William George Horner - Peter Barlow and Continued Fractions

Peter Barlow and Continued Fractions

When Peter Barlow wrote, in 1845, he remembered Davies, but not Horner, asking to borrow a book by Budan (both Davies and Horner were living in Bath at the time). Barlow also had a vague recollection that the material on approximations Horner sent him related to continued fractions rather than what appeared in Phil. Trans.. Horner clearly held Barlow in high regard and it would have been natural for Horner to approach him to request both books and critical advice as Horner draws attention to Barlow's article in issue No. 12 of the New Series of the Mathematical Repository (bound up in Vol. 3 in 1814) in his survey of approximation methods in the following volume of the Repository (bound up in 1819). The anonymous reviewer for The Monthly Review in the issue for December, 1820 writes that he has sighted Horner's letter to Barlow and that the letter confirms that Horner already had his method of approximation at that date (1818).

The methods of both Barlow and Horner use a nesting of expressions akin to continued fractions. Horner was aware of Lagrange's use of continued fractions at least through his reading of Bonnycastle's Algebra which is also mentioned in the survey article in the Repository. Horner may have rewritten his paper either under guidance or of his own volition, with an eye to publication in Phil. Trans.. However, Horner went on to write on the use of continued fractions in the summation of series in Annals of Philosophy in 1826 and on their use in improvements they yield in the solution of equations in Quarterly Journal of Science, Literature and the Arts running over into 1827; he explicitly cites work of Lagrange. Barlow's memory may have been confused by this later work.