MECHANICS. 
631 
as 4. to 3. As be Fort proved, the time in N k is to the 
time in N V as N k to N V, or as 1 to 2 ; but, k V being 
horizontal, the motion in k\ mull be uniform, and it 
will be defcribed by that uniform motion in half the time 
the body falls from N to k ; therefore, if the time in which 
k V is defcribed uniformly be called T, the time in which 
N k is defcribed will be 2T, and the time in which the 
chord N V will be defcribed will be^-T; and confequently 
the time in which a body would fall along the two tan¬ 
gents, is to the time in which it would defcribe the chord, 
as 3 to 4. 
Of the Motion of Projectiles. 
Prop. LXIV. Bodies thrown horizontally or obliquely, have 
a curvilinear motion, and the path which they defcribe is a pa¬ 
rabola ; the air's refinance not being confidercd. —II a body 
be thrown in the direction A F, (tig. 80, 81.) and afted 
upon by the projectile force alone, it will continue to 
move on uniformly in the right line A F, and would 
defcribe equal parts of the line A F in equal times, as 
A C, C D, DE,&c. But if, in any indefinitely-fmall por¬ 
tion of time, in which the body would by the projeCtile 
force move from A to C, it would, by the force of gra¬ 
vity, have fallen from A to G; by the competition of 
thefe forces, it will, at the end of that time, be found in 
H, the oppofite angle of the parallelogram A C G H. In 
two fuch portions of time, whilft it would have moved 
from A to D by the projectile force, it would by gravi¬ 
tation fall through four times A G, that is, A M; and 
therefore, thefe forces being combined, it will be found 
at the end of that time in I, the oppofite angle of the pa¬ 
rallelogram D M. In like manner, at the end of the third 
portion of time, it would by the projeCtile force be car¬ 
ried through three equal divifions to E, and by the force 
of gravitation over nine times AG to N; and confe¬ 
quently, by both thefe forces acting jointly, it will be 
carried to K, the oppofite angle of the parallelogram E N. 
Therefore the lines C H, D I, E K, that is, AG, AM, 
A N, which are to each other as the numbers 1, 4, 9, are 
as the fquares of the lines A C, A D, A E ; that is, G II, 
M I, N K, which are as 1, 2, 3. And, becaufe the aftion 
of gravitation is continual, the body in palling from A 
to H, &c. is perpetually drawn out of the right line in 
■which it would move if the force of gravitation were fuf- 
pended, and therefore moves in a curve ; and II, I, and 
K, are any points in this curve in which lines let fall 
from points equally diltant from A in the line AB meet 
the curve. Therefore the body moves in a parabola, the 
property of which is, that the abfeiffa AG, AM, AN, 
are to each other as the fquares of the ordinates G H, 
M I, N K. 
Remark. Very denfe bodies moving with fmall veloci¬ 
ties defcribe the parabolic track fo nearly, that any devia¬ 
tion is fcarcely difcoverable ; but with very conliderable 
velocities the refinance of the air will caufe the body pro¬ 
jected to defcribe a path altogether different from a para¬ 
bola, which will not appear furpriling when it is known 
that the refinance of the air to a cannon-ball of two 
pounds weight, with the velocity of 2000 feet per fecond. 
Is more than equivalent to Co times the weight of the 
ball. 
Frop. LXV. The path which a body thrown perpendicularly 
upwards deferibes in rifing and falling is a parabola. —A Itone 
lying upon the furface of the earth, partaking of the tpo- 
tion of the earth (here f'uppofed) round its axis, this mo¬ 
tion which it has with the earth will not be dellroyed by 
throwing it in a direction perpendicular to the furface of 
the earth. After the projection, therefore, the ftone will 
be moved by l;wo forces, one horizontal, the other per¬ 
pendicular, and will rife in a direction which may be 
Ihown, as in the laft Prop, to be the parabolic curve ; in 
which it will continue till it reaches the higheft point, 
from whence it might be fliown, as in the laft Prop, that 
it will defeend through the other fide of the parabola. 
