# 1911 Encyclopædia Britannica/Cycloid

**CYCLOID** (from Gr. κύκλος, circle, and εἶδος, form), in geometry, the curve traced out by a point carried on a circle which rolls along a straight line. The name cycloid is now restricted to the curve described when the tracing-point is on the circumference of the circle; if the point is either within or without the circle the curves are generally termed *trochoids*, but they are also known as the *prolate* and *curtate* cycloids respectively. The cycloid is the simplest member of the class of curves known as roulettes. No mention of the cycloid has been found in writings prior to the 15th century. Francis Schooten (*Commentary on Descartes*) assigns the invention of the curve to René Descartes and the first publication on this subject after Descartes to Marin Mersenne. Evangelista Torricelli, in the first regular dissertation on the cycloid (*De dimensione cycloidis*, an appendix to his *De dimensione* *parabolae*, 1644), states that his friend and tutor Galileo discovered the curve about 1599. John Wallis discussed both the history and properties of the curve in a tract *De cycloide* published at Oxford in 1659. He there shows that the cycloid was investigated by Carolus Bovillus about 1500, and by Cardinal Cusanus (Nicolaus de Cusa) as early as 1451. Honoré Fabri (*Synopsis geometrica*, 1669) treated of the curve and enumerated many theorems concerning it. Many other mathematicians have written on the cycloid—Blaise Pascal, W. G. Leibnitz, the Bernoullis, Roger Cotes and others—and so assiduously was it studied that it was sometimes named the “Helen of Geometers.” The determination of the area was the subject of many investigations and much controversy. Galileo attempted the evaluation by weighing the curve against the generating circle; this rough method gave only an approximate value, viz., a little less than thrice the generating circle. Torricelli, by employing the “method of indivisibles,” deduced that the area was exactly three times that of the generating circle; this result had been previously established in 1640 in France by G. P. de Roberval, but his investigation was unknown in Italy. Blaise Pascal determined the area of the section made by any line parallel to the base and the volumes and centres of gravity of the solids generated by revolving the curve about its axis and base. Before publishing his results he proposed these problems for public competition in 1658 under the assumed name of Amos Dettonville. John Wallis in England, and A. la Louère in France, accepted the challenge, but the former could only submit incorrect solutions, while the latter failed completely. Having established his priority, Pascal published his investigations, which occasioned a great sensation among his contemporaries, and Wallis was enabled to correct his methods. Sir Christopher Wren, the famous architect, determined the length of the arc and its centre of gravity, and Pierre Fermat deduced the surface of
the spindle generated by its revolution. A famous period in the
history of the cycloid is marked by a bitter controversy which
sprang up between Descartes and Roberval. The evaluation
of the area of the curve had made Roberval famous in France,
but Descartes considered that the value of his investigation had
been grossly exaggerated; he declared the problem to be of
an elementary nature and submitted a short and simple solution.
At the same time he challenged Roberval and Fermat
to construct the tangent; Roberval failed but Fermat succeeded.
This problem was solved independently by Vicenzo
Viviani in Italy. The cartesian equation was first given by
Wilhelm Gottfried Leibnitz (*Acta eruditorum*, 1686) in the form
*y* = (2*x* − *x*^{2})½ + ∫(2*x* − *x*^{2})½*dx*. Among other early writers on the
cycloid were Phillippe de Lahire (1640–1718) and François Nicole
(1683–1758).

The mechanical properties of the cycloid were investigated
by Christiaan Huygens, who proved the curve to be tautochronous.
His enquiries into evolutes enabled him to prove that
the evolute of a cycloid was an equal cycloid, and by utilizing
this property he constructed the isochronal pendulum generally
known as the *cycloidal pendulum*. In 1697 John Bernoulli
proposed the famous problem of the *brachistochrone* (see
Mechanics), and it was proved by Leibnitz, Newton and several
others that the cycloid was the required curve.

Fig. 1. |

Fig. 2. |

Fig. 3. |

The method by which the cycloid is generated shows that it
consists of an infinite number of cusps placed along the fixed line
and separated by a constant distance equal to the circumference
of the rolling circle. The name cycloid is usually restricted to the
portion between two consecutive cusps (fig. 1, curve *a*); the fixed
line LM is termed the base, and the
line PQ which divides the curve
symmetrically is the *axis*. The
co-ordinates of any point R on the
cycloid are expressible in the form
*x* = *a*(θ + sin θ); *y* = *a*(1 − cos θ),
where the co-ordinate axes are the
tangent at the vertex O and the
axis of the curve, *a* is the radius of
the generating circle, and θ the
angle R′CO, where RR′ is parallel to LM and C is the centre of the
circle in its symmetric position. Eliminating θ between these two
relations the equation is obtained in the form *x* = (2*ay* − *y*^{2})½ + *a*
vers-¹ *y*/*a*. The clumsiness of the relation renders it practically
useless, and the two separate relations in terms of a single parameter
θ suffice for the deduction of most of the properties of the curve.
The length of any arc may be determined by geometrical considerations
or by the methods of the integral calculus. When measured
from the vertex the results may be expressed
in the forms *s* = 4*a* sin ½θ and *s* = √(8*ay*); the
total length of the curve is 8*a*. The intrinsic
equation is *s* = 4*a* sin ψ, and the equation to the
evolute is *s* = 4*a* cos ψ, which proves the evolute
to be a similar cycloid placed as in fig. 2, in
which the curve QOP is the evolute and QPR
the original cycloid. The radius of curvature
at any point is readily deduced from the
intrinsic equation and has the value ρ = 4 cos ½θ, and is equal to
twice the normal which is 2*a* cos ½θ.

The *trochoids* were studied by Torricelli and F. van Schooten,
and more completely by John Wallis, who showed that they possessed
properties similar to those of the common cycloid. The cartesian
equation in terms similar to those used above is *x* = *a*θ + *b* sin θ;
*y* = *a* − *b* cos θ, where *a* is the radius of the generating circle and *b*
the distance of the carried point from the centre of the circle. If
the point is without the circle, *i.e.* if *a* < *b*,
then the curve exhibits a succession of
nodes or loops (fig. 1, curve *b*); if within
the circle, *i.e.* if *a* > *b*, the curve has the
form shown in fig. 1, curve *c*.

The *companion to the cycloid* is a curve so
named on account of its similarity of construction,
form and equation to the common
cycloid. It is generated as follows: Let ABC be a circle having AB
for a diameter. Draw any line DE perpendicular to AB and meeting
the circle in E, and take a point P on DE such that the line DP = arc
BE; then the locus of P is the companion to the cycloid. The curve
is shown in fig. 3. The cartesian equation, referred to the fixed
diameter and the tangent at B as axes may be expressed in the
forms *x* = *a*θ, *y* = *a*(1 − cos θ) and *y* − *a* = *a* sin (*x*/*a* − 12π); the latter
form shows that the locus is the harmonic curve.

For epi- and hypo-cycloids and epi- and hypo-trochoids see Epicycloid.

References.—Geometrical constructions relating to the curves
above described are to be found in T. H. Eagles, *Constructive Geometry*
*of Plane Curves*. For the mechanical and analytical investigation,
reference may be made to articles Mechanics and Infinitesimal Calculus.
A historical bibliography of these curves is given in
Brocard, *Notes de bibliographie des courbes géométriques* (1897). See
also Moritz Cantor, *Geschichte der Mathematik* (1894–1901).