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

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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 or
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

i- i i 11 i , f mathematics

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 ex-
amined 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 ma-
king them available in classifying the facts, and
examining the organic laws which characterize
his particular department of knowledge.

Thirdly. Mathematical knowledge differs from

edge.

every other kind of knowledge in this : it is, as c

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- in, extent

22 INTRODUCTION.

ment and of practical utility. And if I am per-
my mitted to extend the figure, I may add, that the
web of the spider, though perfectly simple, if we
pluce ' see the end and understand the way n 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-
^Ki<w* acting, and to begin at the right place and pro-
constructed. cee( j j n t ^ e r jg ht wav> j g a jj tnat j s nec essary to

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

been air ,

tempted, beginning, and to indicate the way to the math-
ematical student. I am aware that he is start-
ing 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,
|oct the whole field to be gone over is at once sur-
veyed: by calling attention to the faculties of
Advantages the mind which the science brings into exercise,

of consider-
ing the men- we are better prepared to note the intellectual

Ul boilties : . i-ii

operations which the processes require ; and by

PLAN OF THE WORK.

a knowledge of the laws of reasoning, and an ofaknowi-

f . edge of the

acquaintance with the tests of truth, we are en- ia W80 rrea-
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 way clearly marked out, and will recognise, Advantul -' rt

J J to the stu-

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 combina-
tions, and guided by simple laws, he will go for- vvnere

he 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 omer

of progress.

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 or
and lifting the mind towards the same eminence.

tion of the work, has been, to afford substantial

aid to the professional teacher. The nature ol teacher -

24 INTRODUCTION.

Htodu ue,: his duties-their inherent difficulties-the per-
plexities which meet him at every step-the wanl
of sympathy and support in his hours of discour-
agement (and they are many) are circum-
stances which awaken a lively interest in tne
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 road-
side, and there, removed from the highways of
life, spends his days in raising the feeble mind
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
manhood and the girl become a matron before

iU?^ 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 ex-
tent, the moral development of a people. He

The impcr- holds in his hands the keys of knowledge. If
the first moral impressions do not spring into
life at his bidding, he is at the source of the
stream, and gives direction to the current. Al-
though himself imprisoned in the schoolhouse,
his influence and his teachings affect all condi-
tions of society, and reach over the whole hori-

ta

PL AN OP THE WORK. 20

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 i,u.
a vigorous stream. In the hope of doing some-
thing thai may be useful to the professional
teacher, I have attempted a careful and full ob J ect8for

which the

analysis of mathematical science. I have spread *rk 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 ; j^^MiLc,

for, the principles of imparting instruction are

the same for all branches of knowledge.

The system which I have indicated is com- system,
plete in itself. It lays open to the teacher the
entire skeleton of the science exhibits all its ""niu it

presents.

parts separately and in their connection. It
explains a course of reasoning simple in itself, what
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 Bdence
art, no longer regards Arithmetic as a mere combined

with art:

treadmill of mechanical labor, but as a means

20 INTRODUCTION

me dran- and the simplest means of teaching the art and
science of reasoning on quantity and this is
the logic of mathematics. If he would accom-
plish well his work, he must so instruct his

riptit instruc-

tion. pupils that they shall apprehend clearly, think
quickly and correctly, reason justly, and above
all, he must inspire them with a lore of knowl-
edge.

BOOK I.

LOGIC,

CHAPTER I.

SEFDirnOXS - OPERATIONS OK THE MIND TERMS DEFI5KD.
DEFINITIONS.

\$ 1. DEFINITION is a metaphorical word, which Definition
literally signifies " laying down a boundary." metaphorical
All definitions are of names, and of names only ;

Some

but in some definitions, it is clearly apparent, definitions

that nothing is intended except to explain the ^^

meaning of the word; while in others, besides Word8:
explaining the meaning of the word, it is also

implied that there exists, or may exist, a thing corre8 P nd -

* ing to the

corresponding to the word. words.

2. Definitions which do not imply the exist- or definitions
ence of things corresponding to the words de- not 'imply
fined, are those usually found in the Dictionary
of one's own language. They explain only the

28

LOGIC. [BOOK i.

The meaning of the word or term, by giving some
explain equivalent expression which may happen to be

words by . . .

equivalents, better known. Definitions which imply the ex-
istence of things corresponding to the words de-
fined, do more than this.

