tens, four-tens, five-tens, etc. By this principle were derived the following ordinary names:

Twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety. 43. A group of ten tens is called a hundred; a group of ten hundreds, a thousand; the next group receiving a new name consists of a thousand thousands, called a million; the next group of a thousand millions, called a billion, etc.

44. After a thousand the two intermediate groups between those receiving a distinct name, are numbered by tens and hundreds, as ten thousand and hundred thousand.

NOTES.-1. The above shows the principle by which the names of numbers were formed. The names, however, were not derived from the particular expressions given, but originated in the Saxon language.

2. Eleven is from the Saxon endlefen, or Gothic ainlif (ain, one, and lif, ten); twelve is from the Saxon twelif, or Gothic tvalif (tva, two, and lif, ten). Some think eleven meant one left after ten, and twelve, two left after ten.

3. Twenty is from the Saxon twentig (twegen, two, and tig, a ten); thirty is from the Saxon thritig (thri, three, and tig, a ten), etc.

4. Hundred is a primitive word; thousand is from the Saxon thusend, or Gothic thusundi (thus, ten, and hund, hundred); million, billion, etc., are from the Latin.

NOTATION.

45. Notation is the method of writing numbers. Numbers may written in three ways:

1st. By words, or common language.

2d. By figures, called the Arabic Method. 3d. By letters, called the Roman Method.

ARABIC NOTATION.

46. The Arabic System of Notation is the method of expressing numbers by characters called figures.

47. In this system numbers are expressed according to the following principle:

Principle. We represent the first nine numbers by characters, and then use these characters to number the groups, indicating the group numbered by the position of the charac ter.

48. Figures. Figures are characters used in expressing numbers. There are ten figures used, as follows: FIGURES. 1, 2, 3, 4, 5, 6, 7, 8, 9,

0,

naught,

NAMES .} one, two, three, four, five, six, seven, eight, nine, cipher or zero.

AND VALUES.

49. By the combination of these figures all numbers may be expressed; hence they are appropriately called the alphabet of arithmetic.

50. Combination.

These figures are combined accord

ing to the following principles:

1. A figure standing alone, or in the first place at the right of other figures, expresses UNITS or ONES.

2. A figure standing in the second place, counting from the right, expresses TENS; in the third place, HUNDREDS; in the fourth place, THOUSANDS, etc.; thus:

51. Periods. For convenience in writing and reading numbers, the figures are arranged in periods of three places each. The first three places constitute the first or units period; the second three places, the second or thousands period, etc.

52. The terms of each period are considered respectively as the units, tens and hundreds of that period.

53. The name of each of the first eight periods is represented by the following

54. This table enables us to read a numerical expression of twenty-four figures. The succeeding periods are Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tertio-decillions, Quarto-decillions, Quinto-decillions, Sexto-decillions, Septo-decillions, Octo-decillions, Nono-decillions, Vigillions, etc.

55. The combination of figures to express a number forms a numerical expression.

56. The different figures of a numerical expression are called terms. Terms are also used to indicate the numbers represented by the figures.

NOTE.-The use of the word term to indicate both the figures and numbers represented by them enables us to avoid the error of using the word figure for the word number. Thus, instead of saying, "add the figures," which is an absurdity, we can say, add the terms, meaning the numbers denoted by the figures.

EXERCISES IN NUMERATION.

57. The pupils are now prepared to learn to read numbers when expressed by figures. From the preceding explanations, we have the following rule for numeration:

Rule.-I. Begin at the right hand and separate the numerical expression into periods of three figures each.

II. Then begin at the left hand and read each period in succession, giving the name of each period except the last. NOTE. The name of the last period is usually omitted, it being understood.

1. What number is expressed by 5468217?

SOLUTION. Separating the numerical expression into OPERATION. periods of three figures each, beginning at the right hand, we have 5,468,217. The third period is 5 mil

5,468,217

lions, the second period is 468 thousands, and the first 217 units; hence the number is 5 million, 468 thousand, 217.

