be brought to conform to any one of these
Apropos!- tests, it is regarded as proved, and declared to
tlon : when
proved. D6 true.
CHAP. I.] MIXED MATHEMATICS. 115
103. When general formulas have been When a
framed, determining the limits within which may be re-
the deductions may be drawn (that is, what ^proved* 8
shall be the tests of truth), as often as a new
case can be at once seen to come within one
of the formulas, the principle applies to the new
case, and the business is ended. But new cases Trains of
are continually arising, which do not obviously whynecesl
come within any formula that will settle the
questions we want solved in regard to them,
and it is necessary to reduce them to such for-
mulas. This gives rise to the existence of the They give
. -I i i L ri^e to the
science ot mathematics, requiring the highest science of
scientific genius in those who contributed to its
creation, and calling for a most continued and
vigorous exertion of intellect, in order to appro-
priate it, when created.
MIXED MATHEMATICS.
104. The Mixed Mathematics embraces the Mixed
applications of the principles and laws of the tice.
Pure Mathematics to all investigations in which
the mathematical language is employed and to
the solution of all questions of a practical na-
ture, whether they relate to abstract or concrete
quantity.
114 MATHEMATICAL SCIENCE. [BOOK II.
QUANTITY MEASURED.
Quantity. 105. Quantity has been defined, " any thing
which can be increased, diminished, and meas-
increased ure d." The terms increased or diminished, are
and
diminished, easily understood, implying merely the property
defined.
of being made larger or smaller. The term
measured is not so easily comprehended, because
it has only a relative meaning.
Measured. The term "measured," applied to a quantity,
implies the existence of some known quantity
what H of the same kind, which is regarded as a stand-
ard. With this standard, the quantity to be meas-
ured is compared with respect to its extent or
standard: magnitude. To such standard, whatever it may
is called be, we give the name of unity, or unit of
measure; and the number of times which any
quantity contains its unit of measure, is the
numerical value of the quantity measured. The
Magnitude: extent or magnitude of a quantity is, therefore,
merely rela-
Uve . merely relative, and hence, we can form no idea
of it, except by the aid of comparison. Space,
space: for example, is entirely indefinite, and we meas-
indefinite.
ure parts of it by means of certain standards,
Measure- called measures ; and after any measurement is
ment ascer-
tains reia- completed, we have only ascertained the relation
tion:
or proportion which exists between the stand-
ard we adopted and the thing measured. Hence,
CHAP. I.] QUANTITY MEASURED. 115
measurement is, after all, but a mere process A process of
comparison.
of comparison.
106. The quantities, Number and Space, are but Number and
vague and indefinite conceptions, until we compare knowiTby
them with their units of measure, and even these compari
units are arrived at only by processes of comparison.
Comparison is the great means of discovering the comparison
relations of things to each other, as well in general ^thod.
logic, as in the science of mathematics, which
develops the processes by which quantities are
compared, and the results of such comparisons.
107. Besides the classification of quantity Quantity,
into Number and Space, we may, if we please,
divide it into Abstract and Concrete. An ab- Abstract,
stract quantity is a mere number, in which the
unit is not named, and has no relation to mat-
ter or to the kind of things numbered. A con- concrete,
crete quantity is a definite object or a collec-
tion of such objects. Concrete quantities are
expressed by numbers and letters, and also by Howrepre-
lines, straight and curved. The number " three"
Example
is entirely abstract, expressing an idea having ofthe
no connection with things; while the number
"three pounds of tea," or "three apples," pre- Example
sents to the mind an idea of concrete objects.
So, a portion of space, bounded by a surface, all
116 MATHEMATICAL SCIENCE. [BOOK II.
the points of which are equally distant from a
sphere certain point within called the centre, is but a
mental conception of form ; but regarded as a
defined, portion of space, gives rise to the additional idea
of a named and defined thing.
COMPARISON OF QUANTITIES.
