times be convenient to resolve a given divisor into three or more factors. In such cases the following rule may often be used with advantage. RULE V. If the divisor is given as, or is resolved into, more than two factors, none of them exceeding 12, and each being an integer, the dividend may be divided by one of them by short division, and the result by another, and so on till all of them have been used; and the true remainder, if there be any remainder, will be found by successive applications of the method stated in Rule IV. thus:-(a.) Confine attention to the last pair of remainders and the divisor which gave the earlier of them: call that divisor, for the present, the initial divisor; and multiply the last of the pair of remainders by this initial divisor, and add to the product the first of these remainders. (b.) Now use the result just found, taking it as the last remainder in a new pair of remainders now to be dealt with, and taking as the first in this pair the remainder next before those already used; call now the divisor which gave this first remainder the initial divisor. Then multiply the last of the pair of remainders by the initial divisor, and add to the product the first of these remainders. (c.) The result just found will be the required remainder, if the divisor was resolved into three factors; or, what is the same, if all the remainders have now been used: but if the number of factors has been more than three, proceed again according to (b.), and so go on till all the remainders have been used. Then the result finally obtained will be the required remainder. Remark in reference to Rules IV. and V.-It is to be observed that if any of the factors leaves no remainder, a remainder zero is to be counted as being left by it for the due application of Rules IV. and V. Exam. 7. Divide 53217 by 8 × 6 × 7. 8)53217 6)6652...1 7)1108...4 Here, under Rule V., the last pair of remainders are 4 and 2, and the initial divisor to be used with that pair is 6 and we multiply the remainder 2 in this pair by the initial divisor 6, and obtain the product 12, and to this we add 4, the first remainder of the pair, and obtain 16. Next, we use this result 16 as the last remainder of a new pair in which the first is 1, and for which the initial divisor is 8: and we multiply 16 by 8, finding 128, and to this product we add 1; and so obtain the required remainder 129. The quotient then may be 158...2 stated as 158 times with a remainder 129; or, in the other ordinary sense of the term quotient, it may be stated as 15812; as the divisor 8 x 6 x 7 is equal to 336. It will be well for the pupil to work this example by taking the given factors in various orders; and also to find the result by long division, taking 336 as the divisor. The same result ought to be obtained in all cases. = 722772461841 727 29. 347382600435÷727 .......= 477830261688 839 681555129113 1263 * The pupil should prove the work of the several exercises in division by multiplication. Every operation in long division, thus proved, affords him an exercise in each of the four fundamental rules; as, in finding the quotient, he employs division and subtraction, and in the proof, multiplication and addition. Hence, perhaps the exercises here given will not seem, on due consideration, too numerous, though at first sight they might appear to be so. It is of the first importance that the pupil should have acquired both accuracy and despatch in performing the operations in the fundamental rules before he proceeds to apply them in the more advanced parts of arithmetic. In proving the operation in this way, the mode which is generally best for adding in the remainder, is to write it above the first of the partial products. Thus, the proof of exercise 23 will be as in the margin. 591862 907 403 4143034 5326758 536819237 173)436951 (2525129 When the pupil has had some practice in the methods of division already explained, he may omit writing the products in performing long division, which will at least save room in his operations. This method will be understood from the following example :Here the first figure put in the quotient is 2; then we say, twice 3 are 6; 6 from 6, and nothing remains; twice 7 are 14; 14 from 23, and 9 remain; twice 1 are 2, and 2 (carried) are 4; 4 from 4, and nothing remains. We next bring down 9, and place 5 in the quotient; then 5 times 3 are 15; 15 from 19, and 4 remain; 5 times 7 are 35, and 1 (carried) are 36; 36 from 40, and 4 remain; 5 times 1 are 5, and 4 (carried) are 9; 9 from 9, and nothing remains, &c. 909 991 126 854 35. 1111111111111÷854.........= = 1301066874715 = 7384 88262118303 8593 = 6290495 32 328 = - 1221226503715 4444 3517846143470 7539 6784 10782159984321 9387 - 3351628685921 793 44. 3146173847837÷9387 593 493 393 51. 582390171945110÷293......1987679767730229 53. 3333333333333333÷483 55. 1000000000000000÷81......=1234567901234555 56. 1000000000000000÷729 57. 1000000000000000÷111 = 900900900900911 65. 467817938473÷2100......... 72. Divis, the highest mountain in the neighbourhood of Belfast, 73. If it be supposed, as in common circumstances is found to be nearly true, that as many persons die in 33 years as are equal to the entire population, it is required to find how many persons die annually, at an average, out of every million? Answ. 30,303, nearly. 74. How many lessons of ninety-five lines each, are contained in Virgil's Eneid, the number of lines contained in that poem being nine thousand, eight hundred, and ninety-two? Answ. 1041. 