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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 arrangement 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 praca
tical questions. The analysis is printed in sniall type, occupies but little space, and may be omitted by those who wish to use rules without undirstanding them.
Addition and Multiplication, both involving the same principles, are presented in connexion, and alsu Subtraction and Divisiun. A knowledge of deciinals being necessary to a good under. standing 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 solution of questions and the transaction of business.
Reduction asrending 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
I 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 the 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.portanı arithmetical operation perform. ed with whole numbers and decimals. The nature of roots and powers has been more fully explained in the present edition, and several 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 jhat when these are well understood, he will final no difficulty of applying them to the particular cases which may occur,
The third part is mostly practical, and composed of snch rulet
Secr. 2.- Simple number: 51 Of Common Divisors
Compound Addition* 401 Position
57 Sect. 1.-Exchange of our
Equation of Payments 701 Fall of Heavy Bodies
Assessment of Taxes 8! .. and Tables
SECTION ka 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 excep tions, are derived from the names of the nine first numbers, and from the names given to the collections of ten, a huna dred, 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, fify, sixty, sexenty, 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 twenly-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 thout 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 trillions, quarillions, &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,
75. 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 nota. tion, 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 norv 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,
two, 3.three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 ninc. The above characters, taken one at a time, denote all the finahers 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 s0, and so on to ninety, 90; and the intermediate numibers are expressed by writing ibe excesses of simple units in place of the cipher; thus for fourteen 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 units place, and the second place, which contains units of a
NUMERATION. 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 tenzi and units places, instead of the ciphers. Two liundred and twenty-two is written, 222. Here we have the figure 2 repeated three times, and each time with a differeni 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
Tens of Trill.
Tens of Mill.
Tens of Thou.
By this table it will be seen that 2 in the first place deuotes 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 name 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