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THIs ARITHMETICK has been compiled with a view to facilitate the progress of pupils, and lessen the labour of teachers. The questions preceding and following the rules, are designed to lead young learners into habits of thinking and calculating; and thus, to prepare them for practical operations. Experience has demonstrated, that, in the instruction of children in any science, it is necessary to excite their entire attention to the subject before them. The latent energies of their minds must be roused up, and called forth into action. When this can be effectually done, success is rendered certain.— To accomplish this important object, the best method has been found in the frequent use of well selected questions. Though it is a successful course in all juvenile studies, it is particularly so in the science of numbers; and the progress of pupils must be slow with. out it. The questions in the following pages are thought to be sufficiently numerous for the purposes intended; the rules have been arranged according to the plan of some of the best authors on this subject, and the work is offered to the publick with the hope that it will be useful in the schools of our country.

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ARITHMETICK is the science which treats of the nature and properties of numbers: and its operations are conducted chiefly by five principal rules. These are, Numeration, Addition, Subtraction, Multiplication, and Division. |. Numbers in Arithmetick are expressed by the following ten digits or characters, namely : 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, 0 |cypher.

An Integer signifies a whole number, or certain quan tity of units, as one, three, ten. A Fraction is a broken number, or part of a number, as # one half, 8 two-thirds, # one-fourth, 3 three-fourths, five-eigths, &c.

Numeration teaches the different value of figures by their different places, and to express any proposed numbers either by words or characters; or to read and

write any sum or number. -

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8 NUMERATION,

Here any figure in the place of units, reckoning from right to left, denotes only its simple value; but that in the second place denotes ten times its simple value; and that in the third place, one hundred times its simple value ; and so on, the value of any figure in each successive place, being always ten times its former value. Thus in the number 6543, the 3 in the first place denotes only three ; but 4 in the second place signifies four tens or 40; 5 in the third place, five hundred ; and six in the fourth place, six thousand ; which makes the whole number read thus—six thousand five hundred and fortythree. The cypher stands for nothing when alone, or when on the left hand side of an integer; but being joined on the right hand side of other figures, it increases their value in the same ten fold proportion : thus, 50 denotes five tens; and 500 is read five hundred. Though the preceding numeration table contains only twelve places, which render it sufficiently large for young students, yet it may be extended to more places at pleasure. - EXAMPLE.

2-—o-\ ,-\ ,—o-, -oQuatrillions. Trillions. Billions. . Millions. Units. 987,654; 321,234; 567,898; 765,432; 123,456 Here note, that Billions is substituted for millions o millions: Trillions, for millions of millions of millions: Quatrillions, for millions of millions of millions of millions. From millions, to billions, trillions, quatrillions, and other degrees of numeration, the same intermediate denominations, of tens, hundreds, thousands, &c. are ised, as from units to millions. And thus, in ascertaining the amount of very high numbers, we proceed from Millions to Billions, Trillions, Quatrillions, Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, |Octodecillions, Novemdecillions, Vigintillions, &c. all of which answer to miliions so often repeated, as their indices respectively require, according to the above proportion. -

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