclaims that he must try it with counters. CLO. Let me see ; Every 'leven wether tods ; every tod yields pound and odd shilling; fifteen hundred shorn, What comes the wool to? . . . I cannot do't without counters. The Roman method is now chiefly used to denote the vol- umes, chapters, sections and lessons of books, the pages of pre- faces and introductions, to express dates, to mark the hours on clock and watch faces, and in other places for the sake of prom- inence and distinction. CHAPTER IX. PALPABLE ARITHMETIC. earliest methods of representing numbers in arithmetical JL calculation were by means of counters and other palpable emblems. The objects most generally used among all primitive nations were little stones or pebbles, from which we derive our word calculation. Beginning with pebbles or some such sim- ple objects, as they advanced in civilization these were found to be insufficient for their purposes, and they invented instruments to represent numbers, by means of which they were enabled to calculate with great rapidity and correctness. The Japan- ese and Chinese at the present day, with their arithmetical instruments, can add, subtract, multiply and divide as rapidly and correctly as we can with the Arabic system of notation. So extensively was this method used by the early nations before the method of calculating by figures was adopted, that Leslie, in his treatise on arithmetic, gives it a distinct and detailed explanation under the head of Palpable Arithmetic. The sub- ject is so full of interest, both for its own ingenuity and its relation to our present system, that I think it proper to devote a chapter to it, and finding a clearer statement of it in Leslie and Peacock than I could hope to give myself, I have tran- scribed their description, sometimes word for word. The early Egyptians performed their computations mainly by the help of pebbles, and so did the early Greeks and Romans. In the schools of ancient Greece, the boys acquired the elements of knowledge by working on the ABAX, asmooth (H7) 148 THE PHILOSOPHY OF ARITHMETIC. board with narrow rim, so named evidently from the combina- tion of the first three letters of their alphabet, and resembling the tablet on which children were formerly accustomed to begin to learn the art of reading. Pupils were taught to calculate by forming progressive rows of counters, which consisted of round bits of bone or ivory, or even silver coins, according to the wealth or fancy of the individual. The same board, strewed with fine green sand, a color soft and agreeable to the eye, served equally for teaching the rudiments of writing and the principles of geometry. The ancient writers make frequent allusions to these calculat- ing boards. Solon, the great Athenian statesman, used to compare the passive ministers of kings to the counters or pebbles of arithmeticians which, according to the place they hold, are sometimes most important, and sometimes utterly insignificant. The Grecian orators, in speaking of balanced accounts, picture the settlements by saying that the pebbles were cleared away and none left. It thus appears that the ancients, in keeping their accounts, did not arrange the debits and credits separately, but set down pebbles for the former, and took up pebbles for the latter. As soon as the board became cleared, the opposite claims were exactly balanced. It may be observed that the common phrase to clear one's scores or accounts, meaning to settle or adjust them, still preserved in the popular language of Europe, was suggested by the same prac- tice of reckoning with counters, which prevailed, indeed, until a comparatively late period. The Romans borrowed their Abacus from the Greeks, and seem never to have aspired higher in the pursuit of numerical science. To each pebble or counter required for the board they gave the name of calculus, meaning a small white stone, and applied the verb calculare to express the operation of com- bining or separating such pebbles or counters. The use of the Abacus, called also the Mensa Pythagorica, formed an essential part of the education of every noble youth. A small PALPABLE ARITHMETIC. 149 box or coffer, called a Loculus, having compartments for hold- ing the calculi, or counters, was considered as a necessary appendage. Instead of carrying a slate and satchel to school, the Roman boy was accustomed to trudge to school loaded with those ruder implements, his arithmetical board and his box of counters. In the progress of luxury and refinement, dice made of ivory, called tali, were used instead of pebbles, and small silver coins came to supply the place of counters. Under the Em- perors, every patrician living in a spacious mansion and indulging in all the pomp and splendor of Eastern princes, generally entertained, for various functions, a numerous train of foreign slaves or freedmen in his palace. Of these, the librarius, or miniculator, was employed in teaching the children their letters, the notarius registered expenses, the rationarius adjusted and settled accounts, and the tabularius or calculator, working with his counters and board, performed what computations might be required. To facilitate the working by counters, the construction of the Abacus was afterwards improved. Instead of the perpendic- ular lines, or bars, the board had its surface divided by sets of parallel grooves, by stretched wires, or even by successive rows of holes. It was easy to move small counters in the grooves, to slide perforated beads along the wires, or to stick large knobs or round-headed nails in the different holes. To diminish the number of marks required, every column was surmounted by a shorter one, wherein each counter had the same value as five of the ordinary kind. The Abacus, instead of wood, was often, for the sake of convenience and durabil- ity, made of metal, frequently brass, and sometimes silver. Two varieties of this instrument seem to have been used by the Romans. Both of them are delineated from antique monuments the first kind by Ursinus, and the second by Marcus Velserus. In the former, the numbers are represented by flattish perforated beads, ranged on parallel wires; and in 150 THE PHILOSOPHY OP ARITHMETIC. the latter, they are signified by small round counters, moving in parallel grooves. These instruments contain each seven capital divisions, expressing in regular order units, tens, hun- dreds, thousands, ten thousands, hundred thousands, and millions, and as many shorter divisions, of five times the rela- tive value of the larger ones. With four beads on each of the long grooves or wires, and one on each corresponding short one, it is evident that any number could be expressed up to ten millions. The Roman Abacus also contained grooves to mark ounces, half-ounces, quarter-ounces, and thirds of an ounce. The Romans likewise applied the same word Abacus to an article of furniture resembling in shape the arithmetical board, but often highly ornamented, which was destined for the amusement of the opulent. It was used in a game apparently similar to that of chess, in which the infamous and abandoned Nero took particular delight, driving over the surface of the Abacus with a beautiful ivory quadriga or chariot. The Chinese have, from the remotest ages, used in all their computations, an instrument similar in shape and construction to the Roman Abacus, but more complete and uniform. It is admirably adapted to the decimal system of weights, meas- ures, and coins, which prevails throughout the empire. The whole range includes ten bars, and the calculator may begin at any one and reckon upwards or downwards with equal facility, treating fractions exactly like integers an advantage of the utmost consequence in practice. Accordingly these arithmetical machines, of various sizes, have been adopted by all ranks, from the man of letters to the humblest shopkeeper, and are constantly used in all the bazaars and booths of Can- ton and other cities, being handled, it is said, by the native traders with a rapidity and address quite astonishing. Among the various nations which regained their independ- ence by the fall of the Roman Empire, it was found convenient in all transactions where money was concerned, to follow the PALPABLE ARITHMETIC. 151 procedure of the Abacus, in representing numbers by counters placed in parallel rows. During the Middle Ages, it became the usual practice over Europe for merchants, auditors of accounts, or judges appointed to decide in matters of revenue, to appear on a covered bank or bench, so called from an old Saxon or Franconian word signifying a seat. The term scaccarium, a Latinized Oriental word, from which was derived the French and then the English name for the Exchequer, anciently indicated merely a chess-board, being formed from scaccum, one of the pieces in that game. The Court of Exchequer, which takes cognizance of all questions of revenue, was introduced into England by the Norman Conquest. Fitz-Nigel, in a dialogue on the subject, written about the middle of the twelfth century, says that the scaccarium was a quadrangular table about ten feet long and five feet broad, with a ledge or border about four inches high, to prevent anything from rolling over, and was surrounded on all sides by seats for the judges, the tellers, and other officers. It was covered every year, after the term of Easter, with fresh black cloth, divided by perpendicular white lines or distinc- tures, at intervals of about a foot or a palm, and again parted by similar transverse lines. In reckoning accounts, they pro- ceeded according to the rules of arithmetic, using small coins for counters. The lowest bar exhibited pence, the one above it shillings, the next pounds, an d the higher bars denoted suc- cessively tens, twenties, hundreds, thousands, and ten thou- sands of pounds; though, in those early times of penury and severe economy, it very seldom happened that so large a sum as the last ever came to be reckoned. The teller sat about the middle of the table ; on his right hand, eleven pennies were heaped on the first bar, and nineteen shillings on the second, while a quantity of pounds was collected opposite to him, on the third bar. For the sake of expedition he might employ a different mark to represent half the value of any bar, a silver penny for ten shillings, and a gold penny for ten pounds. 162 THE PHILOSOPHY OF ARITHMETIC. In early times, a checkered board, the emblem of calculation, was hung out, to indicate an office for changing money. It was afterwards adopted as the sign of an inn or hostelry, where victuals were sold, or strangers lodged and entertained. It is said that traces of this ancient practice may be found even at the present day. The use of the smaller Abacus in assisting numerical com- putation was not unknown during the Middle Ages. In England, however, it appears to have scarcely entered into actual practice, being mostly confined to those few individuals who, in such a benighted period, passed for men of science and learning. The calculator was styled, in correct Latin, abacista ; but in Italian, abbachista, or abbachiere. The Arabians, having adopted an improved species of numeration, to which they gave the barbarous name of algarismus or algo- rithmus, from their definite article al, and the Greek word for number, this compound term was adopted by the Christians of the West, in admiration of their superior skill, to signify calculation in general, long before the peculiar method of per- forming it had become known and practiced among them. The term algarism was converted in English into augrim or awgrym, and applied even to the pebbles or counters used in ordinary calculation. The same word, algorithm, is now applied by mathematicians to express any peculiar sort of notation. The Abacus had been adopted merely as an instrument for facilitating the process of computation. It became necessary, however, to adopt some simpler and more conveni ent method of expressing numbers. A very ancient practice consisted in employing the various articulations and disposi- tions of the fingers and the hands, to denote the numerical series. On this narrow basis, the Romans framed a system of considerable extent. By the inflexion of the various fingers of the left hand, they proceeded as far as ten, and by combin- ing these with some other given inflexions, as changes in the PALPABLE ARITHMETIC. 153 position of the thumb, they could advance to a hundred ; and using the right hand in a similar manner, they proceeded as far as a thousand and ten thousand. This is as far as the system appears to have been carried by the ancients ; but the venerable Bede, by referring these signs to the various parts of the body, as the head, the throat, the side of the chest, the stomach, the waist, the thigh, etc., has shown how they could be again multiplied a hundred times, and raised to the extent of a million. In this numerical play, the Romans especially had acquired great dexterity. Many allusions to the practice are made by their poets and orators, and without some knowl- edge of the principle adopted, many passages of the classics would lose their whole force. A species of digital arithmetic seems to have existed among nearly all the Eastern nations. The Chinese have a system of indigitation by which they can express on one hand all num- bers less than 100,000 The thumb nail of the right hand touches each joint of the little finger, passing first up the external side, then down the middle, and afterwards up the other side of it, in order to express the nine digits; the tens are denoted in the same way on the second finger ; the hun- dreds on the third; the thousands on the fourth; the tens of thousands on the thumb. It would be only necessary to pro- ceed to the right hand in order to be able to extend this system of numeration much further than could be required for any ordinary purposes. The Bengalese count as far as 15 by touching in succession the joints of the fingers ; and merchants in concluding bargains, the particulars of which they wish to conceal from the by slanders, put their hands beneath a cloth and signify the prices they offer or take by the contact of the fingers. The same custom is prevalent also in Barbary and Arabia, where they conceal their hands beneath the folds of their cloaks, and possess methods which are probably pecu- liar and national, of conveying the expression of numbers to each other. 154 THE PHILOSOPHY OF ARITHMETIC. Juvenal states it as a peculiar felicity of Nestor that he counted the years of his age on his right hand. The image of Janus was represented, according to Pliny, with the fingers so placed as to represent 365, the number of days in the year. Some authors have supposed that Solomon in the passage, "Length of days is in her right hand, and in her left hand riches and honor," referred to this practice. The common phrases, ad digitos redire, in digitos mittere, have the same meaning as computare, and distinctly refer to digital numera- tion ; and the phrase micare digitis, of frequent occurrence, alludes to a game extremely popular among the Romans, and which was probably the same as the morra of modern Italy. This noisy game is played by two persons, who stretch out a number of their fingers at the same moment, and instantly call out a number; and he is the winner who expresses the sum of the number of fingers thrown out. The same game is found amongst the Sicilians, Spaniards, Moors, and Persians, and under the name tsoimoi, is practiced also in China. These signs were merely fugitive, and it became necessary to adopt other marks of a permanent nature for the purpose of recording numbers. But of all the contrivances adopted with this view, the rudest undoubtedly is the method of registering by tallies, introduced into England along with the Court of Exchequer, as another badge of the Norman Conquest. These consist of straight, well-seasoned sticks of hazel or willow, so called from the French verb tattler, to cut, because they are squared at each end. The sum of money was marked on the side with notches, by the cutter of tallies, and like wise in scribed on both sides in Roman characters, by the writer of the tallies. The smallest notch signified a penny, a larger one a shilling, and one still larger a pound; but other notches, increasing suc- cessively in breadth, were made to denote ten, a hundred, and a thousand. The stick was then cleft through the middle by the deputy-chamberlains, with a knife and mallet, the one por- tion being called a tally, or sometimes the scachia, stipes, or PALPABLE ARITHMETIC. 155 kancia, and the other portion named the counter-tally or folium. This strange custom might seem the practice of untutored Indians, and can be compared only to the rude simplicity of the ancient Romans, who kept their diary by means of lapilli or small pebbles, casting a white pebble into the urn on fortu- nate days, and dropping a black one when matters looked unprosperous ; and who sent, at the close of each year, the Praetor Maximus, with great solemnity, to drive a nail in the door of the right side of the temple of Jupiter, next to that of Minerva, the patron of learning and inventor of numbers. The use of counters was general throughout Europe as late as the end of the 15th century: about that period they were no longer used in Italy and Spain, where the early introduction of the Arabic figures and the number of treatises on the use of these figures had rendered them unnecessary. Recorde, in his Ground of Arts, prefaces his second dialogue, entitled " The Accounting by Counters," by observing, "Now that you have learned Arithmetic with the pen, you shall see the same art in counters, which feat doth not onely serve for them that cannot write and read, but also for them that can do both, but have not at the same time their pen or tables with them." We shall now proceed to give some account of the method of performing operations by this palpable or calcular arithmetic. They commenced by drawing seven lines with a piece of chalk, - on a table, board, or slate, or by a pen on paper, as in the margin ; the counters, which " were usually of brass, on the lowest line represented units, on the next tens, and so on as far as millions on the uppermost line; _ a counter placed between two lines repre- sented five counters on the line next below - it; thus, the number represented in the margin is 3629638, and the number of lines * * may evidently be increased so as to represent any number. 166 THE PHILOSOPHY OF ARITHMETIC. , A A A V A A m To add two numbers, such as 788 and 383, we divide tho lines as in the margin, so as to form three columns, writing the first number in the first column, numbering from the left, the second in the second, and the result in the third column. The sum of the counters on the lowest line in the first two columns is 6; we therefore place one on that line in the third column, and carry one to the space above which, added to the one already there, makes one on the second line ; adding this counter to the six already there, we have 7, and therefore place 2 on the line and carry one to the space above ; adding the counters on that space, we find there are 3, hence we leave one in the space and carry one to the next line, in which the sum of the counters is six ; we leave one on the line and carry one to the space above, and adding to the counter already there we have two counters, hence we leave no counter there, but place one on the fourth line ; the sum thus obtained will be 1171. The principle of this operation is extremely simple, and the process could, with a little practice, be performed with much rapidity. In practice, the last column would not be used, as the counters on each line would be removed as the addition proceeded, and replaced by those which denoted their sum. We will illustrate the method of subtraction by taking 682 from 1375. The two count- ers on the first line have none to correspond from which theycau be subtracted ; we therefore bring down the counter from the space above and replace it by 5 counters on the line ; we shall then have 3 counters left on the line and none on the space ; we then bring down 1 counter from the second space, leaving _ -- --- --- ---- PALPABLE ARITHMETIC. 157 a remainder of 4 counters on the line ; then bring down 1 counter from the third line to the second space, and we have 1 counter left ; and so we proceed until the subtraction is com- plete, and we shall have as a remainder 693. Recorde writes the smaller number in the first column, and commences sub- tracting at the upper line. To illustrate the process .of multiplication, let us find the product of 245T by 43. We express the multiplicand in the first column and the multiplier in the second ; multiply first by 3, and place the product in the third column and the product by 4 in the fourth; add the numbers in these two columns, and the sum is the product required. Division may be illustrated by dividing 12832 by 608. Since six hundreds is contained in 12 thousands 2 tens times, we place two counters on the second line of the quotient; multiply- ing 6 hundreds by 2 tens and subtract- ing, we have no re- mainder; multiply- ing 8 by 2 tens, we have 16 tens; but since 16 tens equal 1 hundred and 6 tens, we take off 1 from the 3 in the third or hundreds line, leaving 2 remaining; then take off 1 of those 2 and replace it by 2 in the second space, and then take 1 from the second space and 1 from the second line; then transfer the remaining counters A A -- --- ~* -A A A-* ft.* A . * - * 158 THE PHILOSOPHY OF ARITHMETIC. to the column of the first remainder, and we have as a re- mainder 672. The operation is repeated, placing the quotient 1 on the lowest line of the quotient column ; and in this case we merely subtract the divisor from the first remainder, obtain- ing 64 for the last remainder, and 21 for the quotient. This process may evidently be repeated to any extent ; but in prac- tice it was much simplified by removing the counters of the dividend to form the first remainder, and so on until the opera- tion was complete. Recorde mentions two different ways of representing sums of money by means of counters, one of which he calls the merchant's and the other the auditor's account. In the margin, 198 19s. lid. is expressed by the first method, the low- est line being pence, the second shillings, the third pounds, and the fourth scores of pounds; the spaces represent half a unit of the next superior line, and the detached counters at the left are equivalent to five counters at the right. The operations of addition, subtraction, etc. would be per- formed in a manner similar to those already given. The same sum would be represented by the auditor's account as in the margin; the first group to the right being pence, the second shillings, the next pounds, and the left hand group scores , of pounds; the two lower lines denote units of their respective classes, while in the third line those on the left denote one quarter and on the right one half of the next superior class. The Chinese Computing Table or Swan-Pan, previously mentioned, is represented by the accompanying engraving. It consists of a small oblong board surrounded by a frame or ledge, and parted downwards near the left side by a similar ledge. It is then divided horizontally by ten smooth and PALPABLE ARITHMETIC. 169 slender rods of bamboo, on which are strung two small balls of ivory or bone in the narrow compart- ment, and five such balls in the wider compartment ; each of the latter on the several bars denoting one, and each of the former expressing five. The progressive bars, descending after the Chinese manner of writing, have their values increased ten fold at each step. The arrangement here figured denotes, reckoning downwards, the number 5,804,712,063. The Swan- Pan advances to the length of ten billions, or a thousand times further than the Roman Abacus. But the most admirable feature of the in- strument is, that by beginning the units at any particular bar the decimal subdivisions of the unit may be represented. The Japanese make use of a similar instrument, and the facility with which they perform arithmetical operations is truly surprising. Several persons of eminence, during our own times, have advocated the revival of the practice of calculation by means of counters. Prof. Leslie considers this method as better cal- culated than any other to give a student a philosophical knowl- edge of the classification of numbers, and the theory of their notation ; and he has given, in great detail, examples of the representation of numbers in different scales of notation by counters, and of operations by means of them. There are other species of Palpable Arithmetic, some of which have been adapted especially for the use of blind people: the celebrated Saunderson invented an instrument for this purpose with which he is said to have worked arith- metical questions with extraordinary rapidity. Arithmetical instruments of this kind possess considerable interest and im- portance from their use in lessening the privations consequent upon one of the greatest human calamities. 160 THE PHILOSOPHY OF ARITHMETIC. Among other arithmetical machines for shortening the work of calculation or relieving the operator from any troublesome or difficult exercise of the memory, are Napier's virgulae, or rods, which were formerly much celebrated and generally used. The work in which they were first described was published in 1617, under the title of Babdologia. In the dedication to Chancellor Seton, he says, that the great object of his life had been to shorten and simplify the business of calculation ; and the invention of logarithms, which he had just promulgated, was a noble proof that he had not lived in vain. These virgu- lae, rods, or bones, as they were ofteu called, were thin pieces of brass, ivory, bone, or any other substance, about two inches in length and a quarter of an inch in breadth, distributed into ten sets, generally of five each ; at the head of each of these, in succession, was inscribed one of the nine digits or zero, and underneath them in each piece the products of the digit at the top with each of the nine digits in succession, in a series of eight squares divided by diagonals, in the upper part of which were put the digits in the place of tens, and in the lower the digits in the place of units. In order to multiply any two num- bers together, such as 3469 and 574, those rods are to be placed in contact which are headed by the digits 1, 3, 4, 6, 9, and the successive products of the terms of the multiplier into the multiplicand are found by adding successively the digit on the upper half of the square to the right to that in the lower half of the square to the left, in the line of squares which are oppo- site to the figure of the multiplier which is used ; thus, to mul- tiply 3469 by 4, we take the 13469 line of squares opposite 4, represented in the margin, and the product is 13876, being found by writing 6, the sum of 4 and 3, of 6 and 2, etc.. carrying when necessary. In case of division, those rods are arranged in contact which are headed by the figures of the divisor, and we are thus enabled to obtain the products formed by the divisor and successive terms of the quotient. PALPABLE ARITHMETIC. 161 16 8 64 8 In the case containing these rods, which Napier calls mul- tiplicationis promptuarium, there are usually found also two pieces with broader faces, one consisting of three longitudinal divisions, and the other of four ; one of which is adapted to the extraction of the square, and the other of the cube root ; in the first, one column contains the nine digits, the second their doubles, and the third their squares ; in the second, the first column contains the digits, the second their squares, and the third and fourth their cubes, two columns being necessary for this purpose when the cube consists of three places ; thus, the last division but one in each of these rods is represented as in the margin, the digits occupying the right-hand column. In our times, when the multiplication table is so much more perfectly learned than formerly, the eagerness with which this invention was welcomed will excite some surprise, considering that its only object was to relieve the memory of so light and trivial a burden; but it is in accordance with some of the pro- cesses elsewhere noticed, by which early authors endeavored to simplify arithmetical operations. Pascal, in 1642, at the age of 19, invented the first arith- metical machine, properly so called. It is said to have cost him such mental efforts as to have seriously affected his health, and even to have shortened his days. This machine was im- proved afterwards by other persons, but never came into prac- tical use. In 1673, Leibnitz published a description of a machine which was much superior to that of Pascal, but more complicated in construction and too expensive for its work, since it was capable of performing only addition, subtraction multiplication and division. But these machines are entirely eclipsed by those of Babbage and Scheutz. In 1821, Mr. Babbage, under the patronage of the British government, began the construction of a machine, and in 1833 a small portion of it was put together, and was found to perform its work with the 11 162 THE PHILOSOPHY OF ARITHMETIC. utmost precision. In 1834 he commenced to design a still more powerful engine, which has not yet been constructed. The expense of these machines is enormous, $80,000 having been spent on the partial construction of the first. They are designed for the calculation of tables or series of numbers, such as tables of logarithms, sines, etc. The machine pre- pares a stereotype plate of the table as fast as calculated, so that no errors of the press can occur in publishing the result of its labors. Many incidental benefits have arisen from this invention, among which the most curious and valua- ble was the contrivance of a scheme of mechanical notation by which the connection of all parts of a machine, and the precise action of each part, at each instant of time, may be rendered visible on a diagram, thus enabling the contriver of machinery to devise modes of economizing space and time by a proper arrangement of the parts of his own invention. A machine invented by G. and E. Scheutz, of Stockholm, and finished in 1853, was purchased for the Dudley Observa- tory, at Albany. The Swedish government paid $20,000 as a gratuity towards its construction. The inventors wished to attain the same ends as Mr. Babbage, but by simpler means. It can express numbers decimally or sexagesirnally, and prints by the side of the table the corresponding series of numbers or arguments for which the table is calculated. It has already calculated a table of the true anomaly of Mars for each -fa of a day. In size, it is about equal to a boudoir piano. Other attempts have been made, but so far nothing has been accom- plished which is entirely satisfactory, though the utility of some such engine in the calculation of astronomical and other tables is so great, that it is quite probable that efforts will be continued until complete success is attained. SECTION III. ARITHMETICAL REASONING. I, TUEEE is REASONING IN ARITHMETIC. II. NATURE OF ARITHMETICAL REASONING. III. REASONING IN THE FUNDAMENTAL OPERATIONS IV ARITHMETICAL ANALYSIS. V. THE EQUATION IN ARITHMETIC. VI. INDUCTION IN ARITHMETIC. CHAPTER I. THERE IS REASONING IN ARITHMETIC. ALL reasoning is a process of comparison ; it consists in comparing one idea or object of thought with another. Comparison requires a standard, and this standard is the old, the axiomatic, the known. To these standards we bring the new, the theoretic, the unknown, and compare them that we may understand them. The law of correct reasoning, there- fore, is to compare the new with the old, the theoretic with the axiomatic, the unknown with the known. This process, simple as it seems, is the real process of all reasoning. We pass from idea to truth, and from lower truth to higher truth, in the endless chain of science, by the simple process of comparison. Thus the facts and phenomena of the material world are understood, the laws of nature interpreted, and the principles of science evolved. Thus we pass from the old to the new, from the simple to the complex, from the known to the unknown. Thus we discover the truths and principles of the world of matter and mind, and construct the various sciences. Comparison is the science-builder ; it is the architect which erects the temples of truth, vast, symmetrical, and beauti- ful. In mathematics this process is, perhaps, more clearly exhib- ited than in any other science. In geometry, the definitions and axioms are the standards of comparison ; beginning in these, we trace our way from the simplest primary truth to the profoundest theorem. In arithmetic we have the same basis, (166) 166 THE PHILOSOPHY OF ARITHMETIC. and proceed by the same laws of logical evolution. Defini- tions, as a description of fundamental ideas, and axioms, as the statement of intuitive and necessary truths, are the foundation upon which we rear the superstructure of the science of num- bers. These views, though admitted in respect of geometry, have not always been fully recognized as true of arithmetic. The subject, as presented in the old text-books, was simply a col- lection of rules for numerical operations. The pupil learned the rules and followed them, without any idea of the reason for the operation dictated. There was no thought, no deduc- tion from principle; the pupil plodded on, like a beast of burden or an unthinking machine. There was, in fact, as the subject was presented, no science of arithmetic. We had a science of geometry, pure, exact, and beautiful, as it came from the hand of the great masters. Beginning with primary conceptions and intuitive truths, the pupil could rise step by step from the simplest axiom to the loftiest theorem ; but when he turned his attention to numbers, he found no beautiful relations, no inter- esting logical processes, nothing but a collection of rules for adding, subtracting, calculating the cost of groceries, reckoning interest, etc. Indeed, so universal was this darkness, that the metaphysicians argued that there could be no reasoning in the science of numbers, that it is a science of intuition ; and the poor pupil, not possessing the requisite intuitive power, was obliged to plod along in doubt, darkness, and disgust. Thus things continued until the light of popular education began to spread over the land. Men of thought and genius began to teach the elements of arithmetic to young pupils ; and the necessity of presenting the processes so that children could see the reason for them, began to work a change in the science of numbers. Then came the method of arithmetical analysis, in that little gem of a book by Warren Colburn. It touched the subject as with the wand of an enchantress, and it began to glow with interest and beauty. What before THERE IS REASONING IN ARITHMETIC. 167 was dull routine, now became animated with the spirit of logic, and arithmetic was enabled to take its place beside its sister branch, geometry, in dignity as a science, and value as an educational agency. Before entering into an explanation of the character of arith- metical reasoning, it may be interesting to notice the views of some metaphysicians who have touched upon this subject. It has been maintained, as already indicated, by some eminent logicians, that there is no reasoning in arithmetic. Mansel says, " There is no demonstration in pure arithmetic," and the same idea is held by quite a large number of metaphysicians. This opinion is drawn from a very superficial view of the sub- ject of arithmetic, a not uncommon fault of the metaphysician when he attempts to write upon mathematical science. The course of reasoning which led to this conclusion, is probably as follows : First, addition and subtraction were considered the two fun- damental processes of arithmetic ; all other processes were regarded as the outgrowth of these, and as contained in them. Second, there is no reasoning in addition ; that the sum of 2 and 3 is 5, says Whewell, is seen by intuition ; hence subtrac- tion, which is the reverse of addition, is pure intuition also; and therefore the whole science, which is contained in these two processes, is also intuitive, and involves no reasoning. This inference seems plausible, and by the metaphysicians and many others has been considered conclusive. That this conclusion is not only incorrect but absurd, may be seen by a reference to the more difficult processes of the science. Surely, no one can maintain that there is no reason- ing in the processes of greatest common divisor, least common multiple, reduction and division of fractions, ratio and pro- portion, etc. If these are intuitive with the logicians, it ia very certain that they require a great deal of thinking on the part of the learner. These considerations are sufficient to dis- prove their conclusions, but do not answer their arguments; it 163 THE PHILOSOPHY OF ARITHMETIC. becomes necessary, therefore, to examine the matter a little more closely. Whether the uniting of two small numbers, as three and two, involves a process of reasoning, is a point upon which it is admitted there may be some difference of opinion. The differ- ence of two numbers, however, may be obtained by an infer- ence from the results of addition, and, as such, involves a process of reasoning. The elementary products of the multi- plication table are not intuitive truths: they are, as will be shown in the next article, derived, as a logical inference, from the elementary sums of addition. The same is also true in the case of the elementary quotients in division. Even admitting, then, that there is no reasoning in addition or subtraction, it can clearly be shown that the derivation of the elementary results in multiplication and division does require a process of reasoning. Passing from small numbers, which may be treated independently of any notation, to large numbers ex- pressed by the Arabic system, we see that we are required to reduce from one form to another, as from units to tens, etc., which can be done only by a comparison, and also that the methods are based upon, and derived from such general princi- ples, as " the sum of two numbers is equal to the sum of all their parts," etc. The great mistake, however, in their reasoning, is in assum- ing that all arithmetic is included in addition and subtraction. If it could be proved that addition and subtraction, and the processes growing immediately out of them, contain no rea- soning, a large portion of the science remains which does not find its root in these primary processes. Several divisions of arithmetic have their origin in and grow out of comparison, and not out of addition or subtraction ; and since comparison is reasoning, the divisions of arithmetic growing out of it, it is natural to suppose, involve reasoning processes. Ratio, the comparison of numbers ; proportion, the comparison of ratios ; the progressions, etc., certainly present pretty good examples THERE IS REASONING IN ARITHMETIC. 169 of reasoning. These belong to the department of pure arith- metic. A proportion is essentially numerical, as is shown in another place, and belongs to arithmetic rather than to geometry. If, in geometry, the treatment of a proportion involves a reasoning process, as the logicians will surely admit, it must certainly do so when presented in arithmetic, where it really belongs. It must, therefore, be admitted that there is reasoning in pure arithmetic. Again, if there is no reasoning in arithmetic there is no science, for science is the product of reasoning. If we admit that there is a science of numbers, there must be some reason ing in the science. And again, arithmetic and geometry are regarded as the two great co-ordinate branches of mathematics. Now it is admitted that there is reasoning in geometry, the science of extension ; would it not be absurd, therefore, to sup- pose that there is no reasoning in arithmetic, the science of numbers ? Mansel, as already quoted, says : " Pure arithmetic contains no demonstrations." If by this he means and I presume he does that pure arithmetic contains no reasoning, he is answered by the previous discussion. If, however, ne meam that arithmetic cannot be developed in the demonstrative form of geometry that is, by definition, axiom, proposition, and demonstration he is also in error. Though arithmetic has never been developed in this way, it can be thus developed. The science of number will admit of as rigid and systematic a treatment as the science of extension. Some parts of the sci- ence are even now presented thus; the principles of ratio, proportion, etc., are examples. I propose, at some future time, to give a complete development of the subject, after the manner of geometry. The science, thus presented, would be a valua- ble addition to our academic or collegiate course, as a review of the principles of numbers. Assuming, then, that there is reasoning in arithmetic, in the next chapter I shall consider the nature of reasoning, as employed in the fundamental opera- tions of arithmetic. CHAPTER II. NATURE OF ARITHMETICAL REASONING. IN" order to show the nature of the reasoning of arithmetic, a brief statement of the general nature of reasoning will be presented. All forms of reasoning deal with the two kinds of mental products, ideas and truths. An idea is a simple notion which may be expressed in one or more words, not forming a proposition ; as, bird, triangle, four, etc. A truth is the comparison of two or more ideas which, expressed in language, give a proposition ; as, a bird is an animal, a triangle is a polygon, four is an even number. The comparison of two ideas directly with each other, is called a judgment; as, a bird is an animal, or five is a prime number. Herejfrue is one idea, and a prime number is another idea. Judgments give rise to propositions ; a proposition is a judgment expressed in words. Nature of Reasoning. If we compare two ideas, not directly, but through their relation to a third, the process is oalled reasoning. Thus, if we compare A and B, or B and C, and say A equals B or B equals C, these propositions are judgments. But if, knowing that A equals B, and B equals C, we infer that A equals C, the process is reasoning. Rea- soning may, therefore, be defined as the process of comparing two ideas through their relation to a third. Judgment is a process of direct or immediate comparison ; reasoning is a pro- cess of indirect or mediate comparison. (170) NATURE OP ARITHMETICAL REASONING. 171 In thus comparing two ideas through their relation to a third, it is seen that we derive one judgment from two other judgments; hence we may also define reasoning as the pro- cess of deriving one judgment from two other judgments; or as the process of deriving an unknown truth from two known truths. The two known truths are called premises, and the derived truth the conclusion; and the three propositions together constitute a syllogism. The syllogism is the simplest form in which a process of reasoning can be stated. Its usual form is as follows : A equals B ; but B equals C ; therefore A equals C. Here "A equals B" and "B equals C" are the pre- mises, and "A equals C" is the conclusion. The premises in reasoning are known either by intuition, by immediate judgment, or by a previous course of reasoning. In the syllogism "All men are mortal ; Socrates is a man ; therefore, Socrates is mortal" the first premise is derived by induction, and the second by judgment. In the syllogism "The radii of a circle are equal; R and R' are radii of a cir- cle ; therefore R and R' are equal " the first premise is an intu- ition, and the second is a judgment. In the syllogism "A equals B, and B equals C ; therefore A equals C" both pre- mises are judgments. It should also be remarked that truths drawn from the first steps of the reasoning process, do themselves become the basis of other truths, and these again the basis of others, and so on until the science is complete. This method of reasoning is called Discursive (discursus) ; it passes from one truth to another, like a moving from place to place. We start with the simple truths which are so evident that we cannot help seeing them ; and travel from truth to truth in the pathway of science, until we reach the loftiest conceptions and the profoundest principles. Reasoning, as we have stated, is the comparison of two ideas through their relation to a third; or it may be defined as the derivation of one judgment from two other judgments 172 THE PHILOSOPHY OF ARITHMETIC. These two judgments are not always both expressed ; indeed, in the usual form of thought, one is usually suppressed ; but both are implied, and may be supplied if desired to show the validity of the conclusion. Every truth derived by a pro- cess of reasoning, may be shown to be an inference from two propositions which are the premises or ground of inference, and this is the test of the validity of the truth derived. There are two kinds of reasoning, inductive and deductive. Inductive reasoning is the process of deriving a general truth from several particular ones. It is based upon the principle that what is true of the many is true of the whole. Thus, if we see that heat expands many metals, we infer, by induction, that it will expand all metals. Deduction is the process of deriving a particular truth from a general one. It is based upon the axiom, that what is true of the whole is true of all the parts. Thus, if we know that heat will expand all metals, we infer, by deduction, that it will expand any particular metal, as iron. Mathematics is developed by the process of deductive rea- soning. The science of geometry begins with the presentation of its ideas, as stated in its definitions, and its self-evident truths, as stated in its axioms. From these it passes by the process of deduction to other truths; and then, by means of these in connection with the primary truths, proceeds to still other truths ; and thus the science is unfolded. In arithmetic, no such formal presentation of definitions and axioms is made, and the truths are not presented in the logical form, as in geometry. From this it has been supposed that there is no reasoning in arithmetic. This inference, however, is incorrect; the science of numbers will admit of the same logical treat- ment as the science of space. There are fundamental ideas in arithmetic as in geometry; and there are also fundamental, self-evident truths, from which we may proceed by reasoning to other truths. In this chapter I shall endeavor to show the nature of the reasoning in the Fundamental Operations of Arithmetic. NATURE OF ARITHMETICAL REASONING. 173 Arithmetical Ideas. The fundamental ideas of arithmetic, as given in the process of counting, are the successive numbers one, two, three, etc. These ideas correspond to the different ideas of geometry, and the definitions of them will correspond to the definitions of geometry. In geometry, we have the three dimensions of extension, giving us three distinct classes of ideas, lines, surfaces, and volumes; in arithmetic there is only one fundamental idea of succession, giving us but one fundamental class of notions. The primary ideas of arithmetic are one, two, three, four, Jive, etc., which correspond to the idea of line, angle, triangle, quadrilateral, pentagon, etc., in geometry. These ideas may be defined as in the cor- responding cases in geometry. Thus, two may be defined as one and one; three as two and one, etc.; or, in the logical form three is a number consisting of two units and one unit. There are other ideas of the science growing out of relations, such as factor, common divisor, common multiple, etc. Arithmetical Axioms. The axioms of arithmetic are the self-evident truths that relate to numbers. There are two classes of axioms in arithmetic as in geometry, those which relate to quantity in general, that is, to numbers and space ; and those which belong especially to number. Thus, " Things that are equal to the same thing are equal to each other," and " If equals be added to equals the sums will be equal," etc., belong to both arithmetic and geometry. In geometry we have some axioms which do not apply to numbers, as "All right angles are equal," "A straight line is the shortest dis- tance from one point to another," etc. There are also axioms which are peculiar to arithmetic, and which have no place in geometry. Thus, "A factor of a number is a factor of a mul- tiple of that number," "A multiple of a number contains all the factors of that number," etc. These two classes of axioms are the foundation of the reasoning of arithmetic, as they arc of the science of geometry. Arithmetical Reasoning. The reasoning of arithmetic is 174 THE PHILOSOPHY OF ARITHMETIC. deductive. The basis of our reasoning is the definitions and axioms; that is, the conceptions of arithmetic, and the self- evident truths arising from such conceptions. The definitions present to us the special forms of quantity upon which we reason ; the axioms present the laws which guide us in the reasoning process. The definitions give the subject-matter of reasoning ; the axioms give the principles which determine the form of reasoning, and enable us to go forward in the discovery of new truths. Thus, having defined an angle, and a right angle, we can by comparison, prove that "the sum of the angles formed by one straight line meeting another, is equal to two right angles." Having the definition of a triangle, by comparison we can determine its properties, and the relation of its parts to each other. So in arithmetic, having defined any two numbers, as four and six, we can determine their relation and properties ; or having defined least common mul- tiple, we can obtain the least common multiple of two or more numbers, guiding our operations by the self-evident and neces- sary principles pertaining to the subject. Axioms in Reasoning. In this explanation of reasoning, it is stated that reasoning is a process of comparing two ideas through their relations to a third, and that axioms are the laws which. guide us in comparing. This view of the nature of axioms differs from the one frequently presented. Some logi- cians tell us that axioms are general truths which contain par- ticular truths, and that reasoning is the process of evolving these particular truths from the general ones. The axioms of a science are thus regarded as containing the entire science; if one knows the axioms of geometry, he knows the general truths in which are wrapped up all the particular truths of the science. All that is necessary for him to become a profound geometer is to analyze these axioms and take out what is con- tained in them. The incorrectness, or at least inadequacy of this view of the nature of axioms and their use in reasoning, I cannot now NATURE OF ARITHMETICAL REASONING. 175 Btop to consider. Its fallacy is manifest in the extent of the assumption. It may be very pleasant for one to suppose that when he has acquired the self-evident truths of a science, he has potentially, if not actually, in his mind the entire science; such an expression may do as a figure of speech, but does not, it seems to me, express a scientific truth A general formula may be truly said to contain many special truths which may be derived from it; thus Lagrange's formula of Mechanics embraces the entire doctrine of the science ; but no axiom can oe, in the same sense, said to contain the science of arithmetic or geometry. But whatever may be thought of this view of the nature and use of axioms, it cannot be denied that the explanation of reasoning which I have given is correct. Reasoning is the comparison of two ideas through their relation to a third, the comparison being regulated by self-evident truths. This is the view of Sir William Hamilton, and it has been adopted by sev- eral modern writers on logic. Even if the other view is right that the axioms may be regarded as general truths, from which the particular ones are evolved by reasoning their practical use in reasoning coincides with the explanation of the nature of the reasoning powers which I have presented ; and this idea of the subject will be found to be much more readily under- stood and applied. The simpler view is that the axioms are laws which guide us in the comparison, or they are the laws of inference. Thus, if I wish to compare A and B: seeing that they are each equal to C, I can compare them with each other, and determine their equality by the law that things which are equal to the same thing are equal to each other. So, if I have two equal quantities, I may increase them equally without changing their relation, according to the law enun- ciated in the axiom that if the same quantities be added to equals, the results will be equal. This view of the subject of axioms and of their use in the process of reasoning, may be supported by various considerations, and will be found to 176 THE PHILOSOPHY OF ARITHMETIC. throw light upon several things in logic upon which writera are sometimes not quite clear. In the following chapter I shall apply this view of reasoning to the fundamental operations of arithmetic. CHAPTER III. REASONING IN THE FUNDAMENTAL OPERATIONS. QCIENCB, as already stated, consists of ideas and truths. O Truths are derived either by intuition or reasoning. Intu- itive truths come either by the intuitions of the Sense or the Reason ; derivative truths by the discursive process of induction or deduction. The primary ideas of arithmetic are the individual numbers, one, two, three; its primary truths are the elementary sums and differences of addition and subtrac- tion. How these primary truths are derived, is a question upon which opinion is divided. On the one hand it is claimed that they are intuitive ; on the other, that they are derived by rea- soning. Thus, tivo and one are three, three and two are Jive, etc., are regarded by some as pure axioms, neither requiring nor admitting of a demonstration ; while others regard them as deductions from the primary process of counting. Let us ex- amine the subject somewhat in detail, and also consider the process of deriving other truths growing out of these. Addition. It is generally assumed that the primary sums of the addition tables are axioms. They are intuitive truths growing out of an analysis of our conceptions of a number into its parts, or a synthesis of these parts to form the number. Thus, given the conception of nine, by analysis we see that it consists or is composed of four and ./rue; or given four and^/Jue, by synthesis we immediately see that it gives a combination of nine units, or is equal to nine. This view is maintained by some eminent logicians. "Why is it," says Whewell, "that three 12 ( 177 ) 178 THE PHILOSOPHY OF ARITHMETIC. and two are equal to four and one? Because if we look at five things of any kind we see that it is so. The five are four and one ; they are also three and two. The truth of our asser- tion is involved in our being able to conceive the number five at all. We perceive this truth by intuition, for we cannot see, or imagine we see, five things, without perceiving also that the assertion above stated is true." The other view makes counting the fundamental process, and derives the judgments expressed in the elementary sums by inference. Thus, the process of finding the sum of five and/bur may be stated as follows: The sum of five and/our is that number which is four units after five; By counting we find that the number four units after five is nine; Hence, the sum of five and four is nine. This is a valid syllogism, and shows that the sums might be thus obtained, whether they are actually so obtained or not. It may be objected, however, that they can be obtained only in one way ; and if intuitive, then it is not possible to derive them by any process of reasoning. This does not necessarily follow, for we can often obtain, by a process of reasoning, a truth which we could also derive in some other way. If we discover a new metal, it can be immediately inferred that heat will expand it, since heat expands all metals, which is a pro- cess of deductive reasoning. This truth may also be obtained by direct experiment. Many examples may be given to show that a truth may be derived by reasoning, which might also be derived in some other way. These fundamental truths may be used in obtaining the rela- tions of different combinations of numbers, and such an operation will be a process of reasoning. Thus, it is not evi- dent to the learner, neither is it intuitive with any one, that 7 plus 2 equals 4 plus 5 ; or, what is less readily seen, that 25 plus 37 equals 19 plus 43. These are not axioms, since they cannot be seen to be true without an examination of the grounds of thr> relation. The process of reasoning to prove REASONING IN THE FUNDAMENTAL OPERATIONS. 179 the propositions is as follows : 7 plus 2 equals 9 ; but 4 plus 5 equals 9 ; therefore, 7 plus 2 equals 4 plus 5 ; or, as Whewell puts it, thus : 7 equals 4 and 3, therefore 7 and 2 equals 4 and H and 2 ; and because 3 and 2 are 5, 7 and 2 equals 4 and 5. In the former case the result depends on the axiom, " Things that are equal to the same thing are equal to each other ;" in the latter case, the reasoning process is based upon the axiom, " When equals are added to equals the results are equal." It will be noticed that Whewell's method of proof is very similar to the ordinary demonstration of the theorem that "When one straight line meets another straight line, the sum of the two angles equals two right angles." That this is a valid process of reasoning is evident from its similarity to the geometrical process A plus B equals C ; but D plus E equals C ; therefore, A plus B equals D plus E. It is readily seen that many such cases will arise in which the operations are entirely independent of the notation employed, from which it cannot be doubted that there is reasoning in addition in pure arithmetic. When we proceed to the addition of large numbers, expressed by the Arabic system, which may not be regarded as pure arithmetic, we base the operation upon the axiom that the sum of several numbers is equal to the sum of all the parts of those numbers. That the derivation of a result from this general axiomatic principle is a process of rea- soning, cannot be doubted by any one who is competent to understand in what reasoning consists. Subtraction. Subtraction, like addition, embraces two cases, the finding of the difference between numbers independently of the notation employed to express them, that is, the elementary differences of the subtraction table, and the finding of the dif- ference between large numbers expressed in the Arabic system. The elementary differences in subtraction may be obtained in two ways. First, we may find the difference between two numbers by counting off from the larger number as many units as are contained in the smaller number. Thus, if we wish to 180 THE PHILOSOPHY OF ARITHMETIC. subtract four from nine, we may begin at nine and count back- ward four units, and find we reach five, and thus see that four from nine leaves five. The other method consists in deriving the elementary differences by inference from the ele- mentary sums. The former method is regarded by some as intuitive, although it admits of a syllogistic statement; the latter method, without doubt, involves a process of reasoning. To illustrate, suppose we wish to find the difference between nine and five. The ordinary process of thought is as follows: Since four added to five equals nine, nine diminished by five equals four. This process, put in the formal manner of the syllogism, is as follows: The difference between two numbers is a number which added to the less will equal the greater ; But four added to five, the less, equals nine, the greater ; Therefore, four is the difference between nine and five. This, of course, is too formal for ordinary language, but is all implied in the practical form, "five from nine leaves four, since five and four are nine." In subtracting large numbers expressed by the Arabic system of notation, we proceed upon the principle that the difference between the parts of numbers equals the difference between the numbers themselves, which shows that the process is one of deduction. Multiplication. Multiplication, like addition and subtrac- tion, embraces two cases the finding of the elementary pro- ducts of the multiplication table, and the use of these in ascertaining the product of two numbers expressed by the Arabic system. The elementary products are obtained by deduction from the elementary sums of addition. Thus, in obtaining the product of three times four, the logical form of thought is as follows: Three times four are the sum of three fours; But the sum of three fours is twelve; Hence, three times four are twelve. The first premise is an immediate inference from the defini- REASONING IN THE FUNDAMENTAL OPERATIONS. 181 tion of multiplication ; the second premise we know to be true from addition ; the conclusion is a deductive inference from the two premises. In the common form of thought we omit one of the premises, saying, "three times four are twelve, since the sum of three fours is twelve." The multiplication of large numbers depends on these elementary products thus derived by deduction, and also employs the principle, that the sum of the products of the parts equals the whole product. Division. The reasoning in division is similar to that in multiplication. The elementary quotients of the division table may be obtained in two distinct ways by subtraction or reverse multiplication, but in either case they are an inference from things already known, and are thus derived by a process of reasoning. By the method of subtraction we say, "four is contained in twelve three times, since four can be subtracted from twelve three times ; by the method of reverse multiplica- tion we say, "four is contained in twelve three times, since three times four are twelve." Each of these may be expressed in the form of a syllogism, as in multiplication. The division of larger numbers is based on these elementary quotients, and also upon the principle that the sum of the partial quotients equals the entire quotient. The view here given concerning the origin of the elementary products and quotients may be presented in another way. When we begin addition we have no idea of multiplication ; by and by the idea of a product arises in the mind, and it is immediately seen that the product of the number is the sum arising from taking one number as many times as there are units in another. Suppose then we wish to know the product of 3 times 4, we reason as follows : The product of 3 times 4 equals the sum of 4 taken 3 times ; ut the sum of 4 taken 3 times we find is 12; Hence, the product of 3 times 4 equals 12. Primary quotients may be obtained in a similar manner, and both art valid forms of reasoning. But whatever view may 182 THE PHILOSOPHY OF ARITHMETIC. be taken of the origin of the elementary truths of the funda- mental operations and the fact of a difference of opinion indi- cates a reason for it it certainly cannot be denied, by one who will examine, that there is reasoning in the processes growing out of these fundamental operations, and also in those which have their origin in comparison. These fundamental judgments of the tables of the four "ground rules" are committed to memory, and are employed in the reasoning processes by which we derive other truths in the science. Other Forms. As we leave the fundamental operations, however, the processes of reasoning grow more and more dis- tinct. As each new idea is presented, new truths arise intui- tively, which become the basis for the derivation of other truths, the same as in geometry. To illustrate, take the sub- ject of Greatest Common Divisor. As soon as the idea of a common divisor is clearly apprehended, several truths are per- ceived as growing immediately out of this conception. These truths are intuitively apprehended, and are the axioms pertain- ing to the subject. From these self-evident truths, we proceed to other truths by a process of reasoning usually called demon- stration. Thus, in the subject of greatest common divisor we have these axioms : 1. A divisor of a number is a divisor of any number of times that num- ber. 2. A common divisor of several numbers is the product of some of the common factors of these numbers. 3. The greatest common dinsor of several numbers is the product of all the common prime factors of these numbers. 4. The greatest common divisor of several numbers contains no factors but those which are common to all the numbers. These truths are self-evident and necessary, and are seen to be so as soon as a clear idea of the subject is attained. They may be illustrated, but cannot be demonstrated. They bear precisely the same relation to the arithmetical concep- tion of greatest common divisor that the axioms of geometry REASONING IN THE FUNDAMENTAL OPERATIONS. 183 do to some of the geometrical conceptions. Thus, in geometry, as soon as we have the conception of a circle, it is intuitively seen that all the radii are equal to each other ; or that the radius is equal to one-half of the diameter, etc. Such truths are made the basis of the reasoning by which we derive the other truths relating to the circle. If the process of obtaining these derivative truths in geometry is regarded as reasoning, surely the similar processes in arithmetic are also reasoning. Having a clear conception of the idea of greatest common divisor, and of the self-evident truths or axioms, belonging to it, we are prepared to derive other truths relating to the sub- ject, by the process of reasoning. As an example of a truth derived by demonstration, take the following: The greatest common divisor of two quantities is a divisor of their sum and their difference. In order to demonstrate this theorem, take any two numbers, as 20 and 12. We see that the greatest common divisor is 4. We also know that 20 is 5 times 4 and 12 is 3 times 4. We then reason as follows : The sum of the two numbers equals 5 times 4 plus 3 times 4 or 8 times 4; But 4, the G. C. D., is evidently a divisor of 8 times 4 ; Hence, 4, the G. C. D., is a divisor of the sum of the two numbers. In this syllogism " 8 times 4" is the middle term, the " sum of the two numbers" the major term, and " 4, the greatest common divisor," the minor term; and the syllogism is entirely valid. In a similar manner we may prove that the greatest common divisor is a divisor of the difference of the two num- bers. The method of reasoning with 20 and 12 is seen to be applicable to any two numbers having a common divisor ; hence the truth is general It should be remarked that a large portion of the reasoning in arithmetic consists in changing the form of a quantity, so that we may see a property which was concealed in a previous form, and then inferring that it belongs also to the quantity in 184 THE PHILOSOPHY OP AKITHMETIC. its first form, since the value of the quantity is not changed by changing its form. It is thus seen that the science of arithmetic, like geometry, consists of ideas and truths; that some of these truths are self-evident, and others are derived by a process of reasoning ; and that the process of reasoning in the two sciences is simi- lar. We proceed now to consider some of these forms of rea- soning, and especially the subject of arithmetical analysis, which will be treated in the next chapter. CHAPTER IV. ARITHMETICAL ANALYSIS. 