## Folios 14-15: ADM to AAL

**[14r] ** My dear Lady Lovelace

Your inquiries were received just after I

had dispatched the receipt for Lord L's subscription to the Hist.Soc.

While I think of it (the Hist. Soc. reminds me) Nicolas Occam, or

Ockham, or of Ockham, who flourished about 1350, took his name I

rather think, from the same place as your little boy. He was a mathe-

matician, and one of the most remarkable English metaphysicians

before Locke. It is very likely that the late Ld King may on both

accounts, Ockham and metaphysics, [', Ockham and metaphysics,' inserted] have collected something about him, or that Lord Lovelace

may be in possession of something relating to him. If so, it can

certainly be made useful. His __logic__ was printed very early but

is so scarce that I have never been able to get sight of a copy.

Now to your queries. __Festina lente__, and above all never estimate

progress by the number of pages. You can hardly be a judge of the

progress you make, and I should say that it is more likely you

progress rapidly upon a point that makes you think for an

hour, than upon an hour's quick reading, even when you

feel satisfied. That which you say about the comparison of what

you do with what you see can be done was equally said by Newton

when he compared himself to a boy who had picked up a few pebbles

from the shore; and the last words of Laplace were 'Ce que nous

connaissons est peu de chose; ce que nous ignorons est immense'

So that you have respectable authority for supposing that you

will never get rid of that feeling; and it is no use trying

to catch the horizon.

**[14v] ** Peacocks examples will be of more use than any

book.

As to the functional equation. You must distinguish

in algebra questions of __quantity__ from questions of__form__. For example ''given \(x+8=10\), required \(x\),'' is

a question of quantity but ''given \(x\), an arbitrary

variable, required a function of \(x\) in which if the function

itself be substituted for \(x\), \(x\) shall be the result''

is a question of __form__, independent of __value__, for it is to

be true for all values of \(x\) . One solution is

\(\frac{1-x}{1+x}\)

for \(x\) substitute the function itself, this gives \(\frac{1+\frac{1-x}{1+x}}{1+\frac{1-x}{1+x}}\)

or \(\frac{1+x-(1-x)}{1+x+(1-x)}\) or \(\frac{2x}{2}\) or \(x\) .

Another solution is \(1-x\), since \(1-(1-x)\) is \(x\);

a third is \(-x\), since \(-(-x)\) is \(x\) .

Now suppose \(\varphi(x+y)=\varphi x+\varphi y\) \[\begin{array}{cl} x^2 \:{\small \text{does not satisfy this} }&\overline{x+y}\,\vert^2 \:{\small\text{is not}} \: x^2+y^2\\

ax \:{\small\text{does}}& a(x+y) \:\text{is}\: ax+ay\\ &~\\

\hline\end{array}\]

Let \(\varphi x=x^a\) \(\varphi(xy)=(xy)^a=x^ay^a=\varphi x\times\varphi y\)

\(\varphi x=a^x\) \(\varphi x\times\varphi y=a^x\times a^y=a^{x+y}=\varphi(x+y)\)

**[15r] ** \(\varphi x=ax+b\) \(\frac{\varphi x-\varphi y}{\varphi x-\varphi z}=\frac{ax+b-(ay+b)}{ax+b-(az+b)}=\frac{ax-ay}{ax-az}=\frac{x-y}{x-z}\)

A functional equation is one which has for its [something crossed out]__unknown__ the form proper to satisfy a certain condition

Example. What function of \(x\) is that which is not

altered by changing \(x\) into \(1-x\), let \(x\) be what it

may. Or, required \(\varphi x\) so that \(\varphi x=\varphi(1-x)\) \[\begin{align}{\rm One \:solution \:is}\: \varphi x&=1-2x+2x^2\\{\rm for} \:\varphi(1-x)&=1-2(1-x)+2(1-x)^2 \\&=1-2+2x+2-4x+2x^2\\&=1-2x+2x^2\:\: {\rm as\:before}\\ ~\\&\hline\end{align}\]

The equation of a curve means that equation

which must necessarily be true of the coordinates of

every point in it, and obviously depends upon 1. The

point chosen from which to measure coordinates 2.

The direction chosen for the coordinates. 3. The nature

and position of the curve. For example let the curve

be a circle, the point chosen its center, and the axes

of coordinates two lines at right angles. Let the

[diagram in original] radius be \(a\); then at every point \(x\) and

\( y\) must be the two sides of a right angled

triangle whose hypotenuse is \(a\); or

\(x^2+y^2=a^2\)

which being an equation true at every point**[15v] ** of the circle, is called the equation of the

circle__ __

My wife returns to day from Highgate.

M^{rs} Frend continues very comfortable, and

neither mends nor grows worse. I hope Ld Lovelace

and the little people are well. The old Ockham

will be a poor example for the young one, though

he was a monk, as I suppose. I would have

been nothing else had I lived in his day

Yours very truly

__ADeMorgan__

69 Gower St.

Monday Sep^{t }15/40

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