rojj XVI.'] PHILOSOPHICAL TRANSACTIONS. 20S 



the gravity to be equal throughout: and the descents of projectiles to be in pa- 

 rallel lines, which in reality are towards the centre, the difference being so small, 

 as by no means to be discovered in practice. The opposition of the air, it is true, 

 is considerable against all light bodies moving through it, as likewise against small 

 ones ; but in large and ponderous shot, this impediment is found by experience 

 but very small, and may safely be neglected,* 



Propositions concerning the Descent of heavy Bodies, and the Motion oj" Projects. 



Prop. I. The velocities of falling bodies are proportional to the times, from 

 the beginning of their falls. — For the action of gravity being continual, in every 

 space of time the falling body receives a new impulse, equal to what it had 

 before, in the same space of time, received from the same power; for instance, 

 in the 1st second of time, the falling body has acquired a velocity, which in 

 that time would carry it to a certain distance, suppose 32 feet, and if there were 

 no new force, it would descend at that rate with an equable motion ; but in the 

 next second of time, the same power of gravity continually acting on it, super- 

 adds a new velocity equal to the former ; so that at the end of 2 seconds, the 

 velocity is double to what it was at the end of the first: and after the same 

 manner may it be proved to be triple at the end of the 3d second, and so on. 

 Therefore the velocities of falling bodies are proportional to the times of their 

 falls. Q. E. D. 



Prop. II. The spaces described by the fall of a body, are as the squares of 

 the times, from the beginning of the fall. — Let AB (fig. 1, pi. 9,) represent 

 the time of the fall of a body ; BC, perpendicular to AB, the velocity acquired 

 at the end of the fall; and draw the line AC ; then divide the line AB, repre- 

 senting the time, into as many equal parts as you please, as b, b, b, b, &c. and 

 through these points draw the liries be, be, be, be, &c. parallel to BC. It is 

 manifest that the several lines, be, represent the several velocities of the falling 

 body, in such parts of the time as Ab is of AB, by the former proposition. 

 It is likewise evident, that the area ABC is the sum of all the lines be, being 

 taken, according to the method of indivisibles, infinitely many ; so that the area 

 ABC represents the sum of all the velocities, between none and BC, supposed 

 infinitely many ; which sum is the space descended in the time represented by 



• How erroneous this notion is, we shall have occasion to notice hereafter, when we arrive at 

 those parts of the Transactions containing some of Mr. Robins's papers relating to this subject; where 

 it is found that the effect of the air's resistance, even on heavy iron cannon balls, instead of being 

 inconsiderable, is in fact enormously great ; so much so indeed that without that resistance, they 

 would in many cases range 10 times, or 20 times as far, as they are now found to do; so that, 

 instead of an extreme range of a mile or two, as is now the case, they would be capable of ranging 

 ,lo the distance of 10, 20, or 30 miles ! 



VOL. III. M M 



