May 19, 2018 Saturday
Bedtime Story
Note E of Ada Lovelace - Part 10
Tonight we shall resume with the Note E of
Ada Lovelace from where we had left last night.
She is continuing to explain how the Engine
would compute the nth function using cycles of operations.
It is interesting to note that notation
used by Ada Lovelace for integral function is quite different from the one used
today.
The symbol she used is in modern mathematics
notation for summation function ∑ that can produce either one fixed sum as the
product or in cases of infinite sequence of values, a series.
“It might make the substitution either
wherever x occurs in the original (5), or it might similarly make it wherever x
occurs in the first function itself which is the equivalent of (5).
In some cases the former mode might be
best, and in others the latter.
Whichever is adopted, it must be understood
that the result is to appear arranged in a series following the law originally
prescribed for the development of the nth function.
This result constitutes the second
function; with which we are to proceed exactly as we did with the first
function, in order to obtain the third function, and so on, n-1 times, to
obtain the nth function.
We easily perceive that since every
successive function is arranged in a series following the same law, there would
(after the first function is obtained), because (for reasons on which we cannot
here enter) the first function might in many cases be developed through a set
of processes peculiar to itself, and not recurring for the remaining functions.
We have given but a very slight sketch of
the principal general steps which would be requisite for obtaining an nth
function of such a formula as (5).
The question is so exceedingly complicated,
that perhaps few persons can be expected to follow, to their own satisfaction,
so brief and general a statement as we are here restricted to on this subject.
Still it is a very important case as
regards the engine, and suggests ideas peculiar to itself, which we should
regret to pass wholly without allusion.
Nothing could be more interesting than to
follow out, in every detail, the solution by the engine of such a case as the
above; but the time, space and labor this would necessitate, could only suit a
very extensive work.
To return to the subject of cycles of
operations: some of the notation of the integral calculus lends itself very
aptly to express them: (2) might be thus written:-
(6)
(÷),∑(+1)p(x,
-) or (1),∑(+1)p(2, 3),
where p stands for the variable; (+ 1)p
for the function of the variable, that is, for Φp; and the limits are from 1 to
p, or from 0 to p-1, each increment being equal to unity.
Similarly, (4) would be, -
(7)
∑(+1)n{(÷),∑(+1)p(x,
-)}
the limits of n being from 1 to n, or from
0 to n-1,
(8)
or ∑(+1)n{(1),∑(+1)p(2, 3)}
Perhaps it may be thought that this
notation is merely a circuitous way of expressing what was more simply and as
effectually expressed before; and, in the above example, there may be some
truth in this.”
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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