there is no limit to this increasing operation; and that whatever number of objects be collected together, more may be still added; so that we can never reach the highest possible number, nor approach near to it. As by collecting objects, we are led to understand and add numbers, so by removing the objects collected, we learn to diminish numbers. It is very difficult to form any adequate conception of very high numbers. We frequently speak of numbers, of the extent of which we have no adequate idea. If a person were to reckon a hundred balls every minute, and were to continue at work so reckoning, ten hours a day, he would spend seventeen days in reckoning one million; and a thousand men, reckoning at the same rate, would take forty five years, to number out a billion. All numbers are represented by the ten following characters. 1. 2. 3. 4. 5. 6. 7. 8. 9. one, two, three, four, five, six, seven, eight, nine, O. cypher, or nought. The nine first are called significant figures or digits, and accordingly, as they are placed, represent units, tens, hundreds, or higher classes of numbers. When placed singly, they denote the simple numbers signified by the characters. When several are placed together, the first, or right hand figure only, is to be taken for its simple value; the second, signifies so many tens; the third, so many hundreds; and the others, so many higher classes, according to the order in which they stand. And as it may sometimes be requisite to express a number consisting of tens, hundreds, or higher classes, without any units or QUADRILLIONS. Hundred Thousands of Trillions.. Hundreds of Thousands of Millions... Ten Thousands of Millions.. Thousands of Millions..... Hundreds of Millions.. classes of a lower rank annexed; and as this caonly be done by figures standing in the second, third, or higher place, while there are none to fill up the lower places; therefore, an additional character or cypher (0) is necessary, having no signification of value, when standing alone, but serving to supply the vacant places, and bring the figures to their proper station. The rule of arithmetic, which teaches how to read these characters expressive of numbers, however they may be combined, is called notation, or numeration. The following Table shows the names and divisions of the classes: 2, 3 5 6, 3 4 9, 7 5 3, 2 4 5, 8 9 3, 2 5 4, 7 5 4, 1 2 3 The first six figures from the right hand, are called the unit period; the next five, the million period; the next five, the billion period; then the trillion, quadrillion, quintillion, sextillion,' septillion, oxtillion, and nonillion periods, follow in their order. Before any number is reckoned, it is proper to divide it into periods and half periods, by different marks. Then, begin at the left hand, and read the figures in their order, with the names of their places, from the tables. In writing any number, it is necessary to mark the figures in their proper places, and supply the vacant places with cyphers. Addition, subtraction, multiplication, and division, form the foundation of all arithmetical operations. As the idea of number is acquired by observing several objects collected, so the idea of fractions is acquired by observing an object divided into many parts. As we sometimes meet with objects broken into two, three, or more parts, we may consider any, or all of those divisions promiscuously. This operation is performed by the rule, called Vulgar Fractions. And since the practice of collecting units into parcels of tens has prevailed universally, it has been found convenient to follow a like method in the consideration of fractions, by dividing each unit into ten equal parts, and each of these into ten smaller parts; and so on. Numbers, divided in this manner, are called Decimal Fractions. The following characters are frequently used by way of abbreviation : Characters, expressing The sign of equality. The sign of addition, as, 9+9-18, that is, nine added to nine, are equal to eighteen. Minus, or less. Sign of subtraction, as, 9-5-4, nine less five equal to four. X Multiplied by. Sign of Multiplica tion, as 8×8-64, eight, multiplied by eight, equal to sixtyfour. Divided by. Sign of division, as 28-7-4, twenty eight, divided by se ven, equal to four. Proportion. Sign of Proportion, as, 8:16 16:32, eight is to sixteen, as sixteen to thirty-two.. QUESTIONS. What is the science of arithmetic, and what is its apparent antiquity? How did the Greeks represent their numbers? What are the Roman characters for numbers? What is sexagesimal arithmetic ; when, and by whom was it invented? Whence came the characters and mode of notation of numbers now in use? How are the ideas of addition gained? Is it easy to acquire a distinct idea of very high numbers? How many characters are employed to represent all numbers? Which are the significant figures? How is notation or numeration, performed? What rules 6 constitute the foundation of all arithmetical operations? How is the idea of fractions obtained? What are vulgar fractions? And what are decimal fractions? What signs are adopted for the sake of abbreviation. CHAP. XI. ARITHMETIC- continued. ADDITION teaches how to find the amount of any given numbers. Rule. According to the principle of notation, or numeration, place the figures to be added under each other, according to their value; units under units, tens under tens, hundreds under hundreds, and so on. Begin by adding together the units; set down the unit figure of their sum, and carry on the tens to the next column; continue the operation till you come to the last column, under which put down its whole amount. Example 1. Add together the following sets of numbers, 94141, 394, 29, 7451, and 35120. The sets of numbers being thus prepared by the arrangement of units under units, tens under tens, hundreds under hundreds, thousands under thousands, and tens of thousands under tens of thousands, they are added together, as follows. |