January 23, 2018 Tuesday
Bedtime Story
Continuing with Menabrea
We are continuing with the treatise of
Menabrea:
“Sketch of the Analytical Engine” that was
translated into French by Ada Lovelace and published along with her notes in
1842.
“The two dials, C, B, shall have in
addition a registering hammer, which is to give a number of strokes equal to
that of the divisions indicated by the needle.
For each stroke of the registering hammer
of the dial C, the needle B shall advance one division; similarly, the needle A
shall advance one division for every stroke of the registering hammer of the
dial B.
Such is the general disposition of the
mechanism.
This being understood, let us, at the
beginning of the series of operations that we wish to execute, place the needle
C on the division 2, the needle B on the division 5, and the needle A on the
division 9.
Let us allow the hammer of the dial C to strike;
it will strike twice, and at the same time the needle B will pass over two
divisions.
The latter will then indicate the number 7,
which succeeds the number 5 in the column of first differences.
If we now permit the hammer of the dial B
to strike in its turn, it will strike seven times, during which the needle A
will advance seven divisions; these added to the nine already marked by it will
give the number 16, which is the square number consecutive to 9.
If we now recommence these operations, beginning
with the needle C, which is always to be left on the division 2, we shall
perceive that by repeating them indefinitely, we may successfully reproduce the
series of whole square numbers by means of a very simple mechanism.
The theorem on which is based the
construction of the machine we have just been describing, is a particular case
of the following more general theorem: that if in any polynomial whatever, the
highest power of those variable is m, this same variable be increased by equal
degrees; the corresponding values of the polynomial then calculated, and the
first, second, third etc. differences of these be taken (as for the preceding
series of squares); the mth differences will all be equal to each
other.
So that, in order to produce the series of
values of the polynomial by means of a machine analogous to the one above
described, it is sufficient that there be (m+1) dials, having the mutual
relations we have indicated.
As the differences may be either positive
or negative, the machine will have a contrivance for either advancing or
retrograding each needle, according as the number to be algebraically added may
have the sign plus or minus.
If from a polynomial we pass to a series
having an infinite number of terms, arranged according to the ascending powers
of the variable, it would at first appear, that in order to apply the machine
to the calculation of the function represented by such a series, the mechanism
must include an infinite number of dials, which would in fact render the thing impossible.
But in many cases the difficulty will
disappear, if we observe that for a great number of functions the series which
represent them may be rendered convergent; so that, according to the degree of
approximation desired, we may limit ourselves to the calculation of a certain
number of terms of the series, neglecting the rest.”
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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