E 289^] 
drantal arc of another elliphs, wliofe femi-tranfverfe and 
femi-conjugate axes are^p+^rJ and^ 
-■fn 
Before Mr. maclaurin publifhed his excellent Trea- 
tife of Fluxions, fome very eminent mathematicians ima- 
gined, that the elajlic curve could not be conftru61:ed by 
the quadrature or re61;ification of the conic fedli'ons. But 
that gentleman has fliewn, in that treatife, that the faid 
curve may in every cafe be conftru6ted by the redtifica- 
tion of the hyperbola and elliplis ; and he has obferved, 
that, by the fame means, we may conftru6l the curve 
along which, if a heavy body moved, it would recede 
equally in equal times from a given point. Which lafl; 
mentioned curve Mr. james Bernoulli conftru6led by 
the re6lification of the elaftic curve, and Mr. leibnitz 
and Mr. john Bernoulli by the rectification of a geo- 
metrical curve of a higher kind than the conic feCtions. 
It is obfervable, that Mr. maclaurin’s method of con- 
ftruCtion juft now adverted to, though very elegant, is 
not without a defeCt. The difference between the hy- 
perbolic arc and its tangent being necefiTary to be taken, 
the method alw'ays fails when fome principal point in 
the figure is to be determined ; the faid arc and its tan- 
gent then both becoming infinite, though their differ- 
ence be at the fame time finite. The contents of this 
paper, properly applied, will evince, that both the elajlic 
curve and the curve of equable recefs from a given point 
(with many others) may be conftruCted by the rectifica- 
tion of the ellipfis only, without failure in any point 
R r 2 
XXVII. AJiro- 
