Charles Davies.

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THE

Nature and Utility

OF

MATHEMATICS,

WITH THE BEST METHODS OF INSTRUCTION EXPLAINED
AND ILLUSTRATED

BY

CIIAKLES DA VIES, LL.D.,

EMERITUS PKOFESSOR OP HIGHER MATHEMATICS IN COLUMBIA COLLEOB.

loavoH cQii^^^ ]:-^^'

CHESTNUT HUX, MA.

NEW YORK:

Ill AND 113 William Street.
1873.

BAVIES' MATHEMATICS.

IN THRKE PARTS.

I-GOMMOIT SCHOOL COURSE.

©avies' Primary Aritliiiietic— The fundamental principles clisplaycd in
Object Lessons.

Eavios' Iiitellectsial Aritliiiietic— Eereniii,<j all operations to the
luut 1 as the only tangible basis for logical development.

Davie*' Elements ol" \yritten Aritlimetic.— A practical inlroductioii
lo the wliolc subject. Tlieojy subordinated to Practice.

I5avic«i' IPraotieal Aritlimetic— The combination ofTheory and Practice,
inieuded to be clear, exact, brief, and comprehensive.

Davies' University Aritlimetic— Treating the subject exhaustively as
a 6ci«?ice, in a logical series of connected propositions.

Daviesi' "Elementary Algebra,— A connecting link, conducting the pupil
t-asily from arithmetical processes to abstract analysis.

BJavles' University Algebra. — For institutions desiring a more complete
but not the fullest course in pure Algebra.

Bavies' Practical IHatliematics.— The science practically applied to the
useful arts, as Drawing, Architecture, Surveying, Mechanics, etc.

©avies' Elementary Geometry.— The important principles in simple
lor.n, but with all the exactness of rigorous reasoning.

©avics' Elements of Surveying.— Re-\vritten in 18T0. A simple and
prj,ctical presentatiou of the suljject for the scholar and surveyor.

IIL-GOLLEGIATE COUESE.

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

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

Bavies' ILegendre's fieonietry.— The original is the best Geometry of
Europe. The revised edition is well known.

Davics' Analytical CJeometry. —Being a full course, embracing the
equation of surfaces of the second degree.

Davies' Bifflerential and Integral Calculns.— Constructed on the
basis ot Coutinuous Quantiiy and Consecutive Diflercnces.

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

BJavies' Bescriptive Geometry.— With application to Spherical Trigo-
nometry, Spherical Pi ejections, and Warped Surfaces.

of the mathematical principles involved.

Davies & Peck's JWatliematical Dictionary.— Embracing tlie defini-
tions of all the terms, and also a Cyclopedia of Mathematics.

Davies' Nature and Utility of IWatliematics.— Embracing a con-
densed Logical Analysis of the entire Science, and of its General Uses.

Entered according to Act of Coiiprc.ss. in the yc.ir Eigliteen Hundred and Seventy-three, by

CHARLES DAVIES,

In the Office of the Librarian of Congress, at Washington.

i?4359

PREFACE.

The following work is not a series of speculations. It is
but an analysis of that system of mathematical instruction
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 jorinciples 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
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 oppor. unity for showing the
thousand ways in which science — the highest class of knoAvl-
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

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 cfeemed 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 biit little re-
semblance to the work of either author, yet very much has
been taken from both ; and in all cases Avhere 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, 1873. J

CONTENTS

INTRODUCTION.

FAGS

Objects and Plan op the Wokk. ....................... 11 — 37

B K L

LOGIC.

CHAPTER 1.

Definitions — Operations of the Mind — Terms defined. 27 — 41

SECTION

Definitions 1 — 6

Operations of the Mind concerned in Reasoning 6 — 13

Abstraction 13 —14

Generalization 14 — 33

Terms — Singular Terms — Common Terms 15

Classification. 16—30

Nature of Common Terms 30

Science 21

Art.... 23

CONTENTS.

CHAPTER II.

