MECHANICS. 
Gst 
bodies defeend are as the fqhares of the times; if we fay, 
as one fecond is to i6 - i feet, fo is the fquare of the 
number of feconds which exprefs the time of the defcent 
of the projeCfile, to a fourth proportional, we (hall have 
the number of feet through which the projectile feli, 
which, being doubled, will give us the number of feet 
■which the projeCtile would defcribe in the fame time with 
that of the fall, fuppofing it moved with an uniform ve¬ 
locity, equal to that which it acquired by the end of the 
fall; which lalt-found number of feet, being divided by 
the number of feconds which exprefs the time of the pro¬ 
jectile’s defcent, will give a quotient, exprefling the num¬ 
ber of feet through which the projectile would move in 
one fecond of time with a velocity equal to that which it 
acquired in its defcent, which velocity is equal to the 
velocity with which the projeCtile was thrown up ; con- 
fequently, this velocity is difcovered. 
Prop. LXVII. The fquares of the velocities of a projeElile in 
different points of its parabola , are as the parameters belonging 
to thofe points.■ —For (by the laft Prop.) the velocities in 
the feveral points of the parabola are equal to the veloci¬ 
ties acquired in falling through the fourth parts of the 
parameters of the points. Therefore, the fquares of thefe 
velociiies being as the fpaces deferibed, the fquares of the 
velocities in the feveral points of the parabola are as the 
fourth parts of the parameters of thofe points: but the 
whole parameters are as their fourth parts: therefore the 
fquares of the velocities at the feveral points of the para¬ 
bola are as the parameters of thofe points. 
Cor. Hence, fetting afide any difference which may arife 
from the refinance of the air, a projecfile will ftrike a 
mark as forcibly at the end as at the beginning of its 
courfe, if the two points be equally diftant from the prin¬ 
cipal vertex : for, the parameters belonging to thefe points 
being equal, the velocities in thefe points mull alfo be 
equal. 
Prop. LXVIIr. When a body is thrown obliquely with a 
given velocity, if the [pace through which it mujl have fallen 
■perpendicularly to acquire that velocity is made the diameter of 
a circle, the height to which the body will rife is equal to the 
verfed fine of double the angle of elevation. —Let a body be 
thrown in the direction BE, (fig. 83.) with the fame ve¬ 
locity which any body would acquire by falling perpen¬ 
dicularly through AB; if AB is made the diameter of 
a circle, the greatelt height to which the body will rife 
will be B D. 
Let I L be a right line drawn in the plane of the hori¬ 
zon, touching the circle in B, and making with the line 
BE, which is the direction in which the body is'thrown, 
the angle I BE, or angle of elevation. Becaufe IL 
touches the circle, and EB drawn in the circle meets 
it in the point of contact, the angle EBI is equal to 
the angle E A B ; and E C B is double of E A B ; there¬ 
fore E C B is double of EBI, the angle of elevation; 
and B D is the verfed fine of ECB, that is, of double the 
angle of elevation. 
Let B E reprefent the velocity with which the body is 
thrown. Then, fince this velocity is, by fuppofition, 
luch as might be acquired by falling down A B, if the 
body were thrown perpendicularly upwards with the fame 
velocity B E, it would rife to the height B A. Let the 
oblique motion B E be reiolved into two others, one in the 
direction B D perpendicular to the horizon, and the other 
in the direction DE parallel to it: then theafeending ve¬ 
locity will be to the horizontal velocity as B D to D E, and 
to the whole velocity as B D to BE. But the part of the 
velocity B D is the only part which is employed in raifing 
the body, fince the other part DE is parallel to the plane 
of the horizon. Now, the height of a body afeending 
perpendicularly with the whole velocity B E, will, be to 
the height when it aftends with the part B D as the fquare 
of B E to the iquare of B D. But, becaufe the triangle 
E D B is fimilar to the triangle A E B, B D is to E B as 
E R is to BA; and, BD, BE, BA, being continued 
proportionals, B D is to B A as the fquare of B D is to 
the fquare of B E. And the perpendicular heights to 
which the velocities BE and BD will make the body 
afeend have been fhown to be as the fquare of B E to the 
fquare of B D ; the heights are therefore as B A to B D. 
