55« 



AN ESSAY oy CYCLOIDAL CURVES. 



eolations on the same subjects, the superior 

 perspicuity and conciseness of this method 

 Tvill readily appear. 



ON CYCLOIDAt CCKVES. 



Definition t. When a circle is made to 

 rotate on a rectilinear basis, the figure de- 

 scribed -on the plane of the basis by any 

 point in the plane of the circle, is called a 

 trochoid. A circle concentric witli the ge- 

 nerating circle, and passing through the de- 



attributed by Wallis, (Ph. tr. 169?, n. 2i29,) to Cardinal 

 Cusanus, who wrote about the year 1450 ; but it seems to 

 be at least as probable that the curve, which appears in Cu- 

 sanus's figure, was meant for the semicircle employed in 

 finding a mean proportional. Bovillus, in 1501, has a 

 juster claim to the merit of the invention of the cycloid and 

 trochoid, if it can be any merit to have merely imagined 

 such curves to exist. In 1599, Galileo gave a name to the 

 common cycloid, and attempted its quadrature, but hav- 

 ing been accidentally misled by repeated experiments on 

 the weight of a flat substance, cut into a cycloidal form, he 

 fancied tliat the area bore an incommensurable ratio to that 

 of the circle, and desisted from the investigation. Mer- 

 scnnus described the cycloid, in 1615, under the name of 

 la trochoide, or la roulette, but he went no further. Ro- 



SCribing point, may be called the xlcscnbing be.r\al seems to have first discovered the comparative qua- 



i'ircle. 



Definition ii. If the describing point 

 is in the circumference of the rotating circle, 

 the two circles coincide, and the curve is 

 called a cycloid. 



Definition hi. If a circular basis be 

 substituted for a rectilinear one, the trochoid 

 will become an epitrochoid, and the cycloid 

 an epicycloid. 



Scholium l. These terms have "hitherto bten too pro- 

 miscuously employed ; the terms cycloid and trochoid have 

 been used indifferently ; and the term epicycloid has com- 

 prehended the epitrochoid, the terms prolate and contracted 

 being sometimes added, to imply that the desciibing point 

 is within or without the generating circle. The interior 

 epicycloid and epitrochoid may very properly be distin- 

 guished by the names hypocycloid and hypotrochoid, when- 

 ever they are the separate objects of consideration. The dif- 

 ferent species of epicycloids maybe denominated according 

 to the nurobe;r of their cusps, combined with that of the en- 

 tire revolutions wiich they comprehend: for instance, the 

 epicycloid described by a circle on an equal basis is a simple 

 unicuspidate epicycloid ; and if the diameter of the generat- 

 ing circle be to that of the basis as 5 to 2, the figure will be 

 a quintuple bicuspidate epicycloid. If the describing .circle 

 cf a trochoid or cycloid be so placed as to touch the middle 

 /)f the curve, and each of the ordinates parallel to the basis 

 be diminished by the corresponding ordinate of the circle, 

 the curve thus generated has been denominated the com- 

 panion of the trochoid or cycloid, the figure of sines, and 

 the harmonic curve. 



Sc^ioLiUAi 2. The invention of the cycloid lias been 



drature and rectification of the cycloid, and the content of 

 a cycloidal solid, about the year 1635, but hit treatise was 

 not printed until 16>)5, Torricelli, in 1644, first pub- 

 lished the quadrature and the method of drawing a tangent, 

 both of which had been investigated by Descartes in I63'g. 

 Wallis gave, in 1670, a perfect quadrature of a portion of 

 the cycloid. The epicycloid is said to have been invented 

 by Roemer : its rectification and evolute were investigated 

 by Newton in the Trincipia, publisVied in 1&87. In 1695 

 Mr. Caswell showed the perfect quadrabrliiy of a portion of 

 the epicycloid, and Dr. Halley immediately published an 

 extension of Caswell's discovery, together with a compari- 

 son of all epitrochoidal with circular areas. M. Varignon 

 is also said to have reduced the rectification of the epitro- 

 choid to that of the ellipsis, in the same year. Nicole, De- 

 lahire, Pascal, Reaumur, Maclaurin, the Bernoullis, the 

 commentators on Newton, and many others, have contri- 

 buted to the examination of cycloidal curves, both in 

 planes and in curved surfaces; and Waring, the most pro- 

 found of modern algebraists, tias considerably extended his 

 researches upon the nature of those lines which are gene- 

 rated by a rotatory progression of other curves. In the pre- 

 sent essay, the most remarkable properties of cycloidal 

 curves are deduced io a simpler and more general manner 

 than appears to have been hitherto done, the equations o 

 several species are investigated, and a singular property o 

 the quadricuspidate hypocycloid is demonstrated. Those 

 who wish for further information respecting the history of 

 these curves, may consult either Carlo Dati's essay on the 

 subject, or Montucla's Historyofthe mathematics. 



PitoposiTioN 1. Theokem. (Plate7.Fig- 

 57.) In any curve generated by the rotation of 

 antjther on any basis, the right Hnejoiningthe 