Mathematics 108. We have seen that the pure mathe-
a . matics are concerned with the two quantities,
Number and Space. We have also seen, that rea-
Reasoning soning necessarily involves comparison : hence,
comparison, mathematical reasoning must consist in com-
paring the quantities which come from Number
and Space with each other.
TWO quanti- 19. Any two quantities, compared with
ties can eus- g^jj o flj er must necessarily sustain one of two
tain but two *
relations, relations: they must be equal, or unequal. What
axioms or formulas have we for inferring the
one or the other?
AXIOMS FOR INFERRING EQUALITY.
1. Quantities which contain the same unit an
Formula* equal number of times, are equal.
Equality ^ g Things which being applied to each other
coincide, are equal in all their parts.
CHAP. I.] COMPARISON OP QUANTITIES. 117
3. Things which are equal to the same thing
are equal to one another.
4. A whole is equal to the sum of all its parts
5. If equals be added to equals, the sums are
equal.
6. If equals be taken from equals, the remain-
ders are equal.
AXIOMS FOB INFERRING INEQUALITY.
1. A whole is greater than any of its parts.
2. If equals be added to unequals, the sums Formniat
for
are unequal. inequality.
3. If equals be taken from unequals, the re-
mainders are unequal.
110. "We have thus completed a very brief what fea-
tures hY8
and general analytical view of Mathema- been
tical Science. We have named and defined
the subjects of which it treats and the forms
of the language employed. We have pointed
out the character of the definitions, and the na-
ture of the elementary and intuitive proposi-
tions on which the science rests; also, the kind
of reasoning employed in its creation, and its
divisions resulting from the use of different
symbols and differences of language. We shall
now proceed to treat the brandies separately.
rHAP. II.] ARITHMETIC FIRST NOTIONS. 1 A 9
CHAPTER II.
ARITHMETIC SCIENCE AND ART OF NUMBERS.
SECTION I.
1MTIBKAL UNITS.
FIRST NOTIONS OF NUMBERS.
111. THERE is but a single elementary idea But one ei
mentary idea
in the science of numbers : it is the idea of the in numbers.
UNIT ONE. There is but one way of impressing Howim-
this idea on the mind. It is by presenting to Remind!
the senses a single object; as, one apple, one
peach, one pear, &c.
112. There are three signs by means of Three dgm
for express-
which the idea of one is expressed and commu- ing it.
nicated. They are,
1st. The word ONE. A word
2d. The Roman character 1. Roman
o, m, character-
3d. The figure 1.
MATHEMATICAL SCIENCE [BOOK II
New ideas 113. If one be added to one, the idea thus
b> adding arising is different from the idea of one, and is
one- complex. This new idea has also three signs ;
viz. TWO, II., and 2. If one be again added,
that is, added to two, the new idea has likewise
three signs; viz. THKEE, III., and 3. These
Collections collections, and similar ones, are called num-
are num-
ber, bers. Hence,
A NTJMBEE is a unit or a collection of units.
IDEAS OF NUMBERS GENERALIZED.
ideas or 114. If we begin with the idea of the num-
numbers ,
generalized. e r on e, and then add it to one, making two ;
and then add it to two, making three ; and then
to three, making four ; and then to four, making
HOW formed, five, and so on ; it is plain that we shall form a
series of numbers, each of which will be greater
unity the by one than that which precedes it. Now, one
or unity, is the basis of this series of numbers,
Three ways
xpressing and each number may be expressed in three
them.
ways :
1st way. 1st. By the words ONE, TWO, THREE, &c., of our
common language ;
wway. 2d. By the Roman characters; and,
ad way. 3d. By figures.
CHAP. II.] ARITHMETIC UNITY. 121
115. Since all numbers, whether integral or
fractional, must come from, and hence be con- ""^e" 1
nected with, the unit one, it follows that there
is but one purely elementary idea in the science
of numbers. Hence, the idea of every number, Hence but
regarded as made up of units (and all numbers u purely eu>-
except one must be so regarded when we ana-
lyze them), is necessarily complex. For, since AU other
the number arises from the addition of ones, the nollonsare
complex.
apprehension of it is incomplete until we under-
stand how those additions were made ; and there-
fore, a full idea of the number is necessarily
complex.
