PROJECTILES. 



PROJECTILES, are such bodies as, be- 

 ing put in a violent motion by any great 

 force, are then cast ofF or let go from the 

 place where they received their quantity 

 of motion ; as a stone thrown from a 

 sling, an arrow from a bow, a bullet from 

 a gun, Sec. It is usually taken for grant- 

 ed, by those who treat of the motion of 

 projectiles, that the force of gravity near 

 the earth's surface is every where the 

 same, and acts in parallel directions ; and 

 that the effect of the air's resistance up- 

 on very heavy bodies, such as bombs and 

 cannon-balls, is too small to be taken into 

 consideration. 



Sir Isaac Newton has shown, that the 

 gravity of bodies which are above the su- 

 perficies of the earth, is reciprocally as 

 the squares of their distances from its 

 centre ; but the theorems concerning the 

 descent of heavy bodies, demonstrated by 

 Gallileo, and Huygens, and others, are 

 built upon this foundation, that the action 

 of gravity is the same at all distances ; 

 and the consequences of this hypothesis 

 are found to be very nearly agreeable to 

 experience. For it is obvious, that the 

 error arising from the supposition of gra- 

 vity's acting uniformly, and in parallel 

 lines, must be exceedingly small ; be- 

 cause even the greatest distance of a pro- 

 jectile above the surface of the earth, is 

 inconsiderable, in comparison of its dis- 

 tance from the centre, to which the gra- 

 vitation tends. But then, on the other 

 hand, it is very certain, that the resist- 

 ance of the air to very swift motions, is 

 much greater than it has been commonly 

 represented. Nevertheless, (in the appli- 

 cation of this doctrine to gunnery) if the 

 amplitude of the projection, answering to 

 ^>ne given elevation, be first found by ex- 

 periment (which we suppose) the ampli- 

 tudes in all other cases, where the eleva- 

 tions and velocities do not very much dif- 

 fer from the first, may be determined, to 

 a sufficient degree of exactness, from the 

 foregoing hypothesis ; because, in all such 

 cases, the effects of the resistance will be 

 nearly as the amplitudes themselves ; and 

 were they accurately so, the proportions 

 of the amplitudes, at different eleva- 

 tions, would then be the very same as in 

 vacuo. 



Now, in order to form a clear idea of 

 the subject here proposed, the path of 

 every projectile is to be considered as 

 depending on two different forces ; that 

 is to say; on the impellant force, whereby 

 the motion is first begun, (and would be 

 continued in a right line) and on the 

 force of gravity, by which the projectile, 



during the whole time of its flight, is con 

 tinually urged downwards, and made to 

 deviate more and more from its first di- 

 rection. As whatever relates to the track 

 and flight of a projectile, or ball, (neg- 

 lecting the resistance of the air) is to be 

 determined from the action of these two 

 forces, it will be proper, before we pro- 

 ceed to consider their joint effects, to 

 premise something concerning the nature 

 of the motion produced by each, when 

 supposed to act alone, independently of 

 the other ; to which end we have premis- 

 ed the two following lemmata. 



Lemma I. Every body, after the im- 

 pressed force whereby it is put in mo- 

 tion ceases to act, continues to move 

 uniformly in a right line ; unless it be 

 interrupted by some other force or im- 

 pediment. 



This is a law of nature, and has its de- 

 monstration from experience and matter 

 of fact 



Corollary. It follows from hence, that 

 a ball, after leaving the mouth of the 

 piece, would continue to move along the 

 line of its first direction, and describe 

 spaces therein proportional to the times 

 of their description, were it not for the 

 action of gravity; whereby the direc- 

 tion is changed, and the motion inter- 

 rupted. 



Lemma II. The motion, or velocity, 

 acquired by a ball, in freely descending 

 from rest, by the force of an uniform gra- 

 vity, is as the time of the descent ; and 

 the space fallen through, as the square of 

 that time. 



The first part of this lemma is ex- 

 tremely obvious : for since every motion 

 is proportional to the force whereby it is 

 generated, that generated by the force of 

 an uniform gravity must be as the time 

 of the descent ; because the whole effort 

 of such a force is proportional to the time 

 -rA of its action ; that is, as the 

 time of the descent. 



To demonstrate that the 

 d distances descended are pro. 

 portional to the squares of tlie 

 e times, let the time of falling 

 through any proposed dis- 

 tance A B, be represented by 

 .the right line P Q; which 

 J conceive to be divided into 

 an indefinite number of very 

 small, equal particles, repre- 

 g sented, each, by the symbol 

 m ; and let the distance de- 

 scended in the first of them 

 be A c ; in the second c d ; in 

 the third d e ; and so on. 



P T 



B 







