Rules occasionally useful to men in particular employments of life. PROBLEMS 1st. To find the circumference of a circle, the diameter being given 2d. To find the area of a circle, the diameter being given 3d. To measure the solidity of an irregular body THE SCHOLAR'S ARITHMETIC. INTRODUCTION. ARITHMETIC is the art or science which treats of numbers. The THEORY of Arithmetic explains the nature and quality of numbers, and demonstrates the reason of practical operations. Considered in this sense, Arithmetic is a Science. PRACTICAL ARITHMETIC shews the method of working by numbers, so as to be most useful and expeditious for business. In this sense Arithmetic is an Art. DIRECTIONS TO THE SCHOLAR, Be DEEPLY impress 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 sloth; they are great enemies to improvement. Remember that youth, like the morning, will soon 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 study; make it your inclination; "A desire accomplished is sweet to the soul." not in a hurry to get through your book too soon, Much instruction 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 is to be perused with care. There is not a word inserted in this Treatise, but with a design that it should be studied by the Scholar. As much as possible, endeavour to do every thing of yourself; 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 seeming difficulties, but rather strive to overcome such by patience and application; so shall your progress be easy and the object of your endeavours sure, On entering upon this most useful study, the first thing which the Scholar has to regard, is NOTATION. NOTATION is the art of expressing numbers by certain characters or figof which there are two methods. 1. The Roman method, by Letters. 2. The Arabic method, by Figures. The latter is that of general use. ures In the Arabic method all numbers are expressed by these ten characters or figures. 1 2 3 4 5 Unit; or one 7 8 9 0 6 two; three; four; five; six; seven; eight; nine; cypher [or nothing. The nine first are called significant figures, or digits, each of which standing by itself or alone, invariably expresses a particular or certain number; thus, 1 signifies one, 2 signifies two, 3 signifies three, and so of the rest, until you come to nine, but for any number more than nine, it will always require two or more of those figures set together in order to express that number. This will be more particularly taught by NUMERATION. Numeration teaches how to read or write any sum or number by figures. In setting down numbers for arithmetical operations, especially with beginners, it is usual to begin at the right hand, and proceed towards the left. EXAMPLE. If you wish to write the sum or number 537, begin by setting down the seven, or right hand figure, thus 7, next set down the three, at the left hand of the seven, thus 37, and lastly the five, at the left hand of the three, thus 537, which is the number proposed to be written. In this sum thus written you are next to observe that there are three places, meaning the situations of the three different figures, and that each of these places has an appropriated name. The first place, or that of the right hand figure, or the place of the 7, is called unit's place; the second place, or that of the figure standing next to the right hand figure, in this the place of the 3, is called ten's place; the third place, 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 thousand's place, and so on to what length we please, there being particular names for each place. Now every figure signifies 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 sum or number, is just what the figure expresses standing alone or by itself; but every other figure in the sum or number, or those to the left hand of the first figure, have a different signification 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, expresses so many times ten, as the same figure signifies units or ones when standing alone, that is, it is ten times its simple primitive value; and so on, every removal from the right hand figure, making the figure thus removed ten times the value of the same figure when standing in the place immediately preceding it. co Hund. Tens. Units. EXAMPLE. Take the sum 3 3 3, made by the same figure three times repeated. The first or right hand figure, or the figure in the place of units, has its natural meaning or the same meaning as if standing alone, and signifies three units or ones; but the same figure again towards the left hand in the second place, or place of tens, signifies not three units, but three tens, that is thirty, its value being increased in a tenfold proportion; proceeding on still further towards the left hand, the next figure or that in the third place, or place of hundreds signifies 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 signify ten times more than it did the last time. A CYPHER standing alone is no signification, yet placed at the right hand of another figure it increases the value of that figure in the same tenfold proportion, as if it had been preceded by any other figure. Thus 3, standing slone, signifies three; place a cypher before (30) and it no longer signifies three, but thirty; and another cypher (300) and it signifies three hundred. The value of figures in conjunction, and how to read any sum or number agreeably to the foregoing observations, may be fully understood by the following TABLE. Billions. -Hund. of Thous. of Mill. Teus of Millions. 