## Folios 127-129: AAL to ADM

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 w__hat}__ 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 t__hese}__,

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 r

__esults__

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 q__uantities__ 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 i__mpossible__

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 i__mpossible____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 R__esultant 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.

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