4 RITHMETICAL Analysis is the process of developing -Q- the relation and properties of numbers by a comparison of them through their relation to the unit. All numbers con sist of an aggregation of units, or are so many times the single thing ; and hence bear a definite relation to the unit. This relation the mind immediately apprehends in the conception of a number itself. From this evident relation to the unit, all numbers may be readily compared with each other, and their properties and relations discovered. Let us examine the pro- cess a little more in detail. Unit the Basis. The basis of this analysis is the Unit. The Unit is the primary and fundamental idea of arithmetic. It is the basis of all numbers, a number being a repetition of the Unit, or a collection of units of the same kind. The relation of a number to the Unit, or of the Unit to a number, is consequently immediately seen from the conception of a number itself. The collection is intuitively conceived to be so many times the Unit, or the Unit such a part of the collection. The import- ance of the Unit, as the base of the comparison of numbers, is thus apparent. Integers may be readily compared with each other, through their relation to the fundamental elements out of which they are formed. A Unit is one of the several things considered; and, since a fraction is a number of equal parts of a Unit, it is seen tbax we have a second class of units which we may call fractional (185) 186 THE PHILOSOPHY OF ARITHMETIC. units. These two classes of units may be distinguished as the Unit and the fractional unit. A number of fractional units gives rise to a class of numbers called fractions. The same principle of comparison obtains in the comparison of these as in the comparison of integral numbers. A fractional unit being one of several equal parts of the Unit, its relation to the latter is simple and immediately apprehended. We can thus compare different fractional units by their relation to the Unit, as we did integral numbers by their relation to it. The com- parison of fractions, which at first might have seemed difficult, thus becomes simple and easy. From this consideration we are enabled to see the import- ance of the Unit in the process of arithmetical analysis. As the basis of numbers, it becomes the basis of reasoning with numbers. We compare number with number or fraction with fraction by their intermediate relation to the Unit. The Unit thus becomes the stepping-stone of the reasoning process, the central point around which the circle of logic revolves. Comparison of Integers. Numbers are compared, as has already been remarked, by their relation to the Unit. In the comparison of numbers, the relation between them is not imme- diately apprehended; but knowing the relation that each sustains to the Unit, we can ascertain their relation to each other by this simple intermediate relation. To illustrate this, suppose we wish to compare any two numbers, as 3 and 5 ; let the problem be " What is the relation of 3 to 5 ?" or "3 is what part of 5?" We would reason thus : One is 1 fifth of 5, and if one is 1 fifth of 5, 3, which is three times one, is three times 1 fifth, or 3 fifths of 5. Hence, 3 is 3 fifths of 5. In this example we cannot compare 3 directly with 5; we therefore make the comparison indirectly, by considering their intermediate relation to the Unit, which is readily apprehended. Again, take the problem, "If 3 times a number is 12, what is 5 times the number?" Here, it may be remarked, 3 times the number is the known quantity, and 5 times the number is the unknown quantity, ARITHMETICAL ANALYSIS. 187 which we wish to find by comparing it with the known quan- tity. How shall we make this comparison, and thus pass from the known to the unknown? We cannot compare them directly, since the relation between them is not readily per- ceived ; we must compare them indirectly by means of their relation to the Unit. The process of reasoning is as follows: If 3 times a number is 12, once the number is of 12 or 4; and if once a number is 4, five times the number is 5 times 4, or 20. Thus we readily pass from three times the number to five times the number from the known to the unknown first passing from three to one and then from one to five. In the same manner all numbers may be compared with each other, their relation being determined by this intermediate relation to One, the Unit, the basis of all numbers. Comparison of Fractions. Fractions are also compared by means of their relation to the Unit. A Fraction is a number of fractional units. The fractional unit is one of several equal parts of the Unit; hence the relation between it and the Unit is simple and readily perceived. When we have a num- ber of fractional units that is, a Fraction in comparing it with the Unit, we must first pass from the number of fractional units to the fractional unit itself, and then from the fractional unit to the Unit. From this we can readily pass to a num- ber, or to any other fractional unit, and then to any number of such fractional units, that is, to any fraction. This will be more clearly seen by its application to a problem. Take the problem, " If | of a number is 24, what is f of the Dumber?" We reason thus: If two-thirds of a number is 24, one-third of the number is of 24, or 12; and Mree-thirds, or once the number, is 3 times 12, or 36. If once the number is 36, OHe-fourth of the number is \ of 36, or 9 ; and three-fourths of the number is 3 times 9, or 27. In this problem we compare the two fractions -f and f , by passing from two-thirds down to one-third, then rising up to the Unit, then passing down to one- fourth, and then up to Mree-fourths. In other words, we pasa 188 THE PHILOSOPHY OF ARITHMETIC. from a number of fractional units to the fractional unit, then to the Unit, then to another fractional unit, and then to a number of those fractional units. We first go down, then up, Jien down again, and then up again to the required point. Another excellent example of this method of comparison is given in the solution of the following problem: What is the relation of f to f ? Here 4 is the basis of comparison with which it is required to compare . This relation cannot be immediately seen, but it can readily be determined by the method of analysis. The solution is as follows: One-fifth is \ of f , and if one-fifth is ^ of f , ^ue-fifths, or One, is 5 times \ or | of f . If One is f of , one-third is of or -% of , and two-thirds is 2 times ^, or ; hence |- is of f. In this prob- lem we see the same law of comparison, and this law runs through the entire subject. Having given this general idea of the process, I will state the several simple cases of arithmetical analysis, and illustrate the process of thought by means of a diagram. The central relation of the Unit to the thought process, and the transition from the Unit and to the Unit, will be readily seen. CASE I. To pass from the Unit to B any number. Take the problem : If 1 apple costs 3 cents, what will 4 apples cost? If 1 apple costs 3 cents, 4 apples, which are 4 times 1 apple, will cost 4 times 3 cents, or 12 cents. In this problem the mind starts at the Unit A, and ascends 4 steps to B. CASE II. To pass from any number to the Unit. Take the problem : If 4 apples cost 12 cents,'what will 1 apple cost? The solution is as follows: If 4 apples cost 12 cents, 1 apple, which is 1 fourth of 4 apples, will cost 1 fourth of 12 cents, or 3 cents. In this problem the mind starts at the num- Init. Unit. ARITHMETICAL ANALYSIS. 189 her 4, four steps above the basis, and steps down to the Unit, or basis of numbers. CASE III. To pass from a number to a number. Take the problem: If 3 apples cost 15 cents, what will 4 apples cost? The solution is : If 3 apples cost 15 cents, 1 apple will cost ^ of 15 cents, or 5 cents, and 4 apples will cost 4 times 5 cents, or 20 cents. In this case we are to pass from the collection three to the collection four. In comparing three and four, their relation is not readily seen ; but knowing the relation of three to the Unit, and of the Unit to four, we make the transi- tion from three to four by passing through the Unit. This may be illustrated as follows: Suppose one standing at A and wishing to pass over to C. Unable to step directly from A to C, he first steps down to the starting point, B, and then ascends to C. So in compar ing numbers, when we cannot pass directly from the one to unit. the other, we go down to the Unit, or starting-point of numbers, and then go up to the other number. These relations are intuitively apprehended, being presented in the formation of numbers. In the given problem we stand three steps above the Unit, and we wish to go four steps above the Unit. To do this we first descend the three steps, and then ascend the four steps. CASE IV. To pass from a unit to a fraction. Take the problem : If one ton of hay cost $8, what will f of a ton cost ? The solution is as follows: If one ton of hay costs $8, one-fourth of a ton will cost \ of $8, or $2, and three-fourths of a ton will cost 3 times $2, or $6. In this problem we pass from the Unit to the fourth, one of the equal divisions of the unit, and then to a collection of such equal divisions. In other words, we descend from the integral B 190 THE PHILOSOPHY OF ARITHMETIC. Unit. 4th. n r Cult. Unit to the fractional Unit, and then ascend among the fractional units. It is as if we were standing at A, and wished to pass to C ; we first take four steps down to B, and then three steps up to C, instead of trying to step immediately over from A to C. CASE V. To pass from a fraction to a unit. Take the problem: If f of a ton of hay cost $6, what will one ton cost? The solution is as follows: If tfiree-fourths of a ton of hay cost $6, one-fourth of a ton will cost | of $6, or $2; and /owr-fourths of a ton, or one ton, will cost 4 times $2, or $8. In this problem we pass from a collection of frac- tional units to the frac- tional unit, and then to the integral Unit. It is as if we were standing at A, and wished to pass to C. We cannot make the transi- tion directly, so we step three steps down to B, and then four steps up to C. CASE VI. To pass from a fraction to a fraction. Take the problem If f of a number is 15, what is 4 of the number? The solution is as follows: If //iree-fourths of a number is 15, one-fourth of the number is^of 15 or 5, and /owr-fourths of the number, or once the number, is 4 times 5 or 20. If the number is 20, one-fifth of the number is 1 tifth of 20, or 4 ; and /bur-fifths of the number is 4 times 4 or 16. In this problem we wish to compare the two fractions f and ; but since we cannot perceive the relation of them directly, we must compare them through their relation to the Unit. To do this we first go from three-fourths to one-fourth, then from 4th ARITHMETICAL ANALYSIS. 191 one-fourth to the Unit, then from the Unit to one-fifth, and then to four-fifths. In other words, we first go down from the collection of fractional units to the fractional Unit and then up to the integral Unit ; we then descend to the other frac- tional unit, and then ascend to the number of fractional units required. It is as if we were standing at A and wished to pass to E ; we cannot step directly over from one point to the c Unit. ' "L r D other so we pass from A three steps down to B, then four steps up to C, then five steps down to D, and then four steps up to E. These diagrams, it is believed, present a clear illustration of the subject, and enable one to understand the process of thought in the elementary operations of arithmetical analysis. The Unit is thus seen to lie at the basis of the process, the mind running to it and from it in the comparison of numbers. It will be remembered, however, that these are merely illustra- tions, and are not designed to convey a complete idea of the process in all of its details. This can only be seen by a care- ful analysis of the process itself. Analysis Syllogistic. The process of arithmetical analysis is a process of mediate comparison, and is consequently a reasoning process. This will appear from the fact that it may be presented in the syllogistic form. Take the simplest case: If 4 apples cost 12 cents, what will 5 apples cost? Expressed in the form of a syllogism, we have the following: The cost of 1 apple is \ of the cost of 4 apples; But \ of the cost of 4 apples is \ of 12 cents, or 3 cents ; Hence the cost of 1 apple is 3 cents. 192 THE PHILOSOPHY OP ARITHMETIC. The cost of five apples is 5 times the cost of 1 apple; But, 5 times the cost of 1 apple is 5 times 3 cents, or 15 cents ; Hence, the cost of five apples is 15 cents. It is thus seen that the process of analysis is purely syllo- gistic, and is, consequently, a reasoning process. It is not usually presented in the syllogistic form, since it would be too stiff and formal, and moreover would be more difficult for the young pupil to understand. Direct Comparison. The comparison of numbers, so far as explained, is indirect and mediate, that is, through their relation to the Unit. After becoming familiar with this process, the mind begins to perceive the relations between numbers them- selves, and is thus enabled to reason by comparing the numbers directly, instead of employing their intermediate relations to the common basis. To illustrate, take the problem : If 3 apples cost 10 cents, what will 6 apples cost? We may reason thus: If 3 apples cost 10 cents, 6 apples, which are two times 3 apples, will cost two times 10 cents, or 20 cents. Primarily we would have gone to the Unit, finding the cost of one apple ; but now we may omit this and compare the numbers directly. With integral numbers this direct comparison is simple and easy ; but with fractions it is much more complicated and diffi- cult. Thus, if f of a number is 20, it is difficult to see directly that 4 of the number is f of 20 ; that is, that the relation of to | is ; hence, though we should avail ourselves of the direct relation of integral numbers, it will be found much sim- pler to compare fractions by their intermediate relations to the Unit. CHAPTER V. THE EQUATION IN ARITHMETIC. THE comparison of mathematical quantities is mainly con- cerned with the relations of equality. The relation of equality gives rise to the Equation, one of the most important instruments of mathematical investigation. The Equation lies at the basis of mathematical reasoning; it is the key with which we unlock its most hidden principles ; the instrument with which we develop its profoundest truths. The equation is a universal form of thought, and is not restricted to any one branch of mathematics. In its simple form it belongs to arithmetic and geometry, as well as to algebra. The simplest process of arithmetic, one and one are two (1-4-1=2), is really an equation, as much as x*+ax=b. In the higher departments of the subject of arithmetic, the equational form of thought and expression becomes indispensable. Much of the reasoning of arithmetic, which is not formally thus expressed, may be put in the form of the equation. As an example, take the question, " If of a number is 24, what is the number?" The solution of this maybe expressed as follows: Since of the number = 24, of the number = 12, and 4 of the number=36. Here, "the number" is the un- o known quantity, which is ascertained by comparing it with the known quantity, 24 ; and then, by the analysis, passing from two-thirds of the number to once the number. The illustra- tion given is of a very simple case, but the same principle 13 ( 193 ) 194 THE PHILOSOPHY OF ARITHMETIC. holds in the most complicated processes of arithmetical analy- sis. If, instead of a number, we had value, cost, weight, labor, etc., the method of comparison and analysis would be the same. We can thus see the use of the equation, the great instrument of analysis, even in the elementary processes of arithmetic. Here it begins that wondrous career which ends in the deepest analysis and the broadest generalization. Here we find the germ of that power which, in its higher develop- ment, comprehends the whole science of Mechanics in a single formula, thus holding, potentially, in its mighty grasp, the mathematical laws of the universe. The equation in arithmetic assumes several different forms. We begin by comparing quantities the comparison of equal quantities giving an equation. A comparison of unequal quan- tities gives us ratio, and a comparison of equal ratios gives us another kind of equation, an equation of relations, usually called a proportion. The proportion 4 : 2 : : 6 : 3, is in reality an equa- tion, as much so as 2=2, for it really means 4-i-2=6-7-3. The treatment of the equation gives rise to several special forms of logical procedure, such as transposition, elimination, etc. The equation, I have said, belongs to arithmetic ; and this thought I desire to impress. The equation is a formal compar- ison of two equal quantities. This comparison is being made continually; all of our reasoning involves it; we cannot think without it ; hence, the equation must enter into the reasoning of arithmetic. We compare one thing with another, the known with the unknown, and thus attain to new truths ; and all such forms of comparison involve the equation, and are only possible by means of it. The simplest arithmetical pro- cess, 1 + 1 = 2, is as much an equation as Du=6u-\-du, though the latter may express one of the profoundest generalizations to which the human mind has attained. Substitution. A prominent element of arithmetical reason- ing, accompanying the equation, is substitution. By this we mean the using of one quantity in place of another, to which THE EQUATION IN ARITHMETIC. 195 it is equal. The object of this is that if we have an expression consisting of a combination of several different quantities, and know the relation of these quantities, we may so substitute their values that the expression for the combination may be obtained in terms of one single quantity, the value of which may much more readily be determined ; and then the values of the other quantities, from their relation to this quantity, may also be found. To illustrate, suppose we have the two conditions, twice a number plus three times another number equals 48, and three times this second number equals four times the first. We can readily solve this by substituting for one of these numbers its value in terms of the other, thus obtaining a number of times a single quantity, equal to the known quantity 48. The operation maybe exhibited thus: 2 times the first number + 3 times the second = 48 ; but, 3 times the second number = 4 times the first number; hence, 2 times the first number -f 4 times the first number = 48 ; or, 6 times the first number = 48, and, once the first number = 8, and from this we may easily find the second number. Substitution is a form of deductive reasoning, as may be seen by an analysis of the process. Take the simple example, A-fB=24, and B=3A. We usually reason as follows: If A+B = 24, and B 3A, then A+3A=24, or 4A = 24, eta That the logical character of the process may appear, we should reason thus: If B=3A, A-j-B will equal A-f3A, from the axiom, " If equals be added to equals the sums will be equal." And since A+B=24, and A+B A + 3A, A+3A must equal 24, from the axiom, " Things that are equal to the same thing are equal to each other." Substitution is thus seen to be strictly a deductive process. In practice these log- ical steps are omitted for brevity and conciseness, the argument being sufficiently clear to be readily understood. Substitution is almost an essential accompaniment of the 196 THE PHILOSOPHY OF ARITHMETIC. equation. The comparison of two equal quantities without some other truth, would often be of little value in attaining new truth. By substituting one value for another, we can often so change the equation that it expresses a relation which will immediately lead to some new relation of the known to the unknown, by which we can attain to the value of the un- known. Substitution has been supposed to be restricted to algebraic reasoning; but this is not correct. It is extensively employed in geometrical reasoning, and is just as appropriate in arithmetic as in algebra. Transposition. In the equational form of thought, so con- stantly recurring in arithmetic, it sometimes occurs that we have a multiple of a quantity compared with another multiple of the same quantity, increased or diminished by some other quantity. In such cases it is natural to desire to unite these two multiples into one, which is done by so changing them as to bring them on the same side of the equation. This is what is known as transposition. It is consequently seen that trans- position is a process not foreign to arithmetic, but one entirely legitimate and natural in the comparison of arithmetical ideas. Other processes of thought analogous to those which occur in algebra are employed in arithmetical reasoning. The mind here takes the first step in equational thought, which, when generalized, leads it to the high altitudes of mathematical sci- ence. Here it plumes its wings to follow the master minds in their lofty flights in a region of thought far beyond that of which the mere arithmetician could even dream. The object of this chapter is not to give a philosophical discussion of the equation in general, but to show that it has a place even in arithmetical reasoning, which has sometimes been doubted or denied. CHAPTER VI. INDUCTION IN ARITHMETIC. "Tl MATHEMATICS is a deductive science, and all of its -1YJL truths, not axiomatic, may be derived by a deductive pro- cess of reasoning. Is it possible, however, to obtain any of these truths by Induction ? This is a disputed question; it will therefore, it is thought, be of interest to enter somewhat into details in its discussion. I believe it can be shown that there are many truths in mathematics that can be proved by induction ; and, furthermore, that many of its truths were 'originally obtained by an inductive process ; and still further, that induction is, in many cases, a legitimate method of math- ematical investigation. Induction, as is generally known, is a process of thought from particular facts and truths to general ones. It is the logical process of inferring a general truth from particular facts or truths. Thus, if I observe that heat will expand the sev- eral metals, iron, tin, zinc, lead, etc., I may infer, since these are representatives of the class of metals, that heat will ex- pand all metals. It is thus seen to be a process of reasoning, based upon the principle that what is true of the individuals is true of the class. The basis of Induction is the general proposition that what is true of the many is true of the wiiole; or, as Esser states it, " What belongs or does not belong to many things of the same kind belongs or does not belong to all things of the same kind." That this method of reasoning can be employed in arithme- (197) 198 THE PHILOSOPHY OF ARITHMETIC. tic appears evident a priori. It is certainly not unreasonable to suppose that we may, upon finding a truth which holds in several particular cases in arithmetic, infer that it will hold good in all similar cases. This conclusion is strengthened by the fact that arithmetic is somewhat special in its nature, par- ticularly so as compared with algebra. Its symbols represent special numbers, and dealing thus with special symbols, it is to be expected that we would discover some truths which hold in particular instances, before we know of their general applica- tion. That it is not only possible to reason inductively in arithmetic, but that we do reason thus, may be shown by act- ual examples. First, take the property of the divisibility of numbers by nine. Suppose that, not knowing this property, I divide a number by 9, and then divide the sum of the digits by 9, and thus see that both remainders are the same. Suppose I should try this with several different numbers, and seeing that it holds good in each case, infer that it is true in all cases ; should I not have entire faith in my conclusion, and would not this inference be well founded ? This is an inductive inference, and is as legitimate as the inference that heat expands all metals, because we see that it expands the several particular metals, iron, zinc, tin, etc. Second, take a number of two digits, as 37 ; invert the digits, and take the difference between the two numbers, and we have 73 37 equal to 36, in which the sum of the two digits, 3 and 6, equals 9. If we take several other numbers of two digits and do the same, we shall find the sum of the two digits to be also 9 ; and observing that this is true in several cases, we may infer that it is true in all cases, in which we again have a true inductive inference. Third, take a proportion in arithmetic, and, by actual mul- tiplication, we shall see that the product of the means equals the product of the extremes. Examining several proportions, we shall see that the same is true in each case, and from these INDUCTION IN ARITHMETIC. 199 we can infer that it is true in all cases, in which we again arrive at a general truth by induction. This is not only legit- imate inference, but it is actually the way in which pupils naturally derive the truth before they understand how to demonstrate it. Now, of course each of the above principles will admit of rigorous demonstration by deduction; what I hold, and what I think is clearly shown, is, that they can also be derived by induction. Deduction would prove that they must be so ; in- duction merely shows that they are so. Many other examples from arithmetic might be given in illustration of the same thing. But the use of induction in mathematics is not con- fined to arithmetic; if we go to algebra we shall find that the same method of reasoning may be, and indeed is, employed there. The theorem, x n y n is divisible by x y, may be proved by pure induction. Try the several cases x* y 2 , a; 3 y 3 , x 4 y*, etc., and seeing that the division is exact in the several cases, it is entirely legitimate to infer that it will be exact in all similar cases, or that x n y n is divisible by xy. The same thing may be shown in many other cases, but it is needless to multiply examples. Even in geometry the same method may be applied. I knew a young person who, before he studied geometry, derived by trial and induction the fact that there may be a series of right-angled triangles, whose sides are in the proportion of 3, 4, and 5 ; and there is no doubt that the ancients knew that the square of the hypothenuse equaled the sum of the squares on the other two sides, long before Pythagoras demonstrated it. I have said that some of the truths of mathematics were discovered by induction; among these the most prominent, perhaps, is Newton's Binomial Theorem. Newton discovered this theorem by pure induction. He left no demonstration of it, and yet it was considered of so much importance that it was engraved upon his tomb. His first principles of Calculus were somewhat inductive in their origin, as may be seen in his Principia. THE PHILOSOPHY OF ARITHMETIC. The following formula is used for finding the number of primes up to the number x, when a; is a large number : , A log x B ' in which N denotes the number of primes, and A and B are constants to be determined by trial. This formula was derived by a process of induction. It is found to satisfy the tables of prime numbers, but no deductive demonstration of it has yet been given, and it must therefore be regarded as empir- ical. In the theory of numbers we have the following remarkable property: Every number is the sum of one, two, or three triangular numbers; the sum of one, two, three, or four square numbers ; the sum of one, two, three, four, or five pentagonal numbers, and so on. This law, though known to be entirely general, has never been demonstrated except for the triangular and square numbers. It was discovered by Fermat, who intimates, in his notes on Diophantus, that he was in possession of a demonstration of it; which, however, is doubtful, since such mathematicians as Lagrange, Legendre, and Gauss have failed to demonstrate it. The general law is at present accepted on the basis of induction. It is thus clearly seen that many of the truths of mathemat- ics can be derived by induction; that is, by inferring general truths from particular cases. It is not claimed, however, that this changes the nature of the science. I have before said that mathematics is a deductive science; my object has been merely to show the error of those who hold that it is impos- sible to derive any of the truths of mathematics by induction. I have called especial attention to this subject, on account of the obscure and conflicting views which seem to exist con- cerning it. Several authors speak of the inductive methods of treating arithmetic, while others as positively assert that there can be no inductive treatment of the science. The logicians lead us to infer that induction cannot be applied to mathe- INDUCTION IN ARITHMETIC. 201 matics, and not a few of them distinctly assert it. Dr. Whewell says, in speaking of mathematics : " These sciences have .... no process of proof but deduction." Prof. Podd wrote several pamphlets to prove that there can be no such thing as inductive reasoning in arithmetic ; and several of those whom he criticised in these articles, have acknowledged the correctness of his views, and consequently, their own mistakes. These views, I have already shown, are only partially true. Arithmetic is a deductive science ; all of its truths may prob- ably be derived by deduction ; but it is equally true that some of them may also be obtained by induction, as has been shown above ; and also, that some of them are accepted alone on induction, having never been demonstrated. Great care should be exercised, however, in the use of induction in mathematics. Several supposed truths which were derived by induction were subsequently found to be untrue. Fermat asserted that the formula, 2 m -f 1 is always a prime, when m is taken any term in the series 1, 2, 4, 8, 16, etc., but Euler found that 2 : "-f 1 is a composite number. Lagrange tells us that Euler found by induction the following rule for determining the resolvability of every equation of the form # 2 +At/=B, when B is a prime number: the equation must be possible when B shall have the form, 4An-fr 2 , or 4An+r i .A. This proposition holds good for a large number of cases, and was thought by many mathematicians to be entirely general, but the equation, x 1 t9y 2 =101, Lagrange proves to be an exception to it. The danger of inductive inference in mathematics is also seen in some of the formulas which have been presented for finding prime numbers. Several of these hold good for many terms, and were supposed to be general, but were at last found to be only special. Thus, the formula x l -J-&+41 holds good for forty values of x. The formula x' l +x+ 17 gives seventeen of its first values prime, and 2ar-}-29 gives twenty-nine of its first values prime. 202 THE PHILOSOPHY OF ARITHMETIC. Having shown that mathematics, though a deductive sci- ence, will admit, in some instances, of an inductive treatment, it may be remarked that such treatment is especially adapted to young pupils in the elementary processes of arithmetic. It is difficult for them to draw conclusions from the principles estab- lished by a deductive demonstration ; hence, in some cases, it may be well for them to employ the inductive method. The rules for working fractions may be derived by an inductive inference from the solution of a particular example ; and this method will be much more readily understood than the deriva- tion of them from general principles deductively established. The method is to solve a particular problem by analysis, and then derive a general method by an inductive inference from such analysis. Thus analysis and induction become, as it were, golden keys with which we unlock the complex combina- tions of numbers. It will be well, however, to lead the pupils to the deductive method as soon as possible. Most students will make the transition naturally. The better reasoners among them will themselves rise from this inductive method, being satisfied only with a deductive demonstration; and in this they should be encouraged. They will often see the deductive, or necessary idea, behind the inductive process, and thus pass spontaneously from the particular fact to the general truth. They will some- times discover a truth by trial and inference, that is, by induc- tion, and then learn to demonstrate it deductively ; and it will be a useful exercise for pupils to have some special drill in this manner. They will thus see the relation of the two methods of reasoning, and be impressed with the deductive nature of the science of arithmetic, and the necessary character of its truths. PART II. SYNTHESIS AND ANALYSIS. SECTION I. FUNDAMENTAL OPERATIONS. SECTION EL DERIVATIVE OPERATIONS. SECTION I. FUNDAMENTAL OPERATIONS OF SYNTHESIS AND ANALYSIS I. ADDITION. II. SUBTRACTION. III. MULTIPLICATION IV. DIVISION. CHAPTER I. ADDITION. rp HE fundamental synthetic process of arithmetic is Addi- JL tion. Beginning at the Unit as the primary numerical idea, numbers arise by a process of synthesis. By it we pass from unity to plurality; from the one to the many. This mental process which gives rise to numbers, we naturally extend to the numbers themselves, and thus synthesis becomes the primary operation of arithmetic. This general synthetic process is called Addition. Definition. Addition is the process of finding the sum of two or more numbers. The sum of two or more numbers is a single number which expresses as many units as the several numbers added. The sum is often called the amount. Addition may also be defined as the process of uniting sev- eral numbers into one number which expresses as many units as the several numbers united. This last definition includes both of the previous ones, and avoids the use of the word sum. The former definition is, however, preferred on account of its conciseness and simplicity, and is the one usually adopted by arithmeticians. Principles. The process of addition is performed in accordance with certain necessary laws which are called prin- ciples. The most important of these are the following: I. Only similar numbers can be added. Thus, we cannot find the sum of 4 apples and 5 peaches, for if we unite the numbers we shall have neither 9 apples nor 9 peaches. It has (207) 208 THE PHILOSOPHY OF AKITHMETIC. been claimed, that the sum is & apples and peaches ; in proof of which it is said we speak properly of " 12 knives and forks." meaning 6 knives and 6 forks. Such a combination is, how- ever, popular rather than scientific ; it is not what we mean by a strict use of the word addition. It may also be observed that dissimilar numbers may be brought under the same name and thus become similar, when they can be united in one sum. Thus, 4 sticks and 5 stones may be regarded as so many objects or things, and their sum will be 9 objects or 9 things. So in writing units and tens in the Arabic system ; they cannot be combined directly, but by reducing both to tens or both to units, the addition can be effected. II. The sum is a number similar to the numbers added. This is evidently an axiomatic truth. The sum of 4 cows and 5 cows is 9 cows, and cannot be horses or sheep, or anything besides cows. An apparent exception which will be under- stood by what is said above is, that the sum of 3 horses and 5 cows is 8 animals. III. The sum is the same in whatever order the numbers are added. This is evident from the consideration that in any case we have the combination of the same number of units, and consequently the same sum. Cases. Addition is divided philosophically into two gen- eral cases. The first case consists in finding the sums of numbers independently of the notation used to express them. The second case consists in finding the sum of numbers as expressed in written characters, and thus grows out of the use of the Arabic system of notation. The former deals with small numbers which can be united mentally, and may be called mental addition ; the latter is used with large numbers as expressed with written characters, and may be called written addition. The former is a process of pure arithmetic; the latter is incidental to the system of notation which may be employed, and is not essential to number in itself considered ADDITION. 209 The former method is an independent process, complete in itself; the latter is dependent upon the former for the elements with which it works. By the former case we obtain what we may call the primary sums of addition, or what is generally known as the Addition Table, which we make use of in adding large numbers expressed by the Arabic method of notation. Treatment. The primary synthetic arithmetical process is that of increasing by units. This process is presented in the genesis of numbers where, by counting, we pass from one num- ber to another immediately following it, by the addition of a unit; and it also lies at the foundation of the method by which we find the sum of any two or more numbers. By it we obtain the elementary sums of the first case, and then we use these sums in solving the problems of the second case. The method of treating both of these cases will be presented somewhat in detail. CASE I. To find the primary sums of arithmetic. The primary sums of arithmetic are found by the same process of counting by which our ideas of numbers are generated. The sum of two numbers is primarily determined by beginning at one number and counting forward from it as many units as are in the number to be added to it. Thus, to find the sum of any two numbers, asjfrue and four, we begin o-tftve and count four successive numbers, six, seven, eight, nine, and seeing we reach nine, we know that^iue and four are nine. In this way we obtain the sums of all small numbers, and then commit them to memory, that we may know them when we wish to use them without passing through the steps by which they were obtained. To be assured that this is the real method, we have but to watch young children when adding, and we shall see that they do actually find the sums of numbers in the manner explained. They may often be seen counting their fingers, or marks on the slate, in performing addition. The elementary sums thus found are the basis of addition. We fix them in the memory 14 210 THE PHILOSOPHY OF ARITHMETIC. as we do the elementary products of the multiplication table, and employ them in finding the sums of larger numbers. These primary sums may be regarded as the axioms of addition. They are intuitive truths, that is, truths which can- not be demonstrated, but are seen by intuition. "Why is it," says Whewell, "that three and two are equal to four and one? Because if we look at five things of any kind, we see that it is so. The Jive are four and one; they are also three and two The truth of our assertion is involved in our being able to conceive the number five at all. We perceive this truth by intuition, for we cannot see, or imagine we see, five things, without perceiving also that the assertion above stated is true." CASE II. To add numbers expressed by the Arabic system of notation. The principle by which we find the sum of larger numbers expressed by the Arabic system, is that of adding by parts. Having learned the sums of small numbers, we separate larger numbers into parts corresponding to these small numbers, and then find the sum of these parts which, united, will give the entire sum. Thus in practice we first add the units group, then the tens group, and thus continue until all the groups are added. If the sum of any group amounts to more than nine units of that group, we incorporate the tens term of the sum with the sum of the next higher group. Solution Thus, in adding the two numbers 368 and 579, are write the numbers so that similar terms stand in the same column, and begin at the OPERATION. right to add. 9 units and 8 units are 17 units, or 1 ten and 7 units; we write the 7 units, and q A * add the 1 ten to the sum of the next column. 7 tens and G tens are 13 tens, and 1 ten are 14 tens, or 1 hundred and 4 tens; we write the 4 tens, and add the 1 hundred to the next column. 5 hundreds and 3 hundreds are 8 hundreds, and 1 hundred are 9 hundreds, which we write in hundreds place The entire sum is therefore 947. ADDITION. 211 This method of adding by parts is the result of the beautiful system of Arabic notation, whereby figures in different positions express groups of different value. It is peculiar to this method of expressing numbers, and illustrates its great convenience and utility. In adding large numbers, it would be exceedingly dif- ficult, if not impossible, for the mind to unite them directly into one sum ; but by adding the groups separately, the process is simple and easy. Rule. One of the most common errors of arithmetic is found in the statement of the rules of the fundamental operations. This error consists in confounding the meaning of the words figure and number. Thus, it is usual to speak of "adding the figures," of "carrying the left-hand figure to the next column,'' etc. This is a mistake involving a looseness of thought that ought not to be permitted to remain in the text-books. We cannot add figures, we can add only the numbers which they express. This error can be avoided in several ways. The method here suggested is the use of the word term for figure. The word term is already employed in a similar manner in algebra. It may be used in a dual sense, embracing both the figure and the number expressed by the figure. Numbers and figures have a definite signification, and one cannot be used for the other without a mistake ; but it will be both correct and con- venient to use one word for both. No ambiguity will be occa- sioned by it, as the particular meaning may be determined by the application. In this way we may avoid the error of speak- ing of "adding figures," and also the inconvenient expression sometimes employed of "adding the numbers denoted by the figures." Why do we write the numbers as suggested, and why do we begin at the right hand to add, are questions very fre- quently asked of the arithmetician. In adding numbers we write them one under another, so that figures of the same order stand in the same vertical column, for convenience ID 212 THE PHILOSOPHY OF ARITHMETIC. adding. We begin at the right hand to add as a matter of convenience also, so that when the sum of any column exceeds nine units of that column, we may unite the number denoted by the left hand term to the next column. We can also add by beginning at the left, but it will be seen on trial to be much less convenient. We commence at the bottom of a column to add as a matter of custom ; in practice it is sometimes more con- venient to begin at the bottom and at other times at the top. Were the scale any other than the decimal, the principle and method of adding would be the same. In addition of denom- inate numbers, where the scales are irregular, the same general principle is employed. We find the sum of a lower order of units, reduce this to the next higher order, etc. The difference in practice is that, with the decimal scale, the reduction is evi- dent from the notation, while in the irregular scales we must divide to make the reduction. The general principle of thought in the two cases is, however, identical. CHAPTER II. SUBTRACTION. rpHE fundamental analytical process of arithmetic is Sub- JL traction. This process arises from the reversing of the fundamental synthetic process. The primary operation of arithmetic, as previously seen, is synthesis. Every synthesis implies a corresponding analysis; hence, the second operation of arithmetic, as a logical consequence, must be the oppo- site of the primary synthetic process. In the former case we united numbers to find a sum ; here we separate numbers to find a difference. This general analytic process has received the name of Subtraction. Definition. Subtraction is the process of finding the differ- ence between two numbers. The difference between two num- bers is a number which added to the less will give a sum equal to the greater. The greater number is called the Minuend,' the less number is called the Subtrahend. Subtraction may also be defined as the process of finding how much greater one number is than another; or, as the process of finding a num- ber which, added to the smaller of two numbers, will equal the greater. The definition first presented is, however, pre- ferred. Gases. Subtraction is philosophically divided into two general cases, like addition. The first case consists in finding the difference between two numbers, independent of the nota- tion used to express them. The second case consists in find- ing the difference between numbers as expressed in written (213) 214 THE PHILOSOPHY OF ARITHMETIC. characters, and thus grows out of the use of the Arabic nota don. The first is a case of pure arithmetic, independent of any notation ; the latter is incidental to the notation adopted to express numbers. The former deals with small numbers, and the process being wholly in the mind maybe called. Mental Subtraction; the latter is employed in subtracting large num- bers expressed with written characters, and may be called Written Subtraction. The former is an independent process complete in itself; the latter has its origin in the Arabic system of notation, and is dependent upon the former for its elementary differences. In the ordinary text-books, the second case is usually divided into two separate cases, depending upon the size of the terms in the minuend and subtrahend; but such division is designed to simplify the subject in instruction, and is, therefore, a practical rather than a logical division df the -abject. Principles. The operations in subtraction depend upon some general laws called principles. The most important of the fun- lamental principles of subtraction are the following: 1. Similar numbers only can be subtracted. Thus, we can- jot find the difference between 9 apples and 4 peaches, for if we take the difference between the numbers 9 and 4, which is 5, it will be neither 5 apples nor 5 peaches. Suppose, how- ever, that we have 9 apples and peaches, consisting of 5 apples and 4 peaches; can we then subtract 4 peaches, and will not the remainder be 5 apples? Or suppose we have a collection of knives and forks consisting of half a dozen of each, which are sometimes spoken of as "12 knives and forks;" can we not take away 6 forks and leave remaining 6 knives ? In reply, we remark that such a " taking away" is not what we mean by subtraction, which is defined as the process of finding the difference of two numbers. It is also manifest, as in addition, that if we regard the dis- similar numbers as having the same generic name, they will then become similar and we can subtract them. Thus, 9 SUBTRACTION. 216 apples and 4 peaches may be regarded as 9 objects and 4 objects, the difference of which is 5 objects. So in subtracting the different orders of units in the Arabic scale, we cannot sub- tract them directly as different orders, but by reducing them to the same denomination, the subtraction is readily performed. 2. The difference is a number similar to the minuend and subtrahend. This is a necessary truth intuitively apprehended. Thus 4 men subtracted from d men, leaves 5 men, and not 5 girls, or 5 women. If we have a group consisting of 9 persons, 5 men and 4 women, and take away 4 women, there will remain 5 men ; hence we might infer that 4 women taken from 9 persons leaves 5 men ; but this is not a universal truth; neither, as stated above, is such a taking away, what we mean by subtraction. 3. If the minuend and subtrahend be equally increased or diminished, the remainder will be the same. This is in- cluded in the axiom that the difference between two numbers equals the difference between them when equally increased or diminished. The truth of such a proposition is seen to be necessary as soon as the proposition is clearly apprehended by the mind. 4. The minuend equals the sum of the subtrahend and remainder; the subtrahend equals the difference between the minuend and remainder. These two principles flow from the conception of subtraction, and the relation of the several terms to one another. Given a clear idea of the process of subtraction, and the relation of the three terms in the process, and these truths immediately follow. Method. The two cases of subtraction, as of addition, require distinct methods of treatment. In the former case wo subtract directly as wholes, finding the difference by reversing the process of addition. In the latter case we subtract by parts, using the elementary differences to find the differences of the corresponding parts. An explanation of both cases will be presented. 216 THE PHILOSOPHY OF ARITHMETIC. CASE I. To find the primary differences in arithmetic. The elementary differences are obtained by a reversion of the process of finding the elementary sums. This may be done in two distinct ways. First, we may find the difference between two numbers by counting off from the larger number as many units as are contained in the smaller number. Thus, if we wish to subtract four from nine, we may begin at nine and count backward four units: thus, eight, seven, six, five ; and finding that we reach five, we know that four from nine leaves five. This is the reverse of the process by which we obtained the elementary sums in addition. In one case we count on for the sum ; in the other we count off for the differ- ence. The other method consists in finding the elementary differ- ences by deriving them by inference from the elementary sums. Thus, in finding the difference between five and nine, we may proceed as follows: since four added to five equals nine, nine diminished by five, equals four. This process, put in a formal manner, is as follows : The difference between two numbers is a number which, added to the less, will equal the greater; but, four added to five, the less, equals nine, the greater ; hence, four is the difference between nine and five. In other words, we know that five from nine leaves four, because four added to five equals nine. The difference between these two methods is radical. By the former method we derive the difference by direct intuition, as we obtained the sums in addition. We see that the differ- ence is five. By the second method we infer that the differ- ence \sfive, without directly seeing it. The latter is a process of reasoning, and will admit of being reduced to the form of a syllogism, as is shown above. The point made here is an important one, and will throw some light on the nature of the science of arithmetic, which, by the metaphysicians, has been somewhat imperfectly understood. The second method is preferred in practice to the first, as we SUBTRACTION. 217 can make use of the elementary sums in finding the elementary differences. If the first method is used, it will be necessary to commit the elementary differences as well as the elementary sums. By making the differences depend upon the sums, this labor will be avoided. CASE II. To subtract numbers expressed by the Arabic scale of notation. With large numbers we cannot subtract the one directly from the other as with small numbers ; we there- fore divide the labor, subtracting by parts; that is, we find the difference between the corresponding groups of each term. By this means the labor of subtracting is greatly facilitated, so that with large numbers, which it would be almost, if not quite impossible otherwise to subtract, the operation becomes simple and easy. In the subtraction of numbers expressed in the Arabic scale of notation, two distinct cases arise; first, when the number of each group of the subtrahend does not exceed the correspond- ing number of the minuend ; second, when the number of a group in the subtrahend exceeds the corresponding number in the minuend. In the first case we readily subtract each group in the subtrahend from the corresponding group in the minu- end. In the second case a difficulty arises, for which we have two distinct methods of explanation, called respectively the Method by Borrowing, and the Method by Adding Ten. To illustrate these methods, suppose it be required to sub tract 526 from 874. First Method. Having the numbers writ- OPERATION. ten as in the margin, we commence at tho right to subtract, and reason thus: we cannot take 6 units from 4 units, we will therefore take 1 ten from the 7 tens, and add it to the four units, which will give 14 units. We then subtract 6 units from 14 units, which gives 8 units. We then subtract 2 tens from the 6 tens which remain after taking away the 1 ten, which leaves 4 tens. We also subtract 5 hundreds from 8 hundreds, leaving 3 hundreds; hence the difference is 348. 218 THE PHILOSOPHY OF ARITHMETIC. Second Method. By the second method we reason thus: "We cannot subtract 6 units from 4 units, hence we add 10 to the 4, making 14 units, and then say, G units from 14 units leave 8 units. Now, since \ve have added 10 to the minuend, that the remainder may be correct we must add one ten to the subtrahend ; hence we have 3 tens from 7 tens leave 4 tens, and also as before, 5 hundreds from 8 hundreds, 3 hundreds This solution is founded upon the principle that the difference between two numbers equals the difference between the two numbers equally increased. The first method seems preferable on account of its simpli- city of thought, as it merely changes the form of the minuend. Pupils see the reason of the process by this method more readily than by the method of adding ten. The second method, however, is preferred by some teachers for at least two reasons. First, it is the method generally used in practice; nearly all persons increasing the next lower term after " borrowing," instead of diminishing the upper one. Second, it is, in many cases which arise, much more convenient than the other method, as in subtracting 12345 from 20000. By the second method, the solution of this problem will be much simpler than by the first. Another Method. There is still another method of subtract- ing, which, if not of any practical value, is at least of sufficient interest to be worthy of mention. It 74682 *7 Q C *\ consists in subtracting the terms of the subtrahend from 10, and adding the difference to the corrcs- 46817 ponding terms of the minuend. Thus, in subtracting 27865 from 74682, we say 5 from 10 leaves 5, and 2 are 7; G and 1 to carry arc 7, and 7 from 10 leaves 3, and 8 arc 11 ; set down the 1; 8 from 10 leaves 2, and G arc 8; 7 and 1 to carry arc 8, and 8 from 10 leaves 2, and 4 arc 6, etc. Rule. In the rule for subtraction, arithmeticians make the same mistake as in the rule for addition. Thus, they say, " Subtract each figure of the subtrahend from the figure above SUBTRACTION. 219 it in the minuend," or " take each figure of the subtrahend from the figure above it," or, "if a figure in the lower number is larger than the one above it," etc. These errors are almost inexcusable. We cannot subtract figures, we subtract num- bers. If we "take one figure from another" the other figure will be left, not the difference of the numbers expressed by them. A figure is larger or smaller according to the kind of type in which it is printed. The figure two may be large (2) or small (2). One figure may be larger than another, and express a smaller number; as, 3 and 8. This error may be avoided by the use of the word term for the number expressed by the figure. The rule will then read, "Begin at the right and take each term of the subtrahend from the corresponding term of the minuend," etc. "If a term of the subtrahend is greater than the corresponding term of the minuend," etc. Remarks. We write terms of the same order in the same vertical column for convenience in subtracting, since only num- bers of the same group can be subtracted. We commence at the right, so that when a term of the subtrahend expresses more units than the corresponding term of the minuend, we may take it from the next higher group of the minuend ; or, if we use the other method of subtracting, that we may add 10 of a group to the minuend, and 1 of the next higher group to the subtrahend; in other words, we commence at the right as a matter of convenience, as will be seen in the attempt to sub- tract by commencing at the left. The taking one from the next term of the minuend is called "borrowing," and the adding one to the next term of the sub- trahend is called "carrying." The accuracy of these words has been questioned. To borrow is to obtain that which we expect to return to the one from whom we borrow. It does not seem much like " borrowing" to take from one thing and return what we take to another. It is something like "robbing Peter to pay Paul." In regard to the term "carrying," it 220 THE PHILOSOPHY OF ARITHMETIC. may be asked in what it is carried ; though we may answer, as the boy did, " we carry in the head." Notwithstanding these objections, the terms borrowing and carrying have been sanctioned by good usage ; and, since custom is the lawgiver in language, we may accept them as correct. Their use is a matter of convenience, also, as they indicate operations for which we have no other technical terms. It may be remarked that it required many years for the people of Europe to become familiar with the processes of borrowing and carrying. In a work on arithmetic by Bernard Lamy, published at Amster- dam in 1692, the author states that a friend sends him the mode of using the carriage in subtraction, he having previ- ously borrowed from the upper line ; and this is presented as a novelty. CHAPTER III. MULTIPLICATION. THE general process of synthesis is Addition. Having become familiar with this general synthetic process in ac- cordance with the law of thought, from the universal to the particular, we begin to impose certain conditions upon it. The numbers primarily united were of any relative value; if, now, we impose the condition that the numbers united shall be all equal, with the new idea of the times the number is used, we have a new process of synthesis, which we call Multiplica- tion. Multiplication is thus seen to be a special case of addition, in which the numbers added are all equal. The idea of mul- tiplication is contained in addition, and is an outgrowth of it. They are both synthetic processes one being a general, and the other a more special synthesis. Multiplication, however, involves the idea of " times," which does not appear in addi- tion. This notion of "times," originating in multiplication, is one of the most important in mathematics, and is itself the source of a large portion of the science. Thus, in involution there is no apparent trace of the idea of addition, and the same is true in respect of other processes. If, however, we follow these processes back far enough, we shall find they have their origin in the primary process of addition. Even involution may be performed by successive additions. Definition. Multiplication is the process of finding the product of two numbers. The Product of two numbers is (221) 222 THE PHILOSOPHY OF ARITHMETIC. the result obtained by taking one number as many times as there are units in the other. The number multiplied is called the Multiplicand. The number by which we multiply is called the Multiplier. This definition of multiplication, introducing the word Pro- duct, makes it similar to the definitions of addition and sub- traction, in which the terms sum and difference are used. Defining Division in a similar manner by using the word Quo- tient, we shall have a harmony in the definitions of the four fundamental rules, which has not hitherto existed. I have adopted this method in my Higher Arithmetic, and shall intro- duce it into my other mathematical works. Multiplication is usually defined as the process of taking one number as many times as there are units in another. This definition is not entirely satisfactory. It says nothing about finding a result, which is specified in the definitions of addition and subtraction, and which seems to be necessary also here. To supply this omission, I have previously defined multiplica- tion as the process of finding the result of taking one number as many times as there are units in another. After a very careful consideration of the subject, however, I have concluded to adopt the method of defining multiplication as the process of finding the product, thus securing a uniformity in the defini- tions of the fundamental operations. Principles. The operations of multiplication are founded upon certain necessary truths called principles. The most important of the principles of multiplication are those which follow : 1. The multiplier is always an abstract number. For, the multiplier shows the number of times the multiplicand is taken, and hence must be abstract, since we cannot take any- thing yards times or bushels times, etc. From this it follows that such problems as "Multiply 25 cts. by 25 cts.," or "2s. 6d. by itself" are impossible and absurd. In finding areas and volumes, we speak of multiplying feet by feet for square feet, MULTIPLICATION. 223 square feet by feet for cubic feet, etc. It should be remem- bered, however, that this is merely a convenient expression, which does not indicate the actual process. In finding the area of a rectangle, we multiply the number of square feet on the base by the number of such rows; the multiplicand being square feet and the multiplier an abstract number. 2. The product is always similar to the multiplicand. This is manifest from the fact that the product is merely the sum of the multiplicand used as many times as there are units in the multiplier. Thus, 3 times 4 apples are 12 apples, and cannot be 12 pears or peaches. 3. The product of two numbers in the same, whichever is made the multiplier. This may be seen by placing # * # * 3 rows of 4 stars each in the form of a rectangle, * . * * as in the margin. Now these may be regarded * # * * as 3 rows of 4 stars each, or 4 rows of 3 stars each ; hence 3 times 4 is the same as 4 times 3 ; and the same may be shown for any other two numbers. 4. If the multiplicand be multiplied by all the parts of the multiplier, the sum of all the partial products will be the true product. This grows out of the general principle that the whole is equal to the combination of all of its parts. It is applied in finding the product of two numbers expressed by the Arabic system. 5. The multiplicand equals the quotient of the product divided by the multiplier ; the multiplier equals the quotient of the product divided by the multiplicand. These two principles are manifest to the mind as soon as it attains a clear idea of the processes of multiplication and division, and the relation of the two to each other. Cases. Multiplication is philosophically divided into two general cases. The first case consists in finding the products of numbers independently of the method of notation used to express them. The second case is that which grows out of the use of the Arabic system of notation. The former deals with 224 THE PHILOSOPHY OF ARITHMETIC. small numbers mentally, and may be called Mental Multiplier tion; the latter deals with large numbers, expressed by means of written characters, and may be called Written Multiplica- tion. The former is an independent process complete in itself, and belongs to pure number; the latter has its origin in the Arabic system, and is dependent upon the former for its ele- mentary products. Method. The general method is to find the product of small numbers by addition, and then use these in the multiplication of large numbers. The first case is thus made to depend upon addition, and the second case upon the first case. Both cases will be formally presented. CASE I. To find the elementary products of arithmetic. The first object in multiplication is to find the elementary pro- ducts. By the elementary products are meant the products of small numbers which, arranged together, constitute what is called the Multiplication Table. These elementary products are derived by addition. Thus, we ascertain that four times five are twenty, by finding, by actual addition, that the sum of four fives is twenty. In this manner all the elementary products of the table were originally obtained. This table is committed to memory in order to save labor and facilitate the process of calculation. We are thus able to tell immediately the product of two small numbers, which otherwise we should be obliged to obtain by an actual addition. The elementary products are not derived by intuition, and are therefore not axioms; they are the result of a process of reasoning. Thus, in order to find the product of three times four, we may reason as follows: Three times four is equal to the sum of three fours; but the sum of three fours, we find by addition, is twelve ; hence, three times four is twelve. This is as valid a syllogism as "A is equal to B ; but B is equal to C ; hence, A is equal to C." The extent of the table, for all practical purposes, is limited by "nine times nine." That is, with our Arabic system of MULTIPLICATION. 225 notation and the decimal method of numeration, it is not neces- sary that the elementary products should extend beyond " nine times." It is not at all inconvenient, however, but quite nat- ural that it should include eleven and twelve times, since the names eleven and twelve are a seeming departure from the dec- imal system of numeration. CASE II. To multiply numbers expressed by the Arabic system of notation. When the numbers are small, as we have seen, we multiply them directly as wholes; when we extend beyond the elementary products, the principle is to multiply by parts. Thus, instead of multiplying the multiplicand as a single number, we multiply first one group, then the next group, and so on, as we united numbers in addition. Also, when the multiplier exceeds nine or in practice, twelve that is, when it is expressed in two or more places, we multiply first by the units term, then by the tens term, etc.; and then take the sum of these partial products. To illustrate, let it be required to multiply 65 by 37. To multiply by thirty-seven as a single number, would be quite a difficult task. We do not attempt this, however, but first mul- tiply by 7 units, one part of 37, and then by 3 tens, the other part of 37, and then take the sum of these products. It is also seen that the number 65 is not multiplied as a single number, but by using its parts, 5 units and 6 tens. The method of explaining the process is as follows: Solution. Thirty-seven times 65 equals 7 OPERATION. times 65 plus 3 tens times 65. Seven times 5 65 units are 35 units, or 3 tens and 5 units ; we 37 write the 5 units, and reserve the 3 tens to add 455 to the product of tens. Seven times 6 tens are 42 tens, which, increased by 3 tens, equals 45 tens, or 5 tens and 4 hundreds, which we write in ita proper place. Multiplying similarly by 3 tens, we have 5 tens 9 hundreds and 1 thousand; and taking the sum of these two partial products, we have 2405. 