> 
648 MECH. 
the planes. For the time of the defcent down AC is to 
the time of the fall down A B, as AC to A B ; and the 
time of the fall down A B is to the time of the defcent 
down A G, as A B to AG; therefore the time of the 
defcent from A to C is to the time of defcent from A to G, 
as AC to AG, that is, the times are as the lengths of 
the planes. 
Prop. L. A body acquires the fame velocity in falling down 
tin inclined plane, which it would acquire by falling freely through 
the perpendicular elevation of the plane .—The fquare of the 
velocity which a body acquires by falling to D (fig. 73.) 
is to the fquare of the velocity it acquires by falling to B, 
as the fpace A D is to the fpace A B ; that is, as the 
fquare of A D is to the fquare of AC ; and, confequently, 
the velocity at D is to the velocity at B, as A D is to A C. 
But, becaufe AD and AC (Prop. XLVII. Cor.) are 
paffed over in the fame time, the velocity acquired at D 
is (by Prop. XLVIII.) to that which is acquired at C, 
as A D to AC, Since then the velocity at D has the 
fame ratio to the velocities at B and at C, namely, the 
ratio of A D to A C, the velocities at B and C are equal. 
Cor. x. Hence the velocities acquired by bodies falling 
down planes differently inclined are equal, where the 
heights of the planes are equal. The velocities acquired 
in falling from A to C, and from A to G (fig. 74.) are each 
equal to the velocity acquired in falling from A to B, and 
therefore equal to one another. 
Cor. 2. Hence, if bodies defeend upon inclined planes, 
whofe heights are different, the velocities will be as the 
fquare roots of their heights. 
Prop. LI. A body falls perpendicularly through the diameter, 
and obliquely through any chord, of a circle, in the fame time .— 
In the circle A D B, (fig. 74.) let A B be a diameter, and 
AD any chord; draw BC a tangent to the circle at B; 
produce AD to C, and join D B. Becaufe A D B is a 
right angle, a body by (Prop. XLVII.) will fall from A 
to D on the inclined plane in the fame time in which it 
■will fall from A to B perpendicularly. In like manner 
let the chord A E he produced to G ; and, becaufe A E B 
is a right angle, a body will fall from A to E in the in¬ 
clined plane in the fame time in which it would fall from 
A to B. 
Cor. Hence all the chords of a circle are deferibed in 
equal times. Hence alfo the velocities, and accelerating 
forces, will be as the lengths of the chords. 
Prop. LII. If a body defends along feveral contiguous 
planes, the velocity which it acquires by the whole defcent, pro¬ 
vided it loft no motion in going from one to another, is the fame 
which it would acquire, if it fell from the fame perpendicular 
height along one continued plane ; and this velocity will be the 
fame with that which would be acquired by the perpendicular fall 
from the elevation of the planes. —Let AB, B C, C D, (fig. 75.) 
be feveral contiguous planes; through the points A and D 
draw HE, D F, parallel to the horizon, and produce the 
contiguous planes C B, CD, to G and E. By Prop. L. 
Cor. 1. the fame velocity is acquired at the point B, whe¬ 
ther the body defeends from A to B, or from G to B. 
Therefore, the line B C being the fame in both cafes, the 
velocity acquired at C mutt be the fame, whether the body 
defeends through A B, B C, or along G C. In like man¬ 
ner, it will have the fame velocity at D, whether it falls 
through AB, B C, CD, or along ED, that is (by Prop L.) 
its velocity will be equal to the velocity acquired by the 
perpendicular fall from H to D. 
Cor. Hence, if a body defeends along any arc of a circle, 
or any other curve, the velocity acquired at the end of the 
defcent is equal to the velocity acquired by falling down 
the perpendicular height of the arc ; for fuch a curve may 
be confictered as confining of indefinitely-fmall right lines, 
reprefenting contiguous inclined planes. 
Schol. The velocity of a body palling from one inclined 
plane to another is diminilhed in the ratio of ratlins to the 
cofine of the angle between the directions of the planes. 
Let B C or B m (fig. 76.) reprefent the velocity acquired 
at B, and relblve B C into B n and C n, by letting fall the 
1NIC?. 
perpendicular Cn ; m n will be the velocity loft ; therefore 
the velocity at B is to the velocity diminifhed by parting 
from AB to BD as B C to Bn, or as radius to the co- 
fme of the angle between the directions of the planes. 
