Charles Davies.

The nature and utility of mathematics; with the best methods of instruction explained and illustrated online

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THE LIBRARY

OF

THE UNIVERSITY
OF CALIFORNIA

LOS ANGELES

GIFT OF

Mrs, George I. Cochran



George I. Corhran











t,

*r /



THE



NATURE AND UTILITY



OP



MATHEMATICS,



WITH THE BEST METHODS OF INSTRUCTION EXPLAINED
AND ILLUSTRATED



BY

CHAELES DAVIES, LL.D.,

EMERITUS PROFESSOR OF HIGHER MATHEMATICS IN COLUMBIA COLLJ5OB.



NEW YORK:
PUBLISHED BY A. S. BARNES & CO.,

Ill AND 113 WILLIAM STREET.



DAVIES' MATHEMATICS.



IN THRF.K PARTS.



L-COMMON SCHOOL COUESE,

Davlox' Primary Arithmetic. The fundamental principles displayed in
Object Lts^otis.

Davlcft' Intellectual Arithmetic. -Referring all operations to the
unit 1 a* the only tangible basis for logical development.

Davleft' Element* of Written Arithmetic.- A practical introduction
to the whole subject. Theory subordinated to Practice.

Davle' Practical Arithmetic. The combination of Theory and Practice,
intended to be clear, exact, brief, and comprehensive.

IL-AOADEMIO COUESE.

Davleft 1 University Arithmetic. Treating the subject exhaustively as
a tcience, in a logical series of connected propositions.

Davlei' Elementary Algebra. A connecting link, conducting the pupil
easily from arithmetical processes to abstract analysis.

Davleft' University Algebra. For institutions desiring a more complete
but not the fullest course m pure Algebra.

Davlee' Practical Mathematics. The ecience practically applied to the
useful arts, as Drawing, Architecture, Surveying, Mechanics, etc.

Davle' Elementary Geom etry. The important principles in simple
form, but with all the exactness of rigorous reasoning.

Davie*' Elements of Surveying. Re-written in 1870. A simple and
practical presentation of the subject lor the scholar and surveyor.

m.-COLLEGTATE COUESE.

Davies' Bourdon'* Algebra. -Embracing Sturm's Theorem, and a most
exhaustive and scholarly course.

Davle' University Algebra. A shorter course than Bourdon, for Insti-
tutions having less time to give the subject.

Davies' Legendre's Geometry. The original is the best Geometry of
Europe. The revised edition is well known.

Davieft' Analytical Geometry. Being a full course, embracing the
equation of surfaces of Ihe second degree.

Davleft' Differential and Integral Calculus. Constructed on the
basis of Continuous Quantity and Consecutive Differences.

Davies' Analytical Geometry nnd Calculus. The shorter treatises,
combined in one volume, as more available for American courses of study.

Davle*' DcMcrlptivc Geometry. With application to Spherical Trigo-
nometry, Spherical Projections, and Warped Surfaces.

Davies' Shades, Shadows, and Perspective. A succinct exposition
of the mathematical principles involved.

Davies & Peck's Mathematical Dictionary. Embracing the defini-
tions of all the terms, and ateo a Cyclopedia of Mathematics.

Davleft' Nature and Utility of Mathematics. Embracing a con-
densed Logical Analytic of the entire Science, and of its General Uses.



Entered according to Act of Conprpss. In the ycnr Eighteen Hundred and Seventy-three, by

CHARLES DAVIES,
In the Office of the Librarian of Conuress. at Washington.



-ering &
Mathematical
Sciences
Library



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
nearly half a century, and which has given to that institu-
tion 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 princi-
ples 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 generaliza-
tion, 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 liuve
been sought for wherever science of the highest grade has
been 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 war
with Mexico afforded ample, opportunity for showing the
thousand ways in which science the highest class of knowl-
edge may be made available in practice.

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



1GGC



4 PREFACE.

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 em-
braces all the particulars.

Although my official connection with the Military Aca-
demy was terminated many years since, yet the general
system of Mathematical instruction has not been changed.
Younger and able professors have extended and developed
it, and it now forms an important element in the education
of the country.

The present work is a modification, in many important
particulars, of the Logic and Utility of Mathematics, pub-
lished in the year 1850. The changes in the Text, seemed
to require a change in the Title.

