AN ESSAY ON QUANTITY

ished 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. 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 adopted into mathematics, and its quantity and relations may be measured with mathematical accuracy and certainty.

*Sec. 2. Of proper and improper quantity.*

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.

I call that proper 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 measure of it. Thus a line is measured by known lines, as inches, feet, or miles; and the length of a foot being known, there can be no question about the length of two feet, or of any part or multiple of a foot. And this known length, by being multiplied or divided, is sufficient to give us a distinct idea of any length whatsoever.

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 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. And having once assigned this measure to it, we can then, and not, 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, I think, be reduced to these four: extension, duration, number, and proportion. Though proportion be measurable in its own nature, and