[127r]

 Ashley-Combe
         Sunday.  19^th Sep^r ['1841' added by later reader]

 Dear M^r De Morgan.  I have more to say to you
than ever; ( beginning with many thanks for your
bountiful replies to my last packet).
I will begin with the Article on Negative & Impossible
Quantities, on which I have a good deal to remark.
I have finished it; & I think with on the whole
great success.  I need scarcely say that I like it
parenthetically. I enclose you the demonstration of the
formula \((a+bk)^{m+nk}=\varepsilon^A.cos B+k.\varepsilon^A.\sin B\), which I
found exceedingly easy, after your observations.  

I should tell you that the allusions to the
Irreducible Case of Cubic Equations in this Article,
has so excited my curiosity on the subject, that I
have attacked the chapter on Cubic Equations in ['page 47' inserted] of
R.~Murphy's treatise on the Theory of Equations (Library
Useful Knowledge), hoping there to gain some light
on the subject.  For I know not to what} exactly this
alluded, (my Algebra wits, as you say, not having
been quite proportionally stretched with some of my other
wits).  I have got thro' the first two pages;  and
[127v] shall have to write you some remarks upon these},
either in this letter, or in one as soon as possible.
But as yet I meet with nothing about

 \(\sqrt[3]{}(a+b\sqrt{-1})+\sqrt[3]{}(a-b\sqrt{-1})\) 
I hope I shall be able to understand the rest of the
Chapter.
At the bottom of my demonstration of \((a+bk)^{m+nk}\), you
will find a memorandum (simple as to the workin}g
out}) of the formula \(\cos.(a+bk)\), see page 137 of the
Cyclopedia.  You there say that such a formula
may be interpreted by it's [sic] identical expression on the
Second Side.  That is to say I imagine that the
meaning of \(\cos(a+b\sqrt{-1})\), which (as before pointed out in
the case of the line \(h\) ) is a misapplication of symbols,
may be got at thro' an examination of the results
arrived at by ['the application of' inserted] symbolical rules to this unmeaning
or mis-meaning expression.  That if in a calculation,
such an answer as \(\cos(a+b\sqrt{-1})\) were worked out,
the answer means in fact
[something crossed out]
the remaining side of a parallelogram in which
\( \cos\alpha\frac{\varepsilon^b+\varepsilon^{-b}}{2}\) is a diagonal, and \(\sin\alpha.\frac{\varepsilon^b-\varepsilon^{-b}}{2}.k\) the
other side : the diagonal itself being a 4^th Proportional
to \(1\), \(\cos\alpha\), \(\frac{\varepsilon^b+\varepsilon^{-b}}{2}\), inclined to \(1\) ['(that is to the Unit-Line})' inserted] as the \(\cos\alpha\) is; &
the remaining side being a 4^th Proportional to \(1\), \(\sin\alpha\),
\( \frac{\varepsilon^b-\varepsilon^{-b}}{2}\) inclined to \(1\) at an angle equal to the sum of a
[128r] Right-Angle and the angle made by \(\sin.\alpha\) with the Unit-Line.

I enclose you an explanation I
have written out (according to the Definition of this
Geometrical Algebra), of the two formulae for the Sine
and Cosine.  I am at work now on the Trigonometrical
Chapters of the Differential Calculus.
I do not agree to what is said in page 119 ['(of the Calculus)' inserted] that
results would be the same whether we worked [something crossed out] algebraically
with forms expressive of quantities or not.  It is true
that ['in' inserted] the form \(a+\sqrt{m}-\sqrt{n}\), if \((-1\) be substituted for
\( m\) and \(n\), the results come out the same as if we
work with \(a\) only.  but were the form \(a+\sqrt{m}\),
\( a-\sqrt{m}\), \(a\times\sqrt{m}\), or fifty others one can thin of,
surely the substitution of \((-1)\) for \(m\) will not bring
out results the same as if we worked with \(a\) only;
and in fact can only do so when the impossible
expression is so introduced as to neutralize itself,
if I may so speak.  I think I have explained
myself clearly.
It cannot help striking me that this} extension of
Algebra ought to lead to a further extension
similar in nature, to Geometry in Three-Dimensions;
& that again perhaps to a further extension into
some unknown region, & so on ad-infinitum possibly.
And that it is especially the consideration of
an angle \(=\sqrt{-1}\), which should lead to this;  a symbol,
which when it appears, sees to me in no way more
[128v] satisfactorily accounted for & explained than was
formerly the appearances ['which' inserted] Bombelli in some degree
cleared up by showing that at any rate they
(tho' in themselves unintelligible) led to intelligible
& true results.  You do hint in parts of page 136
at the possibility of something of this sort.

I enclose you also a paper I have
written explaining a difficulty of mine in the
Definitions of this Geometrical Algebra.

It appears to me that there is
no getting on at all without this Algebra.  In
the 3^d Chapter of your Trigonometry (which I have just
been going thro'), though there are no impossible
quantities introduced; yet how unintelligible are
such formulae as \(2ac.\cos B\), \(a.\sin B\), or any
in short where lines are multiplied into lines, if
one only takes the common notion of a line into a line
being a Rectangle.

I cannot send more today; but I have many
other matters to write on; especially the
Logarithmic Theory at the end of the Article.
I am considering it very carefully; & studying
at the same time the Article on Logarithms in the
Cyclopedia.  And I believe I shall have much to
say on it all.
The passage I wanted to ask you about in Lamé's
[129r] 1^st Vol, is pages 54, 55, 56, on the Resultant of the
pressures of a liquid on a vase.  I want to know
if I ought to understand these three pages, or if they
entail some knowledge of mathematical (especially of
trigonometrical) application to Mechanics, which I
do not yet possess.

I hope you receive game regularly.

 Yours most truly

 A. A. Lovelace

P.S.  Did you ever hear of a Science called
Descriptive Geometry?  I think Monge is the
originator of it.