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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.

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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

relation between a pound and a franc, or the number of any one of the two species of coin which is equal to a known number of the other species.

In the simple operations of common arithmetic this necessity for having among the data a representative of the quantity sought, seems so obvious a matter, as not to require being stated. But when we come to the more intricate parts of mathematics, and especially when we come to mixed problems-or things to be done-in which mathematics form only a part, it becomes a consideration of considerable difficulty as well as importance. The difficulty increases, too, in proportion as the mathematical part becomes smaller in respect of the whole case; and thus it is of great importance not in matters of calculation only, but in all matters generally, to make ourselves sure at the outset, that the data, or terms and conditions, by means of which we attempt to arrive at any result, contain elements sufficient for determining that result. We must bear in mind that any one of the conditions which are involved in the data by means of which we endeavour to find an unknown quantity may become the unknown quantity in a problem of another description; and that thus not only all sorts of quantities, but all sorts of relations of quantity to quantity, may become that element among the data, which is the virtual representative of the result; and we may add, that there are many cases in which the relations of quantities are not only all that we can obtain, but that those relations are often indeterminate, or inexpressible by our ordinary means of notation. This is not the place for entering into any particular explanation of when we have or have not data sufficient for the obtaining of the result which we seek; but we mention the subject at the outset, because it is one of great importance, and one which everybody who wishes to study mathematics easily and profitably must bear constantly in mind. To use a homely

expression, the representative of the quantity sought among the data is of nearly the same importance as money in the purse is to a man who would buy a horse at a ready money market. The man may know exactly the place and hour of the sale, may have the means of arriving there in perfect time, may know who has got to sell exactly the horse which he wishes, he may have the blacksmith ready to put new shoes on the horse, and saddle, bridle, and portmanteau all in perfect order for a journey; but not one mile of that journey shall he ride if he is without the cash, the real element which represents the horse and it is even so in all cases, whether mathematical or not.

In the arithmetic of simple or abstract numbers, viewed in its most elementary form, there are only four general problems, or kinds of results. The first is to find the sum of two or more numbers, and that is nothing more than finding one number which shall contain the number 1, the standard by which we measure all simple numbers, as often as it is contained in all the numbers whose sum is sought. The second is to find the difference between two numbers; and this difference is nothing more than a number which added to the less of the two given ones would make a sum equal to the greater, or which taken away from the greater would leave a remainder equal to the less.

The process by which we find a sum is called ADDITION, and that by which we find a difference is called SUBTRACTION. We shall very shortly notice the leading principles of these operations, before we proceed to the two remaining ones; because addition and subtraction are in some respects, though not altogether, the converse, or opposite each other. The operation of adding, and that of subtracting, are exactly the reverse of each other; but the result of addition is not exactly the opposite of

that of subtraction in all cases, for the result of an addition must be equal to the whole of all the given quantities, and therefore it must be greater or less according as they, taken in their whole amount, are greater or less; whereas the result of a subtraction expresses merely the difference of the quantities, and thus it has no necessary reference to the entire value of the one or the other, or of both, but merely to the difference, or how much the one is greater and the other less. This consideration is worthy of some attention, simple as it is, and it will readily be understood when we consider that the difference of any two numbers, however large, that have all their figures the same with each other except the units, is exactly the same as the difference of these unit figures, and that if their difference is 1, then the difference between the numbers is exactly the same as that between 0 and 1. Thus the difference between 31587926 and 31587925 is equal to the difference between 0 and 1, that is, it is 1.

From this it follows that, when our object is to discover the difference of two quantities, we may take away as much of them as ever we please, if we take exactly the same from each; and that we may add as much to them as ever we please, if we add exactly the same to each. The first part of this very obvious power that we have over them is often of great use when we seek the difference of complicated quantities which contain many elements; and the second is the foundation of that borrowing and paying in common subtraction which is not unfrequently an unexplained puzzle to beginners in the arithmetical art.

The simple process of adding can hardly be made plainer by any explanation, and taken in detail it is not a difficult one, as we never have to add more than 9 at any one step; and therefore the only considerations in simple addition are, to take care