AN ESSAY ON QUANTITY

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. These are the measure 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 anything 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 distinguishes them by different names, as may be seen in the above-mentioned definitions.

In reality, I conceive that the applying of measures to things that properly have not quantity, is only a fiction or artifice of