[100r]

 Ockham Park

 Friday.  19^th Feb^y

 Dear M^r De Morgan.  I have one or two
queries to make respecting the ''Calculus of
Finite Differences'' up to page 82.
Page 80, line 4 from the top, ''remembering \(.\ldots\) 
''\( .\ldots\) that in \(\varphi''(x+\theta\omega)\), \(\theta\) itself is a function
''of \(x\) and \(\omega\) , &c''; Now, neither on examining
\( \theta\) as here used & introduced, nor on
referring to the first rise & origin of \(\theta\) in
this capacity, (see page 69), can I discover
that it is a function of \(x\) and \(\omega\) here, or
a function of the analogous \(a\) and \(h\) in
page 69.  I neither see the truth of this
assertion, nor do I perceive the importance
of it (supposing it is true) to the rest of
the argument & demonstration in page 80.

There is also a point of doubt I have
relating to the conclusion in lines 15, 16 from
[100v] the top of page 79 :
It is very clear that the law for the Co-efficients
being proved for \(u_n\), and for \(\Delta u_n\), follows
immediately & easily for \(u_{n+1}\), or \(u_n+\Delta u_n\) .

But if we now wish to establish it
for \(u_{n+2}\), we must prove it true not
only for \(u_{n+1}\), but also for \(\Delta u_{n+1}\) :

To retrace from the beginning : the
object in the first half of page 79 evidently
is to prove firstly, that any order of \(u\),
say \(u_n\) can be expressed in term ['of,' inserted] or in
a Series of all the Differences of \(u\); \(\Delta u\),
\( \Delta^2u\), \(\Delta^3u\), \(..\ldots\ldots\ldots\) \(\Delta^nu\);
Secondly, that the Co-efficients for this Series
follow the law of those in the Binomial Theorem.

Now the first part is evident from the
law of formulation of the Table of Differences;
Since all the Differences \(\Delta u\), \(\Delta^2u\), \(\Delta^3u\) &c
are made out of \(u\), \(u_1\), \(u_2\) &c, it is
obvious that by exactly retracing & reversing
the process, we can make \(u\), \(u_1\), \(u_2\) &c
[101r] out of \(\Delta u\), \(\Delta^2u\), \(\Delta^3u\) &c.
For the second part of the above; if we
can [something crossed out] show that the law for the Co-efficients
holds good up to a certain point, say \(u_4\);
and also that being true for any one
value, it must be true [something crossed out] for the next
value too; the demonstration is effected for
all values :
Now the fact is shown that it is true up
to \(u_4\) . (I must not here enquire why the
fact is so.  That is I suppose not your
arranging, or any part of your affairs).
It is shown that the two parts \(u_3\), \(\Delta u_3\) of
which \(u_4\) is made up are under this law,
& therefore that \(u_4\) is so.  And next it is
shown that any other two parts \(u_n\), \(\Delta u_n\) 
being under this law, their sum \(u_{n+1}\) 
must be so.  But this proves nothing
for a continued succession.  \(u_{n+1}\) being
under this law does not prove that \(\Delta u_{n+1}\) 
is under it, & therefore that \(u_{n+2}\) is under it.
[101v] There seems to me to be a step or condition
omitted.
I am sorry still to be obliged to trouble
you about \(f\,x\), \(f'x\), \(f''x\), I cannot yet
agree to the assertion that the result\ would
not be affected by discontinuity or singularity
in \(f'x\), \(f''x\), &c.  The result it is true
would not be directly affected; but it surely
would be ['indirectly' inserted] affected, inasmuch as the conditions
of page 69, necessary to prove that result,
could not be fulfilled unless we suppose
\( f'x\), \(f''x\) \(..\ldots\) \(f^{(n+1)}x\) continuous &
ordinary as well as \(f\,x\) .  To arrive at
the equation \(\frac{\varphi(a+h)}{\psi(a+h)}=\frac{\varphi^{(n+1)}(a+\theta h)}{\psi^{(n+1)}(a+\theta h)}\) 
page 69, it is a necessary condition that
\( \varphi x\), \(\varphi'x\), \(\varphi''x\) \(\ldots\ldots\ldots\) \(\varphi^{(n+1)}x\) be all
continuous & without singularity from \(x=a\) to
\( x=a+h\) .  And the \(\varphi'x\), \(\varphi''x\) \(..ldots\) \(\varphi^{(n)}x\), \(\varphi^{(n+1)}x\) 
of page 71, could not fulfil this condition
unless \(f'x\), \(f''x\) \(..\ldots\) \(f^{(n)}x\), \(f^{(n+1)}x\) did so
[102r] also.    I fear I am very troublesome about
this.
I have remarks to make respecting some of
the conclusions of the Chapter on Algebraical
Development; but they will keep, and
therefore I will delay them, as I think
I have send abundance, & I have also
some questions to put on the last 8 pages
of your ''Number & Magnitude'' on Logarithms.

On the Differential Calculus I will only
now further say that on the whole I believe
I go on pretty well with it; and that
I suppose I understand as much about it,
[something crossed out] as I am intended to do;
possibly more, for I spare no pains to do
so.
Now for the Logarithms : I had not till now
read the last pages of your Number & Magnitude,
& there are certain points I do not fully
understand.  The last line of the whole, on
the natural logarithms is one.  I cannot
[102v] identify the constituent quality of the natural
logarithms there given, with the constituent
qualities I am already acquainted with thro'
other relations & means : I know ['for instance' inserted] that the
natural logarithms must have 2.717281828
for their Base; that is to say that the
line \(HL\), or \(A\) (\( OK\), or \(V\) being the linear
unit) should be 2.717281828 \(V\) units.
Now I do not see ['but' inserted] that the condition in the
last paragraph of the book is one that
might perfectly consist with any Base whatever.

To prove that I understand
the previous part, at least to a considerable
degree, I enclose a Demonstration I wrote
out of the property to be deduced by the
Student, (see second paragraph of page 79),
& which I believe is quite correct.

Pray of what use is the Theorem
(page 75, ['& which' inserted] continues in page 76)?  I do not see
that it is subservient to anything that
[103r] follows; and it appears to me, to say the
truth, to be rather a useless & cumbersome
addition to a subject already sufficiently
complicated & cumbersome. The passage I
mean is from line 13 (from the top) page 75, to
the middle of page 76.

 Believe me

 Yours very truly

 A. A. Lovelace