The third part is mostly practical, and composed of such rules 95 30! Sect. 1.-Numeration 1] Frac. ch'ng'd to other Frac.94 Sect. 2.-Simple numbers O Common Divisors Simple Multiplication 8 Common Multiples 97 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 Vulgar Fractions changed to Extraction of Square Root 109 Sect. 1.-Compound numbers 37 Sect.9.-- Miscellaneous rulesi 18 Tables of money,waits &c. 38 Arithmetical Progression 118 43 Geometrical Progression 120 Reduction of Decimals 46 Duodecimals Compound Addition 48 Position 124 Compound Subtraction 50 Perinutation of Quantities 126 Multiplication and Division 52 Periodical Decimals 127 Secr 5.- Per Cent 57 PRACTICAL EXERCIBES. 57 Sect. 1.- Exchange of our Varieties in Interest 62 Currencies. Commission and Insurance 63 Sect. 2.- Mensuration. 64 Mensuration of Superficies132 Compound Interest 67 Mensuration of Solids 136 68|Sect. 3.-Philosophical mat Equation of Payments 70 Fall of Heavy Bodies 138 Single Ruie of Three 74 Mechanical Powers 142 Double Rule of Three 78 Sect. 4.-Miscellaneous Assessment of Taxes 841 and Tables 88 Table of Cylindric m'sure 149 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. Quadrillions. Tens. 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 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. |