AN ESSAY ON QUANTITY; OCCASIONED BY READING A TREATISE IN WHICH SIMPLE AND COMPOUND RATIOS ARE APPLIED TO VIRTUE AND MERIT *. SINCE it is thought that mathematical demonstration carries a peculiar evidence along with it, which leaves no room for further dispute, it may be of some use, or entertainment at least, to inquire to what subjects this kind of proof may be applied. Mathematics contain properly the doctrine of measure; and the object of this science is commonly said to be quantity; therefore quantity ought to be defined, what may be measured. Those who have defined quantity to be whatever is capable of more or less, have given too wide a notion of it, which it is apprehended has led some persons to apply mathematical reasoning to subjects that do not admit of it. Pain and pleasure admit of various degrees, but who can pretend to measure them? Whatever has quantity, or is measurable, must be made up of parts, which bear proportion to each other, and to the whole; so that it may be increased by addition of like parts, and diminished by subtraction, may be multiplied and divided, and, in short, may bear any proportion to another quantity of the same kind, that one line or number can bear to another. That this is essential to all mathematical quantity, is evident from the first elements of algebra, which treats of quantity in general, or of those relations and properties which are common to all kinds of quantity. Every algebraical quantity is supposed capable not only of being increased and diminished, but of being exactly doubled, tripled, halved, or of bearing any assignable proportion to another quantity of the same kind. This then is the characteristic of quantity; whatever has this property may be adopted into mathematics; and its quantity and relations may be measured with mathematical accuracy and certainty. There are some quantities which may be called proper, and others improper. This distinction is taken notice of by Aristotle; but it deserves some explanation. That properly is quantity which is measured by its own kind; or which of its own nature is capable of being doubled or tripled, without taking in any quantity of a different kind as a measure of it. Improper quantity is that which cannot be measured by its own kind; but to which we assign a measure by the means of some proper quantity that is related to it. Thus velocity of motion, when we consider it by itself, cannot be measured. We may perceive one body to move faster, another slower; but we can have no distinct idea of a proportion or ratio *This Essay was originally published in the London Philosophical Transactions, volume xlv. between their velocities, without taking in some quantity of another kind to measure them by. Having therefore observed, that by a greater velocity a greater space is passed over in the same time, by a less velocity a less space, and by an equal velocity an equal space; we hence learn to measure velocity by the space passed over in a given time, and to reckon it to be in exact proportion to that space; and having once assigned this measure to it, we can then, and not till then, conceive one velocity to be exactly double, or half, or in any other proportion to another; we may then introduce it into mathematical reasoning without danger of confusion or error, and may also use it as a measure of other improper quantities. All the kinds of proper quantity we know, may perhaps be reduced to these four, extension, duration, number, and proportion. Though proportion be measurable in its own nature, and therefore has proper quantity, yet as things cannot have proportion which have not quantity of some other kind, it follows, that whatever has quantity must have it in one or other of these three kinds, extension, duration, or number. These are the measures of themselves, and of all things else that are measurable. Number is applicable to some things, to which it is not commonly applied by the vulgar. Thus, by attentive consideration, lots and chances of various kinds appear to be made up of a determinate number of chances that are allowed to be equal; and by numbering these, the values and proportions of those which are compounded of them may be demonstrated. Velocity, the quantity of motion, density, elasticity, the vis insita and impressa, the various kinds of centripetal forces, and different orders of fluxions, are all improper quantities; which therefore ought not to be admitted into mathematics, without having a measure of them assigned. The measure of an improper quantity ought always to be included in the definition of it; for it is the giving it a measure that makes it a proper subject of mathematical reasoning. If all mathematicians had considered this as carefully as Sir Isaac Newton appears to have done, some labour had been saved both to themselves and to their readers. That great man, whose clear and comprehensive understanding appears even in his definitions, having frequent occasion to treat of such improper quantities, never fails to define them, so as to give a measure of them, either in proper quantities, or in such as had a known measure. This may be seen in the definitions prefixed to his Princip. Phil. Nat. Math. It is not easy to say how many kinds of improper quantity may, in time, be introduced into mathematics, or to what new subjects measures may be applied but this I think we may conclude, that there is no foundation in nature for, nor can any valuable end be served by, applying measure to any thing but what has these two properties: first, it must admit of degrees of greater and less: secondly, it must be associated with or related to something that has proper quantity, so as that when one is increased the other is increased; when one is diminished, the other is diminished also; and every degree of the one must have a determinate magnitude or quantity of the other corresponding to it. It