Definition F or example : " A triangle is a rectilineal fig-
ure having three sides." This definition does

two things :

implies. i s t. It explains the meaning of the word tri-
angle; and,

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

ofa 3. To define a word when the definition is

definition

which im- to imply the existence of a thing, is to select

P tote ot ^ TOm ^ tne properties of the thing those which

a thing. are mos t simple, general, and obvious; and the

properties properties must be very well known to us before

j^ro. we can decide which are the fittest for this pur-

pose. 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

supports

truth. kind of definition is not only the best form of ex-
pressing certain conceptions, but also contributes
to the development and support of new truths.

** 4. In Mathematics, and indeed, in all stiict

Mathematics

imply sciences, names imply the existence of the things

CHAP. I.] DEFINITIONS. 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 mathe-
matical term, can be devised, which shall not
express certain attributes of the thing correspond-
ing to the name. Every definition of this class Definition*
is a tacit assumption of some proposition which Ofuuscla8s
is expressed by means of the definition, and propositions.
which gives to such definition its importance.

5. All the reasonings in mathematics, which Reasoning

resting on
definitions ;

rest ultimately on definitions, do, in fact, rest

on the intuitive inference, that things corre-

rests on

spending to the words defined have a conceiv- intuitive
able existence as subjects of thought, and do or
may have proximately, an actual existence.*

* There are four rules which aid us in framing defini- Four rules
tions.

1st. The definition must be adequate: that is, neither toe 1st rale,
extended, nor too narrow for the word defined.

2d. The definition must be in itself plainer than the word 2d rule,
defined, else it would not explain it.

3d. The definition should be expressed in a convenient 3d rule.
number of appropriate words.

4th. When the definition implies the existence of a thing

4th rule,
corresponding to the word defined, the certainty of that

existence must be intuitive.

LOGIC.

[BOOK I.

OPERATIONS OF THE MIND CONCERNED IN REASONING.

Thre opera- g. There are three operations of the mind
tiro or the which are immediately concerned in reasoning.

1st. Simple apprehension ; 2d. Judgment ;
3d. Reasoning or Discourse.

sim iea 7- Simple apprehension is the notion (or

prehension, conception) of an object in the mind, analogous
to the perception of the senses. It is either

incompiox. Incomplex or Complex. Incomplex Apprehen-
sion is of one object, or of several without any
relation being perceived between them, as of a

complex, triangle, a square, or a circle : Complex is ot
several with such a relation, as of a triangle
within a circle, or a circle within a square.

8. Judgment is the comparing together in
the mind two of the notions fur ideas) which

Judgment

defined, are the objects of apprehension, whether com-
plex or incomplex, and pronouncing that they
agree or disagree with each other, or that one
of them belongs or does not belong to, the other :
for example : that a right-angled triangle and an

Judgment equilateral triangle belong to the class of figures
called triangles ; or that a square is not a circle.
Judgment, therefore, is either Affirmative or Neg-

D * atiTe - ative

either
' v?

CHAP. I.] ABSTRACTION. 31

9. Reasoning (or discourse) is the act of Reasoning
proceeding from certain judgments to another
founded upon them (or the result of them).

10. Language affords the signs by which
these operations of the mind are recorded, ex-
pressed, and communicated. It is also an in- Bought:
strument of thought, and one of the principal aJso,^

instrument

helps in all mental operations ; and any imper- of thought
fection in the instrument, or in the mode of
using it, will materially affect any result attained
through its aid.

11. Every branch of knowledge has, to a

ETery branch

certain extent, its own appropriate language ; ofknowiedge

. has its own

and for a mind not previously versed in the language,
meaning and right use of the various words and
signs which constitute the language, to attempt must be
the study of methods of philosophizing, would
be as absurd as to attempt reading before learn-
ing the alphabet.

ABSTRACTION.

12. The faculty of abstraction is that power
of the mind which enables us, in contemplating
any object (or objects), to attend exclusively to

LOGIC. [BOOK I.

s>me particular circumstance belonging to it, and
quite withhold our attention from the rest. Thus,
if a person in contemplating a rose should make
the scent a distinct object of attention, and lay
aside all thought of the form, color, &c., he
would draw off, or abstract that particular part ;
of drawing and therefore employ the faculty of abstraction.
He would also employ the same faculty in con-
sidering whiteness, softness, virtue, existence, as
entirely separate from particular objects.

13. The term abstraction, is also used to

denote the operation of abstracting from one or

Abstraction, more things the particular part under consider-

>iow and.

ation ; and likewise to designate the state of the
mind when occupied by abstract ideas. Hence,
abstraction is used in three senses :
Abstraction lst - To denote a faculty or power of the

2d> To Denote a process of the mind ; and,
or mind. 3d> To denote a state of the mind.

