M E C H A N I C S. 
6jO 
chid V M that remains to he deferibed. —Let LR, MS, be 
perpendiculars to the axis DV, meeting the generating 
circle in O and Q, and draw the chords VO, V Q: then, 
the velocity of the pendulum at M, as before proved, will 
be the fame as would have been acquired by a body directly 
falling from R to S, and the velocity acquired at V will 
be the , fame as would have been acquired by a body 
direCtly falling from R to V; but thefe velocities are to 
one another as yRS to 3/RV ; and, face RV : SV : : 
VO 2 : VQ 2 , and RV : RV—SV (—RS) :: VO 2 : VO 2 — 
VQ 2 :: VL 2 : VL 2 —VM 2 (becaufe VL=aVO and VM= 
aV.Q), it follows that the velocity of the pendulum 
acquired in M is to the velocity acquired in V, as 
V VL 2 —VM 2 to f'VL 2 , or as MX to VZ. 
The force of gravity that is fuppofed invariable, acting 
in the direction of the diameter DV, may be reprefented 
by DV ; and may be refolded into the two forces DQ 
and V Q, whereof the fir ft D Q, parallel to t. M the firing, 
ferves only to ftretch the firing, and does not at all con¬ 
tribute to accelerate the motion of the pendulum ; it is 
only the force reprefented by the chord VQ that accele¬ 
rates the motion of it along the curve M m, and is all em¬ 
ployed to produce that effect, the direction VQ being 
parallel to the tangent of the cycloid at M, by Prop. LVI. 
But VMrrciVQ, by Prop, LV 1 I. therefore the force that 
accelerates the pendulum at M, is as the arc of the 
curve V M. 
Cor. It is obvious from the demonftration, that the 
part of the gravity w’hic'n the firing fuftains in any point 
M, is to the whole weight of the pendulum, as the chord 
D Q to the diameter. 
Prop. LXII. Suppofe that the circle LZ l is deferibed by 
the body X with an uniform motion, by the velocity acquired by 
the pendulum in V ; and any arc of the cycloid, as M N, will 
be deferibed by the pendulum in the fame time as the arc of the 
circle X Y by that uniform motion: taking VN, on theJlraight 
line V L, equal to \ N in the cycloid, and drawing N Y paral- 
Id to VZ, meeting the circle in Y.—Let xm be an ordinate 
very near to X M, and draw Xr parallel to the diameter 
L/, meeting xm in r ; then, fince the triangles Xrx and 
VXM are fimilar, it follows that Xx : Mot (—Xr) :: 
V X : MX, that is, as the velocity of the body X to 
that of the body M ; and conlequently the fpaces Xx and 
M m will be deferibed in the fame time by thefe bodies, 
the times being always equal when the fpaces are taken 
in the fame ratio as the velocities. After the fame man¬ 
ner, the other correfponding parts of the lines M N and 
XY will be deferibed in the fame time; and therefore 
the whole fpace MN will be deferibed in the fame time 
as the arc X Y. 
Cor. Therefore the pendulum will ofcillate from L to 
V, in the fame time as the body X will deferibe the 
quadrant LZ. 
Prop. LXII I. The time of a complete ofcillation in the cy¬ 
cloid is to the time in which a body would fall through the axis 
of the cycloid D V, as the circumference of a circle to its dia¬ 
meter .—The time in which the iemi-circumference LZ/ 
is deferibed by the body X, is to the time in wdiich the 
radius LV could be deferibed with the fame velocity, as 
the circumference of a circle to its diameter. But the 
fame time, in which the femi-circumference LZ / is de¬ 
feribed by the body X, is equal to the time of the com¬ 
plete ofcillation LVP in the cycloid, by the Cor. of the 
lad Prop. The time in which a body falls from O to V, 
along the chord O V, is equal to the time in which LV 
(=zOV) could be deferibed by the velocity acquired at 
the point V; and the time of the fall through the chord 
O V is equal to the time of the fall through the diameter 
DV; confequently the time in which .LV could be de¬ 
feribed by a velocity equal to that of the body X, is equal 
to the time of a fall through the diameter D V, It follows 
therefore, that the time ot the entire ofcillation, LVP, is 
to the time ot' a fall through the diameter i)V, as the cir¬ 
cumference of a circie to its diameter. 
