Thomas Reid

“ In reality, I conceive 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 two first 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. ”

“ 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, I apprehend, has led some persons to apply mathematical reasoning to subjects that do not admit of it. ”

“ We may perceive one body to move faster, another slower; but we can have no distinct idea of a proportion or ratio 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 in a less space, and by an equal velocity in 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. ”

“ Whatever has quantity, or is measurable, must be made up of parts, which bear proportion to one another 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 a word, 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. ”

“ 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. ”

“ 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 upon 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. ”

“ 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. ”

“ I apprehend the account that hath been given of the nature of proper and improper quantity, may also throw some light upon the controversy about the force of moving bodies, which long exercised the pens of many mathematicians, and, for what I know, is rather dropped than ended; to the no small scandal of mathematics, which hath always boasted of a degree of evidence, inconsistent with debates that can be brought to no issue. ”

“ 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 ”

“ 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. ”

“ 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 anything but what has these two properties. ”

“ 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. ”

“ Though proportion be measurable in its own nature, and therefore hath 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. ”

“ Suppose a philosopher has considered only that measure of centripetal force which is proportional to the velocity generated by it in a given time, and from this measure deduces several propositions: another philosopher, in a distant country, who has the same general notion of centripetal force, takes the velocity generated by it, and the quantity of matter together, as the measure of it. ”

“ 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. ”

“ Tastes, smells, the sensation 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, as I apprehend, ever can be. ”

“ 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 ”

“ 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 further from the truth ”

“ Thereupon, a serious controversy is begun, whether centripetal force be as the velocity, or as the velocity and quantity of matter taken together. Much mathematical and experimental dust is raised; and yet neither party can ever be brought to yield ”

“ 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 ”