THE SCHOLAR'S ARITHMETIC. INTRODUCTION. ARITHMETIC is the art or science which treats of numbers. It is of two kinds, theoretical and practical. The THEORY of Arithmetic explains the nature and quality of numbers, and demonftrates the reason of practical operations. Confidered in this sense Arithmetic is a Science. PRACTICAL ARITHMETIC fhews the method of working by numbers, fo as to be most useful and expeditious for bufinefs. In this fenfe Arithmetic is an Art. DIRECTIONS TO THE SCHOLAR. Deeply imprefs your mind with a sense of the importance of arithmetical knowledge. The great concerns of life can in no way be conducted without it. Do not, therefore, think any pains too great to be bestowed for so noble an end. Drive far from you idleness and floth; they are great enemies to improvement. Remember that youth, like the morning, will foon be past, and that opportunities once neglected, can never be regained. First of all things, there must be implanted in your mind a fixed delight in ftudy; make it your inclination ; "A defire accomplished is fweet to the foul." Be not in a hurry to get through your book too foon. Much inftruction may be given in these few words, UNDERSTAND EVERY THING AS YOU GO ALONG. Each rule is first to be committed to memory; afterwards, the examples in illustration, and every remark, are to be perused with care. There is not a word inferted in this Treatife, but with a defign that it should be studied by the Scholar. As much as is poffible, endeavor to do every thing of yourfelf; one thing found out by your own thought and reflection, will be of more real use to you, than twenty things told you by an Instructor. Be not overcome by little feeming difficulties, but rather strive to overcome fuch by pa• tience and application; fo fhall your progrefs be easy, and the object of your endeavors fure. On entering upon this most useful ftudy, the first thing which the Scholar has to regard is NOTATION. NOTATION is the art of expreffing numbers by certain characters or figures; of which there are two methods. 1. The Reman method, by Letters. 2. The Arabic method, by figures. The latter is that of general use. In the Arabic method all numbers are expreffed by these ten characters or figures. B ร 2 3 4 5 6- 7 9 0 Unit, or; two; three; four; five; fix; feven; eight; nine; cypher, or nothing. One' The nine first are called fignificant figures, or digits, each of which standing by itself or alone, invariably expreffes a particular and certain number; thus, 1 fignifies one, 2 fignifies two, 3 fignifies three, and fo of the reft until you come to nine, but for any number more than nine, it will always require two or more of those figures fet together in order to express that number. This will be more particularly taught by NUMERATION. NUMERATION teaches how to read or write any fum or number by figures. In fetting down numbers for arithmetical operations, efpecially with beginners, it is ufual to begin at the right hand, and proceed towards the left. EXAMPLE. If you wish to write the fum or number 537, begin by fetting down the feven, or right hand figure, thus, 7, next fet down the three, at the left hand of the feven, thus, 37, and laftly the five, at the left hand of the three, thus 537, which is the number proposed to be written. In this fum thus written you are next to obferve, that there are three places, meaning the fituations of the three different figures, and that each of these places has an appropriated name. The firft place, or that of the right hand figure, or the place of the 7, is called Unit's place; the fecond place, or that of the figure ftanding next to the right hand figure, in this cafe the place of the 3, is called ten's place; the third place, or next towards the left hand, or place of the 5, is called hundred's place; the next, or fourth place, for we may suppose more figures to be connected, is thousand's place; the next to this, tens of thoufand's place, and fo on to what length we please, there being particular names for each place. Now every figure fignifies differently, accordingly as it may happen to occupy one or the other of these places. The value of the first or right hand figure, or of the figure standing in the place of units, in any fum or number, is juft what the figure expreffes ftanding alone or by itself; but every other figure in the fum or number, or those to the left hand of the firft figure, have a different fignification from their true or natural meaning, for the next figure from the right hand towards the left, or that figure in the place of tens expreffes fo many times ten, as the fame figure fignifies units or ones when standing alone, that is, it is ten times its fimple, primitive value; and fo on, every removal from the right hand figure making the figure thus removed ten times the value of the fame figure when standing in the place immediately preceding it. Hund. Tens. Units.. EXAMPLE. Take the fum 3 3 3, made by the fame figure three times repeated. The first or right hand figure, or the figure in the place of units, has its natural meaning or the fame meaning as if standing alone, and fignifies three units or ones; but the fame figure again towards the left hand in the fecond place, or place of tens, fignifies not three units, but three tens, that is, thirty, its value being increased in a tenfold proportion; proceeding on fill further towards the left hand, the next figure, or that in the third place, or place of hundreds, fignifies neither three