647 
MECHANICS. 
ftant of the defcent; whence (as was fhown concerning 
bodies falling freely) the motion mult be uniformly acce¬ 
lerated. 
Cor. Hence, whatever has been demonftrated concern¬ 
ing the perpendicular defcent of bodies, is equally appli¬ 
cable to their defcent down inclined planes, the motion in 
both cafes being uniformly accelerated by the fame power 
of gravitation. 
Prop. XLVI. The force with which a body defccnds by the 
attraElion of gravitation down an inclined plane, is to that zaith 
which it would defend freely , as the elevation of the plane, to its 
length ; or as the fine of the angle of inclination to radius .— 
Let AB (fig. 73.) be the length of an inclined plane, and 
A C its elevation, or perpendicular height. If the force 
of gravitation with which any body defeends perpendicu¬ 
larly be expreffed by AC, and this force be refolved into 
two forcers, AD, DC, by drawing CD perpendicular 
to AB ; becaufe the force C D is deftroyed by the re-ac¬ 
tion of the plane, the body defeends down the inclined 
plane only with the force AD. And A D is to A C 
as A C to A B ; that is, the force of gravitation down the 
inclined plane is to the fame force acting freely as the ele¬ 
vation of the plane to its length, or as the fine of the angle 
of inclination AB C is to the radius A B. 
Cor. 1. Hence, the force necefl'ary to fuftain a body on 
an inclined plane, is to theabfolute weight of a body, as 
the elevation of the plane to its length : for the force re- 
quifite to fuftain a body muft be equal to that with which 
it endeavours to defeend ; which has been fhown to be to 
that with which it would defeend freely, as the elevation 
of the plane to its length. 
Cor. a. If II be the height of an inclined plane, L its 
length, and the force of gravity be reprefented by unity ; 
the accelerating force on the inclined plane is reprefented 
by ^ • For, by the Prop, the accelerating force is to the 
force of gravity, as II is to L the accelerating force 
_ H 
~T" 
Cor. 3. Hence — varies as the fine of the angle of in- 
JL* 
clination. 
Cor. 4. If a body fall down an inclined plane, the velo¬ 
city V generated in TP" is fuch as would carry it uniformly 
over -r-X^FT feet in 1", where, as before, F is equal 16*1. 
For the velocity varies as the force and time, (i. e.) 
as — xT, and the velocity generated by the force of gra- 
viry in one fecond is 2F, therefore, V=—X2FT. 
Ex. IfL : H :: 2 : 1, a body falling down the plane will, 
of the end of 4", acquire a velocity of | x 32*2X4^:64' + 
feet per fecond. 
Cor. 5. The ipace fallen through in T" from a ftate of 
JLJ _ 
reft, is—x FT 2 , for the (paces deferibed vary as the fquares 
of the times. 
Ex. 1. If H=i, the (pace through which a body falls 
in 5" is ^X i6'i X25=2oia feet. 
Ex. 2. To find tlie time in which a body will defeend 
40 feet down this plane—Since S=-^xFT 2 , therefore, 
_ CITE I 10X2 
feconds - 
Cor. 6. The fpace through which a body muft fall from 
a ftate of reft to acquire a velocity V, is —_ x —;• For S is 
V 
as p~’ therefore the fpace through which the body falls by 
the force of gravity, is to the fpace through which it falls 
dow n the plane, as the fquare of the velocity direffly, and 
as the force inverfely in the former cafe, is to the fame in 
the latter ; and, if F (16• 1) be the fpace fallen through by 
gravity, a nd iF is the velocity acquired in 1"; hence. 
F : S :: - : ~ X V 2 , and S=± *L_ . 
V H H +F 
• }• Ir L-caH, and a body fall from a ftate of reft: 
till it has acquired a velocity of 40 feet per fecond, the 
fpace deferibed is —X -—50 feet nearly, 
1 64-4 J 2 
Ex. 2. If a body fail 40 feet from a ftate of reft down 
this plane, to find the velocity acquired. V 2 =E4FSX — 
=64*4X40x4=1288, and V=35*8 feet per fecond. 
Prop. XLVII. 7 he fpace deferibed in any given time by a 
body defending down an inclined plane, is to the fpace through 
which it would fall perpendicularly in the fame time, as the ele¬ 
vation of the plane to its length. —Let AC (in the preceding 
figure) reprefent the force with which a body would fall 
perpendicularly; CD being drawn from C perpendicular 
to A B ; A D, as was ftiown (Prop. XLVI.) will repre¬ 
fent the force with which the body defeends down the in¬ 
clined plane AB. And, fince the (paces through which 
a body falls in any given time muft be as the forces which 
move them, the (pace through which the body will fall 
down the inclined plane AB, is to that through which 
it will fall perpendicularly in the fame time, as the force 
A D to the force A C. But A D is to AC as AC, the 
elevation, to AB, the length, of the plane; therefore the 
fpace through which the body will fall in a given time 
down the inclined plane A B, will be to the fpace through 
which it would fall perpendicularly in the fame time °as 
the elevation of the plane to its length. 
Cor. 1. A body would fall down the inclined olane from 
A to D in the fame time in which it would fall perpendi¬ 
cularly from A to C. For, the (paces palled through in 
any given time are as A C to AB, that is, as A D to A C ; 
confequently, if A C is the fpace paffed through in any 
given time by the body falling freely, AD will be the 
fpace palled through in the fame time, down the inclined 
plane A B. 
Cor. 2. Having the (pace, through which a body falls in 
a perpendicular direction, we can eafily find the fpace 
which a body will deferibe in the fame time, on planes 
differently inclined, by letting fall perpendiculars, as C D, 
on thofe planes refpeflively. ; 
Prop. XLVIf I. The velocity acquired in any given time by a 
body defending down an inclined plane, is to the velocity ac¬ 
quired in thefame time by a bodyfallingfreely, as the elevation of 
the plane to the length. —In an uniformly-accelerated motion 
the velocities produced in equal times are as the forces 
which produce them; but (by Prop. XLVI.) the force 
with which a body defeends down an inclined plane, is to 
that of its perpendicular defcent, as the height of the plane 
to its length; therefore the velocities produced in equal 
times are in the fame ratio. 
Prop. XLIX. The time in which a body moves down an 
inclined plane, is to that in which it would fall perpendicularly 
from the fame height, as the length of the plane to its elevation. 
—The fquare of the time in which AB (fig. 73.) is palled 
over, is to the fquare of the time in which A D is palled 
over, as A B to A D 5 that is, fince A B, A C, A D, are 
continued proportionals, as the fquare of AB to the 
fquare of AC. Therefore the times themfelves are as the 
lines A B, AC; that is, as the length of the plane to its 
elevation. 
Cor. Hence, if feveral inclined planes (fig. 74.) have 
equal altitudes, the times in which thofe planes are de- 
feribed by bodies falling down them, are as the lengths of 
the 
