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part, it has been our main object to familiarize the pupil with the fundamental principles of the science, believing that when these are well understood, he will find no difficulty in applying them to the particular cases which may occur.

The third part is mostly practical, and composed of such rules and other matters as we conceived would be interesting and useful to the student and the man of business.

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ARITHMETIC.

PART II.

WRITTEN ARITHMETIC.

SECTION I.

NOTATION AND NUMERATION.

70. An individual thing taken as a standard of compari son, is called unity, a unit, or one.

71. Number is a collection of units, or ones.

72. Numbers are formed in the following manner; one and one more are called two, two and one, three, three and one, four, four and one, five, five and one, six, six and one, seven, seven and one, eight, eight and one, nine, nine and one, ten; and in this way we might go on to any extent, forming collections of units by the continual addition of one, and giving to each collection a different name. But it is evident, that, if this course were pursued, the names would soon become so numerous that it would be utterly impossible to remember them. Hence has arisen a method of combining a very few names, so as to give an almost infinite variety of distinct expressions. These names, with a few exceptions, are derived from the names of the nine first numbers, and from the names given to the collections of ten, a hundred, and a thousand units. The nine first numbers, whose names are given above, are called units, to distinguish them from the collections of tens, hundreds, &c. The collections of tens are named ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety.(6). The intermediate numbers are expressed by joining the names of the units with the names of the tens. To express one ten and four units, we say fourteen, to express two tens and five units, we say twenty-five, and others in like manner. The collections of ten tens, or hundreds, are expressed by placing before them the names of the units; as, one hundred, two hundred, and so on to nine hundred. The intermediate numbers are formed by joining to the hundreds the collections of tens and units. Το express two hundred, four tens, and six units, we should

say, two hundred forty-six. The collections of ten hundreds are called thousands, which take their names from the collections of units, tens and hundreds, as, one thousand, two thousand, ten thousand, twenty thousand, one hundred thou sand, two hundred thousand, &c. The collections of ten hundred thousands are called millions, the collections of ten hundred millions are called billions, and so on to trillions, quadrillions, &c. and these are severally distinguished like the collections of thousands. The foregoing names, combined according to the method above stated, constitute the spoken

numeration.

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73. To save the trouble of writing large numbers in words, and to render computations more easy, characters, or symbols, have been invented, by which the written expression of numbers is very much abridged. The method of writing numbers in characters is called Notation. The two methods of notation, which have been most extensively used, are the Roman and the Arabic.* The Roman numerals are the seven following letters of the alphabet, I, V, X, L, C, D, M, which are now seldom used, except in numbering chapters, sections, and the like. The Arabic characters are those in common use. They are the ten following: 0 cipher, or zero, 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine. The above characters, taken one at a time, denote all the numbers from zero to nine inclusive, and are called simple units. To denote numbers larger than nine, two or more of these characters must be used. Ten is written 10, twenty 20, thirty 30, and so on to ninety, 90; and the intermediate numbers are expressed by writing the excesses of simple units in place of the cipher; thus for fourteen we write 14, for twentytwo, &c.(13) Hence it will be see that a figure in the second place denotes a number ten times greater than it does when standing alone, or in the first place. The first place at the right hand is therefore distinguished by the name of unit's place, and the second place, which contains units of a

*A comparison of the two methods of notation is exhibited in the following TABLE.

*

1=1 10-X 1100=C
12=II 20=XX 200-CC
3=III 30=XXX 300=CCC
4-IV 40-XL 400-CCCC
5=V 50=L 500 D orI
|6=VI |60=LX 600-DC 1500-MD
7=VII 70=LXX 700 DCC 2000 MM

