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zeros. So finding the roots of an equation reduces to detecting the all-zero state in the results
column and counting the number of times the handle was cranked, which gives the value of x.
If two consecutive results straddle zero (i.e., if the result passes through zero without stopping)
a root will be signalled by a sign change which can also be detected automatically.
At first the machine automatically rang a bell on the mechanical detection of the all-zero
state to signal to the superintendent to stop the machine. Later designs incorporated mecha-
nisms for halting the machine automatically. If there were multiple roots then the operator
would keep cranking, so cycling the engine to find the remaining roots. If there were no
solutions then the machine continued ad infinitum. Babbage wrote explicitly of the machine
halting when it found a root: halting, as a criterion of solvability, is explicit in these earliest
reflections.
Pre-echoes of Turing’s 1936 breakthrough paper ‘On computable numbers’ are unmistak-
able. Whether or not a machine would properly complete its calculation was later called the
‘halting problem’ which, though not explicitly referred to in this way by Turing, is implied in
his 1936 paper and is intimately associated with him. For those interpreting the behaviour of
Turing’s ‘circular machine’, halting became a central logical determinant as to whether or not
machines could decide if a certain class of problems was soluble. Babbage himself did not claim
any special theoretical significance for the halting criterion. For him it was a practical matter:
if there were a solution, the machine would halt; if there were no solutions, the machine would
grind on indefinitely.
Machine computation offered the prospect of solutions to mathematical problems which
had until then had resisted solution by conventional formal analysis. There were equations, for
example, for which there were no known theoretical solutions. By systematically cycling the
machine to produce each next value of the expression, solutions could be found by detecting
the all-zero state even though the roots that produced this result could not (yet) be found by
mathematicians. Having a computational rule could achieve results that had evaded formal
analysis.
In addition to instances of solving the hitherto unsolved, machine computation offered
a more mundane prospect. There are many important ‘series’ in mathematics. These are
sequences of values defined by a general formula. For example, if you wished to know the value
of the 350th term in the series there was no simple way of finding this unless you had a general
formula for the nth term. In some instances no such expression was known. By cycling the
engine through 350 values the required value could be found, something that was impractical,
or at least prohibitively tedious, to do by hand.
For Babbage, his calculating engines represented a new technology for mathematics, and
machine computation as systematic method offered new prospects for solving certain math-
ematical problems.
The Difference Engine embodied mathematical rule in mechanism, and Babbage’s 1822
model, driven by a falling weight, was the first physical machine to execute a computational
algorithm without human intervention. The 1832 demonstration piece was driven not by a fall-
ing weight but by a manual crank at the top of the machine. In both machines the operator did
not need to understand the mechanism, nor the mathematical principle on which it was based,
to achieve useful results. The steps of the computational algorithm were no longer directed by
human intelligence, but by the internal rules embodied in the mechanism and automatically
executed. By exerting physical energy the operator could achieve results that, up to that point
in time, could be achieved only by mental effort—by thinking. The idea that the machine was