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. Al-
though the general outline of the subject has but little re-
semblance 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, )
January r , 1873. i



CONTENTS



INTRODUCTION.

FAB

OBJECTS AKD PLAN OF THE WORK . 11 27



BOOK I.

LOGIC.

CHAPTER 1.

DEFINITIONS OPERATIONS OF THE MIND TERMS DEFINED. 27 41



SECTION

Definitions 1 6

Operations of the Mind concerned in Reasoning 6 12

Abstraction 12 14

Generalization 14 22

Terms Singular Terms Common Terms 15

Classification 16 20

Nature of Common Terms 20

Science 21

Art



CONTENTS.



CHAPTER II.

PAQK
SOUBCES AND MEANS OF KNOWLEDGE INDUCTION 41 54



SECTION

Knowledge 23

Facts and Truths 2427

Intuitive Truths 27

Logical Truths 28

Logic 29

Induction . 3034



CHAPTER HI.

DEDUCTION NATURE OF THE SYLLOGISM ITS USES AND AP- PA ais
PLICATIONS . 54 97



SECTION

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



CONTEXTS.



BOOK II.

MATHEMATICAL SCIENCE.

CHAPTER I.

QUANTITY AND MATHEMATICAL SCIENCE DEFINED DIFFER-
ENT KINDS OF QUANTITY LANGUAGE OF MATHEMATICS
EXPLAINED SUBJECTS CLASSIFIED UNIT OF MEASURE
DEFINED MATHEMATICS A DEDUCTIVE SCIENCE PAQB 99



SECTION

Quantity Defined 75

Mathematics Defined 76

Kinds of Quantity Number and Space 77 87

Language of Mathematics 87 91

Language of Number Geometry Analysis 91 98

Pure Mathematics 98 104

Mixed Mathematics 104 105

Quantity Measured 105 108

Comparison of Quantities 108 109

Axioms Equality Inequality 109 111



CHAPTER II.

MM

ARITHMETIC SCIENCE AND ART OF NUMBERS 119

SECTION I.

SECTION

First Notions of Numbers Ill 114

Ideas of Numbers Generalized 114 117

Unity and a Unit Defined 117

Simple and Denominate Numbers 118- 120

Alphabet Words Grammar 120

Arithmetical Alphabet 121

Spelling and Reading in Addition 122 127

Spelling and Reading in Subtraction 127 128



CONTENTS.



SECTION

Spelling and Reading in Multiplication 129

Spelling and Reading in Division 130

Units increasing by the Scale of Tens 131 138

Units increasing by Varying Scales 138

Integral Units of Arithmetic 139 141

Different Kinds of Units 141 157

Advantages of the System of Units 157 158

Metric System 158159

System of Unities applied to the Four Ground Rules . 159 163



SECTION II.

Fractional Units changing by the Scale of Tens 163 166

Fractional Units in general 166 169

Advantages of the System of Fractional Units 169 171



SECTION III.
Proportion and Ratio 171 180

SECTION IV.
Applications of the Science of Arithmetic 180 188

SECTION V.

Methods of teaching Arithmetic considered 188

Order of Subjects 188 190

Abstract Units 190 192

Factional Units 192 193

Denominate Units 193 194

Ratio and Proportion 194 196

Arithmetical Language 196 204

Necessity of Exact Definitions and Terms 204 210

How should the Subject be presented 210 213

Text-Books 213 gig

First Arithmetic 218 231

Second Arithmetic 231 235

Third Arithmetic 235 240

Concluding Remarks 240 241



CONTENTS.



CHAPTER III.

GEOMETRY DEFINED THINGS OP WHICH IT TREATS COM-
PARISON AND PROPERTIES OF FIGURES DEMONSTRATION
PROPORTION SUGGESTIONS FOR TEACHING. . PAGE 219

SECTION

Geometry 241

Things of which it treats 24225:3

Comparison of Figures with Units of Measure 253 260

Properties of Figures 260

Marks of what may be proved 261

Demonstrations 262 271

Proportion of Figures 271 27 i

Comparison of Figures 274 277

Recapitulation Suggestions for Teachers 277



CHAPTER IV.

