of numbers, and teaches the methods of applying the principles of the science to practical purposes.

11. An axiom is a self-evident truth.

12. A problem is a question proposed for solution, or something to be done.

13. An operation is the process of finding, from given quantities, others that are required.

14. A sign is a symbol employed to indicate the relations. of quantities, or operations to be performed upon them.

15. A rule is a direction for performing an operation.

16. An example is a particular application of a general principle or rule.

17. The principal or fundamental processes of arithmetic are Notation and Numeration, Addition, Subtraction, Multiplication, and Division.

SIGNS.

18. The sign of equality, two short horizontal lines, =, is read equal, or equal to, and denotes that the quantities between which it is placed are equal to each other. Thus, 12 inches = 1 foot, signifies that 12 inches are equal to 1 foot.

19. The sign of addition, an erect cross, +, is read plus, and, or added to, and denotes that the quantities between which it is placed are to be added together. Thus, 8+6 signifies that 6 is to be added to 8.

20. The sign of subtraction, a short horizontal line, is read minus, or less, and denotes that the quantity on the right of it is to be subtracted from the quantity on the left. Thus, 8 6 signifies that 6 is to be subtracted from 8.

21. The sign of multiplication, an inclined cross, X, is read times, or multiplied by, and denotes that the quantities between which it is placed are to be multiplied together. Thus, 76 signifies that 7 is to be multiplied by 6.

22. The sign of division, a horizontal line between two dots,, is read divided by, and denotes that the quantity on the left of it is to be divided by that on the right. Thus, 426 signifies that 42 is to be divided by 6.

23. The sign of aggregation, a parenthesis, ( ), includ

ing several numbers, or a vinculum,

drawn over them,

indicates that the value of the expression is to be used as a single number. Thus, (173) × 5, indicates that the sum of 17 and 3, or 20, is to be multiplied by 5; and 12 + (9—3) 2, indicates that the difference between 9 and 3 divided by 2, or 3, is to be added to 12.

÷

AXIOMS.

24. Arithmetic, in common with other branches of the mathematics, is based upon axioms, few in number, and universally admitted to be so clearly true, that no process of reasoning can make them plainer; as,

1. If the same quantity, or equal quantities, be added to equal quantities, the sums will be equal.

2. If the same quantity, or equal quantities, be subtracted from equal quantities, the remainders will be equal.

3. If the same quantity, or equal quantities, be added to unequal quantities, the sums will be unequal.

4. If the same quantity, or equal quantities, be subtracted from unequal quantities, the remainders will be unequal.

5. If equal quantities be multiplied by the same quantity, or equal quantities, the products will be equal.

6. If equal quantities be divided by the same quantity, or equal quantities, the quotients will be equal.

7. If the same quantity be both added to and subtracted from another, the value of the latter will not be changed.

8. If a quantity be both multiplied and divided by the same quantity, its value will not be changed.

9. If two quantities be equally increased or diminished, their difference will not be changed.

10. Quantities which are equal to the same quantity are equal to each other.

11. Quantities which are like parts of equal quantities are equal to each other.

12. The whole of a quantity is greater than any of its parts.

13. The whole of a quantity is equal to the sum of all its parts.

NOTATION AND NUMERATION.

NOTATION.

25. NOTATION is the process of representing numbers by letters, figures, or other symbols.

The common methods of expressing numbers are three: by words, written or spoken; by letters, called the Roman method; and by figures, called the Arabic method.

26. In common language, we express numbers by the terms one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, twenty-one, etc., giving a distinct name to each unit as far as ten, when we begin a second ten, and pass on to twenty; a third ten, and pass on to thirty; and so on to forty, fifty, sixty, seventy, eighty, and ninety. Proceeding thus we reach ten tens, which we call one hundred, when we begin a second hundred, and pass to two hundred; a third hundred, and pass to three hundred; and so on as far as ten hundred, which we call one thousand. A thousand thousand we call one million; a thousand million, one billion; a thousand billion, one trillion; and so on with numbers still higher.

NOTE 1. - The term eleven is a contraction of one left after ten; and twelve, of two left after ten. Thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, are derived from three and ten, four and ten, five and ten, etc. Twenty, thirty, forty, fifty, sixty, seventy, eighty, and ninety are contractions of two tens, three tens, four tens, etc.

NOTE 2.- - Billion is a contraction of the Latin bis, twice, and million; and trillion, of the Latin tres, three, and million. In like manner from the Latin numerals, quatuor, four; quinque, five; sex, six; septem, seven; octo, eight; novem, nine; decem, ten; undecim, eleven; duodecim, twelve; tredecim, thirteen; quatuordecim, fourteen; quindecim, fifteen; sexdecim, sixteen; septendecim, seventeen; octodecim, eighteen; novemdecim, nineteen; viginti, twenty, -are formed quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decillions, undecillions, duodecillions, etc.

ROMAN NOTATION.

27. THE ROMAN NOTATION, so called from its having originated with the ancient Romans, employs in expressing numbers seven capital letters, viz. :

I, one,

one thousand.

five, ten, fifty, one hundred, five hundred, All intervening and succeeding numbers are expressed by use of these letters, either in repetitions or combinations. By a letter being written after another denoting equal or less value, the sum of their values is represented; as, II represents two; VI, six. By writing a letter denoting a less value before a letter denoting a greater, their difference of value is represented; as, IV represents four; XL, forty.

A dash (—) placed over a letter increases the value denoted by the letter a thousand times; as, V represents five thousand; IV, four thousand.

NOTE 1.-The Roman method of Notation is now but little used, except in numbering sections, chapters, and other divisions of books; and for indicating the hours on the face of clocks, watches, or dials.

NOTE 2. Formerly CIO was used to represent one thousand, and the prefixing of a C and the annexing of a Ɔ increased the number denoted ten times; thus, CCIDƆ represented ten thousand, and CCCɔɔɔ, one hundred thousand.

EXERCISES.

Represent the following numbers by letters :

1. Forty-nine.

2. Ninety-seven.

3. One hundred and eighty-eight.

4. Two hundred and nineteen.

5. Six hundred and sixty-three.

6. One thousand five hundred and six.

Ans. XLIX.

7. One thousand eight hundred and fifty-seven. 8. Four thousand four hundred and forty-four. 9. Eleven thousand nine hundred and eleven. 10. One hundred fifty thousand and fifty. 11. One million twenty thousand and twenty. 12. Three million one hundred thousand.

ARABIC NOTATION.

28. ARABIC NOTATION, so called from its having been made known through the Arabs, employs in expressing numbers ten characters or figures, viz.:

1, 2, 3, 4, 5, 6, 7, 8, 9, 0. one, two, three, four, five, six, seven, eight, nine, cipher. The first nine are sometimes called digits, and the cipher, naught

or zero.

29. The place of a figure is its particular position with regard to other figures; as in 61 (sixty-one) counting from the right, the 1 occupies the first place and the 6 the second place, and so on for any other like arrangement of figures.

30. The digits have been denominated significant figures, because each of itself expresses a positive value, always representing so many units, or ones, as its name indicates. But the size or value of the units represented by a figure differs with the place occupied by the figure. Thus in 235 (two hundred and thirty-five), each of the figures, without regard to its place, expresses units, or ones; but these units or ones differ in value. The 5 occupying the first place represents