SfiO PHILOSOPHICAL TBANSACTIONS. [ANNO ] 748. 



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, halfed, 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 mathe- 

 matical 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 explana- 

 tion. 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 mea- 

 sured. 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 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 4, 

 extension, duration, number, and proportion. Though proportion be mea- 

 surable 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 3 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 im- 



