TALBOT8926

Llandaff
Wednesday evening Jan^y 4^th

My dear Father,

I do not see the proof of the proposition which you sent me about the centre of gravity of a trapezium. I think I have written Monnie¹ [sic] an account more or less of the balls. There were both very good. The worst of balls is that one is tired after them. I have sent you one Photographic Journal and there is another to follow. I am glad you are exhibiting this year in the photographic exhibition. Our weather here has been rather changeable We have had rather ch both sharp frosts and thaws and mild weather. A very bad scandal has arisen these parts which will be brought into the divorce court but I do not think any proceedings have been taken yet. A certain Major Traherne a married man and very well known & much liked in this county has eloped with M^rs Brogden the wife of an iron master at Tondu. She does not seem to have borne much of a character, but Major Traherne being a person whom almost every body knew, and the last person in the world expected to do anything of the sort, it has caused great commotion. This only began to be known for certain on the evening of the Bridgend ball & then only to a few people. M^rs Traherne the wife of Major Traherne is I believe a first cousin of Archdeacon Bloss. I do not think they have been heard of since. His brother Captain Traherne, was at the Bridgend ball of which I believe he was one of the chief promoters & arrangers. He is shortly to be married.

I enclose a letter which I have received today from Mr Nesbitt ² & I shall be glad of it back again when you have read it.

I send you a theorem proved. The figure is on a separate piece of paper.

I expect to be able to prove from it that the external centre of similitude of 3 [circumscribed angle?]¡¯s are all in one straight line.

Enunciation.

If the corresponding extremities of 3 parallel straight lines be joined by straight lines each pair of which intersect. Then the 3 pts of intersection lie on one straight line.

Let this straight line ABC be the intersection of 2 planes. through a pt. A in this line draw DA in the 1^st plane and EA in the 2^nd. join DE [illegible] ADE being a plane ∆ draw FG parallel to DE. Take any other pt B in this common section. Draw BD in the 1^st plane and BE in the 2^nd ║ to FG. & HK is ║ to FG. but if 2 straight lines are ║ they must be in one plane. ∴ FG & HK are in one plane and GK & FH being joined and produced must ∴ be either ║ or meet and since they lie in planes which have a common section, they either meet at C in that section or they are ║ to the common section ABC.

And DE. FG and HK are any ║ lines whatever.

And this theorem is true whatever angle the 2 planes make with each other¡­it is true where the angle vanishes, & the plane theorem above enunciated is proved.

Your affect son
Charles

[enclosed figure of intersecting planes and resulting triangles]

[envelope:]
H Fox Talbot Esq
13 Gt Stuart St
Edinburgh