A C C 



Thus, if .1 body, by means of this conftant foi-cc, ncqiiii'c 

 a velocity of ■;2i feet in one fcconJ of time, it will acquire 

 a velocity of 6+4 feet in two feconds, of 96! feet in three 

 fccondS ; and all bodies, whatever be their quantity of mat- 

 tei-, "will acquire, by the force of gravity, the fame velocity 

 in the fame time. For ever)- equal particle of matter being 

 er.dued with an equal impelhng force, v-iz. its gravity or 

 w.ight, the fum of all the forces, in any compound mafs 

 of matter, will be proportional to the fum of all the weights 

 or quantities of matter to be moved ; confcquently, the 

 forces and mafTes moved, being thus conflantly incrcafed 

 in the fame proportion, the velocities generated will be 

 the fame in all bodies, great or fmall : /'. c. a double force 

 moves a double mafs of matter, with the fame velocity that 

 the fingle force moves the fingle mafs, &c. or, the whole 

 compound mafs falls altogether with the fame velocity, and 

 in the fame manner, as if its particles were not united, but 

 as if each fell by itfelf, and all were feparated from one 

 another ; and being put into n.otion at once, they 

 would fall together, juil as if they were united into one 

 mafs. 



Galileo, who firft difcovered the above-mentioned law of 

 the defcent of falling bodies, illullrated it nearly in the fol- 

 lowing manner. 



The fpace paffed over by a moving body in a given time, 

 and with a given velocity, may be confidered as a reftangle 

 made bv the time and the velocity. — Suppofe A {Plate i. 

 Mechanics, Jig. I.) a heavy body defcending, and let AB 

 reprefent the time of its defcent ; which line we may fup- 

 pofe to be divided into any number of equal parts, AC, 

 CE, EG, &c. reprefenting the intei-vals, or moments of 

 the given time. — Let the body defcend through the firft of 

 thofc divifions, AC, with a certain equable velocity arifmg 

 from the propofed degree of gravity : this velocity will be 

 reprefented by AD ; and the fpace pafied over, by the reft- 

 angle CAD. 



Now, as the aftlon of gravity in the firft moment pro- 

 duced the velocity AD, in the body before at reft ; in the 

 fecond moment, the fame will produce, in the body fo mov- 

 ing, a double velocity, CF ; in the third moment, to the 

 velocity CF will be added a farther degree, which together 

 therewith will make the velocity EH, which is triple of the 

 firft, and fo of the reft. So that in the whole time AB, 

 the body will have acquired a velocity BK. — Again, takino- 

 the divifions of the line, e. g. AC, CE, &c. for the times, 

 the fpaces gone through will be the areas or reclangles 

 CD, EF, &c. So that in the whole time AB, the fpace 

 defcribed by the moveable body, will be equal to all the 

 rcftangles, i. e. to the dented figure ABK. 



Such woirld be the cafe, if the accefilons of velocity only 

 happened in certain given points of time, e. ». in C, in E. 

 &c. fo that the degree of motion ftiould contin\ie the fame 

 tin the next period of acceleration occurs. — If the divifions 

 er intervals of time were fuppofcd lefs, e. g. by half; then 

 the dentures of the figure would be proportionably fmaller ; 



and it would approach fo much the nearer to a triangle 



If they were infinitely fmall, i. f. if the acceffions of velo- 

 city were fuppofed to be made continually, and in every 

 point of time, as is really the cafe; the reftangles thus fuc- 

 ceffively produced will make an exaft triangle, e. g. ABE 

 (fg. 2.) — Here, the whole time AB confitting of the 

 little portions of time A i, 12, &c. and the area of the 

 triangle ABE, of the fum of all the little triangular fur- 

 faces anfwering to the divifions of the time ; the whole area 

 or triangle expreftcs the fpace moved through in the whole 

 time AB ; and the little triangles A 1 /, &c. the fpaces 

 j[one through in the divifions of time A i , &c. 



