## Folios 33-34: ADM to AAL

**[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}

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