15 226 THE PHILOSOPHY OF ARITHMETIC. This method of multiplication is founded upon, and is only possible with a system of notation similar to the Arabic. Without some such method of expressing numbers in char- acters, the multiplication of large numbers would be exceed- ingly laborious, if not altogether impossible. We are thus continually reminded of the advantages of the Arabic system of notation, and learn almost to venerate the people and country that conferred so great a boon upon the human race by its invention. Rule. The error of confounding the meaning of figure and number is repeated in the rule for multiplication. The rule, as usually given is, "Multiply each figure of the multiplicand by the multiplier," etc., or "Multiply the multiplicand by each figure of the multiplier," etc. This error is easily avoided by the use of the word term for figure. It should be remembered that we have two distinct things, the number and the numerical expression. The parts of the numerical expression are figures ; the parts of the entire number are numbers. The word term may be employed to express both of these, without any obscurity and with much convenience. The rule will then read, " Multiply each term of the multiplicand by the multiplier," etc., or, "by each term of the multiplier," etc. Remark. We write the numbers as indicated above for con- venience in multiplying. The placing of the multiplier under the multiplicand, instead of over it, and multiplying from below, is a mere matter of custom, corresponding with the method of adding and subtracting. We begin at the right hand to multiply so that when any product exceeds nine, we may incorporate the number expressed by the left hand figure with the following product. The convenience of this will be readily appreciated by performing the multiplication by begin- ning at the left. It was formerly the custom, however, to begin at the left, writing the partial products in their order and subsequently Collecting them. CHAPTER IV DIVISION. general process of analysis is Subtraction. After the JL mind becomes familiar with this general process, it begins to extend and specialize it, and thus arises a new process called Division. Division is, therefore, a special case of subtraction, in which the same number is to be successively subtracted with the object of finding how many times it is contained. The idea of Division is thus seen to be contained in that of Subtraction, and is the outgrowth of it. Division may also be regarded as arising from a reversing of the process of multiplication. In multiplication, we obtain the product of two numbers ; and since the product is a number of times the multiplicand, we may regard it as containing the multiplicand a number of times. Thus, since four times five are twenty, twenty may be considered as containing five, four times. Division is thus regarded as an analytic process, arising from reversing the synthetic process of multi- plication. It thus appears that Division may have originated in either of two different ways. In which way it did actually arise, it is impossible for us to decide with certainty. It has generally been supposed, judging from the old definition that " Division is a concise method of Subtraction," that it had its genesis in Subtraction. My own opinion, however, is that it originated by reversing multiplication, for which I state the following reasons : First, as subtraction arose from reversing the pro- (227) 228 THE PHILOSOPHY OP ARITHMETIC. cess of addition, so is it natural to suppose that division, a concise subtraction, would arise from reversing multiplica- tion, a concise addition. Second, division involves, as essen- tial to it, the idea of "times," which had already appeared in multiplication. It seems much more natural to take the idea of times from multiplication, where it already existed, than to originate it from the process of subtraction. Definition. Division is the process of finding the quotient of two numbers. The quotient of two numbers is the number of times that one number contains the other. The number divided is the Dividend ; the number we divide by is the Divisor. The definition usually given is, "Division is the process of finding how many times one number is contained in another." This is regarded as correct, but is less simple and concise than the one above suggested. Defining division in this manner, we have a simple and con- cise definition, easily understood and logically accurate. It follows the method generally adopted for addition and sub- traction, and which I have also suggested for multiplication ; and presents a happy uniformity in the definitions of the four fundamental operations of arithmetic. The objects of these four fundamental processes, as thus presented, will respectively be to find the Sum, the Difference, the Product, and the Quo- tient of numbers. Principles. The operations in division are controlled by certain necessary laws of thought to which we give the name of principles. The following are the most important of the principles of division: 1. The dividend and divisor are always similar numbers. This is true of division scientifically considered, as may be seen by regarding it as originating in subtraction or multipli- cation. Supposing that it has its root in subtraction, and remembering that in subtraction the two terms must be alike, we see that this principle follows of necessity. Thus, if we inquire how many times one number is contained in another, DIVISION. 229 it is evident that these numbers must be similar. We may inquire how many times 4 apples are contained in 8 apples, but not how many times 4 peaches are contained in 8 apples. Neither can we say "How many times is 4 contained in 8 apples?" for 8 apples will not contain the abstract number 4 any number of times. The same conclusion is reached if we regard division as originating in multiplication. If we assume that 4 is con- tained in 8 apples 2 apples times, it would follow that 2 apples times 4 equals 8 apples, which is absurd. Several recent writers take the position that a concrete number may be divided by an abstract number, because in practice we hus divide a concrete number into equal parts. This is a (subordination of science to practice, which is neither philo- sophical nor necessary. The practical case which they thus try to include in the theory of the subject, admits of a scientific and simple explanation, without any modification of the funda- mental idea of division ; and when thus explained it becomes apparent that the two terms are similar numbers. 2. The quotient is always an abstract number. This results from the fundamental idea of division, whether we regard it as originating in subtraction or multiplication. The quotient shows how many times one number is contained in another, and one number cannot be contained in another number yards times, or apples times, etc., from which it follows that the quotient must be abstract. The quotient shows how many times one number may be subtracted from or taken out of another before exhausting the latter, and must therefore be a number of times, and consequently abstract. Or, regarding it as arising from multiplication, the quotient is the number of times the divisor which equals the dividend ; and, as such, is a multiplier,- and must, consequently, be abstract. Suppose it were said that 2 is contained in 8 apples, "4 apples times," and all authors agree as to the quotient denoting the number of times the divisor is contained in the dividend then it would follow that "4 apples times" 2 are 8 apples; which is, of course, absurd. THE PHILOSOPHY OF ARITHMETIC. 3. The remainder is always similar to the dividend. This is evident, since the remainder is an undivided part of the divi- dend. In practice, as above intimated, some of these princi- ples seem to be violated, but if the analysis be given, it will be seen that the violation is merely seeming, and not actual. 4. The following principles show the relation of the terms in division : 1. The dividend equals the product of the divisor and quo- tient. 2. The divisor equals the quotient of the dividend and quotient. 3. The dividend equals the product of the divisor and quo- tient, plus the remainder. 4. The divisor equals the dividend minus the remainder, divided by the quotient. 5. The following principles show the result of multiplying or dividing the terms in division: 1. Multiplying the dividend or dividing the divisor by any number multiplies the quotient by that number. 2. Dividing the dividend or multiplying the divisor by any number divides the quotient by that number. 3. Multiplying or dividing both divisor and dividend by the same number does not change the quotient. Cases. Division is philosophically divided into two general cases. The first case consists in finding the quotient of num- bers independently of the method of notation used to express them. The second case is that which grows out of the use of the Arabic system of notation. The former case deals with small numbers mentally, and may be called Mental Division ; the latter deals with large numbers, expressed by means of written characters, and may be called Written Division. The former is an independent process, belonging to pure num- ber, and is complete in itself; the latter operates by means of the Arabic characters, and is dependent upon the former for its elementary quotients. DIVISION. 231 Method. In division we first find the elementary quotients corresponding to the elementary products of the multiplicatioji table. These may be obtained in two different ways, as will be explained. In the second case we operate by parts, using the elementary quotients as a basis of operation. The two cases will be formally presented. CASE I. To find the elementary quotients of arithmetic. The first object in division is to find the elementary quotients corresponding to the elementary products of the multiplication table. These quotients admit of a double origin ; that is, they may be derived by the method of concise subtraction, or of reverse multiplication. Thus, if we wish to ascertain how many times Jive is contained in twenty, we may find how many times five can be taken out of twenty by subtraction, and this will show how many times twenty contains five. This is the method of subtraction, and as thus viewed, division may be regarded as a method of concise subtraction. Again, since we know that four times five are twenty, we can immediately infer that twenty contains four fives, or that twenty contains five four times. This is the method of multiplication, and as thus viewed, division may be regarded as a method of reverse multiplication. Either of these two methods may be used for finding the elementary quotients, but the method of reverse multiplication is much more convenient in practice. The quotients are imme- diately derived from the products of the multiplication table, and we are thus saved the labor of forming and committing a table of division. If, however, the elementary quotients be derived by subtraction, it will be necessary to construct a division table, and commit the quotients, as we do the products in multiplication. These elementary quotients, whether derived by multiplica- tion or subtraction, are the result of a process of reasoning. The process of thought may be illustrated in the problem, "Fire, is contained how many times in twenty?' 1 and is as follows: 232 THE PHILOSOPHY OF ARITHMETIC. Five is contained as many times in twenty as twenty is times five ; but twenty is four times five ; hence, five is contained in twenty, four times. In ordinary language, this is abbreviated thus : five is contained four times in twenty, since four times five are twenty. By the method of subtraction we reason thus : five is con- tained as many times in twenty as five can be successively sub- tracted from or taken out of twenty ; but five can be suc- cessively subtracted from twenty, four times; hence, five is contained/bur times in twenty. The ordinary form of thought is, five is contained four times in twenty, since it can be sub- tracted from twenty, four times. By "subtracted from," as here used, we mean subtracted successively from until twenty is exhausted. CASE II. To divide when the numbers are expressed in the Arabic scale of notation. When the numbers are small, we divide them, as we have seen, directly as wholes ; when we extend beyond the elementary quotients, the principle is to divide by parts. The dividend is not immediately divided as a whole, but is regarded as consisting of parts or groups; and these are so divided that, when remainders occur, they may be incorporated with inferior groups, and thus the whole number be divided. This method, as in multiplication, is due to the system of Arabic notation, and enables us to divide large num- bers, which would be exceedingly difficult, if not impossible, with a different system of notation. In. Written Division, or division of large numbers, two cases are presented. First, when the divisor is so small that only the elementary dividends and divisors are used ; second, when the divisors and dividends are larger than those employed in obtaining the elementary quotients. The methods of treat- ing these two cases are distinguished as Short Division and Long Division. In Short Division, the partial dividends are not written ; in Long Division, the partial dividends and other necessary work are written. DIVISION. 233 Illustration. To illustrate the method of Short Division, divide 537 by 3. Here we cannot divide the given number as a whole, that is, as Jive hundred and thirty-seven, but by sep- arating it into parts, we can readily divide these parts, as they give only the elementary quotients. Thus, we first divide Jive hundred, reduce the remainder of the group to tens and incorporate with the tens group, making 23 tens, divide this as before, and thus continue until the whole of the number hafa been divided. When the divisor is greater than 12, the division can no longer be performed by using the elementary dividends and quotients. The process then becomes more difficult, although it involves the same principles as when smaller numbers are used. As the elementary quotients were derived from multipli- cation, so in Long Division we determine the quotient by mul- tiplying. We multiply the divisor by some number which we suppose to be the quotient term, and if the product does not exceed the partial dividend, nor the difference between the product and partial dividend exceed the divisor, we know that we have obtained the correct quotient figure. The method described is so common that it need not be illustrated by a problem. Rule. The mistake of using figure for number is also made in stating the rule for division. One author says, " Find how many times the divisor is contained in the fewest figures on the left of the dividend," etc.; another says, "Take for the first partial dividend the fewest figures of the given dividend," etc.; another says, "Take for the first partial dividend the least number of figures on the left that will contain the divisor," etc. Of course, figures will not contain the divisor; the num- ber expressed by the figures is what is intended, and therefore should be expressed. The error may be corrected by saying, "Divide the number expressed by the fewest figures on the left that will contain the divisor," or, " by the fewest terms," etc. 234 THE PHILOSOPHY OP ARITHMETIC. Remark. We write the divisor at the left of the dividend and the quotient at the right as a matter of custom. Some pre- fer writing the divisor at the right and placing the quotient under the divisor. We begin at the left to divide, so that the remainder, when one occurs, may be united with the number of units of the next lower order, giving a new partial divi- dend. If we attempt to divide by beginning at the right, we will see the advantage of the ordinary method. SECTION II. DERIVATIVE OPERATIONS OF SYNTHESIS AND ANALYSIS. I. INTRODUCTION. II. COMPOSITION. III. FACTORING. IV. COMMON DIVISOB. V. COMMON MULTIPLE. VI. INVOLUTION. VII. EVOLUTION. CHAPTER I. INTRODUCTION TO DERIVATIVE OPERATIONS. THE four Fundamental Operations are the direct and imme- diate outgrowth of the general processes of synthesis and analysis as applied to numbers. They are called Fundamental Operations because all the other operations involve one or more of these, and may be regarded as being based upon them. They are the foundation or basis upon which the others are built up, the germ from which they are evolved, the soil out of which they grow. Several of the processes of arithmetic are so intimately related to the fundamental operations that they may be regarded as directly originating in and growing out of them. Such are the processes of Factoring, Common Multiple, Com- mon Divisor, etc. These processes have their roots in the general notions cf the fundamental operations, and are evolved from them by a modification and extension of the pri- mary analytic and synthetic processes. They are developed by the thought process of comparison, though they have not their basis in comparison, like the processes of Ratio, Propor- tion, etc. Being thus derived from the fundamental operations, they may be called the Derivative Operations of synthesis and analysis. Let us notice the origin and nature of these deriva- tive operations. If two or more numbers are multiplied together, and the result is considered with respect to its elements, we have the idea of a Composite Number. The general process of forming composite numbers may be called Composition. The numbers (237) 238 THE PHILOSOPHY OF ARITHMETIC. synthetized in forming a composite number are called Factors of that number. If we form a composite number consisting of two equal factors, we have a square ; of three equal factors, a cube, etc., and the process is called Involution. If we find a composite number which is a number of times each of several numbers, or is so composed that each of them is one of its factors, it is called a common multiple of these numbers, and the process is known as finding Common Multiples. These processes are distinct from Multiplication, though related to it. They employ multiplication and are the out- growth of the general multiplicative idea, but pass beyond the primary idea of multiplication. In multiplication, the main idea is the operation of repeating one number as many times as there are units in another to obtain a result; here the thought is the result of the operation compared with the numbers multiplied together. In the former case, the process is purely synthetic ; here comparison unites with synthesis, and employs it for a particular object. The operation of multiplying is assumed as a fact, and employed for the purpose of attaining a result bearing some relation to the elements combined. Having obtained composite numbers, and the idea of their being composed of factors, we naturally begin to analyze them into their elements in order to discover these factors. This gives rise to an analytic process, the converse of Composition. The general process of analyzing a number into its factors is called Factoring. If we resolve a number into several equal factors for the purpose of seeing what factor must be repeated two, or three, etc., times to produce the number, we have a process known as Evolution. If we have given several num bers, and proceed to find a common factor of these numbers, we have the process known as Common Divisor. These processes, though related to Division, are clearly dis- tinguished from it. They are an outgrowth of the general idea of division, but extend beyond it. In division it is the operation of finding how manv times one number is Contained INTRODUCTION. 239 in another that is the prominent idea; here the idea is the result considered in relation to the number or numbers operated upon. In Factoring, the process of comparison enters as an important element. Division is a process purely analytical ; Factoring is analysis, and more ; it is analysis plus comparison. It has its root in Analysis, and is developed by the thought-process of Comparison. There are, therefore, two general derivative processes, Com- position and Factoring, each of which embraces corresponding and opposite processes. The terms, Composition and Factor- ing, are in practice restricted to the general processes; the special processes are known by their particular names. We have thus three pairs of derivative processes, Composition and Factoring, Multiples and Divisors, and Involution and Evolution. These will be treated in successive chapters. CHAPTER II. COMPOSITION. /COMPOSITION is the process of forming composite num- \J bers when their factors are given. It is a general process which contains several subordinate and special ones. When fully analyzed, it will be seen to present several interesting cases besides the more particular ones of Involution and Mul- tiples. From the previous analysis it is seen that there is a real case of Synthesis, the converse of the analytic process of Factoring. This new generalization, and the term I have applied to it, will, I trust, receive the approval of mathematicians. Its importance as a logical necessity, is seen in its relation to Factoring. In the fundamental operations each synthetic pro- cess has its corresponding analytic process. Thus, addition is synthetic, subtraction is analytic ; multiplication is synthetic, division is analytic. It follows, therefore, that there should be a synthetic process corresponding to the analytic process of Factoring. This process I have presented under the name of Composition, or the process of forming composite numbers. Cases. There are several interesting and practical cases of Composition, some of the most important of which are the following: I. To form a composite number out of any factors. II. To form a composite number out of equal factors. III. To form a composite number out of factors bearing any definite relation to each other. (240) COMPOSITION. 241 IV. To form composite numbers which have one or more given common factors. V. To form several or all of the composite numbers possible out of given factors. VI. To determine the number of composite numbers that can be formed out of given factors. Method of Treatment. The method of treatment is to com- bine these factors by multiplication in such a manner as to attain the result desired. I will briefly state the manner of treating each case. CASE I. To form a composite number out of any factors In Case I. we find the result by simply taking the product of the factors. Thus the composite number formed from the fac- tors 2, 3, and 4 equals 2x3x4, or 24. CASE II. To form a composite number out of equal fac- tors. Case II. may be solved in the same manner as Case I., or we may multiply a partial result by itself or by another partial result, to obtain the entire result. Thus, if we wish to find the composite number consisting of eight 2's, we may multi- ply 2 by 2, giving 4, then multiply 4 by 4, giving 16, and then multiply 16 by 16, giving 256, the number required. CASE III. To form a composite number out of factors bearing any definite relation to each other. In this case we may have given one factor and the relation of the other factors to it ; we first find the factors and then take their product. Thus, required the number consisting of three factors, the first being 4, the second twice the first, and the third three times the second. Here, we first find the second factor to be 8, and the third to be 24, and then take the product of 4, 8, and 24, which we find to be 768. CASE IV. To form composite numbers which have one or more given common factors. This case maybe solved by tak- ing the given common factor, and multiplying it by any other 'actors we choose. If it is required that the factor given be the largest common factor of the numbers obtained, the mul- tipliers selected must be prime to each other. To illustrate, 16 242 THE PHILOSOPHY OF ARITHMETIC. find three numbers whose largest common factor shall be 12. If we multiply 12 by 2, 4, and 6, we will have 24, 48, and 72, three numbers whose common factor is 12; but since the num- bers used as multipliers have a common factor, 12 is not the largest factor common to these three numbers. To find three numbers having 12 as their largest common factor, we may multiply 12 by 2, 3, and 5, which gives us the numbers 24, 3(5, and 60, in which 12 is the largest common factor. CASE Y. To form several or all of the composite numbers possible out of given factors. In this case we may take the factors two together, three together, etc., until they arc taken all together; or we may multiply 1 and the first factor by 1 and the second factor, the products thus obtained by 1 and the third factor, etc., until all the factors are used. To illustrate, form all the possible composite numbers out of 2, 3, 5, and 7. We first find all the possible pro- ducts taking them two together; OPERATION. o v q _ p q v c _ IK then all the products taking them > three together, and then the products 2x7 = 14 5x7=35 taking them four together, as is 2x3x5=30 shown in the margin. Another 2x3x7=42 method, not quite so simple in a i; 7 IAK thought but more convenient in 2x3x5x 7==10 practice, is as follows: Multiplying 1 and 2 by 1 and 3, will give 1, 2, 3, and all the composite numbers that can be formed out of 2 and 3; these multiplied by 1 and 5 will give 1, 2, 3,5, and all the com- posite num- bers that can be formed out 1 1 2 3 OPERATION. 1 1 2 5 3 G 1 1 2 7 3 5 G 10 15 30 1 2 3 6 G 10 15 30 7 14 21 42 35 70 105 210 of 2, 3, and 5 ; these multiplied by 1 and 7 will give 1, 2, 3, 5, COMPOSITION. 243 T, and all the composite numbers that can be formed out of 2, 3, 5, and 7. Omitting 1, 2, 3, 5, and 7 in the last result, and we have all the composite numbers that can be formed out of 2, 3, 5, and 7. If some of the given factors are alike, we have an interesting modification of this case. Thus, suppose we wish to find the composite numbers which can be composed out of 2, 2, OPERATION. 2, 3, and 3. In this problem [ ? 8 since 2 is used three times 1 2 3 4 G 8 9 12 18 24 36 72 we may make the first series 1, 2, 2-, and 2 3 , or 1, 2, 4, and 8; and since 3 is used twice, the second series will be 1, 3, and 3 2 , or 1, 3, and 9 ; and the products of these, omitting 1, 2, and 3, will be the composite numbers required. CASE VI. To determine the number of composite num- bers that can be formed out of given factors. We may solve this case by increasing the number of times each factor is used by unity, take the product of the results and diminish it by the number of different factors used increased by one. Tho reason for this method may be readily shown. Suppose we wish to find how many composite numbers can be formed with three 2's and two 3's. Here we sec that 2 used three times as a factor gives with 1 a scries of four terms; and 3 used twice as a factor gives with 1 a scries of lliree terms; hence the product will give a series of 4x3 or 12 terms, and omitting the unit and 2 and 3, we have nine terms. The inference from this solution will give the method stated above. CHAPTER IIL FACTORING. is the process of finding the factors of com J- posite numbers. It is the reverse of Composition. In Composition we have given the factors to find the number; in Factoring we have given the number to find the factors. Com- position is a synthetic process ; it proceeds from the parts by multiplication to the whole. Factoring is an analytic process ; 'it proceeds from the whole by division to the parts. A Factor, as now generally presented in arithmetic, is regarded as a divisor of a number, rather than a maker or pro- ducer of the number. This I regard as an error. The origin of the word, facio, I make, indicates its original meaning to be a maker of a composite number. The fact of a Factor of a number being a divisor of it is a derivative idea, re- sulting from the primary conception of its entering into the composition of the number. This primary idea of the office of a Factor is the one that should be primarily presented to pupils, rather than the secondary or derivative idea. We should define according to the fundamental, rather than the derivative office. To do otherwise is to invert the logical relation of ideas, and must, as I have known it, tend to confusion. Thus taught, it is seen that the proposition, a factor of a number is a divi- sor of the number, is an immediate inference, which would have to be inverted if the secondary office of a factor is made the fundamental idea. (244) FACTORING. 245 Cases. Factoring presents several cases analogous to those of Composition. Some of the principal ones are the following, which, it will be noticed, are the correlatives of those givei under Composition. I. To resolve a number into its prime factors. II. To resolve a number into equal factors. III. To resolve a number into factors bearing a certain rela- tion to each other. IV. To find the divisors common to two or more num- bers. V. To find all the factors or divisors of a number. VI. To find the number of divisors of a number. Method. The general method of treatment is to resolve the number or numbers into their prime factors, and then combine these factors when necessary so as to give the required result. The prime factors of a number are found by division, and con- sequently it is convenient to know before trial what numbers are composite and can be factored, and the conditions of their divisibility. Hence, the subject of Factoring gives rise to the investigation of the methods of determining prime and com- posite numbers, and the conditions of the divisibility of com- posite numbers. This subject will be treated under the head of Prime and Composite Numbers. The method of treating each of the above named cases of factoring will be briefly stated. CASE I. To resolve a number into Us prime factors. In Case I. we divide the number by any prime number greater than 1 which will exactly divide it; divide the quotient, if composite, in the same manner; and thus continue until the quotient is prime. The divisors and the last quotient will be the prime factors required. Thus, suppose we have given 105 to find its prime factors. Dividing 105 by the prime factor 3, and 8)105 the quotient 35 by 5, we see that 105 is composed 6)35 of the three factors 3, 5, and 7, and since these are 7 prime numbers, its prime factors are 3, 5, aud 7. 246 THE PHILOSOPHY OF ARITHMETIC. CASE II. To resolve a number into equal factors. In Case II. wq resolve the number into its prime factors and then com- bine by multiplication one from each set of two equal factors, when we wish one of the two equal factors of the number ; one from each set of three equal factors when we wish one of three equal factors, etc. Thus, suppose we wish to find the (2x2x2x three equal factors of 216, or one of its = (3x3x3 three equal factors. We first resolve 2x3=6 216 into its prime factors, finding 216=2x2x2x3x3x3. Since there are three 2's, one of the three equal factors will contain 2; and since there are three 3's, one of the three equal factors will contain 3 ; hence one of the three equal factors is 2 x 3, or 6. CASE III. To resolve a number into factors bearing a cer- tain relation to each other In this case we may divide the given number by the product of the numbers representing the relation of the other factors to the smallest factor, then resolve the quotient into equal factors, and then multiply this equal factor by the numbers indicating the relation of the other fac- tors to it. Thus, resolve 384 into three factors, such that the second shall be twice the first and the third three times the first. Since the second factor equals 2 times the first and the third equals 3 times the first, 6)384 the product of the factors-, will equal 2x3, 64=4x4x4 or 6 times the first factor, used three times ; ~^ hence if we divide 384 by 6, the quotient, 64, will be the product of the smallest factor used three times ; therefore, if we resolve 64 into three equal factors, one of these factors will be the smallest of the three factors required. One of the three equal factors of 64, found by the previous case, is 4 ; hence, the smallest factor is 4, the second is 4x 2 or 8, and the third is 4x3 or 12. CASE IV. To find the divisors common to two or more FACTORING. 247 numbers In this case we resolve the numbers into their prime factors, and the common prime factors and all the num- bers which we can form by combining them will be all the common divisors. Thus, find the divisors common OPERATION. to 108 and 144. Resolving the 108=2 2 x3 3 numbers into their prime factors, 144=2 4 x3' we find the common factors to be om - foctor=2'x3' 2'x3*; hence, 1, 2, 4, 3, 9, and 2 4 all the possible products arising i 3 9 o A i 4 10 36 from their combination, will be all the divisors of 108 and 144. CASE V. To find all the factors or divisors of a number In this case we resolve the number into its prime factors, form a scries consisting of 1 and the successive powers of one fac- tor, and under this write 1 and the successive powers of an- other factor, and take the products of the terms of this scries, etc. Thus, find all the different divisors of 108. The factors of 108 are two 2's and three OPERATION. 3's. Since 3 is a factor J ^ 2 ^2 X 3 x 3 X 3 3 times, 1, 3, 3 2 , 3 3 , is ! 2 4 ' the first scries of divis- i 3 9 2 7 2 6 18 54 4 12 36 108 ors ; and since 2 is a factor twice, 1, 2, 2 2 is the second scries of divisors; and the products of. the terms of these two scries will give the prime factors and all possible products of them ; and therefore, all the divisors of the number. CASE VI. To find the number of divisors of a number. In this case we resolve the number into its prime factors, in- crease the number of times each factor is used by 1, and take the product of the results. Thus, find the number of divisors of 108. 248 THE PHILOSOPHY OF ARITHMETIC. Factoring, we find 108 equals OPERATION. 2 2 x3 3 . Now it is evident that 1 108=2 2 x3 3 with the first and second powers of (2 + l)x(3+l)=12 2 will give a series of three divisors; and 1 with the first, second and third powers of 3, will give a series of four divis- ors; hence their products will give a series of three times four, or 12 divisors. CHAPTER IV. THE GKEATEST COMMON DIVISOR. A DIVISOR of a number is a number which will exactly divide it. A number is said to exactly divide another when it is contained in it a whole number of times without a remainder. A Common Divisor of two or more numbers is a divisor common to all of them. The Greatest Common Divi- sor of several numbers is the greatest divisor common to all of them. By using the word factor to denote an exact integral divisor, we may define as follows: A Divisor of a number is a factor of the number. A Com- mon Divisor of two or more numbers is a factor common to all of them. The Greatest Common Divisor of several num- bers is the greatest factor common to all of them. These defi- nitions employ the term factor with a derivative signification. A factor is primarily one of the makers of a number, entering into its composition multiplicatively. From this it follows, however, that a factor is an integral divisor of a number, and as such, it may be conveniently and legitimately used in defin- ing a common divisor. In the subject of greatest common divisor, the term " divisor" is used in a sense somewhat special. It signifies an exact divisor a number which is contained a whole number of times without a remainder. The word measure was formerly used instead of divisor, and is in some respects preferable to divisor. A common divisor of several numbers is appropri- ately called their common measure, since it is a common unit (249) 250 THE PHILOSOPHY OF ARITHMETIC. of measure of those numbers. The term measure, iu this sense, originated in Geometry, where a line, surface, or volume which is contained in a given line, surface, or volume, is called the unit of measure of the quantity. In arithmetic, the term divisor is generally preferred. Gases. There are two general cases of greatest common divisor, growing out of a difference in the method of treatment adapted to the problems. When numbers are readily factored, we employ one method of operation ; when they are not readily factored, we are obliged to employ another method. This dual division of the subject into two cases is thus seen to be founded, not upon any distinctions in the idea of the subject, but upon the method of operation adapted to the numbers given. These two cases arc formally stated as follows: I. To find the greatest common divisor when the numbers are readily factored. II. To find the greatest common divisor when the numbers are not readily factored. Treatment. The general method of treatment in the first case is. to analyze the numbers in^o their factors, and take the product of the common factors. In the second case the num- bers arc operated upon in such a manner as to remove all the factors not common, and thus cause the greatest common divi- sor to appear. These two methods will be made clear by their application. CASE I. To find the greatest common divisor when the numbers are readily factored. This case may be solved by two distinct methods. The first method consists in writing the numbers one beside another, and finding all their common factors by division, and then tak- ing the product of these common factors. To illustrate, re- quired the greatest common divisor of 42, 84, and 126. 1st Method. We place the numbers one beside another as in the margin. Dividing by 2, we see that 2 is a common factor of the numbers. Dividing the quotients by 3, we see THE GREATEST COMMON DIVISOR. 25 J that 3 is a common factor of the OPERATION. numbers. Dividing these quotients 2)42 84 126 by 7, we see that 7 is a common 3)21 42 G3 factor of the numbers; and since 7)7 14 21 the final quotients 1, 2, and 3 arc 123 prime to each other, 2, 3, and 7 are GL C. D. =2x3x7=42 all the common factors of the given numbers. Hence 2x3x7 or 42, is the greatest common divisor required. This method, so far as I can learn, was published first by the author of this work, in 1855. It is now in several different text-books. The second method consists in resolving the numbers into their prime factors, and taking the product of all the common factors. To illustrate, take the problem already solved by the first method. 2d Method. Resolving the nurn- OPERATION. bers into their prime factors, wo 42=2x3x7 ' find that 2, 3, and 7, are factors 84=2x2x3x7 1 Q/_9 vx Q xx tjy 17 common to the three numbers; n y* _o o T_^O .... ._ . \x. U. U. ^XoXT 4J hence their product, which is 42, is a common divisor of the numbers; and, since these are all the common factors, 42 is the greatest common divisor. CASE II. To find the greatest common divisor ivhen the numbers are not readily factored. The second case may be solved by a process which may be entitled the method of suc- cessive division. It consists in dividing the greater number by the less, the less number by the remainder, etc., until the division terminates, the last divisor being the greatest common divisor. To illustrate, suppose it be required to find the great est common divisor of 32 and 5G. OPERATION Method. We first divide 5G by 32, then 32)50(1 divide the divisor, 32, by the remainder, 32 24; then divide the divisor, 24, by the 24)32(1 remainder, 8, and find there is no remain- der; then is 8 the greatest common divi- 8)24(3 sor of 32 and 56. 252 THE PHILOSOPHY OF ARITHMETIC. OPERATION. 3215611 24|32|1 8,24 |24 o This method is applicable to all numbers, and may therefore be distinguished from the methods of the previous case by naming it the general method, those being adapted to only a special case. A more conveni- ent method of expressing the successive divis- ion, and one which I recommend for general adoption, is that represented in the margin. It is observed in this method that the quotients are all written in one column at the right, and that the num- bers in the other columns become divisors and dividends iu turn. Explanation. In the explanation of the rationale of the general method of successive division, there are two distinct conceptions of the nature of the process. These two methods may, for convenience in this discussion, be entitled the Old and the New methods of explanation. By the Old Method of explanation I mean the one generally given in the text-books on arithmetic and algebra. The New Method is the one which is found in my own mathematical works. I will present each, pointing out the difference between them. Both methods are based upon the following general principles of common divisor: 1. A divisor of a number is a divisor of any multiple of that number. 2. A common divisor of two numbers is a divisor of their sum, and also of their difference. The Old Method of explaining the process of successive division is briefly stated in the following propositions: 1. Any remainder which exactly divides the previous divi- sor, is a common divisor of the two given quantities. 2. The greatest common divisor will divide each remainder, and cannot be greater than any remainder. 3. Therefore, any remainder which exactly divides the previous divisor is the greatest common divisor. Whatever the special form of the old method of explanation, THE GREATEST COMMON DIVISOR. 263 and we find it considerably varied by different authors, it involves, more or less distinctly, the principles just stated The New Method of conceiving of the nature of the process and explaining it, may be presented in the following princi- ples: 1. Each remainder is a NUMBER OF TIMES the greatest com- mon divisor. 2. A remainder cannot exactly divide the previous divisor unless such remainder is ONCE the greatest common divisor. 3. Hence, the remainder which exactly divides the previous divisor, is QSQ& the greatest common divisor. The first of these principles is evident from the considera- tion that a number of times the greatest common divisor, sub- tracted from another number of times the greatest common divisor, leaves ^number of times the greatest common divisor. The second of these principles becomes evident from the consideration that of any remainder and the previous divisor, the numbers denoting how many times the greatest common divisor is contained in each are prime to each other ; hence, one cannot divide the other unless one of these numbers is a unit, or the remainder becomes once the greatest common divisor. These principles may be readily seen by factoring the two numbers and then dividing. Thus, in the problem already given, knowing the greatest com- mon divisor to be 8, we may re- OPERATION. solve 32 and 56 into a number of times 8, and then divide. Observ- . .- ing the operations in this factored ' form, we see that each remainder Ix8)3x8t'3 is a number of times the greatest 3x8 common divisor, and that the fac- tors 7 and 4, and also 4 and 3, are respectively prime to each other; and also that the division terminates when we reach a divisor which is once the greatest common divisor, and that it 254 THE PHILOSOPHY OF ARITHMETIC. / cannot terminate until we come to once the greatest common divisor. In arithmetic I find it simpler to pre- OPERATION. sent this New Method, in a manner 32 )^( 1 slightly varied from the above, preserving its spirit, but slightly changing the form 4 to adapt it more fully to the comprchen- "Ihoifq sion of younger minds. To illustrate, let 9 , it be required to find the greatest common divisor of 32 and 5G. Dividing as previously explained, we have the work in the margin. The explanation, showing that this process will give the greatest common divisor, is as fol- lows: I 1st. The last remainder, 8, is a number of limes the great- est common divisor. For, since 32 and 5G arc each a number of times the G. C. D., their difference, 24, is a number of times the G. C. D.; and since 24 and 32 are each a number of times the G. C. D., their difference, 8, is also a number of times the G. C. D. 2d. The last remainder, 8, is ONCE the greatest common divisor. For, since 8 divides 24, it will divide 24-|-8, or 32; and since it divides 32 and 24, it will divide 24-J-32, or 56; and now since 8 divides 32 and 56, and is a number of limes the G. C. D., and since once the G. C. D. is the greatest num- ber that will divide 32 and 56, therefore 8 is once the G. C. D. This second method of conceiving the subject is believed to be the true one. It is simpler than the old method, and reaches the root of the matter, which the other does not. It looks down into the process and sees the nature of the remain- ders, and their relation to each other. All the remainders are seen to be a number of times the greatest common divisor, each being a less and less number of times the greatest com- mon divisor; and consequently, if the division be continued far enough, we will at length arrive at once the greatest common divisor. The object of dividing is thus seen to be a search for THE GREATEST COMMON DIVISOR. 255 a smaller number of times the greatest common divisor, know- ing that eventually we will arrive at once this factor, which will be indicated by the termination of the division. The experience of the class-room, especially in the sudden revelation of the philosophy of the division to those who thought they had a clear idea of the subject by the old method, has frequently demonstrated the superiority of the method now suggested. It is also readily seen, from this conception of the subject, that the secret of the method of finding the greatest common divi- sor is not in the division of the numbers, but in the subtrac- tion of them knowing that when we subtract one number of times a factor from another number of times the factor, the remainder is a less number of times the factor, and that the object is to continue the subtraction until we reach once the required factor. Abbreviation. This view of the subject leads us to discover a shorter process of obtaining the greatest common divisor than that of the ordinary method of dividing. Thus, suppose we wish to find the greatest OPERATION. common divisor of 32 and 116. If we divide in 32'llG 4 the ordinary way, we will find that it requires five 123 123 divisions and five quotients. If we take 4 times 32 and subtract 116 from it, we get a smaller re- mainder than if we subtract 3 times 32 from 116, and hence are nearer once the greatest common divisor. If we then subtract 32 from 3 times 12, we obtain a smaller remainder than if we subtract 2 times 12 from 32, and hence arc nearer once the greatest common divisor, etc. This latter method requires but three multiplications and subtractions, and hence saves two-fifths of the work. In many problems nearly one- half the labor is saved by this method. The method of conceiving and explaining the greatest com- mon divisor here given, is perhaps most clearly exhibited by the use of general symbols. Thus, let A and B be any two numbers, of which A is the greater; let c be their greatest 256 THE PHILOSOPHY OF ARITHMETIC. common divisor, and suppose Aac and B=bc] then dividing the greater by the less, the smaller by the remainder, and thus continuing, we have the operation in the margin, which may be explained b.c.)a. c(q as follows: in, tJdiC* t* (* 1st. Each remainder is a number of times the G. C. D. This is shown r.c) b. f c(q f bv the division, since each remainder r< ^ ' J /7j r fjl\f,- = -^r n is a number of times c, the first being (abq) times c, which we indicate r/ - c ) r - c (<7" r'q".c by r times c, etc. 1 (j r f Q 'f\ c r ff c 2d. A remainder cannot exactly . * ' divide the previous divisor unless such remainder is ONCE the G. C. D. To prove this it must be shown that b and r are prime to each other; also, that r and r' are prime to each other, etc. Now, if b and r are not prime to each other, they have a common factor, and hence, r+bq or a contains this factor of b ; but a and b are prime to each other, since c is the greatest common factor of a and b; there- fore, b and r are prime to each other. In the same way it may be shown that r aud r' are prime to each other, r' and r", etc. Hence, since of two numbers prime to each other one cannot contain the other unless the latter is a unit, a remainder can- not exactly divide the previous divisor unless such remainder is once the G. C. D. 3d. Hence, the remainder which does exactly divide the pre- vious divisor is ONCE the Greatest Common Divisor. CHAPTER V. THE LEAST COMMON MULTIPLE. A MULTIPLE of a number is one or more times the num- ber. A Common Multiple of two or more numbers is a number which is a multiple of each of them. The Least Common Multiple of several numbers is the least number which is a multiple of each of them. This conception of a multiple is that it is a number of time* some number. It regards the subject as a special case of form- ing composite numbers. A common multiple is a synthesis of all the different factors of two or more numbers, giving rise to a number which is one or more times each of those numbers. The relation of the subject to multiplication is also seen in th term multiple itself. The primary idea is, what number is one or more times each of several numbers ? This view of a multiple differs from that usually presented by our writers of text-books. The usual definition is A mul- tiple of a number is a number which exactly contains it. This puts containing as the primary idea, and makes the subject seem to originate in division rather than in multiplication. Indeed, some have gone so far in this direction as to change the name from multiple to dividend, calling it a common divi- dend instead of a common multiple. That this idea is incor- rect is evident both from the term multiple, and the nature of the subject. There can be no question of the subject having its origin in multiplication, and it should certainly be denned in accordance with this view. 17 (257) 258 THE PHILOSOPHY OP ARITHMETIC. It will be observed that the subject of Greatest Common Divisor is placed before that of Least Common Multiple ; that is, a special case of Factoring before a special case of Compo- sition, thus reversing the general order of synthesis before analysis. The reason for this is that Common Multiple is a synthesis of factors, and in some numbers these factors are most conveniently found by the method of greatest common divisor. This order is thus a matter of convenience in per- forming the operation, and not that of logical relation. Cases. There are two general cases of Least Common Multiple, as of Greatest Common Divisor. This distinction of cases, as in the corresponding analytic process, is not founded in a variation of the general idea, but rather in the practical ease or difficulty of finding the factors of the numbers. When the numbers are readily factored we employ one method of operation ; when they are not easily factored we employ an- other method. These two cases are formally stated as follows : I. To find the least common multiple when the numbers are readily factored. II. To find the least common multiple when the numbers are not readily factored. Treatment. The general method of treatment in the first case is to resolve the numbers into their different factors by the ordinary method of factoring, and take the product of all the different factors. In the second case, the different fac- tors are found by the process of determining the greatest com- mon divisor, and are then combined as before. CASE I. To find the least common multiple when the num- bers are readily factored. This case may be solved by two distinct methods. The first method consists in resolving the numbers into their prime factors, and then taking the product of all the different prime factors, using each factor the greatest number of times it appears in either number. Thus, required the least common multiple of 20, 30, and TO. We first resolve the numbers into their prime factors THE LEAST COMMON MULTIPLE. 259 Since the factors of 20 are 2 X 2 x 5, the multiple must con tain the factors 2, 2, and 5 ; since the factors of 30 are ^^ TI ^\ 20=2x 2x5 2, 3, and 5, it must contain 30=2x3x5 the factors 2, 3, and 5; and 70=2x5x7 for a similar reason it must L.C. M.=2x 2x3x5x 7=420. contain the factors 2, 5, and 7 ; hence, the least common multiple of 20, 30, and 70 must contain the factors 2, 2, 3, 5, and 7, and no others; and their product, which is 420, is the least common multiple required. The second method consists in writing the numbers one beside another and finding all the different factors by division, and then taking the product of these factors. To illustrate, find the least common multiple of 24, 30, and 70. Placing the numbers beside one another, and dividing by 2, we find that 2 is a factor of all the numbers ; it is therefore a factor of the least common multiple. Divid- ing the quotients by 3, we see that 3 is a factor OPERATION. 2^24 30 70 of some of the. numbers; it is therefore a factor i of the least common multiple. Continuing to * divide, we find all the different factors of the 5)_4_5_35 417 numbers to be 2, 3, 4, 5, and 7 ; hence, their pro- duct, which is 840, will be the least common multiple required. CASE II. To find the least common multiple when the num- bers are not readily factored. The second case is solved by a method which may be called the method of greatest common divisor. By it, when there are two numbers, we find the greatest common divisor of the two numbers and multiply one of them by the quotient of the other divided by their greatest common divisor. When there are more than two numbers, we find the least common multiple of two of the numbers, and then of this multiple and the third number, etc. To illustrate, required the least common multiple of 187 and 221. 260 THE PHILOSOPHY OF ARITHMETIC. We first find the greatest common divisor to be 17. Now, the least common multiple of OPERATION. 187 and 221 must be composed jg7 221 of all the factors of 187, and all 170 187 the factors of 221 not contained 17 in 187. If we divide 221 by the greatest common divisor, we L Q jyj _ i^^ =2431 shall obtain the factors of 221 not If belonging to 187 ; hence, the least common multiple is equal to 187x221-7-17, which we find is 2431. Another statement for this method is, divide the product of the two numbers by their greatest common divisor. The value of this method may be seen by attempting to find the least common multiple of 1127053 and 2264159 by each method. This method is very clearly OPERATION. exhibited by the following gen- A=axc eral explanation. Let A and B S=bxc . be any two quantities, and let L. C. M.=ax6xc= XT? their greatest common divisor be represented by c, and the other factors by a and b, respectively ; then we shall have the L. C. M.=ox6xc, Case L; but 6Xc= A A B, and a=-; hence, L. C. M.= -XJS. c ~c CHAPTER VI. INVOLUTION. INVOLUTION is the process of forming composite numbers jL by the synthesis of equal factors. It is, as has been pre- viously explained, a special case of Composition. If in the general synthesis of factors, we fix upon the condition that all the factors are to be equal, the process is called Involution, and the composite number formed is called a Power of that factor. Involution may, therefore, be defined as the process of rais- ing numbers to required powers. The power of a number is the product obtained by using the number as a factor any num- ber of times. The different powers of a number are called, respectively, the square, the cube, the fourth power, etc. The square of a number is the product obtained by using the num- ber as a factor twice. The cube of a number is the product obtained by using the number as a factor three times. These definitions, which are beginning to be adopted by authors, are regarded as improvements upon those framed from the usual conception of the subject. Symbol. The power of a quantity is indicated by a figure written at the right, and a little above the quantity. Thus, the third power of 5 is indicated by 5 s . The earlier writers on mathematics denoted the powers of numbers by an abbrevia- tion of the name of the power. Harriot, an eminent math- ematician of the 16th century, repeated the quantity to indi- cate the power; thus, for a fourth power he wrote aaaa. The present convenient system of exponents was introduced by (261) 262 THE PHILOSOPHY OF ARITHMETIC. Descartes, an eminent philosopher and mathematician cele- brated for his " cogilo, ergo sum," and the invention of the method of Analytical Geometry. Cases. To raise a number to each different power is a vari- ation of the general idea, and might be regarded as presenting distinct cases; but the methods of operation in each one of these cases are so similar, that they may all be considered under one head. In raising a number to a given power, we may have two objects in view: first, merely to find the required power ; and second, to ascertain the law by which the different parts of a number, as expressed in the Arabic system, are involved. These two objects require different methods of pro- cedure, and upon this difference of method we may found two distinct cases of involution. In practice, it is convenient to divide the second case into the consideration of the square and the cube, thus making three cases. These cases, formally expressed, are as follows: I. To raise a number to any required power. II. To raise a number to the second power, and ascertain the jaw by which the power is formed. III. To raise a number to the third power, and ascertain the law by which the power is formed. Treatment. The general method of treatment is to involve the factors by multiplication. In the first case a variation occurs for the purpose of abbreviation, giving two methods. In the second and third cases the number is resolved into parts and involved in two different ways, giving also two distinct methods. The treatment of both of these cases will now be presented. CASE I. To raise a number to any required OPERATION. power. This case may be solved by forming a 4 product by using the number as a factor as many times as there are units in the index of the power. Thus, to find the third power of 4, we multiply 4 by 4 giving 16, and then multiply 16 by 4 INVOLUTION. 263 giving 64, which is the cube of 4, since the number is used aa a factor three times. In all powers higher than the cube, we may abbreviate the process by taking the product of one power by another. Thus, in finding the 8th power of 2, we may first find the square of 2, which is 4, then multiply 4, OPERATION. the square, by itself, obtaining 16, the 4tb power of 2, and then multiply 16, the 4th power, by itself, giving 256, the 8th power of 2. This method may be applied to all powers higher j^r than the third, and is much more convenient in 15 practice. Thus, in finding the 5th power, we ~256 may take the product of the 2d and 3d powers, or the product of the square by the square by the first power ; in finding the 6th power, we may cube the 2d power, or square the 3d power, or multiply the 4th power by the square, etc. CASE II. Squaring Numbers and finding the law. This case may be solved by two distinct methods. The first consists in separating the number into its elements of units, tens, etc., and multiplying as in algebra so as to exhibit the law by which the parts are involved. The second method per- forms the process of involution as determined by the building up of a figure in geometry. These two methods may be dis- tinguished as the algebraic and geometric, or the analytic and synthetic methods. The ultimate object of these methods is to derive a law of involution by which we may be able to derive methods of evolution. These two methods apply both to the squaring and cubing of numbers. The synthetic method cannot be extended beyond the cubing of numbers; the analytic method is general and will apply to all powers, but is of no practical use in arithmetic beyond the cube. We will, there- fore, apply these two methods only to the squaring and cubing of numbers. ANALYTIC METHOD. By the Analytic Method of squaring numbers, we separate the number into its units, tens, etc , and THE PHILOSOPHY OF ARITHMETIC. keep these elements distinct in the involution of the number, so that the law of the process becomes apparent. To illustrate, find the square of 25. Twenty-five equals 20+5, or 2 tens and 5 units. Writing the number as in the margin, and multiplying by 5 and by 20, and taking the sum of these products, we have 20 2 +2 (5x20)+5 2 . From this we .see that the square of a OPEBATION. 20+5 20 + 5 5X20 + 5 2 20M-5X20 20 2 +2(5x20)+5 s number consisting of tens and units, equals the tensf+S times tens x units+units*. If we involve in the same manner a number consisting of hundreds, tens, and units, we shall find the following law: The square of a number consisting of hundreds, tens, and units equals hundreds' +2 x hundreds x fens+fens 2 +2 X (hundreds-\- tens) x units+units 1 . SYNTHETIC METHOD. The Synthetic Method of solving the same problem is as follows: Let the line AB represent a length of 20 units, and BH, 5 units. Upon AB construct a square: the area will be 20 2 =400 square units. On the two sides DC and BC con- struct rectangles each 20 units long and 5 broad, the area of which will be 5 X 20=100, and the area of both will be 2x100=200 square units. Now add the little square on CG, whose area is 5 2 =25 square units, and the sum of the different areas, 400+200+25=625, is the area of a square whose side is 25. When there are three figures, after completing the second square as above, we must make additions to it as we did to the first square. When there are four figures there are three addi- . tions, etc. CASE III. Cubing Numbers to find the law. This case B6H INVOLUTION. 265 may also be solved bj two distinct methods, as in squaring numbers, which we distinguish as the analytic and synthetic methods. The former involves the number by the method of algebra ; the latter by the principles of geometry. ANALYTIC METHOD. By the Analytic Method we resolve the number into its elements of units, tens, etc., and keep it in this form as we perform the process of involution, that we may exhibit the law by which the elements of a number entei iuto its cube. To illustrate, find the third power of 25. Resolving the number OPERATION. into its units and tens 25 2 =20 2 + 2(5x20) + 5* and squaring as above, we 20+5 have 20 2 +2(5x20)+5 2 . 5x 20*+2x5 2 X 20+5 3 Multiplying the square by 20 3 +2x5x20'+5 2 x20 5 and then by 20, and 20 3 +3x5x20'+3x5'x20+5 3 taking the sum of the products, we have the cube of 25, as given in the margin. Examining the result, we see that the cube of a number of two digits equals tens 3 +3Xtens*Xunits+ 3x tensXunits*+units 3 . Cubing a number of three digits, we obtain the following law:. The cube of a number of three digits equals hundreds 3 +3x hundreds 2 X tens + 3 X hundreds Xiens 2J rtens 3 +3X(hundreds-{- tensfXunits+3 X(hundreds+tens)X units 2 +units 3 . SYNTHETIC METHOD. By the Synthetic Method we use a cube to determine the process of involution. To illustrate, let us find the cube of 45 by this method. Let A, Fig. 1, represent a cube whose sides are 40 units; its contents will be 40 3 = 64000. We then wish to increase the size of this cube so that its sides will be 45 units. To in- crease its dimensions by 5 units, we must add first the three rectangular slabs, B, C, D, Fig. 2; 2d, the three corner pieces, E, F, G, Fig. 3 ; 3d, the little cube H, Fig. 4. The three slabs B, C, D, are 40 units long and wide and 5 units thick; hence, their contents are 40 2 X 5X3=24000; the contents of the cor- ner pieces, E, F, G, Fig. 3, whose length is 40 and breadth 12 266 THE PHILOSOPHY OF ARITHMETIC. 40 X 5 s X 3 =3000, and Fig. 1. and thickness 5, equal of the little cube H, Fig. 4, equal 5 3 =125; hence the contents of the cube represented by Pig. 4 are 64000+ 24000 + 3000+125= 91125. Therefore, the cube of 45, etc. Here we see that 40 s is the cube of the tens ; 40 2 x 5 X 3 is tens 3 x units x 3 ; 40x5 2 x3is3xtens X units*; and 5 s is units 3 ; hence we have, as before, the cube of a number of tens and units equals fens 3 +3 x tens 2 x units+B x tens J + units 8 . the contents Fig. 2. When there are three figures in the number, we complete the second cube OPEEATION. 40 3 =64000 40 2 X 5x3=24000 40x5 2 x3= 3000 5 3 = 125 Hence, 45 3 =91125 as above, and then make additions and complete the third in the same manner. If there are still some figures, and no more blocks to make additions, let the first cube represent the cube already found, and then proceed as at first. CHAPTER VII. EVOLUTION. "INVOLUTION is the process of finding one of the several J-J equal factors of a number. It is an analytic process, the converse of the process of Involution. Involution is a synthe- sis of equal factors ; Evolution is an analysis into equal factors. The former is a special case of composition; the latter is a special case of factoring. One finds its origin in multiplica- tion ; the other in division. Both are contained in the primary synthetic and analytic ideas, and are the result of pushing for- ward and specializing those notions. Any one of the several equal factors of a number is called a root of that number. The degree of a root depends upon the number of equal factors. The square root of a number is one of its two equal factors. The cube root of a number is one of its three equal factors, etc. These definitions are regarded as an improvement upon the old ones, that the square root of a number is a number which multiplied by itself will produce the number, and similarly for the other roots. Evolution may also be defined as the process of finding any required root of a number. Symbol. The Symbol of Evolution is ^/, called the radical sign. This sign was introduced by Stifelius, a German math- ematician of the 15th century. It is a modification of the letter r, the initial of radix, or root. Formerly, the letter r was written before the quantity whose root was to be extracted, and this gradually assumed its present form, v / - To indicate the degree of the root to be extracted, a figure (267) 268 THE PHILOSOPHY OF ARITHMETIC. is prefixed to the radical sign; thus, jS, $/, J/, etc., denote respectively the square root, cube root, fourth root, etc. This figure is called the index of the root, because it indicates the root required. The index of the square root is usually omitted, perhaps because the symbol was applied to the square root for some time before its use was extended to the higher roots. The roots of numbers are also indicated by fractional expo- nents; as 4^, 8*, etc. (7ases. Each different root might be regarded as constitut- ing a distinct case, but it is most convenient to treat the sub- ject under three general cases, as in Involution. These three cases correspond to those of Involution, and may be formally expressed as follows : I. To extract any root of a number when it can be conven- iently resolved into its prime factors. II. To extract the square root of a number when it can not be conveniently factored. III. To extract the cube root of a number when it can not be conveniently factored. Treatment. The general method of treatment is to analyze the number into the parts required. In the first case, we ana- lyze the number into its prime factors, and then make a syn- thesis of some of these factors. In the second and third cases, we separate the number into parts by several distinct methods, corresponding to those of Involution. CASE I. To extract any root when the number can be readily factored. This case is solved by resolving the number into its prime factors, and then involving the factors so as to obtain the equal factor required. For the square root we take the product of each of the two equal factors ; for the cube root we take the product of each of the three equal factors, etc. Thus, to find the square root of 1225, we resolve the number into its OPERATION. prime factors, 5, 5, 7, 7, and take the l J 86 H^S* X J > l! c 1 A / Sq. rt. =5x7=35 product of one of the two 5's and one of the two 7's, giving us 5 x 7, or 35. EVOLUTION. 269 To find the cube root of 1728 OPERATION. we resolve the number into its ^ 28 = 3 * 3 * 3x4x 4x4 , L-U. rt.=oX 4=iz prime factors, as shown in the margin, and take the product of one of the three 3's, and one of the three 4's, giving 3X4, or 12. In a similar manner we find any root of any perfect power that can be resolved into its prime factors. CASE II. To extract the Square Root of a number. The Square Eoot of a number is one of the two equal factors of the number. The square root of a number may also be deflned to be a number which, used as a factor twice,- will produce the given number. The former definition is somewhat analytic ; the process of thought is from the number to its elements. The latter is rather synthetic ; the process of thought is from the elements to the number. The method of extracting the square root of a number con- sists in analyzing the number into two equal multiplicative parts. This is done by first finding the highest term of the root, taking its square out of the number, and using it, accord- ing to the laws of involution, to determine the next term of the root, etc. The method being found in all the works on arithmetic, need not be stated here. Explanation. There are two 'methods of deriving the rule for square root, or of explaining the reason for the operation. These methods are distinguished as the Analytic and Synthetic methods. The former consists in resolving the number into its elements by the laws obtained by the analytic method of involution ; the latter consists in finding the root by means of a geometrical diagram by reversing the process of the corres- ponding method of involution. The synthetic method will apply to both the square and cube root of numbers, but cannot be extended beyond the cube root. The analytic method is general, and can be applied to the determining of any root of a number. In order to determine how many figures there are in the root, 270 THE PHILOSOPHY OF ARITHMETIC. and where to begin the extraction of the root, we employ the following principles: 1. The square of a number consists of twice as many fig- ures as the number, or of twice as many less one. This principle may be demonstrated as follows: Any integral number between 1 and 10 consists of one figure, and any num- ber between their squares, 1 and 100, con- sists of one or two figures: hence the 1 2 =1 10 2 =100 square of a number of one figure is a num- inn 2 i n nnn ber of one or two figures. Any number io00 2 =l 000000 between 10 and 100 consists of two figures, and any number between their respective squares, 100 and 10,000, consists of three or four figures; hence, the square of a number of two figures is a number of three or four figures, etc. Therefore, etc. 2. If a number be pointed off into periods of two figures each, beginning at units place, the number of full periods, together with the partial period at the left, if there be one, will equal the number of places in the square root. This is evident from Prin. 1, since the square of a number contains twice as many places as the number, or twice as many less one. ANALYTIC METHOD. By the analytic method of explaining the process of extracting the square root of numbers, we re- solve the number into its elements, and derive the method of operation by knowing the law of the synthesis of these elements. It is appropriately named the analytic method, because it ana- lyzes a number into its elements, and operates by reversing the synthetic process of involution. We will illustrate this method by extracting the square root of 625. Explanation. By the principles of involution we see that there will be two figures in the root, hence the number con- sists of the square of the tens plus the units of the root, which equals the square of the tens, plus twice the tens into the units, plus the square of the units. The greatest number of EVOLUTION. 