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

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science employs in the development of its prin-

ciples.

MbMkBcr For example, the definition of multiplication is,
that h is the process of taking one number, called
the multiplicand, as many times as there are
units in another called the multiplier. This defi-
nition, as one of science, requires two things.

1st. That the multiplier be an abstract num-
ber; and,

2dly. That the product be a quantity of the
same kind as the multiplicand.

These two principles are certainly correct,
and relating to arithmetic as a science, are uni-
* versa % true - ^ ut are they universally true, in
j n which thev would be understood by

mleofML

learners, when applied to arithmetic as a mixed
object, that is, a science and an art ? Such an
applicaticwi would certainly exclude a large class
of practical rules, which are used in the appli-
cations of arithmetic, without reference to par-
ticular units.

For example, if we have feet in length to be
multiplied by feet in height, we must exclude the

CHAP. II.] ARITHMETIC APPLICATIONS. 175

question as one to which arithmetic is not appli-
cable ; or else we must multiply, as indeed we
do, without reference to the unit, and then assign
a proper unit to the product.

If we have a product arising from the three VVhen ^e

three factors

factors of length, breadth, and thickness, the areimes.
unit of the first product and the unit of the final
product, will not only be different from each
other, but both of them will be different from
the unit of the given numbers. The unit of the The different
given numbers will be a unit of length, the unit
of the first product will be a square, and that of
the final product, a cube.

186. Again, if we wish to find, by the best other

examples.

practical rule, the cost of 467 feet of boards at
30 cents per foot, we should multiply 467 by
30, and declare the cost to be 14010 cents, or
\$140.10.

Now, as a question of science, if you ask, can considered
we multiply feet by cents ? we answer, certainly
not. If you again ask, is the result obtained
sys, we give you the following :

1 foot of boards : 467 feet : : 30 cents : Answer.

Now, the ratio of 1 foot to 467 feet, is the ab
stract number 467 ; and 30 cents being multi-

176 MATHEMATICAL SCIENCE. [BOOK IL

plied by this number, gives for the product 14010
cents. But as the product of two numbers is

Product of

two numerically the same, whichever number be used

numbers.

as the multiplier, we know that 467 multiplied by

30. gives the same number of units as 30 multi-

The first rule pii e d by 467 : hence, the first rule for finding the

correct

amount is correct.

scientific in- 187. I have given these illustrations to point
out the difference between a process of scientific
investigation and a practical rule.

The first should always present the ideas of
Their differ- the subject in their natural order and connection,
what it con- while the other should point out the best way of
obtaining a desired result. In the latter, the
steps of the process may not conform to the or-
der necessary for the investigation of principles ;
but the correctness of the result must be suscepti-
ble of rigorous proof. Much needless and un-
profitable discussion has arisen on many of the
error. processes of arithmetic, from confounding a princi-
ple of science with a rule of mere application.

rule:

CHAP. II. "J ARITHMETIC OBDEK. 177

SECTION V

METHODS OF TEACHING ARITHMETIC CONSIDERED.

ORDER OF THE SUBJECTS.

188. IT has been well remarked by Cousin, cousin.
the great French philosopher, that " As is the Method
method of a philosopher, so will be his system ; decide9

Philosophy

and the adoption of a method decides the destiny
of a philosophy."

What is said here of philosophy in general, is True m
eminently true of the philosophy of mathematical
science ; and there is no branch of it to which
the remark applies, with greater force, than to
that of arithmetic. It is here, that the first no-
tions of mathematical science are acquired. It Anthmetic -
is here, that the mind wakes up, as it were, to
the consciousness of its reasoning powers Here,
it acquires the first knowledge of the abstract
separates, for the first time, the pure ideal from
the actual, and begins to reflect and reason on ^^
pure mental conceptions. It is. therefore, of the uisrbt

should be

highest importance that these first thoughts be nghiiy
impressed on the mind in their natural and proper

1-2

178 MATHEMATICAL SCIENCE. [BOOK II.

Faculties to order, so as to strengthen and cultivate, at the

be cultivated.

same time, the faculties of apprehension, discrim-
ination, and comparison, and also improve the
yet higher faculty of logical deduction.