116. But if we regard a number as an en-
tirety, that is, as an entire or whole thing, as an
entire two, or three, or four, without pausing to when a
f t- i . . number may
analyze the units of which it is made up, it may ^ regarded
then be regarded as a simple or incomplex idea ; Mincom P lex -
though, as we have seen, such idea may always
be traced to that of the unit one, which forms
the basis of the number.
UNITY AND A UNIT DEFINED.
117. When we name a number, as twenty
feet, two things are necessary to its clear appre- ceasar y tothe
apprehenskm
of a numbci
MATHEMATICAL SCIENCE. LBOOK U
First. 1st. A distinct apprehension of the single-
thing which forms the base of the number ; and,
second. gd. A distinct apprehension of the number of
times which that thing is taken.
The basis of The single thing, which forms the base of the
the number i 11 j T^ 11 i
M i-MTv. number, is called UNITY, or a UNIT. It is called
when u is unity, when it is regarded as the primary base
called UNITY, O f tne num ber ; that is, when it is the final stand-
ard to which all the numbers that come from it
andwhena are referred. It is called a unit when it is re-
NIT ' garded as one of the collection of several equal
things which form a number. Thus, in the ex-
ample, one foot, regarded as a standard and the
base of the number, is called UNITY ; but, con-
sidered as one of the twenty equal feet which
make up the number, it is called a UNIT.
OF SIMPLE AND DENOMINATE NUMBERS.
Abstract 118. A simple or abstract unit, is ONE, with-
""' out regard to the kind of thing to which the term
one may be applied.
Denominate -A- denominate or concrete unit, is one thing
miu named or denominated ; as, one apple, one peach,
one pear, one horse, &c.
Number has 119, Number, as such, has no reference
no reference . ,
to the particular things numbered. But to dis-
CHAP. II. ARITHMETIC - ALPHABET. 123
tinguish numbers which are applied to particular to the thmgt
., i i 11 numbered.
units from those which are purely aostiact, we
call the latter Abstract or Simple Numbers,
and
and the former Concrete or Denominate Num- Denominate,
bers. Thus, fifteen is an abstract or simple
number, because the unit is one; and fifteen Examples,
pounds is a concrete or denominate number,
because its unit, one pound, is denominated or
named.
ALPHABET WORDS GR AM MAR.
120. The term alphabet, in its most general Alphabet,
sense, denotes a set of characters which form
the elements of a written language.
When any one of these characters, or any
combination of them, is used as the sign of a
distinct notion or idea, it is called a word ; and
the naming of the characters of which the word
is composed, is called its spelling.
Grammar, as a science, treats of the estab-
lished connection and relation of words, as the
signs of ideas.
ARITHMETICAL ALPHABET.
121. The arithmetical alphabet consists of
Alphabet
ten characters, called figures. They are,
Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine,
0123466789
124 MATHEMATICAL SCIENCE. [BOOK 11
and each may be regarded as a word, since it
stands for a distinct idea.
WORDS SPELLING AND READING IN ADDITION.
ono cannot 122. The idea of one, being elementary, the
character 1 which represents it, is also element-
ary, and hence, cannot be spelled by the other
characters of the Arithmetical Alphabet (121).
But the idea which is expressed by 2 comes from
spelling by the addition of 1 and 1 : hence, the word repre-
sented by the character 2, may be spelled by
ical spelling of the word two.
Three is spelled thus : 1 and 2 are 3 ; and
also, 2 and 1 are 3.
Four is spelled, 1 and 3 are 4 ; 3 and 1 are 4 ;
2 and 2 are 4.
Five is spelled, 1 and 4 are 5 ; 4 and 1 are 5 ;
2 and 3 are 5 ; 3 and 2 are 5.
Six is spelled, 1 and 5 are 6 ; 5 and 1 are 6 ;
2 and 4 are 6 ; 4 and 2 are 6 ; 3 and 3 are 6.
