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Folios 127-129: AAL to ADM


         Sunday.  19th Sepr ['1841' added by later reader]

 Dear Mr 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

I hope I shall be able to understand the rest of the
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 4th 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 4th 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 3d 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] 1st 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.

About this document

Date of authorship: 

19 Sept 1841

Holding institution: 

Bodleian Library, Oxford, UK


Dep. Lovelace Byron

Box 170