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

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4th. That the truths of Geometry may be di-
vided into three classes : three

1st. Those implied in the definitions, viz. First class,
that things exist corresponding to certain
words defined ;

2d. Intuitive or self-evident truths em- second,
bodied in the axioms ;

3d. Truths deduced (that is, inferred) from ^^
the definitions and axioms, called Demonstra-
tive Truths.

5th. That the examination of the properties of Geometrical
the geometrical magnitudes has reference,

25-i MATHEMATICAL SCIENCE. [BOOK II.

comparison. 1st. To their comparison with a standard

or unit of measure ;

Properties. 2d. To the discovery of properties belong-

ing to an individual figure, and yet common to
the entire class to which such figure belongs ;

proportion. 3d. To the comparison, with respect to mag-

nitude, of figures of the same species with each
other; viz. lines with lines, surfaces with sur-
faces, volumes with volumes, and angles with
angles.

SUGGESTIONS FOE THOSE WHO TEACH GEOMETKr.

Suggestions. 1. Be sure that your pupils have a clear ap-
Firet. prehension of space, and of the notion that Ge-
ometry is conversant about space only.

2. Be sure that they understand the significa-
second. tion of the terms, lines, surfaces, and volumes,

and that these names indicate certain portions
of space corresponding to them.

3. See that they understand the distinction
Third, between a straight line and a curve; between a

plane surface and a curved surface; between a
volume bounded by planes and a volume bounded
by curved surfaces.

4. Be careful to have them note the charac-
Fourth. teristics of the different species of plane figures,

such as triangles, quadrilaterals, pentagons, hexa-
gons, &c. ; and then the characteristic of each

CHAP. III.] GEOMETRY. 253

class or subspecies, so that the name shall recall,
at once, the characteristic properties of each
figure.

5. Be careful, also, to have them note the
characteristic differences of the volumes. Let Fifth,
them often name and distinguish those which

are bounded by planes, those bounded by plane
and cu^ed surfaces, and those bounded by
curved surfaces only. Regarding Volume as a
genus, let them give the species and subspecies
into which it may be divided.

6. Having thus made them familiar with the
things which are the subjects of the reasoning, sixth,
explain carefully the nature of the definitions ;

then of the axioms, the grounds of our belief in
them, and the information from which those
self-evident truths are inferred.

7. Then explain to them, that the definitions

and axioms are the basis of all geometrical *a- s^nth.
soning : that every proposition must be deduced
from them, and that they afford the tests of all
the truths which the reasonings establish.

8. Let every figure, used in a demonstration,

be accurately drawn, by the pupil himself, on a Eighth,
blackboard. This will establish a connection
between the eye and the hand, and give, at the
same time, a clear perception of the figure and a
distinct apprehension of the relations of its parts.

254 MATHEMATICAL SCIENCE. [BOOK II

9. Let the pupil, in every demonstration, first
Ninth, enunciate, in general terms, that is, without the

aid of a diagram, or any reference to one, the
proposition to be proved ; and then state the
principles previously established, which are to
be employed in making out the proof.

10. When in the course of a demonstration,

4p

Tenth. an y truth is inferred from its connection with
one before known, let the truth so referred to be
fully and accurately stated, even though the
number of the proposition in which it is proved,
be also required. This is deemed important.

11. Let the pupil be made to understand that
Etev,tb. a demonstration is but a series of logical argu-
ments arising from comparison, and that the
result of every comparison, in respect to quan-
tity, contains the mark either of equality or
inequality.

12. Let the distinction between a positive
Twelfth, and negative demonstration be fully explained

and clearly apprehended.

13. In the comparison of quantities with each
Thirteenth, other, great care should be taken to impress the

fact that proportion exists only between quan-
tities of the same kind, and that ratio is the
measure of proportion.

14. Do not fail to give much importance to
Fourteenth, the kind of quantity under consideration. Let

CHAP. III.J GEOMETRY. 255

the question be often put, What kind of quantity Fourteenth,
are you considering ? Is it a line, a surface, or
a volume ? And what kind of a line, surface, or
volume ?

