GEOMETRY. 
'44(3 
the bafe AB. Put tbe three given perpendiculars 
DK—a, Dl —i, DGrrc, and put *:=AH or BH, half 
ih.e fide of the -quilateral triangle. Then is AC or 
BC—:.r, anci bv riiiht-angled triangles the perpendi- 
riilar C>!—y'AC^—A1 i^=:v'4a;'''— x-—^ 
Now, fince the area or fpace of a redfangle is expreffed 
b\’ the product of the bafe and height, and that a tri¬ 
angle is equal to half a rectangle of equal bafe and 
height, it follo'vs that, the 
V hole triangle ABC is ::=^ABXCI-f=:A:X.v ^ 3, 
the tritingle ABD—.i|ABxDG—xXC—f-r, 
the tri.ui_;le BCD—-JBCXflE=rXtt^tfx, 
tiic triangle ACD—J AC xOF~’tX^=^.r. 
But ti>e three hilt triangles make up, or are equal to, 
file '.vhoie foru.ier or great triangle; that is, x~\/z—ax 
d ^ j^ -l— ^ 
•i-bx-\-cx ; licnce then vrs.-, half the fide of the 
triangle fought. 
Aiib, tince the whole perpendicular CH is ~vy^3, it 
is titereforc that is, the whole perpendicular 
CII, i.-. Jult etjual to the Ann of all the three Inialler 
perjjendiculars DH-J-DF-j-DG, taken together, tvhere- 
evcr the point D is lltuated. 
ikiviag tlii!^ given the elements of common geome¬ 
try, wc proceed tiow to give fome inftances ot a higher 
f uecies of geotaetry, upon the nature of Cycloidal Curves. 
CYCLOIDAL CURVES. 
On the fubjecl of Cycloidal Curves, an ingenious 
I'rench writer l;as made the following remark: II J'cmble 
qu'me dejliiie'e particulicre altnchee a la cycloide, ltd donne 
preftrnhiement attx aui.res courbes un plv^ grand nombre de pro- 
prietii remarquablcs. “The cycloid feems to be pecu¬ 
liarly favoured in pofielfing more remarkable properties 
than any other curve.” 
Definitions. 
1. When a circle is made to rotate on a re6lilinear 
bafis, the figure defcribed on the plane of the bafis by' 
any point in tb.e plane of the circle is called a trochoid. 
A circle concentric with the generating circle, and 
palling through the defcribing point, may be called the 
defcribing circle. 
2. If the defcribing point is in the circumference of 
the rotating circle, the two circles coincide, and the 
curve is called a cycloid. 
3. If a circular balls be fubftituted for a reftilinear 
one, the trochoid will become an epitrochoid, and the 
cycloid an epicycloid. 
Scholium I. The terms have hitherto been too promif. 
cuoully employed ; the terms cycloid and trochoid have 
been ul'ed indiflerently; and the term epicycloid has 
comprehended liie epitrochoid, the terms prolate and 
contratted being I'oinetimes added to imply that tiie de- 
fcribing-point is within or without tlie generating circle. 
The interior epicycloid and epitrochoid may very pro¬ 
perly be diltinguirtied by the names hypocycloid and 
bypotrochoid, wiienever they are the feparate objedls of 
confuleration. The difterent fpecies of epicycloids may 
be denominated according to the number pf their cufps, 
combined with that of the entire revolutions which they 
comprehend; for inftance, the epicycloid/defcribed by 
a circle on an equal balls is a limple unicufpidate epicy¬ 
cloid; and if the diameter of the generating circle be 
to that of t!ie bafis as 5 to 2, the figure will be a quin- 
" tuple bicufpidate epicycloid. If the defcribing circle 
of a trochoid or cycloid be fo placed as to touch the 
middle of the bafis, and each of tire ordinates parallel 
to tlie balls be diminiflied by the correfuonding ordinate 
of tire circle, the curve thus generated has been deno- 
Hiinated the companion of the trochoid, the figure of the 
fines, and the harmonic curve. 