AH numbers 123. In a similar manner, any number in
spelled in a arithmetic may be spelled; and hence we see
* that the process of spelling in addition consists
simply, in naming any two elements which will
make up the number. All the numbers in ad-
CHAP. II. J ARITHMETIC READINGS.
dition
are therefore spelled with
two syllables.
The reading consists in naming only
which expresses the final idea. Thus,
the word
Rci ling: in
what it con-
fists.
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
E .mples.
One
two three
four fire
six seven
eight
nine ten.
We
may now
read
the words
which
express
the first
hundred combinations.
READINGS.
Read.
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
T> three,
ft IT, &C.
Th ee, four,
fee.
Four, fir*
4.C.
FIT?, six, t<v
Six, seven,
fee.
Seven, eighU
&C.
Eight, nine,
fcfr
Nine, ten, t*
1
2
2
2
3
2
4
2
5
2
6
2
7
2
8
2
9
2
10
2
1
3
2
3
3
3
4
3
5
3
6
3
7
3
8
3
9
3
10
3
1
2
3
4
5
6
7
8
9
10
1
5
2
5
3
5
4
5
5
5
6
5
7
5
8
5
9
5
10
5
1
6
2
6
3
6
4
6
5
6
6
6
7
6
8
6
9
6
10
6
1
7
2
7
3
7
4
7
5
7
6
7
7
7
8
7
9
7
10
7
1
8
2
8
3
8
4
8
5
8
6
8
7
8
8
8
9
8
10
8
126
MATHEMATICAL SCIENCE.
[BOOK n.
Ten, eleven,
&C.
Eleven,
twelve, &c.
Example for
reading in
Addition.
123456789 '0
9999999999
123456789 10
10 10 10 10 10 10 10 10 10 10
124 In this example, beginning
at the right hand, we say, 8, 17, 18,
26 : setting down the 6 and carry-
ing the 2, we say, 8, 13, 20, 22, 29 :
setting down the 9 and carrying
the 2, we say, 9, 12, 18, 22, 30:
and setting down the 30, we have the e
878
421
679
354
764
3096
ntire sum
AII examples 3096. All the examples in addition may be done
BO solved.
in a similar manner.
Advantages
of reading.
lat. stated.
3d. stated.
125. The advantages of this method of read-
ing over spelling are very great.
1st. The mind acquires ideas more readily
through the eye than through either of the other
senses. Hence, if the mind be taught to appre-
hend the result of a combination, by merely see-
ing its elements, the process of arriving at it is
much shorter than when those elements are pre-
sented through the instrumentality of sound.
Thus, to see 4 and 4, and think 8, is a very dif-
ferent thing from saying, four and four are eight.
2d. The mind operates with greater rapidity
and certainty, the nearer it is brought to the
CHAP. II.] ARITHMETIC WORDS. 127
ideas which it is to apprehend and combine.
Therefore, all unnecessary words load it and
impede its operations. Hence, to spell when
we can read, is to fill the mind with words
and sounds, instead of ideas.
3d. All the operations of arithmetic, beyond M.taiu
the elementary combinations, are performed on
paper ; and if rapidly and accurately done, must
be done through the eye and by reading. Hence
the great importance of beginning early with a
method which must be acquired before any con-
siderable skill can be attained in the use of
figures.
126. It must not be supposed that the read- Reading
comes alter
ing can be accomplished until the spelling has spelling,
first been learned.
In our common language, we first learn the same asm
our common
alphabet, then we pronounce each letter m a language,
word, and finally, we pronounce the word. We
should do the same in the arithmetical reading.
WORDS SPELLING AND READING IN SUBTRACTION.
127- The processes of spelling and reading same
which we have explained in the addition of j n subtree-
numbers, may, with slight modifications, be ap-
plied in subtraction. Thus, if we are to subtract
128 MATHEMATICAL SCIENCE. [BOOK II.
2 from 5, we say, ordinarily, 2 from 5 leaves 3 ;
or 2 from 5 three remains. Now, the word,
three, is suggested by the relation in which 2
and 5 stand to each other, and this word may be.
