Lacock
Sunday Dec. 4
My Dear Charles
Your photographic journal did not arrive this week, have you changed the address? I made trial of the Wothlytype <1> and think it is likely to prove a very good process –
I think you will do well to join the club <2> to which your friend Swainson <3> belongs. To be a candidate for the Athenæum <4> is a hopeless affair – Your name must be on the list for about nine years before you are ballotted for, and very few are elected, those who fail have to wait nine years more. Some persons of great eminence are let in by the Committee every year, which keeps the Club in a state of good reputation.
Your solution of the Cambridge problem to inscribe a circle in a Gothic arch, <5> which is equilateral, appears to be correct. Your other problem: to find the Center of the circle inscribed in any Circular triangle, formed by 3 circles, is a case of the problem “to find the center of a circle which shall touch 3 given circles“ This is solved as follows,
Let A, B, C, be the given circles. O the centre of the circle required.
Because the circle is to touch the given circles A, B, the locus of its center is a hyperbola whose foci are the centers of A and B.
And because it is to touch the given circles A, C, the locus of its center is another given hyperbola – therefore its center O is the intersection of the 2 hyperbolas –
But to resolve the same problem by the Euclidian geometry is a matter of difficulty – Gergonne in his Annals des Mathematiques <6> has given a celebrated solution, It depends on the theory of radical axes. The radical axis of 2 circles is that straight line, from every point of which the tangents drawn to the 2 circles are equal. Therefore if the circles intersect the radical axis is the prolongation of their common chord. You say you want the figure back again, therefore I send it.
Your affte Father
Envelope:
C. H. Talbot EsqMr Prichard’s <7>
Architect
Llandaff
Notes:
1. The uranium-based printing-out paper invented by Jacob Wothly of Aachen, 1864, initally created quite a stir. It was considerably more sensitive than the albumen paper then in general use, but the claim for permanence was not fulfilled, and it was soon abandoned.
2. Junior University Club, London. [See Doc. No: 08899].
3. A friend of Charles from Trinity College, Cambridge.
4. Athenaeum Club, Pall Mall, London: WHFT’s club; a gentleman’s club composed primarily of artists and scientists.
5. See Doc. No: 08900.
6. Annals de Mathematiques, a serial publication founded in 1810 by Joseph Diez Gergonne (1771–1859), French mathematician.
7. John Prichard, Welsh architect; Charles Henry Talbot apprenticed to.