AN ELEMENTARY TREATISE ON ARITHMETIC. CHAPTER I. NUMERATION, ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF WHOLE NUMBERS. 1. WHATEVER is capable of increase or diminution is, in general, called magnitude, or quantity. For example, a sum of money is a quantity, since we may increase or diminish it. In like manner, a weight and other things of this nature are quantities. 2. In order to measure or determine any quantity, we must consider some other quantity of the same kind as known, which shall be used as a term of comparison and this quantity is called a unit. If it were proposed, for example, to determine the quantity of a sum of money, we should take some known piece of money, as a dollar, an eagle, a crown, or some other coin, and show how many of these pieces are contained in the given sum. In the same manner, if it were proposed to determine the quantity of a weight, we should take a certain known weight, for example, a pound, an ounce, &c. and then show how many times one of these weights is contained in that which we are endeavouring to ascertain. If we wished to measure any length or extension, we should make use of some known length, such as a foot. So that the determination, or the measure of magnitude of all kinds, is reduced to this; fix at pleasure upon any one known magnitude of the same species with that which is to be determined, and consider it as the measure, or unit; then determine the mutual relation of the proposed magnitude and this measure. B 3. This relation is always expressed by numbers; so that a number is nothing but the relation of one magnitude to another, arbitrarily assumed as the unit. A combination of units gives rise to whole numbers; the manner of forming, expressing, and writing numbers by means of characters invented for that purpose, is the object of numeration; and the science which teaches how to perform different operations upon numbers is called Arithmetic. From this it appears that all magnitudes may be expressed by numbers; and that the foundation of all the mathematical sciences must be laid in a complete treatise on the science of numbers, and in an accurate examination of the different possible methods of calculation. NUMERATION OF WHOLE NUMBERS. 4. In order to form whole numbers, we must begin with unity, or one; unity added to itself gives a number named two; this increased by one composes the number three; and by adding successively unity to each number obtained, we compose the numbers four, five, six, seven, eight, nine. This last increased by unity gives the number ten; the collection of ten units form a new order of units, named tens. And in the same manner as we have counted from unity to nine units, we shall also count from one ten to nine tens; but instead of the words one ten, two tens, three tens, four tens, five tens, six tens, seven tens, eight tens, nine tens, we say, ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety. In order to express the numbers comprised between two consecutive tens, we express successively the tens and the units; thus, the collection of three tens and seven units is named thirty-seven. We must except, from this system, the numbers comprised between ten and twenty; for, instead of the words ten-one, ten-two, ten-three, &c. we say, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. 5. The collection of nine tens and nine units forms the number ninety-nine; this increased by a unit gives |
