dredths; 1500 millionths; 408 ten-thousandths; 94000
ten-millionths; 12 thousandths; 3 billionths; 4756 hun-
dred-thousandths; 599 trillionths; fifteen ten-billionths.
6. Three, and forty-one thousandths.*

7. Fifteen, and one hundred and one millionths.
8. Nine thousand and eight hundred-thousandths.
9. Nine thousand, and eight hundred-thousandths.
10. Twenty-three ten-thousandths.

11. Eight hundred and fifty-two, and one tenth.

12. Thirty, and thirty trillionths.

13. Two hundred and six, and eleven hundred and one billionths. Mention the terms of this number.

14. Ninety-three thousand and twelve ten-billionths. 15. Four billion, and five thousandths.

16. Three trillion, and ten hundred-millionths.

NUMERATION OF DECIMALS.

30. RULE.-Read a decimal as an integer, naming after it the order in which its right-hand figure stands.

EXAMPLE.-Read .011.

As an integer, 011 is eleven. By enumerating the orders,-tenths, hundredths, thousandths,- -we find that the right-hand figure stands in the order of thousandths. We therefore read the decimal eleven thousandths. In like manner,

1. Fill the first five decimal places with 3's, and read the decimal thus formed.

*The comma is used here and in similar cases, to show that what precedes it is an integer.

2. Write as one decimal, 5 hundredths, 6 ten-thousandths, 7 millionths (remembering to fill vacant places with ciphers), and read the decimal thus formed.

3. Write 3 in the first decimal order, 9 in the second, 8 in the fourth, 1 in the fifth, and read the decimal.

4. Write 2 in the 7th decimal order; read the decimal. 5. To what order does 6 preceded by nine ciphers and the decimal point belong?

6. Read the following: analyze the first six:

NOTATION AND NUMERATION OF FEDERAL MONEY.

32. Federal Money is the currency (Art. 290) of the United States. Its scale is decimal (Art. 7), 10 mills making 1 cent, 10 cents 1 dime, 10 dimes 1 dollar.

33. In writing and reading Federal Money, the only denominations used are dollars, cents, and mills. Dollars are written as an integer, and preceded by the dollar-mark, $. They are separated by the decimal point from cents, which occupy the first two decimal places (the place of dimes and their own), mills occupying the third place.

34. RULE I.—In writing Federal Money, make the· dollars integers, the cents hundredths, the mills thousandths.

RULE II.-In reading Federal Money, call the integer dollars, the hundredths cents, the thousandths

1. Write fourteen dollars, fifteen cents.
2. Write ten thousand dollars, ten cents.
3. Write nineteen dollars, seven cents.
4. Write thirty-seven cents, five mills.

5. Write two hundred and three thousand dollars.
6. Write seventy-six thousand dollars, two mills.
7. Four thousand and one dollars, 3 cents, 9 mills.
8. Eighty dollars, twenty cents, five mills.

ENGLISH SYSTEM OF NUMERATION.

36. The separation of numbers into periods of three figures each, as shown in Art. 19, is that followed in the United States, France, and the continent of Europe generally. The English group their orders in periods of six figures each, naming them as follows:

A billion according to the English system is a million of millions, and is therefore 1000 times as great as the billion of the United States, which is a thousand million.

37.-EXERCISE IN ENGLISH NUMERATION.

Write the numbers expressed according to the English system of numeration as follows, and then read them according to our own system:

1. Twenty-seven billion, fifty-three thousand million, six hundred thousand seven hundred and fifty-three.

2. One trillion, three hundred and nine thousand five hundred and twelve billion, forty-two thousand and seventeen.

3. Seven thousand trillion, sixteen thousand and seventeen billion, four thousand million, two hundred and ninety.

4. Forty thousand nine hundred and eighteen million, seven hundred thousand and eleven.

5. One billion, six hundred thousand and twelve million, four hundred and fifteen.

6. Seven hundred and seventeen thousand and six trillion, ninety-one thousand six hundred billion, four thousand four hundred and four million, three hundred and twelve thousand five hundred.

7. Read, according to the English system, the numbers given in Examples 23-42, Art. 22.

The Roman Notation.

38. The Roman Notation, so called because it was used by the ancient Romans, prevailed for centuries throughout a great part of Europe, before the Arabic Notation came into use. It employs seven letters. I. denotes one; V., five; X., ten; L., fifty; C., one hundred; D., five hundred; M., one thousand.

E

V resembles the outline of the hand with the five fingers spread. X is two V's, or fives, joined at their points. C begins the Latin word centum, one hundred. A hundred was originally denoted by this character, and the lower half, afterward written as L, denoted fifty. M begins the Latin word mille, a thousand. A thousand was sometimes written CIO; hence IƆ, afterward changed to D, denoted 500.

39. These letters are combined to express numbers, according to the following principles :—

1. If a letter is repeated, its value is repeated. XX. is twenty; III. is three.

2. A letter of less value, placed after one of greater, unites its value to that of the latter. VI. is six.

3. A letter of less value, placed before one of greater, takes its value from that of the latter. IV. is four.

4. A letter of less value, placed between two of greater, takes its value from that of the other two united. LIV. is fifty-four.

5. A bar over a letter increases its value a thousand times. V. is five thousand.