nor thirty, but three hundred, which is ten times the value of that figure, in the place immediately preceding it, or that in the place of tens. So you might proceed and add the figure 3, fifty or an hundred times, and every time the figure was added, it would fignify ten times more than it did the last time before. A CYPHER standing alone is of no fignification, yet placed at the right hand of another figure it increases the value of that figure in the fame ten fold proportion, as if it had been preceded by any other figure. Thus 3, ftanding alone, fignifies three; place a cypher before it, (30) and it no longer fignifies three but thirty; and another cypher (300) and it fignifies three hundred. The value of figures in conjunction, and how to read any fum or number, agreeably to the foregoing obfervations, may be fully understood by the fol lowing Tens. Units. 35 TABLE.. The words at the head of the Table fhew the fignification of the figures against which they itand; and the figures fhew how many of that fignification are meant. Thus, units in the firit place fignifies ones, and 6 ftanding against it fhew that fix ones, or individuals are here meant ; tens in the fecond place fhew that every figure in this place means fo many tens, and 3 ftanding against it fhews that three tens are here meant, equal to thirty, what the figure really fignifies. Hundreds in the third place fhew the meaning of figures in this place to be Hundreds, and 8 thews that eight hundreds are meant. In the fame manner the value of each of the remaining figures in the Table is known. Having proceeded thro' in this way, the fum of the firit line of figures or thofe immediately against the words, will be found to be Two billions, one hundred fixty feven thoufands, two hundred and thirty five millions; four hundred twenty one thoufands; eight hundred and thirty 4 5 fix. In like manner may be read all the re7 maining numbers in the Table. 34 0 7 6 2 1 4 6 3 1 2 1 3 0 2 5 0 3 7 6 4 5 Those words at the head of the Table are applicable to any fum or num ber, and must be committed perfectly to memory fo as to be readily applied on any occafion. For the greater eafe in reckoning, it is convenient and often practifed in public offices and by men of business, to divide any number into periods and half periods, as in the following manner : 5.3 7 9, 6 3 4. 5 2 1, 7 6 8. 5 3 2, 4 6 7 Tens Units~ Thousands Ten thoufands Hundred thousands o MILLIONS. The first fix figures from the right hand are called the unit period, the next fix the million period, after which the trillion, quadrillion, quintillion, periods, Thus, by the use of ten figures may be reckoned every thing which can be numbered; things, the multitude of which far exceed the comprehenfion of man. "It may not be amifs to illustrate by a few examples the extent of num❝bers, which are frequently named without being attended to. If a perfon "employed in telling money, reckon an hundred pieces in a minute, and con"tinue at work ten hours each day, he will take feventeen days to reckon a "million; a thousand men would take 45 years to reckon a billion. If we "fuppofe the whole earth to be as well peopled as Britain, and to have been "fo from the creation, and that the whole race of mankind had conftantly "fpent their time in telling from a heap confifting of a quadrillion of pieces, "they would hardly have yet reckoned a thoufandth part of that quantity." After having been able to read correctly to his inftructor all the numbers in the foregoing Table, the learner may proceed to write the following numbers out in words. 6 9 8 4 37 60 12 2845 14970 3 9 7 8 3 0 1 6 J 3 7 2 1 6 8 @ SECTION 1. FUNDAMENTAL RULES OF ARITHMETIC. THESE are four, Addition, Subtraction, Multiplication, and Division; they may be either simple or compound; simple, when the numbers are all of one sort or denomination; compound, when the numbers are of different denominations. They are called, Principal or Fundamental Rules, because that all other rules and operations in Arithmetic are nothing more than various uses and repetitions of these four rules. The object of every arithmetical operation, is, by certain given quantities which are known, to find out others which are unknown. This cannot be done but by changes effected on the given numbers; and as the only way in which numbers can be changed is either by increasing or by diminishing their quantities, and as there can be no increase or diminution of numbers but by one or the other of the above operations, it consequently follows, that these four rules embrace the whole art of Arithmetic. § 1. Simple Addition. SIMPLE ADDITION is the putting together of two or more numbers, of the fame denomination, fo as to make them one whole or total number; as 3 dollars, 6 dollars, and 8 dollars added or put together, make 17 dollars. RULE. "Write the numbers to be added one under another, with units under " units, tens under tens, and fo on. Draw a line under the lower number, "then add the right hand column; and if the fum be under ten, write it at "the foot of the column; but if it be ten, or an exact number of tens, write a cy“pher; and if it be not an exact number of tens, write the excefs above tens at "the foot of the column; and for every ten the fum contains, carry one to the "next column, and add it in the same manner as the former. Proceed in “like manner to add the other columns, carrying for the tens of each to the next, and mark down the full fum of the left hand column." |