First point: 189. The first point, then, in framing a

course of arithmetical instruction, is to deter-

methodof mine the method of presenting the subject. Is

presenting . .... .

the subject there any thing in the nature 01 the subject it-
self, or the connection of its parts, that points
out the order in which these parts should be
Laws of studied ? Do the laws of science demand a

science : . .

what do particular order ; or are the parts so loosely
5? connected, as to render it a matter of indiffer-
ence where we begin and where we end ? A
review of the analysis of the subject will aid us
in this inquiry.

Eiaaisofthe 190. We have seen* that the science of

numbers, numbers is based on the unit 1. Indeed, the

in what the whole science consists in developing, explain-

science .

consists, ing, and illustrating the laws by which, and

through which, we operate on this unit. There

Four classes are four classes of operations performed on the

of opera-
tions, unit one.

1st. TO in- i s t. To increase it according to the scale

rn-nap th

crease the
unit.

* Section 104.

CHAP. II.] ARITHMETIC INTEGRAL UNITS. 179

of tens forming the system of common num-
bers.

2d. To divide it, in any way we please, form- M To di _
ing the decimal and vulgar fractions.

3d. To increase it according to the vary- aa. Toin-
ing scales, forming all the denominate num-
bers.

4th. To compare it with all the numbers which 4th. TO com-
come from it; and then those numbers with each
other. This embraces proportions, of which the
Rule of Three is the principal brunch.

There is yet a fourth branch of arithmetic; Fourth
viz. the application of the principles and of the
rules drawn from them, in the mechanic arts Practical ap.
and in the ordinary transactions of business. p
This is called the Art, or practical part, of these the
Arithmetic. (See Arithmetical Diagram facing
page 119.)

INTEGRAL UNITS.

191. We begin first with the unit 1, and Unit one

increased

increase it according to the ecale of tens, form- according to

. v / , i TIT the scale of

nig the common system of integral numbers. We ^6.

then perform on these numbers the operations

of the five ground rules; viz. numerate them, operations

add them, subtract them, multiply and divide P

them.

180 MATHEMATICAL SCIENCE. [BOOK IL

FRACTIONAL UNITS.

192. We next pass to the second class ol
of operations on the unit 1 ; viz. the divisions of it

the unit.

General Here we pursue the most general method, and
I0d " first divide it arbitrarily ; that is, into any num-
ber of equal parts. We then observe that the
Method ac- division of it, according to the scale of tens, is
but a particular case of the general law of divi-
sion. We then perform on all the fractional
units which thus arise, every operation of the
five ground rules.

DENOMINATE UNITS.

193. Having operated on the abstract unit 1,

by the processes of augmentation and division,

Next in- we next increase it according to the varying

cording to scales of the denominate number?, and thus pro-

^ uce the system, called Denominate or Concrete
Numbers; after which, we perform on this class of
numbers all the operations of the live ground rules.
Reasons for By placing the subject of fractions directly
a ^ ter tne ^ ve grouud rules, the two opposite oper-
ations of aggregation and division are brought
into direct contrast with each other. It is thus
seen, that the laws of change, in the two systems
of operation on the unit 1, are the same with
very slight modifications.

CHAP. II.] ARITHMETIC RATIO. 181

This system of classification, has, after expe-
rience, been found to be the best for instruction.

R A T I O, O R RULE OF THREE.

3 194. Having considered the two subjects of subjects

considered.

integral and fractional units, we come next to
the comparison of numbers with each other.

This branch of arithmetic develops all the whatthw

- branch do

relative properties of numbers, resulting irom ve i ops .
their inequality.

The method of arrangement, indicated above, what the m

rangement

presents all the operations of arithmetic in con- doea .
nection with the unit 1, which certainly forms
the basis of the arithmetical science.

Besides, this arrangement draws a broad line whatitdoe-

r . i . , . further.

between the science of arithmetic and its ap-
plications ; a distinction which it is very im-
portant to make. The separation of the prin- Theory and
ciples of a science from their applications, so 8h0 uidba
that the learner shall clearly perceive what is Bepa "
theory and what practice, is of the highest im-
portance. Teaching things separately, teaching GoWenmies

... , . .for teacliing.

them well, and pointing out their connections,
are the golden rules of all successful instruc-
tion.