Readings in read at once. Hence, the reading, in subtrac-
Subtraction . i ji j 7. t
explained. tlon ' ls simply naming the word, which expresses
the difference between the subtrahend and min-
uend. Thus, we may read each word of the
following one hundred combinations.
READINGS.
One from
1
2
3
4
5
6
7
8
9
10
one, tc.
1
1
1
1
1
1
1
1
1
1
Two from
2
3
4
5
6
7
8
9
10
11
two, fee.
2
2
2
2
2
2
2
2
2
2
Three from
3
4
5
6
7
8
9
10
11
12
three, A.C.
3
3
3
3
3
3
3
3
3
3
Four from
4
5
6
7
8
9
10
11
12
13
four, ic.
4
4
4
4
4
4
4
4
4
4
Fire from
5
6
7
8
9
10
11
12
13
14
fire, fee.
5
5
5
5
5
5
5
5
5
5
Sz from six,
6
7
8
9
10
11
12
13
14
15
fee.
6
6
6
6
6
6
6
6
6
6
envoi! from
7
8
9
10
11
12
13
14
15
16
wren, fcc.
7
7
7
7
7
7
7
7
7
7
CHAP. II.] ARITHMETIC SPELLING. 129
8 9 10 11 12 13 14 15 16 17 Eight from
888888888S
9
9
10
9
11
9
12
9
13
9
14
9
15
9
16
9
17
9
18
9
Nine from
niuo, &.C.
10 11 12 13 14 15 16 17 18 19 Tenfromten,
10 10 10 10 10 10 10 10 10 10
128. It should be remarked, that in subtrac-
tion, as well as in addition, the spelling of the spelling pro-
... ,. cedesreading
words must necessarily precede their reading. insubtmc-
The spelling consists in naming the figures with
which the operation is performed, the steps of
the operation, and the final result. The reading Reading,
consists in naming the final result only.
SPELLING AND READING IN MULTIPLICATION.
129. Spelling in multiplication is similar to spelling in
Multiplier
the corresponding process in addition or subtrac- uon.
tion. It is simply naming the two elements
which produce the product ; whilst the reading Reading,
consists in naming only the word which ex-
presses the final result.
In multiplying each number from 1 to 10 by Example* -n
2, we usually say, two times 1 are 2 ; two times
2 are 4 : two times 3 are 6 ; two times 4 are 8 ;
two times 5 are 10; two times 6 are 12; two
9
130 MATHEMATICAL SCIENCE. [BOOK II
times 7 are 14 ; two times 8 are 16 ; two times
fa reading. 9 are 18; two times 10 are 20. Whereas, we
should merely read, and say, 2, 4, 6, 8, 10, 12,
14, 16, 18, 20.
In a similar manner we read the entire mul-
tiplication table.
READINGS.
ODceoneia 12 11 10 987654321
12 11 10 9876 -5 4321
area, tc. Q
Threetimesl 12 11 10 987654321
are 3, Ace. g
Four times i 12 11 10 987654321
are 4, &c.
six times i 12 11 10 9 8 7 6 5 4 3 2 1
are six, Ate.
Eightumesi 12 11 10 987654321
CIIAP. II.J ARITHMETIC READING. 131
12 11 10 9 8 7 6 5 4 3 2 1 Nine time* i
are 9, &c.
12 11 10 987654321 Ten times I
10 awlOjtc.
12 11 10 987654321 Eleven times
|| 1 are 11, ic.
12 11 10 987654321 Twelve times
Iarel2,fcc.
SPELLING AND READING IN DIVISION.
130. In all the cases of short division, the in short um-
. , sion, we may
quotient may be read immediately without nam- read;
ing the process by which it is obtained. Thus,
in dividing the following numbers by 2, we
merely read the words below.
2)4 6 8 10 12 16 18 22
two three four five six eight nine eleven.