PAGE

SotmcES AND Means op Knowledge — Indtjction 41 — 54

SECTION

Knowledge 23

Facts and Truths 24—27

Intuitive Truths 27

Logical Truths 28

Logic 29

Induction 30 — 34

CHAPTER in.

Deduction — Natuke of the Syllogism: — Its Uses and Ap- paob
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

CONTEIS'TS.

BOOK II.

MATHEMATICAL SCIENCE.

CHAPTER I.

Quantity and Mathematical Science defined — Differ-
ent KINDS OP Quantity — Language of Mathematics
EXPLAINED — Subjects Classified — Unit of Measure
defined — Mathematics a Deductive Science page

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 Matliematics 104 — 105

Quantity Measured 105 — 108

Comparison of Quantities 108 — 109

Axioms — Equality — Inequality 109 — 111

CHAPTER II.

PAGB

Arithmetic — Science and Art op Numbers 119

SECTION L

SECTIOH

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

Aritlimetical Alphabet 121

Spelling and Reading in Subtraction 127 — 129

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 158—159

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—193

Fractional Units 192—193

Denominate Units 193—194

Ratio and Proportion 194 — 196

^A^rithmetical Language 196 — 204

Necessity of Exact Definitions and Terms 204 — 210

How should the Subject be presented 210 — 213

Text-Books 213—218

First Arithmetic 218—231

Second Arithmetic 231 — 235

Third Arithmetic 235—240

Concluding Remarks. 240—241

COIvrTENTS.

CHAPTER III.

Geometry defined — Things of which it treats — Com:-
PAnisoN AND Properties op Figures — Demonstration
Proportion — Suggestions for Teaching page 219

SECTION

Geometry 241

Tilings of which it treats 242—253

Comparison of Figures witli Units of Measure 253 — 260

Properties of Figures . 260

Marks of what may be proved 261

Demonstrations 263 — 271

Proportion of Figures 271 — 274

Comparison of Figures 274 — 277

Recapitulation — Suggestions for Teachers 277

CHAPTER IV.

Analysis — Algebra — Analytical Geometry — Differ- page
ential and Integral Calculus 257

SECTION

Analysis 278—284

Algebra 284

Analytical Geometry 285—287

Differential and Integral Calculus 287—290

Algebra further considered 290 — 300

Minus Sign 300 — ^302

Subtraction 302

Multiplication 303—306

Zero and Infinity 306 — 311

Of the Equation 311—315

Axioms 315

Equality — its Meaning in Geometry 316

Suggestions for those who teach Algebra - 319

1*

10 COlsrTEXTS.

CHAPTER V.

PAGE

DiPFERENTIAI, CaLCTOUS 289

SECTION

Foundations of Mathematical Science 320 — 332

Limits — of Discontinuous Quantity 332 — 325

Given Quantity — Continuous Quantity 324 — 326

Consecutive Quantities and Tangents ... 326 — 329

Lemmas of Newton 329 — 337

Fruits of Newton's Theory 337—342

Different Definitions of Limits 342 — 343

What Quantities are denoted by 343

Inscribed and Circumscribed Polygons ... 344

Differential Calculus — Language 344 — 346

APPENDIX.

PAGE

A CoTJESE OF Mathematics — What it shotild be 837

BOOK III.

UTILITY OF MATHEMATICS.

CHAPTER I.

The Utility of Mathematics considered as a Means op
Intellectual Training and Culture 349

CHAPTER II.

The Utility of Mathematics regarded as a Means of
Acquiring Knowledge — Baconian Philosophy 864

CHAPTER III.

The Utility of Mathematics considered as furnishing
those Rules of Art which make Knowledge Practi-
cally Effective 381

Alphabetical Index 89"?

INTRODUCTION

OBJECTS AND PLAN OF THE WORK.

Utility and Progress are the two leading utility
ideas of the present age. They were manifested Progress:
in the formation of our political and social insti- Their infii!-
tutions, and have been further developed ni the emmem;
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 educatioa
of pubhc instruction,

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

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

12 INTRODUCTION.

Preparatory primarily, in those preparatory studies which fit
and quaUfy the mind for such Divine Contem-
plations.

