[33r] My dear Lady Lovelace

I return the papers about series which 

are all right, the old one is as you suppose

With reference to your remarks on the diff^l calculus
1. You observe that

 \(\frac{\varphi(x+n\theta+\theta)-\varphi(x+n\theta)}{\theta}\) 
differs from \(\frac{\varphi(x+\theta)-\varphi(x)}{\theta}\) 
in that while \(\theta\) diminishes, \(x+n\theta\) varies.  So it [second 'it' crossed out] is,
and if \(n\) be finite and fixed, it might be shown that the 
limits of the two are the same.  But if \(n\) increase while
\( \theta\) diminishes, in such manner that \(n\theta\) is either equal to
or approaches the limit \(a\), then the first fraction has
the same limit as \(\frac{\varphi(x+a+\theta)-\varphi(x+a)}{\theta}\) 

To illustrate this, let \(\varphi x\) be the ordinate of a
curve, the abscissa being \(x\) .  If \(x\) remains fixed, the triangle
[diagram in original] (blotted) diminishes without
limit with \(\theta\); but if while 
\( \theta\) diminishes, the point \(A\) moves
in toward \(B\), so as continually to
approach \(B\), and to come as near as
[33v] we please to it, and yet never absolutely to reach
\( B\) as long as \(\theta\) has any value, it is obvious that
the small triangle would ride along the curve,
perpetually diminishing its dimensions, and
continually approaching in figure nearer and
nearer to the figure of as small a triangle at \(B\) .
All this necessarily follows from the notion
of continuity

[diagram in original]

2. You want to extend what I have said
about continuous functions to all possible cases,
not being able to imagine a function which changes
its values suddenly.  But for this you must 
wait till you come to the mathematics of disconti-
nuous quantity.  It is perfectly possible though the
calculation would be laborious, to find an algebraical
function which from \(x=1\) to \(x=2\) increases like the
ordinate of a straight line, from \(x=2\) to \(x=3\) draws the
[diagram in original] likeness of a human profile in a
different place, from \(x=3\) to \(x=4\)  
draws a part of a circle, from
\( x=4\) to \(x=5\) is nothing, and from
[34r] \(x=5\) to \(x=6\) makes any odd combination of lines or
curves, perfectly irregular.  None of the notions inciden-
tal to continuity must be applied to such a function

3. Your proof of the diff.co. of \(x^n\) is correct,
but it assumes the binomial theorem.  Now I endeavor
to establish the diff.calc. without any assumption
of an infinite series, in order that the theory
of series may be established upon the differential
calculus

 Besides, if you take the common proof of the 
binomial theorem, you are reasoning in a circle,
for that proof requires that it should be shown
that \(\frac{v^n-w^n}{v-w}\) has the limit \(nv^{n-1}\) as \(w\) approaches
\( v\) .  This is precisely the proposition which you
have deduced from the binomial theorem.

 Pray send your point about the exponential
theorem.

 And thank Lord Lovelace for pheasants and
hare duly received this morning

 Yours very truly

 ADeMorgan

69 G.S. Wed^y