have read little, and to be the original architect of his own system, and the claim was no doubt on the whole true. But he had read Democritus, and, it is said, Anaxagoras and Archelaus. His works, we learn, were full of repetition, and critics speak of vulgarities of language and faults of style. None the less his writings were committed to memory and remained the text-books of Epicureanism to the last. His chief work was a treatise on nature (Περὶ φύσεως), in thirty-seven books, of which fragments from about nine books have been found in the rolls discovered at Herculaneum, along with considerable treatises by several of his followers, and most notably Philodemus. An epitome of his doctrine is contained in three letters preserved by Diogenes.
Authorities.—The chief ancient accounts of Epicurus are in the tenth book of Diogenes Laërtius, in Lucretius, and in several treatises of Cicero and Plutarch. Gassendi, in his De vita, moribus, et doctrina Epicuri (Lyons, 1647), and his Syntagma philosophiae Epicuri, systematized the doctrine. The Volumina Herculanensia (1st and 2nd series) contain fragments of treatises by Epicurus and members of his school. See also H. Usener, Epicurea (Leipzig, 1887) and Epicuri recogniti specimen (Bonn, 1880); Epicuri physica et meteorologica (ed. J. G. Schneider, Leipzig, 1813); Th. Gomperz in his Herkulanische Studien, and in contributions to the Vienna Academy (Monatsberichte), has tried to evolve from the fragments more approximation to modern empiricism than they seem to contain. For criticism see W. Wallace, Epicureanism (London, 1880), and Epicurus; A Lecture (London, 1896); G. Trezza, Epicuro e l’Epicureismo (Florence, 1877; ed. Milan, 1885); E. Zeller, Philosophy of the Stoics, Epicureans and Sceptics (Eng. trans. O. J. Reichel, 1870; ed. 1880); Sir James Mackintosh, On the Progress of Ethical Philosophy (4th ed.); J. Watson, Hedonistic Theories (Glasgow, 1895); J. Kreibig, Epicurus (Vienna, 1886); A. Goedeckemeyer, Epikurs Verhältnis zu Demokrit in der Naturphil. (Strassburg, 1897); Paul von Gizycki, Über das Leben und die Moralphilos. des Epikur (Halle, 1879), and Einleitende Bemerkungen zu einer Untersuchung über den Werth der Naturphilos. des Epikur (Berlin, 1884); P. Cassel, Epikur der Philosoph (Berlin, 1892); M. Guyau, La Morale d’Épicure et ses rapports avec les doctrines contemporaines (Paris, 1878; revised and enlarged, 1881); F. Picavet, De Epicuro novae religionis sectatore (Paris, 1889); H. Sidgwick, History of Ethics (5th ed., 1902). (W. W.; X.)
EPICYCLE (Gr. ἐπί, upon, and κύκλος, circle), in ancient
astronomy, a small circle the centre of which describes a larger
one. It was especially used to represent geometrically the
periodic apparent retrograde motion of the outer planets, Mars,
Jupiter and Saturn, which we now know to be due to the annual
revolution of the earth around the sun, but which in the Ptolemaic
astronomy were taken to be real.
EPICYCLOID, the curve traced out by a point on the circumference
of a circle rolling externally on another circle. If
the moving circle rolls internally on the fixed circle, a point on
the circumference describes a “hypocycloid” (from ὑπό, under).
The locus of any other carried point is an “epitrochoid” when
the circle rolls externally, and a “hypotrochoid” when the
circle rolls internally. The epicycloid was so named by Ole
Römer in 1674, who also demonstrated that cog-wheels having
epicycloidal teeth revolved with minimum friction (see
Mechanics: Applied); this was also proved by Girard
Desargues, Philippe de la Hire and Charles Stephen Louis
Camus. Epicycloids also received attention at the hands of
Edmund Halley, Sir Isaac Newton and others; spherical
epicycloids, in which the moving circle is inclined at a constant
angle to the plane of the fixed circle, were studied by the
Bernoullis, Pierre Louis M. de Maupertuis, François Nicole,
Alexis Claude Clairault and others.
