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being precisely the same in both examples.

CHAP. II.] ARITHMETIC FRACTIOV3. 155

SECTION II.

FRACTIONAL UNITS.

FRACTIONAL UNITS. SCALE OF TENS.

163. IF the unit 1 be divided into ten equal Fraction one-
parts, each part is called one tenth. If one of J^.
these tenths be divided into ten equal parts,
each part is called one hundredth. If one of the hundredth;
hundredths be divided into ten equal parts, each One
part is called one thousandth ; and corresponding thousandth -
names are given to similar parts, how far soever Generaiia*-
the divisions may be carried.

Now, although the tenths which arise from Fractions are
dividing the unit 1, are but equal parts of 1,
they are, nevertheless, WHOLE tenths, and in this
light may be regarded as units.

To avoid confusion, in the use of terms, we Fractional
shall call every equal part of 1 a fractional unit.
Hence, tenths, hundredths, thousandths, tenths
of thousandths, &c., are fractional units, each
having a fixed relation to the unit 1, from which
it was derived.

156 MATHEMATICAL SCIENCE. [BOOK 11.

Fractional 164. Adopting a similar language to that

unit* of the

first order; used m integral numbers, we call the tenths, frac-
tional units of the first order ; the hundredths,
fractional units of the second order; the thou-
sandths, fractional units of the third order ; and
so on for the subsequent divisions.

Language for Is there anv arithmetical language by which
fractional tnese fractional units may be expressed ? The

UllltS.

decimal point, which is merely a dot, or period,
Whatitfixes. indicates the division of the unit 1, according to

the scale of tens. By the arithmetical language,
Kames of the the unit of the place next the point, on the right,

places.

is 1 tenth ; that of the second place, 1 hun-
dredth ; that of the third, 1 thousandth ; that of
the fourth, 1 ten thousandth; and so on for
places still to the right.
*> The scale for decimals, therefore, is

.111111111, &c.j

in which the value of the unit of each place is
known as soon as we have learned the significa-
tion of the language.

If, therefore, we wish to express any of the
parts into which the unit 1 may be divided, ac-
cording to the scale of tens, we have simply to

Any decimal r

number may se l ec t from the alphabet, the figure that will

l>e expressed

e. express the number of parts, and then write it in

CHAP. II.] ARITHMETIC FRACTIONS. 157

the place corresponding to the order of the unit, where any

figure is

Thus, to express four tenths, three thousandths, written,
eight ten-thousandths, and six millionths, we
write

.403806 J Example.

and similarly, for any decimal which can be
named.

165. It should be observed that while the
units of place decrease, according to the scale of
tens, from left to right, they increase according The units in

cicas6 from

to the same scale, from right to left. This is the right to left
same law of increase as that which connects the
units of place in simple numbers. Hence, simple consequence
numbers and decimals being formed according to
the same law, may be written by the side of each
other and treated as a single number, by merely
preserving the separating or decimal point.
Thus, 8974 and .67046 may be written

8974.67046 ;

since ten units, in the place of tenths, make the
uni* one in the place next to the left.

FRACTIONAL UNITS IN GENERAL.

166. If the unit 1 be divided into two equal
parts, each part is called a half. If it be divided

159

MATHEMATICAL, SCIENCE. [BOOK II.

A third, into three equal parts, each part is called a third :
if it be divided into four equal parts, each part is
A fourth, called a fourth: if into five equal parts, each
A fifth, part is called a fifth ; and if into any number of
equal parts, a name is given corresponding to the
number of parts.

Now, although these halves, thirds, fourths,
fifths, &c., are each but parts of the unit 1, they
are, nevertheless, in themselves, whole things.
That is, a half is a whole half; a third, a whole
third ; a fourth, a whole fourth ; and the same
for any other equal part of 1. In this sense,
therefore, they are units, and we call them frac-
Hare a rei- tional units. Each is an exact part of the unit

lion to unity.

1, and has a fixed relation to it.

Generally.

These units

are whole

things.

Examples.

Language for
fractions.

To express

the number

of equal

pert*.

167. Is there any arithmetical language by
which these fractional units can be expressed ?

The bar, written at the right, is the
sign which denotes the division of the
u^it 1 into any number of equal parts.

If we wish to express the number of equal
parts into which it is divided, as 9. for j
example, we simply write the 9 under |
the bar, and then the phrase means, that some
thing regarded as a whole, has been divided into
9 equal parts.

