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12. Reduce £104; 10 Can. to N. Y. Ans. £167 ; 4. 13. Reduce £300; 10; 4; 2 Can. to Penn.

Ans. £450 ; 15; 6; 3. 14. Reduce £937; 18; 11; 1 N. E. to Geo.

Ans. £721 ; 14; 8. 3. 15. Reduce $224; 60 to Can.

Ans. £56; 3. 16. Reduce £225; 6 N. E. to F. M. Ans. $752.00. 17. Reduce £880 15; 11; 1 Penn. to Sterling.

Ans. 528 9; 6; 3. 18. Reduce £6,750 Irish to Geor.

Ans. £6,461. 19. Reduce £1,846 Ster. to Irish.

Ans. £2,000. 20. Reduce £1,722; 18; 9; 3 N. E. to N. Y.

Ans. £2,298; 5; 1. 21. Reduce £2,114; 1 ; 3 Can. to F. M.

Ans. $8,456.25. 22. Change £784; 5; 6; 2 Penn. to Geor.

Ans. £487; 19; 10; 21. 23. Change £923 Sterling to Irish. 24. Change £,4000 Irish to Sterling. 25. Change £157; 8; 3; 3 N. Y. to N. E. 26. Change £1,654 ; 3 ; 8; 1 Penn. to N. E. 27. Change £947; 9; 4; 2 N. E. to F. M. 28. Change $1,444.66 to N. E.. To N. Y. To Penn. 29. Change $945.22 to N. Y. To Geor. To Can. 30. Change £1,846; 15; 4 N. E. to F. M. To Penn.

To Georgia. 31. Change $4,444,4444 to Sterling. 32. Reduce £1,000,000 Sterling to F. M.

ARITHMETIC.

THIRD PART.

NUMERATION. In the following, Third Part, there will be a review of the preceding subjects, embracing the more difficult ope. rations. The rules and explanations will not be repeated, as the pupils can refer to them in the former part.

ROMAN NUMERATION, Before the introduction of the Arabic figures, a method of expressing numbers by Roman Letters was employed. As this method has not entirely gone out of use, it is im. portant that it should be learned. The following letters are employed to express numbers. I. One.

X. Ten.' II. Two.

L. Fifty. III. Three.

C. One Hundred. IIII. or IV. Four.

D. Five Hundred. V. Five.

M. One Thousand. The above letters, by various combinations, are made to express all the numbers ever employed in Roman Nu. meration.

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RULE FOR WRITING AND READING ROMAN NUMBERS.

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As often as a letter is repeated, its value is repeated. When a less number is put before a greater, the less number is subtracted. But when the less number is put after the great. er, it is added to the greater.

Examples. In IV. the less number, I. is put before the greater number V. and is to be subtracted, making the number four.

What is Roman numeration? What is the rule for writing and reading Roman numbers ?

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In VI. the less number is put after the greater, and it is to be added, making the number six.

In XL the ten is subtracted from the fifty.
In LX the ten is added to the fifty.
The following is a table of Roman Numeration :

TABLE.
One I

Ninety

LXXXX or XC
Two
II

One hundred С
Three III

Two hundred CC
Four IIII or IV Three hundred CCC
Five V

Four hundred CCCC
Six VI

Five hundred D or 15*
Seven VII

Six hundred

DC
Eight VIII

Seven hundred DCC
Nine VIIII or IX

Eight hundred DCCC
Ten X

Nine hundred DCCCC
Twenty XX

One thousand

M or CIO
Thirty XXX

Five thousand 133 or VI
Forty XXXX or XL Ten thousand Cciɔɔ or X
Fifty L

Fifty thousand
Sixty LX

Hundred thousand CCCÍɔɔɔ or ī
Seventy LXX

One million M Eighty LXXX Two million * 15 is used instead of D. to represent five hundred, and for every additional ɔ annexed at the right hand, the number is increased ten times.

+ ci) is used to represent one thousand, and for every C and ɔ put at each end, the number is increased ten times.

| A line over any number increases its value one thousand times.

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36.

Write the following numbers in Roman letters :

5. 7. 3. 9. 8. 16. 4. 14. 5. 15. 6. 16. 26.

306. 1. 11. 111. 7. 17. 77. 777. 1800. 1832. 1789.

Read the following Roman numbers :

VI. XIX. XXIV. XXXVI. XXIX. LV. XLI. LXIV. LXXXVIII. XCIX. MDCCCXVIII.

OF OTHER METHODS OF NUMERATION.

By the common method of numeration, ten units of one order make one unit of the next higher order. But it is equally practicable, to have any other number than ten, to constitute a unit of a higher order. Thus we might

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have six units of one order make one unit of the next higher order. Or twelve units of one order might make one of the next higher order.

The number which is selected to constitute units of the higher orders, is called the radix of that system of numeration.

The radix of the common system is ten, and this num. ber, it is supposed, was selected, because men have ten fingers on their hands, and probably used them in expressing numbers.

Before the introduction of the Arabic figures, Ptolemy introduced a method of numeration, in which sixty was the radix. The Chinese and East Indians use it to this day.

But in Ptolemy's system there were not sixty different characters employed. Instead of this, the Roman method of numeration was used for all numbers as far as sixty, and then for the next higher orders the same letters were used over again, with an accent (') placed at the right. For the third order two accents (") were used, and for the fourth order three accents ('"').

To illustrate this method by Arabic figures, 31' 23 sig. nifies 31 sixties and 23.

We have some remnants of this method in the division of time into 60 seconds for a minute, and 60 minutes for an hour, and also the division of the degrees of a circle, into 60 seconds to a minute, and 60 minutes to a degree.

EXERCISES IN NUMERATION, COMMON, VULGAR, AND DE

CIMAL

(See rules on pages 53, 58, and 65.) 1. Two million, four thousand, one hundred and six. 2. Two hundred thousand, and six tenths.

3. Twenty six billion, six thousand, and fifteen thousandths.

4. Two hundred and sixty thousand millionths.

What other methods of numeration are there? What is the radix ? What the radix of the common system? Of Ptolemy's ?

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5. One sixth of two apples are how much, and how writ. ten ?

6. One ninth of twenty oranges are how much, and how written ?

ls it a proper or improper fraction ?

7. One sixth of four bushels is how much ? how written? is it a proper, or improper fraction ?

8. One tenth of forty bushels, how much ? how written? is it a proper or improper fraction ?

9. One tenth of three oranges, how much ? how express. ed?

10. Three tenths of three oranges, how much? how ex. pressed?

11. Four sixths of twelve apples, how much? how ex. pressed?

12. Three thousand tenths of thousandths. 13. Four billions, six thousand, and five ten thousandths.

14. Sixteen billions, three hundred and six millions, five hundred thousand, and six tenths of millionths.

15. Five trillion, five million, five units, and three hun. dred and sixty five millionths.

16. Sixteen hundred and twenty four, and four tenths of billionths.

ADDITION.

Let the pupil add the following numbers:

1 Two hundred and six million ; twenty four thousand, five hundred and six.

Thirty seven billion, twenty six thousand and three.

Four hundred and seventy nine billion, six hundred and sixty seven million, nine hundred and eighty four thou. sand, six hundred and ninety nine.

Fifteen million, seventy seven thousand, nine hundred. Thirty six trillion, four hundred million, and six.

Four quadrillion, seventeen million, three hundred and six.

Six quadrillion, fourteen trillion, seventeen million, fourteen thousand, three hundred and nine.

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