In a similar manner, all the words, expressing inaiicmsea.
the results in short division, may be read.
READINGS.
2)2 4 6 8 10 12 14 16 18 20 22 24 TWO in 2,
once, SLC.
3)3 6 9 12 15 18 21 24 27 30 33 36 Three i. 3,
once, fee.
4)4 8 12 16 20 24 28 32 36 40 44 48 Four in 4,
once, lie.
132 MATHEMATICAL SCIENCE. [BOOK 11.
Five in 5, 5)5 10 15 20 25 30 35 40 45 50 55 60
once, &c.
sixine, 6)6 12 18 24 30 36 42 48 54 60 66 72
once, &c.
?eyenin7, 7)7 14 21 28 35 42 49 56 03 70 77 84
once, &.c.
Eight in 8, 8)8 16 24 32 40 48 56 64 72 80 88 96
once, &c.
Nine in a, 9) 9 18 27 36 45 54 63 72 81 90 99 108
once, fee.
Ten in lo, 10)10 20 30 40 50 60 70 80 90 100 110 120
once, &c.
Eleven in ii, 11)11 22 33 44 55 66 77 88 99 110 121 132
once, &.c.
Twelve in 12, 12)12 24 36 48 60 72 84 96 108 120 132 144
once, &c.
UNITS INCREASING BY THE SCALE OF TENS.
The idea of a 131. The idea of a particular number is ne-
particular
number is cessarily complex ; for, the mind naturally asks :
1st. What is the unit or basis of the number ?
and,
2d. How many times is the unit or basis
taken ?
what a fljr- 132. A figure indicates how many times a
we indicates. ... _-.
unit is taken. Each of the ten figures, however
written, or however placed, always expresses as
many units as its name imports, and no more ;
nor does the^ttre itself at all indicate the kind
CHAP. II.] ARITHMETIC SCALE OF TENS. 133
of unit. Still, every number expressed by one or Number has
more figures, has for its base either the abstract taws.
unit one, or a denominate unit.* If a denomi-
nate unit, its value or kind is pointed out either
by our common language, or as we shall present-
ly see, by the place where the figure is written.
The number of units w r hich may be expressed
by either of the ten figures, is indicated by the Number ex-
name of the figure. If the figure stands alone, ^le figure?
and the unit is not denominated, the basis of the
number is the abstract unit 1.
133. If we write on the right of j
10, How ten !
1, We have ) written.
which is read ONE ten. Here 1 still expresses
ONE, but it is ONE ten ; that is, a unit ten times
as great as the unit 1 ; and this is called a unit unit of the
Of the Second Order. second order.
Again ; if we write two O's on the .
i /*/* HOW to write
100,
right Of 1, We have ) one hundred.
which is read ONE hundred. Here again, 1 still
expresses ONE, but it is ONE hundred ; that is, a
unit ten times as great as the unit ONE ten, and A unit of th
a hundred times as great as the unit 1.
134. If three 1's are written by ) uw-wheo
I Til flmir*.* ana
the side of each other, thus - -
Laws wher
1 J J figures are
) written by
the side of
each other.
Section 118.
MATHEMATICAL SCIENCE. [BOOK II.
Fin*.
Second.
Third.
the ideas, expressed in our common anguage,
are these :
1st. That the 1 on the right, will either express
a single thing denominated, or the abstract unit
one.
2d. That the 1 next to the left expresses 1 ten
that is, a unit ten times- as great as thejirst.
3d. That the 1 still further to the left expresses
1 hundred ; that is, a unit ten times as great as
the second, and one hundred times as great as the
Jirst ; and similarly if there were other places.
When figures are thus written by the side of
each other, the arithmetical language establishes
when figures a re } at i on between the units of their places : that
arfi an writ.
is, the unit of each place, as we pass from the
right hand towards the left, increases according
to the scale of tens. Therefore, by a law of the
arithmetical language, the place of a figure fixes
its unit.
If, then, we write a row of 1's as a scale,
thus:
What the
are so wrrt-