271 tens whose square is contained in 625 f-f 2fw|-w 2 =: 6*25(25 is 2 tens ; squaring the tens and sub- < 2 = 20 J =400 tracting we hare 225, which equals Ztu+u* =225 twice the tens into the units, plus the 2t=^0 i/ == 5 square of the units. Now, since 2^-4-^=225 twice the tens into the units is usually much greater than the units squared, 225 consists principally of twice the tens into the units; hence if we divide 225 by twice the tens, we can ascertain the units. Twice the tens equals 20x2, or 40; dividing 225 by 40, we find the units to be 5, etc. In the margin the law of the involution of the elements is shown by the use of the letters t and u, the initials of tens and units. This representation of the law of the formation of the number enables us to separate it into its elements. SYNTHETIC METHOD. By the synthetic method we use a geometrical figure and derive the process from the method of forming a square v/hose area shall equal the given number. It is called synthetic because we commence with a smaller square and add parts to it, until we find a square of the required area. The method of forming the square will give us a method of finding the square root. To illustrate, let it be required to ex- tract the square root of 625. Explanation. The greatest number of tens whose square is contained in 625 is 2 tens. Let A, Fig. 1, represent a square whose sides are 2 tens or 20 units, its area will be the square of 20, or 400. Subtract- ing 400 from 625, we have 225, hence our square is not large enough by 225 ; we must therefore increase it by 225. To do this we add the two rectangles B and C, each of which is 20 units long, and since they near- ly complete the square, their area must be nearly 225 units ; hence, if we divide by their length we can find their width. Their length is 20x2=40, hence their width is 225-*- 40 or 5 272 THE PHILOSOPHY OP ARITHMETIC. Now complete the square by the addition of the little corner square whose side is 5 units, and then the entire length of all the additions is 40+5, or 45 units, and multiplying by the width we find their area to be 225 square units. Subtracting, nothing remains; hence, the side of a square which contains 625 square units is 25 units. The same method will apply when there are more than two figures in the root. The methods of operation indicated by both the analytic and synthetic methods of explanation, are the same. These methods give the usual rule for the extrac- tion of the square root. CASE III. The Cube Hoot of Numbers. The Cube Hoot of a number is one of the three equal factors of the number. The cube root of a number may also be defined to be a number which, used as a factor three times, will produce the given number. Again, the cube root of a number may be defined as a number which, raised to the third power, will produce the given number. These definitions are all correct, though they differ in idea. The first is analytic ; the thought is from the number to its elements. The second and third are synthetic; the process of thought is from the elements to the number. The method of extracting the cube root of a number consists in analyzing it and finding one of its three equal multiplicative parts. This is done by first finding the highest term of the root and taking its cube out of the number, then finding the second term by means of the first term, taking their combina- tion out of the number, etc. There are several methods of doing this, the three most important of which may be distin- guished as the Old Method, a New Method, and Homer's Method. There are several other methods, which I do not regard of sufficient importance to consider in this work. Old Method. The Old Method is so called because it is the one which has for a long time been taught and practiced. It may be distinguished by the use of 300 and 30 in finding trial and complete divisors. By a slight modification of the method EVOLUTION. 273 the ciphers of these multipliers may be omitted, and this form of the method is now generally preferred. The method may be stated as follows: RULE. I. Separate the number into periods of three figures each, beginning at units place. II. Find the greatest number whose cube is contained in the left-hand period ; place it at the right and subtract its cube from the period, and annex the next period to the remainder jor a dividend. III. Take 3 times the square of the first term of the root regarded as lens for a TRIAL DIVISOR; divide the dividend by it, and place the quotient as the second term of the root. IV. Take 3 times the last term of the root multiplied by the preceding part regarded as tens; write the result under the trial divisor, and under this write the square of the last term of the root ; their sum will be the COMPLETE DIVISOR. V. Multiply the COMPLETE DIVISOR by the last term of the root; subtract the product from the dividend, and to the remainder annex the next period for a new dividend. Take 8 times the square of the root now found, regarded as tens, for a trial divisor, and find the third term of the root as before; and thus continue until all the periods have been used. Explanation. This process of extracting cube root maybe explained by two distinct methods, distinguished as the ana- lytic and synthetic methods. The analytic method consists in resolving the number into its elements by the laws obtained from the analytic method of involution. The synthetic method consists in ascertaining the different terms of the root by the building up of a geometrical cube. In order to determine the number of figures in the root and with what part of the number to begin the evolution, it is necessary to state and demonstrate the following principle: 1. The cube of a number consists of three times as many figures as the number, or of three times as many less one ot two. 18 274 THE PHILOSOPHY OF ARITHMETIC. a Q00 This principle may be demonstrated as follows: Any inte- gral number between 1 and 10 consists of one figure, and any integral number between their cubes, 1 and 1000, consists of one, two, or three figures; hence the cube of a number of one figure is a number of one, two, or three figures. Any number between 10 and 100 consists of two figures, and any number between their cubes, 1000 and 1,000,000, consists of four, five, or six figures; hence the cube of a number of two figures consists of three times two figures, or three times two, less one or two figures. 2. If a number be pointed off into periods of three figures each, beginning at units place, the number of full periods together with the partial period at the left, if there be one, will equal the number of figures in the root. This is evident from Prin. 1, since the cube of a number eon- tains three times as many places as the number, or three times as many, less one or two. ANALYTIC METHOD. By the analytic method of explaining the process of extracting the cube root of numbers, we resolve the number into its elements and derive the process by knowing the law of the synthesis of these elements in the process of involution. We \yill illustrate the method by the solution of the following problem: Required the cube root of 91125. Solution. Since the cube of a number consists of three limes as many places as the number itself, or of three times as many less one or two, the cube root of 91125 will consist of two places, and hence consist of tens and 91-125(40 40 3 =64 000 _5 40 2 X3=4800|27125l5 40X5X3= 600 5 2 = 25 5425 27125 units, and the given number will consist of the cube of the tens, plus three times the square of the tens into the units, plus three times the tens into the square of the units, plus the cube of the rinitf EVOLUTION. 275 The greatest number of tens whose cube is contained in the given number is 4 tens. Cubing the tens and subtracting, we have 27125, which equals three times the square of the tens into the units, etc. Now, since three times the square of the tens into the units is much greater than all the rest of the ex- pression, 27125 must consist principally of three times the square of the tens into the units ; hence if we divide by three times the square of the tens we can ascertain the units. Three times the tens squared equals 3X40 2; =4800; dividing by 4800 we find the units to be 5. We then find three times the tens into the units equal to 40x5x3=600, and units squared equals 5 2 =25. Taking the sum and multiplying by the units, we have 27125, and subtracting, nothing remains. Hence the cube root of 91125 is 45. From this solution we readily derive the rule given above. SYNTHETIC METHOD. By the synthetic method of explana- tion we use a geometrical figure, a cube, and derive the process from the method of forming a cube whose contents shall equal the number of units in the given number. The number is regarded as expressing the number of cubic units in a cubical block, the number of linear units in whose side will be the cube root of the number. It is appropriately called synthetic, since we begin with a cube and add parts to it until we find a cube of the required contents. The method of forming the cube indicates the process of finding the cube root. This method may be illustrated by the solution of the problem already given: Required the cube root of 91125. Solution. We find the number of figures in the root aa before, and then proceed as follows : The greatest number of tens whose cube is contained in the given number is 4 tens. Let A, Fig. 1, represent a cube whose sides are 40, its con- tents will be 40 3 =64000. Subtracting from 91125 we find a remainder of 27125 cubic units; hence, the cube A is not large enough to contain 91125 cubic units by 27125 cubic units; we will therefore increase it by 27125 cubic units. 276 THE PHILOSOPHY OF ARITHMETIC. Fig. i. Fig. 3. Fig. 4. To do this we add the three rectangular slabs B, C, D, Fig. 2, each of which is 40 units in length and breadth ; and since they nearly complete the cube, their contents must be nearly 27125; hence, if we divide 27125 by the sum of the areas of one of their faces as a base, we can ascertain their thickness. The area of a face of one slab is 40*=1600, and of the three, 3X1600=4800; and dividing 27125 by 4800 we have a quo- tient of 5; hence the thickness of the additions is 5 units. We now add the three corner pieces E, F, and G, each of which is 40 units long, 5 wide, and 5 thick; hence the surface of a face of each is 40X5=200 square units, and of the three it is 200X3=600 square units. We now add the little corner cube H, Fig. 4, whose sides are 6 units, and the surface of a face is 5 2 =25. We now take the sum of the surfaces of the additions, and multiply this by the Common thickness, which is 5, and we have their solid contents equal to (4800+600+25)X 5=27125. Subtracting, nothing remains; hence the cube which contains 91125 cubic units is 40+5 or 45 units on a side. When there are more than two figures we increase the size of the new cube, Fig. 4, as we did the first, or let the first cube, Fig. 1, represent the new cube, and proceed as before. OPERATION. 91-125(40 40 3 =64 000 5 40 2 X3=4800 40X5X3= 600 5 2 = 25 5425 27125 45 27125 EVOLUTION. 277 Z'n'fc Ai\v>riOb CtjMiAAiiD. These two methods of explain- h.^ tfce Droenss ot extracting the square and cube roots of num- bers arc entirely distinct: thsy tiio based upon different ideas, though they give rise to the samt practical operation. The synthetic method is the one gcnerm]} given in the text-books on arithmetic; the analytic method was, until recently, confm-ed to algebra. It has been a question whk-L of these methods of explanation is the better, some preferring tho one and some the other. In my own opinion the analytic method is to be pre- ferred for several reasons, among which the following may be stated : First, it is in accordance with the genius of aiithmetic; we explain an arithmetical subject upon arithmetical principles. By the synthetic method we leave the subject of arithmetic, and bring in geometry to explain arithmetic. Should it be said in reply that by the analytic method we arc explaining arithmetic by algebra, let it be remembered that algebra has been called "universal arithmetic," and that all the algebra that is here used is purely arithmetical. In other words, though we may indicate the analysis of the number by letters, the idea is purely an arithmetical one, and is in no way depend- ent upon the principles of algebra as different from arithmetic. Second, I hold that a full, complete, and thorough insight into the subject can be obtained only by the analytic method. The geometric method indicates the process, as well as the analytic; but the analytic method shows the nature of the pro- cess, it exhibits the law of the formation of the square or cube as a pure process of arithmetic; and this gives a deeper in- sight into the subject than can be obtained by the other method. One who knows evolution only by the synthetic method, does not know it thoroughly. Third, the analytic method is general; it will explain the method of extracting all roots. The geometrical method is special; it enables us to extract the square and cube roots only. Thus, the square root is regarded as the side of a 278 THE PHILOSOPHY OF ARITHMETIC. Fig. 1. equare, the cube root as the side of a cube; but we have no geometrical conception of the fourth root, no figure correspond- ing to the fourth power, and therefore no idea of a fourth root; and so on for the higher roots. In respect of the comparative difficulty of the two methods, it may be remarked that it is generally supposed that the syn- thetic method is much easier than the analytic. This, however, I very much doubt; and this opinion is founded, not only upon theory, but also upon the experience of those who have tried both methods. I believe that a thorough knowledge of the subject can be gained much sooner by the analytic than by the synthetic method. My observation has been that pupils often are able to run over the geometrical explanation without really understanding it. It is, therefore, recommended that the ana- lytic method be introduced into our text-books and systems of instruction. The so-called synthetic methods of evolution may also be presented in an analytic form. Thus, instead of adding to the square A, page 271, we can begin with the large square, take out the square A, then obtain the width of the rectangles and the dimensions of the corner square, and then subtract. Indeed, this seems the more natural method, and is now being adopted by American writers. When thus presented, it would be better to call the two methods the al- gebraic and geometric methods. The same may be illustrated in the extraction of the cube root. Let Fir O I represent a cube which contains 91125 cubic units. Taking out the cube, A (40 3 = 64000), we have a solid, Fig. 2, representing 27125 cubic units. This solid consists principally of the three EVOLUXION. 279 Kig. 3. slabs, B, C, and D, each 40 units in length and breadth. Dividing 27125 by the sum of the areas of a face of each, (3X40* = 4800), we find their thickness is 5 units. Removing the slabs, there remain three solids, Fig. 3, each 40 units by 5 units, hence the surface of a face of the three is 3 X 40 X 5 = 600 square miles. Removing E, F, and G, there re- mains the small cube H, Fig. 4, the surface of one of whose faces is 5 2 = 25 square units. Multiplying the sum of all these surfaces by the common thickness, 5, we have (4800 +600+25) X 5 = 27125 cubic units. NEW METHOD OP CUBE ROOT. I will now present a method of extracting cube root which is much more convenient than the ordinary one. The simplification consists in finding a general method of obtaining the trial and true divisors, so that any one divisor may be used in obtaining the next following divisor. In the operations the trial divisor is indicated by t. d., and the true divisor by T. D., the local value of the terms being distinguished by their position. The reason for the method of obtaining the trial and true divisors may be readily shown by the formula. 1. Extract the cube root of 14706125. Solution We find as before the number of terms in the root and the first term of the root and cube, subtract and bring down the first period. We then find as before the trial divisor, 12, by taking 3 times the square of the first term, and, dividing, find the second term of the root to be 4. We then take 3 times the product of the first and second terms and the square of the second term, and add these to the trial divisor as a correction to obtain the true divisor, 1456. We then multiply 1456 by 4, and subtract and bring down the next period. 260 THE PHILOSOPHY OF ARITHMETIC. 12_ 24 To find the next trial divisor, we take the square of the last term, which is 16, and add it to the previous true divisor and the two corrections (which were added to the previous trial divisor), and we , .... have 1728 as the next trial divisor. jg_ To find the t,ue divisor, we add 1728- 3 times the product of the last 360 term of the root into the previous 25 OPERATION. 14-706-125(245 t. d. T. D. (5706 5824 t. d. 882125 882125 part of the root, and also the square 176425 T. D. of the last term, and have 176425 for the true divisor. Multiplying by 5, we have 882125. The method is indicated in the following formulas, which show the formation of the trial and true divisors. 1. TRUE DIVISOR = TRIAL DIVISOR + PRODUCT + SQUARE. 2. TRIAL DIVISOR=SQUARE+TRUE DIVISOR+CORRECTIONS. To show the method with large numbers, extract the cube root OPERATION. of 145780276447. 145-780-276-447(5263 125 75 30 4_ t. d. 7804 4 T,D. 20780 15608 8112- t. d. Solution We find the first term of the root, the first trial divisor, and the first true di- visor, as before. To find the second trial di- visor, we take the sum of the square of 2, the true divisor, and the previous correction, and we have 8112. We find the next true divisor by adding the usual corrections to the trial divisor, and have 820596. We find the third trial di- visor by taking the sum of the square of 6, the previous true divisor, and the corrections, and have 830028. We find the next true divisor as before. 936 36- 820596 36 T.D. 5172726 4923576 830028- 4734 9- t. d. 830501 49 T. D. 249150447 249150447 EVOLUTION. 281 OPERATION. We present another method involving the principle of using the previous work for obtaining trial and complete divisors. A part of this method is easily remembered by the formulas. COMPLETE DIVISOR = TRIAL DIVISOR + CORRECTION. TRIAL DIVISOR = CORRECTION + COM. DIVISOR -f SQUARE. The method is indicated in finding the cube root of 14706125. Solution. We find the number of figures in the root, and the first term of the root, as in the preceding methods. We write 2, the first term of the root, at the left at the head of Col. 1st ; 3 times its square with two dots an- nexed, at the head of Col. 2d ; its cube under the left- hand period ; then subtract and annex the next period for a dividend, and divide it by the number in Col. 2d, as a trial divisor, for the second term of the root. We then take 2 times 2, the first term, and write the product, 4, in Col. 1st, under the 2, and add ; then annex the second term of the root to the 6 in Col. 1st, making 64, and multiply 64 by 4 for a correction, which we write under the trial divisor ; and adding the correction to the trial divisor, we have the complete divisor, 1456. We then multiply 1456 by 4, subtract the product from 6706, and annex the next period for a new dividend. We then square 4, the second figure of the root, write the square under the complete divisor, and add the correction, the com- plete divisor and the square for the next trial divisor, which we find to be 1728. Dividing by the trial divisor we find the next term of the root to be 5. We then take 2 times 4, the second term, write the product 8 under the 64, add it to 64, and annex the third term of the root to the sum, 72, making 725, etc. A part of this method can be easily remembered by means IST COL. 2 4 2DCOL 14-706-125(245 12. . t. d. o 056 5824 64 8 1456 c. D. 16 LI 20 . . t. Q. 3625 882125 882125 176425 c. D. 282 THE PHILOSOPHY OF ARITHMETIC. of the following formulas, which show the formation of the trial and complete divisors : 1. Trial Divisor+Correction=Complete Divisor. 2. Correction + Complete Di visor +Square= Trial Divisor. To show the application of the method we will extract the cube root of 41673648563. 1ST COL. 3 6 2D COL. A 27 . . t. d. o r 376 n l-673-648-563(3467 7 4673 2304 94 8 3076 c. D. L lfi i 1026 12 3468 . . t. d. r 6156 2369648 211773C 10387 352956 c. D. 36 359148 ..t.d. 72709 251912563 251912563 35987509 c. D. HORNER'S METHOD. Horner's Method of extracting the cube root was derived from a method of solving cubic equa- tions invented by Mr. Homer, of Bath, England. It was first published in the Philosophical Transactions for 1819; Under the title of "A New Method of Solving Numerical Equations of all orders by Continuous Approximations." Its inventor, Mr. W. G. Homer, was a teacher of mathematics at Bath ; he died in 1837. It is considered one of the most remarkable additions made to arithmetic in modern times. DeMorgan says that the first elementary writer who saw the value of florner's method was J. R. Young, who introduced it in an elementary treatise on algebra, published in 1826. Among the first to introduce it into arithmetic in this country was Prof. Perkins, of New York. This method differs from both of those already explained, and possesses merits which strongly recommend it for general EVOLUTION. 283 adoption. It is very concise the root of a large number can be extracted with one-half of the work required by the old method. Its conciseness arises from the fact that it proceeds upon a principle which enables us to make use of work already obtained, while the old method requires new calculations every time we find a trial or true divisor. In other words, it is an organized method by which the work is so economized that no operations are superfluous, but each result obtained is made use of in obtaining a subsequent result. It is entirely general in its character, applying to the extrac- tion of all the higher roots. This method can be explained both analytically and synthetically. It is presented in several of the higher arithmetics, and need not be stated here. It is more difficult to remember than either of the other methods, and this is perhaps the principal objection to its general adop- tion. The " New Methods " for cube root they do not apply to higher roots are, however, preferred to Horner's Method, being quite as concise, and much more readily acquired and remem- bered. APPROXIMATE ROOTS. The invention of rules for approxi- mating to the square and other roots of numbers, where those roots are surds, was a favorite speculation with earlier writers on arithmetic and algebra. These rules will be most readily understood and their relative values seen by stating them in algebraic language. 1. The rule given by the Arabs is expressed by the formula, . y> This approximation gives the root in excess; but to increase its accuracy, we may repeat the process, making use of the root obtained. This is the rule given by Lucas di Borgo, and subsequently by Tartaglia, who derived it in common with the rest of h.is countrymen from Leonard of Pisa. 284 THE PHILOSOPHY OF ARITHMETIC. 2. The rule given by Juan do Ortega, 1534, is expressed by the followin formula: This approximation is in defect, but, generally speaking, more accurate than the former. 3. The third method of approximation was proposed by Orontius Fincus, Professor of mathematics in the university of Paris, and who long enjoyed an uncommon reputation in consequence of his having introduced the knowledge of the mathematics of Italy among his countrymen. His method consisted in adding 2, 4, 6, or any even number of ciphers to the number whose root was required, and then reducing the num- ber expressed by the additional figures of the root resulting from these ciphers, to sexagesimal parts of an integer. Thus, in extracting the square root of 10, he would get 3 1 1G2, which reduced to sexagesimals, became 3. 9'. 43". 12'". This is the most remarkable approximation to the invention of decimals which preceded the age of Stevinus. If the author had stopped short at the first separation of the digits in the root, it would have expressed the square root of 10 to 3 decimal places; but the influence of the use of sexagesimals diverted him from this very natural extension of the decimal notation, and retarded for more than half a century this im- provement in the science of numbers The method of Fineus excited the attention of contempora- neous mathematicians, who in adopting it, however, did not reduce the result to sexagesimals, but merely subscribed, as a denominator to the whole not considered as integral, 1 with half as many ciphers as had been added in the operation, giving x/10=f^^. It is under this form that it is noticed by Tar- taglia and.Recorde. Pclletier also, a pupil of Orontius Fincus, after noticing the second of the two methods of approxima- tion, describes this as more accurate and less tedious than any other. Methods of approximation were also quite numerous for the EVOLUTION. 285 extraction of the cube root. That of Lucas di Borgo may be seen from the formula, which Tartaglia says he got from Leonard of Pisa, who had i*. from the Arabians; and he expresses his surprise that he should have committed so grievous an error, unless he had done so without consideration. The method of Oroutius Fineus is represented by the follow- in formula: which errs as much in excess as that of Di Borgo in defect. The method of Cardan is indicated by the formula, which Tartaglia criticises with great bitterness, as might nat- urally be expected from one who had been so treacherously defrauded by him of an important discovery, the general method of solving cubic equations. His own method is rep- resented by the formula, which, though more accurate than that of Cardan, errs in defect while the other erred in excess. In later times, methods of approximation have been proposed which give results much more accurate than any of the pre- ceding. One of the very best that we have met is the follow- ing, given by Alexander Evans, in the January number of The Analyst, 187G: N r For square root, --f- N 2r For cube root, ^ + ~y N nl For nth root, - -r .n-1 286 THE PHILOSOPHY OF ARITHMETIC. To illustrate these formulas we will extract the square root of 2 and the cube root of 6. Suppose the square root of 2 is nearly 1.4, then r=1.4, and substituting in the formula we have N r 2 1 I* 99 which is the correct root to four places ; and by substituting ffi in the formula we get the root correct to eight places. In extracting the cube root of 6, suppose that r=1.8, then substituting in the formula we have N 2r_ 6 2(i$)_50 6 + -' "- 1 ~~- + - which is true to three decimal places. The method cannot be relied upon, however, to give many correct terms in the ap- proximation. In applying it to the cube root of 3, regarding 1.4 as the value of r, we obtain for the root, 1.44353, which is true to only two places. If we then take 1.44 as the value of r, we shall find the next approximation to be 1.442253, which is true to four places. If we take r=1.5 as the cube root of 4, the formula gives the first approximation 1.5925, which is true to only the first decimal place. If we had taken r=1.6, we would have obtained 1.5875, which is correct to three places. The best method is therefore the general one ; for a person who is familiar with the method which I have given under the name of the New Method will extract the root more rapidly than he can with the approximate methods, and may be always certain of the correctness of his result. PART III. COMPARISON. SECTION I. RATIO AND PROPORTION. SECTION II. THE PROGRESSIONS. SECTION III. PERCENTAGE. SECTION IV. THEORY OF NUMBERS. SECTION I. RATIO AND PROPORTION. 19 I. INTRODUCTION. II. NATURE OF RATIO. III. NATURE OF PROPORTION IV. APPLICATION OF PROPORTION. V. COMPOUND PROPORTION, ETC. VI. HISTORY OF PROPORTION. CHAPTER I. INTRODUCTION TO COMPARISON. A RITHMETIC consists fundamentally of three processeB ; JL\. Synthesis, Analysis, and Comparison. Synthesis and Analysis are mechanical processes of uniting and separating numbers ; Comparison is the thought process which directs the general processes of synthesis and analysis, and unfolds the various particular processes contained in them. Comparison also gives rise to several processes which do not grow out of the general operations of synthesis and analysis, but which have their origin in the thought process itself The principal processes originating in Comparison, are Eatio, Proportion, Progression, Percentage,. Reduction, and the Properties of Numbers. The particular manner in which these processes originate will appear from the following considerations. If two numbers be compared with eac,h other, we perceive a definite relation existing between them, and the measure of this relation is called Eatio. Numbers may be compared in two ways: first, by inquiring how much one number is greater or less than another ; ajid secondly, by inquiring how many times one number equals another. Thus, in comparing 6 with 2, we see that 6 is four more than 2, and also that 6 is three times 2. These relations, expressed numerically, give us the ratio of the numbers. The former is called arithmetical ratio ; the latter, geometrical ratio. The term ratio is generally restricted, how- ever, to a geometrical ratio, and it will be thus used here. The comparison of ratios gives rise to several distinct pro- cesses called Proportion. If two equal ratios be compared, ( 291 ) 292 THE PHILOSOPHY OF ARITHMETIC. the numbers producing the ratios being retained in the com- parison, we have what we call a Geometrical Proportion, or simply a Proportion. When the ratios are simple, we have a Simple Proportion; when one or both of the ratios are com- pound, we have a Compound Proportion. If we wish to divide a number into several equal parts, bear- ing a certain relation to each other, we have a process called Partitive Proportion. If we wish to combine numbers in certain definite relations, we have a process called Medial Pro- portion, usually known as Alligation. If we compare num- bers so that each consequent is of the same kind as the next an- tecedent, we have a process known as Conjoined Proportion. If we have a series of numbers differing by a common ratio, we may investigate such a series and ascertain its laws and principles; thus arises the subject of Progressions. If the ratio is arithmetical, the progression is called an Arithmetical Progression; if the ratio is geometrical, the progression is called a Geometrical Progression. Again, as was shown in the Logical Outline of arithmetic, we may take some number as a basis of comparison, and develop the relations of numbers with respect to this basis. It has been found convenient in business transactions to use one hun- dred as such a basis of comparison, which gives rise to the subject of Percentage. In Fractions and Denominate Numbers we have units of different values under the same general kind of quantity. By comparing these, it is seen that we may pass from a unit of one value to one of a greater or less value, and thus arises the process of Reduction. When we pass from a less to a greater unit the process is called Seduction Ascend- ing ; when we pass from a greater to a less unit, the process is called Reduction Descending. By a comparison of numbers, we may also discover certain properties and principles which belong to numbers per se, and also other properties and principles which have their origin in the Arabic system of notation. Such principles may be em- INTRODUCTION TO COMPARISON. 293 braced under the general head of the Properties of Numbers. It is thus seen that several divisions of the science of numbers are not contained in the original processes of synthesis and analysis that is, of addition and subtraction but have their roots in and grow out of the thought-process of comparison. These several subjects, evolved from the comparison of num- bers, will be considered in the order in which they have beec mentioned. CHAPTER H. NATURE OF RATIO. O ATIO originates in the comparison of numbers. It is the -Lv numerical measure of their relation. From it arise some of the most important parts of arithmetic, as proportion, pro- gressions, etc. Its importance, and the inadequate and diverse views held concerning it, make it necessary to give quite a care- ful and thorough discussion of the subject. Definition. Ratio is the measure of the relation of two similar quantities. This definition differs in one respect essen- tially from that usually given. Ratio is generally defined as "the relation of two quantities" relation and ratio being made equivalent. This is not accurate, or, at least, not suffi- ciently definite. The word ratio is a more precise term than relation, as will appear from the following illustration. If we inquire what is the relation of 8 to 2, the natural reply is " 8 is four times 2 ;" but if we inquire what is the ratio of 8 to 2, the correct reply is "four." Here the ratio four is the num- ber which measures the relation of 8 compared with 2. It is thus seen that ratio is not merely the relation of two similar quantities, but the measure of this relation. This definition, presented in the author's own text-books, has already been in- troduced by one or two writers, and seems not unworthy of general adoption. The Terms. A ratio arises from the comparison of two similar quantities. These quantities are called the terms of the ratio. The first term is called the Antecedent ; the second term is called the Consequent. The antecedent is compared with (294) NATURE OF RATIO. 295 the consequent; the consequent is the basis or standard of comparison. Thus, a ratio indicates the value of the first quantity as compared with the second as a standard. The ratio, therefore, expresses how many times the consequent must be taken to produce the antecedent, or what part the antece- dent is of the consequent. In other words, it answers the question the antecedent is how many times the consequent, or, the antecedent is what part of the consequent ? From this it also appears that the ratio equals the antecedent divided by the consequent. Thus, the ratio of 6 to 3 is 2, and the ratio of 3 to 6 is . Method of Eatio. The question has recently been raised whether the correct method of determining a ratio is to divide the antecedent by the consequent or the consequent by the an- tecedent. An eminent author advocates the division of the consequent by the antecedent, and this method has been adopted by several American mathematicians. The old method some of them call the " English Method ;" the new method, the "French Method." The so-called "French Method" we be- lieve to be incorrect in principle and inconvenient in practice The correct method of finding the ratio of two numbers is to divide the antecedent by the consequent. Several reasons will be given in favor of the correctness of this method, which seem to us conclusive. For convenience in the discussion, let us distinguish the two methods as the Old and the New method. 1. Nature of Ratio. First, I think the correctness of the Old Method will appear from the nature of ratio itself. If we inquire " What is the relation of 8 to 2 ?" the natural reply is, " 8 is four times 2." Here the number four is the measure of the relation; hence the ratio of 8 to 2 is four. If the inquiry is, '^What is the relation of 2 to 8?" the natural reply is "2 is one-fourth of 8 ;" hence in this case, the ratio is one-fourth. From this view of the subject it follows that the correct method cf determining a ratio is to divide the antecedent by the conse- quent, and not the consequent by the antecedent. 296 THE PHILOSOPHY OF ARITHMETIC. If I ask the relation of 8 to 2, it would be illogical to reply, " & is one-fourth of 8," for this does not answer my question. To giving the reply, that number should be used first in making the comparison which was used first in the question, and it would be illogical and absurd to invert the order ; yet this is really what, those who advocate the other method must do. If the ratio of 8 to 4 is one-half, then when I ask the question, "What is the relation of 8 to 4 ?" they must say, "4 is one-half of 8," unless it be supposed that they would say, " 8 is one-half of 4." This may be impressed by an illustration suggested by Prof. Dodd. Of two persons, A and B, suppose A to be the father and B the son. Now if the question be asked, " What is the relation of A to B ?" the correct reply is "A is the father of B," and it would be inconsistent to answer, "B is the son of A," for that is the reply to the question, " What is the relation of B to A ?" The same holds in regard to the comparison of numbers, and with even greater force, since it is necessary to be more explicit in science than in ordinary conversation. Hence, if the question is asked, " What is the relation of 8 to 2 ?" the correct reply is, " 8 is four times 2 ;" from which we see that the ratio is four. It is clear, then, that the ratio of two numbers, which is the measure of the relation of the first to the second, is equal to the first divided by the second. 2. Law of Comparison. The true method of determining a ratio may also be shown by the nature and object of the com- parison. The law of comparison is to compare the unknown with the known; thus, we logically write #=4, and not 4=a;. Now, in a ratio, one number is made the basis of comparison, the object being to comprehend or measure the other number by its relation to the basis. In this sense the basis may be regarded as the known quantity, and the other number as the unknown quantity. Now the unit is the basis of all numbers ; it is the standard by which all numbers are measured ; we un- derstand a number only as we know its relation to the unit. When any number, as 8, is presented to the mind, we compare NATURE OF RATIO. 297 It with the unit, not the unit with it. The inquiry is, 8 is how many times one ? hence 8 is the first number named in the com- parison ; it is, therefore, the antecedent, and the ratio is the quotient of the antecedent by the consequent. The advocates of the new method of ratio would have us compare the 1 with the 8, the unit of measure with the thing to be measured, the known with the unknown. This is not only awkward, but it is directly opposed to the established principles of logical thought. 3. Authority. One of the strongest arguments in favor of the division of the first term by the second is the usage of eminent mathematicians. That signification of scientific terms which custom has fixed should not be changed but for the strongest reasons. From the earliest periods of science, math- ematicians have divided the antecedent by the consequent. It was the method employed by Euclid, Pythagoras, and Archi- medes, the three great mathematicians of antiquity; and by Newton, LaPlacc, and LaG range, the three great mathemati- cians of modern times. The English and German, and nearly all the French mathematicians, employ this method, and have done so from the earliest periods. One or two French, and a few American authors have adopted the New Method; but with these few exceptions, the Old Method is the method of mathematicians at all times and in every country where the ratio of numbers has been employed. But not only is the authority of numbers upon this side of the question, but also the greater weight of the authority of eminence. The practice of all of the great mathematicians of every age is in favor of the Old Method. In its favor we may mention the illustrious names of Euclid, Pythagoras, Archi- medes, and to these add the not less illustrious names of Dio- phantus, Newton, Leibnitz, LaPlacc, LaGrange, the Bernoullis, Lcgendre, Arago, Bourdon, Carnot, Barrow, Ilcrschel, Bow- ditch, Pierce, etc.; names which shed a lustre over their country and age, and which are symbols of grand achievements in the 13* THE PHILOSOPHY OF ARITHMETIC. science. All the great works, the masterpieces which stand as monuments of the loftiest triumphs of genius, are upon this side of the question. The Principia of Newton, the Mecanique Celeste of LaPlace, the Mecanique Analytique of LaGrange, the Theorie des Nombres of Legendre, the Analytical Mechanics of Pierce, all employ the Old Method. Such universal agree- ment among great mathematicians should be regarded as a final settlement of the matter. 4. Inconvenience of the Change. Again, the Old Method cannot be changed without confusion. There are definitions in science which involve the idea of ratio, and a correct appre- hension of these definitions requires a precise idea of ratio. These definitions are founded upon the Old Method of ratio ; hence, if we change the method of determining ratio, we shall either have a wrong idea of the subjects defined, or else the definitions must be changed. The latter would be almost a practical impossibility, since they have become fixed forms in scientific language. Science has embalmed certain definitions, and it would seem almost like sacrilege to disturb them. Among these definitions may be mentioned those of specific gravity, differential co-efficient, index of refraction, and the geometrical symbol -. The specific gravity of a body is defined to be the ratio of its weight to the weight of an equal volume of some other body assumed as a standard. The index of re- fraction is the ratio of the sine of the angle of incidence to the sine of the angle of refraction. The differential co-efficient is the ratio of the increment of the function to that of the varia- ble. The geometrical symbol ?r is the ratio of the circumfer- ence to the diameter. These definitions have the authority of the great masters, and will, without doubt, remain as they are. One or two of them have been changed by the advocates of the New Method, but such changes will hardly extend beyond their own text-books. 5. Origin of Symbol. It may further be remarked that the assumed origin of the symbol of ratio is in favor of the method NATURE OF RATIO. 299 here advocated. It is said that the symbol of ratio is derived from that of division ; that is, that : is the symbol -=- with the horizontal line omitted. The symbol of division indicates that the quantity before it is to be divided by the one following it; hence if the theory of the origin is true, it indicates that prima- rily the ratio of two numbers was the quotient of the first divided by the second ; and this primary method should be followed, unless there are good reasons to the contrary. In this connection I remark that the Old Method of ratio gives us the simplest idea of a proportion. A proportion is an equality of ratios, and this idea is most clearly expressed thus: 6-v-3=8-=-4. With the other method of ratio, this sim- ple idea of a proportion cannot be presented. Whether the symbol : is a modification of -r-, is, I presume, not definitely known. It is so asserted by some authors ; but so far as I can learn, it is not known as a historical fact. It seems very reason- able, however, and in some old German works I have noticed that the symbol of division is used for indicating the ratio of numbers. The "French Method," inappropriately so called. These two methods of ratio have been distinguished by the names "English Method," and "French Method;" the Old Method being called the "English Method," and the New Method the "French Method." These names were first applied, I think, by Prof. Ray, although others had previously stated that the French mathematicians made use of the one and the English mathematicians of the other method. Both of these names are founded in error. The " French Method " is not used by the French; the general custom of the French mathematicians is opposed to it. Lacroix is the only mathematician of any eminence who, so far as I have examined, employs it. The " English Method " is not confined to the English, but it is used by French, Germans, Prussians, and Austrians, in fact, by the mathematicians of all countries, and is, therefore, incorrectly named the English Method. 800 THE PHILOSOPHY OF ARITHMETIC. Nearly all the mathematicians of France, it has been said, employ the so-called English Method, and all of the most emi- nent ones do so. Among these may be mentioned LaPlace, LaG range, Legcndrc, Bourdon, Ycruier, Comte, Biot, Carnot, Arago, etc. In proof of this, I will quote from some of their own works. M. Bourdon, in his Arithmetic, page 222, says, "Par exemple, le rapport de 24 a 6 est %-, ou 4 ; et cehii de 6 a 24 esl T 6 T , ou -. Legcndrc, in his Geometry, Book IV., Prop. XIV., says, "done le rapport de la circortference au diametre desiyne ci-dessus par TT =3.1415926." Vernier, in his Arith- metic, page 118, says, " com me la raison est le quotient qu' on obtient quand on diuise P antecedent par le consequent." Other authors might be quoted, but these are sufficient to show that the so-called French Method is not the method of the French. Legendrc and Bourdon are especially referred to, since some popular American text-books, supposed to be translations from these authors, employ the New Method, and have been instru- mental in leading quite a large number of American authors and teachers to adopt that method. In turning to Lacroix, we sec a departure from the general usage of the French mathematicians. In his Arithmetic, which is the only work of his that I have examined, he says, page 85, in comparing the numbers 13, 18, 130, and 180, we see '-que le deuxieme contient le premier aulant de fois que le quatrieme conlient le troisieme; et Us for men t ainsi ce qu'on appdle une proportion." Notice that he is here discussing the subject of proportion, and not the subject of ratio by itself. On the next page he remarks, "Je conlinuerai de prendre le consequent du rapport pour le numeraleur de la fraction qui exprime le rappori et V antecedent pour le denominaleur." This places Lacroix upon the opposite side of this question; and it is clear from the manner in which he expresses himself, that he is conscious of taking a position not authorized by the general custom of his countrymen. I think it can readily be seen how Lacroix was led into this error He commences the NATURE OF RATIO. 301 subject with a problem in proportion, which he solves by anal- ysis, and then, by a mistake plausibly drawn from the process of analysis, seeming to think that the analysis dictates a divi- sion of consequent by antecedent, he defines his terms and an- nounces his method of ratio. The whole discussion is as illog- ical as the conclusion is incorrect. He begins the subject with proportion instead of ratio, thus inverting the whole problem and getting the method of ratio inverted also. The true method is to begin by comparing numbers, determining their relations; and then comparing their relations, make a proportion ; the first will give the true idea of Ratio, and the second of Proportion. Answer to Arguments in Favor of the New Method. This dis- cussion would be imperfect without an attempt to answer some of the arguments which have been presented in favor of the so-called "French Method." An eminent author and educator, who has done more for the adoption of the New Method than any other person in this country, gives a formal defense of it ; a few of his arguments I will notice. His first argument, which is founded upon the nature of comparison, has already been answered in the previous discussion. He says, in comparing numbers, "the standard should be the first number named;" hence, to comprehend 8, he would compare the basis of num- bers, or 1, with 8, instead of comparing the 8 with 1, that is, the number with the basis. The mistake he makes is in com- paring the standard with the thing measured ; that is, the known with the unknown ; the true law of comparison being just the reverse of this. This will be readily seen in continuous quantity which can be clearly understood only by comparing it with some definite part of itself assumed as a unit. Thus, suppose a period of time is considered ; it is clear that we can get a definite idea of it by comparing it with some fixed unit, as a day, or a week, or a year. In these cases it will be seen that we do not compare the unit with the given quantity, as the author quoted would main- tain, but the quantity to be measured with the unit of measure. 302 THE PHILOSOPHY OF ARITHMETIC. His second argument is that the New Method gives a con- venient rule for Proportion ; the fourth term being equal to the third term multiplied by the ratio of the first to the second. The reply is that the Old Method gives just as convenient a rule, namely, " The fourth term equals the third divided by the ratio of the first to the second." His third argument is, that in a geometrical progression the ratio is the quotient of any term divided into the following term. This is the most plausible argument advanced, and demands special notice. If it be true that the ratio of any term to the following term is the quotient of the second divided by the first, then it is true that we here depart from the general method of ratio ; but still it would not follow that the general method of ratio should be changed to harmonize with this exceptional case. A more sensible conclu- sion would be that the method here used should be changed to correspond with the general method. That the general should control the special and not the special the general, is a fixed law of science. Let us see, however, if the form of writing a geometrical progression does present an exception to the general method of expressing a ratio. In a geometrical progression, the ratio is the measure of the relation that any term bears to the preceding term. In the series 1, 2, 4, 8, etc., we do not compare the 1 with the 2, the 2 with the 4, etc., to determine the ratio, as will appear from the fol- lowing considerations. Suppose, for illustration, that we wish to find any term of the series, as the 5th term, would we not reason thus : the 5th term must bear the same relation to the 4th, that the 4th does to the 3d ; and since the 4th is twice the 3d, the 5th term must be twice the 4th, or 16. Here we follow the law of comparison, by comparing the unknown with the known, and reversing the apparent order, name the 8 first and the 4 after it. Should we write the comparison out in full, we would have 5th : 8 : : 8 : 4. If this is true, then, in a geometri- cal series, we do not compare a term with the following term, but rather with the term preceding it The ratio of the series, NATURE OF RATIO. 303 it thus appears, is the ratio of any term to the preceding term, and not to the term following it. In other words, we compare backward, instead of forward, as in ordinary ratio ; and really divide the antecedent of the comparison by the consequent tc obtain the ratio. Some writers explain this apparent departure from the gen- eral signification of ratio, by saying that in a geometrical series we express the " inverse ratio of the terms." Says one, " It is less troublesome to express the common ratio inversely, as then one number will suffice." Says another, " Whenever we meet with the expression, the 'ratio of a geometrical series,' we are to understand the inverse ratio." It seems clearer to me to say that the order of writing the terms is in opposition to the order of thought. We write one way and compare another way. If the expression of the series were dictated by the idea of ratio, we would write it from the right toward the left. The fact is, however, that in a geometrical progression, it is the rate of the progression that we consider, rather than the ratio of the terms; that is, the rate at which the series pro- gresses, and this term would be preferable to ratio in this con- nection. A series of terms, increasing or decreasing by a common multiplier, although an outgrowth from the idea of ratio, pre- sents an idea not identical with that of ratio. This distinction is actually made by several French writers. They use the different words rapport and raison ; the former to express the ratio of two numbers, the latter to denote the rate of the geometrical series. Thus Bourdon, in his Arith- metic, page 279, says, " On appelle Progression par Quotient une suite de nombres tels que le rapport d'un terme quelconque a celui qui le precede est constant dans toute Velendue de la serie. Ge rapport constant, qui existe entre un terme el celui qui le precede immediatement se nomme la Eaison de la progression." Prof. Hcnkle, who has written several excellent articles upon this subject, quotes Biot to the same effect. He says of a geomet- rical progression, " Le Eapport de chaque terme au precedent se 304 THE PHILOSOPHY OF ARITHMETIC. nomme Baison." It will thus be seen that some of the French writers distinguish between ratio and the constant multiplier of a progression, and should the word rate be adopted with us, we would avoid the objection of this seeming departure from the general signification of ratio. I have devoted so much space to the discussion of this sub- ject, because I think it one upon which there should be uni- formity of opinion and practice. Several of our most popular elementary text-books on mathematics have adopted the so- called " French Method," and are teaching it to the youth of the country. Pupils who have been taught the method can with difficulty relinquish it, and if they proceed to Philosophy and Higher Mathematics they will meet with difficulty in every subject containing definitions involving ratio. It is proper to remark that since this article was written, now some ten or twelve years, several authors who had adopted the new method, have discarded it and now use the old method. CHAPTER III. NATURE OF PROPORTION. P)ROPORTION arises from the comparison of ratios. Com- L parison begins with comparing numbers, giving rise to the idea of relation, the measure of which is ratio. After becoming familiar with the idea of the relations of numbers, we begin to compare these relations ; when eq\ial relations are compared, we attain to the idea of a Proportion. Proportion, it is thus seen, has its origin in comparison; it is a comparison of the results of two previous comparisons. Every proportion involves three comparisons; the two which give rise to the ratios, and a third, which compares or equates the ratios. All of these comparisons are exhibited in the expression of a proportion ; the symbol of ratio in the two couplets showing the first two, and the symbol of equality between the couplets showing the third. A proportion, therefore, involves four numbers, so arranged that it will appear that the ratio of the first to the second equals the ratio of the third to the fourth. Thus, the ratio of 6 to 3 being the same as the ratio of 8 to 4, if they are formally compared, as 6 : 3=8 : 4, we have a pro- portion. Notation. A proportion may be written by placing the sign of equality between the two ratios compared; thus 2 : 4=3 : 6. Instead of the sign of equality, the double colon is generally used to express the equality of ratios, the proportion being written, 2 : 4 : : 3 : 6. The symbol of equality, however, is frequently used by the French and German mathematicians, and is always to be preferred in presenting the subject to 20 ( 305 J 306 THE PHILOSOPHY OF ARITHMETIC. learners. A proportion may be read in several different ways Thus we may read the above proportion, "the ratio of 2 to 4 equals the ratio of 3 to 6;" or "2 is to 4 as 3 is to 6." The latter is the method generally used. Definition.- A. Proportion is the comparison of two equal ratios; or, it is the expression of the equality of equal ratios. In this expression the numbers that are compared to obtain the ratio must be indicated. A proportion is thus seen to be an equation, and should be thus regarded. An equation, as gen- erally used, expresses the relation of equal numbers ; a pro- portion expresses the relation of equal ratios One arises from the comparison of quantities ; the other, from the comparison of the relations of quantities. The former is an equation between equal numbers ; the latter is an equation between equal ratios. The definition of proportion generally given is, "A propor- tion is an equality of ratios." This is true, but it is not suf- ficiently definite to constitute a perfect definition. There must be not only an equality of ratios, but a formal comparison of these ratios, to produce a proportion. This comparison must also exhibit the numbers which were compared to produce the equal ratios. Thus, the ratio .of 6 to 3 is 2, and the ratio of 8 to 4 is 2 ; here is an equality of ratios, but not a proportion. Again, if we compare the ratios 2 and 2, we have the equation 2=2, which is not a proportion, since it does not exhibit the numbers which produce the equal ratios. To give a proportion, it is essential that the ratios be compared, and that the com- parison of the numbers which give the ratios be exhibited. The mere equating of the ratios is not sufficient; the propor- tion must show the numbers which; compared, give rise to the equal ratios. A proportion, then, is not only an " equality of ratios," but it is a comparison of equal ratios, in which the comparison of the numbers compared for a ratio is ex- hibited. This idea of the exhibition of the numbers compared for the NATURE OF PROPORTION. 307 ratios, though not formally stated in the definition which I have presented, may be directly inferred from it. For, if we compare as above, 22, so far as we can see, it is merely a comparison of numbers, and not a comparison of ratios. It is true that every ratio is a number, but the converse is not true; hence 2=2 may or may not be the comparison of two ratios. Such comparison would be indefinite; therefore, to express definitely and clearly the equality of ratios, we must retain the numbers compared, to show that the equation is an expression of equal ratios, and not a mere comparison of numbers. The definition is consequently regarded as sufficiently explicit to prevent any misapprehension. . Should we wish to incorporate this idea in the definition, we might define as follows: A Pro- portion is a comparison of equal ratios, in which the numbers producing the ratios are exhibited. Kinds of Proportion. There are several kinds of propor- tion, resulting from a modification or extension of the pri- mary ideas of ratio and proportion. A comparison of three or more pairs of numbers having equal ratios, is called Continued Proportion. An expression of the equality of compound ratios is called Compound Proportion. An Inverse Proportion is one in which two quantities are to each other inversely as two other quantities. An Harnionical Proportion is one in which the first term is to the last as the difference between the first and second is to the difference between the last and the one preceding the last. We have also Partitive and Medial Proportion, which will be defined subsequently. The propor- tion requiring special consideration is Simple Proportion, or the comparison of two simple ratios. Principles. The principles of Proportion are the truths which belong to it, and which exhibit the relations between the different members. The fundamental principle of Proportion is that the product of the means equals the product of the ex- tremes. From this we derive several other principles by which we can find the value of either of the four terms when the 308 THE PHILOSOPHY OF ARITHMETIC. other three are given. There are many other beautiful princi- ples of Proportion, besides this fundamental one and its imme- diate derivatives, which are not usually presented in arithmetic, but may be found in works on algebra and geometry. They are, however, just as much an essential part of pure arithmetic as of geometry, and can all be demonstrated as easily here as there. Indeed, they belong to arithmetic rather than to geom- etry, since a ratio is essentially numerical, and hence should be treated in the science of numbers. These principles, it will be seen, are not self-evident ; they admit of demonstration. Re- membering this, it may be asked, what then becomes of the assertion of the metaphysicians, that there is no reasoning in pure arithmetic ? Demonstration. The fundamental principle of Proportion may be demonstrated in two ways. The method generally given is the following : Take the proportion 4:2:: 6 : 3. From this we have f=f ; clearing of fractions, we have 4x3 =2x6; and, since 4 and 3 are the extremes, and 2 and 6 the means, we infer that the product of the extremes equals the product of the means. This is the method generally used in algebra and geometry. Although entirely satisfactory as a demonstration, the objection might be made that though it proves that the products are equal, it does not show why they are equal. Another method which, in arithmetic, is preferred to the above, is as follows : From the fundamental idea of ratio and propor- tion, we see that in every proportion we have 2d term x ratio . 2d term : : 4th term x ratio : 4th term. Now, in the product of the extremes, we have 2d term, ratio, and 4th term, and in the product of the means, we have the same factors ; hence the products are equal. This is a simple method, clearly seen, and shows not only that the products are equal, but that they must be so, and why they are so, which the other method does not. The products are seen to be equal because in the very nature of the subject they contain the same factors. NATTJBE OF PEOPORTION. 309 The same demonstration may be put in the more concise language of algebra. Take the proportion a : 6 : : c : d, let r = the ratio, then we have a^-br, hence ab.r, and in the same way c=d.r ; hence the proportion becomes b.r : b :: d.r : d. Now, in the extremes we have 6, r, and d, and in the means we have the same factors ; hence the two products will be equal. CHAPTER IV. APPLICATION OF SIMPLE PROPORTION. QIMPLE PROPORTION is employed in the solution of prob- O lems in which three of four quantities are given, to find the fourth. These quantities must be so related that the required quantity bears the same relation to the given quantity of the same kind that one of the two remaining quantities does to the other. We can then form a proportion in which one term is unknown, and this unknown term can be found by the principles of proportion. Thus, suppose the problem to be, What cost 3 yards of cloth, if 2 yards cost $8 ? Here we see that the OPERATION. cost of 3 yards bears the Cost of 3 yds. : $8 : : 3 yds. : 2 yds ; same relation to the cost Cost of 3 yds.=-^ =$12. of 2 yards that 3 yards bears to 2 yards ; nence we have the proportion given in the margin, from which we readily find the value of the unknown term. In all such problems three terms are given to find the fourth ; from which Simple Proportion has been called the Rule of Three. It was regarded as very important by the old school of arithmeticians, and was by them called " The golden rule of three." It is now falling into disrepute, the beautiful system of analysis having, to a great extent, taken its place. The method of analysis is simpler in thought than that of proportion, and in many cases is to be preferred to the solution by propor- tion, especially in elementary arithmetic; but still the rule of (310) APPLICATION OF SIMPLE PROPORTION. 311 Simple Proportion should not be entirely discarded. The comparison of elements by proportion affords a valuable disci- pline and should be retained for educational reasons ; and moreover it is also valuable, if not indispensable, in the solu- tion of some problems which can hardly be reached by analysis. In algebra, geometry, and the higher mathematics, it is, of course, indispensable. Position of the Unknown Quantity. It is seen that, iu the solution of the preceding problem by proportion, I place the unknown quantity in the first term. This is not in accordance with general custom; other writers place the unknown quan- tity in the fourth term. I have ventured to depart from this custom, and to recommend the general adoption of such a depar- ture, for reasons which seem to me conclusive. These reasons are twofold: first, the method suggested is dictated by the laws of logic; and, second, it is more convenient in practice. Both of these points will be briefly considered. First. The law of correct reasoning is to compare the unknown with the known, not the known with the unknown. The ordi- nary method begins the proportion with the known quantities, thus comparing the known with the unknown, in violation of an established principle of logic. The method I have suggested commences with the unknown quantity, and thus compares the unknown with the known^va. conformity to the laws of thought. It seems therefore that the old method is not logically accu- rate, and that the correct method of solving a problem in Rule of Three is to place the unknown quantity in the first term. Second. The method proposed will be found to be much more convenient in practice. A proportion is more easily stated by beginning it with the unknown term. This will be especially appreciated by those who have taught Trigo- nometry. In stating a proportion so as to get the required quantity in the last term, I have seen pupils try two or three statements before obtaining the right one. It cannot be readily seen how the proportion should begin so that the unknown 312 THE PHILOSOPHY OF ARITHMETIC. quantity shall come in the last term. If, however, the pupil begins the proportion with that which he wishes to find, the other terms will arrange themselves without any difficulty. Suppose, for instance, that we wish to obtain an unknown angle of a triangle. If we reason thus : sine of the required angle is to the sine of the given angle as the side opposite the required angle is to the side opposite the given angle; the pupil will write the proportion without any hesitation. If we reverse this order, it is necessary to go through the whole comparison mentally before beginning to write, so that we may be sure to close the proportion with the required quantity. It is therefore believed that the simplest method of stating a proportion is to place the unknown quantity in the first term. The utility of this change has been frequently illustrated in my own experience. I remember, while visiting a young women's college, hearing a recitation in geometry in which the professor was trying to lead a pupil to state a proportion from which a certain line could be determined. The young lady made several attempts and failed, when I said, " Professor, let her begin with the line she wishes to find." He accepted the sug- gestion, and she immediately stated the proportion correctly. Several authors suggest that the unknown quantity should be placed sometimes in one term and sometimes in another to test the pupil's knowledge of the subject. This is a valuable suggestion ; but any position of the unknown term except in the fourth term they regard not as a general, but as an excep- tional method. Their rule is to place the unknown term last; any other arrangement is the exception. What I claim is that the placing of the unknown quantity in the first term should be the rule, and any other arrangement the exception. It is recommended also that the teacher require the learner to place it in different terms, that he may acquire a clear and complete idea of the subject. Symbol for the Unknown. Some authors employ the letter x in arithmetic as a symbol for the unknown quantity. Thus, APPLICATION OF SIMPLE PROPORTION. 