Cycloid. 
—— 
Pirate 
CCLIIML 
Fig, 8, 
182 
the area contained the circular arc DH, and the 
lines DG, G 
Take a int p in the curve, indefinitely near to P; 
and draw the co-ordinates p Ag and p/: join DH, and 
complete the indefinitely narrow para Gu, 
Lr. Because the indefinitely little arc Pp may be con- 
a a straight linc, which is parallel to DH, 
( ), the triangles Prp, HGD are similar; hence 
rp: Pr: y GD: HG ; that is, Gg: rP:; PL: GH; since 
then the age geen: . 
! are See Gromerry 
Now cae Gv sai tie combiored ered as the incre- 
a of the circular space DHG ; and the rectangle 
Lr as the increment of the cycloida space DLP, cor- 
cependiog a the position of the ordinate 
from PHG to vhg ; bra to th e triangles AH», p Pr; 
they vamsn in 
les: therefore 
the increments of spaces DHG, D ofa pei 
and uentl 
the spaces themselves are opel 
Con. If AM be perpendicular to DM, the whole 
cycloidal space ADM is equal to the semicircle DHC. 
Prop. VIII. 
_ If PHG, an ordinate to the axis, meet the era- 
circle in H, and the chord HC be drawn to the 
of the base, and PK parallel to HC, meeting 
ee base in K; the bounded by the cycloidal ae 
Saeee as PK, KC, CD, shall be trip] 
space bound by the circular are D 
ihe sgh nes HC, cD 
, a tange tangent to the circle, meeti the bese 
in Ne also draw the ordinate phg i teins tee 
PHG, m HC in m; he; draw p L parallel 
tohc, and ks parallel to C or KP. eee 
angles CNH, mh H, are similar, and NC=NH ; there- 
EPICYCLOID. 
—. CH ele 
he. 
a is pir ye 
the pacer ee oy and’ A 
are AY ( 38. ha goo is, to 
TC, or PH, is to the arc 
P is in.a cycloi vot which C CHD ‘is 
circle (Prop. 38. . and therefore it is 
ADB. 
Nore. The rty of the 
proposition was-discovered by Huygens, 
to the motion of a pendulum. Suppose 
to be perpendicular to the horizon, and two 
of metal to be bent into the form of cloide, 
born #02 tions VA, VB; then, if a 
were formed fixing a weight to the end of a 
PXV, and entis to vibrate between the 
wei ht P will, by its motion, describe the cyelc 
is manner of describi erie the eo ADI. 
b Pn emp is unf pe hit ep 
given rise to the theory of involutes evolutes, 
one of the most elegant speculations of modern geo 
oe Pi Fates, ox yee 
or the app cation properties cycloid 
to mechanics, see MECHANICS. 
t 
Or Reitvevehh, Jo 
Il. 
fore hm=h H; and because ph=h H-+are HD=;H 
+PH=h m+tm ; that is, because pt4-th=2 h m+ 
th; therefore pt=2hm. And because / s is parallel 
toom, and kp to ch, therefore ps=hm, 
1. Let AEB be a given fixed circle, and EPF a of Rpicy- 
moveable circle, which vols either on the outside of the cloids. 
pe Caen 10.), -or on the inside (as in Fig. 11.), Pear 
; alsolet p be a given ¥n CCLIIL 
Pig. % 
fod nis 79s 
hence the lelogram  K is double the fb i 
and the quadrilateral pk K ¢ is triple the range’, ks, 
that is hem. Now the former of these is manifestly 
the increment of the SE CD SECeROEOS £2 8 
change of position of chord from CH to CA, and 
the latter is the increment of the space HCD ; there- 
fore the space PKCD is triple the HCD. 
Cor. 1. The cycloidal area DAC is triple the semi- 
circle DHC, 
Con. 2. The interior cycloidal space PDG, is the ex- 
cess of three times the contained by the are HD, 
and the lines HC, CD e the trapezoid PGCK. 
Prop. IX. 
Let AB be the base of a cycloid, ADB and CD its 
axis: In DC produced take CV=CD ; and let a semi- 
cycloid, the same as DB, be put in the position AV ; 
and another semicycloid, the same as DA, in the posi- 
a line drawn from O, the centre of the moveab 
Cuengh 2s a given pointe its circumference ; and at.’ 
the beginning of the motion, let P be at A, the point of 
contact of the two circles, and the point p at a; then, 
while the circle makes one complete revolution, by roll-. 
ing along the are AB, the line ‘Op, will revolve sbout, O 
as a cefitre, and the point p will describe a line. pads 
which is called an epicycloid. 
2. When the generating circle revolves on the 
side of the circumference of the fixed circle, the li 
described is the exterior prerene ‘when the ge 
nerating circle rolls on the iimide of the circumference, 
the line described is the interior epicycloid. 
8. The circle EPF is called the generating circle, 
and the point the ating point. 
A. A t line drawn ugh the centre of the 
fixed circle, and H, the middle nee the base, is called 
the axis; and the point d in which the axis meets the 
pas poke me now that «thread is fastened at V;, curve, is called the vertex. s 
then fit be the curve, so as to terminate at A; 
then, be unfolded, beginning at the point ‘A, its Coxpilat ies to the Definitions. | 
re oqremses * 5a) describe noe epoan cb. Cor. 1. The points a and. b, ome of ee 
w and equal to ; cloid, CA, CB, the radii 
and describe the semiciscle AYR. Let PX, the pat tone para 4 ceseaey™. 
of the thread which has been unwrapped from 
2. This of he epi etl the dram 
meet"AC in T ; drew XZ perpendicalay to AN, meet ference, of the 
ing circle in Y.; and PC ay ay” serenade Scnouium,, areas ot 
ae circle in H ; and join AY, CH. set on the that Chola 
KTP, the part of the thread wifolded, is’ mea csh sevclotinait he abopoe to conti~, 
nue its moti, « eis of pico 
circle ¥i8* 10 
