The philosophy of arithmetic as developed from the three fundamental processes of synthesis, analysis and comparison, containing also a history of arithmetic

Front matter

THE

PHILOSOPHY OF ARITHMETIC

AS DEVELOPED FROM THE

Preface

PROGRESS in education is one of the most striking characteristics 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 multiplying in every section of the country ; improved methods of training have been adopted ; dull routine has given way to a healthy intellectual activity ; instruction has become a science and teaching a profession.

This advance is reflected in, and, to a certain extent, has been pioneered by, the improvements in the methods of teaching arithmetic. 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 interest. The spirit of analysis, suggested and developed in it, runs to-day like a golden thread through the whole science, giving simplicity and beauty to all its various parts.

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IV PREFACE.

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 completeness 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 its mission, has done all that it was possible for it to accomplish, but it is not sufficient for the perfection of a science. There must be synthetic 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 arithmetic,in respect to logical symmetry and completeness, differs widely from its sister branch— geometry. 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 excellence and beauty. It stands as a type of all that is best in the classical culture of the thoughtful mind of Greece. Geometry is the perfection 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 speculations upon their properties, but this did comparatively little for the development of the science. The present system of arithmetic is mainly the product of the thought of the past three or four centuries. 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 presented 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 authorities 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 information are referred to the writers mentioned.

Part First treats of the general nature of arithmetic, embracing the Nature of Number, the Nature of ArithmelUMl Language, and the Nature of Arithmetical Reasoning. The natute 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 written 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.

Vi PREFACE.

Considerable attention is given to the nature of Arithmetical Reasoning, a subject which seems not to have been very clearlyunderstood by logicians and arithmeticians. The effort is made to put this matter upon a logical basis, and to ascertain and present 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 basis; that it is founded upon and grows out of the three logical processes, Analysis, Synthesis, and Comparison. This is a new generalization, 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, separate, and compare numbers.

Proceeding with this idea, it is shown that, regarding Addition, Subtraction, Multiplication, and Division, as the fundamental operations of arithmetic, there will arise from them several other processes of a similar character, which I have called the Derivative Processes of Synthesis and Analysis. It is then seen that for each analytical process there should be a corresponding synthetic process. 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 Algebra, as may be seen in my Elementary Algebra.

Continuing this thought, it is shown that Ratio, Proportion, the

PREFACE. Vii

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 logical 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 being made to exhibit their nature and their logical relation to integers. 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 subject of Fractions admits of two methods of treatment, logically distinct in idea and form, and both treatments are presented. Especial attention is given to the treatment of Circulates, and the most important 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 continuous quantity, in which some artificial unit is assumed as a measure. 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 attract 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 consuited; and, in many cases, their very language has been used, their expression being so clear and concise that I could not hope to improve it. In thus combining with the philosophy of arithmetic its history, which in many cases aids in unfolding its philosophy, I have aimed to present a work especially valuable to students and the younger teachers 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 about twelve years ago. This might be regarded as an advantage; for, according to the recommendation of Horace, publication should not be hurried, but " a work should be retained till the ninth year." Quintilian also remarks 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 presenting 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 freshness and vividness of conception what they may lack in elegance of style.

Cherishing many pleasant remembrances associated with the discussion of these ideas before my pupils in the class-room, to many of whom their publication will prove a reminder of days gone by, 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, 3 filler svi lie, 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 Schools.

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 III— Comparison. Section I. — Ratio and Proportion

Chapter I. Introduction to Comparison 291

" II. Nature of Ratio 294

" III. Nature of Proportion 305

IV. Application of Simple Proportion - . .310

" V. Compound Proportion 318

VI. History of Proportion 326

Section II. — The Progressions.

Chapter I. Arithmetical Progression 341

" II. Geometrical Progression 345

Section III. — Percentage

Chapter I. Nature of Percentage 355

II. Nature of Interest 361

Section IV. — The Theory of Numbers

Chapter I. Nature of the Subject 371

" II. Even and Odd Numbers 375

" III. Prime and Composite Numbers 378

" IV. Perfect, Imperfect, etc., Numbers 383

" V. Divisibility of Numbers 389

" VI. The Divisibility by Seven 397

" VII. Properties of the Number Nine , 404

Section I. — Common Fractions

Chapter I. Nature of Fractions 413

" II. Classes of Common Fractions 420

" III. Treatment of Common Fractions 426

" IV. Continued Fractions 434

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

** VII. 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.

THE Science of Arithmetic is one of the purest products of human thought. Based upon an idea among the earliest which spring up in the human mind, and so intimately associated with its commonest experience, it became interwoven 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 relations, 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 to 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 profoundest thinkers. Over it Pythagoras mused with the deepest enthusiasm ; 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 comes down to us adorned with the offerings of a thousand intellects, and sparkling with the o-ems of thought received from the profoundest 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 fundamental and derivative ideas, what is the law of their evolution, 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 questions 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 fundamental 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 author 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 relation 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 f