CHAP. II. J ARITHMETIC FRACTIONS. 159

If, nj\v, we wish to express any
numbei of these fractional units, as 7,
foi example, we place the 7 above the

To show how

7 many are

9

taken.

168. It was observed,* that two things are TWO things

. necessary to

necessary to the clear apprehension of an mte- apprehend a

number.

gral number.

1st. A distinct apprehension of the unit which First.
forms the basis of the number ; and,

2dly. A distinct apprehension of the number second
of times which that unit is taken.

Three things are necessary to the distinct ap- Three

. necessary to

prehension of the value of any fraction, either apprehend*

- i i fraction.

decimal or vulgar.

1st. We must know the unit, or whole thing,
from which the fraction was derived ;

2d. We must know into how many equal parts second.
that unit is divided ; and,

3dly. We must know how many such parts Third.
are taken in the expression.

The unit from which the fraction is derived, cmt or tno

. Traction l

is called the unit of the fraction ; and one of u, e expre*

sion.

the equal parts is called, the fractional unit.
For example, to apprehend the value of the

* Section 117.

160 MATHEMATICAL. SCIENCE. [iSOOKH

what we fraction f of a pound avoirdupois, or f Ib. ; we

must know.

must know,

First. 1st. What is meant by a pound ;

second. 2d. That it has been divided into seven equaj
parts ; and,

Third. 3d. That three of those parts are taken.

In the above fraction, 1 pound is the unit ot
the fraction ; one-seventh of a pound, the frac-
tional unit; and 3 denotes that three fractional
units are taken.

umt when If the unit of a fraction be not named, it is
taken to be the abstract unit 1.

Every equal 169. By considering every equal part of uni-

partofone,a ... ,., ,

unit ty as a unit m itself, having a certain relation to
the unit 1, the mind is led to analyze a frac-
tion, and thus to apprehend its precise significa-
tion.
Advantages Under this searching analysis, the mind at

of the once seizes on the unit of the fraction as the

analysis. > J

principal base. It then looks at the value of
each part. It then inquires how many such
parts are taken.
Equal unit*, It having been shown that equal integral units

whether In-
tegral or frao can alone be added, it is readily seen that the

same principle is equally applicable to frac- tionai,can
What is necessary in order to make such units
equal ?

It is seen at once, that two things are neces- Two lhi "8

necessary for

1st. That they be parts of the same unit ; and, Fil>t -
2d. That they be like parts; in other words, second.

they must be of the same denomination, and

have a common denominator.

In regard to Decimal Fractions, all that is Decimal

Fractions.

necessary, is to observe that units of the same
value are added to each other, and when the
figures expressing them are written down, they
should always be placed in the same column.

8 170. The great difficulty in the management

the manage-

of fractions, consists in comparing them with ment or free

lions

each other, instead of constantly comparing them
with the unit from which they are derived.
By considering them as entire things, having a
fixed relation to the unit which is their base, obviated,
they can be compared as readily as integral num-
bers; for, the mind is never at a loss when it
apprehends the unit, the parts into which it is

Reasons fot

divided, and the number of parts which are greater *in>

plicity in

taken. The only reasons why we apprehend and integers.

11

162 MATHEMATICAL SCIENCE. [BOOK II.

handle integral numbers more readily than frac-
tions, are,
First l g k Because the unit forming the base is

always kept in view ; and,

Second. 2d. Because, in integral numbers, we have
been taught to trace constantly the connection
between the unit and the numbers which come
from it ; while in the methods of treating frac-
tions, these important considerations have been
neglected.

SECTION III.

PROPORTION AND RATIO.

proportion i7i. PROPORTION expresses the relation which

defined.

one number bears to another, with respect to its
being greater or less.
TWO ways of Two numbers may be compared, the one with

comparing.

the other, in two ways :
1st method. 1st. With respect to their difference, called

Arithmetical Proportion ; and,
w method. 2d. With respect to their quotient, called

Geometrical Proportion.

CHAP. II. J ARITHMETIC PROPORTION. 163

Thus, if we compare the numbers 1 and 8, Example or
by their difference, \ve find that the second ex- ^p^io^
ceeds the first by 7 : hence, their difference 7,
is the measure of their arithmetical proportion,

J Arithmetical

and is called, in the old books, their arithmetical Ratio.
ratio.