313 in the problem previously presented, we may write x : $8 : : 3 : 2. This practice is derived from the French, and is commend- able. It is sometimes objected, that it is introducing algebra into arithmetic ; but such objection, however, is not valid. Al- gebra and arithmetic are not two distinct sciences, but rather branches of the same science. The former, at least in its ele- ments, is but a more general kind of arithmetic ; and it is not at all improper to introduce its concise and general language into arithmetic. I think it well, with younger pupils, to ex- press the unknown term in an abbreviated form as is indicated in the previous solution ; when pupils become familiar with this, I would use the symbol a; as a representative of it. Three Terms Statement. It is seen that in the solution of the given problem in proportion, I use four terms in the state- ment. Many authors, however, use only three terms in stating a proportion. This was the method of the old authors, when rules reigned and principles were ignored, in what might be called "the dark ages" of arithmetic. Several recent writers have broken away from the old usage, and write the proportion with four terms instead of three. It is unnecessary to say that the old method was incomplete and incorrect. An ex- pression is not a proportion unless it has four terms. The old method was merely mechanical, and gave the pupil no idea, or at least a very imperfect idea, of the true nature of proportion. The sooner the new method is generally adopted the better for science and education. Method of Statement. No subject in arithmetic is so illogi- cally presented as Simple Proportion in its application to the solution of problems. In the statement of the proportion, all reasoning seems to be completely ignored, and the whole thing becomes a mere mechanical operation for the answer. The pro- cess is as follows: "Write that number which is like the answer sought as the third term ; then if the answer is to be greater than the third term, make the greater of the two remaining numbers the second term and the smaller the first term," etc. 14 314 THE PHILOSOPHY OP ARITHMETIC. Now, though this might do well enough as a rule for get- ting an answer, to require the pupils to explain the solution by it, as is done in many instances, is to rob the subject of any claims to a scientific process. The pupil thus taught to solve his problems has no more idea of proportion than if the subject were not presented in the book. The whole process becomes a piece of charlatanism, utterly devoid of all claims to science. A better rule would be this : Write the number like the answer ; if the answer is to be greater, multiply by the greater of the other two numbers and divide by the less, etc. This would be the better method, since it makes no claims to be a scientific process, as the other does. Both methods are absurd as a pro- cess of reasoning in Arithmetic ; but the latter less so, since it makes no pretensions to be a reasoning process. What then is the true method ? I answer, if a pupil cannot state a proportion by actual comparison of the elements of the problem, he is not prepared for proportion, and should solve the question by analysis. If he uses proportion, he should use it as a logical process of reasoning, and not as a blind mechan- ical form to get the answer. He should then be required to reason thus: Since the cost of 3 yds. bears the same relation to the cost of 2 yds. that 3 yds. bear to 2 yds., we have the pro- portion, cost of 3 yds. : $8 : : 3 yds. : 2 yds. If this is not evident and cannot be readily seen, then we should dispense with proportion until the pupil is old enough to understand it, and require the problems to be solved by analy- sis. If the unknown quantity be placed in the last term we would reason thus : Since 2 yds. bear the same relation to 3 yds. that the cost of 2 yds. bears to the cost of 3 yds, we have the proportion, 2 yds. : 3 yds. : : $8 : cost of 3 yds. Cause and Effect. A new method of explaining proportion has recently been introduced into arithmetic, which may be called the method of Cause and Effect. All problems in pro- portion, it is said, may be considered as a comparison of two causes and two effects; and since effects are proportional to APPLICATION OF SIMPLE PROPORTION. 315 causes, a problem is supposed to be readily stated in a propor- tion. To illustrate, take the problem, If 2 horses eat 6 tons of hay in a year, how much will 3 horses eat in the same time ? Here the horses are regarded as a cause and the tons of hay as an effect, and the reasoning is as follows: 2 horses as a cause bear the same relation to 3 horses as a cause, that 6 tons as an effect, bears to the required effect ; from which we have a pro- portion and can determine the required term. This method was first introduced into arithmetic by Prof. H. N. Robinson, and has been adopted by several authors. The same idea was presented by an arithmetician of Verona, who distinguished the quantities into agents and patients. It is supposed that it tends to simplify the subject, enabling learners more readily to state a proportion than by a simple comparison of the elements. This supposition, however, is not founded in truth. Instead of simplifying the subject, the method of cause and effect really increases the difficulty and tends to confuse the mind. It lugs into arithmetic an idea foreign to the subject, to explain relations which are much more evident than the relation of cause and effect. Another objection to the method is that the relation of quan- tities as cause and effect is often rather fancied than real. In many cases, indeed, there is no such relation existing at all. Take the problem, "If a man walks 6 miles in 2 hours, how far will he walk in 5 hours ?" Will the pupil readily see which is the cause and which the effect ? Will the advocates of the method, tell us whether the 6 miles or the 2 hours are to be regarded as the cause? Or take the problem, "If 18d. ster- ling equal 36 cts. TJ. S., what are 54d. sterling worth?'' Would not the pupils be puzzled to tell which is the cause and which the effect? Indeed, there is no relation of <cause and effect in a large number of such problems ; and any effort to establish such a relation will confuse that which is simple and easily understood. If anything further is needed to show the incorrectness of 316 THE PHILOSOPHY OF ARITHMETIC. the method, take a problem in what is called Inverse Proportion Thus, "If 3 men do a piece of work in 8 days, in what time will 6 men do it?" Here 3 men and 8 days would be regarded as the first cause and effect, and 6 men and the corresponding number of days as the second cause and effect. Now, if we form a proportion, we have the first cause is to second cause as the second effect is to the first effect; from which we see that in this case like causes are not to each other as like effects, a conclusion which completely contradicts the fundamental prin- ciple of the relation of cause and effect. Inverse Proportion. There is a class of problems which give rise to what is called Inverse Proportion. In this the two quantities of the same kind are to each other, not directly as the other two quantities in the order of their relation, but rather inversely as those quantities. Thus, in the problem, " If 3 men build a fence in 12 days, in what time will 9 men build it?" Here we have the required time is to 12 days, not as 9 men to 3 men, but as 3 men to 9 men; that is, inversely as the order indicated by the order of the terms of the first couplet. This is sometimes called Reciprocal Proportion, since the quan- tities are as the reciprocals of 9 and 3 ; that is as -^ to ^ or 3 to 9 Many problems in Inverse Proportion may, however, be stated in a direct proportion. To illustrate, take the problem just solved. Now, if 3 men do a piece of work in 12 days, in 1 day they will do fa of it, and if a number of men do a piece of work in 4 days, in 1 day they will do \ of it ; hence, since the number of men are to each other as the work done, we have the direct proportion, "the number of men required is to 3 men, as \ to -fa," from which we can readily find the term required. If, in this proportion, we multiply the second couplet by 48, it will become 12 : 4, which gives the same proportion as that which was obtained by the method of inverse proportion. It is thus seen that, in some cases at least, the method of inverse proportion may be avoided, and the problem be expressed by a direct proportion. APPLICATION OF SIMPLE PROPORTION. 317 If, however, in the above problem the number of men in both cases had been given, and the number of days in one case required, the problem could not be conveniently stated in a direct proportion, since to do so would require the reciprocal of the unknown quantity. Should this quantity be represented by an algebraic symbol, however, we could still state the pro- portion directly, and readily find the unknown quantity. Proportion distinctly Arithmetical. The subject of propor- tion is purely an arithmetical process. Ratio is a number, hence proportion, arising from the comparison of ratios, must be numerical. These ratios may arise from comparing con- tinuous or discrete quantities, hence we may have a propor- tion wherein geometrical quantities are compared. Attention is called to the fact, however, that the principles of proportion are only generally true with respect of numbers. A propor- tion in geometry, comparing four surfaces or volumes, may be true, but the principles of a proportion can have no meaning in such a case. In taking the product of the means equal to the product of the extremes, we shall have one surface or one vol- ume multiplied by another, which can mean nothing unless they be regarded as numbers. In geometry we regard the product of two lines as giving a surface, and the product of a line and surface as giving a volume; but what idea can we attach to the product of two surfaces or two volumes ? It is thus seen that Proportion is essentially a process of numbers, and is, therefore, a branch of Pure Arithmetic. Since the principles of Proportion admit of demonstration, we inquire again what becomes of Hansel's assertion that " Pure Arith- metic contains no demonstration ?" CHAPTER Y. COMPOUND PROPORTION. A COMPOUND PROPORTION is aproportion in which one or both ratios are compound. It is employed in the solu- tion of problems in which the required term depends upon the comparison of more than two elements. In Simple Proportion the unknown quantity depends upon a comparison of two ele- ments forming one pair of similar quantities; in Compound Proportion it depends upon the comparison of several elements forming two or more pairs of similar quantities. A Compound Ratio has been defined as the product of two or more simple ratios. The expression of a compound ratio is (2 4) Jfi " 101 ' ^ Suc ^ a ra ^ be compared to an equal simple ratio, or if two such compound ratios be compared with each ( n . r> -\ other, we have a compound proportion. Thus -< ~ | ~ > : : 7 : 56 and I K . i A r : : j 7 i 1 4 r are examples of compound propor- tion. In these expressions we mean that the value of the first couplet equals the value of the second; thus, in the first pro- portion we have f X f or ^ equals ^g- ; in the second, | x y 5 ^ = fxA The subject of Compound Proportion has been even more unscientifically treated, if possible, than Simple Proportion. In no work upon Arithmetic, and indeed in no work upon Algebra, have I seen the subject presented in a really scientific manner. As a general thing, problems are given under the head of com- nound proportion, to be solved either mechanically by rale, or else by analysis, which, of course, is not compound proportion. (318) COMPOUND PROPORTION. 319 The principles of a compound proportion are not developed, and in its application it is regarded, not as a scientific process, but as a machine for working out the answer. This, of course, is not as it should be. Compound Proportion is just as much a scientific process as Simple Proportion, and demands just as logical a treatment. I will enforce what I mean by calling attention to a few of the principles df such a proportion, and then showing its scientific application. Principles. In Compound Proportion we have certain defi- nite scientific principles, as in Simple Proportion. A few of these principles will now be stated. 1. The product of all the terms in the means equals the pro- duct of all the terms in the extremes. To show the truth of this, take the proportion given in the margin. From the prin- OPERATION. ( 9 4.") C R fi~) ciples of compound ratio we 1 * ! i n i :: JT-idi have | x fV=f X ^ ; and clear. ' ' 2 x ^ x ^ ing this of fractions we have 2x5x6x14=3x7x4x10. 2x5x6xH = 3x 7x4x10, which, by examining the terms, we see proves the principle. From this principle we can immediately derive two others. 2. Any term in either extreme equals the product of the means, divided by the product of the other terms in the ex- tremes. 3. Any term in either mean equals the product of the extremes divided by the product of the other terms in the means. Other principles can also be derived, as in Simple Proportion, but the three given are all that are necessary in arithmetic. Application. In the application of Compound Proportion to the solution of problems, we should proceed upon the same principles of comparison employed in Simple Proportion. If we do not, the process is not Compound Proportion, and should not be so regarded. To illustrate the true method, we take the problem, "If 4 men earn $24 in 7 days, how much can 14 men earn in 12 days?" 320 THE PHILOSOPHY OF ARITHMETIC. In the solution of this problem by Compound Proportion, we should reason thus : The sum earned is in proportion to the number of men and the time they labor; hence the sum 14 men can earn is to $24, the sum that 4 men earn, as 14 men to 4 men, and also as OPERATION. 12 days to 7 days; giving the com- Sum : 24 : : -j ^ ! * pound proportion which is presented 24x14x12 in the margin. From this we find the 4~x~7 unknown term to be $144. Or we may enter a little more into detail, and say The sum 14 men can earn in 7 days is to the sum 4 men can earn in 7 days, as 14 men is to 4 men; and also the sum 14 men can earn in 12 days is to the sum that they can earn in 7 days, as 12 is to 7; hence we have the compound proportion given in the margin. By Analysis. The subject of Compound Proportion is some- what difficult, in fact too difficult, for young students in arith- metic. With such the method of analysis should be used instead of proportion. The analytical method is clear and simple, and will be readily understood. It should be borne in mind, however, that when we solve by analysis we are not solving by compound proportion, a fact that seems sometimes to be forgotten. In solving the preceding problems by analysis, it is necessary to pass from the 4 men to 14 men, and from the 7 days to 12 days, the sum earned varying as we make the transposition : to do this we pass from the collection to the unit, and then from the unit to the collection. The solution is as follows, the work being as indicated in the margin. If 4 men earn $24 in 7 days one man will earn of $24, and 14 men will earn 14 times or ^ of OPERATION. $24. If 14 men earn ^x $24 in 7 days, Sum= in one day they will earn ^ of ^ of $24, and in 12 days they will earn 12 times ^ of -^ of $24, which is \f- of 3 of $24, which by cancelling, we find equals $144. In- stead of putting it in the form of a compound fraction, we could PARTITIVE PROPORTION. 321 have made the reduction as we passed along ; but in compli- cated problems the method here used is preferred, as the can- cellation of equal factors will often greatly abridge the process. PARTITIVE PROPORTION The subject of ratio gives rise to several arithmetical processes which have received the name of Proportion. Among these we have Partitive Proportion, Conjoined Pro- portion, Medial Proportion, Geometrical Proportion, etc. Geo- metrical Proportion embraces Simple Proportion, Compound Proportion, Inverse Proportion, etc. The other kinds are distinguished by their special names. When we speak of pro- portion, without any qualifying word, we mean Geometrical Proportion. Geometrical Proportion has been treated in the preceding part of this chapter ; the other varieties of proportion will now be presented. The comparison of numbers gives rise to a division of them into parts which shall bear a given relation to each other. This process has received the name of Partitive Proportion. Parti- tive Proportion is the process of dividing numbers into parts bearing certain relations to each other. To illustrate, suppose it be required to divide 24 into two parts, one of which is twice the other. An equivalent problem is, " Given the sum of two numbers equal to 24, and one of the numbers twice the other ; what are the numbers ?" Origin. Partitive Proportion is a process of pure arithme- tic ; it originated, however, in the application of numbers to business transactions. Partnership is a case of Partitive Pro- portion. But, although the subject had its origin in the appli- cation of numbers, it is now, in accordance with the law of the growth of science, a purely abstract process. Gases. This subject embraces quite a large number of cases, arising from the various relations that may exist among the several parts into which a number is divided. It is evident, also, that the greater the number of the parts the more compli- 21 322 THE PHILOSOPHY OF ARITHMETIC. cated will become the process. The most important cases are the following: 1. When the parts are all equal. 2. When one part is a number more or less than the other. 3. When one part is a number of times the other. 4. When one part is a fractional part of the other. 5. When the parts are to each other as given integers. 6. When the parts are to each other as given fractions. 7. When a number of times one part equals a number of times another. 8. When a fractional part of one equals a fractional part of another. These simple cases, it is evident, may be combined with each other, giving rise to others more complicated than any of these. A little ingenuity will suggest a large number of such cases, some of which will be quite interesting. Method of Treatment. To illustrate the character of one of the simple cases and its treatment, let us take a problem and its solution. Case 8 will give us a problem like the following: Divide 34 into two pans such that | of the first part equals | of the second part. The solution of this case is as follows: If f of the first equals | of the second, of the first equals i of | or | of the second, and f of the first equals f of the second ; then | of the second, which is the first, plus 4 of the second, or O ' i O ' ^ of the second part, equals 34, etc. The other cases are solved in my Mental Arithmetic, and need not be presented here. CONJOINED PROPORTION. The comparison of numbers also gives rise to an arithmetical process which has received the name of Conjoined Proportion. Conjoined Proportion is the process of comparing terms so related that each consequent is of the same kind as the next antecedent. The character of the subject is seen by the follow- ing concrete problem: "What cost 8 apples, if 4 apples are worth 2 oranges, and 3 oranges are worth 6 melons, and 4 melons are worth 12 cents?" CONJOINED PROPORTION. 323 An abstract problem, showing that it is a process of pure arithmetic, is as follows : " If twice a number equals 4 times another number, and 3 times the second number equals 6 times a third number, and 4 times the third number equals 2 times a fourth number, and 5 times the fourth number equals 40 ; what is the first number ?" Method of Treatment. Conjoined Proportion is treated by analysis, and presents a very interesting application of the analytical method of reasoning. The problems may be solved in two ways somewhat distinct ; that is, we may begin at the latter part of the problem, and work back, step by step, to the beginning ; or we may commence at the beginning of the prob- lem and pass from quantity to quantity, in regular order, until we find the value of the first quantity in terms of the last. To illustrate, the problem given may be solved thus: Solution 1. If 5 times the fourth number equals 40, once the fourth number equals of 40, or 8, and twice the 4th, which equals 4 times the 3d, equals 2 times 8, or 16. If 4 times the 3d equals 16, once the 3d equals \ of 16, or 4, and 6 times the 3d or 3 times the 2d equals 6 times 4, or 24 ; and so on until we reach once the 1st number. Solution 2. If twice a number equals 4 times another, once the number equals ^ of 4 times, or two times the 2d ; if 3 times the 2d equals 6 times the 3d, once the 2d equals of 6 times, or 2 times the 3d, and 2 times the 2d, or the 1st, equals twice 2 times the 3d, or 4 times the 3d ; and so on until we find once the 1st in terms of the given quantity. Both of these methods are simple and logical. The first method will probably be preferred for its directness and sim- plicity. It may also be remarked that these problems can be solved by Compound Proportion, and perhaps might have been logically treated under that head. MEDIAL PROPORTION. The comparison of numbers and the combining of them in certain relations, give rise to an arithmetical process which 824 THE PHILOSOPHY OF ARITHMETIC. has received the name of Medial Proportion. Medial Propor- tion is the process of finding in what ratio two or more quan- tities may be combined, that the combination may have a mean or average value. The subject, in its application, is usually called Alligation, from alligo, I bind or unite together, the name being suggested, probably, by the method of solution, which consisted of linking or uniting the figures with a line. It may, however, have been suggested by the nature of the process itself, in which the sev- eral quantities are combined. Origin. Medial Proportion also originated in the concrete, that is, in the application of numbers. Indeed, even now it is difficult to present it as an abstract process ; that is, as a process of pure number. It is so intimately associated with the combi- nation of things of different values, that it is very difficult to apply it to the combination of abstract numbers. Still it is evidently a process of pure arithmetic ; and its importance and distinctive character, even as an application of numbers, lead me to speak of it in this connection. Gases. The subject presents a number of cases, the most important of which are the following: 1. Given, the quantity and value of each, to find the mean value. 2. Given, the mean value and the value of each quantity, to find the proportional quantity of each. 3. Given, the mean value, the value of each, and the relative amounts of two or more, to find the other quantities. 4. Given the mean value, the value of each, and the quantity of one or more, to find the other quantities. 5. Given, the mean value, the value of each, and the entire quantity, to find the quantity of each. Method of Treatment. As formerly treated, the subject was one of the most mechanical in arithmetic. The old "linking process," as presented in the text-books, was seldom understood either by teacher or pupil. Recently, however, Prof. Wood, MEDIAL PROPORTION. 825 formerly of the New York State Normal School, has made a very happy application of analysis to the solution of this class of problems, and poured a flood of light upon the subject, so that it is now oue of the most interesting processes of arithmetic. It has extended the domain of the subject also, so that it includes some of the more difficult cases of Indeterminate Analysis, for an illustration of which see my Higher Arithmetic. The method of treatment is to compare one number above the average with one below it by their relation to the average, finding how much must be taken to gain or lose a unit on the one and balancing it with the loss or gain of a unit on the other. In this way the quantities are balanced around the average, and the proportional parts of the combination derived. For an illustration of the method of treatment, see my written arithmetics. CHAPTER VI. HISTORY OF PROPORTION. l^HE Rule of Three, emphatically called the Golden RUe, -L by both ancient and modern writers on arithmetic, is found in the earliest writings upon the science of numbers. In the Eiilawati the rule is divided, as among modern writers, into direct and inverse, simple and compound, with statements for performing the requisite operations, which are said to be quite clear and definite. The terms of the proportion in the Lilawati are written con- secutively, without any marks of separation between them. The first term is called the measure or argument ; the second is its fruit or produce ; the third, which is of the same species as the first, is the demand, requisition, desire, or question. When the fruit increases with the increase of the requisition, as in the direct rule, the second and third terms must be multi plied together and divided by the first ; when the fruit dimin- ishes with the increase of the requisition, as in the inverse rule, the first and second terms must be multiplied together and divided by the third. No proof of the rule is given, and no reference is made to the doctrine of proportion upon which it is founded. Under compound proportion is given the rule for five, seven, nine or more terms. The terms in these cases are divided into two sets, the first belonging to the argument, and the second to the requisition ; the fruit in the first set is called the produce of the argument ; that in the second is called the divisor of the set ; they are to be transposed or reciprocally brought from one set to the other, that is, the fruit is to be put in the second set and the divisor in the first. (326) HISTORY OF PROPORTION. 327 The Rule of Three Direct may be illustrated by the follow- ing example : If two and a-half palas of saffron be obtained for three- sevenths of a nishca, say instantly, best of merchants, how much is got for nine nishcas ?* Statement : 359 T 2 1 Answer, 52 palas and 2 carshas. Rule of Three Inverse may be illustrated by the following examples: If a female slave, 16 years of age, bring 32 nishcas, what will one aged 20 cost? If an ox, which has been worked a second year, sell for 4 nishcas, what will one which has been worked 6 years cost ? 1st question. Statement : 16 32 20. Answer, 25f nishcas. 2d question. Statement : 2 4 6. Answer, 1 nishcas. In order to understand the solution it must be known that the value of living beings was supposed to be regulated by their age, the maximum value of female slaves being fixed at 16 years of age, and of oxen after 2 years' work; their relative value in the given problem being as 3 to 1. The rule of five terms may be illustrated by the following example : If the in- terest of a hundred for a month be five, what is the interest of sixteen for a year ? Statement : 1 12, or transposing 1 12 100 16 the fruit, 100 16 5 5 the product of the larger set is 960, of the lesser 100 ; the quo- tient is -J^ or ^, which is the answer. The interest of money, judging from the examples in Brah- To understand their problems in rule of three it must be known that a j,ala=i carshas ; a car*/o=16 mashat; uiul u fnathnb gunjas, or 10 grain* of barley. Also, a nishc(t=\t> dramma* ; a dramma=16 panas ; apana=4 cucimt, and u eacini=~X) cowry shells. 328 THE PHILOSOPHY OF ARITHMETIC. megupta and Lilawati, varied from 3^ to 5 per cent, a month, exceeding greatly the enormous interest paid in ancient Rome. It is also very high in modern India, where it is not uncommon for native merchants or tradesmen to give 30 per cent, per annum. The rule of eleven terms may be illustrated by the following example: Two elephants which are ten in length, and nine in breadth, thirty-six in girt, seven in height, consume one drona of grain ; how much will be the rations of STATEMENT. ten other elephants, which are a quarter more in height and other dimensions ? The fruit and ^ ^C denominator being transposed, the answer is gg %$*-. Dr. Peacock remarks that the principle of 7 -^ this very curious example would be rather alarm- 1 ing, if extended to other living beings besides elephants. Lucas di Borgo tells us that at his time it was usual for students in arithmetic to commit to memory one or other of two long rules which he presents. Tartaglia mentions the first of these two rules in nearly the same terms as Di Borgo, and gives also a third, which, however, differs from it only in ex- pression. This rule formed part of the system in the practice of this subject, adapted to those who had not sufficient time to acquire, genius to comprehend, or memory to retain, the rules for the reduction and incorporation of fractious; a system reprobated by Tartaglia, and attributed by him partly to the ignorance of the ancient teachers of arithmetic at Venice, and partly to the stinginess and avarice of their pupils, who grudged the time and expense requisite for attaining a perfect under- standing of the peculiarities of fractions. An arithmetician of Verona, named Francesco Feliciano da Lazesio, objects to the memorial rules of Di Borgo as being too general in assuming that two of the quantities are of one species, and two, including the term to be found, of another species; and shows that in some cases they are all of the same denomina- tion. He wishes to distinguish the quantities into agents HISTORY OF PROPORTION. 329 and patients, and these again into actual, or present, and future. The first term of the proportion is the present agent, and its corresponding patient is the second ; the third term is formed by the future agent, and its patient is the quantity to be deter- mined. This, it will be noticed, is similar to the method of cause and effect adopted by some recent authors, and supposed to be original with them. Di Borgo's method of stating and working a problem may be seen in the following example: "If a hundred pounds of fine sugar cost 24 ducats, what will be the cost of 975 pounds?" via. v a . 100 24_ 975 ~y~ x ~jf~ \ v a . o QH r V 01 040 03400 23400 (234 ducati. 950 10000 23400 100 1 The following example of the same process, with fractions in every term, is given by Tartaglia : " If 3 pounds of rhubarb cost 2 ducats, what will be the cost of 23| pounds?" lire. ducati. lire. 7*T nf x 1 1 1 2 X 1 1 3 99 ( 4 Is 1 Partitor 8' "9! 1 07 49 0590 1330(15 ducati 844 [ . 8 1 000 1680(20 grossi 844 8 665 2 dapartir 1330 The quantities, in Di Borgo's solution, are exhibited under a 330 THE PHILOSOPHY OF ARITHMETIC. fractional form, for the purpose of making the process more gen eral, being equally applicable to fractions and whole numbers. It is sufficiently curious that he should have considered it necessary to construct the galea for the division by 100. Different methods of representing the terms of the proportion were adopted by different authors. We will state a few of them as illustrating the solution of the problem, "If 2 apples cost 3 soldi, what will 13 cost?" Tartaglia states the propor- tion as follows : Se pomi 2 || val soldi 3 I che valera pomi 13. Other Italian authors write the numbers consecutively with mere spaces, and no distinctive marks between them ; thus, Pomi. Soldi. Pomi. 2 3 13 or thus, 1 ma. 2 da. 3 tia. 2 3 13 In Recorde and older English writers, they are written as follows : Apples. Pence. 2 3 13^~ ---.19 Answer. Humfrey Baker, 1562, in speaking of the rule, says, "The rule of three is the chiefest, and the most profitable, and most excellent rule of all Arithmetike. For all other rules have neede of it, and it passeth all others; for the which cause, it is sayde the philosophers did name it the Golden Rule; but now in these later days, it is called by us the Rule of Three, because it re- quireth three numbers in the operation." He writes the terms thus: 2 3 13 The custom which generally prevailed during the 11th cen- tury, was to separate the numbers by a horizontal line, as fol lows: HISTORY OF PROPORTION. 331 Apples. 2 - Pence. - 3 - Apples. - 13 Oughtred, by whom the subject of proportion was very care- fully considered, and from whom the sign, : : , to denote the equality of ratios, seems to have been derived, states a propor- tion as follows : 2. 3 : : 13 In still later times the simple dot which separated the terms of the ratios, was replaced by two dots, as in the form which is now universally employed. Compound Proportion, as has been stated, was formerly included under the rule of five, six, etc., terms, there being no division of the subject into simple and compound proportion. To illustrate, take the problem, " If 9 porters drink in 8 days L2 casks of wine, how many casks will serve 24 porters 30 days ?" In solving such problems Tartaglia usually puts the quantity mentioned once only in the last place but one, instead of in the second place. The statement will appear as follows : Divisor, 9x8. Dividend, 12 X 30 x 24 Quotient, *f|4=120. The example, " Twenty braccia of Brescia are equal to 26 braccia of Mantua, and 28 of Mantua to 30 of Rimini ; what number of braccia of Brescia corresponds to 39 of Rimini ?" given by Tartaglia, is solved as follows: Rimini Mantua Mantua Brescia Rimini 30 28 26 20 39 21840 Answer, 28. 332 THE PHILOSOPHY OF AKITHMETIC. We give another example with its solution derived from the same author. " Six eggs are worth 10 danari, and 12 danan are worth 4 thrushes, and 5 thrushes are worth 3 quails, and 8 quails are worth 4 pigeons, and 9 pigeons are worth 2 capons, and 6 capons are worth a staro of wheat; how many eggs are worth 4 stara of wheat ?" 960 1_6 10 12 4 5 3 8 4 9 2 6 4 622080 Answer, 648. ALLIGATION. The rule for Medial Proportion, or Alligation, is of eastern origin, and appears in the Lilawati, though under a somewhat limited form. It is there called suverna-ganita, or computation of gold, and is applied generally to the determin- ation of the fineness or touch of the mass resulting from the union of different masses of gold of different degrees of fine- ness. The questions mostly belong to what we call Alligation Medial. The only question given in illustration of Alligation Alternate is the following : " Two ingots of gold, of the touch of 16 and 10 respectively, being mixed together, the weight be- came of the fineness of 12 ; tell me, friend, the weight of gold in both lumps." The rule given for the solution is, " Subtract the effected fine- ness from that of the gold of a higher degree of touch, and that of the one of the lower degree of touch from the effected fine- ness ; tell me, friend, the weight of gold in both lumps ? The differences multiplied by an arbitrarily assumed number, will be the weights of gold of the lower and higher degrees of purity respectively." Statement: 16, 10. Fineness resulting, 12. If the assumed multiplier be 1, the weights are 2 and 4 mdshas respectively ; if 2, they are 4 and 8 ; if , they are 1 and 2 : thus manifold answers are obtained by varying the as- sumption. HISTORY OF PROPORTION. 333 This rule, though perfectly distinct and clear, applies to two quantities only, and there is no appearance that it was ever applied to a greater number; it involves, however, the princi- ple of the rule which is now used, recognizes the problem as unlimited, and shows in what manner an indefinite number of answers may be obtained. The extension of the rule is not entirely easy, but much more so than the invention of the orig- inal rule itself; the chief honor of the discovery of the rule belongs therefore to the mathematicians of Hindostan. The general rule was known to the Arabians and was denominated Sekis, a term meaning adulterous, inasmuch as it is not con- tent with a single, and, as it were, legitimate solution of the question. It was sometimes called Cecca by the Italians, who appear to have known nothing further of the word than its Arabic origin ; and it constitutes the alligation alternate of modern books of arithmetic. The earlier Italian writers on arithmetic, in imitation of the practice of their Arabian masters, have confined the applications of this rule almost entirely to questions connected with the mix ture of gold, silver, and other metals, with each other. This union was designated by the term consolare, which probably originated 1 iu the dreams of astrologers and alchemists, whu thought it the peculiar province of the sun to produce and generate gold ; and as the process of the alchemist in transmuting the baser metals into gold was supposed to be under the influence of the sun, this gradual refinement, which they in common tended to pro duce, was designated by the common term consolare. In later times, it was applied to silver as well as gold, and still more generally to the common union of these metals with copper. To illustrate the method of Tartaglia, take the question, "A person has five kinds of wheat, worth 54, 58, G2, TO, 76 lire the staro respectively; what portion of each must be taken, so that the sum may be 100 stara, and the price of the mixture 66 lire the staro ?" THE PHILOSOPHY OF ARITHMETIC. 1st. In the proportion of the numbers 10, 4, 10, 8 and 16. 54 58 62 70 76 10 4 10 8 16 2d. In the proportion of the numbers 14, 14, 14, 24, 24. _76, 12 54, 58, 62, to, 10 To" 10 12 4 4 4 8 ~U TI H 4 "24 24 Tartaglia has given three other solutions of this example aris- ing from a different arrangement of the ligatures. Among the English writers the method gradually assumed the form usually found in modern text-books. The method of explanation and the extension of the process as given in a few modern text- books may be ascribed to DeVolson Wood, formerly of the New York State Normal School. POSITION. Among the most celebrated rules to which Pro- portion was applied in the early text-books were those of Single and Double Position. These rules have been supplanted in this country by the simpler processes of arithmetical analysis, but they are still found in English arithmetics; and it has been suggested by a no less eminent scholar and mathematician than Dr. Hill, that they should be retained in our text-books on account of their disciplinary influences. Some historical facts concerning this old rule will be interesting to the reader. The rule of Single Position is the only one which is found in the Lilawati, where it is called Tshtacarman, or operation with an assumed number. We shall give a few examples from it, which, however, present nothing very remarkable beyond the peculiarities of the mode in which they are expressed. 1. Out of a heap of pure lotus flowers, a third part, a fifth. HISTORY OF PROPORTION. 38o a sixth, were offered respectively to the gods Siva, Yishnu, and the Sun, and a quarter was presented to Bhavani ; the remain- ing six were given to the venerable preceptor. Tell me, quickly, the whole number of flowers. Statement : , -5-, , 4 ; known, 6. Put 1 for the assumed number ; the sum of the fractions , , , , subtracted from one, leaves ^ ; divide 6 by this, and the result is 120, the number required. 2. Out of a swarm of bees, one-fifth part of them settled on the blossom of the cadamba, and one-third on the flower of a isilind'hri ; three times the difference of these numbers flew to the bloom of a cutaja. One bee, which remained, hovered and flew about in the air, allured at the same moment by the pleas- ing fragrance of a jasmin and pandanus. Tell me, charming woman, the number of bees. Statement: i, , ^: known quantity, 1; assumed 30. A fifth part of the assumed number is 6, a third is 10, differ- ence 4 ; multiplied by 3 gives 12, and the remainder is 2. Thee the product of the known quantity by the assumed one, being divided by the remainder, shows the number of bees 15. The following question is from the Manor an j an a: 3. The third part of a necklace of pearls, broken in amorous struggle, fell to the ground; its fifth part rested on the couch, the sixth part was saved by the wench, and the tenth part was taken up by the lover ; six pearls remained strung. Say of how many pearls the necklace was composed. Statement: ^, A, , ^5 remained, 6. Answer, 30. Some authors have attributed the invention of the rules of position to Diophantus, though it is impossible to discover upon what grounds. When we consider the nature and difficulty of the problems solved by him, in those parts of his works which remain, we are fully justified in supposing that the Greeks had some method of analyzing and solving such problems, or they would not have proposed them in such number and variety. The Arabs were in possession of tho rules for both Single 336 THE PHILOSOPHY OF ARITHMETIC. and Double Position, with all their applications, and in this instance had advanced far beyond their Indian masters ; and when we consider how small were the additions which they usually made to the sciences which passed through their hands, we might very naturally be inclined to suppose that their knowledge of these rules was derived from the Greeks. There is, however, a vast gap in the history of the sciences after the time of Theoh, and it is quite impossible to trace with certainty their transmission to the Arabs, or to ascertain through what channels some portion of Greek astronomy, at least, was transmitted to the Hindoos; we must therefore rest satisfied with the few hints to be gathered from authors between the 7th and 12th centuries, who had access to many writings which have since perished. The Italian writers on arithmetic derived the knowledge of these rules directly from the Arabians, distinguishing them by the Arabic name of El Cataym. The questions proposed by Di Borgo and Tartaglia are of immense variety, including every case of single and double position ; and the rules which are given for this purpose are such as would immediately result from the formula given in higher algebras. The following example is given and explained by Pi Borgo : 4. A person buys a jewel for a certain number of fiorini, I know not how many, and sells it again for 50. Upon making his calculation, he finds that he gains 3^ soldi in each fiorino, which contains 100 soldi. I ask what is the prime cost. Suppose it to cost any sum you choose; assume 30 fiorini, the gain upon which will amount to 100 soldi, or \ fiorino: 1 added to 30 makes 31 ; and you say that it makes 50 between capital and gain ; the position is therefore false, and the truth will be obtained by saying, if 31 in capital and gain arises from a mere capital of 30, from what sum will 50 arise. Multiply 30 by 50, the product is 1500; divide it by 31, the result is 48-^-f- ; and so much I make the prime cost of the jewel. Tartaglia says that such questions were frequently proposed HISTORY OF PROPORTION. 337 as puzzles by way of dessert at entertainments, and has mixed up with his other questions a large number of such problems. The practice, from some circumstances, appears to be referable to the Greek arithmeticians of the 4th and 5th centuries, and perhaps to an earlier period. Both Di Borgo and Tartaglia sought to include every possi- ble case of mercantile practice under the Rule of Three, giving numerous examples and classifying them in various ways. The Italians were also the inventors of the rule of Practice, which they regarded as an application of the Rule of Three. Tar- taglia gives some interesting and practical examples, with var- ious ingenious methods of solution. The great convenience of these rules for performing the calculations which were con- tinually occurring in trade and commerce, made them a favor- ite study with practical arithmeticians, and they assumed from time to time a constantly increasing neatness and distinctness of form. Stevinus, though, speaks of them with some contempt as forming "a vulgar compendium of the rule of three, suffi- ciently commodious in countries where they reckon by livres, sous and deniers." John Mellis, in his addition to Recorde's arithmetic, presents the rules of Practice in a very simple and complete form, calling attention to them as " briefe rules called rules of practise, of rare, pleasant, and commodious effect, abridged into a briefer method than hath hitherto been pub- lished." Later works gave them still greater compactness and brevity, and in Cocker's Arithmetic, published in 1671, after his death, and in others printed towards the end of the 17th century, they assumed the form in which they are now found in English arithmetics. The subjects of Partnership and Barter, also treated by an application of Proportion, seem to have originated with the Italians. They grew out of their business transactions, and in many cases were so complicated as to require great skill and judgment in their solution. They are interesting as presenting the type of nearly all the questions of this kind found in modern text-books. 22 SECTION II. THE PROGRESSIONS. I. ARITHMETICAL PROGRESSION. II. GEOMETRICAL PROGRESSION. CHAPTER I. ARITHMETICAL PROGRESSION. IN comparing numbers we perceive that we may have a series of numbers which vary by a common law ; such a series is called a Progression. The more general name for such a suc- cession of terms is Series, which is used to embrace every arrangement of quantities that vary by a common law, how- ever simple or complicated, and whether expressed in numbers or in algebraic or transcendental terms. The term Progression is preferred in arithmetic, and is restricted to the arithmetical and geometrical series. The constant relation existing between two or more succes- sive terms of the series is called the Law of the progression. In the series 1, 2, 4, 8, etc., each term equals the preceding term multiplied by 2, and this constant relation constitutes the law of the series. It is evident that the law which connects the terms of a series may be greatly varied, and that we may thus have a large number of different kinds of series. The only two generally treated in arithmetic are the Arithmetical and the Geometrical series, or progressions. Definition, An Arithmetical Progression is a series of terms which vary by a constant difference ; as 2, 4, 6, 8, etc. The difference between any two consecutive terms is called the common difference. In the series given, the common difference is 2. The common difference is sometimes called an arithmet- ical ratio ; it is better, however, to restrict the use of the word ratio to a geometrical ratio, and call this what it really is, a difference. ( 341) 342 THE PHILOSOPHY OF ARITHMETIC. Special attention is called to the definition of an arithmetical progression here presented. The definition usually found in our text-books is, "An arithmetical progression is a series of numbers which increase or decrease by a common difference." In the definition proposed the word vary is used to include both the increase and the decrease of the terms; and this is re- garded as an improvement upon the old definition. It has already been adopted by two or three authors, and should be generally introduced into our text-books on arithmetic. Notation. The English and American authors express an arithmetical progression by writing the terms one after another with a comma between them. The French, with more pre- cision, employ a special notation for it. They place the sym- bol, -T-, before the progression and the period (.) between the terms. Thus Bourdon writes, -=-2. 7. 12. 17. 22. . . 47. 52. 57. 62. This method has been introduced into one or two American text-books, and may, in time, be generally adopted, though the tendency seems to be to adhere to the common form of expres- sion. Cases. There are five quantities in an Arithmetical Progres- sion ; the first term, the common difference, the number of terms, the last term, and the sum of all the terms. If any three of these are given, the other two can be found from them. This gives rise to twenty different cases, in which any three terms being given, the other two may be found. These cases cannot all be solved by arithmetic, since some of them involve the solution of a quadratic equation ; they are, however, very readily treated by the principles of algebra. The two principal cases in arithmetic are as follows: 1. To find the last term, having given the first term, the common difference, and the number of terms. 2. To find the sum of the terms, having given the first term, the last term, and the number of terms. Method of Treatment. The treatment of Arithmetical Pro- ARITHMETICAL PROGRESSION. 343 gression in arithmetic is very simple. We derive the rule for finding the last term by noticing the law of the formation of a few terms and then generalizing this law. Thus we notice that the second term of an arithmetical progression equals the first term plus once the common difference, the third term equals the first term plus twice the common difference, etc. ; hence we infer that the last term equals the first term plus the product of the common difference by the number of terms less one. In finding the sum of the terms we take a series, then write under this series the same series in an inverted order, then adding the two series we see that twice the sum of the series is the same as the sum of the extremes multiplied by the num- ber of terms ; and generalizing this we obtain the rule for find- ing the sum. In algebra we reason in the same way, except that we employ general symbols, and use a general series instead of a special one. Expressing the two fundamental rules in general formu- lae, we can readily find the rest of the twenty cases by the alge- braic process of reasoning. These two simple cases, I think, should in arithmetic be expressed in the concise language of algebraic symbols. Pupils who have not studied algebra will have no difficulty in understanding them. The two rules of arithmetical progression are briefly expressed thus: 1. J = o+(n l).d; a.* = (0+0.5. History. Of the origin of the progressions and the methods of treatment, but little is known. They were the object of the particular attention of the Pythagorean and Platonic arithme- ticians, who enlarged upon the most trivial properties of num- bers with the most tedious minuteness. Directing their spec- ulations, however, to the mysterious harmonies of the physical and intellectual world, they passed over, as unworthy of no- tice, the solutions of those problems which naturally arise from these progressions, and which appear in such numbers in Hin doo, Arabic, and modern European books on Arithmetic. 344 THE PHILOSOPHY OF ARITHMETIC. Very little is known concerning the origin of the familiar problems usually found under this subject. The problem, " How many strokes do the clocks in Venice strike in 24 hours?" is supposed to be of Venetian origin. The following familiar problem is attributed to Bede : " There is a ladder with 100 steps ; on the first step is seated one pigeon, on the second step two pigeons, on the third step three, and so on increasing by one each step ; tell, who can, how many pigeons were placed on the ladder." The celebrated problem, "If a hundred stones be placed in a right line, one yard apart and the first one yard from a basket, what length of ground must a person go over who gathers them up singly, returning with them one by one to the basket ?" though found in many modern text-books, is very old, but its origin is not known. The extraordinary magnitude of the numbers which result from the summation of a geometrical series is well calculated to excite the surprise and admiration of persons who are not fully aware of the principle upon which the increase of the terms depends; and examples are not wanting among the earliest writers, where the rash and ignorant are represented as being seduced into ruinous or impossible engagements. The most celebrated of these is that which tradition has represented as the terms of the reward demanded of an Indian prince by the inventor of the game of chess ; which was a grain of wheat for the first square on the chess board, two grains for the second square, four for the third, and so on, doubling continually to sixty-four, the whole number of squares. Lucas di Borgo solved the question, and found the result to be 18446744073709551615, which he reduces to higher denominations and finds it equal to 209022 castles of corn. Fie then recommends his readers to attend to this result, as they would then have a ready answer to many of those barbioni ignari de la arithmetica who have made wagers on such questions, and have lost their money. CHAPTER II. GEOMETRICAL PROGRESSION. A GEOMETRICAL PROGRESSION is a series of terms which vary by a common multiplier ; as, 1, 2, 4, 8, 16, etc. The common multiplier is called the rate or ratio of the pro- gression ; thus, in the progression given, the rate is 2. The rate of the progression equals the ratio of any term to the pre- ceding term. When the progression is ascending, the rate is greater than a unit ; when it is descending, the rate is less than a unit. The rate is by most authors called the ratio of the series; the reason for preferring the term rate will be stated presently. Notation. The method of writing a geometrical progression, generally employed by English and American authors, is the same as that for an arithmetical progression. The French authors, however, distinguish it from an arithmetical progression by a special notation. They place the symbol -H- before the series, and separate the terms by a colon (:) ; thus, H- 2 : 4 : 8 : 16 : 32 : 64 : 128. The Rate. The constant multipler, as before stated, is gen- erally called the ratio of the series. The term rate, it is thought, is much more appropriate and precise. The objection to the word ratio is that, in the comparison of numbers, the ratio is the quotient of the first term divided by the second, while the rate of a series is equal to any term divided by the previous term ; hence, there is a seeming contradiction of the correct meaning of the term ratio. This contradiction may be only seeming, but to avoid all difficulty in this respect, it will 16* (345) 346 THE PHILOSOPHY OF ARITHMETIC. be better to use a term which is appropriate and not liable to misconception. Rate seems to be an appropriate word, since we naturally speak of the rate of increase or decrease of any- thing ; and by the rate of a progression, we mean its rate of increase or decrease. The French mathematicians make this distinction between ratio and rate ; they use the word rapport, ratio, in proportion, and raison, rate, in progression. Bourdon says, "The con- stant ratio, which exists between any term, and that which imme- diately precedes it, is called the rate of the progression.* By rapport they seem to mean about what we do by ratio ; it is probably from the idea of produce, the ratio being the product of the division. Their word raison seems to mean the same as rate, taken probably from the idea of cause, the rate being the law or cause of the terms being what they are. The term ratio, as used in relation to a progression, has given rise to a good deal of discussion and misapprehension. Some writers who use the word have taken the pains to tell us that they mean, not a direct, but an inverse ratio. Prof. Dodd says, when we speak of the ratio of a geometrical progression being 2, we mean that "the terms progress in a twofold ratio, which simply means that each term has the ratio of 2 to the preceding term ;" and similar remarks are made by other writers. By using the word rate instead of ratio, all this difficulty and misapprehension will be avoided. It is to be hoped, therefore, that the term rate will be generally adopted in speaking of the law of variation of a geometrical series. Cases. There are five quantities considered, as in arithmet- ical progression ; the first term, the rate, the number of terms, the last term, and the sum of the terms. Any three of these being given the other two can be derived from them, which gives rise to twenty distinct cases. These cannot all be solved * Ce rapport constant, qui existe entre un terme et celui qui le prficecU immediatement, se nomine la RAISON de la progression. BOURDON'S Arith metlc, page 279. GEOMETRICAL PROGRESSION. 347 by arithmetic ; the first fifteen are easily derived by common algebra, and the other five readily yield to the logarithmic cal- culus. The two cases generally given in arithmetic are the following : 1. To find the last term, having given the first term, the rate, and the number of terms. 2. To find the sum of the terms, having given the first term, the last term, and the number of terms. Treatment. The general method of treatment in a geomet- rical progression is the same as in an arithmetical progression ; and having been stated under arithmetical progression, need not be repeated here. Several cases cannot be obtained in arithmetic, since they require the solution of an equation. Four cases cannot be solved by elementary algebra, as they depend upon the solution of an exponential equation ; and in obtaining the numerical results we are obliged to make use of logarithms. The two fundamental cases should, we think, in arithmetic be expressed in the symbolic language of algebra; thus, 1. Z=ar-i; 2. #=^=-. r 1 THE INFINITE SERIES. An Infinite Series is a series in which the number of terms is infinite. In a descending pro- gression the terms are continually growing smaller; hence if the series be continued sufficiently far, the last term must be- come less than any assignable quantity ; and if continued to infinity, the last term must become infinitely small. In treating an infinite series, we regard this infinitely small quantity as zero, or nothing. Thus, in finding the sum of a ft 7** descending series, we use the formula S=~ ; and regarding 1 r the last term as nothing, the term Ir disappears, and we have d jS=, , or the sum of the terms of an infinite series descend- 1 r ing equals the first term divided by 1 minus the rate. 348 THE PHILOSOPHY OF ARITHMETIC. This reduction of the last term to zero presents a difficulty not easily explained. The question arises, how can the last term become zero? At what point does a term become so small that, when multiplied by the rate, the product shall be nothing? To illustrate the difficulty, take the series 1, ^, ^, . etc., in which the rate is i. Now if this series be continued to infinity, the last term is supposed to be zero. This supposition seems to involve the idea that the term just before the last is so small that ^ of it is nothing. Who can conceive of such a term ? Who can trace the series down through all the differ- ent values, until we reach a term so small that one-half of it is nothing? This of course cannot be done. The mind shrinks from the effort ; it is unable to grasp the infinitely small. In- deed, neither the infinitely great nor the infinitely small can be positively conceived ; an infinite quantity and an infinitesimal are both beyond the grasp of the human mind. What shall we do then ? Shall we deny that the last term is infinitely small, or zero ? Certainly not : to assume that it is not infinitely small involves a greater difficulty than the sup- position that it is infinitely small. Fix upon any term, how- ever small, and we see that it can be continually divided, and that the division will continue as long as there is a term to be divided, and can only terminate when the term becomes too small to divide, or zero. Hence, to conceive that the infinite term is not zero, is to suppose that the division stopped when it could have proceeded, which is absurd ; consequently, it is absurd to suppose that the last term is not zero. The question then stands thus: we cannot comprehend that the last term is zero, and to conceive that it is not zero is absurd. We are thus in the dilemma that we must believe either the absurd or the incomprehensible. We cannot believe the absurd; we rather accept the incomprehensible. We are therefore forced to the conviction that the last term is zero, even though we cannot fully conceive it to be so. We believe that which we cannot fully understand, because not to believe it leads to an ab GEOMETRICAL PROGRESSION. 349 nrdity, and the mind is so constituted that it will accept the jicomprehensible sooner than the absurd. We take it upon faith; it is the place in science "where reason falters" and faith accepts. This method of considering the subject presents an excellent illustration of the operation of the intuitive power in many questions of religious faith. I may not be able to comprehend a first cause ; but I know there must be one, or else I am in- volved in an absurdity, and the human mind cannot rest in the absurd. It may be remarked that the point of difficulty here considered, is one that frequently occurs in mathematics. The infinitely small is an important element in mathematical inves- tigations. We make use of it in geometry, and in calculus it is the fundamental idea upon which the science is based. The most satisfactory method of removing any doubt that one may have upon the assumption that the last term reduces to zero, is to take a problem which may be solved by an infinite series, and which can also be solved without it. If the result obtained by supposing the last term to be zero, agrees with the result otherwise obtained, the conclusion that the last term is zero must be accepted, whether we can conceive it or not. Such a problem is the following: "Abound and fox are 10 rods apart, and the hound pursues the fox ; how far will the hound run to overtake the fox, if the latter runs -fa as fast as the hound?" Looking at this problem in one way, we see that when the hound has run the 10 rods the fox has run 1 rod, and they are then 1 rod apart. When the hound runs this rod, the fox has run y 1 ^ of a rod ; hence they are then -j 1 ^ of a rod apart. When the hound runs this -fa of a rod, they are -fa of -fa, or -j-J-y of a rod apart; hence the distance the hound will run to catch the fox is correctly represented by the sum of the series 10-j-l +T^+Tfo+TnVH-io&oo+ etc -> to an incite number of terms The sum of this series, obtained by the method of infinite series, which regards the last term as zero, equals 10-h(l 1^)= 350 THE PHILOSOPHY Otf ABITHMBTIC. =11^ rods. Hence the hound runs 11| rods to catch the fox. The problem may also be solved by the following simple method of analysis : By the conditions, ten times the distance the fox runs equals the distance the hound runs ; and this di- minished by the distance the fox runs, is 9 times the distance the fox runs, which equals what the hound gains on the fox, or 10 rods, the distance they were apart ; then once the dis- tance the fox runs equals ^ of a rod, and 10 times the distance the fox runs, which is the distance the hound runs, equals 10x J ^=- L ^ J -, or 11 rods. Or, we may solve it even more simply thus: the hound gains 9 rods in running 10, hence to gain 1 rod he will run -^ of a rod, and to gain 10 rods, so as to catch the fox, he will run 10 times *-, or 1^=11^ rods. This result corresponds with that obtained by the summation of the infinite series ; hence the supposition involved in that solution, that the last term of the series equals zero, must be correct. This problem is sometimes given as a puzzle, in which it is said that since there is always one-tenth of the previous dis- tance between them, the hound will never catch the fox. The fallacy consists in inferring that because there is an infinite number of successive operations, it must require an infinite length of time to perform them. A problem similar to this is the following : "A ball falls 8 feet to the floor and bounds back 4 feet, then falling bounds 2 feet, and so on; how far will it move before coming to rest?" Solv- ing this, we find the distance to be 24 feet. It is sometimes supposed in this problem, that the body will never come to rest ; this is a mistake, for though there will be, in theory at least, an infinite number of motions, they will be accomplished in a finite period of time. The reason of this is, that the infi- nitely small motions are made in infinitely small periods of time, the sum of which does not exceed a finite period. It should be remarked that some writers maintain that the GEOMETRICAL PROGRESSION. 351 results in the infinite series are not absolutely correct, but are merely approximations ; thus, that the sum of the series a+i+i +etc., is not absolutely 1, but only approximately so; in other words, that all we can affirm concerning it is that it comes nearer and nearer to 1 as we increase the number of terms, though it can never reach 1. Unity is the limit towards which it is always approaching, which it never can exceed, and indeed, which it never can reach. This is the doctrine of limits, and is the one usually preferred by modern mathematicians. By this doctrine, in summing the infinite series descending, we are attempting to find the limit towards which the series is approaching, but which it can never reach. This is regarded as the most logical method of consider- ing the subject. Logic may admit self-evident propositions, but it does not admit conclusions that cannot be logically derived from these self-evident assumptions. Thought cannot follow an infinite series step by step to the zero term ; hence a conclusion based on the assumption of a zero term is regarded as illogical and inadmissible. This doctrine of limits as applied to the infinite series, while apparently logical, is not without its difficulties. It would seem to lead to the conclusion that in the case of the " fox and hound problem," given above, the hound would never catch the fox ; unless, as a boy once remarked, " he gets near enough to grab him." So in respect to the elastic ball dropped upon a pave- ment; if the result is only approximately true, does it not follow that the ball never comes to rest, but continues bounding forever ? Here, as in many other cases, faith in the incomprehensible seems more satisfactory than a timid skepticism. It will be interesting to notice that the two different series, + i~f~3T~^'8T"'~ etc ' an< ^ i+^+TV^TJ"^ 610 -' are eac ^ ec l ua l to tne same fraction ^. It is also an interesting truth that the pum of the series beginning with ^, and decreasing at the rate of ^, is just equal to 1. SECTION IIL PERCENTAGE. 