195. I had supposed, that the place of the

18'2 MATHEMATICAL, SCIENCE. |_BOOK IL

Rule of Three, among the branches of arith-
metic, had been fixed long since. But several
Differences in authors of late, have placed most of the practi-

arrangement; .

cal subjects before this rule giving precedence,

for example, to the subjects of Percentage, In-

in what they terest, Discount, Insurance, &c. It is not easy

consist.

to discover the motive of this change. It is
Ratio pwt of certain that the proportion and ratio of num-

the science.

bers are parts of the science of arithmetic ; and
should pre- the properties of numbers which they unfold,

cede applica-
tions, are indispensably necessary to a clear apprehen-
sion of the principles from which the practical
rules are constructed.

We may, it is true, explain each example in

Percentage, Interest, Discount, Insurance, &c.,

cannot well by a separate analysis. But this is a matter

change the

order. of much labor ; and besides, does not conduct
the mind to any general principle, on which
all the operations depend. Whereas, if the Rule
of Three be explained, before entering on the

Advantages practical subjects, it is a great aid and a pow-
erful auxiliary in explaining and establishing
all the practical rules. If the Rule of Three
is to be learned at all, should it not rather
precede than follow its applications? It is a
great point, in instruction, to lay down a gen-

The great era | p r j nc jpi e as early as possible, and then con-

(irinciple at

(nsu-uction. uec t with it all subordinate operations.

CHAP. II.] ARITHMETIC LANGUAGE. 183

ARITHMETICAL LANGUAGE.

196. We have seen that the arithmetical al- Arithmetical

alphabet.

phabet contains ten characters.* rrom these
elements the entire language is formed ; and we
now propose to show in how simple a manner.

The names of the ten characters are the first Names or the

characters.

ten words of the language. If the unit 1 be

added to each of the numbers from to 9 in- First ten

. ,, - . . combiiia-

clusive, we find the first ten combinations in tions.

arithmetic.! If 2 be added, in like manner,

we have the second ten combinations ; adding second ten,

and 90 on for

3, gives us the third ten combinations ; and so others,
on, until we have reached one hundred com-
binations (page 123).

Now, as we progressed, each set of combina- Each set giv-
tions introduced one additional word, and the uonaiword.

results of all the combinations are expressed by
the words from two to twenty inclusive.

197. These one hundred elementary com- AU that need

be cuinmit-

bmations, are all that need be committed to ted tome-
memory ; for, every other is deduced from them.
They are, in fact, but different spellings of the
first nineteen words which follow one. If we ex-
tend the words to one hundred, and recollect that

* Section 114. f Section 116.

184 MATHEMATICAL SCIENCE [BOOK II

at one hundred, we begin to repeat the numbers,

words tobe we see that we have but one hundred words to

only ten above ten are derivative. To this number,

words primi-
tive, must of course be added the few words which

express the sums of the hundreds, thousands, &c.

subtraction: 198. In Subtraction, we also find one hun-
dred elementary combinations ; the results of
which are to be read.* These results, and all
>7umberof the numbers employed in obtaining them, a/e

words.

expressed by twenty words.

Multiplies- 199. In Multiplication (the table being cai-
ried to twelve), we have one hundred and forty-
four elementary combinations,! and fifty- nine

Number of separate words (already known) to express the

words. ft

results of these combinations.

Division: 200. In Division, also, we have one hundred
, and forty-four elementarv combinations,! but

Number of J

words. use on ] v twelve words to express their results.

Four nun-

igCit 2 L Thus ' we have f ur hundred and ej g h -

eiementary ty-eight elementary combinations. The results

combina-
tions, of these combinations are expressed by one hun-

vvordsused: j^ wor( j s . v j z nineteen in addition, ten in sub-

tion, traction, fifty-nine in multiplication, and twelve

10 in subtrac

linll.

59 in mu'ti-

plication, * Section 127 - t Section 129. \$ Section 130.

