January 22, 2018 Monday
Bedtime Story
Menabrea Explains Newton's Method of Finite Differences
We are continuing with the treatise of
Menabrea:
“Sketch of the Analytical Engine” and study
the column that he constructed to explain the principle of Newton’s method of
finite differences.
A B C
Column of First Second
Square Numbers Differences Differences
1
………….. 3
4 ………….. 2 b
……….. 5
a 9
………….. 2 d
………… 7
c 16 ………….. 2
………… 9
25
………….. 2
…………. 11
36
(Please ignore the alphabets placed on the
table as they are only there to point out the direction of summations as it
will be pointed out and be made clear later).
“From the mode in which the last two
columns B and C have been formed, it is easy to see, that if, for instance, we
desire to pass from the number 5 to the succeeding one 7, we must add to the
former the constant difference 2; similarly, if from the square number 9 we
would pass to the following one 16, we must add to the former the difference 7,
which difference is in other words the preceding difference 5, plus the
constant difference 2; or again, which comes to the same thing, to obtain 16 we
only have to add together the three numbers 2, 5, 9 placed obliquely in the direction
ab.
Similarly, we obtain the number 25 by
summing up the three numbers placed in the oblique direction dc: commencing by
the addition 2 +7, we have the first difference 9 consecutively to 7; adding 16
to the 9 we have the square 25.
We see then that the three numbers 2, 5, 9
being given, the whole series of successive square numbers, and that of their
first differences likewise may be obtained by means of simple additions.”
If you recall, we had gone over this method
of finite difference earlier, but I found this explanation of Menabrea quite
appealing.
So I decided to leave it intact and share
the pleasure with you.
“Now, to conceive how these operations may
be reproduced by a machine, suppose the latter to have three dials, designated
as A, B, C, on each of which are traced, say a thousand divisions, by way of
example, over which a needle shall pass.”
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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