Prop. LIII. If two bodies fall down two or more planes 
equally inclined, and proportional, the times offalling down thefe 
planes will be as the fquare roots of their lengths .—Let the in¬ 
clined planes be AB, B C, D E, E F, fig. 77. let AG, DH, 
be lines drawn parallel to the horizon ; let AB, D E, be 
equally inclined to the plane of the horizon, and alfo BC, 
EF; let AB be to DE as AG to DH, and as BC to EF, 
and draw GB, HE. Becaufe AB G, D E H, are fimilar 
triangles, AB is to DE as BG to EH, and ft AB to 4/DE 
as 4/BG to 4/EH ; alfo AB is to DE as B G-J-BC is to 
HE + EF, and y^AB to y'DE as 4/BG+BC, or ft GC, 
is to ft HE+EF, or ftHF. And fince (by conftruftion) 
AB is toDE asBC to EF, AB is to DE as AB+BCis to DE 
+EF, and 4/AB to -j/DE, as ft AB+BC to ft DE+EF. 
But AB, DE,being planes equally inclined, the accelerating 
force of gravitation will be the fame upon each, and the 
bodies defcending upon them may be confidered as falling 
down different parts of the fame plane. Hence, the time of 
defcent along AB is to that along DE, as 4/ AB to 4/DE ; 
and the time of defcent along GC is to that along HF, as 
4/ GC is 4/HF, that is, as ft AB to 4/DE. Again, the 
time of defcent along GB is to that along HE, as 4/BG is 
4/EH, that is,as 4/AB to 4/DE. Since, therefore, the time 
of defcent along the whole plane G C is to that along the 
whole plane H F, as 4/AB to 4/DE, and that of the part 
G B is alfo to that of the part H E, as 4/AB to 4/DE, the 
time of defcent along the remainder B C is to that along 
the remainder E F, as 4/AB to 4/DE. Confequently, the 
time of defcent down AB+BC is to that down DE-f-EF, 
as ft AB to 4/DE, that is, as ft AB-f-BC to 4/DE4-EF. 
Cor. Hence, if bodies defeend through arcs of circles, 
the times of deferibing fimilar arcs will be as the fquare 
roots of the arcs. For fuch fimilar arcs may be confi¬ 
dered as compofed' of an equal number of proportional 
fides, or planes, having the lame inclination to each other, 
and their elevations equal; whence, by this propofition, 
the times of defcent will be as the fquare roots of the 
lengths of the arcs. 
Prop. LIV. If a body be thrown up along an inclined plane, 
or the arc of a curve, it will, in the fame time, rife to the fame 
height from which with equal force it would have defended ; and 
any velocity will be loft in the fame time in which it would, in 
defcending , have been acquired .—For the fame force of gra¬ 
vitation has?, in every refpeft, the lame efficacy to retard 
the motion of bodies afeending, 3 S to accelerate them 
defcending on an inclined plane .or curve. 
Of the Cycloid, and the Motion of a Pendulum in it. 
Dtf If the circle C D H (fig. 78.) roll on the given 
ftraight line A B, fo that all the parts of the circumfe¬ 
rence be applied to it one after another, the point C that 
touched the line A B in A, by a motion thus compounded 
of a circular and rectilineal motion, will deferibe the curve 
line ACEJ 3 , which is called the cycloid. The ftraight 
line A B is called the bafe\ the line EF perpendicular to 
A B, bifeefing it in F, the axis ; and the point E the vertex 
of the cycloid. The circle by whofe revolution the curve 
line is deferibed, is called the generating circle. The line 
C K, parallel to the bale A B, meeting the circle in C and 
the axis in K, is called an ordinate to the axis; and a line 
meeting the curve- in one point, that produced does not 
fall within the curve, is called a tangent to the curve in 
that point. 
Prop. LV. On the axis EF deferibe the generating circle 
E G F, meeting the ordinate C K in G ; and the ordinate will 
be equal to the fum of the arc E G and its right fine G K ; 
i. e. CK = EG-f G K.—It is plain, from the definition, 
that the’line A B is equal to the whole circumference of 
the generating circle, and therefore A F mult be equal to 
the lemi-circumference EGF. It is alfo obvious,-from 
i the 