CHAP. II.] ARITHMETIC LANGUAGE. 1S.J

in division. Of the nineteen words which are i2in division
employed to express the results of the combina-
tions in addition, eight are again used to express
similar results in subtraction. Of the fifty-nine
which express the results of the combinations
in multiplication, sixteen had been used to ex-
press similar results m addition, and one in
subtraction ; and the entire twelve, which ex-
press the results of the combinations in division,
had been used to express results of previous
combinations. Hence, the results of all the ele-
mentary combinations, in the four ground rules,
are expressed by sixty-three different words ; and sixty-three

different

they are the only words employed to translate -ordsinau.
these results from the arithmetical into our com-
mon language.

The language for fractional units is similar Language

the same for

in every particular. By means ot a language fractions,
thus formed we deduce every principle in the
science of numbers.

202. Expressing these ideas and their com-
binations by figures, gives rise to the language Language of

.... . arithmetic:

ol arithmetic. By the aid of this language we

not only unfold the principles of the science, its value and

but are enabled to apply these principles to

every question of a practical nature, involving

the use of figures.

MATHEMATICAL S C I E X C t . [BOOK H.

203. There is bat one further idea to be
presented: it is this. that there are ver
combinations made among the figures. which
****" change, at all their signification

Selecting any two of die figures, as 3 and 5,
T|| for example, ire see at once that there are but

three ways of writing them, mat will at afi
change their signification.
rss : First, write them by the side of each i 3 5.

i 3
) 5

other .......... 53.

Second, write them, die one over i |.

the other ......... i y

AM. Third., pbce a decimal point before i .8,
each ......... - - I A

Now, each manner of writing gives a differ-

ent signification to bom me figures. Use. bow-

ijMMMfee eTer, has estabbsned that suEnincation. ***** we

r know it as soon as we have learned die lan>

We hare thus rrphJatad what we mean by
die arithmetical hnyia^i..
braces the names of its

, the formation and number of its
words, and the laws by which figures are con-

ieel that there is imylkilj and beauty in this
hope it

CUAP. II.] ARITHMETIC DEFINITIONS. 18?

NECESSITY OF EXACT DEFINITIONS AND TERMS.

204. The principles of every science are Principles o!
a collection of mental processes, having estab-
lished connections with each other. In every

branch of mathematics, the Definitions and Definition

and terms :

Terms give form to, and are the signs of, cer-
tain elementary ideas, which are the basis of
the science. Between any term and the idea
which it is employed to express, the connection
should be so intimate, that the one will always
suggest the other.

These definitions and terms, when their sig- when once

fixed must

nifications are once fixed, must always be used always be

in the same sense. The necessity of this is most
urgent. For, "in the whole range of arithmetical
science there is no logical test of truth, but in Reason.
a conformity of the reasoning to the definitions
.nd terms, or to such principles as have been
established from them."

205. With these principles, as guides, we Definitions

. and terms

propose to examine some 01 the definitions and exam ined.
terms which have, heretofore, formed the basis
of the arithmetical science. We shall not con-
fine our quotations to a single author, and shall
make only those which fairly exhibit the gen-
eral use of the terms

188 MATHEMATICAL SCIENCE. [BOOK U.

It is said,
Number de " Number signifies a unit, or a collection of

fined. .

units.
HOW " The common method of expressing numbers

is by the Arabic Notation. The Arabic method

employs the following ten characters, or figures"

&c.
Names of the "The first nine are called significant figures,

because each one always has a value, or denotes

some number."

And a little further on we have,
Figures have " The different values which figures have, are

values.

called simple and local values."

The definition of Number is clear and cor-
Number rect. It is a general term, comprehending al.

rightly de- , . ,

flned- the phrases which are used, to express, either
separately or in connection, one or more things

AUo figures, of the same kind. So, likewise, the definition
of figures, that they are characters, is also right.

Definition de- But mark how soon these definitions are de-
parted from. The reason given why nine of the
figures are called significant is, because "each
one always has a value, or denotes some num-
ber." This brings us directly to the question,

Haa a figure whether a figure has a value ; or, whether it is
a mere representative of value. Is it a number
or a character to represent number? Is it a

It is merely .

a character: quantity or symbol ? It is denned to be a char-

CHAP. II.] ARITHMETIC DEFINITIONS. 189

acter which stands for, or expresses a number.
Has it any other signification ? How then can
we say that it has a value and how is it possi- Has novaiun
ble that it can have a simple and a local value ?
The things which the figures stand for, may
change their value, but not the figures them-
selves. Indeed, it is very difficult for John to
perceive how the figure 2, standing in the sec- but stands

for value.

ond place, is ten times as great as the same fig-
ure 2 standing in the first place on the right!
although he will readily understand, when the
arithmetical language is explained to him, that
the UNIT of one of these places is ten times as unit orpine*
great as that of the other.

nition or principle which forms the basis of the
arithmetical language. It is in these words :

" Numbers increase from right to left in a or number.
tenfold ratio ; that is, each removal of a figure,
one place towards the left, increases its value
ten times."

