because there are four in the dividend and none in the divisor. We, therefore, place the quotient thus,-0044. and to prove that this is the true quotient, we have only to multiply it by the divisor, and the product being the same as the dividend, the operation must be correct. 21. From the great facility with which decimal fractions may be managed, it is very desirable that we could bring vulgar fractions to the same form, in order that they might more easily be wrought with. Now, this may be done on the principles already laid down :-take the fraction, and, on the principle of No. 4, multiply both terms by 1000, it then becomes 1000, which is equal to ; divide (No. 4) both numerator and denominator by 8; then 8) 1008 (125, which last fraction is expressed in the decimal notation thus, (on the principle of No. 15,) ·125, which, from the way it has been derived, must be equal to . This may, however, be found more immediately thus: add as many ciphers to the numerator as you find necessary, and divide by the denominator thus,-8)1000(125. If we have only to add one cipher before we get a quotient figure, we put a point in the quotient; but if more, then we put as many ciphers in the quotient after the point. Thus, 2; 25)100('04, and is just, or ·04. 22. In many cases the quotient would go on without end; but it is to be observed, that it is not necessary to continue any operation in decimals, at least in mechanical calculations, beyond three or four places, as ten thousandth parts are seldom necessary to be considered in practice. For similar reasons, it is unnecessary to give rules for repeating and circulating decimals: i. e. decimal numbers, when the same figures recur in some order-thus, 3333, or, 142142, &c., carry them to four places, and it is all that is neces sary. Other applications of these principles will be found in the next chapter, on compound numbers. COMPOUND NUMBERS. 23. IN mechanical calculations, we are often concerned with weights and measures, and it is necessary to know how to operate with the numbers which express these. The rule given in books of arithmetic are generally very long, and, therefore, not very easily understood; yet the steps of the operation are simple. We shall therefore show the mode of procedure, in some very easy examples, and the reader will find no difficulty in applying the principles he may thus imbibe to cases more complex. feet inch 9 2 6 2 1 3 8 0 11 20 1 8 24. If we have to add 9 yards 2 feet 6 yds. inches, to 2 yards 1 foot 3 inches, 8 yards 0 feet 11 inches, long measure. Then we must in this, as in all other cases of compound addition, arrange them in order, the greater towards the left hand, and the lesser towards the right; and there must be a column for every denomina. tion of weight or measure, in which column the respective quantities must stand, so that feet will stand under feet, inches under inches, pounds under pounds, and ounces under ounces, &c. Add now the column toward the right, which in this example amounts to 20 inches, or 1 foot 8 inches, we therefore put down the 8 inches under the column of inches, and add the 1 foot to the column of feet, which comes to 4 feet; that is, 1 yard and 1 foot. The 1 foot is put down under the column of feet, and the 1 yard is added, or carried, as it is usually called, to the column of yards, whose sum is 20. If we have to add 2 tons 2 cwt. 1 quar. 17 lbs. 10 oz. avoirdupois, to 12 tons 10 cwt. 2 lbs. 2 oz., 2 cwt. 1 quar. 18 lbs. 3 oz., and 9 lbs. 11 oz.; then, from what was remarked above, they will be put down as in the margin. Then the sum of the right hand column is 26 oz., which is 1 lb. 10 oz., we put down the 10 in the column of oz., and carry the 1 lb. to the column of lbs. which is next; and this when added comes to 47 lbs., that is, 1 quar. and 19 lbs.; the 19 is put in the column of lbs. and the 1 is carried to that of quars., which comes to 3, which not amounting to 1 cwt. we put down the 3 in the column of quars. and carry nothing to the column of cwts., which, when added, amounts to 14, this we put down, and, as it does not amount to 20 cwt. or 1 ton, we carry nothing to the column of tons; and when this column is added, its sum is 14. tons cwt. quar. lbs. 2 2 1 25. In Subtraction the same principle of arrangement is to be observed, and the lesser quantity is to be put under the greater. If we have to subtract 1 ton 13 cwt. 2 quars. 17 lbs denomination, viz. oz.