8=VIII 80=LXxx S00-DCCC 5000=100 or V
9-IX 90 XC 900 DCCCC 6000=vI

1000 M orCIO 10000=xorCC
1100=MC
50000=1000
1200 MCC
60000 LX
1300=MCCC 100000
1400-MCCCC 1000000=M
2000000=MM

1829 MDCCCXXIX

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higher order, is called the ten's place. Ten tens, or one hundred, is written, 100, two hundred, 200, and so on to nine hundred, 900, and the intermediate numbers are expressed by writing the excesses of tens and units in the tens' and units' places, instead of the ciphers. Two hundred and twenty-two is written, 222. Here we have the figure 2 repeated three times, and each time with a different value. The 2 in the second place denotes a number ten times greater than the 2 in the first; and the 2 in the third, or hundreds' place, denotes a number ten times greater than the 2 in the second, or ten's place; and this is a fundamental law of Notation, that each removal of a figure one place to the left hand increases its value ten times.

74. We have seen that all numbers may be expressed by repeating and varying the position of ten figures. In doing this, we have to consider these figures as having local values, which depend upon their removal from the place of units. These local values are called the names of the places: which may be learned from the following

TABLE I.

coSextillions.

Hund. of Quint.
Tens of Quint.
Quintillions.
Hund. of Quad.
Tens of Quad.
Quadrillions.

co Hund. of Trill.

Tens of Trill.
coTrillions.
Hund. of Bill.
Tens of Bill.
Billions.
Hund. of Mill.

Tens of Mill.
Millions.

Hund. of Thou.
Tens of Thou.
-Thousands.

Hundreds.

coTens.
Units.

By this table it will be seen that 2 in the first place denotes simply 2 units, that 3 in the second place denotes as many tens as there are simple units in the figure, or 3 tens; that 2 in the third place denotes as many hundreds as there are units in the figure, or 2 hundreds; and so on. read any number, we have only to observe the following RULE.-To the simple value of each figure join the name of its place, beginning at the left hand, and reading the figures in their order towards the right.

Hence to

The figures in the above table would read, three sextillions, four hundred fifty-six quintillions, seven hundred fifty-four quadrillions, three hundred seventy-eight trillions, four hundred sixty-four billions, nine hundred seventy-four millions, three hundred one thousand, two hundred thirty-two.

75. In reading very large numbers it is often convenient to divide them into periods of three figures each, as in the following

TABLE II.

Duodecillions.

Undecillions.

Decillions.

Nonillions.

Octillions.

Septillions.

Sextillions.

Quintillions.

Quadrillious.

Trillions.

Billions.

Millions.

Thousands.

Units.

532, 123,

410,864,232,012,

345, 862,051,234, 525, 411, 243, 673. By this table it will be seen that any number, however large, after dividing it into periods, and knowing the names of the periods, can be read with the same ease as one consisting of three figures only; for the same names, (hundreds, tens, units,) are repeated in every period, and we have only to join to these, successively, the names of the periods. The first, or right hand period, is read, six hundred seventy-three -units, the second, two hundred forty-three thousands, the the third, four hundred eleven millions, and so on.

76. The foregoing is according to the French numeration, which, on account of its simplicity, is now generally adopted in English books. In the older Arithmetics, and in the two first editions of this work, a period is made to consist of six figures, and these were subdivided into half periods, as in the following

TABLE III. Periods. Sextill. Quintill. Quadrill. Trill. th. un. th. un. th. um. cxt. cxu.

Billions. Millions. Units.

Half per. th. un. th. un. th. un.
Figures. 532,123, 410,864, 232,012, 345,862, 051,234, 525,411,243,673

Ma

These two methods agree for the nine first places; but beyond this, the places take different names. Five billions, for example, in the former method, is read five thousand millions in the latter. The principles of notation are, notwithstanding, the same in both throughout-the difference consisting only in enunciation.

Write the following in figures: Eight. Seventeen. Ninety-three. Three hundred sixty. Five thousand four hundred and seven. Thirty thousand fifty nine. Seven millions. Sixty-four billions. One hundred nine quadrillions, one hundred nine millions, one hundred nine thousand, one hundred and nine.

EXAMPLES FOR PRACTICE.
Enumerate, or write the follow-
ing in words:

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