23 I. NATURE OP PEBCENTAOB. II. NATURE OF INTEREST. CHAPTER I. NATURE OF PERCENTAGE. T)ERCENTAGE is a process of computation in which the -C basis of the comparison of numbers is a hundred. The same idea may also be expressed more briefly in the definition, Percentage is the process of computing in hundredths. The former definition was first presented in one of the author's arithmetical works. Up to this time no definition had been given of Percentage as a process of arithmetic. In the text- books, the word was merely defined as meaning so many of a hundred. Soon after this publication appeared, one or two other authors adopted a definition similar to the one given above, presenting the subject as a department of the science ; and in time, it is presumed, all will define it as a process of arithmetic. It will be readily seen that Percentage has its origin in the third division of the science of arithmetic; namely, Comparison. We may compare numbers and determine their relations with respect to their common unit or basis. This is the first and simplest case of comparison, and gives rise to Ratio and Pro- portion. We may also compare numbers with respect to some number agreed upon as a basis of comparison, and develop their relations with respect to this basis. When this number is one hundred, we have the process of Percentage. It is thus seen that the idea of the subject presented in the definition given above is correct. Percentage originated in the fact of the convenience of esti- mating by the hundred, in a decimal scale. It derives its im- (355) 356 THE PHILOSOPHY OF ARITHMETIC. portauce and has received so full a development, partly at least, from the fact of our having a decimal currency. It occupies a more prominent place in American than in English text-books, where the money system is not decimal. Its principal use is in its application to business transactions relating to money, as will be seen in the various ways in which it is employed. It admits, however, of a purely abstract development, entirely independent of concrete examples ; and is, therefore, a process of pure arithmetic. Quantities. Percentage embraces four distinct kinds of quan- tities, the base, the rate, the percentage, and the amount or difference. The Base is the number on which the percentage is estimated. The Bate is the number of hundredths of the base. The Per- centage is the result of taking a number of hundredths of the base. The Amount or Difference is the sum or the difference of the base and percentage. The Amount and Difference are the same kind of quantities, and it would be well, in Percentage, to have some one term which would include them both. In several of the applications we have such a word ; as selling price in Profit and Loss, proceeds in Discount, etc. The expression Resulting Number has been used, but this is a little awkward and inconvenient. The term Proceeds, meaning that which results or comes forth, I have sometimes thought of adopting, and indeed have adopted in one of my works. Some term, in place of amount and difference as used in percentage, is a scientific necessity, and Proceeds is recommended. The Eate was originally expressed as a whole number, and the methods of operation based upon such expression. Latterly it is becoming the custom to represent the rate as a decimal, and to operate with it as such. This is much the better way, and will probably become universal. It gives greater simplicity to the rules, makes the treatment more scientific, and is quite as readily understood by pupils. It may be remarked that NATURE OF PERCENTAGE. 357 the definition of the rate will vary according to which of these forms is taken. The definition above given regards the rate as a decimal. It will thus appear that there is a slight distinction between the term Rate and the expression the rate per cent. Per cent. means by the hundred ; rate per cent, means a certain number of or by the hundred ; while Hate means a certain number of hundi'edths. When money is loaned at 6 per cent, the rate per cent, is 6; but the Hate is .06. Thus Hate and rate by the hundred, are about identical in meaning. We may conse- quently define the Rate to be the number by which we multiply the base in order to obtain any required per cent, of it; and this is what is intended in the definition, The rate is a num- ber of hundredths of the base. Gases. It has been a question among arithmeticians under how many cases Percentage should be presented. There being four distinct classes of quantities five, if like some authors we regard the amount and difference as distinct any two of which being given, the others may be found, it will be seen that there are quite a large number of possible theoretical cases. What is the simplest and most scientific classification of these various cases? In other words, what are the general cases of Per- centage? It has been quite customary to present the subject under six distinct cases, and this affords a very practical view of the subject. Authors, however, have not been uniform in their treatment. I believe that the best way is to present the subject under three general cases, each of which will contain two or three special cases, as we regard the amount and differ- ence, as one or two classes of quantities. Uniting the amount and difference under one general term, as proceeds, we shall have throe .general cases, each including two special cases, making six cases in all ; regarding the amount and difference as two distinct quantities, we shall have three special cases under cjicli general case, making nine cases in all. These three general cases may be formally stated as follows: 358 THE PHILOSOPHY OF ARITHMETIC. 1. Given, the base and the rate, to find the percentage and the proceeds. 2. Given, the base and either the percentage or the proceeds, to find the rate. 3. Given, the rate and either the percentage or the proceeds, to find the base. Treatment. There are two distinct methods of treatment in Percentage, which may be distinguished as the Analytic and the Synthetic methods. The Analytic Method consists in reducing the rate to a common fraction, and taking a fractional part of the base for the percentage, and operating similarly in the other cases. It differs particularly from the other method in the solution of the second and third cases, as will be seen by the solution of a problem. It is the method for mental analysis, and is especially suited to the subject of Mental Arithmetic. To illustrate the analytic method, take the problem, "What is 25% of 360?" We reason thus: 25% of 360 is -^ or { of 360, which is 90. To find the base take the problem, " 90 is 25% of what number?" The solution is, If 90 is 25%, or \, of some number, of the number is 4 times 90, or 360. The case of finding the rate per cent, is solved in a similar manner. The Synthetic Method consists in preserving the rate in the form in which it is presented, and operating accordingly. In the synthetic method there are two ways of operating : the first consists in using the rate as a whole number, and dividing or multiplying by a hundred ; the second operates with the rate in the form of a decimal, according to the principles of decimal multiplication and division. There has, for several years, been a tendency towards the latter method, and arithme- ticians are now generally agreed in its favor. This latter method is greatly to be preferred on account of its simplicity and scientific character. The difference may be shown by a rule for one of the cases. When the rate is used as a whole number, the rule for finding the percentage is, Multiply the base by the rate, and divide the product by 100 NATURE OF PERCENTAGE. 359 When the rate is used as a decimal, the rule is, Multiply the base by the rate. A similar difference will be found to exist in the rules for all the cases. Another consideration in favor of using the rate as a decimal is the ease with which the rules for the other cases are derived from the first. Assuming that the percentage equals the base multiplied by the rate ; it immedi- ately follows that the base equals the percentage divided by the rate, or the rate equals the percentage divided by the base. To illustrate the method preferred, suppose we have the problem in Case 1., "What is 25% of 360?" We would rea- son thus : Twenty-five per cent, of 360 equals 25 hundredths times 360, or 360 x. 25, which by multiplying we find to be 90. To illustrate Case 2, take the problem, " 90 is 25% of what number?" We would solve this as follows: If 90 is 25% of some number, then some number multiplied by .25 equals 90; hence this number equals 90 divided by .25, or 90-=-. 25, which by dividing we find is 360. To illustrate Case 3, take the problem, " 90 is what per cent, of 360 ?" The solution is as follows : If 90 is some per cent, of 360, then 360 multiplied by some rate equals 90 ; hence the rate equals 90 divided by 360, or 90-^-360, which is .25, or 25%. The solution of problems including the proceeds is quite similar, and need not be presented here in detail. The particu- lar method of explanation will be found in my Higher Arith- metic. Formulas. These synthetic methods and rules may all be presented in general formulas, as follows: CASE I. CASE II. CASE III. 1. bxr=p 1. p-r-r=b 1. p-i-b=r 2. &x(l+r)=4 2. A+(l + r)=b 2. A+b=\+r 3. 6x(l r)=D 3. X>-4-(l r)=b 3. D+b=l r The 2d and 3d formulas of each case may be united in one ; thus, using P for proceeds, P=&x(lr); &=P-*-(lr); r=P-j-6 1, or 1 P^b. 360 THE PHILOSOPHY OF ARITHMETIC. Applications. The applications of Percentage are very ex- tensive, owing to the great convenience of reckoning by the hundred in financial transactions. These applications are of two general classes ; those not including the element of time, and those which include this element. The following are the most important of these two classes of applications : IST CLASS. 1. Profit and Loss. 2. Stocks and Dividends. 3. Premium and Discount. 4. Commission. 5. Brokerage. 6. Insurance 7. Taxes. 8. Duties and Customs. 9. Stock Investments. 2o CLASS. 1. Simple Interest. 2. Partial Payments. 3. Discounting. 4. Banking. 5. Exchange. 6. Equation of Payments. 7. Settlement of Accounts. 8. Compound Interest 9. Annuities. The different cases of the first class are solved as in pure percentage, and the rules are almost identical, the technical terms being substituted for base, percentage, etc. The solutions of the various cases of the second class are somewhat modified by the introduction of the element of time. The development of these various cases would occupy too much space for this work, and moreover does not constitute a part of the philosophy of arithmetic; we shall, therefore, give only a single chapter on the genera] nature of Interest. CHAPTER II. NATURE OF INTEREST. T)ERCENTAGE embraces two general classes of problems, JL those that involve the element of time, and those that do not involve this element. The most important application of percentage into which this element enters is Interest ; and in- deed all such applications may be embraced under this general term. Interest may be defined as money paid, or charged for the use of money. It is usually reckoned as so many units on a hundred, and is thus included under the general process of Per- centage. The sum upon which interest is reckoned is called the Principal, in distinction from the interest or profit, which is subordinate to it. The sum of the interest and principal is called the Amount. Interest is either Simple or Compound. Simple Interest is that which is reckoned or allowed upon the principal only, during the whole time of the loan. Compound Interest is reckoned, not only on the sum loaned, but also on the interest as it becomes due. Interest unpaid is regarded as a new loan upon which interest should be paid. Simple Interest. In considering the subject of simple inter- est, the primary object is to find the interest on a given princi- pal for a given time and rate. Various methods have been devised for the solution of this problem. The simplest in principle and most natural, is to find the interest for one year by multiplying the principal by the rate, and multiplying this interest by the time expressed in years. The objection to this 16 ( 361) 362 THE PHILOSOPHY OF ARITHMETIC. method in practice arises from the fact that the time is often given in months and days, which frequently reduce to an incon- venient fractional part of a year. This difficulty has led to a modification of the rule proposed above, which is known as the method of " aliquot parts." The importance of a method that can be readily applied in bus- iness, has led to the exercise of considerable ingenuity in order to discover the shortest and simplest rule in practice. The method now regarded as the simplest is that known as the "six per cent." method. It is based on the rate of 6%, which is the usual rate in this country, and may be expressed as follows : Gall half the number of months cents, and one-sixth of the number of days mills, and multiply their sum, which will be the interest of $1 for the rate and time, by the principal. Another way of stating this rule is, Regard the months as cents, and one-third of the days as mills, and multiply their sum by one-half of the principal. For short periods a modi- fication of the rule, which may be popularly expressed, Mul- tiply dollars by days and divide by 6000, is the most convenient in practice, and is very generally employed by business men. There are also many other methods of working interest which need not be stated here. The general method of finding the interest of a principal may be expressed in a general formula as in Percentage. The gen- eral formula is i=ptr, which is readily remembered by the sen- tence which it suggests "I equals Peter." The several cases which arise in interest can be readily derived from this funda- mental formula. These several rules may be expressed as fol- lows: 1. i=ptr. 3. t=i-t-pr. 2. p = i-i-tr. 4. r = i-i-pt. It is objected to the "six per cent, method," that it gives too great an interest, since it reckons only 360 days in a year ; and it has been suggested that to compute the interest on a loan by this method would be to take usury, and in some states would NATURE OF INTEREST. 363 result in a forfeiture of the debt, or some other penalty. This seems like putting a very nice point on the matter, though it is true that the six per cent, method gives a little more interest than when we reckon 365 days to the year. To obtain exact interest, we find the interest for the years, multiply the interest of one year by the number of days, and divide by 365, and take the sum of the two results. A full presentation of the applica- tions of interest to business and the latest methods of treatment may be found in the author's Higher Arithmetic. Mates of Interest. It is a noteworthy fact that the propriety of receiving interest for the use of money, has been questioned. Indeed, the practice has been censured in both ancient and modern times as an immorality and a wrong to society. It may seem that so absurd a notion hardly needs a passing no- tice, for it is clear that a similar objection may be made to the charge of rents, or even to profits of any kind. A capitalist may invest his money in business and receive a certain return for it ; and if he chooses to let some one else invest it and have the care of such investment, it is clear that he should receive some remuneration for surrendering to another the profit he might have made himself. Again, the borrower can with cap- ital secure a large return of profit in business, and is not only entirely willing to pay for the use of such capital, but is in equity under obligations to do so. Interest on loans is, there- fore, a benefit to both the borrower and lender; and should therefore be both required and allowed. The rate of interest is determined strictly by the principle of competition. When the capital to be invested exceeds the demands of borrowers, the rate of interest is low ; when the demand is in excess of the capital, the rate will be high. The rate will vary also with the security of the loan ; thus the rate on landed mortgages is usually lower than on property less secure and certain, and consequently state loans are usually made at low rates. A lender assumes that he must be paid something for the risk of a loan, and that the greater the risk 364 THE PHILOSOPHY OF ARITHMETIC. the greater the charge. It is on this principle that high inter est is often said to be synonymous with bad security. A high rate of interest may also be due to large profits on capital. In a community where the returns on capital are large, as in rich mining districts for instance, all who have capital would desire to invest, and consequently the difficulty of obtaining a loan would increase and higher rates would obtain. In such cases the opportunity for large gains by the capitalist and the in- creased demand by the borrower would both conspire to increase the rate of interest. The rates of interest have usually been regulated by govern- ments. This action is founded upon a variety of reasons. It has been argued that lenders are unproductive consumers of part of the profit which is produced by labor. Such a notion leaves out of sight, however, that production is impossible without capital, and that capital is accumulated and employed with a view to profit. It is also held that if the state does not regulate rates, borrowers will be open to fraud and extortion on the part of unprincipled lenders. This is the principal con- sideration in favor of state control of interest rates ; and yet there are valid if not unanswerable objections to it. It is, of course, the duty of the government to protect the citizen against usury and fraud ; but most of the considerations in favor of regulating rates of interest will apply to the regulation of the prices of food, land, wages, etc. It seems to be a growing opinion that capital should seek investment at rates determined by natural laws of demand and supply, as the prices of other property are regulated, and not be controlled by legislative en- actment. Historical. The payment of interest on money has been the custom from very early times. We learn from the New Testament that it was paid on bankers' deposits in Judea, though the Jews were forbidden by the laws of Moses to exact interest from one another. In Europe, interest was alternately prohibited and allowed, the church being generally hostile to NATURE OF INTEREST. 365 the practice. In Italy, the trade in money was recognized, and the custom of borrowing and lending was common. In England, it was first sanctioned by the Parliament in 1546, the rate being fixed at 10 per cent.; but in 1552 it was again pro- hibited. Mary, however, borrowed at 12 per cent., which ap- pears to have been the usual rate at that period at Antwerp. In 1571, it was again made legal at 10 per cent., a rate at which the Scotch Parliament fixed it in 1587. The rate fell at the beginning of the seventeenth century, James I. having borrowed in Denmark at 6 per cent. In 1624, it was reduced to 8 per cent.; in 1651, to 6 per cent.; in 1724, to 5 per cent, at which legal rate it remained until all usury laws were re- pealed, an event which occurred only a few years ago. In 1773, it was limited to 12 per cent, in India. In 1660, the rate in Scotland and Ireland was from 10 to 12 per cent.; in France 7 per cent.; in Italy and Holland 3 per cent.; in Spain from 10 to 12 per cent.; in Turkey 20 per cent.; but the East India Company, while the legal rate was 6 per cent., continued to borrow at 4 per cent. The term Usury, meaning the " use of a thing," was origi- nally applied to the legitimate profit arising from the use of money, and meant merely the taking of interest for money. Laws were established in various countries fixing the amount of interest or usury, and the evasion of these laws by charging excessive usury, led to the present use of the term. By the old Roman law of the Twelve Tables, the rate of interest allowed as legitimate was the usura centesima, which was strictly 1 per cont. a month ; and has been supposed by some to have amounted to 12, and by others to 10 per cent, a year. The Roman laws against excessive usury were frequently renewed and constantly evaded, and the same is true of other countries. In England, during the reign of Henry VIII., 10 per cent, was allowed; by 21 James I., 8 percent.; by 12 Charles II., 8 per cent.; by 12 Anne, 5 per cent. Subsequently to the passage of the latter act, the usury laws wore relaxed by several 866 THE PHILOSOPHY OF ARITHMETIC. statutes, and they were ultimately repealed in 1854. Any rate of interest, however high, may now be legally stipulated for, but 5 per cent, remains the legal interest recoverable on all contracts, unless otherwise specified. Much concern has been shown by governments in attempt- ing to fix rates of interest, and prevent usury. The legislation of Solon relieved the Athenian mortgagors ; and during many years of the Roman Republic, the regulation of loans, the limi- tation of the rate of interest, and the relief of insolvent debtors, formed a perpetual topic of agitation, and finally of legislation. In most of the European countries the administration has busied itself, from time to time, in fixing rates of interest, and in denouncing or forbidding usurious bargains. Such legisla- tion has, however, proved vain ; for while the most stringent laws were in force, high rates of interest on loans were com- mon, the law being incompetent to provide against evasion of the statute. The legal rate of the United States government is 6 per cent. Each State fixes its own rate, and attaches its special penalties for usury. In several of the States the usury laws have been repealed, and the general tendency is to allow an open market to the investment of capital. Origin of Methods. The importance of a knowledge of the principles of interest, discount, etc., led arithmeticians to notice these subjects at an early day. Interest was early divided into Simple and Compound. Compound Interest was properly called usura, and was rarely practised in the transactions of merchants with each other. Stevinus terms compound interest, interest prouffitable, or celuy qu'on ajouste au capital, whilst the corresponding discount is termed interest dommageable, or celuy qu'on soubstrait du capital. Problems in simple interest were by Tartaglia and his pre- decessors, solved by the Rule of Three. In calculating the interest of a sum from one day to another, the determination of the number of days in the interval seemed somewhat embar- NATURE OF INTEREST. 367 rassing, and Tartaglia gives a rule for this purpose of which he seems somewhat proud. In passing from one city of Italy to another an additional source of embarrassment presented itself in the different days on which the year was supposed to com- mence, being reckoned at Venice from the 1st of March, at Florence from the Annunciation of the Virgin, and in most other cities of Italy from Christmas day. Tartaglia has noticed five methods of finding the amount of a sum of money at compound interest. Suppose the question to be to find the amount of L300 for 4 years at 10 per cent, a capo d'anno ; the first method is by the following four state- ments : 100 : 300 : : 110 : 330 100 : 330 : : 110 : 363 100 : 363 : : 110 : 399^ 100 : 399 T % : : 110 : The second method merely replaces 100 and 110 by 10 and 11 in the proportion ; the third, which is his own method, mul- tiplies 300 four times successively by 11, and divides the last product by 10,000 ; the fourth consists in adding four suc- cessive tenths to the principal; the last in calculating the amount for L100, and then finding the amount of L300, or any other proposed sum, by a simple proportion. With the exception of discount at compound interest and its ap- plication to correct in part the conclusion respecting the values of annuities, there are few, if any, other questions of compound interest which Tartaglia and his contemporaries can be said to have resolved. A very natural difficulty arose in the solution of questions of this kind : " What is the interest of 100 for 6 months, interest being reckoned at the rate of 20 per cent, per annum ?" Lucas di Borgo and others made out that this would be 10 ; that is, they calculated that, simple interest only being allowed, it was a matter of indifference into how many por- tions of time the whole period was divided, whether into months or half-years. Lucas di Borgo has an article on calculating tables of inter- 368 THE PHILOSOPHY OF ARITHMETIC. eat in which he speaks of their great utility, thereby showing that such tables were in use in Italy, although no work of that date containing them is known to be extant. The first com- pound interest tables now known are those which are presented by Stevinus in his arithmetic, which give the present worth of 10,000,000 from 1 to 30 years, in sixteen tables, the interest being reckoned successively from 1 to 16 pejr cent., and in eight other tables, where the interest is differently reckoned, accord- ing to the custom of Flanders. The origin of the various modern methods of calculating in- terest is not known. The method by " aliquot parts" is a fav- orite rule of the English arithmeticians, and probably originated with them. The " six per cent, method" has been attributed to a Mr. Adams, author of a work on arithmetic. The partic- ular form of the six per cent, method popularly stated, "multi- ply dollars by days and divide by 6000," was used among business men before it was introduced into any arithmetic, and is presumed to have had its origin in some counting-house, but it is not known where. SECTION IV. THE THEORY OF NUMBERS. I. NATURE OF THE SUBJECT. II. EVEN AND ODD NUMBERS. III. PRIME AND COMPOSITE NUMBERS. IV. PERFECT, IMPERFECT, ETC., NUMBERS. V. DIVISIBILITY OF NUMBERS. VI. DIVISIBILITY BY THE NUMBER SEVEN. VII. PROPERTIES OF THE NUMBER NINE. CHAPTER I. NATURE OF THE SUBJECT. T'HE Theory of Numbers, as generally presented, embraces the classification and investigation of the properties of numbers. This subject has engaged the attention and enlisted the talents of many celebrated mathematicians. The ancient writers, who did little for the development of arithmetic as a science or an art, spent much time in theorizing upon the pro- perties of numbers. The science of arithmetic with them was mainly speculative, abounding in fanciful analogies and mys- terious properties. Pythagoras attributed to numbers certain mystical properties, and seems to have conceived the idea of what are now termed Magic Squares. Aristotle, amongst other numerical specula- tions, noticed the practice, in almost all nations, of dividing numbers into groups of tens, and attempted to give a philo- sophical explanation of the cause. The earliest regular system of numbers is that given by Euclid in the 7th, 8th, 9th, and 10th books of his " Elements," which, notwithstanding the embarrassing notation of the Greeks, and the inadequacy of geometry to the investigation of numerical properties, is still very interesting, and displays, like all other parts of the same celebrated work, that depth of thought and accuracy of demon- stration for which its author is so eminently distinguished. Archimedes, also, paid particular attention to the powers and properties of numbers. His tract, entitled " Arenarius," con- tains a method of multiplying and dividing which bears a con- siderable analogy to that which we now employ in multiplication (371) 372 THE PHILOSOPHY OF ARITHMETIC. and division of powers, and which some modern writers have thought inculcated the principles of our present system of loga- rithms. Before the invention of algebra, however, but little progress could be made in this branch of the science ; accord- ingly we find that comparatively few principles had been dis- covered until the time of Diophantus. This eminent mathema- tician, who is the author of the most ancient existing work on the subject of algebra, presents many interesting problems in the properties of numbers ; but, owing to the difficulties of a complicated notation and a deficient analysis, little progress was made, compared with the advance of modern times. From the time of Diupliantus the subject remained unnoticed, or at least unimproved, until Bachet, a French analyst, under- took the translation of Diophantus into Latin. This work, which was published in 1621, contained many marginal notes of the translator, and may be considered as presenting the first germs of our present theory. These were afterward consider- ably extended by Format, in his posthumous edition of the same work, published in 1670, which contains many of the most elegant theorems in this branch of analysis; but they are gen- erally left without demonstration, which he explains in a note by saying that he was preparing a treatise of his own upon the subject. Legendre accounts for the omission by saying that it was in accordance with the spirit of the times for learned men to propose problems to each other for solution. They generally concealed their own method in order to obtain new triumphs for themselves and their nation ; and there was about this time an especial rivalry between the English and French mathematicians. Thus it has happened that most of the demon- strations of Fcrmat have been lost, and the few that rema'ti only make us regret the more those that are wanting. The most of these theorems remained undemonstrated until the subject was again renewed by Euler and Lagrange. Euler, in his "Elements of Algebra," and some other publications, de- monstrated many of the theorems of Fermat, and also added NATURE OF THE SUBJECT. 373 some interesting ones of his own. Lagrange, in his additions to Euler's Algebra and in other writings, greatly extended the theory of numbers by the discovery of many new properties. The subject has received its largest contributions, however, from the hands of Gauss and Legendre. Legendre, in his great work, " Essai sur la Theorie des Nombres," was the first to reduce this branch of analysis to a regular system. Gauss, in his " Disquisitiones Arithmetic, " opened a new field of inquiry by the application of the proper- lies of numbers to the solution of binomial equations of the form, x" 1 0, 011 the solution of which depends the division of the circle into n equal parts. This solution he accomplished in several partial cases; whence the division of the circle into a prime number of equal parts is performed by the solution of equations of inferior degrees; and when the prime number is of the form 2"-f 1 the same may be done geometrically a prob- lem that was far from being supposed possible before the publi- cation of the work mentioned. The most celebrated English work on the subject is that of Peter Barlow, published in 1811, from the preface of which most of the preceding historical facts have been culled. It pre- sents a clear and concise statement of the principles of the sub- ject, and contains several original contributions, among which may be mentioned a demonstration of Fermat's general theorem, on the impossibility of the indeterminate equation x n y"=z", for every value of n greater than 2. This demonstration, how- ever, has been tacitly ignored by mathematicians; and tho French Institute and other learned societies have continued to propose the problem for solution. Almost every modern mathematician of eminence, however, has contributed more or less to the advancement of the theory. Ln the collected works of Euler, Gauss, Jacobi, Cauchy, Dirichlet, Lagrange, Eisenstein, Poinsot, and others, numerous memoirs on the subject will be found ; whilst the recent mathe- matical journals and academical transactions contain researches 374 THE PHILOSOPHY OF ARITHMETIC. in the same field, by all the ablest living mathematicians. One of the most complete treatises on the subject is that of Prof. H. J. S. Smith in the article entitled, " Reports on the Theory of Numbers," which commenced in the Transactions of the British Association for 1859. It embraces a lucid, critical his- tory of the subject, rendered doubly valuable by copious refer- ences to the original sources of information. It will be seen from this brief statement that the subject of the theory of numbers is one of great magnitude and difficulty requiring the application of the principles of algebra for its de- velopment. It is, therefore, not appropriate to treat of it in this work, except so far as to show its logical relation to the general divisions of the science, and to present a few simple properties that may be readily understood by means of the or- dinary principles of arithmetic. These will be interesting to young arithmeticians, and perhaps the 'means of cultivating a taste for a more thorough study of the subject. The subjects to which the attention of the reader will be briefly directed are the following: 1. Even and Odd Numbers. 2. Prime and Composite Numbers. 3. Perfect, Imperfect, etc., Numbers. 4. Divisibility of Numbers. 5. Divisibility by the Number Seven. 6. Properties of the Number Nine CHAPTER II. EVEN AND ODD NUMBERS. "VTUMBERS have been divided into many different classes, -Li founded upon peculiarities discovered by investigating their properties. The series 1, 2, 3, 4, etc., is called the series of Natural Numbers. The Natural Numbers are classified with respect to their relation to the number two, into Odd and Even numbers. They are also divided into two classes with respect to their composition, called Prime and Composite numbers. Composite Numbers are divided into two classes, Perfect and Imperfect numbers, this classification being based upon the relation of the numbers to the sum of their factors. Imperfect Numbers are also divided into two classes with re- *pect to the numbers being greater or less than the sum of their factors. Numbers which are equal each to the sum of the di- visors of the other, are called Amicable Numbers. A few re- marks will be made on each one of these classes. Of the various classes of numbers, the simplest and most natural division is that of Ei>en and Odd numbers. This di- vision is founded upon the relation of numbers to the number 2. Even numbers are those which are multiples of 2 ; Odd numbers are those which are not multiples of 2. In the series of natural numbers the increase is by a unit ; in the series of even numbers the scale of increase is dual. The former arise from counting by 1's, beginning with the unit; the latter in counting by 2's, beginning with the duad. The even numbers are divided into the oddly even numbers, 2, 6, 10, 14, etc.; and the evenly even numbers, 4, 8, 12, 16, etc. The odd numbers are divided into the evenly odd numbers 1, 5, 9, 13, etc; and the oddly odd numbers, 3, 7, 11, 15, etc. (375) 376 THE PHILOSOPHY OF ARITHMETIC. The formula for the even numbers is 2n ; the formula for the odd numbers is 2/i-fl. In the oddly even numbers n is an odd number ; in the evenly even numbers n is an even number. In the evenly odd numbers n is even; in the oddly odd numbers n is odd. The evenly odd numbers are of the form 4?i-|- 1 ; the oddly odd numbers are of the form 4n-|-3. There are many interesting principles relating to even and odd numbers, a few of which will be stated. 1. Every prime number except 2 is an odd number. 2. The differences of the successive square numbers produce the odd numbers. 3. The sum or difference of two even numbers or two odd numbers is an even number. 4. The sum or difference of an even number and an odd num- ber is odd. 5. The sum of any number of even numbers is even ; the sum of an even number of odd numbers is even, and the sum of an odd number of odd numbers is odd. 6. The product of two even numbers is even ; of two odd nuui- oers is odd ; of an even number and an odd number is even. T. The quotient of an even by an odd number, when exact, is even; the quotient of an odd by an odd, when exact, is odd; the quotient of an even by an even, when exact, is either even or odd. 8. An odd number is not exactly divisible by an even num- ber, and the remainder is odd. 9. If an even number is not exactly divisible by an even number, its remainder is even. 10. If an even number is not exactly divisible by an odd number, then when the quotient is even the remainder is even, and when the quotient is odd, the remainder is odd. 11. If an odd number is not exactly divisible by an odd number, then when the quotient is odd the remainder is even, and when the quotient is even the remainder is odd. 12. If an odd number divides an even number, it will also EVEN AND ODD NUMBERS. 377 divide one-half of it ; if an even number be divisible by an odd number, it will be divisible by double that number. 13. Any power of an even number is even ; and conversely the root of an even number which is a complete power is even 14. Any power of an odd number is odd ; and conversely the root of an odd number which is a complete power is odd. 15. The sum or difference of any complete power and its root is even. These principles can be readily proved by the ordinary meth- ods of arithmetical reasoning. To illustrate, take the third principle, the reasoning of which is as follows : Two even numbers are each a number of 2's, hence their sum will be the sum of two different numbers of 2's, which must be a number of 2's, and their difference will be the difference between two different numbers of 2's, which is also a number of 2's. In add- ing two odd numbers we will have a number of 2's-f-l, added to another number o/2's+l, which will give us a number of 2's + 2, or an exact number of 2's, etc. The simplest method is by using the general notation of al- gebra. Thus in the given principle, these two even number? will be represented by 2ra and 2n' ; their sum will be 2n+2n', or 2 (n-f w'), which is of the form of 2n, and is thus even ; their difference will be 2n 2n', or 2(n n f ), which is of the form of 2n, and is even. The two odd numbers are of the form 2/i-fl and 2n'+ 1, and their sum is 2 (n-fn'-f 1), which is of the form of 2n, and even ; their difference is 2rc 2n', or 2 (n n'), which is evidently even. All the other principles may be demonstrated in a similar manner. CHAPTER III. PEIME AND COMPOSITE NUMBERS. rpHE most celebrated classification of numbers is that of Prime JL and Composite. This classification is with respect of their formation by multiplication or the possibility of their being re- solved into factors. The Composite number is one which can be produced by the multiplication of other numbers ; the Prime number is one which cannot be produced by the multiplication of other numbers. The distinction may be regarded as having reference to the dependence or independence of their existence. The composite number is regarded as deriving its existence from other numbers which make it; the prime number does not derive its being from any other numbers, but is indepen- dent and self-existent. Perhaps no subject in arithmetic has received more attention from mathematicians than that of Prime and Composite Numbers. The object has been to discover some general method of find- ing prime numbers, and of determining whether a given num- oer is prime or composite. Such a method, though laboriously sought for by the best mathematical minds, has not, beyond 8 certain limit, been discovered. The problem of ascertaining prime numbers was discussed as far back as the days of Eratosthenes, a mathematician of Alexandria, distinguished also as having first conceived the plan of measuring the earth. He invented a method of obtain- ing primes by excluding from the series of natural numbers those that are not prime, and thus discovering those that are. This method consisted in inscribing the series of odd numbers upon parchment, and then cutting out the composite numbers, (378) PRIME AND COMPOSITE NUMBERS. 379 and leaving the primes. The parchment, with its holes, resem- bled a sieve ; hence the method is called Eratosthenes' sieve. His method may be illustrated as follows: Suppose we write the series of odd numbers from 1 to 99 in- clusive. Since the series increases by 2, the third term from 3 is 3+3 x 2, which is divisible by 3 ; hence every third term is divisible by 3, and is therefore composite. In a similar manner we see that every fifth term after 5 is divisible by 5, and therefore composite ; and every seventh term after 7 is di- visible by 7, and therefore composite. Cutting out these com- posite numbers, we have all the prime numbers below 100. By this method, assisted by some mechanical contrivance, Vega computed and published a table of prime numbers from 1 to 400,000. This method is, however, very tedious and inconvenient, and mathematicians have earnestly sought for properties of prime and composite numbers to guide them in ascertaining primes. The following principles are useful in discovering or determining prime numbers: 1. All prime numbers except 2 are odd, and consequently terminate with an odd digit. The converse of this, that all odd numbers are prime, is not, however, true. 2. All prime numbers, except 2 and 5, must terminate with 1, 3, 7, or 9 ; all other numbers are composite. This is the series of odd digits with the omission of 5, since any number terminating with 5, can be divided by 5 without a remainder. 3. Every prime number, except 2, if increased or diminished by 1, is divisible by 4. In other words, every prime number, except 2, is of the form 4n 1. This will admit of demonstra- tion. 4. Every prime number, except 2 and 3, if increased or di- minished by 1, is divisible by 6. In other words, every prime number, except 2 and 3, is of the form 6n 1. This may also be demonstrated. 5. Every prime number, except 2, 3, and 5, is a measure of 380 THE PHILOSOPHY OF ARITHMETIC. the number expressed, in common notation, by as many 1's as there-are units, less one, in the prime number. Thus, 7 is a measure of 111,111 ; and 13 of 111,111,111,111. 6. Every prime number, except 2 and 5, is contained with- out a remainder in the number expressed in the common nota- tion by as many 9's as there are units, less one, in the prime number itself. Thus, 3 is a measure of 99 ; 7 of 999,999; and 13 of 999,999,999,999. 7. Three prime numbers cannot be in arithmetical progression, unless their common difference is divisible by 6 ; except 3 be the first prime number, in which case there may be three prime numbers in such progression, but in no case can there be more than three. 8. This last principle is generally true, and may be stated as follows : There cannot be n prime numbers in arithmetical progression unless their common difference be divisible by 2. 3. 5. 7. 11... n; except the case in which n is the first term of the progression, in which case there may be n such numbers, but not more. Though we have no general method for finding prime num- bers, there are several ways of detecting whether an assigned number is or is not a prime. Several remarkable formulas have been discovered which contain a large number of prime num- bers. The formula x*+x+ 41, by making successively #=0, 1, 2, 3, 4, etc., will give a series 41, 43, 47, 53, 61, 71, etc., the first forty terms of which are prime numbers. This formula is mentioned by Euler in the Memoirs of Berlin, 1772. Of the two formulas # 2 +#-|-17, and 2a? 2 -f 29, the former gives seven- teen of its first terms primes, and the latter twenty-nine. Fer- mat asserted that the formula 2-f-l is always a prime when in is taken any term in the series 1, 2, 4, 8, 16, etc.; but Euler found that 2 :i2 +l=641 x 6,700,417 is not a prime. One of the most celebrated theorems for investigating primes is that discovered by Fermat and known as FermaVs Theorem. The theorem may be stated thus: If p be a prime, the (p l)th PRIME AND COMPOSITE NUMBERS. 381 power of every number prime to p will, when diminished by unity, be exactly divisible by p. Expressed in algebraic language, we have the theorem P p ~ ' 1, is a multiple of p when p and P are prime to each other. Thus, 25 s 1 is exactly divisible by 7. Fermat is said to have been in possession of a proof of the theorem, though Euler was the first to publish its demonstra- tion. Euler's first demonstration was a very simple one, and is that usually given in the text-books. Amongst the other demonstrations of the theorem, those given by Lagrange are highly esteemed. It has been demonstrated by Legendre (Essai sur la Theorie des Nombres), that every arithmetical progression, of which the first term and common difference are prime to each other, con- tains an infinite number of prime numbers. It has been also shown by him that if N represents any number, then will the formula N h.logy 1.08366 represent the number of prime numbers that are less than N, very nearly. Another celebrated theorem is that invented by Sir John Wilson, known as Wilson's Theorem. This theorem may be stated as follows : The continued product, increased by unity, of all the integers less than a given prime, is exactly divisible by that prime. The algebraic formula which expresses the the- orem, 1 + 1.2.3... (n 1), is divisible by n , n being a prime number. Thus 1 + 1.2.3.4.5. 6=721, is exactly divisible by 7. This theorem was first demonstrated by Lagrange ; his pro cess of reasoning, as might be expected, was very ingenious. It was afterward demonstrated by Euler, and finally byQausb, who extended the theorem by proving that " The product of all those numbers less than, and prime to, a given number, al, is divisible by a ;" the ambiguous sign being , when a 382 THE PHILOSOPHY OF ARITHMETIC. is of the form p m , or 2p m , p being any prime number greater than 2 ; and, also, when a=4; but positive in all other cases. Wilson's Theorem furnishes us with an infallible rule, in theory, for ascertaining whether a given number be a prime or not ; for it evidently belongs exclusively to those numbers, as it fails in all other cases ; but it is of no use in a practical point of view, on account of the great magnitude of the product even for a few terms. In the later works on the Theory of Numbers it is demon- strated that, No algebraical formula can represent prime numbers only. It is also shown that, The number of prime numbers is infinite. The latter proposition is evident a priori; the former was pretty nearly evident from induction before it received a rigid demonstration. The distribution of prime numbers does not follow any known law; but for a given interval it is found that the number of primes is generally less the higher the beginning of the interval is taken. The whole number of primes below 10,000 is 1,230; between 10,000 and 20,000 it is 1,033; between 20,000 and 30,000 it is 983 ; between 90,000 and 100,000 it is 879. The largest prime which had been verified when Barlow wrote, is 2 31 1 = 2,147,483,647, which was found by Euler. The term prime is also applied to a species of numbers called complex numbers, first suggested by Gauss in 1825. Accord- ing to this theory, a complex integer is of the form a + b^/^j t in which a and b denote ordinary (real) integers. The product a 2 + 6 2 , of a complex number a+frv/IT^ and its conjugate, a fe^/ITf, is called its norm, and is denoted by the symbols N(a + b^ 1)> -N( a &\/ l)- The four associative numbers, a + fcv/^T. a\/~\ b, a b^/^}, and a<Si -f b, as well as their respective conjugates, have all the same norm. A com- plex number is said to be prime when it admits of no divisor except itself, its associatives, and the four units, 1, 1, \/i, and v/ 13 !- Many of the higher theorems, such as that of Fermat, may be extended to the system of complex numbers. CHAPTER IV. PERFECT, IMPERFECT, ETC., NUMBERS. HAVING separated numbers into their factors, the human mind, ever active in the attempt to discover the new, be- gan to compare the sum of the factors or divisors of numbers with the numbers themselves, and thus discovered certain re- lations which gave rise to three new classes of numbers. In some cases it was seen that a number was just equal to the sum of all of its divisors, not including itself, and such num- bers were called Perfect Numbers. Numbers not possessing this property were called Imperfect Numbers ; and were divided into two classes, Defective and Abundant, according as they were greater or less than the sum of their divisors. Pushing the comparison still further, it was also discovered that some numbers were reciprocally equal to their divisors; and this relation was so intimate that such numbers were re- garded as friendly or Amicable Numbers. These several classes will be formally defined in this chapter. Perfect and Imperfect numbers were known by the ancient Greek mathematicians, but their properties have been developed by the mathematicians of modern times. Amicable Numbers were first investigated by the Dutch mathematician Van Schooten, who lived from 1581 to 1640. A Perfect Number is one which is equal to the sum of all its divisors, except itself; thus, 6=1+2+3; 28=1+2+4+7+14 An J in perfect Number is one which is not equal to the sum of all its divisors. Imperfect Numbers are Abundant or De- fective. An Abundant Number is one the sum of whose di- visors exceeds the number itself; as, l+2+3+6+9>18. A (383) 384 THE PHILOSOPHY OF ARITHMETIC. Defective Number is one the sum of whose divisors is less than the number itself; as, 1+2+4+8 < 16. Every number of the form (2"" 1 ) (2 n 1), the latter factor being a prime number, is a perfect number. The only values of n yet found, which make 2" 1 a prime are 2, 3, 5, 7, 13, 17, 19, and 31 ; there are, therefore, only ten perfect numbers known. Substituting 2 for n in the formula, we have 2(2 2 1) =6, the first perfect number; the second is 2 2 (2 3 1)=28. The first eight perfect numbers are, 6, 28, 496, 8128, 3S55033F,, 8589869056, 137438691328, 2305843008139952128. Each number, as is seen, ends in 6 or 28. The difficulty in finding perfect numbers consists in finding primes of the form of 2" 1. The greatest prime number, ac- cording to Barlow, yet ascertained, is 2 31 1 = 2147483647, dis- covered by Euler ; and the last of the above perfect numbers, which depends upon this, is the greatest perfect number known at present, and Barlow remarks that it is probably the greatest that will ever be discovered ; for, as they are merely curious without being useful, it is not likely that any person will attempt to find one beyond it. An author of an arithmetic gives two other numbers which are said to be perfect, 2417851639228- 158837784576, 9903520314282971830448816128, but I do not know his authority. Two numbers are called Amicable when each is equal to the sum of the divisors of the other ; thus, 284 and 220. The for- mulas for finding amicable numbers are A2 n + } d and B= Z n+l bc, in which n is an integer, and b, c, and d are prime numbers satisfying the following conditions: 1st, 6=3 x 2" 1 ; 2d, c=6x2 n -1; 3d, d=18x2 2(l 1. If we make n=l, we find 6=5, c=ll, and d=71; substituting these in the above formulas, we have .4=4x71=284, and 5=4x5x11=220, the first pair of amicable numbers. The next two pairs are 17296, 18416, and 936358, 9437056. The first pair, 220 and 284, were found by E. Van Schooten, with whom the name amicable appears to have originated, though PERFECT, IMPERFECT, ETC., NUMBERS. 385 Rudolphus and Descartes were previously acquainted with this property of certain numbers. A formula for amicable numbers was, in fact, given by Descartes, and afterwards gen- eralized by Euler and others. Figurate Numbers. Figurate Numbers are numbers formed from an arithmetical progression whose first term is unity, and common difference integral, by taking successively the sum of the first two, the first three, the first four, etc., terms of the series; and then operating on the new series in the same man- ner as in the original progression in order to obtain a second series, and so on. For example, take the series of natural numbers in which the common difference is 1, as repre- ented by A in the margin; then the A, 1-2-3-4-5-6-7 series B, derived as stated above, will ^' |"J JQ~JJj"ge " ?1 " gjj be figurate numbers ; series C, derived p' 1.5.15.35.70-126-210 as above from series B and series D, derived from series C, will be figurate numbers. Other seriea could be obtained by beginning with any other arithmetical series whose first term is 1, and common difference an integer. Thus, the series derived from the progression 1, 3, 5, 7, 9, etc., is 1, 4, 9, 16, 25, etc. A more general method of conceiving figurate numbers is to regard them as a series of numbers, the general term of each series being expressed by the formula, n(n+l)(n+2)(n+3) .... (n+m) 1.2.3.4 . . . in which m represents the order of the series, and n represents the place of the required term. Series of figurate numbers are divided into orders; when m = 0, the series is of the 1st order; when m = I, the seriea is of the 2d order ; when m 2, it is of the 3d order, etc. By regarding m equal to in this formula, and substituting successive numbers 1, 2, 3, etc., for n, it will be seen that the general term is n, and we find that the figurate series of the first order is the series of natural numbers, 1, 2, 3, 4, etc., n. S86 THE PHILOSOPHY OF ARITHMETIC. By regarding m equal to 1, the general term of the series becomes -^ - > and substituting the successive values of n, 1 . 2 1, 2, 3, etc., we find the terms to be 1, 3, 6, 10, 15, 21, 28, etc., which is the series of figurate numbers of the second order. Ir a similar manner we find the general term of the figurate series of the 3d and 4th orders to be respectively, n(n + l)(n + 2) n(n + l)(n + 2) (n + 3) and 1.2.3 1.2.3.4 from which we can readily derive those series. These several series of figurate numbers are the same as those represented in the margin above. One of the most remarkable properties of the series of figu- rate numbers is that, if the nth term of a series of any order be added to the (n -f- l)th term of the series of the preceding order, the sum will be equal to the (n-fl)th term of the series of the given order. Thus, in the series marked C, if we add the second term, 4, to the third term, 6, in series B, we shall have the third term, 10, of series C ; the third term of series C plus the fourth term of series B equals the fourth term of series C, etc. If we begin with a series of 1's, all of the series of figurate numbers may be deduced in succession by the application of th : s principle. ORDERS OF FIGURATE NUMBERS. Series of Us 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1st order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 2d order 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 3d order 1, 4, 10, 20, 35, 56, 84, 120, 165, 220 4th order 1, 5, 15, 35, TO, 126, 210, 330, 495, U5 5th order 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002 6th order 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005 1th order 1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440 By inspecting these series, it will be seen that the values PERFECT, IMPERFECT, ETC., NUMBERS. 387 ead diagonally upward are the numerical coefficients of the terras in the development of (a -j- b) with an exponent corre- sponding to the order of the series. It is said that it was this principle which gave rise to a complete investigation of the subject of figurate numbers. In speaking of denning figurate numbers by giving the form of each of the orders, Barlow remarks that it is more simple to deduce the generation of figurate numbers from their form than to deduce their form from their generation. The principle given above, showing the relation of the terms of two succes- sive orders of figurate numbers, is ascribed to Fermat, and is considered by him as one of his most interesting propositions. Polygonal Numbers are figurate numbers which represent the sides of polygons. The second series of figurate numbers, I, 3, 6, 10, etc., are called triangular . numbers, because the number of units that they express can be arranged in * * * the form of a triangle. If we take the series 1, 3, 5, 7, 9, etc., in which the common difference is 2, we obtain .... the figurate series, 1, 4, 9, 16, 25, etc., which are called square numbers, because they can be arranged in a square. The series 1, 4, 7, 10, etc., in which the common difference is 3, gives the series 1, 5, 12, 22, etc., which are called pentagonal numbers, because they can be arranged in the form of a penta- gon. In a similar manner we obtain hexagonal, heptagonal, octagonal, etc., numbers. It will be noticed that the number of the sides of the polygon which they represent is always two greater than the common difference of the series from which they were derived. Common difference=l ; 1, 2, 3, 4, 5, When the common Triangular numbers 1, 3, 6, 10, 15, 21 A -ff f ,1 Common difference=2; 1, 3, 5, 7, 9. 11 difference of the Square numbers 4 | 9> 16 | ^ 36 series in arithmeti- Common difference=3 ; 1, 4, 7,10,13,16 cal progression is Pentagonal numbers I, the sums of the terms give the triangular numbers; when 888 THE PHILOSOPHY OF ARITHMETIC. the common difference is 2, the sums of the terms are the squart. numbers ; when the difference is 3, the sums are the pentagonal numbers, and so on. These numbers are called polygonal from possessing the pro- perty that the same number of points may be arranged in the form of that polygonal figure to which it belongs. Thus the pentagonal numbers 5, 12, 22, 35, 51, etc., may be severally arranged in the form of a pentagon. Thus, 5 points will form one pentagon; 12 points will form a second pentagon enclos- ing the former; 22, a third pentagon enclosing both of the former, etc. The following property of polygonal numbers was discovered by Fermat : Every number is either a triangular number or the sum of two or three triangular numbers; every number is either a square number, or the sum of two, three, or four square numbers ; every number is either a pentagonal number or the sum of two, three, four, or Jive pentagonal numbers ; etc. This property is generally true, although it has been demonstrated for only triangular and square numbers. All the other cases still remain without demonstration, notwith- standing the researches of many of the ablest mathematicians. Permat himself, however, as appears from one of his notes on Diophantus, was in possession of the demonstration, although it was never published, which circumstance renders the theorem still more interesting to mathematicians, and the demonstration of it more desirable. Pyramidal Numbers are those which represent the number of bodies that can be arranged in pyramids. They are formed by the successive sums of polygonal numbers in the same man- ner as the polygonal numbers are formed from arithmetical progressions. The Triangular Pyramidal numbers are the series of figurate numbers derived from the series of triangular numbers. Thus, from the triangular numbers 1, 3, 6, 10, 15, etc., we have the triangular pyramidal numbers 1, 4, 10, 20, etc. The Square Pyramidal numbers are derived from the square numbers. CHAPTER V. DIVISIBILITY OF NUMBERS. IN factoring a composite number, we divide successively by exact divisors of the number till we obtain a quotient which is a prime number. In order to know by what numbers to divide, it is convenient to have some tests of divisibility, other- wise it would be necessary to try several numbers until we hit upon one which is exactly contained. There are certain laws which indicate, without the test of actual division, whether a number is divisible by a given factor, some of which are simple and may be readily applied The investigation of these laws of the relations of the factors of numbers to the numbers them- selves, gives rise to a subject known as" the Divisibility of Num- bers. The laws for the divisibility of numbers, as usually presented, embrace the conditions of divisibility by the numbers 2, 3, 4, etc., up to 12. These laws may be stated as follows: 1. A number is divisible by 2 when the right-hand term is zero or an even digit. For, the number is evidently an even number, and all even numbers are divisible by 2. 2. A number is divisible by 3 when the sum of the numbers denoted by its digits is divisible by 3. It will be shown here- after that every number is a multiple of 9, plus the sum of its digits; hence, since 3 is a factor of 9, the number is divisible by 3 when the sum of the digits is divisible by 3. 3. A number is divisible by 4, when the two right-hand terms are ciphers, or when they express a number which is divisible by 4. If the two right-hand terms are ciphers, the number (389) 390 THE PHILOSOPHY OF ARITHMETIC. equals a number of hundreds, aiid since 100 Is divisible by 4, any number of hundreds is divisible by 4. If the number ex- pressed by the two right-hand digits is divisible by 4, the num- ber will consist of a number of hundreds, plus the number ex- pressed by the two right-hand digits ; and since both of these are divisible by 4, their sum, which is the number itself, is divisible by 4. 4. A number is divisible by 5, when its right-hand term is or 5. If the right-hand term is 0, the number is a number of times 10 ; and since 10 is divisible by 5, the number itself is divisible by 5. If the right-hand term is 5, the entire num- ber will consist of a number of tens, plus 5 ; and since both of these are divisible by 5, their sum, which is the number itself, is divisible by 5. 5. A number is divisible by 6, when it is even and the sum of the digits is divisible by 3. Since the number is even, it is divisible by 2, and since the sum of the digits is divisible by 3, the number is divisible by 3, and since it contains both 2 and 3 it will contain their product, 3x2, or 6. 6. A number is divisible by 7, when the sum of the odd nu- merical periods, minus the sum of the even numerical periods, is divisible by 7. The law for the divisibility by 7 is perhaps of not so much practical importance as the others, being not quite so readily applied, but it is of too much scientific interest to be omitted from the series. Its demonstration will be given in the following chapter. 7. A number is divisible by 8, when the three right-hand terms are ciphers, or when the number expressed by them is divisible by 8, If the three right-hand terms are ciphers, the number equals a number of thousands; and since 1000 is divisible by 8, any number of thousands is divisible by 8. If the number ex- pressed by the three right-hand digits is divisible by 8, the entire number will consist of a number of thousands, plus the number expressed by the three right-hand digits (thus 17368 = 17,000 + 368) ; and since both of these parts are divisible by 8, their sum, which is the number itself, is divisible by 8. DIVISIBILITY OF NUMBERS. 391 8. A number is divisible by 9, when the sum of the digits is divisible by 9. This law is derived from showing that a num- ber may be resolved into two parts, one part being a multiple of 9 and the other the sum of the digits. A complete demon- stration is presented on a subsequent page, to which the reader is referred. 9. A number is divisible by 10, when the unit term is 0. For, such a number equals a number of tens, and any number of tens is divisible by 10; hence the number is divisible by 10. 10. A number is divisible by 11, when the difference between the sums of the digits in the odd places and in the even places is divisible by 11, or when the difference is 0. This law is derived by showing that a number may be resolved into two parts, one part being a multiple of 11, and the other part con- sisting of the sum of the digits in the odd places, minus the sum of the digits in the even places. A complete demonstra- tion will be presented on a subsequent page. 11. A number is divisible by 12, when the sum of the digits is divisible by 3 and the number expressed by the two right- hand digits is divisible by 4. For, since the sum of the digits is divisible by 3, the number is divisible by 3, and since the number expressed by the two right-hand digits is divisible by 4, the number is divisible by 4; hence, since the number is divisible by both 3 and 4, it is divisible by their product, or 12. These laws are simple, and, with the exception of those re- lating to the numbers 7, 9, and 1 1, readily applied. The laws of dividing by 9 and 11 present some interesting points, which will be formally discussed. It will be noticed, upon examining text- books on arithmetic, and also works on the theory of numbers, that the law of divisibility by 7 is omitted. Apparently efforts \vere made to discover such a law, for several writers give some special rules for dividing by 7 ; but it would seem that no general law was known to them. In the principle as above presented, this hiatus is filled up by a law not quite so simp].- as that for the other numbers, but still of scientific interest, if 392 THE PHILOSOPHY OF ARITHMETIC. not of much practical value. Besides the law given, there are several other laws, interesting as showing the development of the subject, and which we therefore present. The methods of demonstration are similar to those used in proving the divisi- bility of numbers by 9 and 11; indeed, one of the laws from which the others were derived was discovered by the applica- tion of that method to the number 7. I shall therefore first present the demonstration of divisibility by 9 and 11, and then state and demonstrate the laws relating to the number 7. Divisibility by Nine. The law of divisibility by nine has been known for a long time. By whom it was discovered has not been ascertained. Its application to testing the correctness of the work in the fundamental rules, called proof by " casting out nines," has been attributed to the Arabs. The law, as pre- viously stated, is that a number is divisible by nine when the sum of the digits is divisible by nine. This principle depends on a more general law which will be first stated, and then the law of exact division, as well as some other interesting princi- ples, will be drawn from it. 1. A number divided by 9 leaves the same remainder as the sum of the digits divided by 9. This theorem can be demonstrated both arithmetically and algebraically. We will first present the arithmetical demonstra- tion. If we take any number, as 6854, and analyze it, as in the margin, r 4_ 4 we will see ft aej J 50=5x10 =5x (9+l)=5x9 +5 thatitcon- J 800=8x100 =8x (99+l)=8x99 +8 sists of two 1 6000=6 x 1000=6 x (999+l)=6 X 999+6 parts: the Multiple of 9 Sumof^igits .-. 6854 = 5x9+8x99+6x999 + 4+5+8+6 first p&Tu CL multiple of 9, and the second part the sum of the digits. The first part is evidently divisible by 9, hence the only re- mainder that can arise from dividing a number by 9 will be equal to the remainder arising from dividing the sum of the digits by 9. When the sum of the digits is exactly divisible DIVISIBILITY OF NUMBERS. 393 by 9, it is evident that the number itself is exactly divisible by 9, which proves the theorem. From this theorem the fol- lowing principles may be readily inferred : 2. A number is exactly divisible by 9 when the sum of it* digits is divisible by 9. 3. The difference between any number and the sum of its digits is divisible by 9. 4. A number divided by 9 gives the same remainder as any one formed by changing the order of the figures. 5. The difference between two numbers, the sums of whose digits are equal, is exactly divisible by 9. The fundamental theorem may also be demonstrated algebra- ically as follows: Let a, b, c, d, etc., represent the digits of any number, aod r the radix of the scale, that is, the number of units in a group ; then every number may be represented by formula (1) below. If we now subtract b, c, d, etc., from one part of this expression, and add them to another part, it will not change the value, and we shall have formula (2) ; and factoring, we obtain formula (3). (1). ^T=a+6r+cr 2 +dr s +er < +etc. (2). N=br 6-Hcr 2 c+dr 3 d+er f e, etc.+a + b + c+d+e +etc. (3). N=b (rl) + c (r 2 1) + d (r s -l) + e (r 4 -!) -fete. +a Now, r 1, r 1 1, r 3 1, etc., etc., are all divisible by r 1 ; hence the only remainder which can arise from dividing the number by j 1, will occur from dividing a+b+c+d+etc., by r 1; that is, any number divided by r 1 leaves the same remainder as the sum of the digits divided by r 1. In our decimal scale r=10, hence r 1=9; and hence any number divided by 9 leaves the same remainder as the sum of the digits divided by 9. This law is the basis of some very interesting properties, and also of the proof of the fundamental rules called "casting out nines." Divisibility by Eleven. The law of the divisibility of num- 17* 394: THE PHILOSOPHY OF ARITHMETIC. bers by 1 1 is quite similar to that of 9. This might have been anticipated, as they each differ from the basis of the scale by unity, the former being a unit below and the latter a unit above the base. The law, as previously stated, is that a number it divisible by 11 when the difference between the sum of the digits in the odd places and the even places is divisible by 11. This principle depends upon a more general one, which will first be stated, and then this, as well as some other interesting prin- ciples, will be derived from it. 1. Every number is a multiple of 11, plus the sum of the digits in the odd places, minus the sum of the digits in the even places. This principle may be demonstrated both arith- metically and algebraically. We will first give the arithmetical proof. If we take any number, as 65478, and analyze it as in- +8 70= 7x10= 7X(H 1)= 7x117 65478= 400= 4x100= 4x(99-fl)= 4x99+4 5000= 5 X 1000=5 X (1001 I)=5xl001 5 L 60000=6xlOOOO=6x(9999+l)=6x 9999+6 Sum of Sum of Multiples of 11. odd digits. even digits. /. 