[96r]

 Ockham

 Sat^dy 6^th Feb^y 

 ['1841' added by later reader]

 Dear M^r De Morgan.  Had I
waited a day or two longer,
I need not have troubled
you with my letter of Wed^dy,
& I can only reproach
myself now with having been
a little too hasty\, in my
examination of the Theorem in
pages 68, 69, and having
sent you an enquiry which
certainly indicates some
negligence.  I fear this letter
[96v] may not be in time to
stop one from you. [something crossed out]
However I will try to
send it by an opportunity this
afternoon.
But, to show you that I
now understand the matter
completely :
In the first place the question
of the Denominator, or the
Numerator, being all of the
same sign, in such [something crossed out] collection of
expressions as
\( \frac{a-b}{m-n}\), \(\frac{c-a}{p-m}\), \(\frac{d-c}{r-p}\), \(\frac{e-d}{q-r}\) &c
has nothing whatever to do
with the letters effacing each
other when the above are
[97r] put into the form,
\( \frac{(a-b)+(c-a)+(d-c)+(e-d)}{(m-n)+(p-m)+(r-p)+(q-r)}\) &c;
whether \((a-b)\), &c be positive
or negative, or some one &
some the other, still
\( \frac{a-b+c-a+d-c+e-d}{m-n+p-m+r-p+q-r}\) &c
must \(=\frac{e-b}{q-n}\) 
In the second place, the
Denominator must be all of
the same sign, in order
to fulfil the conditions of
the Lemma in page 48;
& this is the reason why
the condition is made respectively
\( \psi\ x\) always increasing or
[97v] always decreasing &c.
For \(\varphi\ x\), it matters not
whether it alternately increases
& decreases (provided always
that it be continuous).

I believe I now
have the whole quite clear;
& I shall be more careful
in future.
I enclose a paper upon
pages 70, 71, 72, 73.
It is merely the general
argument, put into my own
order & from; & I send
it in order to know if
you think I understand as
much about the matter as
[98r] I am intended to do.
You know I always have
so many metaphysical
enquiries & speculations which
intrude themselves, that I
never am really satisfied
that I understand anything;
because, understand it as
well as I may, my
comprehension can only be
an infinitesimal fraction of
all I want to understand
about the many connexions
& relations which occur to
me, how the matter in
question was first thought of
[98v] or arrived at, &c, &c.
I am particularly curious
about this wonderful Theorem.
However I try to keep
my metaphysical head in
order, & to remember Locke's
two axioms.
You should receive this about
6 o'clock this evening, if not
before.  I fear you will
have written to me today
however.  Believe me

 Yours most truly

 A. A. Lovelace