## Folios 7-8: ADM to AAL

**[7r]** My dear Lady Lovelace

The Theorem in page 16 can be easily proved when

the following is proved

[\( \frac{a+b}{}\) crossed out] \(\frac{a+a'}{b+b'}\) lies between \(\frac{a}{b}\) and \(\frac{a'}{b'}\)

\( \frac{a+a'}{b+b'}=\frac{a(1+\frac{a'}{a})}{b(1+\frac{b'}{b})}=\frac{a}{b}\times\frac{1+\frac{a'}{a}}{1+\frac{b'}{b}}\)

Now if \(\frac{a}{b}\) be greater than \(\frac{a'}{b'}\)

\(ab'\) \(\cdots\cdots\cdots\cdots\cdot\cdot\) \(a'b\)

\(\frac{b'}{b}\) \(\cdots\cdots\cdots\cdots\cdots\) \(\frac{a'}{a}\) whence \(\frac{1+\frac{a'}{a}}{1+\frac{b'}{b}}\) is less than \(1\)

or \(\frac{a}{b}\times\frac{1+\frac{a'}{a}}{1+\frac{b'}{b}}\) is less than \(\frac{a}{b}\)

or \(\frac{a+a'}{b+b'}\) is less than \(\frac{a}{b}\)

Similarly, it may be shown that if \(\frac{a}{b}\) be less than \(\frac{a'}{b'}\)

\( \frac{a+a'}{b+b'}\) is greater than \(\frac{a}{b}\) . You will now I think, not

have much difficulty in proving the whole. Page 48

contains the general view of this theorem

Page 29. Our conclusions are really the same. To say

that [diagram in original] is a r^{t} angled triangle, is to say that \(OP\) is

straight and not curved. The following however will explain

**[7v] ** [diagram in original] By the tangent of \(\angle POM\) is meant

the fraction \(\frac{PM}{OM}\), which is, by

similar triangles, the same thing for

every point of \(OP\) .

If then \(PM=\frac{2}{3}OM\), always, we have \(\frac{PM}{OM}=\frac{2}{3}\) always, or

the direction \(OP\) is always such as to make the angle \(POM\)

the same, namely that angle which has \(\frac{2}{3}\) for its tangent.

To see all this fully something of Trigonometry and the

application of algebra to geometry is required.

The Differential and Integral Calculus deal in the

same elements, but the former separates one element from

the mass and examines it, the latter puts together

the different elements to make the whole mass.

The examination of \(PQ\,MN\) (p. 29) with a view to the

relation between \(OM\) and \(MP\) is a case of the first:

the __summation__ of the rectangles in page 30, of the

second.

Page 32. The reference is unnecessary.

The first series \(1+4+ \& c\) is finite, the second infinite.

It is not easy to see__ à priori__ why one problem should

be attainable with given means and another not

so. It is stated here with a view to the following

common misapprehension.

**[8r]** It is thought that Newton and Leibnitz had some

remarkable new conception of principles, which is not

true. Archimedes and others [`and others' inserted] had a differential and integral

calculus, but not an algebraical system of sufficient

power to express very general truths.

Many persons before Newton knew, for instance that if

\( \frac{(x+h)^n-x^n}{h}\) could be developed for any value of \(n\),

the tangents of a great many curves could be drawn

and they knew this upon principles precisely the same

as Newton and Leibnitz knew it. But Newton

did} develope \(\frac{(x+h)^n-x^n}{h}\) and did} that which they

could not do}.

It was the additions made to the powers of algebra

in the seventeenth century, and not any new

conceptions of quantity, which made it worth while

to attempt that organization which has been called

the Differential Calculus

I should recommend your decidedly continuing the

Differential Calculus, warning you that you will

have long digressions to make in Algebra and

Trigonometry. I should recommend you to get my

Trigonometry, but not to attempt anything till I

send you a sketch of what to read in it. The

Algebra you must go through at some time or

**[8v] ** other, adding to it the article

``Negative and impossible quantities

in the Penny Cyclopaedia.

I have no doubt of being able to talk this

matter over with you in town when you

arrive

In the mean while, as mechanical expertness

in differentiation is of the utmost consequence,

and as it is the most valuable exercise in

algebraical manipulation which you can

possibly have, I should recommend your

thoroughly acquiring and keeping up the

Chapter you are now upon.

Yours very truly

__ADeMorgan__

3 Grotes' Place

Monday Aug^{st} 17/40

M^{r} Frend is rather better. I will add Lord Lovelace's

name to my list of members.

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