2010 Hundreds of Millions. Millions. Hundreds of Thousands. Tens. Units. 4 2 1 8 3 3 4 0 7 6 2 1 4 6 3 1 1 3 0 2 5 0 3 7 6 4 4 1 3 9 8 2 1 0 6 THE words at the head of the Table shew the signification of the figures against which they stand; and the figures shew how manyof that signification are meant. Thus Units in the first place signify ones, and 6 standing against it, shews that six ones or individuals are here meant; tens in the second place shew that every figure in this place means so many tens, and 3 standing against it, shews that three tens are here meant, equal to thir6 ty, what the figure really signifies. Hundreds 2 in the third place shew the meaning of fig5 ures in this place to be Hundreds, and 8 4 shews that eight hundreds are meant. In 27 0 2 1 3 6 7 5 the same manner the value of each of the re4 6 3 2 7 2 9 1 maining figures in the table is known. Hav1 2 3 4 6 3 2 ing proceeded through in this way, the sum 2 3 4 5 6 7 of the first line of figures or those immedi8 9 0 9 8 ately against the words, will be found to be 76 5 4 Two Billions, one hundred sixty seven thou1 2 3 sands, two hundred and thirty-five Millions ; 4 5 four hundred twenty-one thousands; eight hun7 dred and thirty-six. In the like manner may be read all the remaining numbers in the Table. Those words at the head of the Table are applicable to any sum or number, and must be committed perfectly to memory so as to be readily applied on any occasion. For the greater ease of reckoning, it is convenient and often practised 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 TRILLIONS. Ten billions co Hundreds thous. of bill. BILLIONS. Hundred thous. of mill. Thousand millions MILLIONS. Hundreds Thousands Ten thousands co Hundred thousands o Tens Units -1 B The first six figures from the right hand are called the unit period, the next six the million period, after which the trillion, quadrillion, quintillion periods, &c. follow in their order. Thus by the use of ten figures may be reckoned every thing which can be numbered; things, the multitude of which far exceeds the comprehension of man. "It may not be amiss to illustrate by a few examples the extent of num"bers, which are frequently named without being attended to. If a person employed in telling money, reckon an hundred pieces in a minute, "and continue at work ten hours each day, he will take seventeen days "to reckon a million; a thousand men would take 45 years to reckon a bil"lion. If we suppose the whole earth to be as well peopled as Britain, "and to have been so from the creation, and that the whole race of mankind "had constantly spent their time in telling from a heap consisting of a quadrillion of pieces, they would hardly have yet reckoned a thousandth part of that quantity." After having been able to read correctly to his instructor, all the numbers in the foregoing Table, the learner may proceed to write the following numbers out in figures. Two hundred and sixty-three. Five thousand one hundred and sixty. One hundred thousand, six hundred and four. Five million, eighteen thousand, seven hundred & six. Two million, six hundred and fifty thousand, one hundred and thirty-seven. Seven hundred and ninety-four million, one hundred and forty-nine thousand, five hundred and twenty. Three thousand, nine hundred and forty million, four hundred and two thousand, eight hundred and four. Five hundred thirty six thousand, two hundred and seventy two million, one hundred and three thousand and six. Four billion, six hundred thousand million, seven hundred thousand, two hundred and ninetytwo. Explanation of the Characters made use of in this Work. + X ÷ or )( :: : The sign of equality; as 100 cts=1 Dol. signifies that 100 cents are equal to 1 dollar. Saint George's Cross, the sign of addition; as 2+46, that is, 2 added to 4 are equal to 6. The sign of subtraction; as 6—2—4; that is, 2 taken from 6 leaves 4, Saint Andrew's Cross, the sign of multiplication; as 4 × 6—24; that is, 4 times 6 are equal to 24. Reversed Parentheses, the sign of division; as 3)6(2, that is, 6 divided by 3 the quotient is 2,r 6—3—2. The sign of proportion; as 2: 4 :: 8: 16, that is, as 2 is to 4 so is 8 to 16. |