18 DEFINITION OF QUANTITY. month of hard toil; but a lawyer in first-rate practice would consider it little for speaking some hundred words which cost him no labour at all. A wagon and horses would not only be much for a man to carry, but too much for being carried by a score of the strongest men in any parish. But all the wagons and horses, and all other things, moveable and immoveable, on the surface of the earth, are so very little for the earth to carry, that they do not in the least hinder its motion round the sun, which is at the rate of more than seventy thousand miles in the hour. As the words much and little are thus equally expressive of quantity, the simplest and most general definition of quantity which we can obtain is, that which we can call either much or little. It will follow from this definition that there is almost an endless variety of quantities, not only of individual quantities but of kinds of quantities; and these quantities consist, not only of things, but of the relations of things, and of all sorts of changes and successions, whenever we can call them either much or little. Thus time is a quantity, mere distance from place to place is a quantity, motion is a quantity, and even the change of motion is a quantity: for, in respect of time we can say, "It is much longer;" of distance, "It is much greater;" of motion, "It is much quicker;" and of change of motion, "It is much quicker now than it was before." We can also apply the word "little," or some word having a similar meaning, in each of these cases, and therefore they answer the whole definition of the word quantity. In order to be able to use quantities in practice, it is necessary that we should have the means of answering the question, "How much?" or "How little?" with regard to them; and thus the next consideration is, how this is to be done. In the simplest view of the matter, we may suppose the quantity respecting which we wish to answer the question to be known as a whole; as, for instance, how much money is in the purse, how much measure is in the table, and so of other cases. Now it will be immediately felt that in order to answer these questions, simple as they are, there is an element wanting; for when we put the question, "How much money is in the purse?" another question immediately rises to the mind, and demands an answer before the first one: "What kind of money?sovereigns? shillings? or what?" If we took any quantity whatever, a similar question would arise; so that in all cases where we asked, "how much?" we would be met by the question, "of what?" The answer to this question must be made in something that we know already; for if not, the very same question would arise a second time. Let us take an instance: "How much money is in the purse?" "Of what money?" "British money." And then comes the question, "What kind of British money?" and if the answer be, “sovereigns," or "shillings," or anything else that we name, and know as a kind or denomination of British money, we are in a condition for getting an answer to the first question, but not till then. The shilling or sovereign, or whatever else the denomination may be, is the standard or measure which we are to apply to the money in the purse; and in the case of every quantity, we must have a standard or measure before we can find how much the quantity is, and this measure must be known to us, and must be of exactly the same kind as the quantity. This, though a simple consideration, is an important one, and it may not be amiss to see what would be the effect of referring to a standard not of the same kind with the quantity we intended to measure, Perhaps none is better than the traveller's ironical question to 20 COMPARISON OF QUANTITIES. the Irish ditcher, and the ditcher's reply. The traveller was going toward the town of Mullingar, and asked a ditcher by the road-side, "Pray, my good fellow, how far is it from Mullingar to Martinmas?" "Plase you, yer honour, and it's just as far as from Christmas to the ace of spades." But we have not only to determine how large single quantities are in terms of some known measure, for we often have occasion to compare the whole of one quantity with the whole of another, without any reference to the particular measuring of either of them, and therefore it becomes necessary to have some general means of determining when quantities are of the same kind with each other and when they are not. In the case of quantities which really exist and are palpable to the senses, we are never at much loss to find out, at least in a general way, which are of the same kind with each other and which not; but in our mathematical inquiries, we make use only of the relations of quantities and not of the actual quantities themselves, and therefore it becomes necessary that we should have a standard whereby to determine generally when they are of the same kind and when they are not. Now the simplest test of sameness that we can have is that of being able to say that the quantities are either equal, or that the one of them is greater than the other; and simple as this seems, it is all which we require, only we must be careful to view each of them in its whole character, and not to estimate both in any one quality which is common to the two. Thus if the comparison were one hour of time and four miles of a road, and it were asked whether these were of equal length, or which were the longer, no answer could be given, and the quantities are clearly not of the same kind. If, however, we referred the hour and the four miles to something travelling along the road, they might be equal, or either of them might be the greater. For instance, to a man walking four miles an hour, the hour of time and the four miles of road would be of exactly the same length; to a coach running twelve miles an hour, the hour would be three times as long as the four miles of road; and to a pig getting on at the rate of a mile an hour, the four miles of road would be four times as long as the hour. Thus when we speak of quantities, as being of the same kind or of different kinds, it must always be understood that we speak of them as independent quantities and not as relations of other quantities, even though they can exist only in the latter sense. Thus the strength of a man, the strength of a horse, the motion of the wind, the weight of falling water, and the elastic force of steam, are in one sense all quantities of the same kind, and the effects of them, and consequently they themselves, admit of being measured by the very same standard, though the substantive existences are all different, and no two of them admit of any comparison. All quantities, which can exist, or of which we can form any notion as being either much or little, whether we can express them exactly in terms of any standard or not, can become subjects of mathematics, and so can all those relations of quantities to each other which we can in any way understand or express; and when we speak precisely of a quantity, or name that which we call the value of it, we always name a relation-the relation which it bears to the known standard in which we estimate that kind of quantity. In quantities of daily occurrence, we generally have a considerable number of standards or denominations, as we term them; as in money we have pounds, shillings, pence, and farthings, and any one quantity of money we can express in any one of those denominations with equal accuracy, though for convenience we express large quantities in the larger ones, and small quantities in the smaller, and the relation of any known or measured quantity to the standard in which it is 22 NUMBER AND QUANTITY. measured is expressed by a number. Thus five pounds expresses the relation of a known quantity of money to a pound, considered as the standard; and it is of no consequence as to mathematical value, whether this five pounds be five pieces of gold coin, each equal to a pound, or anything else which would at the public market readily and always exchange for those five pieces. In the case of two such standards, as, for instance, one pound and one shilling, it is evident that we can express the value in terms of either of them, provided we know the relation between them; and from the common way of measuring by means of a standard, with which everybody is acquainted, it will be perceived that the relation of any quantity to any other of the same kind is the number of times that the first is contained in the second. Thus the relation of a shilling to a pound is one to twenty, and the relation of a pound to a shilling is twenty to one; and it must be understood in all cases that the number which results from this comparison is the measure or value of the second, in terms of the first or standard, considered as one whole. So also, upon the same principle, it is evident that if any two quantities of the same kind are in this way compared with a standard, which is the same in the case of both, the results of the comparisons with this standard would accurately express the relation of the two quantities to each other. This is called the proportion or ratio of the two quantities, and though simple when viewed in this light, it is, in a practical point of view, one of the most important principles in the whole range of mathematical science. We must readily admit this, when we consider that we can obtain no knowledge of the value of any thing but by referring it to some standard which we already know; and not only this, but that we can get no knowledge of anything whatever but by comparing it with what we already know. This comparison, this finding of the ratio, or relation, |