Intellectual Arithmetic, or, an Analysis of the Science of Numbers, with Verbal Reference to Mental Training and Development

Front matter

ARITHMETI C,

OB,

AN ANALYSIS OF THE SCIENCE OF NUMBERS* i

Introduction

Every book of instruction should have a specific object to which the entire work, both in matter and method, should strictly conform.

It is the object of this book to train and develop the mind by means of the science of numbers. Numbers are the instruments here employed to strengthen the memory, to cultivate the faculty of abstraction and to give force and vigor to the reasoning powers.

All our ideas of numbers are either of unity or of multiplicity — unity being the elementary idea from which all others are derived.

A true analysis must conform to the nature of the subject analyzed. It must separate all the ideas and principles into their primary elements, and then explain and make manifest the laws by which these elements are connected with each other. Hence, the analysis of numbers must begin with the unit 1, — for this is the foundation, and the science is but the development of the various processes by which all other numbers are derived from 1, as a base, and a comparison of the base 1, with the numbers so derived.

Every number has what we call a base: that is, “ number being a collection of things of the same kind,” one of these things is the base of the number ; and this thing , is called a unit . If we have the mini-

IV INTRODUCTION.

ber 3 hundred, we may consider it in several points of view :

1 st. It is one hundred taken 3 times, and if we regard one hundred as the base, then, the base is taken 3 times to make up the number; and 100 is the unit.

2nd. We may consider the number as made up cf 30 tens, and if we regard 10 as the base, then the base is taken 30 times ; and 10 is the unit.

3rd. W e may also consider the number as made up of 300 ones, in which, the base is 1, and the unit of the number 1.

Again, if we analyze the number,

cwt. qr. lb. oz. dr.

13 2 20 12 4

We see, that 1 cwt. is the base of \3cwt. ; lqr. the base of 2 qr. ; 1 lb. the base of 20 lb. ; 1 oz. the base of 12oz. ; and dr. the base of 4 dr., and all these bases may be referred to 1 dram as a primary base ; hence, as in simple numbers, every base may be referred to the unit 1 : therefore, in every entire number , 1 is the primary base.

Let us see if the same be true in fractional numbers.

If we have the fraction it denotes :

1st. That something regarded as a whole has been divided into 8 equal parts : and,

2nd. That 7 of these parts are taken.

In this collection of 7 things, (each of which is -l), £ is the base of the fractional number ; but it is not the primary base ; for ± implies, either | of 1 or ol some collection of l’s; if a collection of l’s we call tnat collection unity , which may be referred to the primary base 1 : hence, every number , either integral or fractional , has the unit 1 for a primary base.

A fractional number, therefore, is merely a collec-

INTRODUCTION.

V tion of the Qqual parts of unity, and to one of these parts we give the name of fractional unit . The unit which is divided is called the unit of the fraction , and may be a collection of units, (as what is % of 40 ?) or it may be the unit 1.

The term unity, in mathematical science, is applied to any number or quantity regarded as a whole : the term unit, in arithmetic, to any number which is used as the base of a collection. Thus, 10 is a unit of the second order, being the base for the collection of 1 0’s 100 is a unit of the third order, being the base for the collection of hundreds, and similarly for other bases. Thus, also, in the fraction ^ is the fractional unit, being the fractional base, while the primary base is the unit 1.

Every arithmetical process, therefore, has a direct reference to the unit 1 ; and with this view of the subject before him, the pupil always has the means of making a correct analysis.

Addition is the process of finding a number which shall contain as many units, and no more, as are found in all the numbers added. Multiplication is taking one number, called the multiplicand, as many times as there are units in another number, called the multiplier, and the number which shows the result of such taking, is called the product : and similarly for all other arithmetical processes.!

A clear conception of elementary principles, by which we mean, those principles that result from a final analysis, lies at the foundation of all knowledge, [t is not till we get such conceptions, and have learned the laws by which they are connected, that we have acquired any thing deserving the name of science.

1*

VI

INTRODUCTION.

To learn one thing at a time — to learn that thing thoroughly — and to learn its connections with other things are the golden steps that lead to the temple of knowledge.

it will be seen, in Lessons XVI. and XVII., Section VII., that unity has been employed to denote any number entering into an arithmetical question. This use of unity affords a powerful means of solving most questions which otherwise present great difficulties ; and is, it is believed, a link of closer connection between the subjects of arithmetic and algebra, than has before been used.

It has been the author’s aim, in the present work, to treat the subject of number in accordance with these principles, and to give to the wdiole a scientific form, and logical development. That he might not fail in so difficult and delicate an undertaking, he has defined all the terms, and given a full analysis of every process employed.

