EPICYCLOID. 181° 
Con! Avdircle described on the-axis of the commion circumference’of the circle DHC tothe base AB. For  Cycloid: 
get the base is equal to the circumference of the circle. ““\—"—" 
PratTe 
ccuitl. 
Fig. 4. 
Figs. 5. 6. 
cycloid, as a diameter, is equal to the generating circle. 
ie Prop. III. 
Let DHC (Fig. 4.) be the circle described on the 
spoeiendteies and PHG the ordinate; the 
equal to the circular are HD. 
is 
the position of the circle 
when the point in the circumference Set dapecloas the 
baron The \ciecles CRD) RPE axb yeni, se 
lines HG, PI are manifestly the halves of chords 
, however, be in infinite terms, by 
ae 2 
a equation. See Fiuxions. 
i 
" 
i 
be 
F 
i 
fr 
ES] 
a2 
oF 
MP 
FF 
i 
8 
i 
5 
i 
F 
: 
fy 
a 
ih 
be 
Sens bose 
elle fae 
Ue 
af : z br, 
iy 
: 
a 
: 
we 
ll 
=x 
S 
= 
i 
Q 
é 
g 
| 
H 
' 
3 
= 
: 
i 
ad 
. 
t 
5 
= 
straight line PH. Now, from simi- 
TORE ote 
are DH isto the straight line PH as the 
: 
of 
: 
FQE. : 
Cor. 2. Let DG=z, PG=y, are DH=z; and let a. 
be the radius of the circle described on the axis, and 6 
the radius of the generating circle ; then because by 
the theorem Hp=! z, the nature of the prolate and 
curtate cycloids will be expressed by the two equations 
xa — Cos. z, 
CC : z+ Sin. x. 
Prop. V. 
In the common cycloid, if a circle be described on p 
> ecinntir ; and from any point P in CCL. 
the curve, an ordinate PG be drawn to the axis, meet- Fig. 7. 
DC, the axis, as a 
ing the circle in H ; a tangent PV to the cycloid shall 
be parallel to HD, the chord of the arc between the 
he points HD draw the tangents HR, DR, the 
At the points H, D e ts HR, DR, 
latter of which will be el to the ordinate PG: 
draw also another ordinate p h g indefinitely near to the 
former, so that the indefinitely small arcs Pp, Hh may 
be considered as coinciding with the tangents VP, RH ; 
lastly, draw Pq parallel to HA, and join D A meeting 
PGi 
in m. 
Because PH=arc DH, mt heat fo) therefore 
h—PH=arc DH h—are DH, that is, pg = Hh; 
Cie the trimgles 2H, h RD being similar, and AR= 
RD, therefore }H=Hm; hence pq=Hm, and ph= 
Pm; the Pphm is therefore a and 
consequently pV 1s parallel to kD, or to HD. 
Prop. VI. 
The are DP of the common cycloid is double the 
chord DH of the corresponding arc of the generating 
Let the ordinate ph g be indefinitely near to PH ; 
h, meeting in m, and draw Hn perpendicu« 
tohm 
Because the peti br indefinitely small, as in 
proposition, ma’ considered as‘ coincidi 
with tangents to the curves, And because Pp is < 
; Prop. 5.) the figure Pphm isa |. 
m; hence Pp=mh ; but Hm=H/A, as was 
in . 5; and therefore xnm=nh, and mh=2hn; 
therefore Pp=2hn. Now Pp andxh are evidently 
the increments which the oidal are DP and the 
chord DH receive by the ordinate changing its position 
from PG to pg; therefore the increment of the arc is 
always double the increment of the corresponding 
Ss now the arc and chord to be 
i moving _ to itself from the vertex 
, and the increment of the one is double 
the other, the arc will always be double the corre- 
sponding chord. 
Cor. The whole cycloid ADB is four times the 
diameter of the generating circle, or four times the’ 
join D 
z 
Prop. VII. 
If DM be drawn from the vertex of the cycloid pa- pig, g, 
rallel to the base andl Seaman, Beane P in the curve 
be drawn to axis, aiters, © 
geperating cine) in H ; and PL be drawn 
to ; the external cycloidal area D 
is equal to’ 
