38 INTEGERS AND DECIMALS. means of the representations of numbers, instead of the things which we have occasion actually to number. That this is a very great advantage we may see by the following case-suppose that we had no arithmetic beyond simply counting one, two, three, and so on (and even this, as we have already seen, is not attainable without science), and that it were required to find the whole price of any number of pounds of goods at one shilling, one penny, and one farthing for each pound. The only way that we could go about even this very simple case would be to lay out the goods in single pounds in a row, place one shilling, one penny, and one farthing against every single pound, and then count the money thus placed. Decimal numbers, or, as they are shortly named, "decimals," are merely a continuation of the very same scale as integer numbers; and by merely shifting the decimal point we may express the multiplication or the division of any number by 10 as often as may be necessary. Thus, in the following numbers, though each contains the very same figures or characters, in the same order, yet if we take them from the first to the last, each is one-tenth part of the one above it; and if we take them in the opposite order, or from the last to the first, each is ten times the one below it: 879456321. 87945632.1 8794563.21 879456.321 87945.6321 8794.56321 879.456321 87.9456321 8.79456321 .879456321 The first of these numbers is read eight hundred and seventynine millions, four hundred and fifty-six thousand, three hundred and twenty-one; its exponent is, because there are eight figures besides the units. There are nine divisions by 10 in the succeeding lines, and therefore the exponent of the last line is 9 less than 8, or -1. The last line is read eight hundred and seventy-nine millions, four hundred and fifty-six thousand, three hundred and twenty-one, thousand millionth parts. The exponent of the last figure of the first line is, that of the last figure is o; the exponent of the first figure of the last line is -1, and that of the first figure of the same is -9; thus there are 18 different exponents, answering to the 18 different places of figures. We may farther remark, that every number expressed by the common figures or characters of arithmetic, may be considered as expressing a number of times 1 of its right hand figure; and that any number of Os on the left of an integer, or on the right of a decimal number, do not in the least affect the value of the figures, or that of the number itself. Thirdly, EXPONENTIAL NUMBERS.-These are altogether different in their nature from integer and decimal numbers; for, while both of these stand for numbers, or numbers of things, according as they are applied, exponential numbers stand for numbers of times multiplying or dividing, and never can be made to stand for numbers of things, or to be in any way expressive of the value of real quantities or existences. They are thus not numbers, but expressions for the relations of numbers to some one particular number; and this number, in our scale of arithmetic, is the number 10. If the exponent has not the sign - before it, the number of which it is the exponent always contains integers, and always one place more of them than the number of times 1 in the exponent. If the 40 EXPONENTIAL NUMBERS. exponent has the sign - the corresponding number never contains any integers,, but is wholly decimal, and there are always as many Os on the left, or between it and the decimal point, as the number of times 1 in the exponent, wanting one. Thus, if we take the eighteen places of the first and last lines of numbers in the preceding example, with 1 in place of each of the figures, and mark the exponent of each, we shall have the following expression: 108, 107, 106, 105, 104, 103, 102, 101, 10°. 10-1, 10-2, 10-5, 10-4, 10-5, 10—6, 10-7, 10-3, 10–9. It will be seen, from this expression,-which goes as far both ways as there is ever much occasion for in practice, but which may be extended at pleasure both ways, by adding 1 to every succeeding exponent both on the left and the right, and taking care to continue the sign before those on the right; that, read from right to left, this is a regular series of multiplications by 10; but that if we read it from left to right, it is a regular series of divisions by 10; and, as 1 in the exponent when it has not the sign answers to one multiplication by 10, and when it has the sign to one division by 10, it follows that there is no common addition or subtraction of those exponential numbers, for the addition of them is evidently the same as the multiplication of the numbers of which they are the exponents, and the subtraction of them is the same as the division of those numbers. Farther, O in these exponential numbers means 1, and not nothing, as it means in common numbers; and the exponents which have the sign before them do not mean imaginary numbers, that is, numbers less than nothing, they mean numbers of times divided by 10. It is necessary to pay particular attention to the difference in meaning between those exponential numbers and common numbers, as used in ordinary arithmetic; because, though they have exactly the same forms, their meanings are altogether different. Exponential numbers are called LOGARITHMS, which means "the voices of numbers," that is, what they express, or the account which they can give of themselves; and this expression is always the number of times which 10 requires to be multiplied by itself, or divided by itself, in order to produce the common or natural number answering to the logarithm. Those logarithms, or voices of numbers, are of vast use in many of the more elaborate parts of mathematical science, both in the investigation of principles and in the application of those principles to practical cases. But it requires more general views than any upon which we have hitherto entered, fully to explain even as much of their nature as is necessary for popular purposes; and therefore we shall need to revert to them in a future section, after we are in possession of the other elements which are necessary. We shall only add here, that by means of logarithms, calculations which required days before this invention, can be performed in minutes in consequence of it, and that they have enabled us to perform many calculations with ease which without their aid were altogether impossible. We have deemed it necessary to give the general definition, and also some short explanation of the nature of those exponential or logarithmic numbers along with the explanation of the notation and scale of the natural numbers; because when the meanings of the natural numbers are once rooted in the mind without any explanation, it becomes somewhat difficult to convey a clear and distinct notion of the same characters used as exponents. 42 SECTION IV. COMMON OPERATIONS IN ARITHMETIC. THE use of arithmetic, and indeed of all branches of mathematics, consists in enabling us to find that which we wish to know but do not, and to do this by means of that which is already known to us; and the process by which this is obtained is called an operation. Or we may say that an operation is any process by which we are enabled, from known quantities, to arrive at the knowledge of quantities which are not known. In order to do this in any case we must have always one known quantity of the same kind with that unknown one which we are to find by the operation; and in arithmetical operations we must have this known quantity expressed in the very same unit of measure, or denomination, as the one whose value we seek. Thus if our object is to ascertain how many pounds will require to be paid under conditions which are given, we must have a pound, or something expressible in terms of a pound, among the data which we are to use in our operation; and in like manner, if we seek for the value of any quantity whatever, we must have either the unit in which that quantity is to be expressed, or something convertible into this unit, among the data. Thus if length of time were the quantity sought, we could not find it unless a quantity expressing time were given; and the same in all other cases. It is not necessary, however, that the given quantity should be in the same denomination with that which is sought, provided we know the relation between them. For instance, if a certain number of pounds sterling were the given quantity, and a number of French francs the quantity sought, we could find it with little less labour than if francs had been given, provided we knew the |