## Folios 70-73: AAL to ADM

Ockham

Tuesday. 22^{nd} Dec^{r}

Dear M^{r} De Morgan I now see

exactly my mistake. I had

overlooked that the Series in

question is not one in s__uccessive__

Powers of \(x\) ['like that in page 185' inserted], but only in

s__uccessive__ __even__ powers of \(x\) .

I used once to regret these

sort of errors, & to speak of

t__ime__ l__ost__ over them. But I

have materially altered my

mind on this subject. I often

gain more from the discovery

of a mistake of this sort, than

from 10 acquisitions made at**[70v] ** o__nce__ & without any kind of

difficulty. __ __

There is still one little thing in

your Demonstration not perfectly

clear to me. At the end

you remark that ''our result

''gave the 503\) ^\textup{d}\) Term instead of

''the 502\) ^\textup{nd}\), which arose from

''taking the w__hole__ n__umber__ next

''above \(\sqrt{x+\frac{1}{4}}\) instead of an

''intermediate fraction.'' __ __

In examining the equation

\) n=\) next whole number above

\(\frac{5}{4}+\frac{1}{2}\sqrt{1000,000+\frac{1}{4}}\)

I see clearly that \(\frac{5}{4}+\frac{1}{2}(1001)\) is

g__reate__r than \(\frac{5}{4}+\frac{1}{2}\sqrt{1000,000+\frac{1}{4}}\);

that the true answer would be

\( \frac{5}{4}+\frac{1}{2}\left(1000+\frac{1}{a}\right)\), \(\frac{1}{a}\) being some**[71r] ** f__raction__. We should then have

had,

\( n=\) nearest whole number above

\(\frac{5}{4}+500+\frac{1}{2a}\), instead of \(=\)

\( =\frac{5}{4}+500+\frac{1}{2}\)

But, since \(\frac{5}{4}\) is g__reater__ than \(1\),

the result __must__ exceed \(501\) even

if we neglected the \(\frac{1}{4}\) altogether;

and therefore __at any rate__ __\) n\) __

(the next whole number above

\( \frac{5}{4}+\frac{1}{2}\sqrt{Z+\frac{1}{4}}\) ), must be \(502\), &

\( n+1\) consequently \(=503\) .

I do not therefore see that the

fact of taking \(1001\) instead of

the real square part of \(Z+\frac{1}{4}\)

d__oes__ account for the discrepancy

in question . __ __

**[71v] ** I have now some question to

put respecting certain operations

with Incommensurables. Thanks to

your Treatise I think I understand

['this subject' inserted] pretty tolerably now. But there

are still one or two points of

P__ractical__ A__pplication__ which I

am [something crossed out] busy in working up

previous to leaving the subject

altogether as a direct study, &

which I find not quite plain

sailing. __ __

I have been writing out in

the __Mathematical Scrap-Book__,

a full explanation of the

operations with Incommensurables

analogous to those of Multiplication,

Division, Raising of Powers &c,

and a day or two ago I was**[72r] ** about completing it with that

analogous to the extraction of

Roots, when I found I did

not fully understand the

process, that is beyond the

consideration of o__ne__ Mean

Proportional. I have written

out & enclose my explanation

for o__ne__ Mean Proportional,

& my difficulty in the case

of t__wo__ or more Mean

Proportionals. __ __

Also, I wished now to return

to the passage, page 29, lines 8

and 9 from the top, (Trigonometry)

which first suggested to me the

necessity of studying the subject

of Incommensurables; in order

that I might see if I could**[72v] ** n__ow__ demonstrate the Proposition

of (46), for \(\theta\) and \(\sin\theta\) Incom=

=mensurable quantities. But I do

not find that I can. I believe

I understand the example referred

to in (4), the long & short

of which I understand to be

that if in the Right-Angled

Triangle [diagram in original] \(A\), \(B\), \(C\) are

Incommensurables, and \(V\) be any

given linear unit, then the

Ratio compounded of \(A:V\) and \(A:V\)

a__dded to__ the Ratio compounded of

\( B:V\) & \(B:V\), is equal to the

Ratio compounded of \(C:V\) and

\( C:V\) . __ __

With respect to the Ratio of an

Angle with it's [\textit{sic}] Sine, I began to**[73r] ** write it out as follows, after the

manner of pages 68 & 69 of the

Number & Magnitude:

\( \theta\) or \(\theta:1\) is the Ratio of \(\frac{AB}{AO}\)

\( \sin\theta\) or \(\sin\theta:1\) is the Ratio of \(\frac{BM}{AO}\),

\( AB:AO\), \(BM:AO\) being

Incommensurable Ratios, what

then does \(\frac{\theta}{\sin\theta}\) really mean? __ __

In the first place we may

consider it to mean

\(\theta\,\frac{1}{\sin\theta}:1\), or a Ratio

compounded of the Ratio \(\theta:1\) and

\( \frac{1}{\sin\theta}\), or compounded of the

Ratios \(AB:AO\) and \(AO:BM\) . __ __

But further than this I cannot

get, nor see my way at all.

I conclude that in Incommensurable

language, a __Ratio equal to \(1\) __ or

a __Ratio approximately to \(1\) __ can**[73v] ** only mean a __Ratio in which____the Magnitude constituting the____Antecedent__ is e__qual__ to the__Magnitude constituting the__

C__onsequent__, or is constantly__approaching an equality to it,__

and therefore that if we take

the above Ratio compounded

of \(AB:AO\) and \(AO:BM\), or

the Ratio \(AB:BM\), & prove

that \(AB\) constantly approaches

in equality to \(BM\), that is the

desired Demonstration. __ __

__ __I can only end by repeating

what I have often said before,

that I am very troublesome,

& only wish I could do you

any such service as you are

doing me. Yours most truly

A. A. L

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