112 ' Mr. Hellins on the Rectification 
“ tangent, whilst the point of contact is supposed to be car- 
“ ried to an infinite distance from the vertex of the curve, 
“ seeing that, by the help of that limit , the computation would 
“ be rendered practicable in the case wherein, without such 
“ help, the before-mentioned theorems fail. The result of 
“ my endeavours respecting that point appears in this Me- 
“ moir : which, among other matters, contains the investiga- 
££ tion of a general theorem for finding the length of any arc 
££ of any conic hyperbola by means of two elliptic arcs/’— - 
Vol. I. p. 23 and 24. 
And towards the end of the same memoir he has expressed 
himself thus: “ Mr. Maclaurin’s method of construction ” 
[of the elastic curved ££ just now adverted to, though very 
£t elegant, is not without a defect. The difference between the 
£C hyperbolic arc and its tangent being necessary to be taken, 
££ the method (for the reason mentioned at the beginning 
££ of this Memoir) always fails when some principal point in 
“ the figure is to be determined ; the said arc and its tangent 
<£ then both becoming infinite, though their difference be at 
£C the same time finite.” P. 3 6. 
3. Whoever reads the passages here quoted, and knows 
not what was done on the subject before Mr. Landen handled 
it, will undoubtedly conceive that he was the first person who 
solved the problem of computing the difference between the length 
of the infinite arch of an hyperbola and its asymptote. Yet the 
fact is not so. That difference may be computed, in many 
cases, by the first series given by Mr. Maclaurin in Art. 808 
of his Treatise of Fluxions , which series admits of an easy 
transformation into another form, by which the aforesaid dif- 
ference may be computed in all cases; or the fluent may be 