If we compare the same numbers by their E^p^of
quotient, we find that the second contains the Geometrical

Proportion.

first 8 times : hence, 8 is the measure of their
geometrical proportion, and is called their geo-
metrical ratio.*

8 172. The two numbers which are thus corn-
Terms.

pared, are called terms. The first is called the Antecedent.
antecedent, and the second the consequent. consequent.

In comparing numbers with respect to their comparison
difference, the question is, how much is one :
greater than the other ? Their difference affords
the true answer, and is the measure of their pro-
portion.

In comparing numbers with respect to their comparison

, by quotient.

quotient, the question is, how many times is one
greater or less than the other ? Their quotient
or ratio, is the true answer, and is the measure

* The term ratio, as now generally used, means the quo-
dent arising from dividing one number by another. We
shall use it only in this sense.

164 MATHEMATICAL. SCIENCE. [BOOK n -

Example by of their proportion. Ten, for example, is 9

difference. . ,

greater than 1, it we compare the numbers one

and ten by their difference. But if we compare

By quotient, them by their quotient, ten is said to be ten

"Ten times." times as great the language "ten times" having

reference to the quotient, which is always taken

as the measure of the relative value of two

Examples of numbers so compared. Thus, when we say,

thisuseofthe t k at ^ un j ts o f our common system of numbers

term.

increase in a tenfold ratio, we mean that they so

increase that each succeeding unit shall contain

the preceding one ten times. This is a conven-

convenient ient language to express a particular relation of

language. two numDers> an( j [ s perfectly correct, when

used in conformity to an accurate definition.

in what 173. All authors agree, that the measure of
agreeT" tne geometrical proportion, between two num-
bers, is their ratio ; but they are by no means
in what disa- unanimous, nor does each always agree with
gree> himself, in the manner of determining this ratio.
Some determine it, by dividing the first term by
me- the second ; others, by dividing the second term

by the first.* All agree, that the standard, what-

SUuklard the

ever it may be, should be made the divisor.

* The Encyclopedia Metropolitans, a work distinguished
by the excellence of its scientific articles, adopts the latter
method.

CHAP. II.J ARITHMETIC KAT1O. 165

This leads us to inquire, whether the mind what is the

f. ,., ~ best form.

axes most readily on the first or second number
as a standard ; that is, whether its tendency is
to regard the second number as arising from the
first, or the first as arising from the second.

174. All our ideas of numbers begin at origin of

numbers.

one.* This is the starting-point. We con-
ceive of a number only by measuring it with Ho w we coo
one, as a standard. One is primarily in the ^^er*
mind before we acquire an idea of any other
number. Hence, then, the comparison begins ^-here the
at one, which is the standard or unit, and all c ^^'
other numbers are measured by it. When, there-
fore, we inquire what is the relation of one to
any other number, as eight, the idea presented The idea
is, how many times does eight contain the stand- P reaented -
ard?

We measure by this standard, and the ratio is standard.

Ratio.

the result of the measurement. In this view of

the case, the standard should be the first number What **

should be.

named, and the ratio, the quotient of the second
number divided by the first. Thus, the ratio of
2 to 6 would be expressed by 3, three being the
number of times which 6 contains 2.

* Section 111.
11

166 MATHEMATICAL SCIENCE. [BOOK. II

other reasons 175. The reason for adopting this method

for this me- .

ihodofcom- ol comparison will appear still stronger, it we
take fractional numbers. Thus, if we seek the
relation between one and one-half, the mind im-
mediately looks to the part which one-half is of

Comparison one> an( j this is determined by dividing one-half

of unity with

fractions, by 1 ; that is, by dividing the second by the
first : whereas, if we adopt the other method,
we divide our standard, and find a quotient 2.

Geometrical g 176. It may be proper here to observe, tnat

proportion.

while the term "geometrical proportion is used

to express the relation of two numbers, com-

Ageometri- pared by their ratio, the term, "A geometrical

Cftl proper-

tion defined, proportion," is applied to four numbers, in which
the ratio of the first to the second is the same as
that of the third to the fourth. Thus,

Example. 2 : 4 :: 6 : 12,

is a geometrical proportion, of which the ratio
is 2.

vantages which result from regarding the ratio
as the quotient of the second term divided by
the first.

