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56. What is the interest of $135.16 from the 4th June, 1817 to 13th April, 1818, at 5 per cent. ?

57. What is the interest of $35.37 from 13th July, 1815, to 17th Nov. 1818, at 47 per cent. ?

58. What is the interest of $45.87 from 19th Sept. 1819, to 11th Aug. 1821, at 7į per cent. ?

59. What is the interest of $133 from 230 Oct. 1817, to 19th Jan. 1820, at 4 per cent. ?

60. What is the interest of 113£. 14s. for 1 year, 5 months, and 3 days, at 7 per cent. ?

61. What is the interest of 87£. 15s. 4d. for 2 years, 11 months, 3 days, at 7 per cent. ?

62. What is the interest of 43£. 16s. for 9 months and 13 days, at 8 per cent. ?

63. What is the interest of 142£. 19s. for 1 year, 8 months, and 13 days, at 9 per cent. ?

61. What is the interest of $372 for 4 years, 8 months, and 17 days, at 74 per cent. ?

65. What is the interest of 1 dollar for 15 days at 7 per cent. ?

66. What is the interest of $.25 for 13 days, at 7 per cent. ?

67. What is the interest of $.375 for 19 days, at 11 per cent. ?

68. What is the interest of $1147 for 8 hours, at 6 per cent. ?

69. What is the interest of 137£. lls. for 11 days at 9

per cent. ?

70. What is the interest of 15s. for 3 months, at 8 per cent. ?

71. Wliat is the interest of 16.£. 7s. 8d. for 2 months, at 12 per cent. ?

72. What is the interest of 4s. 3d. for 17 years, 3 months, and 7 days, at 8 per cent. ?

73. A man gave a note 13th Feb. 1817, for $753, interest at 6 per cent., and paid on it as follows: 19th. Aug. 1817, $45 ; 27th June, 1818, $143 ; 19th Dec, 1819, $25; 11th May 1819, $100; and 14th Sept. 1820, he paid the rest, principal and interest. How much was the last payment

74. A note was given 17th July, 1814, for $1432, interest at 6 per cent., and payments were made as follows ; 15th Sept. same year, $150; 2d Jan. 1815, $130; 10th. Nov. 1815, $23; 11th April, 1817, $237 ; 15th Aug. 1818, $47. llow much was due on the note, principal and interest, 5th Feb, 1819?

ARITHMETIC.

PART II.

NUMERATION.

one are

I. A single thing of any kind is called a unit or unity.

Particular names are given to the different collections of units. A single unit is called

Onc. If to one unit we join one unit more, the collection is called two; that is, one added to one is called two, or one and

Two. One added to two is called three ; two and one are Threc. One added three is called four; three and one are Four. One added to four is called five ; four and one are Five. One added to five is called six ; five and one are

Siz. One added to six is called seven ; six and one are Seven.

One added to seven is called eight ; seven and one are

Eight. One added to eight is called nine ; eight and one are Nine. One added to nine is called ten ; nine and one are Теп.

In this manner we might continue to add units, and to give a name to each different collection. But it is easy to perceive that, if it were continued to a great extent, it would be absolutely impossible to remember the different names ; and it would also be impossible to perform operations on large numbers. Besides, we must necessarily stop somewhere; and at whatever number' we stop, it would still be possible to add more ; and should we ever have occasion to do so, we should be obliged to invent new names for them, and to explain them to others. To avoid these inconveniences, a method has been contrived to express all the numbers, that are necessary to be used, with

very

fax games.

The first ten numbers have each a distinct name. The collection of ten simple units is then considered a unit: it is called a unit of the second order. We speak of the collections of ten, in the same manner that we speak of simple units ; thus we say one ten, two tens, three tens, four tens, five tens, six tens, seven tens, eight tens, nine tens. These expressions are usually contracted ; and instead of them we say ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety.

The numbers between the tens are expressed by adding the numbers below ten to the tens. One added to ten is called ten and one; two added to ten is called ten and two; three added to ten is called ten and three, &c. These are contracted in common language ; instead of saying ten and three, ten and four, &c., we say thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. These names seem to have been formed from three and ten, four and ten, &c. rather than from ten and three, ten and four, &c., the number which is added to ten being expressed first. The signification, however, is the same. The names eleven and twelve, seem not to have been derived from one and ten, two and ten ; although twelve seems to bear some analogy to two. The names oneteen, twoteen, would have been more expressive ; and perhaps all the numbers from ten to twenty would be better expressed by saying ten one, ten two, ten three, &c.

