— 
“THE 
PHILOSOPHICAL MAGAZINE 
AND JOURNAL. 
30% JUNE 1826. 
LVIII. On the Rectification of Curve Lines. By Tuomas 
BEVERLEY, Esq. 
To the Editor of the Philosophical Magazine and Journal. 
Sir, 
HE following property of curves, which I have recently 
discovered, from its general and extensive application, 
may probably be found very interesting and useful to those who 
sometimes make digressions into the more abstruse parts of 
the higher geometry ; as it may be the meansof rectifying many 
curves which have not yet been found susceptible of rectifica- 
tion. 
The property is the same in all plane curves whatever, 
whether they return into themselves, or proceed on ad injini- 
tum. 
The rectification of the tangential curve is always finite, pro- 
vided the curve round which it is described either returns 
into itself or has asymptotes to the infinite branches; if not, 
it will of course be infinite likewise. 
And the rectification is had without even finding the parti- 
cular equation of the curve to be rectified. 
Proposition 1.—ACM is any 
plane curve, to which CT is a 
tangent at C, join AC (A being 
the vertex), and demit AQ per- 
pendicularly on the tangent; so 
shall the rectification of the curve 
described by Q be represented 
by fAC x d. TAQ. 
Demonstr.—Draw the ordinate CD, and call it y; also draw 
the abscissa AD, and call it z. We have (z representing the 
ry d . 
curve AC), CT= re DT = i , and therefore by trigo- 
, ‘ d ‘ 
nometry = = sin TAQ = sin TCD, at = sin £4 T= 
Vol. 67. No. 338, June 1826. $D cos 