sometimes happens, that we have occasion to apply different measures to the same thing Centripetal force, as defined by Newton, may be measured various ways; he himself gives different measures of it, and distinguished them by different names, as may be seen in the above-mentioned definitions. In reality Dr. M. conceives, that the applying of measures to things that properly have not quantity, is only a fiction or artifice of the mind, for enabling us to conceive more easily, and more distinctly to express and demonstrate, the properties and relations of those things that have real quantity. The propositions contained in the first two books of Newton's Principia might perhaps be expressed and demonstrated without those various measures of motion, and of centripetal and impressed forces which he uses, but this would occasion such intricate and perplexed circumlocutions, and such a tedious length of demonstrations, as would frighten any sober person from attempting to read them. From the nature of quantity we may see what it is that gives mathematics such advantage over other sciences, in clearness and certainty; namely, that quantity admits of a much greater variety of relations than any other subject of human reasoning; and at the same time every relation or proportion of quantities may, by the help of lines and numbers, be so distinctly defined, as to be easily distinguished from all others, without any danger of mistake. Hence it is that we are able to trace its relations through a long process of reasoning, and with a perspicuity and accuracy which we in vain expect in subjects not capable of mensuration. Extended quantities, such as lines, surfaces, and solids, besides what they have in common with all other quantities, have this peculiar, that their parts have a particular place and disposition among themselves: a line may not only bear any assignable proportion to another, in length or magnitude, but lines of the same length may vary in the disposition of their parts; one may be straight, another may be part of a curve of any kind or dimension, of which there is an endless variety. The like may be said of surfaces and solids. So that extended quantities admit of no less variety with regard to their form, than with regard to their magnitude and as their various forms may be exactly defined and measured, no less than their magnitudes, hence it is that geometry, which treats of extended quantity, leads us into a much greater compass and variety of reasoning than any other branch of mathematics. Long deductions in algebra for the most part are made, not so much by a train of reasoning in the mind, as by an artificial kind of operation, which is built on a few very simple principles: but in geometry we may build one proposition on another, a third upon that, and so on, without ever coming to a limit which we cannot exceed. The properties of the more simple figures can hardly be exhausted, much less those of the more complex ones. It may be deduced from what has been said above, that mathematical evidence is an evidence sui generis, not competent to any proposition which does not express a relation of things measurable by lines or numbers. All proper quantity may be measured by these, and improper quantities must be measured by those that are proper. There are many things capable of more or less, which perhaps are not capable of mensuration. Tastes, smells, the sensations of heat and cold, beauty, pleasure, all the affections and appetites of the mind, wisdom, folly, and most kinds of probability, with many other things too tedious to enumerate, admit of degrees, but have not yet been reduced to measure, nor perhaps ever can be. I say, most kinds of probability, because one kind of it, viz. the probability of chances, is properly measurable by number, as observed above. Though attempts have been made to apply mathematical reasoning to some of these things, and the quantity of virtue and merit in actions has been measured by simple and compound ratios; yet Dr. M. does not think that any real knowledge has been struck out this way; it may perhaps, if discreetly used, be a help to discourse on these subjects, by pleasing the imagination, and illustrating what is already known; but till our affections and appetites shall themselves be reduced to quantity, and exact measures of their various degrees be assigned, in vain shall we essay to measure virtue and merit by them. This is only to ring changes on words, and to make a show of mathematical reasoning, without advancing one step in real knowledge. Dr. M. apprehends that the account given of the nature of proper and improper quantity, may also throw some light on the controversy about the force of moving bodies, which long exercised the pens of many mathematicians, and perhaps is rather dropped than ended; to the no small scandal of mathematics, which has always boasted of a degree of evidence, inconsistent with debates that can be brought to no issue. Though philosophers on both sides agree with each other, and with the vulgar in this, that the force of a moving body is the same, while its velocity is the same, is increased when its velocity is increased, and diminished when that is diminished. But this vague notion of force, in which both sides agree, though perhaps sufficient for common discourse, yet is not sufficient to make it a subject of mathematical reasoning; in order to that, it must be more accurately defined, and so defined as to give us a measure of it, that we may understand what is meant by a double or a triple force. The ratio of one force to another cannot be perceived but by a measure; and that measure must be settled, not by mathematical reasoning, but by a definition. Let any one consider force without relation to any other quantity, and see whether he can conceive one force exactly double to another; I am sure I cannot, says he, nor shall, till I shall be endowed with some new faculty; for I know nothing of force but by its effects, and therefore can measure it only by its effects. Till force then is defined, and by that definition a measure of it assigned, we fight in the dark about a vague idea, which is not sufficiently determined to be admitted into any mathematical proposition. And when such a definition is given, the controversy will presently be ended. Of the Newtonian Measure of Force.