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

Front matter

I

INTELLECTUAL

ARITHM E T 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 tilings 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 num*

IV

INTRODUCTION.

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

1st . 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. W e may consider the number as made up of 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 13cwt. ; 1 qr. the base of 2 qr. ; 1 lb. the base of 20 lb. ; 1 oz. the base of 12oz. ; and dr. the base of 4 cfr\, 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 :

ls^. 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 ^), % is the base of the fractional number ; but it is not the primary base ; for implies, either of 1 or f of some collection of l’s; if a collection of l’s we call that 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.

Y tion of the equal 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 10’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 |, J 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. It 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 he 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 whole 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.

Fishkill Landing, February , 1854.

INTELLECTUAL ARITHMETIC.

Section First. Lesson I

Counting.

Suggestions. — There is but one simple idea in Arithmetic — -it is the idea of 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 ? What is the unit in seven pears ? How many units in twelve peaches ? AVhat is the emit that is counted in the lesson ? How many stars in the 4th line ? How many in the 14th ?

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

INTELLECTUAL ARITHMETIC.

[sec. I.

8 9

LESSON II.

Figures from One to Twenty

.

Tjf? ^ ^ W

. ^ ^ ^ TflV "'IF

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 numbers 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.

Figun

Naught . .

One . . .

Two . . .

Three . . .

Four . , .

Five . . .

Seven .

Eight . . .

Nine . . .

Ten . . .

Eleven » .

Twelve . .

Thirteen . .

Fourteen . .

Fifteen . .

Sixteen . .

Seventeen

Eighteen . .

Nineteen . .

Twenty . .

Twenty-one .

Twenty-two .

Twenty-three 23

Twenty-four .

Twenty-five .

Twenty-six .

Twenty-seven 27

Twenty-eight 28

Twenty-nine

Thirty . .

Thirty-one .

Thirty-two .

Thirty- three .

LESSON III. from One to One Iz

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 . .

Fifty-seven .

Fifty-eight .

Fifty-nine

Sixty . . .

Sixty-one

Sixty-two

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 87 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

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

LESSON IV.

Roman Tab'e.

One

XX . .

Twenty

Eighteen

DCCCC

Nine hundred

XIX Nineteen

One thousand

This table is read, one I, one ; two I’s, two ; three I’s, three ; IV, four, &c.

What stands for two? What stands for four? What stands for five ? What stands for eight ? What stands for ten ? WTmt stands for twenty ? What stands for thirty ? What stands for forty ? What stands for fifty? What stands for sixty? What stands for seventy ? What stands for eighty ? What stands for ninety ? What stands for one hun¬ dred ? What stands for five hundred ? What for one- thousand ?

Suggestion. — This lesson is intended to teach that numbers may be expressed by the Roman characters, as well as by words and by figures. Hence, there are three ways of expressing numbers, viz : by words, by figures, and by letters.

LES. V.]

INTELLECTUAL ARITHMETIC.

LESSON Y.

Addition Table from 1 to 3 inclusive.

1. One and one, are how many* ?

2. One and two, are how many '?

3. One and three, are how many ?

4. One and four, are how many ?

5. One and five, are how many ?

6. One and six, are how many ?

7. One and seven, are how many %

8. One and eight, are how many ?

9. One and nine, are how many ?

10. One and ten, are how many %

11. Two and one, are how many ?

12. Two and two, are how many ?

13. Two and three, are how many %

14. Two and four, are how many ?

15. Two and five, are how many?

16. Two and six, are how many ?

17. Two and seven, are how many ?

18. Two and eight, are how many ?

19. Two and nine, are how many ?

20. Two and ten,* are how many ?

21. Three and one, are how many %

22. Three and two, are how many ?

23. Three and three, are how many ?

24. Three and four, are how many ?

25. Three and five, are how many ?

26. Three and six, are how many ?

27. Three and seven, are how many ?

28. Three and eight, are how many ?

29. Three and nine, are how many %

30. Three and ten, are how many %

* Suggestions. — The sum of two or more numbers contains as many units as there are in the numbers added. Thus, 2 is the sum of 1 and 1 ; 4 the sum of 2 and 2, or of 1 and 8.

Addition is the process of finding the sum of two or more numbers. The sign + (plus,) placed between two numbers signifies that they are to be added.

INTELLECTUAL ARITHMETIC.

[sec. I.

QUESTIONS.