Since, therefore, the firlt velocity BE would make the 
body afeend through BA, the other velocity B D, which 
is the part of the whole velocity which aCts to make the 
body thrown in the direction D E to afeend, will carry it 
to the height B D, which is the verfed fine of double the 
angle of elevation. The fame might be fhown in any 
other direction of the body, as B F, or B G. 
Prop. LXIX. When a body is thrown obliquely with a given 
velocity, f the fpace through which it mujl have fallen perpen¬ 
dicularly to acquire that velocity is made the diameter of a circle, 
the random will be equal to four times the fine of double the angle 
of elevation .—If EBI be the angle of elevation, and ECB 
double that angle, D E will be the fine of double the angle 
of elevation. Let a body be thrown from the point B in 
the direction BE, with the velocity which it would ac¬ 
quire in falling through AB;'the random, or horizontal 
diftance at which the body will fall, is equal to four 
times D E. 
For, fince, (as in the laft Prop.) the velocity B E being 
refolved in B D, D E, the afeending velocity is BD, and 
the horizontal D E, if thefe two velocities were to con¬ 
tinue uniform, the fpaces deferibed in equal times would 
be as the velocities; and, in the fame time in which the 
body by the afeending velocity would rife through B D, 
by the horizontal velocity it would be carried forwards 
through D E. Of thefe velocities, the horizontal one DE 
is uniform, becaufe the force of gravity can neither acce¬ 
lerate nor retard a motion in this direction ; but the 
afeending velocity is uniformly retarded ; and therefore 
the body will be twice as long in afeending to its greateft: 
height B D, as it would have been if the firft afeending 
velocity had continued uniform: but on this fuppofition, 
the body would have been carried through BD and DE 
in the fame time : therefore in double the time, that is, 
in the time of afeent through BD with an uniformly-re¬ 
tarded velocity, it would be carried forward through twice 
DE : conlequently, in the times of defcent and alcent to¬ 
gether, it would move forwards through four times DE. 
Therefore, a body thrown from B in the direction B E 
with fuch a velocity as might be acquired by falling down 
A B, the diameter of a circle, will fall at the diftance of 
four times the fine of double the angle of elevation. 
Prop. LXX. The random of a projeElile will be the great Jl 
pojjible , with a given velocity, when the angle f elevation is art 
angle of forty-jive degrees. —The velocity being given, the 
height from whence the body mult have fallen to acquire 
that velocity, or the diameter of the circle A B, is a given 
quantity. And in a given circle the greateft fine is the 
radius or fine of a right angle: therefore four times the 
radius is greater than four times any other fine; and con- 
fequently, the random which is equal to four times the 
radius (which, by Prop. LXVIII. will be the cafe when 
the double angle of elevation is a right one, or the angle 
of elevation forty-five degrees) will be the greateft pof- 
fible random. 
This propofition, and the two following, may be illuf- 
trated by water (pouting from a pipe. 
Prop. LXXI. The random of a projeElile, whofe velocity is 
given, will be the fame at two different elevations, if the one be 
as muck above forty-five degrees as the other is below it .—If 
E B I be an angle ot 30 degrees, and GBI an angle of 
60 degrees, becaufe EBI tails ftiort of halt a right angle 
as much as GBI exceeds it, the double of E B I will tall 
fhort of a right angle as much as the double of G B I will 
exceed it; therefore, from the definition of a fine, thefe 
doubles will have the fame fine. Confequently, four times 
their fines, that is, (by Prop. LXIX.) their randoms, will 
be equal. 
Prop. LXXII. The greatejl random of a projeElile, whofe 
velocity is given, is double the height to which it would rije f 
it were thrown perpendicularly with the fame velocity. —If a 
body be projected in the direction B F, at an angle of 
forty-five degrees, and Us velocity be equal to that which 
a body 
