The logic and utility of mathematics

Front matter

The National Series of Standard School-Hooks.

ORTHOGRAPHY.

SMITH'S SERIES

Preface

The following work is not a series of speculations. It is but an analysis of that system of mathematical instruction which has been steadily pursued at the Military Academy over a quarter of a century, and which has given to that institution its celebrity as a school of mathematical science.

It is of the essence of that system that a principle be taught before it is applied to practice ; that general principles and general laws be taught, for their contemplation is far more improving to the mind than the examination of isolated propositions ; and that when such principles and such laws are fully comprehended, their applications be then taught as consequences or practical results.

This view of education led, at an early day, to the union of the French and English systems of mathematics. By this union the exact and beautiful methods of generalization, which distinguish the French school, were blended with the practical methods of the English system.

The fruits of this new system of instruction have been abundant. The graduates of the Military Academy have been sought for wherever science of the highest grade has been

PREFACE.

needed. Russia has sought them to construct her railroads ;* the Coast Survey needed their aid ; the works of internal improvement of the first class in our country, have mostly been conducted under their direction ; and the recent war with Mexico afforded ample opportunity for showing the thousand ways in which science — the highest class of knowledge — may be made available in practice.

All these results are due to the system of instruction. In that system Mathematics is the basis — Science precedes Art — Theory goes before Practice — the general formula embraces all the particulars.

It was deemed necessary to the full development of the plan of the work, to give a general view of the subject of Logic. The materials of Book I. have been drawn, mainly, from the works of Archbishop Whately and Mr. Mill. Although the general outline of the subject has but little resemblance to the work of either author, yet very much has been taken from both ; and in all cases where it could be done consistently with my own plan, I have adopted their exact language. This remark is particularly applicable to Chapter III., Book I., which is taken, with few alterations, from Whately.

For a full account of the objects and plan of the work, the reader is referred to the Introduction.

Fishkill Landing, June, 1850.

* Major Whistler, the engineer, to whom was intrusted the great enterprise of constructing a railroad from St. Petersburg to Moscow, and Maj or Brown, who succeeded him at his death, were both graduates of the Military Academy.

Introduction

PA.OI

Objects and Plan of the Work ........*••>. 11

Book I . Logic

CHAPTER L Definitions — Operations of the Mind — Terms defined . . 27

Definitions 1 — 6

Operations of the Mind concerned in Reasoning 6 — 12

Abstraction 1 2 — 14

Generalization 14 — 22

Terms — Singular Terms — Common Terms 15

Classification 16 — 20

Nature of Common Terms 20

Art 22

6 CONTENTS.

Chapter II

rxam Sources and Means of Knowledge — Induction 41

Knowledge 23

Facts and Truths 24—27

Intuitive Truths 27

Logical Truths 28

Induction 30—34

Chapter III

Deduction — Nature of the Syllogism — Its Uses and Applications Page 54

«CTION

Deduction 34

Propositions 35 — 40

Syllogism 40—42

Analytical Outline of Deduction 42 — 67

Aristotle's Dictum 54 — 61

Distribution and Non-distribution of Terms 61 — 67

Rules for examining Syllogisms 67

Of Fallacies 68—71

Concluding Remarks 71 — 75

CONTENTS.

Chapter I

Quantity and Mathematical Science defined — Different kinds of Quantity — Language of Mathematics explained — Subjects Classified — L t nit of Measure defined — Mathematics a Deductive Science . . . .Page 99

Quantity. 75—79

Space 81—87

Language of Mathematics 91 — 94

Quantity Measured , 94 — 97

Pure Mathematics 97-101

Comparison of Quantities 101

Axioms or Formulas for inferring Equality 102

Axioms or Formulas for inferring Inequality 102

CHAPTER IT.

PAOX

Arithmetic — Science and Art of Numbers 117

SECTION I. First Notions of Numbers 104 — 107

Ideas of Numbers Generalized 107 — 110

Unity and a Unit Defined 110

Simple and Denominate Numbers 11 1 — 1 1 3

Alphabet — Words — Grammar 113

Arithmetical Alphabet 114

Spelling and Reading in Addition 115 — 120

Spelling and Reading in Subtraction 120 — 122

CONTENTS.

