GEOMETRY. 
443 
Prop. X. Probkm.—To jind a geometrical equation for 
the bicufpidate epicycloid. —Let CP=:PR. Join RMS, 
PM, PD; draw CT 
perpendicular to RT, 
TE to CR, and EG, 
MB, RA, to SC. 
Then the angle DRP 
=|MKP=SCP, and 
by equal triangles, 
RA=:CT, and RD= 
CD, and by fimilar 
triangles RM : RP :: 
RE : RT, and RP : 
RD RT : RC; therefore RM : RD : : RE : RC, 
and ME is parallel to SC, and EG=;BM. Put CP—a, 
BC —x, BM —CM—r, CT=m; then by Prop. 7, 
OJ. av2_,2_^2. RC : CT :: CT : 
CE :: CE ; EG, or jv; hence y—-^^?C=:i6a‘*y^, — 
4aa 64 
z—a '^y~— M— aa^z 
4 
zxx-]^yy — aa^ ; whence by involution 
the equation of th^ fixth order may be had at length. 
Corollary. Since CRM=SCR, a ray in the direftion 
of the tangent MR will be reflefted by a circle FR al¬ 
ways parallel to SC : therefore SM is the cauftic of the 
circle FR when the incident rays are parallel to CS. 
Prop. XI. Problem.— To find a geometrical equation for 
the quadricufpidate hypocycloid .—Let CR=PR, then the 
angle PRM |PKM = 
2PCS, RAC=:ACR, RA 
= RC=RB=:RP, AB= 
SC, and drawing the per¬ 
pendiculars CT, TD, TE, 
and MF, RM=RT, AM 
=:BT, AF=EC, FC=AE, 
andFMzrBD. LetSC—«, 
FCz=x, FM=y, CM=r, 
GT=ir; then AB : AC :: 
AC : AT : : AT : AE, 
whence AT = axx^, and 
in the fame manner Bl'= 
<7175 ; and CT being a mean proportional between AT 
and TB, u^=a^x^y-j, and u^—a^x-y-. But by Prop. 7, 
therefore 2jePxy^=za^=:=s^zzza^=zx^:^y^j'^ 
vyhence the equation may be had at length by involu¬ 
tion. Tliefame refult may be obtained by Dr. Waring’s 
method of redudlion, from axx-^ -^aj^^zzza. 
Corollary. Since the portion of the tangent AB in¬ 
tercepted betyveen the perpendiculars AC, BC, is a 
conllant quantity, this hypocycloid may in that fenfe 
be called an equitangential curve ; and the reclangular 
corner of a palfage muft be rounded off into th.c form 
of this curve in order to admit a beam of a given length 
to be carried round it. 
Ppop. XII. Problem.—To invefiigate thofe cafes in zokick 
the general propofuions either fail or require peculiar modifica¬ 
tions. — Cafe I. If the 
generating circle be con- 
lidered as infinitely 
final I, or the bans as in¬ 
finitely large, I'o as to 
become a Itraight line, 
the epicycloid will be¬ 
come a common cycloid, and the ratio of CP to CK in 
Prop. 3, cor. 2, becoming that of equality, the length 
of the arc SM will be four times tlte verfed fine of half 
PM, and V.M twice the chord RM or ^'X, therefore 
the iquare of the arc VM is alwayj as the abfeifs VZ. 
i lie evolute is^ an equal cycloid, and ilic ciicles in 
Pro]). G, being as i to 4, the area of the cycloid is to 
that of its generating circle as 3 <0 i. The properties 
of the cycloid as an ifochronous’, and as a bracltifio- 
chronous, curve, belong to mechanics ; and it is demonO 
firated by writers on optics that its cauftic is compofed 
of two cycloids. 
Cafe 2. If the generating circle be fiippofed to be¬ 
come infinite while the bale remains finite, the e{ucv- 
cloid will become 
the involute of a 
circle ; and the 
fluxionof thecurve 
being always, by 
Prop. 3, cor. I, to 
that of PM as PM 
to CP, its length 
SM will be a third 
proportional to IP 
and PM. Call CP, a, and PA 4 , x, then the fluxion of 
X V 
SM is — ; but the rectangle contained by half PM and 
the fluxion of SM is the fluxion of the area PSM, or 
PSM ———. The epitrochoid deferibed by the 
.J 2a 6a 
point C of the generating plane will be the fpiral of 
Archimedes, fince CN is always equal to PjM—PS—QV; 
and fince the angular motion of CN and PM are alio 
equal, the area CON^iPSMar;—. Inftead of the el- 
6a 
lipfis of Prop. 3, let PX be a pai'abola of which IP is 
the parameter, and continuing NM to X, the arc PX 
will be equal to CON. For making LQ=CP, it is well 
known that the fluxion of PX varies as XQ, or as PN, 
which-reprefents the fluxion of CON. For the curva¬ 
ture, PY in Prop. 4, becomes —CP, and the radius is a 
third proportional to NZ and NP. 
Cafe 3. Suppofing now the generating circle to be¬ 
come again finite, but to have its concavity turned to¬ 
wards the bafis, the fame curve will be deferibed as 
would be deferibed by the rotation of a third circle on 
the fame bafis in a contrary diredtion, equal in diameter 
to the difference of thofe of the two firff circles. 
Cafe 4. If th.e circles be of the fame fize, with their ■ 
concavities turned the fame way, no curve can be de¬ 
feribed ; but if the generating circle be flill further lef- 
fened, a hypocycloid will be produced, if the fame figure 
as that which would be deferibed by a thiid circle equal 
in diameter to the dift'erence of the two firlh All the 
general propofitions are equally applicable to hypocy- 
cloids w'ith other epicycloids, as might ealily have been 
underflood from an inl'po^tion of the figures, ,if there 
had been room for a double feries.. 
Cafe 5. If the diameter of the generating circle be 
half that of the bafis, 
the hypocycloid will 
become a right line, 
and the hypotroclioid 
an ellipfis. For fince 
the angle PKh 4 = 
2PCS, PCM, being- 
half PKM, coincides 
with PCS, and M is 
always in CS. Let 
GNL be the deferibing circle of the hypotroclioid, and 
join GNO, then NL is parallel, and ON perpendicular 
to SC, anti ON=;HL, which is always to GO'as CL to 
CG ; therefore AN is an ellipfis ; and the centre C will 
evidently deferibe a ciicle. 
GEOMETRY of the COMPASS. 
This ingenious department of geometry, for which 
we are indebted to lignor L. Mafeheroni, fcems not only 
well calculated to exercile the ]io\vers of invention in 
