Gt Stuart Str
Edinb.
Saturday
Jany 7th
My Dear Charles
Your theorem, deduced from geometry of 3 dimensions, is good, and much more luminous than any demonstration derived from the formulæ of analytical geometry.
But the theorem that the 3 external centres of similitude of any 3 circles lie in a straight line, can be demonstrated with the utmost simplicity without the aid of that, or any other Lemma. <1> For, if O be the centre of similitude of the two circles A, B and any line be drawn through O, the perps let fall on it from the centres of A and B are in the ratio of the radii, and therefore in a constant ratio*. Let there be a third circle C and let O’ be the centre of similitude of A and C. Join OO’. Let fall perps on it from the 3 centres. Call them pa pb pc or simply a b c. Then by what precedes because O is ce of sim. of A and B a : b : : rad of it : rad of B and because O’ is ce of sim. of A and C a : c : : rad. of it : rad of C ∴ b : c : : rad of B : rad of C ∴ by the note, line OO’ passes thro the point O” wch is ce of simde of B and C
Q.E.D.
*note, and if the perps upon any line are in the ratio of the radii, that line passes thro’ the centre of similitude.
Mr Lang F.R.S.E. has been reading a paper at the Royal Society <2> on Epicycloidal Curves which seems very curious. He describes the curves with a burning lathe upon type metal and prints from them. Suppose the centre of circle A to be carried by an equable motion round the circumference of a second circle B and a point P in the circfre of the 2d circle, to revolve with an equable motion round its own centre. The curve generated by P will vary in form if you alter the ratio of the radii, or the velocity of rotation. It may be simply undulating on a succession of loops. Mr Lang’s problem is to find under what circumstances the loops touch each other? He handed the printed diagrams about the room.
I have written you another letter about Mr Nesbitt’s <3> photograph. To save you trouble I have written about it on a separate sheet, so that you can send it to him.
Your affte
father
Envelope:
C. H. TalbotMr Prichard’s <4>
Architect
Llandaff
Notes:
1. A prelimonary proposition, a premise taken for granted.
2. Royal Society of Edinburgh.
3. Alexander Nesbitt (1817–1886), archaeologist & ancient glass collector.
4. John Prichard, Welsh architect; Charles Henry Talbot apprenticed to.