February 16, 2018 Friday
Bedtime Story
Note G of Ada Lovelace - 3
We shall continue with the last and the
final Note G of Ada Lovelace.
“The engine may indeed be considered as including
the whole Calculus of Finite Differences; many of whose theorems would be
especially and beautifully fitted for development by its processes, and would
offer peculiarly interesting considerations.
We may mention, as an example the
calculation of the numbers of Bernoulli by means of the Differences of the
Zero.
We remarked in Note B., that any set of
columns on which numbers are inscribed, represents merely a general function of
the several quantities, until the special function has been impressed by means
of the Operation and Variable-cards.
Consequently, if instead of requiring the
value of the function, we require that of its integral, or of its differential
coefficient, we have merely to order whatever particular combination of the
ingredient quantities may constitute that integral or the coefficient.
In axn, for instance, instead of
the quantities
Being ordered to appear on V3 in the combination axn, they would be ordered to appear in that of
anxn-1
They would then stand thus:-
Similarly, we might have
,
the integral of axn.
An interesting example for following out
the processes of the engine would be such a form as
Or any other cases of integration by successive reductions, where an integral which contains an operation repeated n times can be made to depend upon another which contains the same n-1 or n-2 times, and so on until by continued reduction we arrive at a certain ultimate form, whose value has then to be determined.
The methods on Arbogast’s Calcul des
Derivations are peculiarly fitted for the notation and the processes of the
engine.
Likewise the whole of the Combinatorial
Analysis, which consist first in a purely numerical calculation of indices, and
secondly in the distribution and combination of the quantities according to
laws prescribed by these indices.
We will terminate these Notes by following
up in detail the steps through which the engine could compute the Numbers of Bernoulli,
this being (in the form in which we shall deduce it) a rather complicated example
of its powers.”
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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