1. How many fingers have you on one hand, not counting the thumb ? How many on both hands ?

2. Counting the thumb, how many have you on each hand ? How many on both ?

3. One and four are how many ? One and five ? One and nine ? One and ten ?

4. James has one apple and buys five : how many will he then have ?

5. John has six apples and buys one : how many will he then have ?

6. Charles has nine marbles and John gives him one : how many will he then have ?

8. How many are 2 and 7 ? How many are 2 and 8 ? How many are 2 and 9 ?

9. James has two tops and buys four: how many will he then have? Two and four are how many ?

10. John has two apples and William gives him ' six : how many will he then have ?

11. Bought two quills for two cents, and four quills for four cents : how many quills did I buy ?

12. James bought 2 apples for two cents and 8 more for eight cents: how many did he buy in all? Three and six are how many ?

13. If you buy two peaches for two cents and 9 peaches for nine cents, how many peaches do you buy ? How much do you pay for them ?

15. John has three nuts in one hand and five in the

other : how many in both ? 3 and 8 are how many ?

.16. James has three pencils and John five : how many have both ? 3 and six are how many ?

LES. VI. J

INTELLECTUAL ARITHMETIC.

LESSON VI.

Addition Table from 4 to 6 inclusive.

1. Four and one are how many %

2. Four and two are how many ?

3. Four and three are how many 1

4. Four and four are how many ?

5. Four and five are how many %

6. Four and six are how many ?

7. Four and seven are how many %

8. Four and eight are how many %

Q. Four and nine are how many %

10. Four and ten are how many 1

11. Five and one are how many %

12. Five and two are how many %

13. Five and three and how many 1

14. Five and four are how many ?

15. Five and five are how many ?

16. Five and six are how many 1

17. Five and seven are how many %

18. Five and eight are how many 1

19. Five and nine are how many %

20. Five and ten are how many %

21. Six and one are how many 1

22. Six and two are how many ?

23. Six and three are how many 1

24. Six and four are how many 1

25. Six and five are how many ?

26. Six and six are how many %

27. Six and seven are how many ?

23. Six and eight are how many 1

29. Six and nine are how many 1

30. Six and ten are how many 1

QUESTIONS.

1. John has four tops and Charles one : how many have both %

2. William has four apples and James three : how many have both ?

4. John has four chestnuts in one hand and three in the other : how many has he in both ?

5. Charles has four quills and John seven: how many have both ? Four and 7 are how many ?

6. John and James have each four tops : how many have both ? Four and 9 are how many ?

7. William has four birds in one cage and seven in another: how many in both?

8. Jane has four pins in her cushion and puts in six more : how many will she then have ?

9. Mary has four needles and buys eight : how many will she then have? 4 and 10 are how many ?

10. John buys three pears for four cents and six pears for eight cents : how many pears does he buy ?

11. How many are 5 and 1? How many are 6

12. John has five marbles in one hand and eight in the other : how many in both ?

13. Charles has five cents and his father gives him seven : how many has he then ?

14. John has five apples and Reuben gives him nine : how many has he then ?

15. Isaac buys five sheets of paper for five cents, and ten sheets more for ten cents : how many sheets does he buy ?

16. If I buy five oranges for five cents, and six oranges for six cents, how many do I buy ? Five and 5 are how many ?

17. How many are 6 and 1 ? How many are 6

LES. VII.] INTELLECTUAL ARITHMETIC.

18. How many are 6 and 61 How many are 6

19. William carries six apples to school in his basket and Henry four : how many in both baskets 1

20. John has six ‘apples, and his sister Jane gives him five : how many has he then 1

21. Charles has six apples and wins eight from John : how many has he thenl

22. William buys three tops for six cents and eight tops for ten cents : how many tops does he buy 1

23. James buys six. eggs for six cents and eight eggs for nine cents : how many eggs does he buy 1

24. Jane has 6 apples and Mary gives her 9 : how many will she then have 2

LESSON Y II.

Addition Table from *7 to 9 inclusive.

1. Seven and one are how many 1

2. Seven and two are how many 1

3. Seven and three are how many 1

4. Seven and four are how many 1

5. Seven and five are how many 1

6. Seven and six are how many 1

7. Seven and seven are how many 1

8. Seven and eight are how many 1

9. Seven and nine are how many 1

10. Seven and ten are how many 1

11. Eight and one are how manyl

12. Eight and two are how many 1

13. Eight and three are how manyl

14. Eight and four are how many 1

15. Eight and five are how many 1

16. Eight and six are how manyl

17. Eight and seven are how many ?

18. Eight and eight are how many ?

19. Eight and nine are how many ?

20. Eight and ten are how many ?

21. Nine and one are how many ?

22. Nine and two are how many ?

23. Nine and three are how many?

24. Nine and four are how many ?

25. Nine and five are how many ?

26. Nine and six are how many ?

27. Nine and seven are how many ?

28. Nine and eight are how many ?

29. Nine and nine are how many ?

30. Nine and ten are how many ?

QUESTIONS.

