442 
G E O M E 1' R Y. 
Jlie right line AB in A; let the angle BAD be always 
equal to GH(J), and it is evident that AD, AE, and AF, 
will he equal refpeftively to WG, WM, and Vv(l>. 
But the angular motion of V^^G on W being equal to 
tile fum of tlie angular motions of GM on G, and CG 
on C, is to that of AF, or of GM, or half that ot KM, 
in the ratio of CH to CG, or CR to CP; therefore the 
fluxions of the areas SWG, SWM, and SW<J), are to 
thofe of the fegments AD, AE, and AF, in the fame 
ratio ; and that ratio being conftant, the whole areas, 
and their differences, are alfo refpettiveiy to each other 
as CR to CP. 
Scholium. The qiiadrable fpaccs of Halley are t^iofe 
wiiich are comprehended'between the arc of the epitro- 
cJioid, that of the defcriblng circle, and that of a circle 
concentric with the bafis, and cutting the deferibing 
circle at the extremities of its diameter. 
Prop. VII. Problem.—To find a central equation for the 
epicycloid. —Let CT be perpendicular to RT, the tan¬ 
gent at the point M, then 
PMR will be a right angle, 
and PM parallel to CT. 
On the centre C deferibe 
throughM the circleMNO, 
and let MQ be perpendi¬ 
cular to RO. Then tlie 
reftangle OQN = PQR, 
OQ : PQ :: QR : QN, by 
addition OQ : PQ :: OR : 
PN ; hence, by divifion, 
OP : PQ :: IR : PN, and 
OPN or INF 
PQ— ---But 
IR 
PR 
PM^ = PR X PQ = — X 
INP; and by fimilar triangles CT. : CR :: PM : PR, 
whence CTo^i^-^XPMv7=;CRr7X~^- Let MZ and 
’ PRj ‘ ‘ IRP 
RY be tangents to SP, then INP=MZ^, and IRP=: 
RYi^, CT=CRX^Y’ 
conftant ratio of CR to RY. Puttin 
Jss—aa 
QP—a, CR—b, 
CM=s, CT=:2/, then 
-Jb- 
bb — aa ' 
Corollary I. Join FN, and complete tlie parallelogram 
MPNI., then fince F-F=DZ=;EN, FN is perpendicular 
to FfK, and ML to NL; and, NL being always equal 
to I'hM or CK, L is always in a circle deferibed on the 
centre N, LM a tangent to that circle, and ZM a per¬ 
pendicular to that tangent drawn from the point Z- 
Corollary 2. The unicufpidate epicycloM admits of a 
peculiar central equa¬ 
tion, with refpect to 
the point S. Call 
SM, s, and let ST—?/ 
be perpendicular to 
the tangent MT, tlien 
TH4 ■ 
For the tri¬ 
PROr.VIII. Problem.-—To find a geometrical equation for 
the conchoidal epitrochoid .—Let CP=PK. On tlie centre 
C deferibe a circle 
equal to -GM, cut¬ 
ting SC in Z. Join 
MZF, tlien the arc 
DZ—GM, and MZ 
is parallel to CK ; 
therefore EF is alfo 
equal to DZ or GM, 
C F' is parallel to 
KM, and MF:=CK: 
therefore this epitro¬ 
choid, is the curve 
named by Delahire the conchoid of a circular bafis, as 
was firlt obferved by Reaumur in 1708, and afterwards 
by Maclaurin, in 1720. Call CK, ffl; DE, b •, ZH, x ■, 
HM, y ; ZM, s-, and let ZI be perpendicular to CK ; 
then FZ—a — s, -, and, CIZ and ZHM being 
fimilar, CZ : Cl ;: ZM : ZH, or - : ::s:x-. hence 
2 2 
b x~as — ss, bx-F-s^ezzss,. b~x'^-\-xbxs'^-fi‘^~a^s'^-, and by 
fuhfiituting for — a'^x'^-\-b-x'^-\-ix-}--\^2b\y^ 
angles SIP and MTS 
being fimilar, and IP 
being half of SM, 
orj, SP=-v/—, SP^ : 
2 
SMj :: IP^ : ST^, or 
, - - 
— : ;; — ; 2^ and zau-z=.s^. 
2 4 
Corollary 3-. The unicufpidate epicycloid is one of the 
caufiics of a circle. For making the angle CRY— 
MRCczzgSCP, the triangle CRY is ifofceles, and CY 
is confiant; fo that all rays in the diredtion of the tan¬ 
gent MR will be refledled by the circle QR towards Y, 
and confequently SM wdll be the cauflic of a radiant 
point at Y. 
Prop. IX. Problem.—To find a geometrical equation for 
the tricufpidate hypocycloid .—Let PA and MF be perpen¬ 
dicular to CS. foin 
PMB, KM, RMG, 
and PD. 'I'hen the 
angle APB is equal 
to the difference of ^ _ _ _ 
APC and MPR, or SGAJTD C D 
to that of their complements PRM, PCA ; but PRM 
=riPKM=i.PCA, therefore APB:2z-i PC A —ADPrz; 
APS, and the triangles APS and APB are fimilar and 
equal. Let SC=i2, SFiz^x, FM—and SBzccr. Tlien 
SA : SP : : SP : SD, and SPrry/ar. Draw PE per¬ 
pendicular to BP; then BEizziSDcnra, BC—a.— r, EC 
—rj and by fimilar triangles, CP : CR :: EC : CG 
iEC=<i 
therefore 
BP : BM, or 2c 
\l ar 
but BE : BG 
— far—PM.-, again, 
?.a 
BP : BM BA : BF, or x/ar ; —far-.-. 
rr 
6 a’ 
and 
SF=:x, 
rr 
, 6 ax— 6 ar- 
6 a 
-rr, and r=^a±\/ ^aa — 6 a.\'- 
But MF^c^BM^-—BF^, or 
-, and fa-y^z 
4ar^ — r^ 
oa -itaa 
By adding to this the fquare of the former 
equation, and proceeding in the fame manner to exter¬ 
minate r, we obtain an equation of the value of x and 
y, which, w'hen the furds are brought to the fame fide, 
and the fquare of the whole is taken, is at lafi reduced 
to .Y*— -^ix^-fzx^y^ —i2/2Y>'--j»_y^-i-i2a2_)'2—o, a regular 
equation of tlie fourth order. 
S'holium. The equation of the correfponding hypo- 
trochoids may be invefiigated nearly in the fame man¬ 
ner, by dividing PR and PM in a given ratio, but the 
tnocel's will bed'oniewhat piore tedious. 
Prop. 