10. Required the names of the following places:

Sixth; twelfth; eighth; tenth; fourteenth; nineteenth; thirteenth ; seventeenth; twenty-first; eighteenth; twenty-fourth; twenty-second; twenty-ninth; thirty-fourth.

11. Required the names of the following periods:

First; third; fifth; second; seventh; fourth; ninth; sixth; tenth; eleventh; fourteenth; sixteenth.

12. Required the places of the following periods:

Thousands; millions; ten-thousands; hundred-thousands; ten-millions; billions; hundred-trillions; quintillions; octillions; hundredsextillions; ten-quintillions; septillions; hundred-quadrillions.

13. Required the period and place of the following:

Billions; hundred-billions; ten-billions; quintillions; ten-trillions; ten-quadrillions; hundred-quintillions; septillions; hundred-sextillions; ten-nonillions; sextillions; ten-octillions; hundred-quadrillions. NOTE.-After pupils are familiar with reading by dividing into periods, the division may be omitted or performed mentally.

EXERCISES IN NOTATION.

58. Having learned to read numerical expressions, we are now prepared to write them. From the principles which have been explained, we have the following rule:

Rule.-I. Begin at the left and write the hundreds, tens, and units, of each period in their proper order.

II. When there are vacant places, fill them with ciphers. 1. Express in figures the number three thousand eight hundred and six.

SOLUTION. We write the 3 thousands in the 4th place, 8 hundreds in the 3d place, and there being no tens, we write a cipher in the 2d place, and 6 units in the 1st place, and we have 3806.

Express the following numbers in figures:

1. Forty-six million and forty-sixty-five million four thousand seven thousand. and seven.

2. Two hundred and two million

and twenty-two.

3. Six hundred and sixty million five hundred and thirty-seven thousand and three.

4. One billion four million and

eighty.

6. Seventy trillion eight billion one million and six hundred.

7. One hundred and two quadrillion three billion four hundred thousand and fifty.

seven

8. Thirty-five octillion hundred and ten trillion thirty

5. Two hundred and nine billion million six hundred and seventeen.

9. Twenty undecillion six hun- 10. Seventy duodecillion, nine dred nonillion ninety-four septillion octillion five hundred and thirtythree hundred and one billion fifty-two sextillion four hundred trileight thousand three hundred and lion eight million one hundred four. and ten.

59. Orders. Since we may have 2 tens, 3 hundreds, etc., the same as 2 apples, 3 books, etc., these different groups may be regarded as units of different orders; thus, are called Units of the 1st order

UNITS

TENS

HUNDREDS

THOUSANDS

TEN-THOUSANDS

60. From this it is seen that ten units of a lower order make one unit of the next higher order; the system of notation is therefore called the Decimal System, from the Latin decem, ten.

NOTE. The pupil should notice carefully the distinction between periods and orders of units. The first period, called units period, consists of units of the 1st, 2d and 3d order; the second period, called thousands period, consists of units of the 4th, 5th and 6th orders, etc. Periods increase by thousands; orders by tens.

EXAMPLES FOR PRACTICE.

Write and read the following: 1. Two units of the 2d order, and four of the 1st.

5. Eight units of the 8th order, six of the 6th, three of the 3d, and

2. Nine units of the 4th order, one of the 1st. and three of the 1st.

3. Five units of the 7th order, four of the 4th, and eight of the 2d.

4. Three units of the 9th order, five of the 5th, two of the 2d, and four of the 1st.

6. One unit of the 11th order, four of the 10th, nine of the 7th, two of the 6th, and seven of the 3d.

7. Five units of the 10th order, two of the 6th, and three of the 1st. 8. Six units of the 13th order, and four of the 5th.

THE DECIMAL SCALE.

61. In Numeration and Notation we have two classes of units, Simple and Collective.

62. A Simple Unit is a single thing, or one; a collective unit denotes a group or collection, regarded as a whole.