15. In all cases of measurement, the unit of
measure should receive special attention. If

lines are measured, or compared by means of a Fifteenth,
common unit, see that the pupil perceives that
unit clearly, and apprehends distinctly its rela-
tions to the lines which it measures. In sur-
faces, take much pains to mark out on the
blackboard the particular square which forms
the unit of measure, and write unit, or unit of
measure, over it. So in the measurement of
volumes, let the unit or measuring cube be ex-
hibited, and the conception of its size clearly
formed in the mind ; and then impress the im-
portant fact, that, all measurement consists in
merely comparing a unit of measure with the
quantity measured; and that the number which
expresses the ratio is the numerical expression
for that measure.

16. Be careful to explain the difference of the sixteenth
terms Equal and Equal in all their parts, and

never permit the pupil to use the terms as sy-
nonymous. An accurate use of words leads to
nice discriminations of thought.

CHAP. IV ]

AN ALY Sid.

257

CHAPTER IV.

ANALYSIS ALGEBRA ANALYTICAL GEOMETRY.

ANALYSIS.

278. ANALYSIS is a general term, embra-
cing that entire portion of mathematical science
in which the quantities considered are repre-
sented by letters of the alphabet, and the opera-
tions to be performed on them are indicated by
signs.

defined.

279. We have seen that all numbers must Numbers

must be ot

be numbers of something;* for, there is no such thins:
thing as a number without a basis : that is, every
number must be based on the abstract unit one,
or on some unit denominated. But although
numbers must be numbers of something, yet they but may be

of many kind

may be numbers of any thing, for the unit may of thing*,
be whatever we choose to make it.

* Section 112.
17

MATHEMATICAL SCIENCE. '[BOOK. II.

AII quantity 280. All quantity, made up of definite parts,

consists of

parts . can be numbered exactly or approximative^,

and, in this respect, possesses all the properties

of number. Propositions, therefore, concerning

numbers, have the remarkable peculiarity, that

Propositions they are propositions concerning all quantities

in regard to

number whatever. That half of six is three, is equally
* P q U J amity. true > whatever the word six may represent,
whether six abstract units, six men, or six tri-
angles. Analysis extends the generalization stili
further. A number represents, or stands for, that
particular number of things of the same kind,
Algebraic without reference to the nature of the thing ;

symbols . 111 r

more gener- but an analytical symbol does more, for it may
stand for all numbers, or for all quantities which
numbers represent, or even for quantities which
cannot be exactly expressed numerically.

Anything As soon as we conceive of a thing we may

conceived .. . , . . , .

may be di- conceive it divided into equal parts, and may

Vlded- represent either or all of those parts by a or x,

or may, if we please, denote the thing itself by a

or x, without any reference to its being divided

into parts.

Each figure 281. In Geometry, each geometrical hgure
stands for a class ; and when we have demon-
strated a property of a figure, that property ;
considered as proved for every figure of the class.

CHAP. IV.J ANALYSIS. 259

For example: when we prove that the square Example,
described on the hypothenuse of a right-angled
triangle is equal to the sum of the squares de-
scribed on the other two sides, we demonstrate the
fact for all right-angled triangles. But in analy- in analysis
sis, all numbers, all lines, all surfaces, all vol- B tand for
urnes, may be denoted by a single symbol, a or x. ^l^ *
Hence, all truths inferred by means of these
symbols are true of all things whatever, and not
like those of number and geometry, true only
of particular classes of things. It is, therefore,
not surprising, that the symbols of analysis do
not excite in our minds the ideas of particular Hence, the

. . . truths int'i-r-

things. 1 he mere written characters, a, b, c, a, >& ^ ^
x, y, z, serve as the representatives of things in
general, whether abstract or concrete, whether
known or unknown, whether finite or infinite.

282. In the uses which we make of these symbols

come to be

symbols, and the processes of reasoning carried regarded as
on by means of them, the mind insensibly comes
to regard them as things, and not as mere signs ;
and we constantly predicate of them the prop-
erties of things in general, without pausing to
inquire what kind of thing is implied. Thus, Example.
we define an equation to be a proposition in Th eequa .
which equality is predicated of one thing as Uon>
compared with another. For example :

260 MATHEMATICAL SCIENCE. ffiOOK II

a + c = x,
whataxioms ^ s an equation, because x is declared to be

necessary to e g ua l to the gum o f a an( J c J n ^6 solution of
its solution.

equations, we employ the axioms, " If equals be

added to equals, the sums will be equal ;" and,

" If equals be taken from equals, the remainders

They express will be equal." Now, these axioms do not ex-

qualities of . -

things, press qualities of language, but properties of
Hence, in- quantity. Hence, all inferences in mathemat-

ferences re- . i i

late to things, ical science, deduced through the instrumentality
of symbols, whether Arithmetical, Geometrical,
or Analytical, must be regarded as concerning
quantity, and not symbols.

