CYC 
C Y C> 
476 CYC 
number, as deserving to be written in letter's 
of gold. 
The golden numbers are those placed in 
the first column of the calendar, betwixt 
March 21 and April 18, both inclusive, to 
denote the days upon which those full moons 
fall, which happen upon,' or next after, 
March 21, in those years of which they are 
Respectively the golden numbers. See Ca- 
lendar. 
For finding the golden number, add one 
to the current year of our Lord, because 
one year of this cycle was elapsed before the 
Christian sera began, and divide by nineteen, 
the remainder is the current year of this 
cycle, or the golden number; but if nothing 
remains, it shews that it is the last year of the 
cycle, and consequently the golden number 
is' 19. 
Cycle of the Roman indiction, is a 
period of fifteen years, in use among the 
Romans, commencing from the third year 
before Christ. This cycle has no connection 
with the celestial motions ; but was insti- 
tuted, according to Baronins, by Constan- 
tine ; who, having reduced the time winch 
the Romans were obliged to serve to fif- 
teen years, was consequently obliged, every 
fifteen years, to impose, or indicerc accord- 
ing to the Latin expression, an extraordinary 
tax for the payment of those who were dis- 
charged ; and hence arose this cycle. 
To Jind the cycle of indiction for any 
given yhar, add 3 to the given year, and di- 
vide the sum by 15, the remainder is the 
current year of the cycle of indiction ; if 
there be no remainder, it is the fifteenth or 
last year of the , indiction. 
These three cycles multiplied into one an- 
other, that is, 28x19x15, amount to 7980, 
which is called the Julian period, after which 
the three foregoing cycles will begin again 
together. This period had its imaginary 
beginning 710 years before the creation, ac- 
cording to the common opinion among cHro- 
nologers concerning the age of the world, 
and is not yet Complete. It is much used in 
chronological tables. See Chronology. 
CYCLOID. The Cycloid is a curve of such 
importance, that the honour of its invention has 
been much Contested among- mathematicians. 
The French ascribe it to their countryman Ro- 
feerVall; but there is sufficient evidence that he 
was not the original inventor of it. Torricellius 
inform* us, in a treatise containing a demonstra- 
tion of one of the properties of this curve, viz. 
that its area is triple that of the generating cir- 
cle, and published in 1644, that this curve was 
known forty-five years before this period, or in 
1 599, to his master Galileo, and distinguished 
by the appellation of Cycloid. Besides, Dr. Wal- 
lis has discovered among the mathematical works 
of Bovillus, published at several times between 
the years 1501 and 1510, that this curve had 
been considered in his time : and he has also 
found, that it was known to cardinal Cusanus, 
who gave an account of it in a copy of his works 
transcribed in 1451. But its various properties 
have been gradually discovered ; and the me- 
thod of regulating the motion of a pendulum by 
this curve, which is the most useful application 
of it, was the invention of Huygens in the last 
century. 
The cycloid is a curve generated by the mo- 
tion of any point in the periphery of a circle, 
whilst the periphery itself revolves on a right 
line, till that point which touched the line at 
the beginning of the motion be brought to touch 
it again. 
Let the rolling circle be HFI (Plate Mi seel, 
fig. 29); F the point in the periphery, which at 
the beginning of the motion touched the line 
BC in B; and the curve BGC described by that 
point, whilst it is moving from B to C, will be 
the cycloid, or trochoid, so called from the man- 
ner of its formation. The circle HFI, by the re- 
volution of which the curve is formed, is called 
the generating circle ; the right line BC is the 
base ; the line GA bisecting the base in A is the 
axis; and G is the vertex : a line OF parallel to 
the base, and intercepted between the curve and 
the axis, is an ordinate ; and the space included 
by the curve and base, yiz. BGCAB, is the cy- 
cloidal space. 
The curve BGC is called the primary cycloid, 
to distinguish it from the protracted cycloid 
MN, and from the contracted cycloid QR, de- 
scribed by the contemporary motion of the cor- 
responding points/; and », in the circles 0/>N 
less, andO-n-R greater, than the circle OPGrespec- 
tively. The protracted cycloid is applied by Dr. 
Wallis to the solution of Kepler’s problem for 
dividing the area of an ellipse in a given ratio, 
of which an account is given in Keill’s Astro- 
nomy, Lect. 23 and 24. 
We shall now enumerate the chief properties 
of the common cycloid. (1) The whole base BC 
(fig. 30), is equal to the periphery of the gene- 
rating circle ; and any part of the base BH is 
equal to the arc FH of the same circle in the 
position HFI. (2) Any right line FD inter- 
cepted between the curve and the generating 
circle in the position GAD, where AB is — AC, 
is equal to the corresponding arc DG of the ge- 
nerating circle. For AB zz HFI, and BH zz 
FH. v HA zz FI zz DG, and HA — FD ; v 
FD zz DG. (3) The ordinate FO = FD DO 
zz DG -j- DO zz to the sum of the arc inter- 
cepted between the ordinate and the vertex, and 
its right sine. (4) The line LI drawn parallel to 
the chord MG, is a tangent to the cycloid in L. 
