Cyeloia. 
—— 
Of the Cy- 
Cleid. 
PLaTe 
CCLUL. 
Fig. 1. 
Figs. 2, 3. 
180 
same vertical nor the same horizontal ” on ac- 
count of its the attention of the most 
celebrated icians in Europe, who found, that 
turned his attention to the theory of epicycloids, while 
be iform ; and that on this ac- 
diminished. 
of 
the force of gravity 
all its vibrations in equal times, 
it described a greater or a lesser arc. But, by 
the hypothesis, and supposing the force of 
gravity to be directed to the earth’s centre, and to be in 
as the distance from the centre, it became a 
oe on nae gd pcm: Taf: ppm 
, 80 a8 to perform unequal vibrations in equal 
times? Sir Isaac Newton shewed that the curve ought 
be an epi - See Principia, lib. i. prop. 51. 
icycloids was treated by 
Herman, in the — volume of the —— of the 
A . It appears thata ician, 
named O re aagyl this problem, “ to 
pierce a spherical roof with oval windows, the perime- 
a of any one of which may be-absolutely rectifiable.” 
Herman beli 
Academy of Sciences of Paris, 1732, where he shews 
that the rectification of. the curve proposed by Herman 
er or ene oe 
e shall now give a brief view of the properties of 
eycloids and epicycloids. 
I. Or rue Cycrom. 
Definitions. 
1. If a circle, EPF, roll along a straight line AB, 
(Plate CCLIII. Fig. 1.), so that every point. of the cir- 
cumference may touch the line in succession ; and if 
i sa hy ot bape vn eters ea 
tact wi straight line at the beginning of the mo- 
Gli, ils tha ‘die nda vention Complete reveling 
the point P will have described a curve line APDB, 
which is called a common cycloid, also sometimes sim- 
A ae aS along a straight line 
lines, a pendulum moving in a cy- ced, 
EPICYCLOID. 
itis called a curtate cycloid, if the point is without the Cyctoit 
circle. e A balay Apa get 
8. In each of the three cycloids, the circle EF is cal- 
ved Tks waraigi lime AB) which oles tha>ppelesatin 
4. The ne joins the points: 
each cycloid, where the motion of the point that de« 
scribes the curve begins and ends, is the dase of 
the cycloid. : : tid: Lect Soni 
5. A straight line CD which bisects the base at right 
angles, and terminates in ‘the curve, is called the azis; 
the point D, in which it meets the curve, is called 
the verter of each kind of cycloid. 5 Ybeirn 
6. A straight line drawn from any point in the curve, 
perpendicular to the axis, is. an ordinate to the 
axis; and the ent of the axis between 
and an ordinate, is called an abscissa. 
Corollary 
be sw 
the vertex 
nitely. 
Proposition I, ih edatona a 
In any cycloid, the base is equal to the circumference. 
o the generating el PB, (Fig. 1.) every point 
n the common APB, (Fig. 1.) every im p 
the cisensstasuct ee aeipeneslalagiedoveticaatieas CLI 
ly touch the base, without sliding along it, while _ Fig. 1. 
circle makes a complete revolution : therefore, the 
Vidi le ee Se eee 
In the 
being the point of the ci genera- 
F os fae ich touches the line a 6 at the beginni 
d end of the motion; and P being the point in the 
revolving radius OQ, which ger the cycloid 
APDB, it is manifest that at the inning of the mo-. 
tion, the line QP will have the positi a A, a perpen- 
dicular to ab; and at the end, it will have the position 
6B, another icular to a6;*therefore aA, 6B 
ab, which again is manifestly equal to the circumfes 
Pror. II, nal 
In the three kinds of eycloids, the axis is equal to- 
Wie Ee ee eee 
generating ci 
and curtate cycloids, Se een 3.) Figs. 2, 3 