Spelling and Reading in Multiplication 122

Spelling and Reading in Division 123

Units increasing by the Scale of Tens 124 — 131

Units increasing by Varying Scales 131

Integer Units of Arithmetic 132

Abstract or Simple Units 132—134

Units of Currency 1 34—1 36

Units of Weight 136—139

Units of Measure 139—150

Advantages of the System of Unities 150

System of Unities applied to the Four Ground Rules. 151 — 155

SECTION V. Methods of teaching Arithmetic considered 180

Order of the Subjects 180—183

1st. Integer Units 183—185

2d. Fractional Units 185

3d. Comparison of Numbers, or Rule of Three 186 — 188

4th. Practical Part, or Applications of Arithmetic .... 188

Objections to Classification answered 189 — 191

Objections to the new Method 191

Arithmetical Language 192 — 200

Necessity of exact Definitions and Terms 200 — 206

How should the Subjects be presented 206 — 209

Text-Books 209—214

First Arithmetic 214—227

CONTENTS. 9

Second Arithmetic 227-— 231

Third Arithmetic 231—236

Concluding Remarks 236

Chapter III

Geometry defined — Things of which it treats — Comparison and Properties of Figures — Demonstration — Proportion — Suggestions for Teaching .... Page 223

Things of which it treats 238—248

Comparison of Figures with Units of Measure 249 — 256

Properties of Figures 256

Marks of what may be proved 257

Demonstration 258 — 207

Proportion of Figures 267 — 270

Comparison of Figures 270 — 273

Recapitulation — Suggestions for Teachers 273

Chapter IV

Analysis — Algebra — Analytical Geometry — Differential and Integral Calculus Page 261

Analytical Geometry 281 — 283

Differential and Integral Calculus 283 — 286

Algebra further considered 286 — 296

Minus Sign 296 — 298

Subtraction 298

Zero and Infinity « 302—307

Of the Equation 307—311

Axioms 311

Equality — its meaning in Geometry 312

Suggestions for those who teach Algebra ,. 315

10 CONTENTS.

Appendix

A Course of Mathematics — What it should be 341

Alphabetical Index 353

INTRODUCTION

OBJECTS AND PLAN OF THE WORK.

Utility and Progress are the two leading utility ideas of the present age. They were manifested v ro ~ re9S . in the formation of our political and social insti- Their infla-

ence in ^oV

tutions, and have been further developed in the ermneul: extension of those institutions, with their subduing and civilizing influences, over the fairest portions of a great continent. They are now becoming the controlling elements in our systems i» education. of public instruction.

What, then, must be the basis of that system w^t

the basis of

of education which shall embrace within its ho- utility and rizon a Utility as comprehensive and a Progress as permanent as the ordinations of Providence, exhibited in the laws of nature, as made known by science ? It must obviously be laid in the examination and analysis of those laws ; and

Progress.

Preparatory primarily, in those preparatory studies which fit and qualify the mind for such Divine Contemplations.

Bacons When Bacon had analyzed the philosophy of

Philosophy.

the ancients, he found it speculative. The great highways of life had been deserted. Nature, spread out to the intelligence of man, in all the minuteness and generality of its laws — in all the harmony and beauty which those laws develop — had scarcely been consulted by the ancient phi- Phiioso- losophers. They had looked within, and not

phy of the

Ancients, without. They sought to rear systems on the uncertain foundations of human hypothesis and speculation, instead of resting them on the immutable laws of Providence, as manifested in the materia] world. Bacon broke the bars oi this mental prison-house: bade the mind go free. and investigate nature.

Foundations Bacon laid the foundations of his philosophy m of Bacons organ j c i aws an( j explained the several processes

Philosophy : © r l

of experience, observation, experiment, and induction, by which these laws are made known, why op- He rejected the reasonings of Aristotle because

pcsedto Aris-

totie's. they were not progressive and useful ; because they added little to knowledge, and contributed nothing to ameliorate the sufferings and elevate the condition of humanity.

The time seems now to be at hand when the Practical philosophy of Bacon is to find its full development. The only fear is, that in passing from a speculative to a practical philosophy, we may, for a time, lose sight of the fact, that Practice

without Science is Empiricism; and that all its true nature, which is truly great in the practical must be the

application and result of an antecedent ideal.

