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

95

30!

CONTENTS-WRITTEN ARITHMETIC.

Sect. 1.-Numeration 1] Frac. ch'ng'd to other Frac.94

Sect. 2.-Simple numbers O Common Divisors

Simple Addition

5 Fractions Reduced

96

Simple Multiplication 8 Common Multiples 97

Siinple Subtraction

13 Common Denominators 98

Simple Division

16 Reduction of Fractions 100

Sect. 3.- Decimals

24 Addition of Fractions 102

Numeration of Decimals 25 Subtraction of Fractions 103

Addition of Decimals 26 Rule of three in Fractions 103

Multiplication of Decimals 27 Secr. 8.-Roots and Powers 100

Subtraction of Decimals 299 Involution

106

Division of Decirnals

Evolution

108

Vulgar Fractions changed to Extraction of Square Root 109

Decimals

31

of Cube Root 113

Federal Money

33

of Roots in general 116

Sect. 1.-Compound numbers 37 Sect.9.-- Miscellaneous rulesi 18

Tables of money,waits &c. 38 Arithmetical Progression 118

Reduction

43 Geometrical Progression 120

Reduction of Decimals 46 Duodecimals

122

Compound Addition 48 Position

124

Compound Subtraction 50 Perinutation of Quantities 126

Multiplication and Division 52 Periodical Decimals 127

Miscellaneous matters 54

PART III.

Secr 5.- Per Cent

57 PRACTICAL EXERCIBES.
Simple Interest

57 Sect. 1.- Exchange of our

Varieties in Interest 62 Currencies.

129

Commission and Insurance 63 Sect. 2.- Mensuration.

Interest on Notes

64 Mensuration of Superficies132

Compound Interest 67 Mensuration of Solids 136
Discount

68|Sect. 3.-Philosophical mat

Loss and Gain

69 ters

138

Equation of Payments 70 Fall of Heavy Bodies 138

Sect. 6.- Proportion

72! Pendulums

141

Single Ruie of Three 74 Mechanical Powers 142

Double Rule of Three 78 Sect. 4.-Miscellaneous

Fellowship

80 Questions

144

Alligation

81 Sect. 5.- Practical Rules

Assessment of Taxes 841 and Tables

148

Sect. 7.-Fructions

88 Table of Cylindric m'sure 149

Definitions

891

Square Timber 6 150

Integers treated as Fract'us 90 Log Table-Log

153

M'ltipli. and Div. of 91

Board 154

92 Sect. 6.— Book Keeping

155

Division by

93 Practical Forins,

164

by

ARITHMETIC.

PART II.

WRITTEN ARITHMETIC.

SECTION I.

NOTATION AND NUMERATION. 70. An individual thing taken as a standard of comparison, 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 varietv 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 ien, 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. To 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 collec tions of units, tens and hundreds, as, one thousand, two thou sand, 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 frillions, quadrillions, &c. and these are severally distinguished like the collections of thousands. The foregoing pames, combined according to the method above stated, constitute the spoken numeration.

73. To save the trouble of writing large numbers in words, and to render computations inore 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 Notution. 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: O 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 29, thirty 30, and so on to ninety, 30; and the intermediate numhers are expressed by writing the excesses of simple units in place of the cipher; thus for fourteer we write 14, for ntytwo, 22, &c.(13) Hence it will be seen 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 units 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=I 110=X (100=C 11000=M orCl] 10000= orCCIO 2=II 20=XX 200=CC 11100=MC

50000=1020 3=III 30=XXX 300=CCC 1200=MCC

60000=LX 4=IV 40=XL 400=CCCC 1300=MCCC 100000=CCCIɔɔɔ 5=V 50=L 500=D orl] 1400=MCCCC 1000000=M |6=VÍ |60=LX 600=DC 1500=MD

2000000=MM 7=VII 70=LXX 700=DCC 2000=MM 1829=MDCCCXXIX 8=VIII 80=LXXX 300=DCCC 5000=1ɔɔ or v 3=IX 90=XC 900=ncccc 6000=yt

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.

Sextillions.
Hund. of Quint.
cTens of Quint.
oQuintillions.
«Hund. of Quad.
erTens of Quad.

Quadrillions.
wHund. of Trill.
Tens of Trill.
Trillions.
Hund. of Bill.
Tens of Bill.
Billions.
coHund. of Mill.
Tens of Mill.
Millions.
WHund. of Thcu.
o Tens of Thou.
-Thousands.
Hundreds.

Tens.
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. Hence to read any number, we have only to observe the following

Rule.-To the simple value of each figure join the nume of its place, beginning at the left hand, and reading the figures in their order towards the right.

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

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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. Billions. Millions. Units. Half per. th. un. th. un.

th. un. thu ul. thi un. cxt. cau.

th. un.

Figures. 1532,123,410,864, 232,012, 345,862,051,234,525,411, 243,673

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 enunciationi.

EXAMPLES FOR PRACTICE. Write the following in figures: Env:nerate, or write the followEight. Seventeen. Ninety-three. ing in words: Three hundred sixty Five thou

31

7890112 sand four hundred and seven. Thir

65

7-4351234 ty thousand fifty nine. Seven

123

137111055 millions. Sixty-four billions. One

2040 8900000000 hundred nine quadrillions, one hun

60735 30000010010 dred nine millions, one hundred nine

123456 222000111002 thousand, one hundred and nine.

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