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TO THE PUBLIC.
When the improved edition of this work was published, in 1828, it was intended that the Written Arithmetic which forms the sec ond and third parts should always be accompanied by the Mental Arithmetic embraced in the first part. Since that time it has, however, been thought best to transpose such tables from the Mental to the Written Arithmetic, as to render the latter coin. plete without the former, in order to lessen the expense of tho book to those who do not wish to study mental arithmetic, or who have studied some other treatise; and, thus prepared, it is now presented to the public. No alteration has been made from the last edition in the arrangiment of the rules, and the whole of the second part is presented as before, on the inductive plan of Lacroix. The principles are first devolved by the analysis of familiar examples, and the method of applying these principles to the
solution of questions is then expressed in general terms, forming > a Rule, which is still further illustrated by a great variety of prac
tical questions. The analysis is printed in snall type, occupies but little space, and may be omitted by those who wish to use rules without understanding them.
Addition and Multiplication, both involving the same principles, are presented in connexion, and alsu Subtraction and Divisiun. A knowledge of derinals being necessary 10 a good understanding of our federal currency and this knowledge being easily acquired by snch as have learned the noiation of wLole numbers, decimals and Federal money are introduced immediately after the first section on simple numbers. By acquainting ine pupil thus early with decimals, he will be likely to understand ihem better and to avail himself of the facilities they afford in the soJution of questions and the transaction of business.
Reduction ascending and desceniling are arranged in parallel columns and the answers to the questions of one column are found in the corresponding questions of the other. Compound multipli. cation and division are arranged in the same way, and only one general rule for each is given, which was thought better than to perplex the pupil with a multiplicity of cases.
Interest and other calculations by the hundred are all treated decimally, that method being most simple and conformable to the notation of our currency. The nature and principles of proportion are fully developed and the method of applying them to tho solution of questions clearly shown.
The written urithmetic of fractions being, to young pupils, somewhat difficult to be understood, is defcrred till ihey are made familliar with the most im.portani arithmetical operation perform. ed with whole numbers and decimals. The nature of roots and poroers has been more fully explained in the present edition, and Beveral new diagrams introduced for their elucidation. Through. out the second pari, 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 final no difficulty A applying them to the particular cases which may occur,
The third part is mostly practical, and composed of snch rulet
57 Secr. 1.-Exchange of our
68|Sect. 3.-Philosophical mat
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 onc, 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 excepe tions, 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, firy, sixty, seoenty, 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 forly 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 hun dred 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 names, 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 numþers is very much abridged. The method of writing numbers in characters is called Notution. The two methods of nota. tion, which have been most extensively used, are the Roman and the Arabic.* The Roman numerals are the seven following letters of the alphahet, 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 suuhers 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 courteon we write 14, for twentyto, 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 unil'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. i=1 110=X 1100=0 1000=M orCI) 10000=xorcclo 2-1 20=XX 200=CC 1100=MC 50000==
I၁၃၁ 3=lil 30=XXX 300=CCC 1200=MCC =IV 10=XL 400=CCCC 1300=MCCC 100000=CCCI??
50-L 500=Dorlɔ 1400=MCCCC 1000000=M 16-VI 60=LX 600=20 1500=MD
2000000=MM VII 70=LXX 700 DCC '2000=MM 1929=-MDCCCXXIX 12=VIII 0=,XXS SON=DCCC 5000= =!or 2=IX :90=XC 900 =1280cc'c000=i?
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 ? repeated three times, and each time with a different value. The 2 in the second place denotes a number ten times great er 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
Tens of Bill.
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