ANALYSIS ALGEBRA ANALYTICAL GEOMETRY DIFFER- PA GB
ENTIAL AND INTEGRAL CALCULUS . . 2~>7



SECTION

Analysis 278284

Algebra 2*4

Analytical Geometry 285 287

Differential and Integral Calculus 287 290

Algebra further considered 290 300

Minus Sign 300302

Subtraction 302

Multiplication 303 306

Zero and Infinity 306 311

Of the Equation 311315

Axioms 315

Equality its Meaning in Geometry 316

Suggestions for those who teach Algebra 319

1*



10 CONTENTS.



CHAPTER V.

PAGE

DIFFERENTIAL CALCULUS

SECTION

Foundations of Mathematical Science 320 332

Limits of Discontinuous Quantity 322 325

Given Quantity Continuous Quantity 324 326

Consecutive Quantities and Tangents 326 329

Lemmas of Newton 329 337

Fruits of Newton's Theory 337342

Different Definitions of Limits 342343

What Quantities are denoted by 343

Inscribed and Circumscribed Polygons 344

Differential Calculus Language 344 346



APPENDIX.

FAOB

A COURSE OF MATHEMATICS WHAT IT SHOULD BE 837



BOOK III.

UTILITY OF MATHEMATICS.

CHAPTER I.

THE UTILITY OF MATHEMATICS CONSIDERED AS A MEANS OF
INTELLECTUAL TRAINING AND CULTURE 849

CHAPTER II.

THE UTILITY OF MATHEMATICS REGARDED AS A MEANS OF
ACQUIRING KNOWLEDGE BACONIAN PHILOSOPHY 364

CHAPTER III.

THE UTILITY OF MATHEMATICS CONSIDERED AS FURNISHING
THOSE RULES OF ART WHICH MAKE KNOWLEDGE PRACTI-
CALLY EFFECTIVE 381

ALPHABETICAL INDEX 397



INTRODUCTION,



OBJECTS AND PLAN OF THE WORK.



UTILITY and Progress are the two leading utility






ideas of the present age. They were manifested

in the formation of our political and social insti- Their influ-

ence in go*
tutions, and have been further developed in the



extension of those institutions, with their subdu-
ing and civilizing influences, over the fairest por-
tions of a great continent. They are now be-
coming the controlling elements in our systems in education.
of public instruction.

What, then, must be the basis of that system wiwt

/. . . ^ ne I* 88 ' 8 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



1 v! i x r K o in" c T I o N .

primarily, in those preparatory studios which fit
and qualify the mind lor such Divine Contem-
plations.

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-
Pha a - losophers. They had looked w r ithin, and not

phyoftbo

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

Bacon laid the foundations of his philosophy m
. organic law r s, and explained the several processes
of experience, observation, experiment, and in-
duction, by which these laws are made known,
why op- He rejected the reasonings of Aristotle because

pMdtoAfto.

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



PLAN OF THE WORK. 13

The time seems now to be at hand when the Practical
philosophy of Bacon is to find its full develop-
ment. 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 it true na-
ture,
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, what i

the true qr

and the basis of that Practical, which the pres- temofedu-
ent 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 expan-
sion to thought, and stability to action ? What Which will

develop and

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 com-
pact ? What subject of study is, from its na- what are

the subjects

ture, most likely to ensure this training, and O r 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-emi-
nence.



14 INTRODUCTION.



The first impressions which the child receives
f Number and Quantity are the foundations oi



dge * his mathematical knowledge. They form, as it

were, a part of his intellectual being. The laws

Laws of of Nature are merely truths or generalized facts,

Nature.

in regard to matter, derived by induction from

experience, observation, arid experiment. The

laws of mathematical science are generalized

Number truths derived from the consideration of Number

space. anc ^ 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.



The knowledge which mathematical science

teal knowl- .

edge: 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 ar.
acquaintance with physical nature in all its mi-
what it nuteness and in all its grandeur : which survey
doM - the laws of elementary organization, by the mi-
croscope, and weigh the spheres in the balance
of universal gravitation.

what The processes of mathematical science serve

tQ g j ve menta l unhv an(J w l lo ] enesg< Thev im _

part that knowledge which applies the means of



PLAN OF THE WORK. 15

crystallization to a chaos of scattered particulars, Right knowi.
and discovers at once the general law, if there the'L^^f
be one, which forms a connecting link between c| 7 8lalllza '

lion.

them. Such results can only be attained by
minds highly disciplined by scientific combina-
tions. 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



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 IIow l)1 "

educator re-

as the great means of accomplishing his work, ganismati.-

The definitions present clear and separate ideas,

which the mind readily apprehends. The axioms The axiom*

are the simplest exercises of the reasoning fac-

ulty, and afford the most satisfactory results in

the early use and employment of that faculty.

