x = (a + b) cosθ − mb cos(a + b/b)θ, y = (a + b) sinθ − mb sin(a + b/b)θ,
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 ½(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 r² = 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 r² = 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 of Asclepius combined to make Epidaurus a place of no small importance. Its origin was ascribed to a Carian colony, whose memory was possibly preserved in Epicarus, the earlier name of the city; it was afterwards occupied by Ionians, and appears to have incorporated a body of Phlegyans from Thessaly. The Ionians in turn succumbed to the Dorians of Argos, who, according to the legend, were led by Deiphontes; and from that time the city continued to preserve its Dorian character. It not only colonized the neighbouring islands, and founded the city of Aegina, by which it was ultimately outstripped in wealth and power, but also took part with the people of Argos and Troezen in their settlements in the south of Asia Minor. The monarchical government introduced by Deiphontes gave way to an oligarchy, and the oligarchy degenerated into a despotism. When Procles the tyrant was carried captive by Periander of Corinth, the oligarchy was restored, and the people of Epidaurus continued ever afterwards close allies of the Spartan power. The governing body consisted of 180 members, chosen from certain influential families, and the executive was entrusted to a select committee of artynae (from ἀρτύνειν, to manage). The rural population, who had no share in the affairs of the city, were called κονίποδες (“dusty-feet”). Among the objects of interest described by Pausanias as extant in Epidaurus are the image of Athena Cissaea in the Acropolis, the temple of Dionysus and Artemis, a shrine of Aphrodite, statues of Asclepius and his wife Epione, and a temple of Hera. The site of the last is identified with the chapel of St Nicolas; a few portions of the outer walls of the city can be traced; and the name Epidaurus is still preserved by the little village of Nea-Epidavros, or Pidhavro.
The Hieron (sacred precinct) of Asclepius, which lies inland about 8 m. from the town of Epidaurus, has been thoroughly excavated by the Greek Archaeological Society since the year 1881, under the direction of M. Kavvadias. In addition to the sacred precinct, with its temples and other buildings, the theatre and stadium have been cleared; and several other extensive buildings, including baths, gymnasia, and a hospital for invalids, have also been found. The sacred road from Epidaurus, which is flanked by tombs, approaches the precinct through a gateway or propylaea. The chief buildings are grouped together, and include temples of Asclepius and Artemis, the Tholos, and the Abaton, or portico where the patients slept. In addition to remains of architecture and sculpture, some of them of high merit, there have been found many inscriptions, throwing light on the cures attributed to the god. The chief buildings outside the sacred precinct are the theatre and the stadium.