2. James has seven oranges in one basket and six in another: how many in both ?

3. William has seven apples and John gives him nine : how many has he then ?

4. A father has two sons and gives seven cents to each : how many cents does he give to both ?

5. If Henry buys seven apples, and Mary gives him nine : how many will he then have ?

6. If George buys seven quills at one time and 8 at another : how many does he buy in all ?

7. William has 5 marbles and Henry gives him 8 : how many will he then have ?

9. A boy has eight marbles and gains five : how many has he then? 8 and 10 are how many?

10. If he has eight and gains nine, how many will

he have ? 6 and 4 and 5, are how many ?

LES. VII.] INTELLECTUAL ARITHMETIC.

11. If George buys eight marbles for three cents and eight more for four cents, how many will he buy in all ?

12. John has eight marbles and Charles gives him nine : how many has he then ?

13. Eight and four are how many? Eight and seven how many ? Eight and 6 are how many ?

14. How many are 9 and 2? How many are 9

and 4 ? 1 and 3 and 6, are how many ?

15. Charles has nine apples and buys five more:

how many has he then ? 9 and 9, are how many?

16. If he has nine and buys eight, how many will

he have ? 4 and 5 and 6, are how many ?

17. Nine and seven are how many ? 8 and 7 ?

18. If James buys nine oranges for nine cents and eight more for 9 cents, how many will he buy in all?

19. Six sheets of paper cost nine cents and 2 pencils cost one cent: what do the paper and pencils cost? 6 and 7 and 8, are how many?

20. How many are 10 and 2 ? Plow many are 10

21. James has ten pencils and then buys eight:

how many has he then ? 9 and 4 and 3, are how

many ?

'22. John gives ten chestnuts to Henry and nine to William: how many does he give to both?

23. James spends six cents for candy, four cents

for gingerbread, and eight cents for nuts : how much does he spend in all ? 1 and 2 and 4 and 10, are how

many ?

24. Nancy has ten pins on her cushion, and sticks nine more there : how many will she then have ?

25. Jane has ten needles and Lucy gives her seven : how many will she then have ?

26. Oliver buys ten oranges for twelve cents and ten more for eight cents : how many does he buy ?

2*

INTELLECTUAL ARITHMETIC.

[sec. I.

LESSON VIII.

Numbers added with reference to Ten.

1. Five and 5 are how many ? 4 and 6 ? 3 and

2. Ten and 10 are how many ? 15 and 5 are how

many ? 16 and 4 ? 17 and 3 ? 18 and 2 ? 19 and

3. Twenty and ten are how many ? 25 and 5 ?

4. Thirty and 10 are how many? 35 and 5? 36

5. Forty and 10 are how many ? 45 and 5 ? 46

6. Fifty and 10 are how many? 55 and 5? 56

7. Sixty and 10 are how many? 65 and 5? 66

8. Seventy and 10 are how many? 75 and 5?

9. Eighty and 10 are how many ? 85 and 5 ? 89

10. Ninety and ten are how many? 99 and 1 ?

^Suggestions. — The first part of this lesson points out to the pupil the method of adding, by considering what numbers make exact tens. This is very important, and should be much dwelt on.

The second part shows that the first figure of a sum is always derived from the units.

The questions of the lesson should first be put in the order in which they are written, and then promiscuously, until thorough¬ ly learned.

LES. IX.] INTELLECTUAL ARITHMETIC.

11. Two and two are how many? 12 and 2?

12. Three and 3 are how many? 13 and 3 ? 23

13. Four and 4 are how many ? 4 and 14? 24

14. Five and 5 are how many ? 15 and 5 ? 25

15. Six and 6 are how many ? 16 and 6 ? 26 and

16. Seven and 7 are how many ? 17 and 7 ? 27

17. Eight and 8 are how many? 18 and 8? 28 and eight? 78 and 8 ? 88 and 8 ? 98 and 8 ?

18. Nine and 9 are how many ? 19 and 9 ? 29

LESSON IX

Showing the formation of Numbers f rom 11 to 100.

