Ill 
Mr. Hellins on the Rectification, &c. 
more curious tlian useful, as it is circuitous, and requires much 
more calculation than is requisite for that purpose by an appropriate 
theorem. The establishment of this truth is the main design 
of this short paper ; a truth which, to my surprise, has not 
been noticed in any book that has come to my hands.* 
2. But, before I proceed to investigation, it seems proper to 
remark, that Mr. Landen has, in his Memoir' f on the Hyper- 
bola and Ellipsis, expressed himself as if he thought that the 
difference between an hyperbolic arch and its tangent, when 
both are of an immense length, could not be computed before 
he published that work. His words at the beginning of the 
memoir above-mentioned, are these : <s Some of the theorems 
“ given by mathematicians for the calculation of fluents by 
“ means of elliptic and hyperbolic arcs requiring, in. the appli- 
“ cation thereof, the difference to be taken between an arc of 
“ an hyperbola and its tangent ; and such difference being 
“ not directly attainable when such arc and its tangent both 
“ become infinite, as they will do when the whole fluent is 
“ wanted, although such fluent be at the same time finite ; 
sc those theorems therefore in that case fail, a computation 
“ thereby being then impracticable without some farther 
“ help.” 
“ The supplying that defect I considered as a point of some 
“ importance in geometry, and therefore I earnestly wished 
“ and endeavoured to accomplish that business ; my aim being 
“ to ascertain, by means of such arcs as above-mentioned, the 
*' limit of the difference between the hyperbolic arc and its 
* On this occasion, no one has betrayed more ignorance, nor shown a greater want 
of candour, than the writer of Art. XIII. in the Monthly Review for April 1803. 
f This is the second in the 1st Volume of his Memoirs, printed in the year 1780. 
