Folios 100-103: AAL to ADM
Ockham Park
Friday. 19th Feby
Dear Mr 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
About this document
All Ada Lovelace manuscript images on the
Clay Mathematics Institute website are
© 2015 The Lovelace Byron Papers,
reproduced by permission of
Pollinger Limited. To re-use them in
any form, please apply to
katyloffman@pollingerltd.com.
The LaTeX transcripts of the letters
were made by Christopher Hollings
(christopher.hollings@maths.ox.ac.uk).
Their re-use in any form requires his
permission, and is subject to the
rights reserved to the owner of
The Lovelace Byron Papers.
Bodleian Library, Oxford, UK
Dep. Lovelace Byron