[29r] My dear Lady Lovelace

I have made some additional
notes on your papers.

[diagram in original] The meaning of \(\frac{\theta}{\sin\theta}\) is as follows

 \(\theta:1\) and \(1:\sin\theta\) compounded
give it in arithmetic

In fact \(\frac{a}{b}\) in arithmetic is another way of writing
\( a:b\) .

In geometry \(AB:AO\) is \(\theta[:]1\) 

 and \(AO\) or \(OB:BM\) is \(\sin\theta\) 

The compounded ratio is that of \(AB:BM\) 
which approaches without limit to the
ratio of \(1\) to \(1\) as \(AB\) is diminished
Your notion of the ratio approximating to
unity is correct.  The term 'ratio approximating
to \(a\) ' is a mixture of the geometrical and
[29v] arithmetical mode of speaking, it
should be 'ratio approximating to

 \(a:1\) .

I think you have got over the diffi-
culty of that part of the subject

I was sorry to have been out
when Lord Lovelace called, and
could not get down to S\) ^\textup{t}\) James' Square
till you had gone.  With best
remembrances I am
                              Yours very truly
                               ADeMorgan