Prop. LXVI. The velocity with which a body ought to be 
projefled to make it defcribe a given parabola, is fuch as it would 
acquire by falling through a fpace equal to the fourth part of 
the parameter, belonging to that point of the parabola from which 
it is intended to be projefled .—The velocity of the projeCtile 
at the point A (by Prop. LXIV.) is fuch as would carry it 
from A to E, in the fame time in which it would defeend 
from its gravity from A to N. And the velocity acqbired 
in falling from A to N is fuch as in the fame time by an 
uniform motion would carry the body through a fpnee 
double of AN. Therefore the velocity which is acquired 
by the body in falling to N is to that with which the body 
is projected at A, and uniformly carried forwards to E, 
as twice AN is to AE. But lince, from the nature of 
AE 2 
the parabola, — is equal to the parameter of the point 
A, one-fourth part of this parameter will be expreffed by 
XAE 2 
■ — And, becaufe the velocities acquired by falling 
bodies are as the fquare roots of the fp 3 ces they fall 
through, the velocity acquired by a body in falling 
through AN is to the velocity acquired in falling through' 
i AE 2 
or one-fourth part of the parameter of A, as the 
AN 
fquare root of AN to the fquare root of 
lAE 2 
AN 
that is, 
as 3/ AN to 
jAE 
3/ AN 
, or AN to AE, or twice AN to AE. 
Therefore the. velocity acquired by a body in falling from 
A to N has the fame ratio to the velocity with which the 
body is projected or the line AE defcribed, and to the 
velocity acquired by a body in falling through a fourth 
part of the parameter belonging to the point A ; confe¬ 
quently thefe velocities are equal. 
Cor. Hence may be determined the direction in which 
a projeCtile from a given point, with a given velocity, 
muff be thrown to itrike an objeCt in a given fitiiation. 
Let A be the place from which the body is to be thrown, 
and K the fituation of the objeCt, (fig. 82.) Rai'fe AB 
perpendicular to the plane of the horizon, and equal to 
tour times the height from which a body mult fall to ac¬ 
quire the given velocity. Bifect A B in G : through G' 
draw PIG perpendicular to AB: at the point A"raife 
AC perpendicular to A K, and meeting HG in C: on 
C as a centre, with the radius C A, defcribe the circle 
ABDj and through K draw the right line KEI perpen¬ 
dicular to the plane of the horizon, and cutting the circle 
ABD in the points E and I. A E, or A I, will be thg 
direction required. 
For, drawing B I, BE, fince A K is a tangent to the 
circle, and BA, IK, are parallel to each other, the angle 
ABE is equal to the angle EAKj and the alternate 
angles B A E, AEK, are equal: therefore the triangles 
ABE, AEK, are fimilar; and A B is to A E as A E 
to EK. Therefore ABxEK=AE 2 ; and AB: 
AE 
' EK ‘ 
In like manner, the triangles BAI, KAI, being fimilar, 
Al 2 
BA is equal to . Since, then, AB is equal to four 
1 iSu 
times the height from which a body muft fall to acquire 
the velocity with which it is to be thrown ; ——— (or - 
y ’ EK ^ IK 
its equal) is the fame. Confequently (by this Prop.) the 
point K will be in a parabola which the body will de¬ 
fcribe, which is thrown with the given velocity in the 
directions AE or AI, and the body will Itrike an object 
placed at K. 
Schol. If the velocity with which a projeCtile is thrown 
be required, it may be determined from experiments in 
the following manner. By the help of a pendulum, or 
any other exaCt chronometer, let the time of the perpen¬ 
dicular flight be taken j then, fince the times of theafeent 
and defeent are equal, the time of the defeent muft be 
equal to one-half of the time of the flight, confequently, 
that time will be known: and, lince a heavy body de- 
feends from a ftate of reft at the rate of i6'i feet in tbs 
firft fecond of time, and that the fpaces through which 
3 bodies 
