G^t Stuart S^tr
Edinb.
Saturday
Jan^y 7^th

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 perp^s 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 perp^s on it from the 3 centres. Call them ^p_a ^p_b ^p_c or simply a b c. Then by what precedes because O is c^e of sim. of A and B a : b : : rad of it : rad of B and because O’ is c^e 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” w^ch is c^e of sim^de of B and C

Q.E.D.

*note, and if the perp^s upon any line are in the ratio of the radii, that line passes thro’ the centre of similitude.

M^r 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 circ^fre of the 2^d 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. M^r 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 M^r 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 aff^te
father

---

Envelope:

C. H. Talbot
M^r 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.