ARITHMETIC.

Art. 1.—ARITHMETIC is the science of numbers.

It explains their properties, and teaches how to apply them to practical purposes.

Art. 2. The principal, or fundamental rules, are, Notation, Numeration, Addition, Subtraction, Multiplication, and Division. These are called fundamental rules, because all questions in Arithmetic are solved by one or more of them.

Art. 3.-Notation is the expressing of any number or quantity by figures; thus, 1 one; 2 two; 3 three; 4 four; 5 five; 6 six; 7 seven; 8 eight; 9 nine; 0 cipher. The first nine figures are sometimes called digits, from the Latin word digitus, which means a finger. In the early stages of society people counted by their fingers; they were also formerly all called ciphers-hence the art of Arithmetic was called ciphering.

Art. 4.-There are two methods of Notation-the Arabic, as above, and the Roman, which is expressed by the following seven letters of the alphabet:

I, V, X, L, C, D, M.

1 2 3 4 5 6 7 8 9 10 20 30 40 50 I, II, III, IV, V, VI, VII, VIII, IX, X, XX, XXX, XL, L,

LX, LXX, LXXX, XC, C, D, M.

Art. 5.—When a letter of less, is placed before one of a greater value, it diminishes the value of the greater, by the value of itself-thus, X signifies ten, but IX is only nine. When a letter of less, is placed after one of greater value, it increases the value of the greater by the value of itself.

This method is seldom used except in numbering chapters, sections, etc.

QUESTIONS.-1. What is Arithmetic? 2. What are the principal, or fundamental rules? 3. Why so called? 4. What is Notation? 5. What are the first nine figures sometimes called? 6. What were they all formerly called? 7. How many methods of Notation, and what are they? 8. How many are the Arabic characters, or figures? 9. By what is the Roman method expressed? 10. How is a letter affected when one of less value is placed before it? 11. How when one of less value is placed after it? 12. For what is the Roman method of Notation principally used?

NUMERATION.

Art. 6.-Numeration teaches to express in words the value of any number represented by figures. Thus, 365 is read, three hundred and sixty-five.

Art. 7.-Figures have a simple and relative value. When a figure stands alone its value is simply so many units, or ones; as, 2 two; 3 three; 4 four. Their relative value is derived from the place they occupy when joined together, or from their distance from the unit's place. Thus, 2 and 3 express their own value; simply so many units; but they are made to express either 23 or 32; that is, either three units and two tens, or two units and three tens. Hence it appears that the first, or right-hand place, always expresses so many units; it is therefore called the unit's place; the second, the place of tens, expressing always as many tens as the figure contains units. The third place is hundreds; the fourth, thousands, as may be seen by the following

QUESTIONS.-13. What is Numeration? 14. What is the value of a figure standing alone? 15. From what is their relative value derived? 16. What does the first, or right-hand figure, always express, and what is it called? 17. What are the second, third, and fourth places called? 18. What is the value of the cipher, when standing alone, or at the left hand of another figure? 19. What effect has it when placed at the right of another figure?

Art. 8.-The cipher, when standing alone, or at the left hand of another figure, signifies nothing, as 05, 005, is five in either case, because it still occupies the unit's place. But when placed at the right hand of another figure, it increases its value in a tenfold ratio, by removing the figure farther from the unit's place. This may be seen by the following

TABLE II.

0 Nothing. 20 Twenty.

200 Two hundred.

2,000 Two thousand.

20,000 Twenty thousand.

200,000 Two hundred thousand.

2,000,000 Two millions.

Art. 9.-To know the value of any number of figures. RULE.-1. Numerate from the right hand to the left, by saying units, tens, hundreds, &c., as in the Table.

2. To the simple value of each figure join the name of its place, reading from the left hand to the right.

694 542 987 562 714 923 610 78 184 542 365 987 963

HHMMMMM

The first division of the foregoing Table is according to the French method, into periods of three figures each: the name of the period is superadded. The second division is according to the English method, into periods of six figures each. The name of each period is subjoined. The two divisions of the

QUESTIONS.-20. How may the value of any number be found? 21. What are the two methods of numeration in the third table? 22. In what respect do they differ?

Table agree for the first nine figures-beyond that they as sume different names. The principles of Notation in both are the same. In the former method the names, units, tens, hundreds, are repeated in each period; in the latter method, thousands, tens of thousands, hundreds of thousands, are repeated with the name of the period. If the sum be not expressed in figures, it is necessary to know the method of notation employed.

Art. 10.-Let the scholar point the following numbers into periods, and read them.

3445

67891

983452

5437643

67821356

436543897

5678923412

96754329876

1234678901263

Art. 11.-Express the following numbers in figures.

1. Twenty-three.

2. Thirty-five.

3. One hundred and twenty.

4. One hundred and twenty-six.

5. Ten thousand three hundred and twenty.

6. Four millions four thousand and four.

7. One hundred and seventeen millions, one hundred and two.

8. Three billions, three millions, seventeen thousand and ten.

9. One hundred billions, one hundred thousand, two hundred and fifty.

10. Twenty billions and twenty.

11. Seven billions, seven thousand and seventeen.

12. One hundred and seven billions, twenty-seven thousand and one.

13. Five hundred and four trillions, two billions, ten millions, ten thousand and ten.

14. Forty-five trillions, forty billions and thirteen.

15. Two millions, two thousand, three hundred and three. 16. Thirty quadrillions, fifty millions, four thousand, three hundred and forty-eight.

17. Four hundred and four quadrillions, seven hundred and seven thousand, two hundred and two.

18. Four quintillions, thirty-five quadrillions, three trillions, two billions, twenty-seven millions, three hundred and forty thousand, four hundred and seventeen.

ADDITION TABLE.

Art. 12.—Signs.—A cross + is the sign of addition. It shows that the numbers between which t is placed are to be added. Two parallel horizontal lines signify equality. Thus: 3+4=7 is read, 3 added to 4, or 3 plus 4 (plus is a Latin word, which signifies more) is equal to 7.

=

The following TABLE may be read thus: 2 and 0 are two; 2 and 1 are 3, &c.

QUESTIONS.-Two and 0-how many? 2. Two and 1--how many? 3. Two and 2-how many?

The scholar should be questioned in this manner, until he is familiar with the above table.