—12 oz. from 7 oz. we cannot, bu we add a lb. or 16 oz. to the 7, which is supposed to be borrowed from the column of lbs. which stands next it, towards the left; now 16 added to 7 makes 23, and 12 from 23 leaves 11, which is put down in the column of oz. Now we must pay back to the column of lbs. the pound or 16 oz. which we borrowed, therefore, it is 18 from 4. Here we have to borrow from the column of quars., and 1 quar. being 28 lbs. we borrow 28, then 28 and 4 are 32, therefore 18 from 32 leaves 14, which is put down, and the 1 quar. paid back to the column of quars.; 3 from 1, we cannot, and must borrow 1 cwt. or 4 quars., therefore 3 from 5 and 2 remains, which is put down. Add 1 to 13 for the 1 cwt. that was borrowed, then 14 from 8, we cannot, but borrow 20 from the next column, then 14 from 28 and 14 remains. Pay back to the column of tons the 1 ton, or 20 cwt. which we borrowed, then 2 from 9 and 7 remains, which is put down. The same principle holds in other examples, the only va riation being that the numbers to be borrowed from the next higher column, will depend upon the relative values of these columns, which may be known by examining a table of the particular weight or measure to which the example may refer. 26. In Multiplication, which is only a short way of performing addition in particular cases; the principles are nearly similar thus, to multiply 3 tons 2 cwt. 2 quars. 6 lbs. 10 oz. by 3; they are arranged as in margin. Then the first product is 30 oz. or 1 lb. which is carried to the co tons cwt. quars. lbs. Oz. 3 2 2 6 10 3 9 2 19 14 lumn of lbs., and 14 oz., The product of which is put down in the column of oz. the lbs. is 18, and the one lb. carried is 19, which no amounting to 28 lbs. or I quar., nothing is to be carried to the column of quars. The product of the quars. is 6, which is 1 cwt. to be carried and 2 quars. to be put down The product of ewts. is 6, and the one carried from the former column makes 7, nothing being carried; the column of tons is 9. By examining the following examples. and referring to the tables of weights and measures, the general application may be easily inferred. See Appendix to Arithmetic. 27. It may not be out of place here to notice, Duodecimal, or what is commonly called Cross Multiplication; which is very useful to artificers in general, in measuring timber, &c. 5 8 1 7 8 5 4 10 11 13 3 11 The foot is divided into 12 inches, each inch into 12 parts, and each part again into 12 seconds; these last, however, are so sinall, that they are generally neglected in calculation. If we wish to find the surface of a plank, whose breadth is 1 foot 7 inches, and length 8 feet 5 inches, we place the one under the other, feet under feet, inches under inches, &c., as in the margin. Multiply the inches and feet in the upper line, by the feet in the under line, placing the product. of the inches, under the inches, and that of the feet, under the feet. Then multiply the inches and feet, of the upper line, by the inches in the under line, placing the product one place further towards the right, and carry by twelves where necessary; as in this example, 7 times 5 is 35, that is, two twelves and 11 over; the 11 is put down, and the 2 added to the product of the next column,-7 times 8 is 56, and the 2 carried makes 58, that is four twelves and 10 over; the 10 is put down, and the 4 carried to the next column. These are now added, observing again to carry by twelves. The feet in the example are square feet, but the inches are not square, as might be thought at first sight, but 12th parts of a square foot; and also the numbers standing in the third place, are 12th parts of these 12 parts of a foot, and so on. 28. Before we consider the Division of compound nuinbers, it will be necessary to attend a little to the nature of reduction. This is usually thought by beginners to be very perplexing, but a little attention to the principle, will obviate all this apparent difficulty. In every lineal foot there are 12 inches, and therefore there will be 12 times as many inches, in any number of feet, as there are feet; thus, in 8 feet there are 8 times-12, that is, 96 inches. In every lb. avoirdupois there are 16 ounces, therefore in 18 lbs. there are 18 times 16, that is, 288 ounces. So that we multiply the higher denomination, by that number of the lower which makes one of the higher, and the product is the number of the lower contained in the number of the higher, which we multiply. In the previous examples, feet and pounds are the higher denominations, and inches and ounces are the lower. From these remarks it will be easy to see, how we proceed in finding the number of parts of an inch contained in 3 yards 2 feet 7 inches, and parts, long measure. Bring the yards to feet, 3 multiplied by 3 are 9, to which we add the 2 feet, which make 11. This brought to inches, is 11 multiplied by 12, or 132, to which we add the 7 inches, making 139. This brought to parts gives 139, multiplied by 8, that is, 1112, to which we add the 5 eighth parts, making 1117 the answer. The examples subjoined are managed in a like manner; the multipliers varying with the kind of weight or mea |