What, then, are the sources of that Utility, Whal i&

the true sys»

and the basis of that Practical, which the pres- temofeduent generation desire, and aftei which they are so anxiously seeking ? What system of training and discipline will best develop and steady the intellect of the young ; give vigor and expansion to thought, and stability to action? What which will

course of study will most enlarge the sphere of steady the investigation ; give the greatest freedom to the mind without licentiousness, and the greatest freedom to action consistent with the laws of nature, and the obligations of the social compact ? What subject of study is, from its na- what are

,., , , . . . the subjects

ture, most likely to ensure this training, and of study? contribute to such results, and at the same time lay the foundations of all that is truly great in the Practical ? It has seemed to me that math- Mathematics ematical science may lay claim to this pre-eminence.

INTRODUCTION.

Nature.

Founda- The first impressions which the child receives

ematicai °f Number and Quantity are the foundations of

knowledge. his mat h em atical knowledge. They form, as it

were, a part of his intellectual being. The laws

Laws of f Nature are merely truths or generalized facts,

in regard to matter, derived by induction from

experience, observation, and experiment. The

laws of mathematical science are generalized

truths derived from the consideration of Number

and Space. All the processes of inquiry and

investigation are conducted according to fixed

laws, and form a science ; and every new thought

and higher impression form additional links in

the lengthening chain.

and Space.

Mathematical knowledge :

What it does.

The knowledge which mathematical science imparts to the mind is deep — profound — abiding. It gives rise to trains of thought, which are born in the pure ideal, and fed and nurtured by an acquaintance with physical nature in all its minuteness and in all its grandeur : which survey the laws of elementary organization, by the microscope, and weigh the spheres in the balance of universal gravitation.

what ^h e P rocesses of mathematical science serve

the processes to gj ve men t a ] unity and wholeness. They impart that knowledge which applies the means of effect.

crystallization to a chaos of scattered particulars, Right knowiand discovers at once the general law, if there themeans^of be one, which forms a connecting link between "Jfjj" 8 ** them. Such results can only be attained by minds highly disciplined by scientific combinations. In all these processes no fact of science is forgotten or lost. They are all engraved on the great tablet of universal truth, there to be read by succeeding generations so long as the it records

and preserve!

laws of mind remain unchanged. This is stri- truth, kingly illustrated by the fact, that any diligent student of a college may now read the works of Newton, or the Mecanique Celeste of La Place.

The educator regards mathematical science How lh(J

educator re-

as the great means of accomplishing his work, gardsmath- The definitions present clear and separate ideas, which the mind readily apprehends. The axioms The nmmi are the simplest exercises of the reasoning faculty, and afford the most satisfactory results in the early use and employment of that faculty. The trains of reasoning which follow are combinations, according to logical rules, of what has been previously fully comprehended ; and influence of

, . , , , .the study of

the mind and the argument grow together, so mathematics that the thread of science and the warp of the on the miQdi intellect entwine themselves, and become inseparable. Such a training will lay the foundations

1G INTRODUCTION.

of systematic knowledge, so greatly preferable to conjectural judgments.

How the The philosopher regards mathematical science regards as the mere tools of his higher vocation. Look-

mathematics: ^ ^^ & gtea( J y an( J anx i ous e ye tO NgtW*,

and the great laws which regulate and govern all things, he becomes earnestly intent on their examination, and absorbed in the wonderful harmonies which he discovers. Urged forward by its necessity these high impulses, he sometimes neglects that

to him. . ...

thorough preparation, in mathematical science, necessary to success ; and is not unfrequently obliged, like Antaeus, to touch again his mother earth, in order to renew his strength.

The views The mere practical man regards with favor

of the practi- , i • i

cai man. only the results of science, deeming the reasonings through which these results are arrived at, quite superfluous. Such should remember that

instruments the mind requires instruments as well as the hands, and that it should be equally trained in their combinations and uses. Such is, indeed, now the complication of human affairs, that to do one thing well, it is necessary to know the properties and relations of many things. Every

Everything thing, whether existing in the abstract or in the material world ; whether an element of knowledge or a rule of art, has its connections and its to know

. iii • i i tne law ia to

law: to understand these connections and that knowtne

law, is to know the thing. When the principle thm "'

is clearly apprehended, the practice is easj r .