75. The earth's equatorial and polar diameters are 41,847,426 and 41,707,620 feet respectively: divide each of them by their difference. Answ. 299 and 298; rem. in each case 45432 feet. FRACTIONS:-INTRODUCTORY CHAPTER. In the chapter of Introductory Explanations at the commencement of this treatise, some preliminary information has been given in respect to the nature of fractional numbers, or fractional numerical expressions, and their relation to whole numbers, or to numbers properly so called. Also in the explanations on division, the subject of fractions has been touched on in more than one place, as the consideration of the remainders left in division naturally and unavoidably brings fractions into notice for practical expression and treatment. The exposition of the subject of fractions hitherto introduced has, however, been very brief, and has consisted chiefly in some fundamental explanations of the nature of fractions in arithmetic, and of their relations to proper or "whole" numbers; and in some explanations of nomenclature used in reference to them alone, or to them conjointly with proper numbers. In respect to the nature of fractions in arithmetic some notions have been offered for consideration as to fractional things, such as parts of the indefinitely divisible units of quantity (yards, feet, tons, gallons, hours, &c.) used for affording numerical expression for quantities of things, or for answering the question how much (as when we speak of three fifths of a gallon, or of four tenths of a ton); and also notions have been offered of fractions of groups of single indivisible objects, the groups and fractions of groups being used for affording a certain kind of numerical expression a fractional expression-different from an ordinary number, in reply to the question how many of those single objects there are (as when we speak of 2 dozen, or of a million). In the present chapter some of the most simple and essential principles of fractions, and methods of managing them, will be taught, especially those which are most useful to a learner of arithmetic at an early stage of his progress, while the more intricate processes, which can well be dispensed with at first, will be reserved for more advanced chapters in this book. If a piece of tape one yard long is divided equally among eight persons, each person is said to get one eighth of a yard; and this share for each is called a fraction of a yard. If a piece of tape five yards long is divided equally among eight persons, each person is said to get one eighth of the five yards; and this share also is called a fraction of a yard, the yard being regarded as the unit which is broken up into fractions. In the second case it is obvious that the length which each person gets is five times as much as each got in the first case. This may be seen by considering that the length to be divided was five times as much in the second case as in the first, while the number of persons among whom the division was made was the same in both. Or otherwise, the same may be seen by conceiving, in the second case, every yard of the five to be marked out in eight equal parts, one for each person, so that there would be for each person, on the whole, five of those equal parts; or each person would get five eighths of a yard. Thus we see that an eighth of five yards is the same length as five eighths of one yard. The yard here is the unit in which lengths are counted, and the length which each person gets as his share is called a fraction of this unit. The explanations given in the foregoing example will aid the comprehension of the more general statements following. If any unit of quantity (as, for instance, a yard, an inch, a pint, or an hour) be divided into any number of equal parts; and if one or more of those parts be taken, the quantity so obtained is called a fraction of that unit, or simply it is called a fraction, in relation to that unit, or that quantity called one.† Likewise in the case of expressing numbers of objects by taking the objects in groups (as, for instance, in millions, or dozens, or in groups of any other number each; it may be in twenty-fives, or in twenties, or in thirteens, &c., when convenient for any reason), if iny group or number of objects treated as a unit, or considered as one of something, one group, or one whole, be divided into any number of equal parts into which it may be divisible; and if one or more It is to be observed, however, that we may also, if we please, perfectly well treat this eighth part of five yards as a fraction of the length five yards. It is the fraction called one eighth, and denoted, of that length. In that mode of thought we may regard the length five yards as a certain unit, or single whole, of which the part under consideration, 1th of it, is a fraction or broken portion. The two modes of thought, one stated above in the text and the other stated here in the foot note, are quite consistent mutually, and it is proper that both should be understood. In some considerations-often, for instance, in algebra-any portion of a unit may be called a fraction of the unit, whether admitting or not of being arrived at by dividing the unit into a number of parts all equal, and then taking an exact number of these parts. In arithmetic, however, only such quantities can be numerically expressed perfectly, and dealt with exactly, as admit of being arrived at in that way, and only such are called arithmetical fractions. By making the equal parts small enough and numerous enough, we can arrive at any requisite degree of exactitude in the expressing of a quantity which cannot be made up out of any exact number of exactly equal parts of a unit. For instance, it is known to mathematicians that if a square has its diagonal a unit in length, it is impossible to divide that unit into any number of equal parts such that an exact number of those parts will just make up a length equal to a side of the square; but by taking the equal parts small enough, we can have as small an error as we please in stating the length of the side as a certain number of those small equal parts of the diagonal. The side and diagonal of the square are called incommensurable with one another; and there are numberless other incommensurable quantities met with in mathematics. On this subject the reader may refer to remarks in the chapter of Introductory Explanations, page 4, as to a group or assemblage of single things being an object of our conception, which may, when we please, te treated as a single thing, or as a unit. |