February 05, 2018 Monday
Bedtime Story
Continuing with Menabrea - 11
But it seems that Babbage himself may not
have completely worked out the solution of the mechanical implementation of the
process.
In the Note 5 that concerns this point Ada
writes:
“Not having had leisure to discuss with Mr.
Babbage the manner of introducing into his machine the combination of
algebraical signs, I do not pretend here to expose the method he uses for this
purpose; but I considered that I ought myself to supply the deficiency,
conceiving that this paper would have been imperfect if I had omitted to point
out one means that might be employed for resolving this essential part of the
problem in question.”
Let us return back to Menabrea’s treatise
after his having dealt the problem of assigning signs.
“The machine is not only capable of
executing those numerical calculations which depend on a given algebraical
formula, but it is also fitted for analytical calculations in which there are
one or several variables to be considered.
It must be assumed that the analytical
expression to be operated on can be developed according to powers of the
variable, or according to determinate functions of this same variable, such as
circular functions, for instance; and similarly for the result that is to be
attained.
If we then suppose that above the columns
of the store, we have inscribed the powers or the functions of the variable,
arranged according to whatever is the prescribed law of development, the
coefficients of those several terms may be respectively placed on the
corresponding column before each.
In this manner we shall have a
representation of an analytical development; and, supposing the position of the
several terms composing it to be invariable, the problem will be reduced to
that of calculating their coefficients according to the laws demanded by the
nature of the question.”
What Menabrea is emphasizing is that the
analytical engine is not just another simple mechanical calculator like the
previous machines, but a machine capable of solving complex mathematical
equations.
To explain this better Menabrea takes one
specific operation of multiplying (a +bx1) with (A + B Cos1
x) and goes into complete detail how the machine would do this operation.
As always first the primitive data is to be
fed into the columns, in this case writing down the values of x0, x1,
cos0x, cos1x above the columns V0, V1,
V2, V3.
The next step would also include writing
down the initial values of products (x0.cos0x), (x0.cos1x),
(x1.cos0x), (x1.cos1x) above the
columns V4, V5, V6, V7.
So any coefficients for x0, x1,
cos0x, cos1x will be passed on to the columns V0,
V1, V2, V3 by means of the mill.
The engine then with the intervention of
the punched cards will get the final product.
Menabrea explains each and every step that
the machine would perform mechanically using similar table that we had gone
through earlier.
I shall reproduce the table as a card in
the night to come so that if interested, you can see how the engine will
perform the operations for the function (a +bx1).(A + B Cos1
x) using columns of the store and mill with the punched cards directing which
operation to be performed how many times.
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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