it stands with regard to other figures; thus, the first thing to be done is, to ascertain the place, or rank of each figure, in order that, on reading it, he may know whether to call it units, tens, hundreds, thousands, &c. This important point is ascertained in the following easy manner. Thus, we begin with the figure on the right hand of the series, and count on to that on the left; saying units, tens, hundreds, thousands; hereby ascertaining, that the figure on the left represents thousands; we read thus, three thousand, four hundred and fifty-seven. If another figure be added to the series, it raises the highest to tens of thousands; and, if to these yet another be added, we shall have hundreds of thousands, and so on. I will now add two more, and place them at the left hand of the series, thus: 2 6 3 4 5 7 Counting back, from the place of units, as before, we shall find, that the figure 2 is in the place of hundreds of thousands, and we shall read the series thus: two hundred sixty-three thousand, four hundred and fifty-seven; another figure would lead us into millions, as : 3263457 three millions, two hundred sixty-three thousand, four hundred and fifty-seven, and another after that, into tens of millions, as: 43 26 3457 Let us now see how the cipher, or naught, operates in this branch of the subject. We will take out the figure 6 from the above series, and insert, in the place of it, a cipher; we shall then have 203457 which is to be read, two hundred and three thousand, &c., instead of two hundred, sixty-three thousand, &c. If we insert the cipher in the place of the 3, we shall have 260457 which is two hundred and sixty thousand, &c. By these examples the attentive learner will not fail to become fully acquainted with the use of this figure, or cipher. There are besides the denominations of figures of which I have already treated, namely, units, tens, hundreds, and thousands, some of yet higher denominations, as, billions, trillions, &c. These large numbers are not wanted in the affairs of business, nor will they now be worthy of much attention from the learner. But as this division of the work would not be complete without some notice of them, I shall here insert the table of numbers, from units to billions, and the learner can refer to it whenever occasion or curiosity may lead him to do so. NUMERATION TABLE, Showing the Places of 1 Units. 10 Tens. 100 Hundreds. 1000 Thousands. 10000 Tens of Thousands. 100000 Hundreds of Thousands. 1000000 Millions. 10000000 Tens of Millions. 100000000 Hundreds of Millions. 1000000000 Thousands of Millions. We have here, also, the use of the cipher, and the value and importance of the several places of figures, clearly shown; the unit one, being advanced first to ten, then to one hundred, and onward, until, by the addition of nine ciphers, it expresses one thousand millions. Q. How must figures be numerated? Q. In what manner should figures be read? Q. Why do we numerate figures from the right hand to the left? A. Because they increase in value from the right to the left. Q. In what proportion do they increase in value? A. In a tenfold proportion; as, ten units make one ten, ten tens one hundred, ten hundreds one thousand, &c. EXPLANATIONS. To read large amounts requires a kind of spelling, which is to be avoided by dividing the long lines of figures into portions of three figures each, by a comma, as here shown, thus: 1,000,000,000 By this means the scholar will see, at a glance, which are hundreds, which thousands, and which millions; and he can begin at once to name them. SECOND NUMERATION TABLE. 3, 5 7 8, 1 4 6, 4 3 7, 3 4 Millions. Tens of Millions. Hundreds of Thousands of Millions. -Hundreds of Millions. Thousands of Millions. Billions. Note.-To TEACHERS. I have treated thus largely of Notation and Numeration, as they may be justly considered the basis of every Arithmetical operation, and from the belief that their nature and principles are generally not well understood by the learner, if understood at all. In no branch is it more necessary that the learner should be well versed, than in the nature and principles of Notation and Numeration; for, without a thorough knowledge of these, he will be wholly unable to enter upon any subsequent rule of Arithmetick. Having now, as I trust, fully explained to the young learner the nature and use of NOTATION and NUMERATION, I shall now proceed to an explanation OF THE WORKING OF FIGURES. In future, I shall address myself directly to the learner, and proceed to this branch of the subject, by which you will be enabled to accomplish all the purposes, whether of business or of curious inquiry, for which the art of Arithmetick is adapted. This working of figures, this reckoning, and calculating, and adjusting of numbers, consists merely of joining and separating them; that is, of joining them in certain ways, and of separating them into other portions as occasion may require. This is the whole art of Arithmetick. To join and to separate are but two modes of proceeding; and by two modes all the ends of Arithmetick may be accomplished, even to the largest and most important of its purposes. These two modes, by which it is possible to accomplish every object of the art, are, first, ADDITION, or the joining together different numbers; and, second, SUBSTRACTION, or the separating of different numbers. Though every thing may be accomplished by these two processes, it would, however, in most cases, be very tedious, unpleasant, and a great waste of time, to confine ourselves to the use of them merely; so another mode of joining numbers together, than that of Addition, is, in certain cases, adopted; and another mode of separating them than that of Substraction. These other modes are called, the first, MULTIPLICATION, and the other DIVISION. Thus, although every thing may be accomplished by Two modes of proceeding, TWO OTHER modes, for the purpose of abridging the labour, in certain cases, have been adopted; and these four processes, together with Numeration, are called the funda mental or foundation rules of Arithmetick. |