65478=7x11+4x99+5x1001+6x9999 + 8+4+6 5+7 dicated, we shall see that it consists of two parts; the first being a multiple of 11, and the second consisting of the sum of the digits in the odd places, minus the sum of the digits in the even places. The first part is evidently divisible by 11 ; hence the only remainder that can arise from dividing a number by 11 will be equal to the remainder arising from dividing the difference between the sums of the digits in the odd places and the even places by 11. When this difference is exactly divisible by 11, it follows that the number itself is divisible by 11. When the sum of the digits in the even places is greater than the sum in the odd places, we take the difference, divide by 11, and subtract the remainder from 11 to find the true remainder. The reason for this will appear from the above demonstration. From this theorem the following prin ciples can be readily inferred : 2. A number is exactly divisibliTby 11, when the sum of the DIVISIBILITY OF NUMBERS. 395 digits in the odd places is equal to the sum of the digits in the even places. 3 A number is exactly divisible by II, when the difference between the sums of the digits in the odd places and the even places is a multiple of 11. 4. A number increased by the sum of the digits in the even places and diminished by the sum of the digits in the odd places, is exactly divisible by 11. 5. The excess of ll's in any number is not changed by add- ing any multiple of 11 to the sum of the digits of either order. The algebraic demonstration of this property is as follows: Taking the same formula as for the number 9, we add b and then subtract b, we subtract c and "hen add c, etc., the formula becoming (2) below, being the same in value as the first, but changed in form. Then, factoring, we have (3). (1). N " T hr+ / dr +er*+etc. (2). N=br+b+cr 2 c+dr^d+er* e+etc.+a 6+c d+e, etc. (3).A^6(r+l)+c(r 2 1) + d (r s + 1 ) + e (r 4 l)+etc.+(a+ c fe-f-etc.) (6+d+ctc.) Now r-H, r 2 1, r s +l, etc., are each divisible by r+1; hence the only remainder that can arise from dividing this number by r+1 must arise from dividing (o+c+e+etc.) (6+d+etc.) by r+ 1 ; that is, by dividing the difference of the sum of the digits in the even places subtracted from the sum of the digits in the odd places by r+ ! . In the decimal scale, r=10, and r+l=ll; hence we sue that any number divided by 1 1 leaves the same remainder as the difference of the sum of the digits in the even places, subtracted from the sum of the digits in the odd places does when divided by 11. When this difference is exactly divisible by 11, the number itself is divisi- 1)1.', which proves the principle of the divisibility by 11. This principle may also be used for the proof of the fundamental rules, but not quite so conveniently as that of the number 'J. CHAPTER VI. THE DIVISIBILITY BY SEVEN. THE Divisibility of Numbers, as presented by different authors, embraces the conditions of divisibility by the numbers 2, 3, etc., up to 12, with the omission of the num- ber 7. This omission leads us to inquire whether there is any general law for the divisibility of numbers by 7. A few of our text-books present some special truths in regard to this subject, among which are the following : 1. A number is divisible by 7 when the unit term is one-half or one-ninth of the part on the left. Thus 21, 42, 63, 126, and 91, 182, 273, etc. 2. A number is divisible by 7 when the number expressed by the two right-hand terms is five times the part on the left, or one-third of it. Thus 525, 840, 1995, and 602, 903, 3612, etc. 3. A number consisting of not more than two numerical periods is divisible by 7 when these periods are alike. Thus 45045, 235235, 506506, etc., are divisible by 7. There are, however, some general laws for the divisibility by 7, which seem to have been overlooked by most writers on the theory of numbers, and which, though of not much practical importance, are interesting in a scientific point of view. The first and least simple of these laws is as follows : 1. A number is divisible by 7, when the sum of once the first, or units digit, 3 times the second, 2 times the third, 6 times the fourth, 4 limes the fifth, 5 times the sixth, once the seventh, 3 times the eighth, etc., is divisible by 7. It will be (396) THE DIVISIBILITY BY SEVEN. 397 seen that the series of multipliers is 1, 3, 2, 6, 4, 5. To illus- trate the law, take the number 7935942, and we have for the sum of the multiples of the digits, 1 x 2+3 x 4+2 x 9+6 x 5+4 x 3+5 x 9+1 x 7= 126, which is exactly divisible by 7 ; and if we divide the number itself by 7, we find there is no remainder. Assuming this principle it will be demonstrated on page 398 we can derive several other principles of divisibility from it. In this law we see that the second half of the series of mul- tipliers, 6, 4, 5, equals respectively 7 minus the first half, 1, 3, 2; hence, instead of adding the multiples of the second series, 6, 4, 5, we may subtract the respective multiples of the terms of the second period by the first series of multipliers, 1, 3, 2, which will give rise to the following principle : 2. A number is divisible by 7, when the number arising from the sum of once the first digit, 3 times the second, 2 times the third, minus the sum of the same multiples of the next three digits, plus the sum of the same multiples of the next three digits, etc., is divisible by 7. It will be seen that the series of multipliers is 1, 3, 2, the first products additive, the second products subtractive, etc. ; the odd numerical periods being additive and the even periods subtractive. If we take the number 5439728, we have 1x8+ 8 x 2+2 x 71 X 98 X 32 X 4+1 x 5=7, which isdivisible by 7. Upon trial we find the original number is also exactly di- visible by 7. This second principle may also be stated thus: A number is divisible by 7 when the sum of the multiples expressed by the numbers, 1, 3, 2, of the terms of the odd numerical periods, minus the sum of the same multiples of the terms of the evert numerical periods, is divisible by 7. Now, if we add exact multiples of 7 to the multiples of the terms which are united in the test of divisibility, it will not change the remainder. Thus, taking the number 5439728, if we add 7 X 2 to 3 x 2, we have 10 x 2, or 20 ; and adding 98 X 1 398 THE PHILOSOPHY OF ARITHMETIC. X) 2x7 we have 100x7, or 700; hence we may use in place )f 1x8+3x2+2x7, 8+20+700, or 728, the first numerical period ; and in the same way it may be shown that we may use the second period subtractively in the test, etc. Hence from Principle 2 we may derive the following principle: 3. A number is divisible by 7, when the sum of the odd nu- merical periods, minus the sum of the even numerical periods, is divisible by 7. To illustrate, take the number 5,643,378,762; we have for the sum of the odd numerical periods 762+643=1405; for the sum of the even periods, 378+5=383; the difference is 1022, which is exactly divisible by 7 ; and if we divide the number itself by 7, we find that there is also no remainder. If we apply the same reasoning to Principle 1, by which we derived Principle 3 from Principle 2, we shall derive from it the following principle : 4. A number is divisible by 7, when the sum of the numbers denoted by the double numerical periods is divisible by 7. Thus, in the number 5,643,378,762, we have 5,643+378,762= 384,405, which is divisible by 7, and the number is also divisi- ble by 7. The first principle, from which I have derived the other three, may be demonstrated arithmetically and algebraically. Let us take any number as 98765432 and analyze it thus : 2= 1X2 30= 3X10= 3x(7+3)= 3x7+3x3 400= 4x100= 4 X (98+2)= 4x98+2x4 5000= 5X1000= 5x (994+6)= 5x994+6x5 60000= 6x10000= 6 X (9996+4)= 6x9996+4x6 700000= 7x100000= 7x(99995+5)= 7x99995+5x7 8000000= 8x1000000= 8x (999999+1)= 8x999999+1x8 90000000=9 X 10000000=9 X (9999997+ 3) =9 X 9999997+3 X 9 Here 98765432=a multiple of 7 plus once the 1st term, plus three times the second term, plus two times the third term, plus six times the fourth term, plus four times the fifth term, plus five times the sixth term, plus once the seventh term, plus three times the eighth term. Hence the only remainder that can occur must arise from dividing the sum of the multiples of the terms THE DIVISIBILITY BY SEVEN. 399 by 7 ; hence when the sura of these multiples is divisible by 7, the number is divisible by 7, which proves the principle. The second principle, which is readily derived from the first, may be demonstrated independently, as follows: 2= 1x2 30= 3x10= 3x(7+3)= 3x7+3x3 400= 4x100= 4x(98+2)= 4x98+2x4 5000= 5X1000= 5 X (1001 1)= 5x10011x5 60000= 6x10000= 6x(10003 3)= 6x100033x6 700000= 7x100000= 7x (100002 2)= 7x1000022x7 8000000= 8x1000000= 8x (999999+1)= 8x999999+1x8 90000000=9xlOOOOOOO=9x(9999997+3)=9x 9999997+3x9 Here 98765432=a multiple of 7, plus once, the first digit, plus three times the second, plus twice the third, minus once the fourth, minus three times the fifth, minus twice the sixth, plus once the seventh, plus three times the eighth. Hence the only remainder that can occur must arise from dividing the difference between the additive and subtractive multiples of the digits by 7 ; therefore, when this difference is divisible by 7, the number is divisible by 7, which proves the principle. When the sum of the subtractive multiples of the digits is greater than the sum of the additive, we take the difference, divide by 7, and subtract the remainder from 7 to find the true remainder. To demonstrate the third principle, take any number, as 7,946,- 321,675 and analyze it, and it will be seen to consist of parts which are multiples of 7, plus the periods in the odd places, minus the periods in the even places. 675= 675 7946321675= - 321000= 321 X (1001 1)= 821x1001321 946000000= 946X (999999+1) =946 X 999x 1001+940 . 7000000000=7x (1000000001 I)=7x999001x 1001- 7 Multiples of 7. Odd periods. Even periods. 321 X 1001+946x999999+7x1000000001 +"675+946 - 321+7 Now 1001 is a multiple of 7, 999999 is 999 times 1001, and 1000000001 is also a multiple of 1001, and if we continue the number to still higher periods, we shall find a constant series of multiples of 1001, alternately 1 more and 1 less than the number represented by one unit of the period. Hence 7,946,321,675 is composed of the sum of three multiples of 7, plus (675 + 946) (321 + 7), or the difference between t.hp nms 400 THE PHILOSOPHY OF ARITHMETIC. of the even and odd periods. The first part is evidently divisi- ble by 7, therefore the divisibility of the number depends on the divisibility of the difference of the sums of the odd and even periods ; and when this difference is divisible by 7, the number itself must be divisible by 7, which proves the prin- ciple. From this demonstration, we can immediately derive the fol- lowing principle, more general than the one stated and from which that may be derived: 3. Any number divided by 7 gives the same remainder as is obtained when the sum of the odd numerical periods, minus the sum of the even numerical periods, is divided by 7. If the sum of the even periods is the greater, we find the difference, divide by 7, and subtract the remainder from 7 for the true remainder. This investigation leads to a still more general principle of divisibility, derived from the fact that 1001, which maybe con- sidered as the basis of the above demonstration, is the product of 7, 11, and 13; hence what we have just proved for 7, is also true of 11 and 13. The most general form of the principle then is as follows: 6. Any number divided by 7, II, or 13 gives the same re- mainder as is obtained when the sum of the odd numerical periods, minus the sum of the even numerical periods, is divided by 7, 11, or 13 respectively. A special truth growing out of this general principle, had been previously given in the rule that any number of not more than two periods, when those two periods are alike, is divisible by 7, 11, or 13. All such numbers, on examination, will be found to be multiples of 1001, and, of course, divisible by its factors. It may seem surprising that those who were familiar with this special truth, and were thus on the very brink of a dis- covery, did not extend it and reach the general law above pre- sented. The fourth Principle, which was derived from the first, maj also be demonstrated independently by a method similar to that used in proving the third Principle. The algebraic demon THE DIVISIBILITY BY SEVEN. 401 siration of Principle 1, which is the foundation of the other principles, is as follows: Take the same general formula as used in demonstrating the divisibility by 9 and 11, add and subtract 36, 3 z e, B 3 d, etc., and the formula is readily reduced to the form of (5). (1). N=a+br+cr*+dr 3 +er'+fr 6 +gr*+hr''+etc. (2). N=br 36+cr 2 3'c+dr s 3 s d+er* tfe+fr 6 3 5 /, etc. +a+36+9c+27d+81e+243/+ete. (3). #=6(r-3)+c(r 2 -3 2 )+d(r 3 3 3 )+e(V 4 -3 4 )+/(r 5 3 5 ) +g(r* 3 6 ) etc.+a+36+9c+27d+81e+243/+7290, etc. (4). N=b(r 3)+c(r 2 3'0+d(r 3 3 3 )+e (r 4 3 4 ) +/(r 3 5 ) (5). N= \ b (V_3) + c (V 3') + d (r 3 3 s ) + e (r 4 3 4 ) +/ (^S^+grCr 6 S^+etc. +7c+2W+TTe+238/+ 7280+ etc. j -fa +36+ 2c+6d + 4e+5/+ 1 gr+etc. Now the first part of this expression is exactly divisible by r 3, or 7 ; hence the only remainder that can arise must occur from dividing a+36+2c+6d, etc., by r 3, or 7 ; that is, by dividing by 7 the sum of once the first digit, three times the second, two times the third, six times the fourth, four times the fifth, five times the sixth, and so on in the same order; and when this sum is exactly divisible by 7, the number is divisi- ble by 7. By a slight change in the terms of the formula, the theorem as stated in the second form may also be derived. Several years after the discovery of the law expressed in Principle 2, I learned that Prof. Elliott had employed the same property as early as 1846. Whether it was known to any mathematicians previous to this date, I am not able to ascertain. Laws for Other Numbers. In a similar manner we may find a law for the divisibility of numbers by 13, 17, etc. The law 26 402 THE PHILOSOPHY OP ARITHMETIC. for 13 may be stated as follows: A number is divisible by 13 when ONCE the first term, MINUS the sum of 3 times the second 4 times the third and 1 time the fourth, PLUS the sum of the same multiples of the next three terms, MINUS the sum of the same multiples of the next three terms, etc., is divisible by 13. It will be noticed that after the first term, the series of num- bers by which we multiply is 3, 4, 1, which is easily remem- bered and readily applied. To illustrate, take the number 8765432; we have 2 (3x3+4 x 4+1 x5)+(3x 6+4x7+1 x 8) =26, which is divisible by 13; and on trial we find the num- ber itself is also divisible. This law is derived from the more general principle that any number divided by 13 will give the same remainder as that ob- tained by dividing the result arising from the above multiples by 13. This principle may be demonstrated by taking any number, as 4987654, and analyzing it as in the previous case. 4= +1x4 50= 5x10= 5x(13 3)= 5x13-3x5 600= 6X100= 6X (104-4)= 6x104-4x6 4987654=-! 7000= 7x1000= 7x (1001-1)= 7x1001-1x7 80000= 8x10000= 8 X (9997+ 3)= 8x9997+3x8 900000= 9X100000= 9x (99996+4)= 9x99996+4x^ L 4000000 =4x 1000000 =4x(999999+l)=4x 999999+1x4 Laws for the divisibility of numbers by 17, 19, 23, etc., may be obtained in a similar manner. We present a few of then) below, including 7, 11, and 13, already given. ( 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, etc. ' (orl, 3, 2, 6, 4, 5. 1, 3, 2, 6, 4, 5, etc. 1, -1, 1, -1, 1, -1, 1, -1, 1, -I, 1, -I, etc. orl, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, etc. ( 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, etc. ' (orl, 10, 9, 12, 3, 4, 1, 10, 9, 12, 3, 4, etc. ., (1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, etc. (orl, 10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, etc f 1, 10, 18, 16, -4, 1, 10, 18, 16, -4, etc. ' ' (orl, 10, 18, 16, 37, 1, 10, 18, 16, 37, etc. ( . THE DIVISIBILITY BY SEVEN. 403 ( 1, 10, 27, -22, -1, -10, -27, 22, 1, 10, 27, etc. "jorl, 10, 27, 51, 72, 63, 46, 22, I, 10, 27, etc. 99. . . 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, etc. , ( 1, 10, -1, -10, 1, 10, -1, -10, 1, 10, -1, -10, etc. "(or 1, 10, 100, 91,1,10,100, 91,1,10,100, 91, etc. The laws for 99 and 101, it is seen, are very simple and readily applied. CHAPTER VII. PKOPERTIES OF THE NUMBER NINE. rpHE number Nine possesses the most remarkable .pro- JL perties of any of the natural numbers. Many of these properties have been known for centuries and have excited much interest among both mathematicians and ordinary scholars. So striking and peculiar are some of these properties that the number nine has been called " the most romantic " of all the numbers. On account of its relation to the numerical scale, if we get the factor 9 into a number it will cling to the expression and turn up in a variety of ways, now in one place and now in another, in a manner truly surprising. It reminds one of a mountain streamlet which ripples along its pathway, now buried beneath the ground and for awhile hidden from our sight, but presently gurgling to the surface at the most unexpected moment. It is no wonder that the property has been regarded as magical, and the number been called the "magical number." A few of these interesting properties will be here presented. 1. The first property of this number which attracts our at- tention is, that all through the column of "nine times" in the multiplication table, the sum of the terms is nine or a multiple of nine. Begin with twice nine, 18; add the digits together, and 1 and 8 are 9. Three times 9 are 27 ; 2 and 7 are 9. So it goes on up to eleven times nine, which gives 99. Add the digits; 9 and 9 are 18; 8 and 1 are nine. Go on in the same manner to any extent, and it is impossible to get rid of the figure 9. Multiply 326 by 9, and we have 2934, the sum of whose digits is 18, the sum of whose digits is 9. Let the ( 404) PROPERTIES OF THE NUMBER NINE. 405 number nine once enter any calculation involving multiplica- tion, and whatever you do, "like the body of Eugene Aram's victim," it is sure to turn up again. This curious property is explained by the principle of divisibility of numbers presented in the previous chapter. All these numbers being divisible by 9, the sums of their digits must be 9, or a multiple of 9. 2. Another curious property of the number nine is that if you take any row of figures and change their order as you please, the numbers thus obtained, when divided by 9, leave the same remainder. Thus, 42378, 24783, 82734, etc., when divided by 9 all give the same remainder, 6. The reason of this is, that the sum of the digits is the same, in whatever order they stand ; and, as previously shown, the remainder from dividing a number by 9, is the same as the remainder from dividing the sum of its digits by 9. 3. An interesting principle is presented in the following puzzle, which, to the uninitiated, seems very singular. Take a number consisting of two places, invert the figures, and take the difference between the resulting number and the first number, and tell me one figure of the remainder and I will name the other. The secret is that the sum of the two digits of the remainder will always equal 9. Thus take 74, invert the terms, and wo have 47; take the difference of the two num- bers and we have 27, in which we see that the sum of 7 and 2 equals 9. In this case, suppose I had not known what number was taken ; if the person had named one digit, say 2, I could have immediately named the other digit 7, since I know that the sum of the two digits is always 9. The reason for this is that both numbers, having the same digits, are multiples of 9 with the same remainder; hence their difference is an exact multiple of 9, and consequently the sum of the two digits will equal 9. When the digits of the number are equal, the difference will be 0; and when they differ by unity, the difference will be 9. 4. There is another interesting puzzle, based upon theso 406 THE PHILOSOPHY OP AKITHMETIC. principles, which is very curious to one who does not see the philosophy of it, and interesting to one who does. You tell a person to write a number of three or more figures ; divide by &, and name the remainder; erase one figure of the number; divide by 9, and tell you the remainder ; and you will tell what figure was erased. This is readily done when the principle is understood. If the second remainder is less than the first, the figure erased is the difference between the remainders; but if the second remainder is greater than the first, the figure erased equals the difference of the remainders subtracted from 9. The reason for this is that the remainder, after dividing a number by 9, is the same as the remainder after dividing the sum of the digits by 9, and hence the sum of the digits being diminished by the number erased, the remainder will also be diminished by it. If there is no remainder either time, then the term erased must be either or 9. To illustrate, suppose the number selected were 45T; divid- ing by 9 the remainder is T; erasing the second term and dividing, the remainder is 2 ; hence the term erased is 7 less 2 or 5. If the number were 461, dividing by 9, the remainder is 2; erasing the second term and dividing, the remainder is 5; hence the term erased must be the difference between 5 and 2, or 3, subtracted from 9, which is 6. 5. The following puzzle also arises from the principle of the divisibility by 9. Take any number, divide it by 9, and name the remainder; multiply the number taken by some number which I name, and divide the product by 9, and I will name the remainder. To tell the remainder, I multiply the first remainder by the number which I named as a multiplier, and divide this product by 9. The remainder thus arising will evidently be the same as the remainder which the person obtained. 6. If we take any number consisting of three consecutive digits and, by changing the place of the digits, make two other PROPERTIES OF THE NUMBER NINE. - 407 numbers, the sum of these three numbers will be divisible by 9. This depends on the principle that the sum of any three consecutive digits is divisible by 3; and consequently each number, if not an exact multiple of 9, is a multiple of 9 plus 3, or of 9 plus a multiple of 3 ; and therefore the sum of three numbers is a multiple of 9 plus three 3's, and thus an exact multiple of 9. If we permutate the digits, making five other numbers, the sum of the six numbers will be divisible by twice 9 ; which may also be readily explained. 7. From the law of the divisibility by nine, several other properties, especially interesting to the young arithmetician, may be derived. Among these may be mentioned the follow- ing: 1. If we subtract the sum of the digits from any number the difference will be exactly divisible by 9. 2. If we take two numbers in which the sums of the digits are the same, the difference of the two numbers will be divisible by 9. 3. Ar- range the terms of any number in whatever order we choose, and divide by 9, and the remainder in each case is the same. Such properties as these must have seemed exceedingly curioua to the early arithmeticians, and fully entitle the number nine to be regarded as a magical number. All of these properties, it is proper to remark, would have belonged to the numbw eleven, if our scale had been duodecimal instead of decimal. PART IV. FRACTIONS. 18 SECTION I. SECTION II. DECIMAL FRACTIONS. SECTION I. COMMON FRACTIONS I. NATURE OF FRACTIONS. II. CLASSES OF COMMON FRACTIONS III. TREATMENT OF COMMON FRACTIONS IV. CONTINUED FRACTIONS. CHAPTER I. NATURE OF FRACTIONS. fPHE Unit is the fundamental idea of arithmetic. Prom it JL arise two great classes of numbers Integers and Frac- tions. Integers have their origin in the multiplication of the Unit; Fractions arise from the division of the Unit. One is the result of an immediate synthesis; the other, of a primary analysis. Fractions have their origin in the analysis of the Unit, as integers arise from the synthesis of units. When the Unit is divided into equal parts, each part is seen to bear a certain relation to the Unit, and these parts may be collected together and numbered. This complex process of di vision, relation, and collection, gives us a fraction. The con ception of a fraction, therefore, involves three things: 1st, a division of the unit; 2d, a comparison of the part with the unit ; 3d, a collection of the equal parts considered. When a unit is divided into a number of equal parts, the comparison of the part with the unit gives the fractional idea, and the col- lection of the parts gives the fraction itself. Herein is clearly seen the distinction between an integer and a fraction. The former is an immediate synthesis; the latter involves a process of division, an idea of relation, and a synthesis of the parts. A fraction is, therefore, a triune product a result of analysis, comparison, and synthesis. Fractions, as has been stated, have their origin in a division of the Unit ; they may also be derived from the comparison of numbers. Thus the comparison of one with two, or of two with four, may give the idea of one-half ; and in a similar manner (418) 414 THE PHILOSOPHY OF ARITHMETIC. other fractions may be obtained. This, however, is a possible rather than the actual origin ; fractions really originated in the division of the Unit. When the Unit is divided into equal parts, these parts are collected and numbered as individual things ; they may, there- fore, be regarded as a special kind of units. To distinguish them from the Unit already considered, we call them fraction al units. This gives us two classes of units, integral units and fractional units. The integral unit is known as the Unit ; when fractional units are meant we use the distinguishing term fractional. The definite conception of an integer requires a clear idea of the Unit ; the definite conception of a fraction requires a clear idea both of the integral and the fractional unit. The character of the thing divided, and the nature of the divis- ion, must be kept clearly before the mind, in order to obtain a distinct conception of the fraction. From this brief statement of the nature of the fraction we are prepared to define it. Definition. A fraction is a number of the equal parts of a Unit. This definition is an immediate inference from the con- ception of a fraction above presented. We divide the Unit into equal parts, and then take a number of these equal parts, and this is the fraction. A definition quite similar to this is, a fraction is one or more of the equal parts of a unit. This is not incorrect, though it is preferred to use the word "num- ber " for " one or more." It is believed that the idea is thus expressed in the most concise and elegant form, and that it will meet the approval of mathematicians. Several other definitions of a fraction have been presented by different authors, some of which are correct, while others are liable to serious objections. One writer says, "A fraction is a part of a unit." This is only part of the truth, for a fraction may be not only one part but several parts of a unit. Another writer says, "A fraction is an expression for one or more of the equal parts of a unit." In this definition the expression, the written or printed symbols, is made the fraction, which is NATURE OF FRACTIONS. 415 evidently incorrect, as we have fractions previous to and inde- pendent of the expression of them. The expressions are not subjects of mathematical calculation, and hence they cannot be fractions. The same distinction holds between a fraction and its expression, as between a number and its expression. Thus we have the number four and the figure 4 ; so we have the fraction three-fourths, and the expression , as two distinct things. Another definition of a fraction is that it is an " unexecuted division." Says one writer, "A fraction is nothing more nor less than an unexecuted division." Says another, "A fraction may be regarded as an expression of an unexecuted division." This conception of a fraction is incorrect, as the idea of a frac- tion, and the idea of the division of one number by another, are entirely distinct. The fraction i (4 fifths), means four of the equal parts which are obtained by dividing a unit into five equal parts. The division of 4 by 5 will give the expression |, but the idea of 4 divided by 5 is entirely distinct from the fractional idea; and hence the assertion, that a fraction is nothing more nor less than an unexecuted division, is absurd. A fraction has also been defined to be the relation of a part of anything to the whole. This was the idea of Sir Isaac Newton, and is correct, though it is rather too abstract for a popular definition. Another form of stating the same idea is that " a fraction is that definite part which a portion is of the whole." Thus, if we divide an apple into two equal por- tions, one of these is one-half of the whole, and this definite part, one-half, is the fraction. This form of statement is not incorrect, though, like Newton's, it is too abstract for a popular definition. Notation. A fraction being a number of equal parts of a unit, it is natural that, in the notation of a fraction, we should indicate the number of parts used, by a figure. It would also, at first thought, seem natural to represent the name of the frac- tional unit by the words, half, third, etc., as 2 thirds, 3 fourths, 416 THE PHILOSOPHY OF ARITHMETIC. etc.; or by their abbreviations, as 2-3ds, 3-4ths, etc. The let ters would be finally omitted altogether, and the expressions become 2-3, 3-4, etc. This probably was the primary form, as is indicated by the expressions, 2-3 for 2 thirds, 3-4 for 3 fourths, which we meet in some of the older books. It has been found more convenient, however, not to express directly the name of the part, but rather to represent the num- ber of parts into which the unit is divided, from which the name of the part is inferred. This might have been done by writing one figure after another, 2-3, the 3 denoting the number of equal parts of the unit, and the 2 the number of parts con- sidered. In practice it has been agreed, however, to write the figure denoting the number of parts into which the unit is divided, under the other, separating them by a line, as in divi- sion. The number expressed by the figure below the line is called the denominator of the fraction, the number expressed by the figure above the line is called the numerator of the fraction. The primary object of the figure below the line is not to name the fractional unit, but to denote the number of equal parts of a unit ; from this the name of the fractional unit is in- ferred. Primarily, then, in our present notation, the denomi- nator of the fraction is not the denomination of the fraction, though from the denominator the denomination is inferred. The denominator thus serves the double object of showing directly the number of equal parts into which the unit is divided, and, indirectly, the name or denomination of the fraction. This distinction should be carefully noted. In integers we have one word to indicate the thing itself, and another to indicate the expression of it. Thus, number means the how many, or thing itself; and figure, the expres- sion of it; the thing and its symbol being distinguished by in- dependent names. In fractions there are no such terms to distin- guish the expression of a fraction from the fraction itself. We are therefore obliged to use the same word fraction to designate both. This we are authorized to do by a figure of rhetoric NATURE OF FRACTIONS. 417 called Metonymy, in which the name of an object is sometimes given to the symbol, or expression of the object. It is conse- quently allowable to use the word fraction when we mean the expression of a fraction, though this frequently occasions con- fusion and calls for particular care on the part of the teacher to prevent it. We are sometimes obliged to make the same dual use of the terms numerator and denominator, but should always do so with extreme caution to avoid confusioa. The expression of a fraction in its relation to the fraction itself, is seen, when analyzed, to be a more complicated thing than at first appears. To illustrate; first, we have the fraction itself, as so many parts of a unit ; then we have the two figures to represent the fraction ; and then we have the numbers, which these two figures denote ; all of which should be carefully dis- tinguished, if we would have a clear idea of the relation of a fraction to its notation. If we begin with the unit and com- pare it with the fraction as expressed, the matter becomes still more complicated. Thus, first we have the Unit; then the equal parts into which the Unit is divided; then the relation of these parts to the Unit; then the expression for a number of these parts, consisting of two figures ; and then the numbers which these figures denote. It is therefore not entirely sur- prising that writers should have been careless and confused in their use of the terms relating to fractions. History. Before proceeding to the classification and treat- ment of Fractions, attention is called to a few points concerning their origin and history. The treatment of fractions by Ahmes is shown in the chapter en the origin of our system of arith- metic. In the Lilawati, fractions are denoted by writing tin- numerator above the denominator, without any line between them. The introduction of the line of separation is due to the Arabs; and it is found in their earliest manuscripts on arith- metic. To denote a fraction of a fraction, as $ of |, the two fractions are written consecutively, without any symbol between them. To represent u number increased by u fraction, the 27 418 THE PHILOSOPHY OF ARITHMETIC. fraction is written beneath the number ; and when the fraction is to be subtracted from the number a dot is prefixed to it; thus, 2 3 2 is denoted by A and 3 by -JL 4 4 In other cases, their notation is not intelligible without ver- bal explanation, and the same is true of the Arabs and earlier European writers, who were singularly deficient in artifices of notation. In the solution of a problem in the Lilawati, in which " the fourth i i STATEMENT. ^ ^ of a sixteenth of the fifth of three 12345164 quarters of two-thirds of a moiety" is required, the work is written as indicated in the margin ; which gives r ^ f , or y^rr- In solving the problem, " Tell me, dear woman, quickly, how much a fifth, a quarter, a third, a half and a sixth STATEMENT. 11111 29 make when added together," the work 54326 20 appears in the Lilawati as indicated in the margin. In solving the problem, " Tell me what is the residue of three, subtracting these frac- tions ;" they expressed the work as in- STATEMENT. dicated, which it is apparent could not 1^1326 20 be understood without an explanation. The Lilawati contains four rules for the reduction and as- similation of fractions, as well as the application of their eight fundamental rules of arithmetic to them. These rules are clear and simple, and differ very little from those used in modern practice. That the author regarded fractions as somewhat difficult, is apparent from the following problem : " Tell me the result of dividing five by two and a third, and a sixth by a third, if thy understanding, sharpened into confidence, be com- petent to the division of fractions." The notation of compound fractions varied with different authors; thus with Lucas di Borgo of , or f xf, v a was represented as in the margin, where v a denotes I ? via, or times. Stifel denoted three-fourths of two- NATURE OF FRACTIONS. 419 thirds of one-seventh by writing the fractions nearly under one another as in the margin ; and the same operation was indicated by Gemma Frisius thus : ^ i im a notation simple and convenient. In the writings of Lucas di Borgo, when two fractions are to be added together or subtracted one from another, the operations to be performed are indicated as follows : 8 9 fXlAH^-iV 12 where those quantities are to be multiplied together which are connected by the lines. There seems to be very little difference between the operations in fractions in ancient and modern text- books. In the works of Di Borgo and Tartaglia, the number of cases and their subdivisions are unnecessarily multiplied, and the reader is frequently more perplexed than instructed by the minuteness of their explanations. It may be remarked that the early writers seem to have been extremely embarrassed by the usage and meaning of the term multiplication in the case of fractions, where the product is less than the multiplicand; and some of their methods of explaining the seeming inconsistency are curious and ingenious. CHAPTER II. CLASSES OF COMMON FRACTIONS. TT^R ACTIONS are divided into two general classes Com J- mon and Decimal. A Common Fraction is a number of equal parts of a unit, without any restriction as to the size of those parts. A Decimal Fraction is a number of the decimal divisions of a unit ; that is, a number of tenths, hundredths, etc. This distinction of fractions originated in a difference in the notation, rather than in any essential difference in the fractions themselves. It was seen that the decimal scale of notation, when extended to the right of the units place, was capable of expressing tenths, hundredths, etc., and that there would be a great advantage in such an expression of them ; and thus the decimal fraction came to be regarded and treated as a distinct class. A brief discussion of each will be given. Common Fractions are variously classified, according to dif- ferent considerations. The primary division is that based upon their relative value compared with the Unit. Classifying them in reference to this relation, we have Proper Fractions and Improper Fractions. A Proper Fraction is one whose value is less than a unit ; that is, one which is properly a fraction ac- cording to the primary conception of a fraction. An Improper Fraction is one which is equal to or greater than a unit ; that is, one which is not properly a fraction in the primary meaning of the term. Another division of common fractions arises from the idea of dividing a fraction into equal parts. A fraction originated (420) CLASSES OP COMMON FRACTIONS. 421 in the division of the Unit into equal parts; now, if we ex- tend this idea to obtaining a number of equal parts of a fraction, we get what is called a Compound Fraction. The Compound Fraction, it is thus seen, originated in the extension of the primary idea of division, which gave rise to the simple fraction This idea of a compound fraction leads to the division of frac- tions into two classes Simple Fractions and Compound Frac- tions. A Compound Fraction is technically denned as a fraction of a fraction. If we extend the fractional idea a little further, and suppose the numerator, or denominator, or both, to become fractional, we have what arithmeticians call a Complex Fraction. The Complex Fraction may be defined as a fraction whose numera- tor, or denominator, or both, are fractional. Whether the com- plex fraction agrees with the definition of a fraction, or with the functions ascribed to the numerator or the denominator of a fraction, is a point which will be considered a little further on ; but its origin was a natural outgrowth of the principle of pushing a notation to its limits. It should be noticed that the complex fraction may also have originated in the expression of the division of one fraction by another by writing the divisor under the dividend with a line between them ; but the proba- bilities are that it originated as first indicated, by an extension of the fractional idea. Fractions, therefore, are divided with regard to their value, as compared with the Unit, into Proper and Improper Fractions; with regard to their form, into Simple, Compound, and Com- plex. There is also another form of expressing fractional rela- tions, so closely connected with the common fraction that it may be embraced under the same general head. I refer to the Continued Fraction, which will be treated with the general subject of common fractions. Improper Fractions. According to the primary idea, a fraction is regarded as a part of a unit, and hence as less than a unit. But since we can speak of any number of fractional 4:22 THE PHILOSOPHY OF ARITHMETIC. units as we do of integral units, there arises a fractional expression whose value is greater than a unit. Thus we may speak of 5 fourths, T fourths, etc., although in a unit there are only 4 fourths. These we call improper fractions; that is they are improperly fractions from the primary idea of a fraction. The improper fraction presents several points of difficulty and interest, which will be briefly considered. Take the expression $|; is this strictly a fraction? That it is properly a fraction, appears from the definition of a fraction and from the discussion just given. How, then, shall it be read? If we read it "f of a dollar," some one will object, that there are only four fourths in a dollar, and hence you cannot speak of five fourths of a dollar. If it be read " dollars," we will object, since there are not enough to make dollars, the plural meaning two or more. But, says some one, the gram- mars tell us that "the plural means more than one," and since $| is more than one, we may use the plural form and say "| dollars." This, we reply, is a mere quibble, as the grammar- ians contemplate only integers when they say "more than one," and really mean "two or more." The reading "% dol- lars" is, therefore, not strictly correct. How, then, should it be read ? I think the correct reading is "^ of a dollar." We mean by it five of such parts as are obtained by dividing a dollar into four equal parts. It is true there are not five fourths in one dollar, and the reading does not assume that there are. No one will object to saying of 100 cents equals 125 cents, which is equivalent to saying |- of a dollar equals a dollar and a quarter. The fractional units, are fourths of a dollar, and the number of fractional units is five; hence the fraction is "five-fourths of a dollar." It is an improper fraction improperly a fraction from the primary idea of a fraction and in the name "improper fraction" we apparently enter a tittle protest against the absolute correctness of the reading in view of the primary idea of the fraction. If we have $| or $-^, we can then say |- dollars or ^ dollars, CLASSES OF COMMON FRACTIONS. 423 since we then have ''two or more." This discussion seems to have been called for from the fact that the question is often raised and debated as to what is the correct reading of the improper fraction. Complex Fraction. According to the strictest definition of a fraction, the complex fraction is an impossibility. This is rendered evident from a consideration of the functions ascribed to the denominator by the definition. The denominator shows the number of equal parts into which the unit is divided ; hence, in the complex fraction ~, the denominator, , denotes that the unit is divided into f equal parts. This is an impos- sibility, as may be seen at least in two ways. First, we can divide a unit into three or two equal parts, but not into one part, since there will be no -division; and if we cannot divide it into one equal part, it is evident that we cannot divide it into less than one equal part. Secondly, if any one doubts the conclusion from this reasoning, let him take an apple and endeavor to divide it into f equal parts. The effort I have sometimes known to be in a high degree amusing, and always conclusive of the correctness of the position assumed above. A somewhat plausible argument in favor of the correctness of the complex fraction is the following: In the algebraic frac- tion r , the numerator and denominator are general expressions, o and hence may represent fractions as well as integers. If then 6=| we shall have a complex fraction. This method of reason- ing is too general for arithmetic ; even in algebra it would prove fl *7* f* that clearing the equation, -=-, of fractions, does not clear it CL of fractions, since in adx=bc, each term may be a fraction. The expression =- means a divided by 6, and is a fraction only so far as it coincides with our arithmetical idea of a fraction We conclude, therefore, that strictly speaking, the complex fraction is an impossibility. It is merely a convenient expres- sion that one fraction is to be divided by another. 424 THE PHILOSOPHY OP ARITHMETIC. Should the idea and expression of a complex fraction, there- fore, be discarded from arithmetic? This does not follow, and is not recommended. It is a convenient form of expressing the division of one fraction by another, and may thus be retained. Those who use it, however, should understand that it is not strictly a fraction, according to the primary idea of a fraction, but a representation of the division of a fraction by a fraction, or of a whole number and a fraction when only one term is fractional. Is a Fraction a Number ? It has been stated by some writers, and seems frequently to be the idea of pupils, that a fraction is not a number. This, however, is a mistake, as will appear from a slight consideration of the matter. Newton's definition of number provides for the fractional number when the object measured is a definite part of the measure ; it consequently ap- pears that the fraction is a number, if we accept his definition as correct. The definition, "A Fraction is a number of equal parts of unity," also makes it clear that a fraction is a num- ber. Again, if it is not a number, what kind of a quantity is it ; and why should it be treated in arithmetic, the science of numbers ? Five inches is certainly a number ; hence its equiv- alent, five-twelfths of a foot, is also a number. Numbers are of two classes, -integers and fractions ; and fractions are num- bers, as much so as integers. The fractional number, it will be noticed, involves two ideas first, the integral unit; and second, the fractional unit. In an integer we have the idea of a num- ber of units ; in the fraction we have, not only an idea of a number of units, but also the relation of the fractional unit to the integral unit. Is a Fraction a Denominate Number ? It has been affirmed by some authors that "fractions are a species of denominate numbers." This, however, is true only in a very limited or partial sense. Three quarts is not precisely the same as three- fourths of a gallon, though they are equal in value. In the latter case, there is a direct and necessary relation of a part to CLASSES OF COMMON FRACTIONS. 425 a unit ; in the former case, no such relation is implied. To un- derstand the fraction, three-fourths of a gallon, the idea of the unit, gallon, must be in the mind ; in three quarts no such con- dition is necessary. In one case there are two units considered, the gallon and the fourth; in the other case but one unit, the quart, not considering the unit of the pure numbers, three and four themselves. Fourths have reference to the integral unit, and always imply this relation ; quarts have no reference to gallons, and do not imply gallons. Again, the fraction three-fourths may be used entirely dis- tinct from any denominate unit, and in this case it must be an abstract, not a denominate number. Two is one-fourth of eight ; here the measure of this relation, one-fourth, cannot but be ab- stract. It is evident, therefore, that a fraction is not a denom- inate number. There are abstract and denominate fractions, aa there are abstract and denominate integers. CHAPTER III. TREATMENT OF COMMON FRACTIONS. A FRACTION has been defined as a number of the equal parts of a unit. The parts into which the unit is divided arc called fractional units. A fraction may, therefore, also be defined as a number of fractional units. Fractions are divided, as previously stated, into Common and Decimal Fractions. A Common Fraction is a number of fractional units expressed with a numerator and a denominator ; as two-thirds, written f . The denominator of a fraction denotes the number of equal parts into which the unit is divided. The numerator of a frac- tion denotes the number of fractional units in the fraction. A common fraction is usually expressed by writing the numerator above the denominator with a line between them. Care should be taken not to define the denominator as the "figure below the line," and the numerator as the "figure above the line ;" and then speak of multiplying the numerator and denominator. This will lead one to suppose that figures may be multiplied, rather than the numbers which they represent. It is surpris- ing that so many writers upon arithmetic should have fallen into this error. Gases. Fractions admit of th same general treatment as integers; we therefore have the same fundamental cases in fractions as in whole numbers. These cases are all embraced under the general processes of Synthesis, Analysis, and Com- parison. The cases of synthesis and analysis are the same as in whole numbers. To perform the synthetic and analytic processes, we need to change fractions from one form to another ; (426) TREATMENT OF COMMON FRACTIONS. 427 hence Reduction enters largely into the treatment of fractions. The comparison of fractions gives rise to several cases called the Relation of Fractions, which do not appear in whole num- bers. The various cases of fractions then are ; Reduction, Ad- dition, Subtraction, Multiplication, Division, Relation, Com- position, Factoring, Common Divisor, Common Multiple, In- volution, and Evolution. A complete view of the fundamental processes is presented in the following logical outline. Composition, Factoring, Invo- lution, and Evolution, presenting no points different from those of whole numbers, are omitted in the treatment. The other cases arising out of Comparison apply equally to integers and fractions, and do not require a distinct treatment. 1. Number to a Fraction. 2. Fraction to a Number. 3. To Higher Terms. 4. To Lower Terms. 5. Compound to Simple. 6. Dissimilar to Similar. (1. The denominators alike. (2. The denominators unlike. (1. The denominators alike. (2. The denominators unlike. 1. Fraction by a Number. Outline of the Cases of Fractions. 1. REDUCTION. 2. ADDITION. 3. SUBTRACTION. 4. MULTIPLICATION. 2. Number by a Fraction. 3. Fraction by a Fraction. 1. Fraction by a Number. 2. Number by a Fraction. 3. Fraction by a Fraction. 1. Number to a Number. 2. Fraction to a Number. 3. Number to a Fraction. 4. Fraction to a Fraction. The " Relation of Fractions" is a new division of the subject of fractions: it was first published in the Normal Written Arithmetic, in 1863, arid has since been introduced into several other works on written arithmetic, and will probably be gen erally adopted. 5. DIVISION. 6. RELATION. 4:28 THE PHILOSOPHY OF ARITHMETIC. Methods of Treatment. There are two methods of develop- ing the subject of common fractions, which may be distinguished as the Inductive and Deductive methods. These two methods are entirely distinct in principle and form; and the distinction, being new, seems worthy of special attention. By the Inductive Method, we solve each case by analysis, and derive the rules, or methods of operation, from these anal- yses, by inference or induction. The method is called induc- tive, because it proceeds from the analysis of particular problems to a general method which applies to all problems of a given class. The solutions, it will be noticed, are independent of any previously established principles of fractions, each case being treated by the method of arithmetical analysis which reasons to and from the Unit. To illustrate the method we will take the problem, "In f how many twentieths?" We analyze this as follows: One equals f$, and equals of 20 twentieths, or 5 twentieths; and f equals 3 times 5 twentieths, or 15 twentieths; hence f equals |-. Now, by examining this solution, we see that we multiply the numerator of f by the number which denotes how many times four, the given denominator, equals the required denom- inator, twenty, which is the same as multiplying both terms of | by the same number, five ; hence we derive the rule, " to reduce a fraction to higher terms, multiply both terms by the same number." For another illustration, take the converse of this problem, "In |f how many fourths?" The solution is as follows : One equals -f$, and ^ equals of f$, which is -fa; hence - of the number of 20ths equals the number of 4ths ; of 15 is 3, hence |-jj equals f. This is the analysis of the problem; we then proceed to derive a rule by which all such problems may be solved. By examining this analysis, we see that we take the same part of the numerator for the numerator of the required fraction that the denominator of the required fraction is of the denominator of the given fraction ; hence we derive the rule, TREATMENT OF COMMON FRACTIONS. 429 "to reduce a fraction to lower terms divide both numerator and denominator by the same number." This rule is thus obtained by analyzing the analysis ; it may also be obtained by compar- ing the two fractions. Thus, comparing f and |-f, we see that 3 equals 15 divided by 5, and 4 equals 20 divided by 5 that is, both divided by the same number and seeing that thif principle holds good in several cases, we infer the rule. By the Deductive Method we first establish a few general principles by demonstration, and then derive the rules, or methods of operation, from these principles. The method is called deductive because it proceeds from the general principle to the particular problem. To illustrate this method, let us solve the same problem, " Reduce f to twentieths." By a gen- eral proposition which we assume has been demonstrated, we have the principle, " Multiplying both terms of a fraction by any number does not change its value;" hence we may reduce | to twentieths by multiplying both terms by 5, which will give the required denominator, and we have f equal to ^-|. For another illustration, we will solve the converse problem, " Reduce | to fourths." By a general proposition, which we assume has been demonstrated, we have the principle, " Divid- ing both terms of a fraction by the same number does not change its value;" hence we may reduce -^ to fourths by dividing both numerator and denominator by any number which will give the required denominator. This number, we see, is 5; hence, dividing both numerator and denominator by 5, we have ^ equal to . We will illustrate the difference of these two methods still further by a problem in compound fractions. Take the ques- tion, "What is f of $?" The analysis is as follows: ^ of | is one of the three equal parts into which may be divided ; if each 5th is divided into 3 equal parts, $ or the Unit will be divided into 5 times 3, or 15 equal parts, and each part will be-j^; hence ^ of -^ is 3^, and of is 4 times -j^, or fa and of $ is 2 times &' or & Examining this analysis, we see that we have mul- 430 THE PHILOSOPHY OF ARITHMETIC. tiplied the two denominators together and the two numerators together, from which we derive the rule for the reduction of compound fractions. By the deductive method we would reason as follows : By a principle previously demonstrated, ^ of f, which is the same as dividing 4 by 3, is T 4 -; and f of f by another principle, is T 8 ^. It will be noticed that the deduc- tive method is much shorter than the inductive method, because while the former explains every point involved, the latter makes use of principles previously demonstrated. If in the deductive solution, we should stop and demonstrate the principles we are to use, it would make the solution much longer. The difference of the two methods can also be clearly illustrated in the divi- sion and relation of fractions. In my higher arithmetic the two methods are presented in each case, where a full comparison may be made of them. The distinction between these two methods is broad and emphatic. By the Inductive Method the problem is solved without any reference to any previously established principle ; by the Deductive Method, the solution is derived from a gen- eral principle supposed to have been previously demonstrated. Both of these methods may be used in the development of frac- tions, and it is a question worthy of consideration which is to be preferred. The Inductive Method is believed to be simpler and more easily understood by young pupils. It is especially adapted to beginners, since it proceeds according to the simple steps of analysis, or the comparison of the collection with the unit. It also follows the law of the development of the young mind "from the particular to the general." It is especially suited to the subject of Mental Arithmetic, on account of its simplicity and the mental discipline it is calculated to afford. The Deductive Method is more difficult in thought than the Inductive Method. Young pupils always find a difficulty in founding a process of reasoning upon previously established principles. It is not natural for the youthful mind to reason from TREATMENT OF COMMON FRACTIONS. 481 generals to particulars. Besides, the demonstrations of these general principles are not readily understood by young pupils. With much experience as a teacher, I state that it is a rare thing to find a pupil who can give a good logical demonstration of these principles, and text-books and teachers often do no better. The so-called demonstrations in many of our text-books are mere explanations or illustrations, and not logical proofs of the propositions. To say that " multiplying the denominator of a fraction increases the number of parts of the fraction, and diminishes their size in the same proportion," is a loose sort of statement that comes very far short of scientific demonstration. We will consider these principles and their demonstration. Fundamental Principles. In the Deductive Method, we have stated, we first establish several general principles, and then derive the rules or methods of operation from them. These principles relate to the multiplication of the numerator and denominator of a fraction. They may be demonstrated in two distinct ways. One of these is founded upon the princi- ples of division ; the other upon the nature of the fraction and the functions of the numerator and denominator. All the various methods in our text-books on arithmetic may be em- braced under these two general methods. The Method of Division is employed by a large majority of our writers on arithmetic. This method consists in regarding the fraction as an expression of an unexecuted division, the numerator representing the dividend, and the denominator the divisor, and the value of the fraction being the quotient. Then, by principles of division presumed to have beeu previously established, since dividing the dividend divides the quotient, dividing the numerator divides the fraction; and since multi plying the divisor divides the quotient, multiplying the denom inator divides the fraction, etc. The Fractional Method of demonstrating these fundamental principles is based upon the nature of the fraction itself. It regards the fraction as a number of equal parts of a unit, an* 432 THE PHILOSOPHY OF ARITHMETIC. determines the result of these operations by comparing the fractional unit with the Unit. Thus, if we multiply the de- nominator of a fraction by any number, as three, the Unit will be divided into three times as many equal parts, hence each part will be one-third as large as before; and the same number of parts being taken, the value of the fraction will be one-third as large as before. In a similar manner all the principles may be demonstrated. The Fractional Method is undoubtedly the correct one. The Method of Division is liable to several objections, and should be discarded in teaching and in writing text-books, as appears from several considerations. First, it is illogical to leave the conception of a fraction and pass to that of division, to establish a principle of a fraction. A fraction and an expression of division are two distinct things, and should not be confounded. The fraction f is three-fourths, and does not mean 3 divided by 4. It is true that the expression | does also mean 3 divided by 4; but when we regard it as a fraction we have and should have no idea of the division of three by four. It is, therefore, illogical, I say, to convert a fraction into a division of one number by another to attain to a principle of the fraction. Secondly, it is not only illogical to treat the subject in this manner, but it does not give the learner the true idea of it. He may see that multiplying the denominator does divide the value of the fraction, but he will not see down into the core of the matter, why it does so. The method, to say the least, gives but a superficial view of the subject, and is therefore objectionable. If the fraction will admit of a simple treatment as a fraction, it is absurd to transform it into something else o to prove its principles. It may be said in favor of the method of division, that it is simpler and more easily understood by . Earner; but this both theory and experience in instruction will disprove. I believe that the pupil can quite as readily see thut dividing TREATMENT OF COMMON FRACTIONS. 433 the numerator of a fraction divides the value of the fraction, as he can see that dividing the dividend divides the quotient ; and the same holds for the other principles. This method may sometimes seem a little easier to the learner, because it depends upon an assumed principle ; but require the pupil to prove that principle, and he will find it quite as difficult as to prove the fractional principle itself. For the method of demon- strating these theorems which the author prefers in arithmetic, the reader is referred to his arithmetical worka 28 CHAPTER IV. CONTINUED FRACTIONS. 17^ VERY new idea, when once fixed, becomes a starting point -I-J from which we pass to other new ideas. The mind never rests satisfied with the old; it is always reaching out beyond the known into the unknown. " Still sighs the world for some- thing new," is as true in science as in society. Given a new conception, and the tendency is to push it forward until it leads us to other ideas and truths not anticipated in the original con- ception. Thus, from the original idea of a simple fraction originated the compound and complex fractions ; and thus also by extending the original conception, arose the Continued Frac- tion. Definition. A Continued Fraction is a fraction whose nu- merator is 1, and denominator an integer plus a fraction whose numerator is also 1 and denominator a similar fraction, and so on. Thus, Several recent authors, for convenience, write a continued fraction with the sign of addition between the denominators; 1111 Origin. Continued Fractions were first suggested to the world in a work by Cataldi, published in 1613, at Bologna. Cataldi reduces the square roots of even numbers to continued fractions, and then uses these fractions in approximation, though without the modern rule by which each approximation is educed from the preceding two. Daniel Schwenter, according to Fink, was the first to make any material contribution (1625) towards deter- ininingthe convergents of continued fractions. Continued fractions were also proposed about the year 1670, by Lord Brouncker, President of the Royal Society. It is known that in order to ex- press the ratio of the circumscribed square to the circumference (434) CONTINUED FRACTIONS. 435 of the circle, he derived the following con- j_f.i tinued fraction given in the margin ; but ^^\-2f. by what means he was led to it, has not been ascertained. He was the first to investigate and make any use of their properties. Dr. Wallis subsequently added to and improved the subject, giving a general method of reducing all kinds of continued fraetions to common fractions. The complete development of these fractions, with their ap- plication to the solution of numerical equations and problems in indeterminate analysis, is due to the Continental mathemati- cians. Huygens is said to have explained the manner of form- ing the fractions by continual divisions, and to have demon- strated the principal properties of the converging fractions which result from them. John Bernoulli made a happy and useful application of the continued fraction to a new species of calculation which he devised for facilitating the construction of tables of proportional parts. The most complete development of continued fractions was given by Euler, who introduced the term f radio continua. Treatment. The subject of continued fractions is most con- veniently treated by the algebraic method, and may be fouud quite fully presented in some of the works on higher algebra. In this place we shall briefly consider: 1. Reducing common fractions to continued fractions; 2. Reducing continued frac- tions to common fractions; 3. Their application; 4. Their prin- ciples. We shall first show how a common fraction may be reduced to a continued fraction. Take the common fraction ^ 7 . Dividing both numerator and denominator by 68, we have the first expression in. the mar- J_^2i *_j_ i gin; dividing the numerator and denom- "" inator of the second fraction by 21, we 9, have the second expression in the margin ; dividing again by 5, we have the third expression in the margin ; which finishes the division, a* 4:36 THE PHILOSOPHY OF ARITHMETIC. the numerator of the last fraction is unity. The terms -, ^. , etc., are called the first, second, third, etc., partial fractious. The same result may be obtained by dividing as in finding the greatest common divisor, and taking the several quotients for the successive denominators. Taking j^ T , and dividing as if to find the greatest common divisor of its terms, we see that the resulting quotients are the same as 68 157 the denominators of the partial fractions. Hence we _ ." derive the followin rule for reducin common fractions to continued fractions: Find the greatest common divisor of the terms of the given fraction ; the reciprocals of the successive quotients will be the partial fractions which constitute the continued fraction required. Let us now see how a continued fraction may be reduced to a common fraction. This reduction may be effected in two ways ; by beginning at the last fraction and working up, or by beginning at the first fraction and working down. If we take the continued fraction given in the margin and reduce the complex fraction formed by the last two terms to a simple frac- tion, we shall have ^ 5 T . Taking this result and the preceding partial fraction together, 15 21 we have - , which reduced equals . Joining this to the 2 + 21 47 preceding term, we have - , which equals . Finally, 1+47 68 =3 , the value of the fraction. 0+68 251 By beginning at the first fraction, approximate values of the continued fraction may be obtained by respectively reducing two, three, or more of the partial fractions to simple fractions. Thus, in the fraction given above, the first approximate value 111 111 is ; the second - , or ; the third is - - -^-, or q I q /o -- ; the fourth - : the fifth -7^7- 11 48 251 CONTINUED FRACTIONS. 437 By exhibiting this process in an analytic form, a law may be discovered which presents a simpler and easier method of find- ing approximate values than either of the others. Let us take the fraction in the margin and find 2 . i its successive approximate values, and notice the ~^ la^ A the derivation of one approximation from the previous ones. The work may be written as follows: 2 =i, 1st approx. val. =f, 2d 3x2+1 1 3x5+1 3x5+l _ _ 3 X 5+1 (3 x 2+l)x5+2 -7x5+2 -. (3X5+l)X4+3 3x(5+i)+l ' {(3 x 2+l) x 5+2}x 4+3x2+1 16x4+3 37x4+7 T * 5 . We take ^, the first term of the continued fraction, for the first approximate value. Reducing the complex fraction formed by the first two terms of the continued fraction, we have ^ for the second approximate value. Continuing the reduction, we obtain ^f and -fJ^ for the remaining values. Examining the last two reductions, we find that the third approximate value is obtained by multiplying the terms of the second approximate fraction by the denominator of the third partial fraction, and adding to these products the corresponding terms of the first approximate fraction. We see also that the fourth approximate value is equal to the product of the terms of the third approxi- mate value by the denominator of the fourth partial fraction, plus the corresponding terms of the second approximation. Hence we derive the following rule : For the first approximate value take the first partial fraction; for the second value, reduce the complex fraction formed by the first two terms of the continued fraction ; for each succeeding approximate value, multiply both terms of 438 THE PHILOSOPHY OF ARITHMETIC. the approximation last obtained by the next denominator of the continued fraction, and add to the products the corre- sponding terms of the preceding approximation. We will now show the application of continued fractions by the solution of several practical questions. 