NOTE. In the new French system, the values of the base of each measure-viz. Metre, Litre, Stere, Are, and Gramme-are increased or decreased by the following words prefixed to them. Thus: SYNOPSIS OF ARITHMETIC. The following synopsis of several of the rules of arithmetic, often referred to in elementary books on mechanical science, are here inserted for the convenience of reference. These rules and examples are given merely to refresh the memory, it being taken for granted that the reader has already acquainted himself with the principles of common arithmetic. They will, however, be found serviceable, both as a convenience of reference, and to give some insight to the subjects on which they treat. DECIMAL FRACTIONS. A decimal fraction derives its name from the Latin decem, "ten," which denotes the nature of its numbers, representing the parts of an integral quantity, divided into a tenfold proportion. It has for its denominator a UNIT, or whole thing, as a gallon, a pound, a yard, &c., and is supposed to be divided into ten equal parts, called tenths; those tenths into ten equal parts, called hundredths, and so on, without end. The denominator of a decimal being always known to consist of a unit, with as many ciphers annexed as the numerator has places, is never expressed, being understood to be 10, 100, 1000, &c., according as the numerator consists of 1, 2, 3, or more figures. Thus: 16 24 125 T&c., the numerators only are written with a dot or comma before them, thus 2 24 125. The use of the dot (*) is to separate the decimal from the whole numbers. The first figure on the right of the decimal point is in the place of tenths, the second in the place of hundredths, the third in the place of thousandths, &c., always decreasing from the left towards the right in a tenfold ratio, as in the following A cipher placed on the left hand of a decimal, decreases its value in a tenfold ratio by removing it farther from the decimal point. But annexing a cipher to any decimal, does not alter its value at all. Thus, 0.4 is ten times the value of 0.04, and a hundred times 0.004. But 0.7=0.70=0-700=0-7000, &c., as above remarked. 0-375 66 66 0.1876 66 66 twenty-five hundredths. three hundred and seventy-five thousandth. Mixed numbers consist of a whole number and a decimal; as 4.25 and 3.875. ADDITION OF DECIMALS. Rule. Arrange the numbers so that the decimal points shall be directly over each other, and then add as in whole numbers, and place the decimal point directly below all the other points. Example. Add the following units and fractions : SUBTRACTION OF DECIMALS. Rule. Place the numbers directly under each other, according to their several values, as in addition; then subtract as in whole numbers, and point off the decimals, as in the last rule. Example. Subtract 7.75 from 15.125. 5.125 7.375 remainder. MULTIPLICATION OF DECIMALS. Rule. Place the factors under each other, and multiply them together as in whole numbers; then point off as many figures from the right hand of the product as there are decimals places in both factors, observing, if there be not enough, to annex as many ciphers to the left hand of the product as will supply the deficiency. Example. Multiply 3.625 by 2.75 3-625×2.75-9-96875. Ans. DIVISION OF DECIMALS. Rule. Prepare the decimal as directed for multiplication; divide as in whole numbers; cut off as many figures for decimals in the quotient as the number of decimals in the dividend exceeds the number in the divisor; and if the places in the quotient be not so many as the rule requires, supply the deficiency by annexing ciphers to the left hand of the quotient. Example 1. Divide 173.5425 by 3.75. 3.75)173-5425(46-27. 1500 2354 2250 1042 750 2925 2625 300 Example 2.-Divide 63.50 by 425. 4:25)63-50(14.94 425 2100 1700 4000 3825 1750 A cipher placed on the left hand of a decimal, decreases its value in a tenfold ratio by removing it farther from the decimal point. But annexing a cipher to any decimal, does not alter its value at all. Thus, 0.4 is ten times the value of 0.04, and a hundred times 0.004. But 0.7=0.70=0-700=0.7000, &c., as above remarked. 0.375 66 66 0.1876 66 66 twenty-five hundredths. three hundred and seventy-five thousandth. Mixed numbers consist of a whole number and a decimal; as 4.25 and 3.875. ADDITION OF DECIMALS. |