The trains of reasoning which follow are com-

binations, according to logical rules, of what

has been previously fully comprehended, and influence ot

, the study of

the mind and the argument grow together, so mathematic*
that the thread of science and the warp of the or
intellect entwine themselves, and become insep-
arable. Such a training will lay the foundations



1({ INTRODUCTION.



of systematic knowledge, so greatly preferable
to conjectural judgments.

HOW the The philosopher regards mathematical science

regardV* as tne mere tools of his higher vocation. Look-

mmht-mmics: ^ ^^ ft stea( jy an( j an xious eye to Nature,

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

to him.

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



The mere practical man regards with favor

of the practi-
cal man. only the results of science, deeming the reason-
ings through which these results are arrived at,
quite superfluous. Such should remember that

iMtnimenta the mind requires instruments as well as the

of the mind

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
Ewy thing thing, whether existing in the abstract or in the

kMalaw.

material world; whether an element of knowl-



PL AN OF THE WORK.



IT



edge or a rule of art, has its connections and its TO know

, the Inw is to

law: to understand these connections and that knowt h e
law, is to know the thing. When the principle
is clearly apprehended, the practice is easy.



analyzed.



HOW.



With these general views, and under a firm

, . ,

conviction that mathematical science must be-
come the great basis of education, I have be-
stowed much time and labor on its analysis, as
a subject of knowledge. I have endeavored to
present its elements separately, and in their con-
nections ; to point out and note the mental fac-
ulties 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 acccmplish what was

deemed n

these ends, in the way that seemed to me most
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 gov-
ern 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

2



18 INTRODUCTION.



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-
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 out-
line of the nature of that reasoning by means
of which all inferred knowledge is acquired.

BooklL Book II. treats of Mathematical Science.

Here I have endeavored to explain the nature of

or what it the subjects with which mathematical science is

treats. i-i i i

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

treating.

language used, and an examination of the reason-
ings employed in the methods of proof.

Book UL Book III. explains and illustrates the Utility of

rtility of

Mathematics Mathematics : First, as a means of mental disci-
pline and training ; Secondly, as a means of ac-
quiring knowledge; and, Thirdly, as furnishing
those rules of art, which make knowledge prac-
tically effective.



PLAN OF THE WORK. 19



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 ad-
vantages and benefits which each class may re-
ceive from its perusal and study.

There are four classes of readers, who may, Fourciasse*
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 ; -mini.

4th. Professional Teachers. Fourth.

First. The general reader, who reads for im. Advantage*
provement, and desires to acquire knowledge, eral reader,
must carefully search out the import of language.
He must early establish and carefully cultivate
the habit of noting the connection between ideas comm-
and their signs, and also the relation of ideas to VOT ^^A
each other. Such analysis leads to attentive
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,



20 INTRODUCTION.

in the study in the construction of the mathematical science.

math ^ (iUc3 where the relation between the sign (or language)
and the thing signified (or idea expressed), is un
mistakable, gives precision and accuracy, leads
to right arrangement and classification, and thus
prepares the mind for the reception of general
knowledge.

Advantage* Secondly. The increase of knowledge carriea
^Tm^ ^ with it ^e necessity of classification. A limited
number of isolated facts may be remembered, or
a few simple principles applied, without tracing
out their connections, or determining the places
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 stimu-
lating elements of the present, there is occasion
for the constant exercise of our highest facul-
Knowiedge ties. Knowledge reduced to order ; that is,
order J knowledge so classified and arranged as to be
sdeuce - easily remembered, readily referred to, and ad-
vantageously applied, will alone suffice to sift
the pure metal from the dust of ages, and fashion
it for present use. Such knowledge is Science.



PLANOFTHEVVORK. 21



Masses of facts, like masses of matter, are ca- Knowledge
pable of very minute subdivisions ; and when \ve ^ed io\u
know the law of combination, they are readily



Online LibraryCharles DaviesThe nature and utility of mathematics; with the best methods of instruction explained and illustrated → online text (page 1 of 30)