AN ESSAY ON CYCLOIDAL CURVES. 



557 



tautoclironous curves in a resisting medium. 

 In 1736 it was the subject of a dispute be- 

 tween MM. Clairaut and Fontaine : it is not 

 yet entirely forgotten on that spot of acade- 

 mic Giroiind which srave birth to the discove- 

 ries of Newton; and its equation is to be 

 found in a work no less common than F.nier- 

 son's Fluxions, nearly in the same form as 

 that which is published as new in the Philoso- 

 phical Transactions for 17'98. We find in 

 the same paper a new method of dividing an 

 elliptic area in a given ratio ; but the curve 

 which the author calls a cycloid is the' com- 

 panion of a trochoid, and is only a distortion 

 of the figure by which Newton had very simply 

 and elegantly solved the same problem. It 

 is unnecessary to compare the altenijit tode- 

 mon'^trate the incommensuruhility of an oval 

 with the Newtonian method ; since Dr. 

 Waring's proof, deduced from the nature of 

 the equation of limits, is decidedly more sa- 

 tisfactory than any other hitherto made 

 known. On the whole, it appears that this 

 ingenious gentleman has been somewhat un- 

 fortunate in the choice of those problems 

 which he has selected as sf)ecimens of the 

 elegance of the modern mode of demonstra- 

 tion; whether those, which he has l^rought 

 forwards without proof, would have furnished 

 him with a more favourable opportunity for 

 the display of neatness and accuracy, may 

 be more easily determined, whenever he may 

 think proper to lay before the public their 

 analysis, construction, and demonstration at 

 full length. But, allowing the superiority of 

 the modern calculations in many cases, their 

 great advantage appears to be derived from 

 the methods of series and approximations ; 

 indeed, however we may wish to adhere to 

 the rigour of the ajicient demonstrations, it 



is absolutely necessary for the purposes of the 

 higher geometry to extend, in some measure, 

 the foundations which the ancients laid in 

 their postulates. Perhaps the most material 

 addition may be comprehended in this form: 

 " Let it be granted that any curve line may 

 be drawn whenever an indefinitely great 

 number of points maybe geometrically found 

 in, or indefinitely near to, that line." No. 

 doubt it is lAathematically impossible to 

 comply with this postulate; but it must be 

 remembered, that it is also impossible to 

 dr.iw, with strictly mathematical accuracy, 

 a right line or a circle ; but in both cases we 

 can approach sufficiently near to the truth for 

 practice: and itappcars to be more convenient 

 to consider such curves as arc thus described 

 as belonging to geometry, than to limit the 

 number of geometrical curves, according to 

 Descartes, to those of which the ordinate and 

 absciss are comparable by an algebraical 

 cquiition. This postulate forms the connect- 

 ing link between rational and irrational quan- 

 tities, between the infinite an<I the indefinite, 

 between perfect resemblance and identity; 

 and the irrational "eometrv, which has lonij, 

 been tacitly built on it, exhibits the principal 

 advantages of analytical calculus in a more 

 elegant form. The groundwork of this irra- 

 tional geometry is found in the method of 

 exhaustions of Euclid and Archimedes, and 

 it has been employed more or less generally 

 by Descartes, Newton, Cotes, Roberval, Va- 

 rJgnon, Delahire, Maclaurin, and many 

 other mathematicians. In the annexed es- 

 say on cycloidal curves, the geometrical form 

 of fluxions, or more properly speaking, the 

 Newtonian method of ultimate ratios, has 

 principally been adopted; and it is presumed, 

 that by a comparison witli algebraical cal- 