Quantity As analytical symbols are the representatives
wH^res- f quantity in general, there is no necessity of
enttothe k ee p m g the idea of- quantity continually alive in
the mind; and the processes of thought may,
without danger, be allowed to rest on the sym-
bols themselves, and therefore, become to that
extent, merely mechanical. But, when we look
rue reason- back and see on what the reasoning is based, and
based on the now ^ ne processes have been conducted, we shall
supposition fl n( j t j iat ever y ste p was taken on the supposition

of quantity.

that we were actually dealing with things, and
not symbols ; and that, without this understand-
ing of the language, the whole system is without
signification, and fails.

CHAP. IV. J ANALYSIS. 261

283. There are three principal branches of Three

. | . branches

Analysis :

1st. Algebra. Algebra,

2d. Analytical Geometry. Analytical

Geometry,

3d. Differential and Integral Calculus.

ALGEBRA.

284. Algebra is, in fact, a species of uni- Algebra:
versal Arithmetic, in which letters and signs are universal
employed to abridge and generalize all processes
involving numbers. It is divided into two parts, TWO pans
corresponding to the science and art of Arith-
metic :

1st. That which has for its object the investi- Firetpart
gation of the properties of numbers, embracing
all the processes of reasoning by which new
properties are inferred from known ones ; and,

2d. The solution of all problems or questions scond part
involving the determination of certain numbers
which are unknown, from their connection with
certain others which are known or given.

ANALYTICAL GEOMETRY.

285. Analytical Geometry examines the Analytical

.. 11.' c i Geometry.

properties, measures, and relations of the geo-
metrical magnitudes by means of the analytical its nature.

262 MATHEMATICAL SCIENCE. [BOOK II

symbols. This branch of mathematical science
originated with the illustrious Descartes, a ceie-

the original

founder of brated French mathematician of the 17th cen-

tiiis science. TT i i i i r

tury. He observed that the positions ot points.

\\ hat be

observed, the direction of lines, and the forms of surfaces,

could be expressed by means of the algebraic

AH position symbols ; and consequently, that every change,

expressed by

symbols, either in the position or extent ot a geometrical
magnitude, produced a corresponding change in
certain symbols, by which such magnitude could
be represented. As soon as it was found that,
to every variety of position in points, direction
in lines, or form of curves or surfaces, there cor-
responded certain analytical expressions (called
their Equations), it followed, that if the processes
were known by which these equations could be
The equation examined, the relation of their parts determined,

develops the

properties and the laws according to which those parts
nit^de.^" varv > relative to one another, ascertained, then
the corresponding changes in the geometrical
magnitudes, thus represented, could be imme-
diately inferred.

Hence, it follows that every geometrical ques-
Power over tion can be solved, if we can resolve the corre-

the magni- 11-

uide extend- spending algebraic equation ; and the power ovei
the geometrical magnitudes was extended just in
proportion as the properties of quantity were
brought to light by means of the Calculus. The

CHAP. IV.] ANALYSIS. 203

applications of this Calculus were soon made to TO what sub.

,, , . , . ject applied.

the subjects ot mechanics, astronomy, and in-
deed, in a greater or less degree, to all branches
of natural philosophy ; so that, at the present iu> present
time, all the varieties of physical phenomena,
with which the higher branches of the science
are conversant, are found to answer to varieties
determinable by the algebraic analysis.

286. Two classes of quantities, and conse- Quantities

which enter

quently two sets of symbols, quite distinct from into the ci
each other, enter into this Calculus ; the one
called Constants, which preserve a fixed or given constants.
value throughout the same discussion or investi-
gation ; and the other called Variables, which variables,
undergo certain changes of value, the laws of
which are indicated by the algebraic expressions
or equations into which they enter. Hence,

Analytical Geometry may be defined as that Analytical
branch of mathematical science, which exam- defined,
ines, discusses, and develops the properties of
geometrical magnitudes by noting the changes
thai take place in the algebraic symbols which
represent them, the laws of change being deter-
mined by an algebraic equation or formula.