With these general views, and under a firm Mathematics conviction that mathematical science must become the great basis of education, I have bestowed much time and labor on its analvsis, as a subject of knowledge. I have endeavored to present its elements separately, and in their con- flow, nections ; to point out and note the mental faculties which it calls into exercise ; to show why and how it develops those faculties ; and in what respect it gives to the whole mental machinery greater power and certainty of action than can be attained by other studies. To accomplish what was

deemed no-

these ends, in the way that seemed to me most cessary. desirable, I have divided the work into three parts, arranged under the heads of Book I., II.. and III

Book I. treats of Logic, both as a science and Logic an art ; that is, it explains the laws which govern the reasoning faculty, in the complicated processes of argumentation, and lays down the Explanation, rules, deduced from those laws, for conducting such processes. It being one of the leading

For what objects to show that mathematical science is the

best subject for the development and application

of the principles of logic ; and, indeed, that the

science itself is but the application of those prin-

The necessity ciples to the abstract quantities Number and

of treating it.

Space, it appeared indispensable to give, in a manner best adapted to my purpose, an outline of the nature of that reasoning by means of which all inferred knowledge is acquired.

Book ii. Book II. treats of Mathematical Science.

Here I have endeavored to explain the nature of Ofwhatit the subjects with which mathematical science is conversant; the ideas which arise in examining and contemplating those subjects; the language employed to express those ideas, and the laws of their connection. This, of course, led to a class- Manner of ification of the subjects; to an analysis of the language used, and an examination of the reasonings employed in the methods of proof.

treating.

Book in. Book III. explains and illustrates the Utility of Mathematics. Mathematics : First, as a means of mental discipline and training ; Secondly, as a means of acquiring knowledge ; and, Thirdly, as furnishing those rules of art, which make knowledge practically effective.

Having thus given the general outlines of the classes of work, we will refer to the classes of readers for whose use it is designed, and the particular advantages and benefits which each class may receive from its perusal and study.

There are four classes of readers, who may, Four classes it is supposed, be profited, more or less, by the perusal of this work :

1st. The general reader ; First class.

2d. Professional men and students ; second.

3d. Students of mathematics and philosophy ; Third.

4th. Professional Teachers. Fourth.

First. The general reader, who reads for im. Advantages provement, and desires to acquire knowledge, era i re rder. must carefully search out the import of language. He must early establish and carefully cultivate the habit of noting the connection between ideas connecand their signs, and also the relation of ideas to ^ordtlnT 3 each other. Such analysis leads to attentive ldeas * reading, to clear apprehension, deep reflection, and soon to generalization.

Logic considers the forms in which truth must Logic be expressed, and lays down rules for reducing all trains of thought to such known forms. This habit of analyzing arms us with tests by which its value* we separate argument from sophistry — truth from falsehood. The application of these principles, in the study in the construction of the mathematical science, ^atblaacs. where the relation between the sign (or language) and the thing signified (or idea expressed), is unmistakable, gives precision and accuracy, leads to right arrangement and classification, and thus prepares the mind for the reception of general knowledge.

Advantages Secondly. The increase of knowledge carries

to profession- . . . , . r ^ • r • a i • • 1

aimen. W1 *" ]t the necessity ol classification. A limited number of isolated facts may be remembered, or a few simple principles applied, without tracing out their connections, or determining the placet which they occupy in the science of general knowledge. But when these facts and principles are greatly multiplied, as they are in the learned The reason, professions ; when the labors of preceding gen* erations are to be examined, analyzed, compared; when new systems are to be formed, combining all that is valuable in the past with the stimulating elements of the present, there is occasion for the constant exercise of our highest facul- Knowiedso ^ es - Knowledge reduced to order ; that is, reduced to i alow } eo v e so classified and arranged as to be

order is ° °

science, easily remembered, readily referred to, and advantageously applied, will alone suffice to sift the pure metal from the dust of ages, and fashion it for present use. Such knowledge is Science.