The numbers between twenty and thirty, and between thirty and forty, &c. are expressed by adding the numbers below ten to these numbers; thus one added to twenty is called twenty-one, two added to twenty is called twenty-two, &c.; one added to thirty is called thirty-one, two added to thirty is called thirty-two, &c.; and in the same manner forty-one, forty-two, fifty-one, fifty-two, &c. All the numbers are expressed in this way as far as ninety-nine, that is nine tens and nine units.

If one be added to ninety-nine, we have ten tens. We then put the ten tens together as we did the ten units, and this collection we call a unit of the third order, and give it a

It is called one hundred. We say one hundred, two hundreds, &c. to nine hundreds, in the same manner, as we say one, two, three, &c.

The numbers between the hundreds are expressed by adding tens and units. With units, tens, and hundreds we

name.

can express nine hundreds, nine tens, and nine units; which is called nine hundred and ninety-nine. . If one unit be added to this number, we have a collection of ten hundreds ; this is also made a unit, which is called a unit of the fourth order; and has a name. The name is thousand.

This principle may be continued to any extent. Every collection of ten units of one order is made a unit of a higher order ; and the intermediate numbers are expressed by the units of the inferior orders. Hence it appears that a very few names serve to express all the different numbers which we ever have occasion to use. To

express

all the numbers from one to nine thousand, nine hundred, and ninety-nine, requires, properly speaking, but twelve different names. It will be shown hereafter, that these twelve names express the numbers a great deal farther.

Various methods have been invented for writing numbers, which are more expeditious, than that of writing their names at length, and which, at the same time, facilitate the processes of calculation. Of these the most remarkable is the one in common use, in which the numbers are expressed by characters called figures. This method is so perfect, that no better can be expected or even desired. These figures are supposed to have been invented by the Arabs; hence they are sometimes called Arabic figures. The figures are nine in number. They are exactly accommodated to the manner of naming numbers explained above.*

* Next to the Arabic figures, the Roman method seems to be the most convenient and the most simple. It is very nearly accommodated to the mode of naming numbers explained above. A short description of it may be interesting to some; and it will often be found extremely useful to explain this method to the pupil before the other. The pupil will understand the principles of this, sooner than of the other, and having learned this, he will more easily comprehend the other. He will perfectly comprehend the principle of carrying, in this, both in addition and subtraction, and the similarity of this to the common method is so striking that he will readily understand that also.

The pupil may perform some of the examples in Sects. I, II, and VIII, Part I, with Roman characters.

THE ROMAN NOTATION.

1

One was written with a single mark, thus,
Two was written with two marks
Three was written
Four was written

One is written

1 Two is written

2 Three is written

3 Four is written

4 Five is written

5
Siz is written

6
Seven is written
Eight is written

8
Nine is written

9 These nine figures are sometimes called the 9 digits. By

Five was written
Six was written
Seven was written
Eight was written
Nine was written
Ten, instead of being written with ten marks,

was expressed by two marks crossing cach
other, thus,

Х
which expressed a unit of the second order.
Two tens or twenty were written

XX
Three tens or thirty were written

XXX And so on to ten tens, which were written with ten crosses. But as it was found inconvenient to express numbers so large as seven or eight, with marks as represented above, the X was cut in two, thus , and the upper part v was used to express one half of ten, or five, and the numbers from five to ten were expressed by writing marks after the V, to express the number of units added to five. Six was written

VI
Seven was written

VII
Eight was written

Vill
Nine was written

VILLE
The intermediate numbers between the tens were expressed by
writing the excess above even tens after the teps.
Eleven was written

XI
Twelve was written

Xll, &c.
Twenty-seven was written

XXVII, &c. To express ten Xs, or ten tens, that is, one unit, of the third order, or one hundred, three marks were used, thus, C. And to avoid the in eonvenience of writing seven or eight Xs, the C was divided, thus C and the lower part L used to express five Xs, or fifty:

To express ten hundreds, four dashes were used, thus, M. This lası was afterwards written in this form CĐ and sometimes Ci], and was then divided, and I was used to express five hundreds.

These dashes resemble some of the letters of the alphabet, and those letters were afterwards substituted for them.

The I resembles the 1; the V resembles the V; the X resembles the X, the L reseinbles the L; the C was substituted for the [; the 12 resembles the D; and the M resembles the M.

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