January 21, 2018 Sunday
Bedtime Story
Menabrea Begins with Difference Engine
We are continuing with the treatise of
Menabrea:
“Sketch of the Analytical Engine”
“The illustrious inventor (Babbage) having
been kind enough to communicate to me some of his views on this subject during
the visit he made at Turin, I have, with his approbation, thrown together the impressions
that have left on my mind.
But the reader must not expect to find a
description of Mr. Babbage’s engine; the comprehension of this would entail
studies of much length; and I shall endeavor merely to give an insight into the
end proposed, and to develop the principles on which its attainment depends.
I must first premise that this engine is
entirely different from that of which there is a notice in the “Treatise on the
Economy of Machinery’, by the same author.
But as the latter gave rise to the engine
in question, I consider it will be a useful preliminary briefly to recall what
were Mr. Babbage’s first essays, and also the circumstances in which they
originated.
It is well known that the French government,
wishing to promote the extension of decimal system, had ordered the
construction of logarithmical and trigonometrical tables of enormous extent.
M. de Prony, who had been entrusted with
the direction of the undertaking, divided it into three sections, to each of
which was appointed a special class of persons.
In the first section the formulae were so
confined as to render them subservient to the purposes of numerical
calculation; in the second, these same formulae were calculated for the values
of the variable, selected at certain successive distance; and under the third
section, comprising about eighty individuals, who were most of them only
acquainted with the first two rules of arithmetic, the values which were
intermediate to those calculated by the second section were interpolated by means
of simple additions and subtractions.
An undertaking similar to that just
mentioned having been entered upon in England, Mr. Babbage conceived that the
operations performed under the third section might be executed by a machine;
and this idea he realized by means of mechanism, which has in part been put
together, and to which the name Difference Engine is applicable, on account of
the principle upon which its construction is founded.
To give some notion of this, it will
suffice to consider the series of whole square numbers 1, 4, 9, 16, 25, 36, 49,
64 etc.
By subtracting each of these from the
succeeding one, we obtain a new series, which we will name the series of First
Differences, consisting of the numbers 3, 5, 7, 9, 11, 13, 15 etc.
On subtracting from each of these the
previous ones, we obtain the Second Differences, which are all constant and
equal to 2.
We may represent this succession of
operations, and their results, in the following table.”
If you recall, I had explained to you Newton’s
method of finite differences in one of my bedtime stories which is the
mathematics behind the workings of difference engine and this is exactly what
Menabrea is explaining here.
I shall draw out the table in the nights to
come.
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:
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