Questions in Every question in the Rule of Three is a

the Rule of

Three- geometrical proportion, excepting only, that the

CHAP. 11. J ARITHMETIC RATIO. 167

last term is wanting. When that term is found, Their mture.

the geometrical proportion becomes complete.

In all such proportions, the first term is used as

the divisor. Further, for every question in the

Rule of Three, we have this clear and simple

solution : viz. that, the unknown term or an- HOW solved.

swer, is equal to the third term multiplied by

the ratio of the first two. This simple rule, for

finding the fourth term, cannot be given, unless Thisrule<le -

pends on the

we define ratio to be the quotient of the second definition oi

Ratio.

term divided by the first. Convenience, there-
fore, as well as general analogy, indicates this as
the proper definition of the term ratio.

178. Again, all authors, so far as I have
consulted them, are uniform in their definition '
of the ratio of a geometrical progression : .viz.
that it is the quotient which arises from divid-
ing the second term by the first, or any other
term by the preceding one. For example, in
the progression

2 : 4 : 8 : 16 : 32 : 64, &c., E^^

all concur that the ratio is 2 ; that is, that it is in which
the quotient which arises from dividing the sec- \^e c .
ond term by the first : or any other term by the
preceding term. But a geometrical progression
differs from a geometrical proportion only in

168

MATHEMATICAL SCIENCE. [BOOK II

The same this : in the former, the ratio of any two term?

should lake

place in every is the same ; while in the latter, the ratio ot tne

a^ fi rst an d se cond is different from that of the sec-
mi the tame. ond and tnird There is, therefore, no essential
difference in the two proportions.

Why, then, should we say that in the propor-

tion

2 : 4 :: 6 : 12,

Examples.

Wherein

the ratio is the quotient of the first term divided
by the second ; while in the progression

2 : 4 : 8 : 16 : 32 : 64, &c.,

the ratio is defined to be the quotient of the sec-
ond term divided by the first, or of any term di-
vided by the preceding term ?

As far as I have examined, all the authors
wno have defined the ratio of two numbers to
ed from their be the quot i ent o f the first divided by the sec-

deflnitions : J

ond, have departed from that definition in the
case of a geometrical progression. They have
HOW used there used the word ratio, to express the quo-
tient of the second term divided by the first,
and this without any explanation of a change
in the definition.

Most of them have also departed from then
definition, in informing us that " numbers in-
crease f rom r jght to left in a tenfold ratio," in

other in-

CHAP. II.] ARITHMETIC PROPORTION. IG'J

which the term ratio is used to denote the quo- Ratio a not

tient of the second number divided by the first.

The definition of ratio is thus departed from,

and the idea of it becomes confused. Such consequen-

discrepancies cannot but introduce confusion

into the minds of learners. The same term

should always be used in the same sense, and

have but a single signification. Science does what science

demands.

not permit the slightest departure from this rule.
I have, therefore, adopted but a single significa-
tion of ratio, and have chosen that one to which Thedeflm-

all authors, so tar as 1 know, have given their
sanction ; although some, it is true, have also
used it in a different sense.

179. One important remark on the subject importam

,. . . T i Remark.

of proportion is yet to be made. It is this :

Any two numbers which are compared togeth- Number)

T/T comparrd

er, either by their difference or quotient, must must be of

be of the same kind: that is, they must either kind '
have the same unit, as a base, or be susceptible
of reduction to the same unit.

For example, we can compare 2 pounds with Example*

6 pounds : their difference is 4 pounds, and their Arithmetics

ratio is, the abstract number 3. We can also
compare 2 feet with 8 yards : for, although the Uon '
unit 1 foot is different from the unit 1 yard, still
6 yards are equal to 24 feet. Hence, the differ-

170 MATHEMATICAL SCIENCE. [ HOOK II.

ence of the numbers is 22 feet, and their ratio
the abstract number 12.

Q n the other hand, we cannot compare 2 dol-

with (liffwut

units cannot lurs with 2 yards of cloth, for they are quantities

be compared. , . , ,

ot dinerent kinds, not being susceptible of reduc-
tion to a common unit.
Abstract Abstract numbers may always be compared,

numbers may

be compared, since they have a common unit 1.

SECTION IT.

APPLICATIONS OF THE SCIENCE OF ARITHMETIC.

180. ARITHMETIC is both a science and an
Arithmetic: art. It is a science in all that relates to the

In what a

science, properties, laws, and proportions of numbers.