Masses of facts, like masses of matter, are ca- Knowledge pable of very minute subdivisions ; and when we duced t0 ita know the law of combination, they are readily elemeats - divided or reunited. To know the law, in any case, is to ascend to the source ; and without that knowledge the mind gropes in darkness.

It has been my aim to present such a view objects of

tVif* work

of Logic and Mathematical Science as would clearly indicate, to the professional student, and even to the general reader, the outlines of these subjects. Logic exhibits the general formula Logic and applicable to all kinds of argumentation, and mathematics is an application of logic to the abstract quantities Number and Space.

When the professional student shall have examined the subject, even to the extent to which certainty of it is here treated, he will be impressed with the clearness, simplicity, certainty, and generality of its principles ; and will find no difficulty in making them available in classifying the facts, and examining the organic laws which characterize his particular department of knowledge.

Thirdly. Mathematical knowledge differs from Matnematievery other kind of knowledge in this : it is, as c ede ^ ' it were, a web of connected principles spun out from a few abstract ideas, until it has become one of the great means of intellectual develop- its extent.

ment and of practical utility. And if I am per-

Necessity mitted to extend the figure, I may add, that the

attheright° web of the spider, though perfectly simple, if we

pa( see the end and understand the way in which

it is put together, is yet too complicated to be

unravelled, unless we begin at the right point,

and observe the law of its formation. So with

mathematical science. It is evolved from a few

— a very few — elementary and intuitive princi-

How pies : the law of its evolution is simple but ex-

mathemati- . , .

cai science is acting, and to begin at the right place and proconstructed. cee j j n t j ie r jgh t wa y ? j s a ]j ttia.t is necessary to

make the subject easy, interesting, and useful, what has I have endeavored to point out the place of

been attempted, beginning, and to indicate the way to the mathematical student. I am aware that he is starting on a road where the guide-boards resemble each other, and where, for the want of careful observation, they are often mistaken ; I have sought, therefore, to furnish him with the maps and guide-books of an old traveller.

Advantages By explaining with minuteness the subjects the whole about which mathematical science is conversant, subject. t ^ e w h ] e fj e j(j t b e g 0ne over i s a t once surveyed: by calling attention to the faculties of Advantages the mind which the science brings into exercise,

of considering tlie men- We are better prepared to note the intellectual

tal faculties : . 1 • 1 . i • 11

operations which the processes require ; and by a knowledge of the laws of reasoning, and an ofaknowiacquaintance with the tests of truth, we are en- ia WSOfri . a . abled to verify all our results. These means have been furnished in the following work, and to aid the student in classification and arrangement, diagrams have been prepared exhibiting separ- what has

been done.

ately and in connection all the principal parts of mathematical science. The student, therefore, who adopts the system here indicated, will find his wav clearlv marked out, and will recognise, Advanta ^ e *

J ° totbestu-

from their general resemblance to the descrip- dent, tions, all the guide-posts which he meets. He will be at no loss to discover the connection between the parts of his subject. Beginning with first principles and elementary combinations, and guided by simple laws, he will go for- *vnere

be begins.

ward from the exercises of Mental Arithmetic to the higher analysis of Mathematical Science on an ascent so gentle, and with a progress so Drde r

of progrtjs*

steady, as scarcely to note the changes. And indeed, why should he ? For all mathematical processes are alike in their nature, governed by the same laws, exercising the same faculties,, unity of and lifting the mind towards the same eminence.

tbe subject.

o

Fourthly. The leading idea, in the construe- Advantages tion of the work, has been, to afford substantial prof essionai aid to the professional teacher. The nature ot teacher -

His duties: his duties — their inherent difficulties — the per- Diecourage- plexities which meet him at every step — the want difficulties: of sympathy and support in his hours of discouragement — (and they are many) — are circumstances which aw T aken a lively interest in the hearts of all who have shared the toils, and been themselves laborers in the same vineyard. He takes his place in the schoolhouse by the roadside, and there, removed from the highways of Remoteness life, spends his days in raising the feeble mind Htei ' of childhood to strength — in planting aright the seeds of knowledge — in curbing the turbulence of passion — in eradicating evil and inspiring good. The fruits of his labors are seen but once in a generation. The boy must grow to Fruits of manhood and the girl become a matron before

his efforts, . .

whenseeu. he is certain that his labors have not been in vain.