Cor. r. Hence the ofcillqtions in the cycloid are all per¬ 
formed in equal times; for they are all in the fame ratio, 
to the time in which a body falls through the diameter 
DV. If therefore a pendulum ofcillates in a cycloid, 
the time of the ofcillation in any arc is equal to the time 
of the ofcillation in the greatelt arc B VA, and the time 
in the lead arc is equal to the time in the greatelt. 
Cor. a. The cycloid may be conlidered as coinciding, 
in V, with any final! arc of a circle deferibed from the 
centre C, palling through V; and the time in a fmall arc 
of fuch a circle will be equai to the time in the cycloid 
and hence is underftood why the times in very little arcs 
are equal, becaufe thefe little arcs may be conudered as, 
portions of the cycloid as well as of the circle. 
Cor. 3. The time of a complete ofcillation in any little 
arc of a circle, is to the time in which a body would fall 
through half the radius, as the circumference of a circle 
to its diameter; and, fince the latter time is half the time 
in which a body would fall through the whole diameter, 
or any chord, it follows that the time of an ofcillation in 
any little arc, is to the time in which a body would fall 
through its chord, as the femicircle to the diameter. Sup¬ 
pofe N V a frnall arc of the circle deferibed from the centre 
C; then the time in the arc N V is fo far from being equal 
to the time in the chord N V, even when they are fup¬ 
pofed to be evanefeent, that the lalt ratio of thefe times 
is that of the circumference of a circle to four times the 
diameter: and hence an error in feveral mechanical wri¬ 
ters is to be correfted, who, from the equality of the eva¬ 
nefeent arcs and their chords, too rathly conclude the 
time of a fall of a body in any of thefe arcs equal to the 
time of the fall of q body in their chords. 
Cor. 4. The time of the ofcillations in cycloids, or in 
fmall arcs of circles, are in a fubduplicate ratio of the 
length of the pendulums. For the time of the ofcillation 
in the arc LVP is in a given ratio to the time of the fall 
through D V; which time is in the fubduplicate ratio of 
the fpace DV, or of its double C V, the length of the 
pendulum. 
Cor. 5. But, if the bodies that ofcillate be afted on by 
unequal accelerating forces, then the ofcillations will be 
performed in times that are to one another in the ratio 
compounded of the direct fubduplicate ratio of the lengths 
of the pendulums, and inverfe fubduplicate ratio of the 
accelerating forces: becaufe the time of the fall through 
D V is in the fubduplicate ratio of the fpace DV direttly, 
and of the force of gravity inverfely ; and the time of the 
ofcillations is in a given ratio to that time. Hence it ap¬ 
pears, that, if ofcillations of unequal pendulums are per¬ 
formed in the fame time, the accelerating gravities of thefe 
pendulums mult be as their lengths; and thus we con¬ 
clude that the force of gravity decreafes as you go to¬ 
wards the equator; lir.ee we find that the lengths of pen¬ 
dulums that vibrate feconds are always lefs at a lefs dif- 
tance from the equator. 
Cor. 6, From this Propofition we learn how to know 
exaitly what fpace a falling body defcribes in any given 
time ; for, finding by experiment what pendulum ofcil¬ 
lates in that time, the half of the length of the pendulum 
will be to the fpace required, in the duplicate ratio of the 
diameter to the circumference ; becaufe fpaces deferibed 
by a falling body, from the beginning of its motion, are 
as the fquares of the times in which they are deferibed ; 
and the ratio of the times, in which thefe fpaces are de¬ 
feribed, is that of the diameter to the circumference: and 
tints Mr. Huygens demonltrates that falling bodies, by 
their gravity only, deferibe 15 Parifian feet and 1 inch in 
a fecond of time. 
Schol. That it may be nnderltood how the time in a 
fmall arc is not the lame with that in its chord, though 
the evanefeent arc is equal to its chord; we may here 
demonftrate, that, if V k and N k be two planes touching 
the arc NV in V and N, though the evanefeent chord 
N V be equal to the fum of thefe tangents V k and N k, 
yet the time in the chord is to ths time in thefe tangents 
as 
