same ten-fold proportion; as in the following Scale or Table of Notation. ADDITION OF DECIMALS. SET the numbers under each other according to the value of their places, like as in whole numbers; in which state the decimal separating points will stand all exactly under each other. Then, beginning at the right-hand, add up all the columns of numbers as in integers; and point off as many places, for decimals, as are in the greatest number of decimal places in any of the lines that are added; or place the point directly below all the other points. EXAMPLES. 1. To add together 29.0146, and 31465, and 2109, and 62417, and 14-16. 29.0146 3146.5 ⚫62417 14.16 5299-29877 the Sum. Ex. 2. What is the sum of 276, 39-213, 72014-9, 417, and 5032 ? 957-13 3. What is the sum of 7530, 16-201, 30142, 957 13, 6-72119 and 03014. 4. What is the sum of 312-09, 3.5711, 71956, 71 498, 9739-215, 179, and ⚫0027 ? SUBTRACTION SUBTRACTION OF DECIMALS. PLACE the numbers under each other according to the value of their places, as in the last Rule. Then, beginning at the right-hand, subtract as in whole numbers, and point off the decimals as in Addition. EXAMPLES. 1. To find the difference between 91.73 and 2.138. 91.73 Ans. 89-592 the Difference. 2. Find the diff. between 1-9185 and 2-73. Ans. 0-8115. Ans. 209-90858. Ans. 2713.084. * MULTIPLICATION OF DECIMALS. PLACE the factors, and multiply them together the same as if they were whole numbers.-Then point off in the product just as many places of decimals as there are decimals in both the factors. But if there be not so many figures in the product, then supply the defect by prefixing ciphers. 126 4332 = *The Rule will be evident from this example :-Let it be required to multiply 12 by 361; these numbers are equivalent to and the product of which is ; 04332, by the na1881 ture of Notation, which consists of as many places as there are ciphers, that is, of as many places as there are in both numbers. And in like manner for any other numbers. 100000 EXAMPLES. COMPOUND PROPORTION. COMPOUND PROPORTION shows how to resolve such questions as require two or more statings by Simple Proportion; and these may be either Direct or Inverse. In these questions, there is always given an odd number of terms, either five or seven, or nine, &c. These are distinguished into terms of supposition, and terms of demand, there being always one term more of the former than of the latter, which is of the same kind with the answer sought. The method is thus : SET down in the middle place that term of supposition which is of the same kind with the answer sought.-Take one of the other terms of supposition, and one of the demanding terms which is of the same kind with it; then place one of them for a first term, and the other for a third, according to the directions given in the Rule of Three.-Do the same with another term of supposition, and its corresponding demanding term; and so on if there be more terms of each kind; setting the numbers under each other which fall all on the left-hand side of the middle term, and the same for the others on the right-hand side.-Then, to work By several Operations.-Take the two upper terms and the middle term, in the same order as they stand, for the first Rule-of-Three question to be worked, whence will be found a fourth term. Then take this fourth number, so found, for the middle term of a second Rule-of-Three question, and the next two under terms in the general stating, in the same order as they stand, finding a fourth term for them. And so on, as far as there are any numbers in the general stating, making always the fourth number, resulting from each simple stating, to be the second term in the next following one. So shall the last resulting number be the answer to the question. By one Operation-Multiply together all the terms standing under each other, on the left-hand side of the middle term; and, in like manner, multiply together all those on the right-hand side of it. Then multiply the middle term by the latter product, and divide the result by the former product; so shall the quotient be the answer sought. VOL. I. H EXAMPLES. may fall in a column straight below each other; but observing to increase the first figure of every line with what would arise from the figures omitted, in this manner, namely 1 from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, &c; and the sum of all the lines will be the product as required, commonly to the nearest unit in the last figure. EXAMPLES. 1. To multiply 27.14986 by 92-41035, so as to retain only four places of decimals in the product. 2. Multiply 480.14936 by 2-72416, retaining only four de cimals in the product. 3. Multiply 2490.3048 by 573286, retaining only five decimals in the product. 4. Multiply 325-701428 by 7218393, retaining only three decimals in the product. DIVISION OF DECIMALS. DIVIDE as in whole numbers; and point off in the quotient as many places for decimals, as the decimal places in the dividend exceed those in the divisor*. • The reason of this Rule is evident; for, since the divisor multiplied by the quotient gives the dividend, therefore the number of decimal places in the dividend, is equal to those in the divisor and quotient, taken together, by the nature of Multiplication; and consequently the quotient itself must contain at many as the dividend exceeds the divisor. Another Another way to know the place for the decimal point, is this: The first figure of the quotient must be made to occupy the same place, of integers or decimals, as doth that figure of the dividend which stands over the unit's figure of the first product. When the places of the quotient are not so many as the Rule requires, the defect is to be supplied by prefixing ciphers. When there happens to be a remainder after the division, or when the decimal places in the divisor are more than those in the dividend; then ciphers may be annexed to the dividend, and the quotient carried on as far as required. 178) 48520998 (00272589 2639) 27·00000 (102-3114 WHEN the divisor is an integer, with any number of ciphers annexed: cut off those ciphers, and remove the decimal point in the dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers if need be; then proceed as before.* *This is no more than dividing both divisor and dividend by the same number, either 10, or 100, or 1000, &c. according to the number of ciphers cut off, which, leaving them in the same proportion, does not affect the quotient. And, in the same way, the decimal point may be moved the same number of places in both the divisor and dividend, either to the right or left, whether they have ciphers or not. EXAMPLES. |