Yet, to the teacher is committed the high trust of forming the intellectual, and, to a certain extent, the moral development of a people. He

Theimpor- holds in his hands the keys of knowledge. If

lance of his a! n -, . . , . .

labors. the " rst mora l impressions do not spring into life at his bidding, he is at the source of the stream, and gives direction to the current. Although himself imprisoned in the schoolhouse, his influence and his teachings affect all conditions of society, and reach over the whole hori-

PLAN OF THE WORK.

zon of civilization. He impresses himself on The influence the young of the age in which he lives, and lives again in the age which succeeds him.

All good teaching must flow from copious sources of

good teach-

knowledge. The shallow fountain cannot emit m g. a vigorous stream. In the hope of doing something that may be useful to the professional teacher, I have attempted a careful and full

Objects for which the

analysis of mathematical science. I have spread work was

. undertaken.

out, in detail, those methods which have been carefully examined and subjected to the test of long experience. If they are the right meth- principles ods, they will serve as standards of teaching; . ,, te '

3 J o ' mg, thes

for, the principles of imparting instruction are the same for all branches of knowledge.

The system which I have indicated is complete in itself. It lays open to the teacher the entire skeleton of the science — exhibits all its parts separately and in their connection. It explains a course of reasoning simple in itself, and applicable not only to every process in mathematical science, but to all processes of argumentation in every subject of knowledge.

The teacher who thus combines science with art, no longer regards Arithmetic as a mere treadmill of mechanical labor, but as a means —

Sygtea

What it present?.

What it explains.

Science combined with art :

The advan- and the simplest means — of teaching the art and intromit, science of reasoning on quantity — and this is

the logic of mathematics. If he would accom-

Resuitsof plish well his work, he must so instruct his

tion. pupils that they shall apprehend clearly, think

quickly and correctly, reason justly, and open

their minds freely to the reception of all know!

edge.

book r.

LOGIC.

tome tions

C II A PTER I.

DEFINITION'S — OPERATIONS OF THE MIND TERMS DEFINED.

DEFINITIONS.

§ 1. Definition is a metaphorical word, which Definition

a

literally signifies " laying down a boundary. " metaphorical All definitions are of names, and of names only ;

but in some definitions, it is clearly apparent, defimti

that nothing is intended except to explain the ^J^

meaning of the word; while in others, besides word3:

explaining the meaning of the word, it is also ° thin™™

implied that there exists, or mav exist, a thins; corres P° nd -

r J ' ° ingtothe

corresponding to the word. *«*l

§ 2. Definitions which do not imply the exist- of definitions

r . . , . . , , which do

ence of things corresponding to the words de- not implv fined, are those usually found in the Dictionary thiDgs a"*"

J J sponding

of one's own language. They explain only the *>wonk.

Th meaning of the word or term, by giving some explain equivalent expression which may happen to be equivalents, better known. Definitions which imply the existence of things corresponding to the words defined, do more than this.

Definition F or example: "A triangle is a rectilineal fig-

of a

triangle; ure having three sides." This definition does

what .1 •

two things : it o

implies. i s t j t explains the meaning of the word tri-

angle ; and,

2d. It implies that there exists, or may exist, a rectilineal figure having three sides.

or a § 3. To define a word when the definition is

definition

which im- to imply the existenee of a thing, is to select P iltence o( h° m *H the properties of the thing those which athing. are most s j M1 p] e general, and obvious; and the Properties properties must l;e very well known to us before known. we can decide which are the fittest for this purpose. Hence, a thing may have many properties besides those which are named in the definition a definition of the word which stands for it. This second truth# kind of definition is not only the best form of expressing certain conceptions, but also contributes to the development and support of new truths.

in § 4. In Mathematics, and indeed, in all strict

Mathematics m ...

names imply sciences, names imply the existence ot the things

CHAP. I.] DLflNITIONS. 29 which they name; and the definitions of those things names express attributes of the things ; so that express no correct definition whatever, of any mathematical term, can be devised, which shall not express certain attributes of the thing corresponding to the name. Every definition of this class Definitions is a tacit assumption of some proposition which ofthlsclas3 is expressed by means of the definition, and propositions. which gives to such definition its importance.