VOL. XXXV.J PHILOSOPHICAL TRANSACTIONS. 221 



In a body in motion, there may be considered distinctly, the quantity of the 

 matter, and the velocity of the motion. The force arising from the quantity 

 of the matter, as its cause, must necessarily be proportional to the quantity of 

 the matter: and the force arising from the velocity of the motion, as ics cause, 

 must necessarily be proportional to the velocity of the motion. The whole 

 force therefore arising from these two causes, must necessarily be proportional 

 to these two causes taken together. And therefore in bodies of equal size and 

 density, or in one and the same body, the quantity of matter continuing always 

 the same, the force must necessarily be always proportional to the velocity of 

 the n)Otion. If the force were as the square of the velocity, all that part of 

 the force, which was above the proportion of the velocity, would arise either 

 out of nothing, or, according to Mr. Leibnitz's Philosophy, out of some 

 living soul essentinlly belonging to every particle of matter. 



Whenever any effect whatever is in a duplicate proportion, or as the square 

 of any cause; it is always either because there are two causes acting at the same 

 time, or that one and the same cause continues to act for a double quantity 

 of time. 



The resistance made to a body moving in any fluid medium, is in a duplicate 

 proportion to the velocity of its motion ; because, in proportion to its velocity, 

 it is resisted by a greater number of particles in the same time ; and again, in 

 proportion to its velocity, it is resisted by the same particles singly with a 

 greater force, as being to be moved out of their places with greater velocity. 



Light decreases in a duplicate proportion of its distance from the sun ; be- 

 cause the rays divaricate according to two dimensions ; according to the dimen- 

 sion upwards or downwards, and according to the dimension sideways. But 

 according to the third dimension forwards from the sun, a ray of light under- 

 goes no alteration ; because the particles, of which it consists, being emitted 

 all of them witli an equal velocity, continue every where at an equal distance 

 from each other. 



One and the same cause, acting in a double quantity of time, produces the 

 same effect, as two equal causes acting in a single quantity of time. One and 

 the same force, in two parts of time, will cause a body in motion to describe 



And as the infinitesimal of an infinitesimal (that is, a second fluxion, or the second power of infini- 

 tesimal) is to ], so is 1 to infinity of infinites, or the second power of infinite; that is, for instance, 

 it is as a finite physical line to an infinite surface, or as a finite physical surface to an infinite solid. 

 And as (which is beyond all proportion lower than the infinit'th power of an infinitesimal) is to 1, 

 so is 1 to that which is beyond all proportion higher than the infinit'th power of infinite. Which 

 clearly removes the foundation of all the ridiculous consequences, which have been drawn from the 

 supposition of the forementioned false proportion. — Orig. 