Draw an ordinate LMN, and another ordi- 
nate Pw indefinitely near to it ; and draw L <p 
and Mm parallel to GA. From the centre 
O draw OM. Then Yp -f- LM — mn — 
P« zz MG -f- M«, and LM zz MG ; • * P 'p — 
mn — M«, and P 'p zz Mn -{- mn. But if the or- 
dinate P*- be supposed to approach to LN, and 
at last to coincide with it, when M« and Mm 
vanish, the triangles Mnm and MNO will be 
similar ; v Mn * mn * * OM * ON, and M« -J- 
rnn : Mn : • OM’-f ON zz A*N ( OM, i. e. Yp 
: m« : ; an : om. But Mn : Mm (L/o • ; 
OM : MN; v Yp \ L f ("AN; MN) ” MN 
* NG. Consequently the triangles YLp and 
MNG, having the sides about the equal angles p 
and N proportional, are similar, and PLI is pa- 
rallel to MG ; be. the tangent to the cycloid at 
the point L is parallel to the chord MG. 
(5) The arc of a cycloid is equal to twice the 
chord of the corresponding- arc of the generating- 
circle : i. e. GQ — 2GS. 
Let ST be an infinitely small arc of the circle, 
and let the ordinates NO and irR, infinitely near 
to each other, be drawn. Produce the chord 
GS to meet wR in V ; let Tv be perpendicular 
to SV ; and let the tangents to the circle at S 
and G, viz. SW and GW, meet each other in W. 
Tite arc ST, being infinitely small, may be con- 
sidered as a right line, or as part of the tangent 
WS produced; and therefore the triangles GSW 
and STV will be similar. But GW — WS ; 
See Simpson’s Gometry, B. 3. v ST zz TV, 
and the perpendicular Tr bisects the base SV ; 
consequently SV zz 2Sr. But QR is paral- 
lel to SV, and equal to it ; and therefore 
equal to 2Sr : and QR is the increment of the 
curve GQ, generated whilst the chord GS in- 
creases by Sr, because Gr zz GT when T and 
S coincide. . Since then the cycloidal curve, and 
the chord of the corresponding arc of the circle, 
begin to increase together at G, and the curve 
I increases with a velocity double of that with 
which the chord increases, and this is every 
where the case, the arc of the cycloid GQ wilt 
be always double of the chord GS. 
Corollary. Hence the semicycloid GC is equal 
to 2GA, or double of the diameter of the gene- 
rating circle; and the whole cycloid BGC zz 
4GA zz four times the diameter of the gene- 
rating circle. 
(6) If GX be parallel to BAC, the base of the 
cycloid, and LX parallel to its axis, GA 5 the 
space GXL, terminated by the cycloidal arc 
GL, and the right lines GX and LX, will be 
equal to the circular area GMN. 
Let P.v be parallel to LX ; and by what has 
been shewn above, Yp * Lp * ’ MN * NG ; v 
NG x Yp — MN x Lp, i.e. Xr X LX zz MN 
X N«, or the smali space LX.vP — the .small 
space MNr». Consequently the area* GXL ami 
GMN increasing by equal increments, are equal. 
Hence, if BY be perpendicular to the base BC 
at B, and meet GX produced at Y, the space 
GYBPG will be equal to the semicircle GMA: 
and the rectangle GABY under the diameter 
GA and BAzz 4 the circumference of the gene- 
rating circle, will be four times the semicircle 
GMA; and therefore the area GI.BAG zz three 
times the area of the generating semicircle 
GMA, and the whole cycloidal area BLGCAIi 
will be equal to three times the area of the ge- 
nerating circle GMAT. Also, if GB he drawn, 
the area intercepted between the cycloid GLB, 
and the right line GB,will be equal to the semi- 
circle GMA ; for the area GLBAG zz 8GMA, 
and the triangle GBA zz ^GA X AB zz: the 
rectangle of the radius and semi circumference 
zz the area of the generating circle GMAT zz 
2GMA ; and therefore GLBAG — GBA — 
3GMA - 2 GMA zz GMA. 
(7) The cycloid is the curve of swiftest de- 
scent ; or a heavy body, descending by the force 
of its own gravity, will move from one point of 
thi s curve to any other point in less time, than it 
will move by any other line joining these points. 
(8) A body falls through any arc, FG, LG, 
&c. (fig. 30), of an inverted cycloid, whether it 
be great or small, in the same time. Hence, if 
a pendulum be made to vibrate in the arc of a 
cycloid, all the vibrations will be performed in 
the same time. 
CYCLOPTERUS, the sticker, in ichthy- 
ology, a genus belonging to the order of am- 
phibia nantes. The head is obtuse, and fur- 
nished with saw-teeth ; there are four rays 
in the gills; and the belly-fins are connected 
together in an. orbicular form. There are 
three species: 
1. The lumpus, or lump-fish, grows to the 
length of nineteen inches, and weighs seven 
pounds. The shape of the body is like tlrat 
of the; bream, deep and very thick, and it 
swims edge-ways. The back is sharp and 
elevated ; the belly flat, of a bright crimson 
colour. Along the body there run several 
rows of sharp and bony tubercles, and the 
whole skin is covered with small ones. The 
tail and vent-fins are purple. The pectoral 
fins are large and broad, almost uniting at 
their base. Beneath these is the part by 
which it adheres to the rocks, &c. It con- 
sists of an oval aperture, surrounded with a 
fleshy, muscular, and obtuse substance; 
edged with many small threaded appen- 
dages, which concur as so many clasp ers. 
By means of this part it adheres with vast 
force to any thing it pleases. As a proof of 
its tenacity, it has been known, that in Hing- 
ing a lish of this species into a pail of 
water, it fixed itself so firmly to the bottom, 
that on taking the fish by the tail, the whole 
pail by that means was lifted, though it held 
some gallons, without once making the .£$h 