GENERALIZATION.
Generaliza- \$ 14 Generanzation IS the prOCCSS of COn-

"r^^f tem P latm g tne agreement of several objects in
contempiar certain points (that is, abstracting the circum-

1 1 ni: the

stances of agreement, disregarding the differ-

CHAP. I.]

TERMS.

33

ences), and giving to all and each of these ob- ofseveraj
jects a name applicable to them in respect to thing8 -
this agreement. For example ; we give the
name of triangle, to every rectilineal figure hav-
ing three sides : thus we abstract this property

.. ., Generalira-

trom all the others (lor, the triangle has three t i<m
angles, may be equilateral, or scalene, or right-
angled), and name the entire class from the prop-
erty so abstracted. Generalization therefore implies
necessarily implies abstraction ; though abstrac-
tion does not imply generalization.

A term.

TERMS SINGULAR TERMS COMMON TERMS.

15. An act of apprehension, expressed in
language, is called a Term. Proper names, or
any other terms which denote each but a single
individual, as " Caesar," " the Hudson," " the
Conqueror of Pompey," are called Singular singular

terms.

Terms.

On the other hand, those terms which denote
any individual of a whole class (which are form-
ed by the process of abstraction and generaliza-
tion), are called Common or general Terms. For common

1-1 i terma.

example ; quadrilateral is a common term, appli-
cable to every rectilineal plane figure having
four sides ; River, to all rivers ; and Conqueror,
to all conquerors. The individuals for which a
common term stands, are called its Significates.

3

34 LOGIC. [BOOK L

CLASSIFICATION.

16. Common terms afford the means of clas-
sification ; that is, of the arrangement of objects
into classes, with reference to some common and
distinguishing characteristic. A collection, com-
prehending a number of objects, so arranged, is
called a Genus or Species genus being the
more extensive term, and often embracing many

For example: animal is a genus embracing
every thing which is endowed with life, the pow-
er of voluntary motion, and sensation. It has
many species, such as man, beast, bird, &c. It
we say of an animal, that it is rational, it be
longs to the species man, for this is the charac-
teristic of that species. If we say that it has
wings, it belongs to the species bird, for this, in
like manner, is the characteristic of the species
bird.

A species may likewise be divided into classes,
or subspecies; thus the species man, may be
divided into the classes, male and female, and
these classes may be again divided until we reach
the individuals.

17. Now, it wfll appear from the principles
which govern this system of classification, that

AP. I.] CLASSIFICATION 33

the characteristic of a genus is of a more exten-
sive signification, but involves fewer particu-
lars than that ot a species. In like manner, the
characteristic of a species is more extensive, but
less full and complete, than that of a subspecies
or class, and the characteristics of these less full
than that of an individual.

For example ; if we take as a genus the Quadri-
laterals of Geometry, of which the characteristic
is. that they have four sides, then every plane
rectilineal figure, having four sides, will fail under
this class. If, then, we divide all quadrilaterals
into two species, viz. those whose opposite sides,
taken two and two, are not parallel, and those
whose opposite sides, taken two and two, are
parallel, we shall have in the first class, all irreg-
ular quadrilaterals, including the trapezoid (1 and
2) ; and in the other, the parallelogram, the rhom-
bus, the rectangle, and the square (3,4, 5, and 6).

If, then, we divide the first species into two
subspecies or classes, we shall have in the one, the
irregular quadrilaterals (1), and in the other, the
trapezoids (2) ; and each of these classes, being
made up of individuals having the same char-
acteristics, are not susceptible of further division.
If we divide the second species into two
classes, arranging those which have oblique an-
gles in the one, and those which have right

36 LOGIC. [BOOK i

angles in the other, we shall have in the first,

two varieties, viz. the common parallelogram

s e and the equilateral parallelogram or rhombus (3

111x1 and 4) ; and in the second, two varieties also,

classes.

viz. the rectangle and the square (5 and 6).
Now, each of these six figures is a quadn-

Each Indi-

vidual falling lateral; and hence, possesses the characteristic

enjo s f tne g enus > an( l eacn variety of both species
aii the enjoys all the characteristics of the species to

characteris-

tics. which it belongs, together with some other dis-
tinguishing feature ; and similarly, of all classi-
fications.