Now, it must be remembered, that number DOW not
has been defined as signifying " a unit, or a thTde'eni-
collection of units." How, then, can it have a Uon ber "

given.

right hand, or a left ? and how can it increase
from right to left in a tenfold ratio?" The
explanation given is, that "each removal of a

190

MATHEMATICAL SCIENCE. [BOOK II.

Explanation.

Increase of
numbers baa
no connection
with figures.

Ratio.

"Tenfold
ratio:"

notions of
numbers.

First.

Second.

Third.

Fourth,

figure one place towards the left, increases its
value ten times."

Number, signifying a collection of units, must
necessarily increase according to the law by
which these units are combined ; and that law
of increase, whatever it may be, has not the
slightest connection with the figures which are
used to express the numbers.

Besides, is the term ratio (yet undefined),
one which expresses an elementary idea? And
is the term, a " tenfold ratio," one of sufficient
simplicity for the basis of a system ?

Does, then, this definition, which in substance
is used by most authors, involve and carry to
the mind of the young learner, the four leading
ideas which form the basis of the arithmetical
notation ? viz. :

1st. That numbers are expressions for one or
more things of the same kind.

2d. That numbers are expressed by certain
characters called figures ; and of which there
are ten.

3d. That each figure always expresses as
many units as its name imports, and no more.

4th. That the kind of thing which a figure
expresses depends on the place which the figure
occupies, or on the value of the units, indicated
in some other wnv

CHAP. II.] ARITHMETIC DEFINITIONS.

191

PLACE is merely one of the forms of language
by which we designate the unit of a number,
expressed by a figure. The definition attributes
this property of place both to number and fig-
ures, while it belongs to neither.

207. Having considered the definitions and
terms in the first division of Arithmetic, viz. in
notation and numeration, we will now pass to Definition* in

The following are the definitions of Addition,
taken from three standard works before me :

" The putting together of two or more num- First,
bers (as in the foregoing examples), so as to
make one whole number, is called Addition, and
the whole number is called the sum, or amount."

" ADDITION is the collecting of numbers to- second,
gether to find their sum."

" The process of uniting two or more num- Thud.
kfs together, so as to form one single number,

" The answer, or the number thus found, is
called the sum, or amount."

Now, is there in either of these definitions Defect
any test, or means of determining when the
pupil gets the thing he seeks for, viz. " the sum
of two or more numbers?" No previous defi- Reason
nition has been given, in either work, of the

192 MATHEMATICAL SCIENCE.

term SUM. How is the learner to know whal
he is seeking for, unless that thing be defined ?
Noprin- Suppose that John be required to find the sum.

ciple as a

standard. Oi the numbers 3 and 5, and pronounces it to
be 10. How will you correct him, by showing
that he has not conformed to the definitions and
rules ? You certainly cannot, because you have
established no test of a correct process.

But, if you have previously defined SUM to be
a number which contains as many units as there
are in all the numbers added : or, if you say,

uorrectdefl- "Addition is the process of uniting two or
more numbers, in such a way, that all the units
which they contain may be expressed by a sin-
gle number, called the sum, or sum total ;" you
will then have a test for the correctness of the

<;iveatest. process of Addition; viz. Does the number,
which you call the sum, contain as many units
as there are in all the numbers added ? The
answer to this question will show that John is
wrong.

Definitions rf 208. I will now quote the definitions of
Fractions from the same authors, and in the
same order of reference.

First " We have seen, that numbers expressing whole
things, are called integers, or whole numbers ;
but that, in division, it is often necessary to

CHAP. II. J ARITHMETIC EEFINIT1ONS. 193

divide or break a whole thing into parts, and
that these parts are called fractions, or broken

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