Scholium 2. 'I'he invention of the cycloid luis been 
attribuced by Wallis, Pliil.Tr. for 1C97, No. 229, to car¬ 
dinal Cufanus, who wrote about the year 1+50; Vriit it 
feems to be at leall as probable that tlie curve which 
appears in Cufanus’s figure was meant for the femicircle 
employed in finding a mean proportion-;!. Bovillus, in 
1501, has a juflerclaim to tb.e merit of the invention ot 
the cycloid and trochoid, if it can be any merit to have 
merely imagined fucli curves to exilL In 1599, Galileo 
gave a name to the common cycloid, and attempted its 
quadrature ; but having been accidentally milled by re¬ 
peated experiments on the weigiit of a flat fublhince cut 
into a cycloidal form, he fancied that the area bore an 
i.iu'ommenfurable ratio to that of tlie circle, and delilted 
from the invelligati 'u. Merfennus defcribed the cycloid, 
ill 1615, under the name of la trochoidc, or la roulette, but 
he went no further. Roberv'al feems to have firfl dif- 
covered the comparative quadrature and redhfication of 
the cycloid, and the content of a cycloidal folid, about 
the year 1635, but his ircatife was not printed until 1695. 
Tonicelli, in 1644, firil publilhed the quadrature and 
tlie method of drawing a tangent. Wallis gave, in 1670, 
a perfefl: quadrature of a portion of the cycloid. '1 lie 
epicycloid is faid to have been invented by Roemer ; its 
redtification and evolute were inveftigated by Newton 
in the Principi-a, publilhed in 1687. In 1693, Mr. Cafwell 
fliewed the perfeiSf quadrability of a portion ot the epi¬ 
cycloid, and Dr. Halley immediately publiflied an ex- 
tenfion of Calwell’s dilcovery, together with a conipa- 
rilbn of all epitrochoidal with circular areas. M. \ a- 
rignon is all'o laid to have reduced the lettification ot 
tlie epitrochoid to that of the eilipfis, in the fame year. 
Nicole, Delahire, Pafeal, Reaumur, Maclaurin, the 
Bernoullis, the commentators on Newton, and many 
others, have contributed to the examination ot cycloi¬ 
dal curves, both in planes and in curved Airfaces ; and 
Waring, I he moft profound of modern algebraifts, has 
conliderably extended his refearches upon the nature of 
thole lines which are generated by a rotatory progiel- 
fioif of other curves. In the prelent elfay^ the moft re¬ 
markable properties of cycloidal curves are deduced in 
a fiinpler and more general manner than appears to have 
been hitherto done, the equations of I'everal fpecies are 
inveftigated, a fingular property of the quadricufpidate 
hypocycloid is demonflrated, and the peculiarities of 
the fpiral of Archimedes are derived from its generation 
as an epitrochoid. 
Prop. I. Theorem. —In any curve generated by the ro¬ 
tation of another on any bafis, the right line joining tb« 
defcribing point, and 
the point of contadt 
of the generating curve 
and tlie bafis, is al¬ 
ways perpendicular to 
the curve defcribed. 
It may by I'ome be 
deemed Aifficieut to 
confider the generating 
curve as a redlilinear 
polygon of an infinite 
number of Tides; fince, in this point of view, the pro- 
pofition requires no further demonftration; and, indeed, 
Newton and others have not f'cnipled to take it for 
granted : but it is prefumed that a more rigid proof 
will not be confidered as fuperfluous. Let M be the 
defcribing point, and P the point of contadl; and lef 
LO, MP, and NQ, be I'ucceffive pofitions of the fame 
chord af the generating curve at infinitely fmall dif- 
tances; then it is obvious, and cafily demonfirable, 
that the arcs OP and PQ, defcribed by the point P of 
the generating curve in its paifage from O to P, and 
from P to Q, will be perpendicular to the bafis al P, 
and will therefoi'e touch each other. Let the are's L, 
IMK, and N, be defcribed with the radius PM, on the 
centres O, P, and Q. Then the curve defcribed by 'M 
will touch IMK; for fince O and Q lie ultimately in 
the fame direftion from P, if L be- above IMK, N will 
a alfo 