A C C 



Cut tliefe triangles being fimilar, their areas are to nrn 

 another, as the fquares of their homologous fides A B, A i, 

 &c. and confcquently, the fpaces moved are to each other 

 as the fquares of the times. 



If the velocity were unifoi-m, the fpace would be equal 

 to the product of the velocity and time ; i. e. by an ob- 

 vious notation S=VxT ; but, in this cafe, the velocity in- 

 creafes from o till it becomes equal to V, and therefore the 

 fpace defcribed muft be equal to half the above produdl ; 

 /. e. S = i VxT, and s=i vxt, and S : s : : * VxT : ♦ 

 V X t : : VT : vt. But V : v : : T : t, and VT : vt : : 

 T X T : t X t : confcquently S : s : : T-' : t-. 



Hence we may eafily infer the great law of acceleration, 

 viz. " That a defcending body uniformly accelerated, de- 

 " feribc!, in the whole tune of its defcent, a fpace which 

 " is juft half of what it would have defcribed in the fame 

 " time, with the accelerated velocity it has acquired at tlie 

 " end of its fall." For, the whole ipace the falling body 

 has moved through in the time Wi, we have already fliewn, 

 will be reprefented by the triangle ABE ; and the fpace 

 the fame body would move through in the fame time with 

 the velocity BE, will be reprefented by the rectangle ABEF. 

 — But tlie triangle is known to be equal to juft half the 

 rectangle. — Therefore the fpace moved is juft half of what 

 the body would have moved with the velocity acquired at 

 the end of the fall. Hence we infer, that the fpace moved 

 with the laft acquired velocity BE, in half the time AB, 

 is equal to that really moved by the falling body in the 

 whole time AB. 



From the preceding principles and reafoning we deduce the 

 following general laws of uniformly accelerated motions: viz. 



1. That the velocities acquired are conftantly propor- 

 tional to the times. 



2. That the fpaces are proportional to the fquares of the 

 times ; fo that if a falling body dcfcribe any given length in 

 a given time, in double that time it will defcribe four times 

 that length, in thrice the time nine times the length, &c.; 

 and univerfally, if the times be in arithmetical proportion, i, 

 2, 3, 4, S:c. the fpaces defcribed will be i, 4,9, 16, &c. 

 Thus, a body, which falls by gravity through 1 6 -J, feet in the 

 firft fecond of time, will fall through four times as much, or 

 64 I feet in 2 feconds, &c. And fince the velocities ac- 

 quired in faUing are as the times, the fpaces will be as the 

 fquares of the velocities: and both tlie times and velocities 

 will be in a lubduplicate ratio, or as the fquare roots, of the 

 fpaces. 



3. The fpaces defcribed by a falling body in a feries of 

 equal moments or intervals of time, will be as the odd num- 

 bers I, 3, 5, 7, 9, &c. which are the differences of the 

 fquares or whole fpaces, /'. e. a body which has fallen through 

 1 6-pV feetin the firft fecond, will fall in the next fecondthrougk 

 48 5 feet, and in the third fecond through 80 y- feet, &c. 



Retaining the above notation, S : s : ; T" : t' or : : V* ■; 

 v=; and V : V or T : t : : ^ S : v/ s '• ^- : : S| : s i : and 

 the times will be reciprocally as the velocities, and diredlly as 

 the fpaces; for S : s : : TV : tv, and Stv=sT V : con- 

 fcquently T : t :: Sv : sV; orT=|-. \Mien the ac 



celerating forces are different, but conftant, the fpaces will 

 be a? the produAs of the forces into the fquares of the times; 

 and the times will be in the fubduplicate ratio of the fpaces di- 

 rectly, and of the forces inverfely. For when the force is given, 

 the velocity (V) is as the time (T) ; when the forces are diffe- 

 rent, but conftant, and the time is given, the velocity (V) will 

 be as the force (F). But v/hen neitherthe force northe time 

 is given, the velocity (V ) will be parily as the time and partly 

 as the force, or as their produd ( F x T). Thus, V : v : ': 

 5 ¥xT 