-You say, the force of a body in motion is as its velocity: either you mean to lay this down as a definition, as Newton himself has done; or you mean to affirm it as a proposition capable of proof. If you mean to lay it down as a definition, it is no more than if you should say, I call that a double force which gives a double velocity to the same body, a triple force which gives a triple velocity, and so on in proportion. This he entirely agrees to: no mathematical definition of force can be given that is more clear and simple, none that is more agreeable to the common use of the word in language. For since all men agree, that the force of the body being the same, the velocity must also be the same; the force being increased or diminished, the velocity must be so also, what can be more natural or proper than to take the velocity for the measure of force? Several other things might be advanced to show that this definition agrees best with the common popular notion of the word force. If two bodies meet directly with a shock, which mutually destroys their motion, without producing any other sensible effect, the vulgar would pronounce, without hesitation, that they met with equal force; and so they do, according to the measure of force above laid down; for we find by experience, that in this case their velocities are reciprocally as their quantities of matter. In mechanics, where by a machine two powers or weights are kept in equilibrio, the vulgar would reckon that these powers act with equal force, and so by this definition they do. The power of gravity being constant and uniform, any one would expect that it should give equal degrees of force to a body in equal times, and so by this definition it does. So that this definition is not only clear and simple, but it agrees best with the use of the word force in common language, and this is all that can be desired in a definition. But if you are not satisfied with laying it down as a definition, that the force of a body is as its velocity, but will needs prove it by demonstration or experiment; I must beg of you, before you take one step in the proof, to let me know what you mean by force, and what by a double or a triple force. This you must do by a definition which contains a measure of force. Some primary measure of force must be taken for granted, or laid down by way of definition; otherwise we can never reason about its quantity. And why then may you not take the velocity for the primary measure as well as any other? You will find none that is more simple, more distinct, or more agreeable to the common use of the word force; and he that rejects one definition that has these properties, has equal right to reject any other. I say then, that it is impossible, by mathematical reasoning or experiment, to prove that the force of a body is as its velocity, without taking for granted the thing you would prove, or something else that is no more evident than the thing to be proved. Of the Leibnitzian Measure of Force.-Let us next hear the Leibnitzian, who says, that the force of a body is as the square of its velocity. If he lays this down as a definition, I shall rather agree to it than quarrel about words, and for the future shall understand him, by a quadruple force to mean that which gives a double velocity, by nine times the force, that which gives three times the velocity, and so on in duplicate proportion. While he keeps by his definition, it will not necessarily lead him into any error in mathematics or mechanics. For however paradoxical his conclusions may appear, however different in words from theirs who measure force by the simple ratio of the velocity; they will in their meaning be the same: just as he who would call a foot twenty-four inches, without changing other measures of length, when he says a yard contains a foot and a half, means the very same as you do, when you say a yard contains three feet. But though I allow this measure of force to be distinct, and cannot be charged with falsehood, for no definition can be false, yet I say, in the first place, It is less simple than the other; for why should a duplicate ratio be used where the simple ratio will do as well? In the next place, This measure of force is less agreeable to the common use of the word force, as has been shown above; and this indeed is all that the many laboured arguments and experiments, brought to overturn it, do prove. This also is evident, from the paradoxes into which it has led its defenders. We are next to consider the pretences of the Leibnitzian, who will undertake to prove by demonstration, or experiment, that force is as the square of the velocity. I ask him first, what he lays down for the first measure of force? The only measure I remember to have been given by the philosophers of that side, and which seems first of all to have led Leibnitz into his notion of force, is this; the height to which a body is impelled by any impressed force, is, says he, the whole effect of that force, and therefore must be proportional to the cause: but this height is found to be as the square of the velocity which the body had at the beginning of its motion. In this argument I apprehend that great man has been extremely unfortunate. For, first, whereas all proof should be taken from principles that are common to both sides, in order to prove a thing we deny, he assumes a principle which we think farther from the truth; namely, that the height |