In the annexed figure, there are shown various examples of the curves named above, when the radii of the rolling and fixed circles are in the ratio of 1 to 3. Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself. In the particular case when the radii are in the ratio of 1 to 3 the epicycloid (curve a) will consist of three cusps external to the circle and placed at equal distances along its circumference. Similarly, the corresponding epitrochoids will exhibit three loops or nodes (curve b), or assume the form shown in the curve c. It is interesting to compare the forms of these curves with the three forms of the cycloid (q.v.). The hypocycloid derived from the same circles is shown as curve d, and is seen to consist of three cusps arranged internally to the fixed circle; the corresponding hypotrochoid consists of a three-foil and is shown in curve e. The epicycloid shown is termed the “three-cusped epicycloid” or the “epicycloid of Cremona.”
The cartesian equation to the epicycloid assumes the form
when the centre of the fixed circle is the origin, and the axis of x passes through the initial point of the curve (i.e. the original position of the moving point on the fixed circle), a and b being the radii of the fixed and rolling circles, and θ the angle through which the line joining the centres of the two circles has passed. It may be shown that if the distance of the carried point from the centre of the rolling circle be mb, the equation to the epitrochoid is
The equations to the hypocycloid and its corresponding trochoidal curves are derived from the two preceding equations by changing the sign of b. Leonhard Euler (Acta Petrop. 1784) showed that the same hypocycloid can be generated by circles having radii of 12(a ± b) rolling on a circle of radius a; and also that the hypocycloid formed when the radius of the rolling circle is greater than that of the fixed circle is the same as the epicycloid formed by the rolling of a circle whose radius is the difference of the original radii. These propositions may be derived from the formulae given above, or proved directly by purely geometrical methods.
The tangential polar equation to the epicycloid, as given above, is p = (a + 2b) sin(a/a + 2b)ψ, while the intrinsic equation is s = 4(b/a)(a + b) cos(a/a + 2b)ψ and the pedal equation is r 2 = a2 + (4b·a + b)p2/(a + 2b)2. Therefore any epicycloid or hypocycloid may be represented by the equations p = A sin Bψ or p = A cos Bψ, s = A sin Bψ or s = A cos Bψ, or r 2 = A + Bp2, the constants A and B being readily determined by the above considerations.
If the radius of the rolling circle be one-half of the fixed circle, the hypocycloid becomes a diameter of this circle; this may be confirmed from the equation to the hypocycloid. If the ratio of the radii be as 1 to 4, we obtain the four-cusped hypocycloid, which has the simple cartesian equation x2/3 + y2/3 = a2/3. This curve is the envelope of a line of constant length, which moves so that its extremities are always on two fixed lines at right angles to each other, i.e. of the line x/α + y/β = 1, with the condition α2 + β2 = 1/a, a constant. The epicycloid when the radii of the circles are equal is the cardioid (q.v.), and the corresponding trochoidal curves are limaçons (q.v.). Epicycloids are also examples of certain caustics (q.v.).
For the methods of determining the formulae and results stated above see J. Edwards, Differential Calculus, and for geometrical constructions see T. H. Eagles, Plane Curves.
EPIDAURUS, the name of two ancient cities of southern
Greece.
1. A maritime city situated on the eastern coast of Argolis, sometimes distinguished as ἡ ἱερὰ Ἐπίδαυρος, or Epidaurus the Holy. It stood on a small rocky peninsula with a natural harbour on the northern side and an open but serviceable bay on the southern; and from this position acquired the epithet of δίστομος, or the two-mouthed. Its narrow but fertile territory consisted of a plain shut in on all sides except towards the sea by considerable elevations, among which the most remarkable were Mount Arachnaeon and Titthion. The conterminous states were Corinth, Argos, Troezen and Hermione. Its proximity to Athens and the islands of the Saronic gulf, the commercial advantages of its position, and the fame of its temple