264 MATHEMATICAL SCIENCE [BOOK IL

DIFFERENTIAL AND INTEGRAL CALCULUS.

8 ^87 In this branch of mathematical science,

considered.

as in Analytical Geometry, two kinds of quan-
vanabies, fay are considered, viz. Variables and Constants ;

Constants.

and consequently, two distinct sets of symbols
The science, are employed. The science consists of a series
of processes which note the changes that take
place in the value of the Variables. Those
changes of value take place according to fixed
laws established by algebraic formulas, and are
Marks, indicated by certain marks drawn from the va-
Differen- riable symbols, called Differentials. By these
marks we are enabled to trace out with the
accuracy of exact science the most hidden prop-
erties of quantity, as well as the most general
and minute laws which regulate its changes of
value.

Analytical 288. It will be observed, that Analytical

and ' Geometry and the Differential and Integral Cal-

Caicuius : cu j us treat of quantity regarded under the same

general aspect, viz. as subject to changes or va-

Howtiey nations in magnitude according to laws indica-

regard quan-
tity : ted by algebraical formulas; and the quantities,

whether variable or constant, are, in both cases,

by what represented by the same algebraic symbols, viz.

ted ' the constants by the first, and the variables by

CHAP. IV. J ALGEBRA. 265

the final letters of the alphabet. There is, how- Difference;
ever, this important difference : in Analytical
Geometry all the results are inferred from the in what it

consists.

relations which exist between the quantities
themselves, while in the Differential and Integral
Calculus they are deduced by considering what
may be indicated by marks drawn from variable
quantities, under certain suppositions, and by
marks of such marks.

289. Algebra, Analytical Geometry, the Dif- Analytical

Science.

ferential and Integral Calculus, extended into the
Theory of Variations, make up the subject of
analytical science, of which Algebra is the ele-
mentary branch. We shall, in this chapter,
limit our remarks to the subject of Algebra; Algebra,
reserving a separate chapter for the Differential Differential

Calculus.

and Integral Calculus. This subject embraces a
very remarkable class of quantities.

ALGEBRA.

290. On an analysis of the subject of Alge- Algebra,
bra, we think it will appear that the subject itself
presents no serious difficulties, and that most of Difficulties,
the embarrassment which is experienced by the
pupil in gaining a knowledge of its principles, as H ow over-
well as in their applications, arises from not at

266 MATHEMATICAL SCIENCE. [BOOK II.

Language, tending sufficiently to the language or signs of

the thoughts which are combined in the reason-

ings. At the hazard, therefore, of being a little

diffuse, I shall begin with the very elements of

the algebraic language, and explain, with much

minuteness, the exact signification of the char-

charactere acters that stand for the quantities which are the

Hut'quaTitit^ subjects of the analysis ; and also of those signs

signs. which indicate the several operations to be per-

formed on the quantities.

Quantities. 291. The quantities which ai'e the subjects
HOW divided, of the algebraic analysis may be divided into
two classes : those which are known or given,
and those which are unknown or sought. The
known are uniformly represented by the first

letters of the alphabet, a, b, c, d, &c. ; and the
unknown by the final letters, x, y, z, v, w, &c.

May be in-

\ 292. Quantity is susceptible of being in-
creased or crease( j or diminished ; and there are six oper-

diminisbed.

Five opera- ations which can be performed upon a quantity
tions ' that will give results differing from the quantity

it&<;lf, viz. :

RnjU 1st. To add it to itself or to some other quan-

tity;
second. 2d. To subtract some other quantity from it;

CHAP. IV J ALGEBRA 26?

3d. To multiply it by a number ; ThinL

4th. To divide it; Fourth.

5th. To raise it to any power ; and Fifth -

6th. To extract a root of it. sixth.

The cases in which the multiplier or divisor

is 1, are of course excepted ; as also the case Exception.

where a root is to be extracted of 1.

293. The six signs which denote these oper- sign.
ations are too well known to be repeated here.
These, with the signs of equality and inequality, Elements

of the

the letters of the alphabet and the figures which Algebraic
are employed, make up the elements of the alge-
braic language. The words and phrases of the its words

and phraaoa

algebraic, like those of every other language, are
to be taken in connection with each other, and

are not to be interpreted as separate and isolated H * inter-
preted,
symbols.