18. In special classifications, it is often not

necessary to begin with the most general char-

subaitem acteristicsj and then the genus with which we

enus - begin, is in fact but a species of a more extended

classification, and is called a Subaltern Genus.

For example ; if we begin with the genus Par-
allelogram, we shall at once nave two species,
viz. those parallelograms whose angles are oblique
and those whose angles are right angles ; and in
each species there will be two varieties, viz. in the
first, the common parallelogram and the rhom-
bus ; and in the second, the rectangle and square.

Highest 19- A genus which cannot be considered
as a species, that is, which cannot be referred

CHAP. I.] NATURE OF COMMON TERMS. 37

to a more extended classification, is called the Highest

genus.

highest genus ; and a species which cannot be
considered as a genus, because it contains only species
individuals having the same characteristic, is
called the lowest species.

NATURE OF COMMON TERMS.

\$ 20. It should be steadily kept in mind, that
the " common terms" employed in classification, A commiv .
have not, as the names of individuals have, any to te ^, h t ^ uo
real existing thing in nature corresponding to coTespo 1 * 1 -
them ; but that each is merely a name denoting

. i . is an

have formed of an individual. But as this name

does not include any thing wherein that indi- does not

include anf

vidual differs from others of the same class, it thing in
is applicable equally well to all or any of them. ^^4^^
Thus, quadrilateral denotes no real thing, dis- differ:
tinct from each individual, but merely any recti-
lineal figure of four sides, vie wed inadequately ;
that is, after abstracting and omitting all that
is peculiar to each individual of the class. By

J but to

this means, a common term becomes applicable applicable to

many

alike to any one of several individuals, or, taken individuals,
in the plural, to several individuals together.

Much needless difficulty has been raised re- Needle
specting the results of this process: many hav
ing contended, and perhaps more having taken

38 LOGIC. [BOOK i.

Difficulty in it for granted, that there must be some really
existing thing corresponding to each of those

common common terms, and of which such term is the

MOML

name, standing for and representing it. For ex-

ample ; since there is a really existing thing cor-

Noone responding to and signified by the proper and

^ singular name "./Etna," it has been supposed

ii to ech. tnat tne common term " Mountain" must have

some one really existing thing corresponding to

it, and of course distinct from each individual

mountain, yet existing in each, since the term,

being common, is applicable, separately, to every

one of them.

The fact is, the notion expressed by a common
term is merely an inadequate (or incomplete)
notion of an individual ; and from the very cir-
cumstance of its inadequacy, it will apply equally
8i Iultin well to any one of several individuals. For ex-

the thing.

ample ; if I omit the mention and the consider-

ation of every circumstance which distinguishes

./Etna from any other mountain, I then form a

notion, that inadequately designates ./Etna. This

Mountain" notion is expressed by the common term " moun-

ppilLbie tam " which does not imply any of the peculiar-

ities of the mountain ./Etna, and is equally ap-

plicable to any one of several individuals.

In regard to classification, we should also bear
in mind, that we may fix, arbitrarily, on the

CHAP. 1 J SCIENCE. 39

characteristic which we choose to abstract and May fix on

r -c T attributes

consider as the basis of our classification, disre- arbitrarily
earding all the rest : so that the same individual .

ciMBtnc&iioii

may be referred to any of several different spe-
cies, and the same species to several genera, as
suits our purpose.

SCIENCE.

21. Science, in its popular signification,
means knowledge.* In a more restricted sense, science

in its general

it means knowledge reduced to order ; that is,

knowledge so classified and arranged as to be
vantageously applied. In a more strict and g^^tion
technical sense, it has another signification.

"Every thing in nature, as well in the in- VieW80f
animate as in the animated world, happens or Kimt -
is done according to rules, though we do not
always know them. Water falls according to
the laws of gravitation, and the motion of walk- Generallaw *
ing is performed by animals according to rules.
The fish in the water, the bird in the air, move
according to rules. There is nowhere any want
of rule. When we think we find that want, we

Nowhere

can only say that, in this case, the r*les are un- any warn of

rule.

known to us. f

Assuming that all the phenomena of nature

* Section 23. f Kant.

40 LOGIC. [BOCK i

are consequences of general and immutable laws,

a technical we may define Science to be the analysis of

MDwdeOned: those j awg> comprehending not only the con-

an analysis nec ted processes of experiment and reasoning

of the laws

of nature, which make them known to man, but also those
processes of reasoning which make known their
individual and concurrent operation in the de-
velopment of individual phenomena.

ART.

22. Art is the application of knowledge to

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