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to knowledge of fractions, in a great degree, contributes to a right understanding of the Rule of Proportion, on which the remaining part of the science entirely depends. In the Rule of Proportion he has ventured to differ from most of his predecessors, in making the ratio between the first and second numbers, in place of the first and third; and it is surprising that this was not earlier done by other writers on the subject,” especially as many of them were men of extensive erudition and highly talented, and though some of them have seen this matter in the same point of view that he does, yet they have excused themselves, by stating the difficulty that arises when long established custom is attempted to be laid aside ; this is a reason so powerful and of such magnitude, that he is ready to shrink from the effort of endeavouring to reform an erroneous system, but as he believes that many who are competent judges agree with him in opinion, he humbly hopes his attempt will not be eensured except by those who are willing to give a better reason than his deviation from the old beaten track. - In the choice of the questions and examples, much attention has been paid to select such as are useful and likely to occur in businessor life, most of them are entirely new, and many of them, or cases of like nature, have actually taken place. They are: much more numerous than in most books of arithmetic, consisting of nearly four thousand, of which more than three hundred are wrought at full length, perhaps in as concise a manner as possible, many not taking up half the figures required in similar eases, by the rules and examples given, even in some of the most celebratewpublications. The answers to those questions are not inserted at the end thereof, except to those which are worked at length, which he has found of advantage, as every one of them can be proved by a different operation, for which
* While writing the present work, and before it was ready for press, he has the pleasure of seeing this part of his plan anticipated by the publication of a treatise on arithmetic in theory and practice, by James Thompson, M.A.
there are directions given in the different rules; thus every question will have a double effect; but, for the use of the teacher, the answers to such questions to which it might be. useful to have answers are given under their proper heads at the end of the book, and so placed, as that they may be: removed or inserted or not as the teacher may require without, injury to the work. ... " The demonstrations and reasons of every rule, so much insisted on by some writers, except to the first rules, and in some other particular cases, are designedly omitted, as it is an incontestible truth, that, for the most part, they are far above. the comprehension of learners; but for the benefit of the more advanced student, such references are given, as will enable: him where to find the best, that in the author's opinion have yet been offered to public inspection. Notwithstanding that great” pains have been taken, both by the author and his printer, to render this work as perfect as possible, there are a few errorsin it, which are duly noticed; they are chiefly typographical, and were occasioned by the author's distance from the press; should any others have escaped his notice, he will be glad to receive communications from any intelligent teacher, and will: in another edition endeavour to have them corrected, or make: any real improvement they may have the kindness to suggest.
: CoRK, - * * * * * 7th Month, 1824, - - - *
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ARITHMETIC is that part of pure Mathematics which treats
of the science of Numbers. Number is either one, or a collection of units put together. A Unit is that by which any thing is considered as one,
and is the beginning of every number. Numbers are whole, broken, or mixed.
A whole number represents one, or a multitude of entire units, as–one man-three yards-seven crowns—one hundred pounds. A broken number, or fraction, is a part or parts of a unit, consisting of two parts, the numerator and denominator. The Denominator of a fraction, is the number of parts into which an integer, or whole number is supposed to be divided; and the Numerator, the number of those parts taken. A mixed number, is a whole number with a part or parts of a unitannexed thereto, as one and one-half, two and one-fourth, &c. The use of Arithmetic is to represent by written characters all sorts of numbers proposed, and to know their value—to add or collect numbers together—to subtract or take one number from another—to multiply one by the other—and, to divide or separate any number into as many equal parts as shall be assigned. The rule by which we read or express numbers, is called Numeration; and the other operations are performed by four B
rules or precepts, which are called Addition, Subtraction, Multiplication, and Division; by the proper application of which may be resolved all questions of possible solution. Strictly speaking, Arithmetic admits of but two operations, for in every change made on number it must be either increased or diminished—and Multiplication is but a compendious addition, and Division a compendious subtraction.
NUMERATION teaches to express numbers by words or figures, or to read or write any sum or number.
The characters or figures by which all numbers are expressed or denoted, are the following ten, lone, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 mine, 0 cipher.
These characters are also called Digits, from a latin word which signifies a finger, by the aid of which we are enabled to designate any number whatever, with the utmost facility and distinctness, and in a form, which is most convenient for arithmetical computation; beside the value which each figure has taken separately they have another, which depends on their place when taken together, as may be seen by the following
1 2 3 4 5 6 78 9 Hundreds of Millions 100000000
1 2 3 4 5 6 7 8 Tens of Millions 10000000
The figure in the first place reckoning from the right to the left, always denotes only its simple value; that in the second place ten times its simple value, that in the third place one hundred times its simple value, and so on ; each successive place being always ten times its former value, thus, in the number 9999, the nine in the first place signfies only nine, that in the second place nine tens or ninety, that in the
third place nine hundred, and that in the fourth place nine thousand; which several values collected together may be thus read, nine thousand nine hundred and ninety-nine. * The cipher stands for nothing by itself, and placed at the lefthand of another figure alters motits value; but affixed at the right hand of another figure, or figures, it increases their value in the same ten-fold proportion, thus, 6 signifies only six, but 60 signifies sixty, 600 six hundred, and 6000 six thousand. To facilitate numeration it is common to mark off by a comma every six figures, commencing from the right hand, every such division is called a period; and sometimes into divisions of half periods, each consisting of only three figures. The name of the first period is called Units; of the second, Millions; of the third, Billions; of the fourth, Trillions; of the fifth, Quadrillions, &c. and as the name of Million is given to ten hundred thousand, so ten hundred thousand millions is called a Billion; the place of billions therefore commences at the thirteenth column or place,f in like manner the name of Trillion, Quadrillion, &c. The first part of any period is so many units of it, and the latter so many thousands. - * "The following TABLE exhibits a summary of the whole doctrine.
The names of the other periods are Quintillions, Sextillions, Septillions, Octillions, and Nonillions, each period consisting as above of six places of figures.
• For the further illustration of the beautiful simplicity and convenience of this method of designating numbers, and its superiority to the literal notation which had anciently been in use for the purpose, I refer the reader to John Walker's “Philosophy of Arithmetic,” where he will find this matter treated in an ingenious and elegant manner.-Pages lto3.
+ Here it may be useful to some to remark, that the facility with which we can express, multiply and divide large numbers, makes us insensible of the enormity of their bulk when they attain the higher periods, for example, a billion, as there is not this number of seconds in thirty thousand years, nor would there be a billion of §: of corn produced on 60000 acres of good land, though there were 8000 grains in a pint, and that 4 bushels make a barrel, and that each acre produced eight barrels of grain at an average.