294. The symbols of quantity are designed symboiaof
to represent quantity in general, whether abstract
or concrete, whether known or unknown ; and
the signs which indicate the operations to be General,
performed on the quantities are to be interpreted
in a sense equally general. When the sign plus
is written, it indicates that the quantity before Examples,
which it is placed is to be added to some other signs plus

. . . . ,. . . and minus.

quantity ; and the sign minus implies the exist-

2G8 MATHEMATICAL SCIENCE. [BOOK II.

ence of a minuend, from which the subtrahend

is to be taken. One thing should be observed in

agna have regard to the signs which indicate the operations

no effect on

ihe nature of that are to be performed on quantities, viz. they

a quantity.

do not at all affect or change the nature of the
quantity before or after which they are written
but merely indicate what is to be done with the.

Examples: quantity. In Algebra, for example, the minus

Algcbra< sign merely indicates that the quantity before

which it is written is to be subtracted from

Analytical some other quantity ; and in Analytical Geom-

Geometry.

etry, that the line before which it falls is esti-
mated in a contrary direction from that in which
it would have been reckoned, had it had the sign
plus ; but in neither case is the nature of the
quantity itself different from what it would have
been had it had the sign plus.

interpret*- The interpretation of the language of Algebra

Cg^agef i s the fi rst thing to which the attention of a pupil

should be directed ; and he should be drilled on

the meaning and import of the symbols, until

their significations and uses are as familiar as

.u necessity, the sounds and combinations of the letters of the
alphabet.

Elements 295. Beginning with the elements of the
language, let any number or quantity be desig-
nated by the letter a, and let it be required to

CHAP. IV.J ALGEBRA. 269

add this letter to itself, and find the res Jt or sum.
The addition will be expressed by

a + a = the sum.

But how is the sum to be expressed ? By simply signification
regarding a as one a, or la, and then observing
that one a and one a make two a's or 2 a : hence,

a -\-a = 2a;

and thus we place a figure before a letter to in-
dicate how many times it is taken. Such figure
is called a Coefficient. coefficient.

296. The product of several numbers is in- Product-,
dicated by the sign of multiplication, or by sim-
ply writing the letters which represent the num-
bers by the side of each other. Thus,

a x b x c x d xf, or dbcdf, ho

ted

indicates the continued product of a, b, c, d, and
/, arid each letter is called a factor of the prod-
uct : hence, a factor of a product is one of the
multipliers which produce it. Any figure, as 5,
written before a product, as

5 abcdf,

is the coefficient of the product, and shows that coefficient oi
the product is taken 5 times.

270 MATHEMATICAL SCIENCE. [BOOK II.

Equal fac- 297. If in the product alcdf, the numbers

tors:

what the re P resen t e d by a, b, c, d, and / were equal to each
product other, they would each be represented by a single

becomes. J

letter a, and the product would then become

HOW that is, we indicate the product of several equal

factors by simply writing the letter once and

placing a figure above and a little at the right

of it, to indicate how many times it is taken as

Exponent: a factor. The figure so written is called an ex-

where writ- ponent. Hence, an exponent denotes how many

equal factors are employed. The result of the

multiplications, is called the 5th Power of .

Division: 298. The division of one quantity by an-
how other is indicated by simply writing the divisoi
below the dividend, after the manner of a frac-
tion ; by placing it on the right of the dividend
with a horizontal line and two dots between them;
or by placing it on the right with a vertical line
between them : thus either form of expression,

, I -f- a, or b \ a,

forms. d

indicates the division of b by a.

Boots: 299. The extraction of a root is indicated
how indi- by the sign */ This sign, when used by itself

cated.

indicates the lowest root, viz. the square root.

CHAP. IV.] \LGEBRA. 271

If any other root is to be extracted, as the third,
fourth, fifth, &c., the figure marking the degree index;

of the root is written above and at the left of where writ-
ten,
the sign ; as,

V cube root, \$~ fourth root, &c.

The figure so written, is called the Index of the
root.

We have thus given the very simple and gen- Language
eral language by which we indicate every one
of the six operations that may be performed on
an algebraic quantity, and every process in Al-
gebra involves one or other of these operations.

MINUS SIGN.

300. The algebraic symbols are divided into Algebraic

language :

two classes entirely distinct from each other,

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