PROJECTILES. 
nerated, tliai 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 
Pt 
tA of its action ; that is, as the 
time of the descent. 
XcL To demonstrate that the 
distances descended are pro- 
I g portional to the squares of the 
times, let the time of falling 
through any proposed dis- 
^ tance A B, be represented by 
T*' the right line P Q ; which 
conceive to be divided into 
an indefinite number of very 
small, equal, particles, repre- 
sented each, by the symbol 
TO ; and let the distance de- 
scended in the first of them 
be A c ; in the second c d ; in 
the third de ; and so on. 
Then the velocity acquired being always 
as the time from the beginning of the 
descent, it will at the middle of the first of 
..the said particles be represented by one-half 
' to ; at the middle of the second, by li 
TO ; at the middle of the third, by 3i to, 
&c. which values constitute the series 
TO hm 7to 9to „ 
2 > 2 ’ 'i ’ 2 ’ 2 ’ 
But since the velocity, at the middle of 
any one of the said particles of time, 
is an exact mean between the velocities 
of the two extremes thereof, the corres- 
ponding particle of the distance, A B, 
may be therefore considered as described 
with that, mean velocity : and so, the 
spaces Ac, c d, d e, ef, &c. being re- 
spectively equal to the above-mentioned 
. . TO .‘jto 5to 7to „ 
quantities—,^,—,—, &c. it follows, 
by the continual addition of these, that the 
space Ac, Ad, Ac, A/, &c. fallen through 
from the beginning, will be expressed by 
m Attn 9iW 
2’ T’ T’ 
25m 
&c. which are evi- 
dently to one another in proportion, as, 
1 , 4, 9, 16, 25, &c. that is, as tlie squares^ of 
the times. Q. E. D, 
Corollai-y. Seeing the velocity acquired 
in any number (n) of the aforesaid equal 
particles of time (measured by the space 
that would be described in one single parti- 
clfe) is represented by (n) times to, or nm ; 
it will therefore be, as one particle pf time, 
is to n such particles, so is nm, the said dis- 
tance answering to the former time, to the 
distance, n‘m, corresponding to the latter, 
with the same celerity acquired at the end 
of the said n particles. Whence it appears 
that the space (found above) through 
which the ball falls, in any given time n, is 
just the half of that (»’m) which might be 
uniformly described with the last, or greatest 
celerity in the same time. 
Scholium. It is found by experiment, 
that any heavy body, near the earth’s sur- 
face (where the force of gravity may be 
considered as uniform) descends about 16 
feet from rest, in the first second of time. 
Therefore, as tlie distances fallen through, 
are proved above to be in proportion as 
the squares of the time. It follows that, as 
the square of one second is to the square of 
any given number of seconds, so is 16 
feet to the number of feet, a heavy body 
will freely descend in the said number of 
seconds, inience the nulnber of feet 
descended in any given time will be found, 
by multiplying the square of the number of 
seconds by 16. Thus the distance descended 
in 2, 3, 4, 5, &c. seconds, will appear to be 
64, 144, 256, 400 feet, &c. respectively. 
Moreover, from hence, the time of the 
descent through any given distance will be 
obtained, by dividing the said distance in 
feet, by 16, and extracting the square root 
of the. quotient; or, which comes to the 
same thing, by extracting the square root of 
the whole distance, and then taking one- 
half of that root for the number of seconds 
required. Thus, if the distance be supposed 
2,640 feet ; then, by either of the two 
ways, the time of the descent will come out 
12.84,'or 12.50 seconds. 
It appears also (from the corol.) that the 
velocity per second (in feet) at the end of 
the fall, will be determined by multiplying 
the number of seconds in the fall by 32. 
Thus it is found that a ball, at the end of 
ten seconds, has acquired a velocity of 320 
feet per second. After the same manner, 
by having any two of the four following 
quantities, viz. the force, the times, the 
velocity, and distance, the other two may 
be determined : for let the space freely de- 
scended by a ball, in the first secqud of 
time (which is as the accelerating force) be 
denoted by F ; also let T denote the num- 
ber of seconds wherein any distance, D, 
is descended ; and let V be tlie velocity per 
second, at the end of the descent; then 
will . 
v= 2 F T = 2 = gp 
T 
li 2 