Bacon's When Bacon had analyzed the philosophy oi

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 im-
mutable laws of Providence, as manifested in
the material world. Bacon broke the bars ol
this mental prison-house: bade the mind go free,
and investigate nature.

Foundations Bacou laid the foundations of his philosophy m
of Bacon's Qy„g^y;^[Q Jaws, and explained the several processes

Philosophy : o ' r i

of experience, observation, experiment, and in-
duction, by which these laws are made known.
Why op- He rejected the reasonings of Aristotle because

pisedtoAris- , • t r ^ ^

totie's. they were not progressive and useiul ; 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 its 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 is

the true sys

and the basis of that Practical, which the pres- temoiedu-
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, anid stability to action? What which wm

develop and

course of study will most enlarge the sphere of steady tuo
investigation ; give the greatest freedom to the
mind without licentiousness, and the greatest
frieedom 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 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- Mathematica
ematical science may lay claim to this pre-emi-
nence.

14

INTRODUCTION

Founda- The fii'st jmpressions which the child receives

tionsofmath- r- tv- i i r\ • j r i • l

ematicai ^^ iNumbcr and (4uantity are the loundations oi
Lnovviedge. j^j^ 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,
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
the lengthening chain.

Nature.

Number

and
tspace.

Mathemat-
ical kiiowl-
ndge :

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 mi-
nuteness and in all its grandeur : which survey
the laws of elementary organization, by the mi-
croscope, and weigh the spheres in the balance
of universal gravitation.

What '^^^ processes of mathematical science serve

the processes j-q gjyg mental uuity and wholeness. They im-
part that knowledge which applies the means of

PLANOFTHEWORK. 15

crystallization to a chaos of scattered particulars, Right knowi-
and discovers at once the general law, if there t,,g ,^^^^3 ^^^
be one, which forms a connectino; link between "J's'a"'2a-

' o 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 it records
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 uu^

educator re-

as the great means of accomplishing his work, gardsmatii-
The definitions present clear and separate ideas,
which the mind readily apprehends. The axioms xheaxioma.
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 bfeen previously fully comprehended; and influence ot

. , the study of

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

10 INTRODUCTION.

of systematic knowledge, so greatly preferable
to conjectural judgments.

How the The philosopher regards mathematical science

philosopher i r i • i • i • t i

regards ^^ the mCrC tOOlS 01 hlS higher vocation. Look-
mathematics: • •,1 iJ J • iAT.

ing with a steady and anxious eye to JNature,
and the great laws which 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
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 mci'c practical man regards with favor

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

Instruments the mind rcquircs instruments as well as the

of the mind i i i i • i •

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

has a law.

material world; whether an element of knowl-

PLANOFTHEWORK. 17

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

, , the law is to

law : to understand these connections and that know the
law, is to know the thing. When the principle ^^"^'
is clearly apprehended, the practice is easy.

With these general views, and under a firm Mathemanw

, , . , . , analyzed.

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- How.
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 accomplish whatwaa

deemed ne-

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 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-

Tiie necessity ciplcs 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.

Book II. Bool- ji treats of Mathematical Science,

Here I have endeavored to explain the nature of
Of what it 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

treating. . . ^ •

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

Book tii. Book III. explains and illustrates the Utility of

utility of T., , . _,. , ,. .

Mathematics. Mathematics : r irst, 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.

PLANOFTHEWORK. 19

Having thus given tiie 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 naay re-
ceive 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 ; ThM.

4th. Professional Teachers. Fourth.

1 1 • ^ -1 11 to the gen-

provement, and desires to acquire knowledge, erai reader,
must carefully search out the import of language.
He must early establish and carefully cultivate
the habit of noting the connection between ideas connec-
and their signs, and also the relation of ideas to ^ordsaud"
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 Logir.
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 itsvaiue:
we separate argument from sophistry — truth from
falsehood. The application of these principles,

20 INTRODUCTION.

fn the study in the Construction of the mathematical science.

mathematics, wherc 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