The science is a collection of those connected

science de- processes which develop and make known the

fined*

laws that regulate and govern all the operations
performed on numbers.

181. Arithmetic is an art, in this : the sci-
ence lays open the properties and laws of num-
bers, and furnishes certain principles from which

CHAP. II.] A HIT II M I'. Tl r A IT I.IC A T I <> N h I.I

practical and useful rules arc formed, a|>|>lira.U-
in the iiii-cliaiiic. arts and in business transac-
tions. The art of Arithmetic consists in the it.
judicious and ski.ful application oi the princi-
ples of the science ; and the rules contain the
directions for such application.

182. In explaining the science of Arithmetic, in explaining
great care should be taken that the analysis of what ..*
every question, and the reasoning by which the
principles are proved, be made according to the
strictest rules of mathematical logic.

Every principle should be laid down and ex- HOW each
plained, not only with reference to its subsequent ^ "kn*,
use and application in arithmetic, but also, with stated
reference to its connection with the entire mathe-
matical science of which, arithmetic is the ele-
mentary branch.

183. That analysis of questions, therefore, what
where cost is compared with quantity, or quan- ^^"y.
tity with cost, and which leads the mind of the
learner to suppose that a ratio exists between
quantities that have not a common unit, is, with-
out explanation, certainly faulty as a process of
science.

For example : if two yards of cloth cost 4 dol-

Example.

lars, what will 6 yards cost at the ;ame rate ?

"

172 MATHEMATICAL SCIENCE. [BOOK II

Analysis: Analysis. Two yards of cloth will cost twice
as much as 1 yard : therefore, if two yards of
cloth cost 4 dollars, 1 yard will cost 2 dollars.
Again : if 1 yard of cloth cost 2 dollars, 6 yards,
being six times as much, will cost six times two
dollars, or 12 dollars,
satisfactory Now, this analysis is perfectly satisfactory to

to a child. . ., , .. . , . ,

a child. He perceives a certain relation between
2 yards and 4 dollars, and between 6 yards and
12 dollars : indeed, in his mind, he compares
these numbers together, and is perfectly satisfied
with the result of the comparison.

Advancing in his mathematical course, how-
ever, he soon comes to the subject of propor-
tions, treated as a science. He there finds,
Reason why greatly to his surprise, that he cannot compare

k is defective.

together numbers which have different units ;
and that his antecedent and consequent must be
of the same kind. He thus learns that the whole
system of analysis, based on the above method of
comparison, is not in accordance with the prin-
ciples of science.
True What, then, is the true analysis ? It is this :

analysis : i / i i i

6 yards of cloth being 3 times as great as 2
yards, will cost three times as much : but 2 yards
cost 4 dollars ; hence, 6 yards will cost 3 times
4, or 12 dollars. If this last analysis be not

More scien-

tiflc - as simple as the first, it is certainly mote strictly

CHAP. II.J ARITHMETIC APPLICATIONS. 173

scientific ; and when once learned, can be ap- iu
plied through the whole range of mathematical
science.

184. There is yet another view of this ques- Reasons in

.... , .- favor of the

tion which removes, to a great degree, if not

entirely, the objections to the first analysis. It is
this:

The proportion between 1 yard of cloth and
its cost, two dollars, cannot, it is true, as the
units are now expressed, be measured by a ratio,
according to the mathematical definition of a
ratio. Still, however, between 1 and 2, regard-
ed as abstract numbers, there is the same relation N Umber8
existing as between the numbers 6 and 12, also mustbere -

garded as ah-

regarded as abstract. Now, by leaving out of 8tract:
view, for a moment, the units of the numbers,
and finding 12 as an abstract number, and then the analyst*
assigning to it its proper unit, we have a correct
analysis, as well as a correct result.

185. It should be borne in mind, that practi- HOW the

. . rules of arith

cal arithmetic, or arithmetic as an art, selects meticare
from all the principles of the science, the mate-
rials for the construction of its rules and the
proofs of its methods. As a mere branch of What
practical knowledge, it cares nothing about the P"**' * 1

knowledge

forms or methods of investigation it demands

J74 MATHEMATICAL BCIEKCE. [jJOOK U.

the fruits of them all, in the most concentrated

Be* raie cf and practical form. Hence, the best rule of art,

" L which is the one most easily applied, and which

reaches the result by the shortest process, is not

always constructed after those methods which

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