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Brian Randell, in his article The COLOSSUS, explains how computing ushered in the 20th Century, by the development of the Colossus computer. He cites a passage from the Passages from the Life of a Philosopher, illustrating how Charles Babbage described the Analytical Engine :

That the whole of the conditions which enable a finite machine to make calculations of unlimited extent are fulfilled in the Analytical Engine.

I couldn't get how calculations of unlimited extent can be performed by the Analytical engine as described by Babbage. An example would be helpful too.

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  • $\begingroup$ If you have extra questions then please post them as separate questions. $\endgroup$
    – Ubi.B
    Commented Jan 12, 2018 at 7:13
  • $\begingroup$ A small point: you wrote about calculations being fulfilled, but what Babbage was saying was that certain conditions were fulfilled, namely those enabling a finite machine to perform calculations of unlimited extent. $\endgroup$
    – PJTraill
    Commented May 13, 2018 at 21:06
  • $\begingroup$ @PJTraill: Yeah that's right. I first thought that he was mentioning about the engines power but as you said he was referring to the device flexibility. $\endgroup$
    – justin
    Commented May 15, 2018 at 14:45

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tl;dr- The bit about "unlimited extent" is referring to being able to perform general-purpose computation rather than infinite computation. They probably thought that it was trippy for a finite device to be able to pursue any sort of computation.


It seems that Brian Randell was referring to how the Analytical Engine was meant to be a general-computing device that could, in principle, perform any sort of computation.

The fuller quote discusses the notions of Turing completeness and "unlimited extent" as two different perspectives on this topic:

Turing thus was the first to arrive at an understanding of the universal nature of a (conceptual) digital computer which matches and indeed surpasses the philosophic understanding that I believe Babbage had attained, a century earlier, of the universality of his planned (mechanical) Analytical Engine. Babbage’s phrasing was “that the whole of the conditions which enable a finite machine to make calculations of unlimited extent are fulfilled in the Analytical Engine” [2, p.28], where the term “extent” encompassed both the amount and accuracy of the data to be processed, and the length and logical complexity of the algorithm to be performed. Central to the Universal Turing Machine is the idea of having data, and input data in particular, represent a program (called a “table” in Turing’s paper). A hitherto little known manuscript by Babbage [1] which has recently been published for the first time makes it clear that Babbage had reached an almost similar level of understanding. In the manuscript he points out that a fully detailed sequence of “formula cards” might be prepared by the Analytical Engine from a more abstract sequence. However, this is not to say that Turing’s work was in any way derived from Babbage’s – indeed there is no evidence that Turing even knew of Babbage at this time, but this topic will be returned to later.

-"The COLOSSUS", Brian Randell, page 4

This seems to be referring to the property of the Analytical Engine to perform general-purpose computation, which the above quote refers to as "universal nature" and "universality".

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  • $\begingroup$ Could you show me an example to show how Babbage's Analytic engine, made calculations of unlimited extent. $\endgroup$
    – justin
    Commented Jan 5, 2018 at 9:06
  • $\begingroup$ @justin Wikipedia describes the argument for Turing completeness (unlimited extent). Basically it sounds like Babbage didn't explicitly write down the instruction set but rather example states programs might go through, so Turing completeness was inferred rather than proven. Their inference seems pretty reasonable, but the correctness of their claim to "unlimited extent" would seem to hinge on whether or not their inferences were on-target. $\endgroup$
    – Nat
    Commented Jan 5, 2018 at 10:32
  • $\begingroup$ @justin Looks like some later work attempted to construct an instruction set for it. I haven't checked that out yet, but if we assume that it's valid, we could use that as a basis for making an argument for Turing completeness. $\endgroup$
    – Nat
    Commented Jan 5, 2018 at 10:38
  • $\begingroup$ To be honest I still couldn't get how did Analytic engine made calculations of unlimited extent. I know that the Analytic engine made by Babbage could perform operations that range from addition to division but how can his machine make calculations of unlimited extent in a single run when they are just about a finite number of parts listed here. $\endgroup$
    – justin
    Commented Jan 8, 2018 at 5:10
  • $\begingroup$ @Justin I have already covered that part. Unlimited extent is possible on theory. If you have unlimited memory then you can certainly run and print infinite sequence of number. Also, you need to see that machine shouldn't break in between if you are feeding unlimited memory to it and running instruction for infinite time. Actually, that frequently happened. Machine use to break even in limited calculation. Some or an other tooth wheel use to break or had a malfunction. $\endgroup$
    – Ubi.B
    Commented Jan 10, 2018 at 4:57
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By conceiving the idea of how the analytical engine works this statement will become much understandable.

Working of the Analytical Engine:
Analytical Engine mainly contains two parts. The Column and the Mill which are equivalent to the modern concepts of Registers and the CPU. The Analytical Engine will accept any input function ranging from basic arithmetic operations (+,-,*,/), extraction of root, exponential, polynomials, circular functions, log and even can able to solve simultaneous equations. Most of the complex function can be approximated by the polynomial functions. It will solve these input functions through intermediate steps which will be provided to it through cards (mechanical way of coding). It accepts three kind of cards

  1. operators (the operations need to be performed sequentially)
  2. Variables (the column number on which the above mentioned operations need to store the value)
  3. Constants ( the input constants of the equation is mentioned with on which column it needs to store the value).

The two basic principles of Analytical engine are that every arithmetical calculation ultimately depends on four principal operations — addition, sub- traction, multiplication, and division ; the second, in the possi- bility of reducing every analytical calculation to that of the co- efficients for the several terms of a series. If this last principle be true, all the operations of analysis come within the domain of the engine.

Thus, with finite operations from the cards it can achieve computation to the unlimited extent.

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