be, and the first decimal multiplication consequently 3%, this divided by 4 would become 430(7 or 7 and a half tenths, and therefore would still involve a common fraction; but if the decimal multiplier is made 100, the second power of 10, instead of 10 the first power, the new fraction is 30%, and 4)300/75 this divided by 4 is which reduced to lower 4)400 100 terms by dividing both 75 and 100 by 25, gives 2, the original fraction, and, therefore, is a true decimal equivalent of it. § 50. In many cases, it is necessary to multiply the terms of the vulgar fraction by adding still more than two cyphers to them, to be able to divide by the denominator without leaving a remainder; and in some cases a perfectly exact division is not at all attainable; but, as after three decimal cyphers are used, the question concerns but thousandth parts of the value of the fraction, further division is seldom important. § 51. The great advantage of the decimal fraction is, that it becomes unnecessary to write down the denominator of the fraction, because it is known to be ten or a power of ten, having as many cyphers following the commencing figure, as there are decimals in the nominator; thus 85 means 8,5; 8-624 means 8+ + 100 + 100, or 8 The figures of the decimal numerator, therefore, can be placed in the same line with the figures of preceding integers, and are distinguished from these only by a point called the decimal point, placed between the integers and them. Thus, instead of writing the compound number 48, with the denominator of the fraction appearing, the same value is signified by the decimal expression 48.9. Instead of 48,75 stands the decimal 48.75. Instead of 100 48+0 1000 § 52. It will be seen immediately, that decimal figures so written may in almost all computations be treated as simple integers, their exact signification being settled in the end by the required placing of the decimal point. And thus the operations of addition, subtraction, multiplication, and division, producing fractional expressions, may be carried on as if no fractions were present. § 53. It is important to remark, that the adding or subtracting of cyphers at the end of a decimal expression, does not at all alter its value, although the appearance and name are changed. Three tenths (36) is an equivalent fraction of 30% or 100%, and 100 so forth; and therefore 8.3, and 8.30, and 8.300 are all equivalent expressions. The computer who adds a visible cypher to the numerator-part of the fraction, virtually adds a cypher also to the unwritten and unseen denominator below. § 54. By shifting, in a row of figures, the decimal point one move to the right hand, the whole expression, and every figure in it, is multiplied by ten. By similarly shifting the point one move to the left hand, the mass of figures is divided by ten. Thus, the expression 28.46 means twenty-eight wholes and fortysix hundredths. The same figures, with the point shifted one move to the right (284-6), means 284 wholes and six tenths, and the same figures with the point moved to the left (2·846), means two wholes, and eight hundred and forty-six thousandths. Shifting the decimal point two moves multiplies or divides all by 100; three moves multiplies or divides all by 1000, and so on. § 55. Any decimal fraction may be converted into an equivalent common fraction, by writing the figures of the decimal denominator under the decimal numerator-figures, thus 20-5205, 20:52 205, 20-627 20,627, &c. = = 10009 Decimal fractions may be added, subtracted, multiplied, and divided on the same principles as common fractions. § 56. ADDITION OF DECIMALS. Rule. Place the sums as if they were ordinary integers, only with their decimal points in the same vertical line, and add them as integers. To all decimal sums there may be given, without altering the value, the same number of figures as the longest has, by adding cyphers at the end of any one shorter than the rest. § 57. SUBTRACTION OF DECIMALS. Rule.--Prepare and work as for subtraction among integers, paying attention to the placing of the decimal point as for addition. § 58. MULTIPLICATION OF DECIMALS. Decimal fractions, although their denominators do not appear written down, bear the same relations to integers that their equivalent common fractions do, and the arithmetical operations among them are in principle identical. The same paradoxical use of the word multiplication (in reality a subdivision) is made with respect to them as with respect to common fractions. For multiplication of common fractions, the rule given is to multiply together the numerators for a new numerator, and the denominators for a new denominator (see § 43). This may be done in decimals by so multiplying together the visible numerators for a numerator, and the unwritten denominators, consisting of the figure 1 and as many cyphers as there are decimal places in both denominators, as a denominator. But a shorter rule is, to multiply the two decimal expressions together, as if they were whole numbers, and then to mark off in the product, as many decimal places as there are in the multiplicand and multiplier together. This is explained by the subjoined example. Multiply 14.86 By disregarding in the upper factor or multiplicand 14.86 the decimal point for 100, that factor is multiplied by 100 (see § 54), and therefore the product becomes 100 times too great. Then by disregarding the decimal point for 10 in the second factor or multiplier 4.5, the product becomes still further, ten times too great, and, therefore, to be rendered true, must be divided both by 100 and by 10, that is, by 1000 in all. This is done by placing the decimal point where it makes three decimal figures in the final product. § 59. DIVISION OF DECIMALS. It has been explained that common fractions having the same denominator, are divided by one another through merely dividing their numerators as common numbers, omitting altogether the denominators. Decimals have already like denominators, which become exactly the same by equalising the cyphers in them. If a common fraction be left after such division, it can be converted into a decimal by the rule already Here is a simple example :— given in § 47, 49. § 60. INVOLUTION AND EVOLUTION. These are the names of important forms of multiplication and division, both among integers and fractions. Involution is the successive multiplication of a number by itself, producing what are called its powers, of which the second is named the square, and the third the cube, as explained at page 703, and at § 23 of this supplement. Thus, in regard to the number 2 multiplied by itself, there come 2, 4, 8, 16, 32, 64, &c. and in regard to 10 there come— 10, 100, 1000, 10,000, &c. Evolution is the reverse operation, to find any root of any given number. For small numbers the operation is simple and easy. In books of arithmetic, tables are given of the first ten powers of the first ten numbers, saving vast labour to ordinary computers. It would be a departure from the promised simplicity of this popular Introduction to Physics to enter into the details of these operations, which would include the subjects of progressions, logarithms, &c. But essential to the purposes of the work is the consideration of the subject of proportion and the rule of three, which leads to the formation and use of arithmetical formulæ, and which can be satisfactorily explained to common apprehension. § 61. PROPORTION. In the whole field of mathematical labour, there is no operation more valuable than that which concerns proportions. It relates, for instance, to the finding a fourth number or quantity, having a certain relation to any one of three others which may present themselves, or are already known. It has been called, therefore, the rule of three, and, from its singular utility, has received also the name of the golden rule. It is, in reality, an easy application of the simple doctrines of fractions, already explained. § 62. Different numbers in pairs have different relations to one another. Among the most simple of these, and the most familiar, from having to be so often referred to, are, the cases where one is the half of the other, as in the instances of or where one is the third part of the other, as or, placing the larger number first, when one is double, triple, quadruple, &c., of the other. § 63. The most clear and precise written expression for related numbers is to place them in the form of fractions, as, (3) two thirds, (4) four sixths, (2) nine tenths, &c., in which form the lower term declares into how many equal parts some number is supposed to be divided, and the upper term tells how many of these parts are taken to form the fraction. The pairs of numbers presented in the last paragraph may be written down as fractions, thus,,,,, ; and the upper terms of all these are halves or thirds of the lower terms. All those in each set, therefore, are of equal fractional value, that is to say, are equal fractions. The relation of numbers is called their ratio, or proportion to each other. In the first set presented above, the ratio is called that of one to two, or of half to whole. In the second set it is that of one to three, or of a third to the whole. § 64. Two pairs of numbers such that between the terms of the |