1. Let it be required to express approximately, in the fraction of a day, the difference between a solar year and 365 days. By the old reckoning, the excess of the solar year over 365 days was 5 hours, 48 minutes, 48 seconds. Reducing, we find this excess equals 20,928 seconds, and 24 hours equals 86,400 seconds. Therefore, the true value of the fraction =-||m=-^|f. Now, converting |4^ into a continued fraction, we have the expres- sion given in the margin, from which, by the last rule, we obtain the approx- iinr" imate values J, ^ &, & *. Hih The fraction agrees with the correction introduced into the calendar by Julius Caesar, by means of bissextile or leap year. The fraction -^ is the cor- rection used by the Persian astronomers, who add 8 days in every 33 years, by having f regular leap-years, and then de- ferring the eighth for 5 years. 2. Required the approximate ratio of the English foot to the French metre containing 39.371 inches. The true ratio is ff^-. Reducing to a continued fraction, we find some of the first approximate values to be , T 3 7 , T \, fa, |f, yVg-- Hence the foot is to the metre as 3 to 10, nearly; a more correct ratio is 32 to 105. 3. To find some of the approximate values of the ratio of *,he circumference of a circle to the diameter. Taking the value of the circumference of the circle whose diameter is 1, to 10 places of decimals, the ratio of the diameter to the circumference will be expressed by the common fraction Hif IHHHHHnr- Reducing to a continued fraction, some of the first approximate values are, \, -fa, i||, li-f . Inverting these tractions, we have the ratio of the circumference to the diame- CONTINUED FRACTIONS. 439 ter, which is the ratio commonly used. The second gives ty which is the ratio said to have been found by Archimedes ; and the fourth gives f-f-f , which is the same as that determined by Metius, which is more exact than 3.141592, from which it is de- rived. Continued fractions have been employed for obtaining elegant approximations to the roots of surds. Thus, let it be required to find the square root of , or the ratio of the side of a square to its diagonal. The square root of i?, or ^/\, equals . Dividing both terms by the numerator we have o~ = ~i o T* Multi- >/ 2 l~r\/2 1 1 ^2 i plying both terms of the fraction by x/2-fl, it will be come r~7~ = S~I To i"- Substituting, we have v/ 2 + 1 2 + \/ 2 1 1 1 1 ""TFi 24V2 1* n~ /9 1 1 Again, the fraction S = - becomes, as before, equalto / 2 1 1 and by thus continuing the process, we find to equal the following continued fraction : Some of the first approximate values of this fraction are |, f , 7 TT lr T9 2ffT e f Continued fractions are also applied to the solution of inde- terminate problems, as may be seen in Barlow's Thec/ry of Numbers, or Legendre's Thforie des Nombre*. 440 THE PHILOSOPHY OF ARITHMETIC. There are several beautiful principles belonging to the ap- proximate values of continued fractions, a few of which we present in this place. The values just obtained for the ratio of the side of a square to its diagonal are used as illustrations. 1. The approximate fractions are alternately too small and too large. Thus, , ff-, $$, are too small, while \, %, f^, and | are too large. 2. Any one of these fractions differs from the true value of the continued fraction by a quantity which is less than the reciprocal of the square of its denominator. Thus, j-|, which is the ratio much used by carpenters in cutting braces, differs from the true ratio by a quantity less than ( T V) 2== ^i^- 3. Any two consecutive approximate fractions, when re duced to a common denominator, will differ by a unit in their numerators. Thus f and \%, when reduced to a common de- nominator, become -ffy and T 8 T \. 4. All approximate fractions are in their lowest terms. If they were not, the difference of the numerators of two consec- utive approximate fractions, when reduced to a common denom- inator, would differ by more than unity. For each numerator is multiplied by the denominator of the other fraction, hence one derived numerator contains the original numerator, and the other the original denominator of either fraction. If then there were a common factor, it must be a factor of the difference of the numerators; and this difference would be greater than unity, which is contrary to the previous principle. The successive approximate values are called the convergents of the fraction. The numerator or denominator of the convergent is called, by Sylvester, a cumulant. A non-terminating contin- ued fraction whose quotients recur, is called a periodical or recurring continued fraction. Its value can be shown to be equal to one of the roots of a quadratic equation. It can also be shown that every quadratic surd gives rise to an equivalent periodic continued fraction. SECTION II. DECIMAL FRACTIONS. I. ORIGIN OP DECIMALS. II. TREATMENT OP DECIMALS. III. NATURE OP CIRCULATES. IV. TREATMENT OP CIRCULATES. V. PRINCIPLES OF CIRCULATES. VI. COMPLEMENTARY REPETENDS. VII. A NEW CIRCULATE FORM. CHAPTER I ORIGIN OP DECIMALS. rpHE invention of the Decimal Fraction, like the invention of JL the Arabic scale, was one of the happy strokes of genius. The common fraction was expressed by a notation quite dis- tinct from that of integers, and required not only a different treatment, but one much more complicated and difficult. The expression of the decimal divisions of the unit in the same scale with integers, and the possibility of reducing common fractions to the decimal form, wrought quite a revolution in the science of arithmetic, and has greatly simplified it. This new method of expressing fractions gave rise to a much simpler method of treating them ; and has elevated the decimal fraction into dis- tinction, and gained for it an independent consideration. Origin. The Decimal Fraction had its origin within the last three centuries. Theoretically it may have originated in either of two ways. There may have been a transition from the com- mon fraction to the decimal, by noticing that a number of tenths, hundredths, etc., might be expressed by the decimal scale. This is the manner in which the subject is usually presented in the text-books of the present day. Thus, after the pupil is made familiar with the fractions fa, y^, etc., it is stated that -fa may be expressed thus, .1 ; -^ thus, .01, etc. The decimal fraction could also have arisen directly from the decimal scale. Thus, since the law of the scale is, that terms diminish in value from left to right in a ten-fold ratio, the idea of carrying the scale on to the right of the unit would naturally present itself, and such a continuation would give rise to the decimal. As the unit (443) 444 THE PHILOSOPHY OF ARITHA1ETIC. was one-tenth of the tens, the first place to the right of the unit would be one-tenth of the unit ; the second place, one-tenth of one-tenth, or one-hundredth, etc. These two methods of conceiving the origin of decimals are entirely distinct ; indeed, they are the converse of each other. In one case we pass from the common fraction to its expression in the decimal scale; in the other we pass from the expression in the decimal scale to the fraction. This distinction, it may be remarked, has a practical bearing upon the method of teaching the subject. In which way it did actually originate is not definitely known, though De Morgan holds that the table of compound interest suggested decimal fractions to Stevinus. History. The introduction of decimal fractions was formerly ascribed to Regiomontanus, but subsequent investigations have shown this to be incorrect. The mistake seems to have arisen from the confused manner in which Wallis stated that Regio- montanus introduced the decimal radius into trigonometry in place of the sexagesimal. Decimal fractions were introduced so gradually that it is difficult, if not impossible, to assign their origin to any one person. The earliest indications of the deci- mal idea are found in a work published in 1525 by a French mathematician named Orontius Fineus. In extracting the square root of 10, he extracts the approximate root of 10000000 and obtains 3162. He then separates 162, which with him is not a fraction, but only a means of procuring fractions, and converts it, after the scientific custom of the times, into sexa- gesimal fractions (having as base 60), so that the square root of 10 would be expressed 3 9' 43" 12'", or a-t-^+yffor+iiiSSoo- He concludes that chapter of his book by stating that in this 162, 1 is a tenth, 6 is six hundredths, etc., so that it would seem that he had quite a clear notion of decimals. Tartaglia, in 1556, gives a full account of the metnod of Orontius, but prefers the common fractional form 3 T WV In Recorde's Whetstone of Witte, 1557, the same rule is copied; but after obtaining three decimal places of the square root, the ORIGIN OF DECIMALS. 445 remainder is written as a common fraction. Peter Raums, in an arithmetic published in Paris in 1584 or 1592, also quotes the rule of Orontius. In 1585, Stevinus wrote a special treatise in French, called " The DISME, by the which we can operate with whole numbers without fractions." It was first published in Dutch about the year 1590, and describes in very express and simple terms the advantages to be derived from this new arithmetic. Decimals are called nombres de disme: those iii the first place whose sign is (1) are called primes, those in the second place whose sign is (2) are called seconds, and so on ; whilst all integers are char- acterized by the sign (0), which is put over the last digit. The following are some of his arithmetical operations by means of decimals, representing multiplication and division. (0) (1) (2) (0) (1) (2) (3) (4) (5) (1) (2) 3257 344352 (9 6 8946 1 8 6 5114 7637 29137122 3 4 4 3 5 2(3 5 8 7 96666 999 It will be seen that he employs the " scratch method " of division. The following is an example of indefinite division found in his work : (0) (1) (2) (3) f=l 333 In this treatise Stevinus proposed to supersede fractions by c^.s/m-.s, or decimals. He enumerates the advantages which would result from the decimal subdivision of the units of length, urea, capacity, value, and lastly of a degree of the quadrant, in the uniformity of notation, and the increased facility of per- forming all arithmetical operations in which fractions of such units were involved. It is remarkable, however, that though while he confines himself to the matter of his computation he 446 TUB PHILOSOPHY OF ARITHMETIC. admits his dismes, when he passes to their form he converts them into integers. Still, he must be regarded as the real in- ventor and introducer of the system of decimals. De Morgan says "The Disme is the first announcement of the use of deci- mal fractions ;" and Dr. Peacock also remarks that " the first notice of decimals, properly so called, is to be found in La This work of Steviuus was translated into English in 1608, by Richard Norton, under the title, "Diame, the arte of tenths, or decimal Arithnietike, teaching how to perform all computa- tions whatsoever by whole numbers without fractions, by the four principles of common Arithmetike : namely, addition, sub- traction, multiplication, and division, invented by the excellent mathematician, Simon Stevin." In this work the notation is changed to (1) (2) (3) (4) 3, 7, 5, 9. The introduction of decimals into works on arithmetic was slow, even after their use had been shown by Stevinus. One of the earliest English works in which decimal fractions are really used, is that of Richard Witt, 1613, containing tables of half-yearly and compound interest. These tables are con- structed for ten million pounds ; seven figures are cut off, and the reduction to shillings and pence, with a temporary decimal separation, is introduced when wanted. Thus, when the quar- terly table of amounts of interest at ten per cent, is used for three years, the principal being 100Z., in the table stands 1372- 66420, which multiplied by 100 and seven places cut off, gives tne first line of the following citation : " The Worke (1 1372 Facit < sh 13 66429 2858 4296." (d 3 Giving 1372Z. 13s. 3d. for the answer. The tables are expressly stated to consist of numerators, with 100... for a denominator Napier's work, published in 1617, contains a treatise on deci- ORIGIN OF DECIMALS. 447 mals, though he does not use the decimal point, except in one or two instances, but rather indicates the place of the decimal figures by primes, seconds, etc., according to the method of Stevinus. The author expressly attributes the origin of dec- imals to Stevinus. In 1619 we find the contents of Norton's treatise embodied in an English work entitled, "The Art of Tens, or Decimall Arithmetike, wherein the art of Arithmetike is taught in a more exact and perfect method, avoyding the intricacies of fractions. Exercised by Henry Lyte, Gentleman, and by him set forth for his countries good. London, 1619." It is dedicated to Charles, Prince of Wales, and he tells us that he has been requested for ten years to publish his exercises in decimall Arithmetike. After enlarging upon the advantages which attend the knowledge of this arithmetic to landlords and tenants, merchants and tradesmen, surveyors, gaugers, farmers, etc., and all men's affairs, whether by sea or land, he adds, "if God spare my life, I will spend some time in most cities of this land for my countries good to teach this art." This author was one of the earliest users of decimal fractions In the year 1619 there appeared, at Frankfort, a work on decimal arithmetic by Johann Hartman Beyern, in which the author states that he first thought upon the subject in the year 1597, but that he was prevented from pursuing it for many years by the little leisure afforded him from his professional pursuits. He makes no mention of Stevinus, but assumes throughout the invention as his own. The decimal places ure indicated by the superscription of the Roman numerals, though the exponent corresponding to every digit in the decimal places is not always put down. Thus, 34.1426 is written 34.1 I 4 II 2 III 6 IV , or 34.14 II 26 IV , or 34.1426 IV . The author must have been acquainted with the liabdologia of Napier, as one chapter of his work is devoted to the explanation of the construction and use of these rods, which enjoyed a most extraordinary popularity at that period; and 448 THE PHILOSOPHY OF ARITHMETIC. he could not, consequently, have been ignorant of Napier's notation or of the work of Stevinus ; and we may therefore doubt the truth of his pretensions to being the originator of the system of decimals. Albert Girard published an edition of the works of Stevinus in 1625, and in the solution of the equation x 3 3x 1 by a table of sines, of which method he was the author, we find the three roots as in the margin. On 1,532} o itr [ another occasion, he denotes the separation of the *; ' f integers and decimals by a vertical line. He does not always adhere to this simple notation, as we after- wards find the square root of 4^ expressed by 20816(4) ; and on another occasion we find similar vestiges of the original notation of Stevinus. Oughtred is said to have contributed much to the propaga- tion and general introduction of decimal arithmetic. In the first chapter of his Clavis, published in 1631, we find an explanation of decimal notation. The integers he separates from the decimal, or parts, by a mark, L, which be calls the separatrix, as in the examples, 0^56, 48^, for .56 and 48.5; and in giving examples of the common operations of arithmetic he unites them under common rules. His view of the theory of decimals was generally adopted, and in some cases hi* notation also, by English writers on arithmetic for more than thirty years after this period. In " Webster's tables for simple interest," etc., 1634, decimals seem to be treated as a thing generally known, though no decimal point is used. During the same year, 1634, Peter Herigone, of Paris, published a work in which he introduces the decimal fraction of Stevinus, having a chapter " des nombres de ia dixme." The mark of the decimal is made by marking the place where the last figure comes. Thus when 137 livres 16 sous is to be taken 23 years 7 months, the product of 1378' and 23583'" is found to be 32497374"", or 3249 liv. 14 sous, 8 deniers. In 1633, John Johnson (Survaighor) published a ORIGIN OF DECIMALS. 449 work, the second part of which is called "Decimall Arithmatick wherby all fractionall operations are wrought, in whole num- bers," etc. In his decimal fractions Johnson has the rudest form of notation ; for he generally writes the places of decimals 1.2.3.4.5. over the figures; thus, 146.03817 would be 146103817. In 1640, the "Arithmetica Practica" of Adrian Metius contains sexagesimal fractions, but not decimal ones ; and a work by Job. Henr. Alsted, in 1641, containing a slight treatise on arithmetic and algebra, says nothing about decimal fractions. About this time the subject of decimals must have been pretty generally understood; for in "Moore's Arithmetick," 1650, the subject of decimals is quite thoroughly presented and the contracted methods of multiplication and division are given. Noah Bridges, in his "Arithmetick Natural and Deci- mal, "has an appendix on decimals, though the author expresses his disapproval of the use which some would make of decimals, averring that the rule of practice is more convenient in many eases. John Wallis, 1657, uses the old decimal notation 12 345, but he afterwards adopts the usual point in his algebra; and subsequently decimals seem to have been no longer regarded as a novelty, but took their place along with the other accepted subjects and methods of arithmetic. It may be supposed that the publication of the tables of log- arithms was necessarily connected with the knowledge and use of decimal arithmetic ; but this, Dr. Peacock thinks, is not so. Tho theory of absolute indices, in its general form at least, was at that time unknown ; and logarithms were not considered as the indices of the base, but as a measure of ratios merely. Under this view of their theory, it was a matter of indifference whether we assumed the measure of the rntio of 10 to 1 to be one, ten, a hundred, ten millions, or ten billions, the number assumed by Briggs in his system of logarithms. Thus, whether tin- logarithms are expressed by decimals or integers, they will have the same characteristics, and their use in calculation is 29 450 THE PHILOSOPHY OF ARITHMETIC. exactly the same. It is under the integral forms that the loga- rithms are given in the earlier tables, such as those of Napier, Briggs, Kepler, etc. This statement will sufficiently explain the reason why no notice is taken of decimals in the elaborate explanations which are given of the theory and construction of logarithms by Na- pier, Briggs, and Kepler ; and indeed we find no mention of them in any English author between 1619 and 1631. In that year the Logarithmical Arithmetike was published by Gellibrand, a friend of Briggs who died the year before, with a much more detailed and popular explanation of the doctrine of logarithms than was to be found in Briggs's Arithmetica Logarithmica. It is there stated that the logarithms of 19695, 1969 -fa, 19 T Vinr are respectively 4,29435 etc., 3,29435 etc., 1,29435 etc., differing merely in their characteristic; and ^, r < WiJ) are called decimal fractions. Rules are also given for the reduction of vulgar fractions to decimals, by a simple proportion; and, lastly, a table for the reduction of shillings, pence, and farthings to deci- mals of a pound sterling. The Decimal Point. The final and greatest improvement in the system of decimal arithmetic, by which the notations of decimals and integers are assimilated, was the introduction of the decimal point, and much labor has been spent to ascertain its author. According to Dr. Peacock, the decimal point was introduced by Napier, the illustrious inventor of logarithms. In writing decimals Napier seems to have generally employed the method of Stevinus, which was to indicate the decimal places by primes, seconds, etc. ; but there are at least two in- stances in which he used a character as a decimal separatrix. The first is an example of division in which he writes 1993,273, using a comma, and then presents his answer in the form 1993 2/ >j// 3/// The other instance occurs in a problem in multi- plication, in which he draws a line down through the places of the partial products that would be occupied by the decimal point; but in the sum he uses the exponents of Stevinus, ORIGIN OF DECIMALS. 451 which thus combines both methods, and stands 1994 | 9' 1" 6'" a"". The problems in which these occur are found in the Rabdol- ogia, published in 161*7, in which he mentions the invention of Stevi- , f uus in terms of highest praise, and \^ explains his notation without notic- 402 ing his own simplification of it. 429 The use of the comma, above re- 861094,000(1993,273 ferred to, is presented in the ac- 3888 companying solution, in which it 3888 is required to divide 861094 by 1296 432. I present but a part of the etc. process of division. ...... The quotient is 1993,273, The use of the vertical line Qr jgg 3 %, >,,} ^//J is found in an example of ab- breviated multiplication which occurs in the solution of the following problem: "If 31416 be the approximate value of the circumference of a circle whose diameter is 10,000, what is the numerical value of the circumference of a circle whose diame- ter is 635?" This solution is said to be the first example found of this abbreviated multiplication ; the use of it, how- ever, became very popular in a short time afterward, being ee- pecially useful in the multiplication of the large numbers which were made use of in the construction of the tables of sines, etc. This seems like a very near approach to the decimal point, if it is not indeed the introduction of it ; but De Morgan main- tains that Napier only used his comma or line as a rest in the process, and not as "a final and permanent indication, as well as a way of pointing out where the integers end and the frac- tions begin." It must be admitted that the use of the separatrix was merely incidental, and not the practice of Napier ; but he seems to be the first to use a mark for this purpose, even in- cidentally, and there can be no doubt that even this incidental use had very great influence in leading to the general adoption of a decimal point. 452 THE PHILOSOPHY OF ARITHMETIC. De Morgan thinks that Richard Witt, who published a work four years before Napier, " made a nearer approach to the dec- imal point" than Napier; yet he says, "I can hardly admit him to have arrived at the notation of the decimal point " Witt, in a work published in 1613, presents some tables of compound interest, in which decimal fractions are used. The tables are constructed for ten millions of pounds, seven figures are cut off, and the reduction to shillings and pence with a temporary decimal separatrix, in the form of a vertical line, is introduced when wanted, as may be seen on page 446. But though his tables are distinctly stated to contain only numerators, the denominator of which is always unity followed by ciphers, and though he had arrived at a complete and permanent command of the decimal separator, and though he always multiplies or divides by a power of 10 by changing the place of the decimal separator, which is a vertical line, yet De Morgan thinks he gave no "meaning to the quantity with its separator inserted." He thinks that if Witt had been "asked what his 123 | 456 was, he would have answered: It gives 123^. not Jt 12 3 T 4 7nrV" Briggs, the author of the common system of logarithms, was a disciple of Napier, and might have been expected to adopt Napier's method of writing decimals. We find, however, that in 1624, instead of using a decimal point he draws a line under the decimal terms, omitting the denominator; thus, 5 9321. A work by Albert Girard, published in 1629 at Amsterdam, is remarkable as using the decimal point on a single occasion. Oughtred, in his Clavis, published in 1631, uses both the vertical and sub-horizontal separatrix, thus shutting up the numerator in a semi-rectangular outline, as 23 456 for 23.456. William Webster's work, published in 1 634, treats of decimals as a thing generally known ; but does not make use of the decimal point, using the partition line to separate integers and decimals. In 1657 John Wallis pub lished a work in which the old notation, 12 345, was used ; ORIGIN OF DECIMALS. but he subsequently adopted the decimal point in his algebra. 12345 la 1643, the notation used in Johnson's arithmetic is 3 2291 9, and 312500, and 34,625, and sometimes 358149411 fifths. Kav- anagh says that the present notation was, for the first time, clearly set forth in some editions of Wingate's arithmetic, 1650. On the Continent the notation used was 12 345 or 12[345, even in works of the highest repute, up to the beginning of the 18th century. The following summary presents some of the different methods of writing decimals which are found among the early writers on arithmetic, both in England and on the Continent: 34. 1'. 4". 2'". 6"" 34 1426 (1) (2) (3) (4) 34. 1 . 4 . 2 . 6 34 1426 34. 1 . 4 . 2 . 6 34'1426 34.1426"" 34,1426 34.1426W It is believed that Gunter, who was born in 1581, did more for the introduction of the decimal point than any one of his cotemporaries. He first adopted the notation of Briggs, but gradually dropped it and substituted the decimal point. In one of his works, De Morgan tells us, Briggs's notation appears without explanation, and 116 04 is given as the third proportional to 100 and 108. On a subsequent page a dot is added to Briggs's notation in one instance; thus 100J. in 20 years at 8 per cent, becomes 466.095Z. At the bottom of the same page, Briggs's notation disappears thus: "It appeareth before, that 100/. due at the yeares end is worth but 92 592 in ready money . If it be due at the end of two yeares, the present worth is 85Z.733; then adding these two together, wee have 178/.32G for the present worth of 100 pound annuity for 2 yeares, and so forward." After this change, thus made without warning in the middle of a sentence, Briggs's notation does not again occur in the part of the work which relates to numbers. In a pre vious work on the sector, etc., the simple point is always used; 454 THE PHILOSOPHY OF ARITHMETIC. but in explanation the fraction is not thus written, but described as parts./ Thus, 32.81 feet used in the operation is, in the de- scription of the answer, 32 feet 81 parts. Fink says that decimal fractions were known by Rudolff, who in the division of integers by powers of 10 cut off the required number of places with a comma. He also attributes the intro- duction of the decimal point to Kepler, while Cantor says it is found in the trigonometric tables of Pitiscus, published in 1612. It was some time after this, however, before the decimal point was fully recognized in all its uses, even in England. As long as Oughtred was widely used, which was until the end of the seventeenth century, there must have been a large school of those who were trained to the notation 23 I 456. The complete and final victory of the decimal point must be referred to the first quarter of the eighteenth century. It may seem surprising that the decimal fraction should have been introduced so late in the history of the science ; this delay, however, admits of explanation. The decimal division of the unit would be of no value until after the Arabic system of notation was adopted. Even then the introduction was necessarily slow. Simple as they now appear, the development of decimal fractions was too great an effort for one mind, or even one age. The idea of their use dawned gradually upon the mind, and one mathe- matician taking up what another had timidly begun, added an idea or two, until the subject was at length fully conceived and developed. The advantages of the decimal notation of fractions are so ob- vious that they hardly need to be specified. Many of the opera- tions upon fractions are thereby greatly simplified, and others are entirely avoided. The fundamental operations of addition, sub- traction, multiplication and division, are the same as in integers, and the cases of reduction to lower terms, common denominator, etc., do not occur at all. The advantages would have have been still greater if the basis of the numeral scale had been twelve in- stead of ten, as appears from a previous discussion. CHAPTER II. THE TREATMENT OF DECIMALS. A DECIMAL FRACTION is a number of the decimal divisions of a unit; or it is a number of tenths, hundredths, etc. Some authors define it as a fraction whose denominator is ten or some power of ten ; and others as a fraction whoso denominator is one followed by one or more ciphers. Both of these definitions are correct, but seem less satisfactory than the one first presented. They are objectionable on account of not expressing the kind of fractional unit, but rather indicating its nature by describing the denominator of the fraction. A Decimal Fraction may be expressed in two ways in the form of a common fraction, or by means of the decimal scale. When expressed by the scale it is distinguished from the general meaning of the term decimal fraction by calling it a Decimal. A Decimal may thus be defined as a decimal fraction expressed by the decimal method of notation. Thus &> iVff' e ^ c -' are decimal fractions, but not decimals; while .5, .45, etc., are both decimal fractions and decimals. This distinction is convenient in practice, and is believed to be strictly logical. It has not been generally adopted, but then; seems to be a growing tendency towards such a distinction. In popular language, however, we use the term "decimal fraction" as equivalent to a decimal. Notation. The decimal fraction, as expressed by the decimal scale, has no denominator written, the denominator being indicated by a point before the numerator. This notation, as already seen, arises from that of integers, and is merely un 456 THE PHILOSOPHY OF ARITHMETIC. extension of it. Beginning at units' place, by a beautiful generalization, numbers are regarded as increasing toward the left and decreasing toward the right, in a ten-fold ratio, the result of which is a decimal division of the unit, corresponding to each decimal multiple of it. In order to distinguish between the integral and fractional expression and locate each term properly, a point or separatrix is used. Various marks have been employed for this purpose, at different times, but the period is now generally adopted. The origin of this use of the decimal separatrix is discussed in the previous chapter. Sir Isaac Newton held that the point should be placed near the top of the figures, thus, 3'56, to prevent it from being confounded with the period used as a mark of punctuation. Cases. The cases in decimals, it is evident, must be nearly the same as in whole numbers. The relation of common fractions to decimals would, it is natural to suppose, give rise to one or more new processes. A new method of notation having been agreed upon for a special class of common fractions, the inquiry naturally arises, Can other common fractions be expressed as decimals, and how? We thus begin to pass from common fractions to decimals; and, reversing tbis process, pass back from decimals to common fractions. This gives rise to a process known as the Reduction of Fractions, embracing the two cases of reducing common fractions to deci- mals, and its converse, decimals to common fractions. The reduction of common fractions to decimals gives rise to a par- ticular kind of decimals called circulates, which require an independent treatment. The other cases of decimals are the same as in whole numbers. Method of Treatment. The method of treating decimals is quite similar to that of whole numbers. Indeed, they so closely resemble integers that many authors have been of the opinion that they should be presented with them. It is claimed that there is but one principle in the expression of integers and TREATMENT OF DECIMALS. 457 decimals, and that the processes and reasoning are the same, whether the scale is ascending or descending. It is therefore concluded that the notation of decimals should be presented with that of integers, and that the fundamental processes of addition, subtraction, etc., should be applied to them both in the same connection. There are, however, valid objections to this seemingly plausi- 'ole inference It will be admitted that the mechanical opera- tions are the same ; but the reasoning processes, in at least two of the fundamental operations, are not identical. The fixing of the decimal point in multiplication and division, would be entirely too difficult to be presented along with the fundamental operations of integers. Besides, it would be illogical to separate one class of fractions from the general subject of fractions ; and moreover, one process, namely the reduction of decimals, could not be considered until after common fractions had been dis- cussed. These considerations have been sufficient to prevent authors of arithmetic from uniting the treatment of decimals with that of integers, and will, I doubt not, continue to sepa- rate them. Numeration. In the treatment of decimals, the first thing to be considered is the method of reading and writing them, or their Numeration and Notation. These processes present sev- eral points worthy of notice, points which seem to have escaped the attention of the writers on arithmetic. Having introduced the subject of decimals by explaining that the first place to the rig-lit of units is tenths, the second place hundredths, etc., it im- mediately follows that .45 is read "4 tenths and 5 hundredths," but it does not immediately follow, as many arithmeticians are in the habit of assuming, that it is read "45 hundredths." If, however, it is first explained that -^ is written .4, and y*^, .45. then it does not immediately follow that .45 is read "4 tenths and 5 huudredths." The usual method of presenting decimals is to explain that the first place to the right of the decimal point is tenths, tho second place hundredths, etc.; it should thon b 20 458 THE PHILOSOPHY OF ARITHMETIC. shown that the decimal can be otherwise read. Thus, suppose we have the decimal .45: this expresses primarily 4 tenths and 5 hundredtlis ; and since 4 tenths equals 40 hundredths, and 40 hundredths and 5 hundredths are 45 hundredths, the expression 45 may also be read 45 hundredths. This must be explained if we desire to preserve the chain of logical thought in our treatment From this it is seen that in practice there are two methods of reading decimals, which may be expressed as follows : 1. Begin at the decimal point and read in succession the value of each term belonging to the decimal, or 2. Bead the decimal as a whole number, and annex the name of the right-hand decimal place. It will be noticed that in reading a large decimal we should numerate from the decimal point to derive the denominator, and toward the decimal point to determine the numerator. Notation. The writing of decimals, when conceived or read to us, presents several points of interest. When the decimal is conceived analytically, that is, as so many tenths, hundredths, etc., we write it by the following rule : 1. Fix the decimal point and write each term, in its proper decimal place. If the decimal is conceived synthetically, that is, as a number of ten-thousandths, or a number of millionths, etc., we write it by the following rule : 2. Write the numerator as an integer, and then place the decimal point so that the right-hand term shall express the de- nomination of the decimal. In writing a decimal in which the numerator does not occupy the required number of decimal places, it is not readily seen where to place the decimal point, and how many ciphers to pre- fix. The best practical rule in this case is the following. 3 Write the numerator as an integer, and then begin at. the right and numerate backward, filling vacant places with ciphers, until we reach the required denomination, and to the expression thus obtained, prefix the decimal point. TREATMENT OF DECIMALS. 459 Thus, to write 475 millionths, we first write 475 ; then be- ginning at the 5, we numerate toward the left, saying tenths, hundredth*, thousandths, ten-thousandths (writing a cipher), hundred-thousandths (writing a cipher), millionths (writing a cipher), and then place the decimal point. Several other methods have been suggested for writing decimals, among which is the following, by Prof. Henkle. It is seen that the tens of any number of tenths, the hundreds of any number of hundredths, the thousands of any number of thousandths, etc., each fall in the order of units when the decimal is expressed. Thus 56 tenths, is 5.6, the 5 tens falling in units' 1 place ; 2345 hundredths is 23.45, the 3 hundreds falling in units' place, etc. Hence the rule, 1. Begin at the left and write the term corresponding to the denominator of the decimal in the place of units. Reduction. The methods of treating the two cases of reduc- tion are very simple. In reducing a common fraction to a decimal fraction, we reduce the different terms of the numerator to tenths, hundredths, etc., and divide by the denominator. In reducing a decimal to a common fraction, we express the deci- mal in the form of a common fraction, and then reduce it to its lowest terms. Fundamental Operations. Addition and subtraction are treated exactly as in integers, the same rules applying to both. The mechanical processes of multiplication and division are the same as in whole numbers ; the only difference being the placing of the point in the product and quotient. There are two .methods of explaining the location of the decimal point in multiplication and division, based upon the different concep- tions of the origin of the decimal. One locates the point by the principles of common fractions; the other derives the method from the pure decimal conception. The latter is the simpler and more practical method. These two methods are explained in my works on written arithmetic, and need not IK> presented here. CHAPTER III. NATURE OF CIRCULATES. adoption of the method of expressing fractions by the decimal scale opened up a new avenue of thought in the science of numbers. Decimals were treated without writing the denominator, and common fractions were frequently thrown into the decimal form and operated upon by means of the rules for whole numbers. The process of changing common fractions into the decimal scale led to the discovery of an interesting class of decimals called Circulating Decimals. These new forms soon attracted the attention and called forth the ingenuity of mathematicians; and, when investigated, were found to possess some remarkable and interesting properties. Origin. Circulating Decimals have their origin in the reduction of common fractions to decimals. In making this reduction, we annex ciphers to the numerator, and divide by the denominator. This division sometimes terminates with an exact quotient, and sometimes would continue on without ending. When it does terminate, the common fraction can be exactly expressed in a decimal ; when it does not terminate, if the division be carried sufficiently far, a figure or set of figures will begin to repeat in the same order. Such a decimal is called a circulating decimal, or simply a Circulate. It is thus seen that Circulates have their origin, not in the nature of number itself, but in the method of notation adopted to express numbers. They are an outgrowth of the Arabic system of notation and the decimal scale upon which it is based. If the scale of this system were duodecimal instead of (460) NATURE OF CIRCULATES. 461 decimal, the subject of Circulates would be greatly modified. Thus %, , ^, etc., which now give circulates, would then give finite decimals; while i, |, J^, etc., would give circulating decimals. Notation. The part of the circulate which repeats is called .1 Repetend A Repetend is indicated by placing one or two periods or dots over it. A repetend of one figure is expressed oy placing a point over the figure which repeats ; thus .3 expresses .333, etc. A repetend of more than one figure is expressed by placing a period over the first and the last figure; thus, 6.345 expresses 6.345345, etc. Sometimes the first part of a decimal does not repeat, while the latter part does repeat. Such a decimal is called a mixed circulate. The part which repeats is called the repeating part ; the part which does not repeat is called the non-repeating or finite part of the circulate. Thus 4.536 is a mixed circulate in which 5 is the finite, or non-repeating part, and 36 the repeating part. In an expression consisting of a whole number and a circulate, if the whole number contains terms similar to those of the repetend, the repetend may be indicated by placing one of the dots over a term in the whole number. Thus, suppose we have the circulate 54.234234, etc. ; this is usually expressed thus, 54.234; but, since the term just before the decimal point is the same as the last term of the repetend, it may also be expressed thus, 54.23. This indicates that 423 repeats; and. expanding the expression, we have 54.23423 etc., which, expressed in the ordinary way, becomes 54.234. In the same way, G.04 denotes 6.046 ; 20.12 denotes 20.1220. The reading of a repetend is a matter which often puzzles voung teachers. Thus, in the case of the repetend .3, since the denominator is 9, we cannot say "the decimal 3 tenths;" neither will it answer to say "the decimal 3 ninths;" how, then, shall it be read? The true reading is "the circulate 3 tenths." Calling it a circulate distinguishes it from tho decimal fraction 3 tenths, and also indicates that it is equal to 3 ninths. 4:62 THE PHILOSOPHY OF ARITHMETIC. Again, how shall we read 436 ? It is not sufficiently explicit to say "the mixed circulate 436 thousandths," or "the mixed circulate 4 tenths and 36 thousandths," since neither of these expresses the idea exactly. The correct reading is, "the mixed circulate 436 thousandths, whose non-repeating part is 3 tenths and repeating part 36 thousandths." There may be other read- ings equally correct ; the one suggested is given to lead teachers to avoid the adoption of those which are erroneous. Definitions. A Circulate is a decimal in which one or more figures repeat in the same order. A Repetend is the term or series of terms which repeat. This distinction between a cir- culate and a repetend should be carefully noted, as it is not always clearly understood. Circulates are Pure and Mixed; Repetends are Perfect, and Imperfect, Similar and Dissimi- lar, and Complementary. A Perfect Repetend is one which contains as many decimal places, less one, as there are units in the denominator of the equivalent common fraction. Thus, -^= .142857, and ^=.0588235294117647 are each perfect repe- tends. Similar Repetends are those which begin and end respec- tively at the same decimal places; as .427 and .536. Dissimi- lar Repetends are those which begin or end at different decimal places. Especial attention is called to this definition of simi- lar repetends, as it is a departure from the view usually taken Repetends which begin at the same place have usually been re- garded as similar; and those which end at the same place, conterminous. It is thought, however, to be much more pre- cise to regard repetends beginning and ending respectively at the same places as similar. Repetends are surely not quite similar if they end at different places ; to be similar they should both begin and end at the same place. This view makes it necessary to employ some other term to indicate a similarity of beginning. There being no word thus used, the term cooriginous, expressing a coorigin, is suggested. Its appro- priateness may be seen by comparing it with conterminous, de- NATURE OF CIRCULATES. 46<J noting a Determination, which has already been adopted to denote a similarity of endings. Cases. Since circulates have their origin in the reduction of common fractions to decimals, it follows that the first case in the treatment of circulates is Reduction. The Reduction of Circulates embraces three distinct cases: 1. The reduction of common fractions to circulates; 2. The reduction of circulate^ to common fractions; 3. The reduction of dissimilar repeteurt.- to similar repetends. We have also Addition, Subtraction, Multiplication, and Division of Circulates. I have also recently introduced in my Higher Arithmetic the Greatest Common Divisor and Least Common Multiple of Circulates, subjects not heretofore treated in any arithmetical work. The comparison of circulates with common fractions gives rise to a number of interesting truths, which will be presented under the head of Principles of Circulates. Method of Treatment. The method of reducing common fractions to circulates is the same as that of reducing them to ordinary decimals. An abbreviation, based upon a principle of repetends, is sometimes employed. The method of reducing circulates to common fractions differs considerably from the method of reducing decimals to common fractions. In the finite decimal, the denominator understood is 1 with as many ciphers annexed as there are places in the decimal ; in the circulate the denominator of the repetend is as many 9's as there are places in the repetend. There are three methods of explaining this reduction, as will be shown in the treatment. Circulates can be added, subtracted, multiplied, and divided, by first reducing them to common fractions ; or they may be expanded sufficiently far so that the repeating figures may appear in the result. Both of these methods are objectionable on account of their length, and are therefore not usually employed. In the addition and subtraction of circulates, it i better to reduce them to similar repetends and then perform the operation. In the multiplication and division of circulates, a slight modification of this method is employed. CHAPTER IT. TREATMENT OF CIRCULATES. rpHE Treatment of Circulates embraces the operations of JL Reduction, Addition, Subtraction, Multiplication, Division, Greatest Common Divisor, Least Common Multiple, etc., and the Principles of Circulates. Attention will be called to the treatment of several of these subjects, and a distinct chapter will be devoted to the Principles of Circulates. Reduction of Circulates. The Reduction of Circulates is conveniently treated under four cases : 1. To reduce common fractions to circulates. 2. To reduce a pure circulate to a common fraction. 3. To reduce a mixed circulate to a common fraction. 4. To reduce dissimilar repetends to similar ones. 1. To reduce common fractions to circulates. The gen- eral method of reducing common fractions to circulates is to annex ciphers to the numerator of the common fraction, and divide by the denominator, continuing the division until the figures of the circulate begin to repeat. Thus, to reduce -^ to a circulate, we annex ciphers to the numerator 5, divide by the denominator 12, indicate the repeating figure by placing a period over it ; and we have the circulate .416. When the circulate consists of many figures, the process of reduction may be abbreviated by employing some of the prin- ciples of repetends. Thus, suppose it be required to reduce -^ to a repetend. By actual division to five places, we find ^=0.03448^. Now - is 8 times $, hence multiplying this by 8 we have ^=0.27586^-. Substituting this value of -^ in the expression for the value of -^y, and we have ^=0.0344827586^. (464) TREATMENT OF CIRCULATES. 465 This, multiplied by 6, gives ^=0.20689655 17^; which, sub- stituted in the second expression for , gives 3^=0.03448275862068965517^. Multiplying by 7, we get ^=0.24137931034482758620f ; which, substituted in the third expression for -fa, gives 3V=0.0344827586206896551724137931034482758620f&. As the terms have begun to repeat, it is unnecessary to continue the process any further. It will be seen, on examina- tion, that the repetend consists of 28 figures, or one less than the denominator of -fa, and therefore is a perfect repetend. 2. To reduce a pure circulate to a common fraction. There are three distinct methods of explaining this case, as has already been stated. In order to illustrate these methods, we will solve the problem, Reduce .45 to a common fraction. In the first method, having proved by actual division that 1=1, .01=^, .001=^-^, etc., we derive the denominator of any circulate from its relation to these given circulates. To illustrate, reduce the circulate .45 to a common fraction. The method is as follows: Since .61=^, as shown by OPERATION. actual division, .45, which is 45 times .61, equals ,oi=A 45 times fa, or ff, which, reduced to its lowest .45 |j? & terms, equals -fa. By the second method, we multiply the circulate by 1 with. as many ciphers annexed as there are places in the repetend, which makes a whole number of the repeating part of the circulate. We then subtract the two circulates, and have a certain number of times the given circulate equal to the differ- ence, from which the given circulate is readily found. We will illustrate by the solution of the same problem. Let C represent the common fraction which equals the circulate ; we will then ^ ] have C=4545 etc. ; multiplying by 100 iooC=45i4545 etc'. to make a whole number of the repeating qon 45 part, we have 100 times the common C=M=-A fraction equal to 45.4545 etc. ; subtract- ing once the common fraction from 100 times the common PO 4:66 THE PHILOSOPHY OF ARITHMETIC. fraction, we have 99 times the common fraction equal to 45.4545 etc., minus .4545 etc., which equals 45; hence once the common fraction equals H, or T 5 T . By the third method, the repetend is regarded as an infinite series, the ratio being a fraction whose numerator is 1, and denominator 1 with as many ciphers annexed as there aro places in the repetend. The solution is as follows: The repetend .45 may .. OPERATION. be regarded as an infinite series, y 4 ^ +nnnnr^ etc - The f rmu la f r the Bum of an infinite series is S=- 1 r Substituting the value of a~-fis> aa( l r ~Ttt> we ^ ave S=-nnj -f- y^g-, which equals |-f , or y^-. 3. To reduce a mixed circulate to a common fraction. There are three distinct methods of reducing mixed circulates to common fractions, as in the preceding case. To illustrate these methods we will solve the problem, Reduce .3i8 to a common fraction. By _ OPERATION. the first method, we reason thus: The -318=yV of 3.18 mixed circulate.3 18 equals y 1 ^ of 3.1 8, which = -__U =._LL by the preceding case equals y 1 ^ of 3^-f, or _ 35 _ 7 j- 1 ^ of 3 T 2 r , which equals T *y%, or -fa. lllf -'*' By the second method, we reason as follows: Let C repre- sent the common fraction, then we u 11 u n 01010 !* i OPERATION. shall have C .31818 etc.; multiply- Q_ 31818 etc ing by 10 to make a whole number etc. of the non-repeating part, we have 10000=318.1818 etc. 10 times the fraction equals 3.1818 etc.; multiplying this by 100 to make a whole number of the repeating part, wt have 1000 times the fraction equals 318.1818 etc.; subtract- ing 10 times the fraction from 1000 times the fraction, we have 990 times the fraction equals 315, from which we find the fraction equals |^|, or -fa. TREATMENT OF CIRCULATES. 467 In the previous method we see that we subtract OPERATION. the finite part from the entire circulate, and divide by as many 9's as there are figures in the repe- tend, with as many ciphers annexed as there are decimal places before the repetend; hence, by r j 816 7 generalizing this into a rule, we may perform the operation as in the margin. This is the method preferred in practice. This case may also be solved by regarding the repetend as an infinite scries, and finding its OPERATION. sum by geometrical progression, ,3i and then adding it to the finite part. The solution is presented in the margin, in which it is seen that we regard y^j- as the first term of the series, and T -^ as the rate. 4. To reduce dissimilar repetends to similar ones. To solve this case it is necessary to understand the following principles: 1. Any terminate decimal may be considered interminate, its repetend being ciphers; thus, .45 .450, or .45000, etc. 2. A simple repetend may be made compound by repeating the repeating figure; thus, .3=. 33=. 3333, etc. 3. A compound repetend may be enlarged by moving the right-hand dot towards the right over an exact number of periods ; thus, .245=.24545, etc. 4. Both dots of a repetend may be moved the same number of places to the right; thus, .5378=.53783 or .537837, etc., for each expression developed will give the same result. 5. Dissimilar rcpetends may be made cobriginous by moving both dots of the repetend to the right until they all begin at the same place. 0. Dissimilar repetends may be made conterminous by mov- ing the right-hand clots of each repetend over an exact number of periods of each repetend until they end at the same place. The method of treating this case may be illustrated by the 468 THE PHILOSOPHY OF ARITHMETIC. OPERATION. .45 =.45454545454545 .4362 =.43628623623623 .813694=.81369436943694 following example: Make .45, .4362, and .813694 similar. The solution is as follows: To make these repetends similar, they must be made to begin and end at the same place. To do this, we first move the left-hand dots so that they begin at the same place, and then move the right-hand dots over an exact number of periods, so that they will end at the same place. Now the number of places in the periods are re- spectively 2, 3, and 4 ; hence the number of places in the new periods must be a common multiple of 2, 3, and 4, which is 12 ; we therefore move the right-hand dot so that each repetend shall contain 12 places. Divisor and Multiple. The Greatest Common Divisor of two or more decimals is the greatest decimal that will exactly divide them. Such a divisor can be found by reducing tho decimals to common fractions, and applying the method for common fractions ; but it can also be found by keeping them in the decimal form ; and the latter method is generally less tedious and more direct. To illustrate the method, let us find the greatest common divisor of .375 and .423. We make tho two circulates similar, and sub- tract the finite part, which re- duces them to fractions having a common denominator. We then find the greatest common divisor of their numerators, 1638, which is the numerator of the greatest common divisor, the denominator being of tho same denomination as the simi- lar decimals ; hence the greatest common divisor is 5 aWA ff> or ,0001638. OPERATION. .3751575 .4234234 3757572 3813264 4234230 3757572 1 8 9 2 4 4 . C 476658 501228 55692 49140 24570 26208 6552 6552 snnnp=-00 1638 01638, G D TREATMENT OF CIRCULATES. 469 The method, it is seen, consists in reducing the decimals to a common denominator, finding the greatest common divisor of their numerators, writing this over the common denominator, and reducing the resulting fraction to a decimal. The Least Common Multiple of two or more decimals is the least number that will exactly contain each of them. Such a multiple can be found by reducing the decimals to common fractions and applying the method for common fractions; but it can also be found by keeping them in their decimal form ; and the latter method is preferred, as being generally more direct and less laborious. To illustrate the method, let us find the least common mul- tiple of .327, i.Oll and .075. We reduce the circulates to frac- tions having a common denominator, as in the previous case. The least common multiple of these numerators is 275699700, which is the numerator of the least common multiple, the denominator being the common denomina- OPERATION. 3 4 25 101 .32727 3 1.01110 10 .07575 32724 101100 07575 10908 33700 2525 2727 8425 2525 2727 337 101 27 337 1 tor of the Reducin fractions. 3 x 4 x 25 X 101 x 27 X 337=275699700 iL5^||^.fiJL=2757.2727, L. C. M. =2757.2 the least common mul- tiple, to whole numbers and decimals, we have 2757.2, the least common multiple. It will be seen that the method consists in reducing the dec- imals to a common denominator, finding the least common multiple of their numerators, writing this over the common denominator, and reducing the resulting fraction to a decimal. T CHAPTER V. PRINCIPLES OF CIRCULATES. investigation of the relation of circulate forms to com- mon fractions has led to the discovery of some very inter- esting and remarkable properties. These will be considered under the head of Principles of Circulates, and Complemen- tary Itepetends. The subject being rather briefly treated in the text-books, will be presented here somewhat in detail. A brief and simple explanation will be given in connection with each principle. 1. A common fraction whose denominator contains no other prime factors than 2 or 5, can be reduced to a simple decimal. For, since 2 and 5 are factors of 10, if we annex as many ciphers to the numerator as there are 2's or 5's in the denominator, the numerator will then be exactly divisible by the denominator. 2. The number of places in the simple decimal to which a common fraction may be reduced, is equal to the greatest num- ber of 2's or 5's in the denominator. For, to make the numer- ator contain the denominator, we must annex a cipher for every 2 or 5 in the denominator, and the number of places in the quotient, which is the decimal, will equal the number of ciphers annexed. 3. Every common fraction, in its lowest terms, whose denom- inator contains other prime factors than 2 or 5, will give an inter minate decimal. For, since 2 and 5 are the only factors of 10, if the denominator contains other prime factors, the nu- merator with ciphers annexed will not exactly contain the denominator; hence the division will not terminate, and the result will be an interminate decimal. (470) PRINCIPLES OF CIRCULATES. 471 i. Every common fraction which does not give a simplf decimal, gives a circulate. For, in reducing a common frac- tion to a decimal, there cannot be more different remainders than there are units in the denominator; hence, if the division be continued, a remainder must occur which has already been used, and we shall thus have a series of remainders and divi- dends like those already used, therefore the terms of the quo- tient will be repeated 5. The number of figure* in, a repetend cannot exceed the number of units in the denominator of the common fraction which produces it, less I. For, in reducing a common fraction to a decimal, when the number of decimal places equals the number of units in the denominator, less 1, all the possible different remainders will have been used, and hence the divi- dends, and therefore the quotients which constitute the circu- late, will begin to repeat. In many cases the remainders begin to repeat before we have as many as the denominator less 1. 6. The number of places in a repetend, when the denominator of the common fraction producing it is a prime, is always equal to the number of units in the denominator, less 1, or to some factor of this number. For, the repetend must end when it reaches the point where it has as many places less 1 as there are units in the denominator of the producing fraction; hence, if it ends before this, the number of places must be an exact part of the number of places in the denominator less 1, that it may terminate when it has as many places* as the denominator less 1. This is not generally true when the denominator is composite, as in J T , -fa, ^, ^ etc - 1. A common fraction whose denominator contains 2'x or Vs iinlh other prime factors, will give a mixed circulate, and the number of places in the non-repeating part will equal the rireatest number of 2's or 5's in the denominator. Dividing first by the 2's and 5's, we shall have a decimal numerator containing as many places as the greatest number of 2's or 5'a 472 THE PHILOSOPHY OF ARITHMETIC. (Prin. 2). If we now divide by the other factors, the dividends consisting of the terms of the decimal numerator will not give the same series of remainders as when we have a series of dividends with ciphers annexed ; hence the circulate will begin directly after the last place of these decimal terms. To illustrate, take -^Q, and factor the denominator, and we have dividing by the 2 and the 5's we have -^, in which it is evident the circulate must begin in the third decimal place, just as the circulate from -f- begins in the first decimal place. 8. When the reciprocal of a prime number gives a perfect repetend, the remainder which occurs at the close of the period is 1. For, since the reduction of the fraction to a circulate commenced with a dividend of 1 with one or more ciphers annexed, that the quotients may repeat we must begin with the same dividend, and therefore the remainder at the close of the period must be 1. 9. When the reciprocal of any prime number is reduced to a repetend, the remainder which occurs when the number of decimal places is one less than, the prime, is 1. For, since the number of decimal places in the period equals the denominator less 1, or is a factor of the denominator less 1, at the close of a period consisting of as many places as the denominator less 1, there will be an exact number of repeating periods, and .therefore che remainder will be 1. 10. A number consisting of as many 9's as there are units tn any prime less 1, is divisible by that prime. For, if we divide 1 with ciphers annexed by a prime, after a number of places 1 less than the prime, the remainder is 1; hence 1 wilh the same number of ciphers annexed, minus 1, would be exactly divisible by the prime; but this remainder will be a series of 9's, therefore such a series of 9's is divisible by the prime. Thus 999999 is divisible by 7. 11. A number consisting of as many 1's as there are units PRINCIPLES OF CIRCULATES. 473 in any prime (except 3), less 1, is divisible by that prime For the prime is a divisor of a series of 9's (Prin. 10), which is equal to 9 times a series of 1's; and since 9 and the prime are relatively prime, and the prime is a divisor of 9 times a series of 1's, it must be a divisor also of a series of 1's. Thus 111111 is divisible by 7 ; also 1111111111 is divisible by 11. 12. A number consisting of any digit used as many times as there are units in a prime (except 3), less 1, is divisible by that prime. For, since such a series of 1's is divisible by the prime, any number of times such a series of 1's will be divisible by the prime. Hence 222222, 333333, 444444, etc., are each divisible by 7. 13. The same perfect repetend will express the value of all proper fractions having the same prime denominator, by starting at different places. Thus, |=.14285714285 etc. But tf-=.lf, hence the part that follows 1 in the repetend of ^ is the repetend off; that is, ^=.428571. Again, ^--.14f; hence the part that follows .14 in the repetend of \ is the repetend of ; that is, -f=. 285714. In a similar manner we find ^=.857142, $=.571428 ; and the same thing is generally true. 14. In reducing the reciprocal of a prime to a decimal, if we obtain a remainder 1 less than the prime, we have one- half of the repetend, and the remaining half can be found by subtracting the terms of the first half respectively from 9. Tn k<> }, and let us suppose in decimating we have reached a remain- der of 6; now what follows will be the repetend of , and the repetend of $ added to the repetend of ^ must equal 1, since -|-|=l; hence the sum of these two repetends must equal .999999 etc., since .999999 etc. equals 1. Now in adding the terms of these two repetends together, that the sum may bo a series of 9's, there must be just as many places before the point where 6 occurred as a remainder, as after; hence G occurred as a remainder when we were half through the scries. Again, since the sum of the terms of the latter and the for- mer half of the repetend equals a series of 9's. each term of 474 THE PHILOSOPHY OF ARITHMETIC. the Grst half of the repetend, subtracted from 9, will give the corresponding term of the latter half of the series. All perfect repetends possess this property, and a large num- ber of those which are not perfect. Repetends possessing this property are called complementary repetends. The last two properties are of great practical value in reducing common fractions to repetends. 15. Any prime is an exact divisor of 10 raised to a power denoted by the number of terms in the repetend of the prime, less 1 ; or of 1 raised loa power denoted by any multiple of the number of terms, less 1. For, by Prin. 6, the number of places in the repetend must equal the number of units in the prime, or some factor of that number ; hence the dividend used in ob- taining a period must be 10 raised to a power equal to the number of terms in the period; and since the remainder at the end of the period is 1, the prime will exactly divide 10 raised to a power equal to the number of terms in the period, less 1. Both this and principle 6 depend on Fermat\~> Theorem, that pp-i i i s divisible by p when p and P are prime to each other." For 10, the base of the decimal system, is prime to any prime number except 2 and 5; hence 10 P ~ J 1 is always exactly divisible by p, when p is any prime except 2 and 5. It thus follows that in the division of 1 with ciphers annexed, the remainder is always 1 when the number of places in the quotient is equal to the number of units in the prime. From this we can readily derive the second part of principle 6, and also principle 15. 16. Any prime is an exact divisor of a number when it will divide the sum of the numbers formed by taking groups of the number consisting of as many terms as there are figures in the repetend of the reciprocal of that prime. We will show this for a prime whose reciprocal gives a repetend of three places. The number 47,685,672,856, may be put in the form 856 + 672xl0 3 + 685xl0 6 + 47xl0 9 , or 672x(10 3 1) + 685 X (10 l)-f47x(10 9 1) + 856+ 672 + 685 + 47; but these PRINCIPLES OF CIRCULATES. 475 different powers of 10, diminished by 1, are all divisible by any number whose reciprocal gives a number of three places, as 37; hence if the sum of the groups, 47 -f 685+672 + 856, is divisible by 37, the entire number is also divisible by 37. The same may be illustrated with any other number, and the principle is therefore general. The principle admits, also, of a general demonstration. From this general proposition we derive the following special principles embraced under it: 1. Since the reciprocals of 3 and 9 give a repetend of one place, they will divide a number when they divide the sum of the digits. 2. Since the reciprocals of 11, 33, and 99, give a repetend of two places, they will divide a number when they divide the sum of the numbers found by taking groups of two places. 3. Since the reciprocals of 27, 37, and 111, give repetends of three places, they will divide a number when they divide the sum of the numbers formed by taking groups of three places. 4. Since the reciprocal of 101 gives a repetend of four places, it will divide a number when it divides the sum of the numbers formed by taking groups of four places. 5. Sinco the reciprocals of 41 and 271 give repetends of five places, they will divide a number when they divide the sum of the numbers formed by taking groups of five places. 6. Since the reciprocals of 7, 13, 21, and 39 give repetends of six places, they will divide a number when they divide the sum of the numbers formed by taking groups of six places. PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY Physical & Applied Sci.