The work is complete in itself. It is a mental ana¬ lysis of the science of numbers, designed to be accessible to the youngest pupils because of its simple gradations, and useful to the advanced pupil because of its scientific arrangement, its logical connections and its higher analysis of the properties and relations of numbers.

In the preparation of this work, many valuable suggestions and methods have been furnished by practical teachers. They were cheerfully offered and thankfully adopted.

JFisiikill Landing, February , 181)4

INTELLECTUAL ARITHMETIC.

SECTION FIRST. LESSON i

Counting.

Three, . ###

Eight, .

Suggestions. — There is but one simple idea in Arithmetic— -it is the idea cf the unit one.

Any collection of units is a number. Hence, every number is derived from one, and consequently has one for a base. Count¬ ing is merely naming numbers. In this lesson, the names of numbers are written opposite the collection.

How many units in four ? In six apples, what is the unit f What is the unit in seven pears ? How many units in twelve peaches t What is the unit that is counted in the lesson ? How many stars in the 4th line ? How many in the 14tli ?

From what are all numbers derived ? What is the base of every number ?

B

INTELLECTUAL ARITHMETIC.

[bec. L

(j

S

15 10

LESSON II.

Figures f rom One to Twenty.

. . .

. m mm m

. mmmmm

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. . mmmmmmm

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. mmmmmmmmm

. mmmmmmmmmm

. mmmmmmmmmmm

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. mm mm mm mm mm mm mm

. %mmmm mm mm mmmmmm

. . . . %%%mmmmmmmmmmmmm

• . •mmmmmmmmmmmmmmmmm

Which figure stands for two ? Which figure stands for four? Which figure stands for nine? Which stands for eight? What stands for ten? What stands for twelve ? What stands for fourteen ? What stands for sixteen ? What stands for eighteen ? What stands for twenty ? What stands for seven¬ teen ? What stands for fifteen ? What stands for nineteen ? What stands for thirteen?

Suggestions. — This lesson is intended to teach that number* may be expressed by figures, as well as by words. The teacher should explain to the pupil that the figure 2 and the word two, have the same meaning, and similarly for every figure and its corresponding' word. Either the figure or the word, denotes as many units as its name points out.

LES. III.] INTELLECTUAL ARITHMETIC.

LESSON III.

Figures from One to One Hundred.

Naught One" .

Two .

Three .

Four .

Five .

Six . .

Seven .

Eight .

Nine .

Ten .

Eleven Twelve .Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty .

Twenty-one . 21 Twenty-two . 22 Twenty-three 23 Twenty-four. 24 Twenty-five . 25 Twenty-six . 26 Twenty-seven 27 Twenty-eight 28 Twenty -nine 29 Thirty . .30

Thirty-one . 31 Thirty-two . 32 Thirty- three . 33

Thirty-four .

Thirty -five .

Thirty-six

Thirty -seven

Thirty-eight .

Thirty -nine .

Forty .

Forty -one .

Forty- two

Forty-three .

Forty-four .

Forty-five

Forty-six. .

Forty-seven .

Forty-eight .

Forty-nine .

Fifty . . .

Fifty -one . .

Fifty -two

Fifty-three .

Fifty -four

Fifty -five

Fifty-six . .

Fift y -seven .

Fifty-eight .

Fifty-nine

Sixty .

Sixty-one

Sixty-twro

Sixty-three .

Sixty-four

Sixty-five

Sixty-six .

Sixty-seven .

Sixty-eight . 68 Sixty -nine . 69 Seventy . .70

Seventy-one . 71 | Seventy-two . 72 ; Seventy-three 73 ! Seventy -four 74 | Seventy -five . 75 Seventy -six . 76 Seventy -seven 77 Seventy -eight 78 Seventy-nine 79 Eighty . . 80

Eighty-one . 81 Eighty -two . 82 Eighty-three 83 Eighty-four . 84 Eighty-five . 85 Eighty-six . 86 Eighty-seven 81 Eighty-eight 88 Eighty -nine . 89 Ninety . . 90

Ninety-one . 91 Ninety-two . 92 Ninety-three 93 Ninety -four . 94 Ninety-five . 95 Ninety-six . 96 Ninety-seven 97 Ninety -eight 98 Ninety-nine . 99 One hundred 100 Two hundred 200

. 0 . 1

. 2

o

. o

. 4

Suggestion. — This lesson is intended to teach that the words and the figures are simply two forms of language.

10 INTELLECTUAL ARITHMETIC. [SEG. i.

LESSON IV.

Roman Table.

. One

XX . .

Twenty

. Eighteen

DCCCC

Nine hundred