1. Eleven and 1 are how many ? Eleven and 2? Eleven and 3 ? Eleven and 4 ? Eleven and 5 ? Eleven and 6 ? Eleven and 7 ? Eleven and 8 ? Eleven and 9 ? Eleven and 10 ? Eleven and 11 ?*

* Suggestion. — This lesson indicates how all the numbers may be formed from 1 1 to include one hundred and nine. After the questions have been put in the order in which they are written, they should be put promiscuously.

20 ARITHMETICAL ARITHMETIC. [SEC. I.

2. Twenty-two and 1 are how many ? Twenty-two and 2 ? Twenty-two and 3 ? Twenty-two and 4 ? Twenty-two and 5 ? Twenty-two and 6 ? Twen¬ ty-two and 7 ? Twenty-two and 8 ? Twenty-two and 9? Twenty-two and 10? Twenty-two and 11?

3. Thirty-three and 1 are how many ? Thirty-three and 2 ? Thirty -three and 3 ? Thirty-three and 4 ? Thirty-three and 5 ? Thirty-three and 6 ? Thirty-three and 7 ? Thirty-three and 8? Thirty-three and 9? Thirty-three and 10? Thirty-three and 11?

4. Forty -four and 1 are how many ? Forty-four and 2? Forty-four and 3? Forty-four and 4? Forty-four and 5? Forty-four and 6 ? Forty-four and 7 ? Forty-four and 8 ? Forty-four and 9? Forty-four and 10 ? Forty-four and 11 ?

5. Fifty-five and 1 are how- many ? Fifty-five and

2? Fifty-five and 3 ? Fifty-five and 4? Fifty-five

and 5 ? Fifty -five and 6 ? Fifty-five and 7 ? Fifty-

five and 8? Fifty-five and 9 ? Fifty-five and 10 ? Fifty -five and 11 ?

6. Sixty-six and 1 are how many? Sixty-six and

2 ? Sixty-six and 3 ? Sixty-six and 4 ? Sixty-six

and 5 ? Sixty -six and G ? Sixty-six and 7 ? Sixty-six and 8? Sixty -six and 9? Sixty-six and 10? Sixty-six and 11 ?

7. Seventy-seven and 1 are how many ? Seventy-seven and 2 ? Seventy-seven and 3 ? Seventy-seven and 4 ? Seventy-seven and 5 ? Seventy-seven and G? Seventy-seven and 7? Seventy-seven and 8? Seventy-seven and 9 ? Seventy -seven and 10? Sev¬ enty-seven and 11?

8. Eighty-eight and 1 are how many? Eighty-eight and 2 ? Eighty-eight and 3 ? Eighty-eight and 4 ? Eighty-eight and 5 ? Eighty-eight and 6 ? Eighty-eight and 7 ? Eighty-eight and 8 ? Eighty-

INTELLECTUAL ARITHMETIC.

LES. X.]

eight and 9 ? Eighty-eight and 10? Eighty-eight and 11 ?

9. Ninety-nine and 1 are how many? Ninety-nine and 2? Ninety-nine and 3? Ninety-nine and 4? Ninety-nine and 5 ? Ninety-nine and 6 ? Nine¬ ty-nine. and 7? Ninety-nine and 8? Ninety -nine and 9 ? Ninety -nine and 10?

LESSON X.

Practical Questions.

1. Let each of the following combinations be given as a separate example. How many are

10 and 20 and 4 ? 10 and 30 and 9? 10 and 40 and 6 ? 10 and 50 and 3 ? 20 and 30 and 9 ? 15 and 20 and 6 ? 25 and 15 and 4? 35 and 12 and 3 ? 40 and 60 and 9 ? 8 and 20 and 10 ?

2. Jane has 13 pins in her cushion and Mary 27 : how many pins have both ?

3. John has a number of pears : he gives 8 to William, 12 to Charles, 9 to James and has 1 left: how many had he at first ?

4. There are 4 bags of coffee : the first contains 16 pounds, the second 14 pounds, the third 7 pounds, and the fourth 3 pounds : how many pounds in all the bags ?

5. A farmer has 4 pastures containing sheep. In the first there are 3 sheep, in the second there are 6, in

INTELLECTUAL ARITHMETIC. [SEC. I.

the third there are 7, and in the fourth there are 8 : how many are there in the four pastures ?

6. James gave 18 cents for a squirrel, 82 cents for a cage, and 15 cents for nuts : how many cents did he pay in all ?

7. A man bought a cow for 25 dollars, a calf for 5 dollars, 3 lambs for 8 dollars, and a pig for 2 dollars : what did he pay for all ?

9. How many are 3 and 4 and 16 and 5 and 4 and 5 ?