C 3 
VI. On the Rectification of the Hyperbola by Means of Two Ellipses; 
proving that Method to be circuitous, and such as requires much 
more Calculation than is requisite by an appropriate Theorem: 
in which Process a new Theorem for the Rectification of that 
Curve is discovered. 
To which are added some further Observations on the Rectification 
of the Hyperbola: among which the great Advantage of descend- 
ing Series over ascending Series, in many Cases, is clearly 
shown ; and several Methods are given for computing the con- 
stant Quantity by which those Series differ from each other. By 
the Rev. John Hellins, B. D. F. R. S. and Vicar of Potter’s- 
Pury, in Northamptonshire. 
Being an Appendix to his former Paper on the Rectification of the 
Hyperbola , inserted in the Philosophical Transactions for the 
Tear 1802. Communicated by Nevil Maskelyne, D.D. F.R.S . 
Astronomer Royal. 
Read January 10, 1811. 
i. IT he rectification of the hyperbola by means of two el- 
lipses is one proof, among many others, of the great sagacity 
of the late Mr. John Landen, F. R.S. ; and the ingenuity which 
he displayed on that occasion has obtained the notice and called 
forth the praises of eminent mathematicians both in this island 
and on the Continent. Yet, while the great ingenuity of the 
device is thus generally and justly allowed, this method of 
rectifying the hyperbola has always appeared to me to be 
