ADDITION. The figures are to be carefully written, so that these points fall exactly under each other, as you will see that they do in each of the following examples. In adding up these figures in the usual •77 •469 manner, you find that the pile on the left .354 amounts to 16, which is 16; you know 062 also that 16 is an improper fraction, or that 1.655 it is one and six-tenths; you therefore write the 1 below and the 6 after the decimal point, ind your work is complete. You see the same in this sum, Add 7.463 the one carried to and added in with 17.056 49.6 the whole numbers; and in this sim 57.5 ple manner can mixed numbers be 131.619 added. Knowing as you do the crouble of adding Vulgar Fractions, I need not point out the power and ease in calculations that a proper knowledge of Decimals gives us. SUBTRACTION OF DECIMALS. As to Subtraction, it is equally simple ; you write the numbers as before described, and then proceed as in whole numbers. These three examples will suffice to teach the pupil the rule observed. In the first example, the decimal subtracted is of course less than the one subtracted from. In the se cond example, the lesser number has the largest deci. mal, to being more than 32%,we have therefore one to carry to the whole number in the bottom line. In the third example, the decimal being less in the lesser number, there is nothing to carry to the whole number. MULTIPLICATION OF DECIMALS. We pay no attention in this rule to arranging the decimal point, but proceed exactly as in whole numbers. Division, as I have said before, being the reverse of Multiplication, or I may so say, exactly undoing what Multiplication has done, let us reverse the above sums, by taking the product for a dividend, and the multiplier for a divisor. You see by the exam 3.407) 157331853 (46.179 ple, you proceed exactly 13628 as in Division of whole 21051 numbers ; and when the 20442 work is complete, count 6098 the number of decimal 3407 places in your dividend, 26915 and the number in the 23849 divisor, and you cut off 30663 for decimal places on 30663 the right of the quotient as many figures as is the difference between the dividend and divisor. See the decimals of the above example. Divisor. Dividend. Difference. 3 places. DUODECIMALS, TWELFTHS, OR CROSS MULTIPLICATION. Is a calculation of measurements, taken in feet and inches, and is used to calculate quantity in superficial extent and in bulk; as the measurement of boards, the work of plasterers, painters, glaziers, and the like. Any substance measured by feet can be calculated by this rule. For instance, a painter painting wainscot ten feet long and six feet high, if he wanted to ascertain how many feet he painted, would multiply the ten by the six, and its amount 60 would be the number of feet he had painted. Imagine the figure on the right to be the 2 3 4 5 6 7 8 9 10 wainscoting describ 11 12 13 14 15 16 17 18 19 20 ed as that which was 21 22 23 24 25 26 27 28 29 30 painted, and that the 31 32 33 34 35 36|37|38|39 40 painter had drawn 41 42 43 44 45 46 47 48 49 50 lines, as it is done in 151 52 53 54 55 56 57 58 59 60 the figure, and numbered them, he would, as you see, have sixty squares of a foot each. He therefore adopts the readier method of multiplying the length by the breadth, which produces the same result. But if a mason had a block of stone the same length and breadth and four feet thick, then the same rule would not be sufficient for him, he having another dimension ; but we know that if the mason sawed his stone in slabs of a foot thick each, he would have four slabs, and if he further sawed each of these slabs as the painter drew his lines, that he would have 60 blocks of a foot square in each, or 240 in the whole, so that by this rule the pupil is aware that to multiply the length and breadth gives square feet, which answers the painter's purpose ; and that the length, breadth, and thickness, if multiplied together, gives cubic feet, which is just what the mason, or any one measuring work that has length, breadth, and thickness to be considered, would do. And if our measurements were always in feet or yards, no more need be said on the subject ; but as we know that they are kept in feet and inches, which are the twelfths of feet, and in some of those trades whose work is more expensive, into parts which are the twelfths of an inch, hence it is that duodecimals are required, and we will now proceed with the rule. Find the contents in feet of a surface 8 feet 5 inches broad and 4 feet 7 inches high. In the above work we have feet feet. inches. and inches; these inches are not 8 5 4 7 square inches, they are therefore twelfths, and parts are144ths, of a 33 8 4 10 11 square fod Thus inches mul 38 6 11 tiplied by feet are inches, feet multiplied by feet are feet, but inches multiplied together are parts or twelfths of an inch. Therefore in our example we begin by multiplying by the four feet in the bottom line, saying 4 times 5 are 20, these are twenty inches or one foot eight inches; we write the 8 and carry one to the feet, saying 4 times 8 are 32 and 1 makes 33, which being the highest denomination required, we write under the feet. We have now to multiply by 7 inches, and, as I have said, inches multiplied together produce parts, we therefore say times 5 are 35 parts, or two inches and eleven parts; we proceed next to multiply the feet, which when multiplied by inches produce inches, we therefore say 7 times 8 are 56 and 2 are 58; fiftyeight divided by twelve, the inches contained in a foot, gives 4 feet 10 inches; these sums we write under their proper denominations, and add both lines together; the amount is the dimensions sought. To sum up the process in the form of a rule, you state your sum by writing the different denominations exactly under one another; you then begin to multiply the lowest denomination in the top line by the highest in the bottom line, and bear in mind that any denomination you multiply by feet produces the same denomination; thus parts multiplied by feet produce parts, inches produce inches, and feet produce feet. But parts multiplied by inches only produce twelfths of parts, inches twelfths of inches, and feet twelfths of feet. If we go still lower, and multiply parts by parts, they produce twelfths of twelfths of parts, or two denominations below parts, and each succeeding denomination is reduced in like manner. The different denominations are distinguished by marks, thus, ft. (°) (") (") ("''') |