VOL. XLV.J PHILOSOPHICAL TRANSACTIONS, 56l 



pressa, the various kinds of centripetal forces, and different orders of fluxions, 

 are all improper quantities ; which therefore ought not to be admitted into ma- 

 thematics, without having a measure of them signed. The measure of an im- 

 proper 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 ap- 

 pears, 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 2 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 difl^erent 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 Dr. M. conceives that the applying of measures to things that pro- 

 perly 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 ex- 

 pressed and demonstrated without those various measures of motion, and of cen- 

 tripetal and impressed forces which he uses : but this would occasion such intri- 

 cate 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 sucn 

 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 distin- 

 guished from all others, without any danger of mistake. Hence it is that we are 



VOL. IX. 4 C 