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. Beginning, 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 fundamental 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 primary operations are Synthesis and Analysis.

To determine when and how to unite, and when and how to separate, we employ a process of reasoning called Comparison. This process compares numbers and determines their relations. Synthesis and Analysis are mechanical processes , Comparison is the thought process. Comparison directs the original processes, modifies them so as to produce from them new ones, and also gives rise to other processes not contained in the original ones. It is, in other words, by this thought process working upon the idea of number, that the original processes 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 arithmetic is developed. Comparison, therefore, in arithmetic as in geometry, 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 three things; Synthesis, Analysis and Comparison. Synthesis and Analysis are fundamental mechanical operations, suggested in the formation of numbers; Comjiarison is the fundamental thought process which controls these operations, unfolds their potential ideas, and also gives rise to other divisions of the science growing immediately out of itself. In other words, the science of arithmetic has a triune basis; it has its roots in, and grows out of, the three processes, Synthesis Analysis, and Comparison. Let us examine these processes and see the number, nature, and relations of the divisions growing out of the fundamental operations, and thus determine the logical character of the science of arithmetic.

Synthesis. — A general synthesis is called Addition. A special case of the synthetic process of Addition, in which the numbers added are all equal, their sum receiving the name of product, is called Multiplication. The forming of Composite Numbers by a synthesis of factors, which may be called Composition ; Multiples, formed by a synthesis of particular factor? ; and Involution, by 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 process 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. Factoring 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 required ; 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 Factoring, 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 Proportion. 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 numbers. In comparing numbers, we may assume some number as a basis of reference and develop their relations in regard to this basis ; — when this basis is a hundred, we have the process called Percentage.

Thus we obtain a complete outline of the science of numbers, and perceive more clearly the logical relations of the divisions of the science. Arithmetic is conceived as based upon the two fundamental operations, synthesis and analysis, these operations being controlled by comparison, which develops new processes from these and also from itself. The whole science of Pure Arithmetic is the outgrowth of this triune basis, Synthesis, Analysis, and Comparison. The rest of arithmetic consists of the solution of problems, either real or theoretic, and may be included under the head of Applications of Arithmetic.

This conception of the subject is new and important. It has been heretofore held that addition and subtraction comprehended the entire science of arithmetic; that all other processes are contained in them, and are an outgrowth from them. This is a fallacy, which, among other things, has led logicians to the absurd conclusion that there is no reasoning in arithmetic. Assuming that there is no reasoning in the primary processes of synthesis and analysis, and that these primary processes contain the entire science, they naturally conclude that there is no reasoning in the science itself. The analysis of the subject here given dispels this error and exhibits the subject in its true light. Synthesis and Analysis are seen to be the primary mechanical processes; Comparison, the thought process, touches them with her wand of magic, and they germinate and bring forth other processes, having their roots in these primary ones. Comparison also becomes the foundation of processes distinct from those of synthesis and analysis, processes which cannot be conceived as growing out of synthesis and analysis, but 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, independent of the language of arithmetic. These

A LOGICAL OUTLINE OF ARITHMETIC.

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 " carrying 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 processes 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 CTTT,fV,^o,-a /Addition. ( Composition.

±. byntnesis. ^ Multiplication> J 7 Common Multiple.

( \ Involution.

Logical

Outline

of

Arithmetic.

„ f Subtraction II. Analysis. {Division.

III. Comparison.

{Factoring, f Common Divisor. \ Evolution.

1. Ratio.

2. Proportion.

3. Progression.

4. Reduction.

5. Percentage.

6. Propertied of Numbers.

THE PHILOSOPHY OF AKITHMETIC.

W t-H

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Chapter II. Origin And Development Of 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 history 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 uncertainty. History can speak positfvely 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 generally 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 counting 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)

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 numbers 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 pointing 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 principle 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 :