150 6. CALCULATIONdepended on turning a crank a certain number of times in order to find a sum.
Leibniz used a variant of this machine with a removable set of wheels that would
multiply, provided that the user kept count of the number of times the crank was
turned.
Machines designed to calculate for a specific purpose continued to be built for
centuries, but all were doomed to be replaced by the general-purpose information
processor that has spread like wildfire around the world in the past two decades.
The first prefiguration of such a machine was Charles Babbage's difference engine,
designed in the 1830s but built only partially many decades later. Although only one
such machine seems to have been built, and only part of Babbage's more ambitious
analytical engine was constructed, the idea of a general-purpose computer that
could accept instructions and modify its operation in accordance with them was a
brilliant innovation. Unfortunately, the full implementation of this idea could not
be carried out by mechanical devices with moving parts. It needed the reliability of
electronics, first thermionic valves (vacuum tubes) and then transistors, to produce
the marvels of technology that we all use nowadays. That technology was developed
in Britain and the United States, greatly stimulated by the needs of code breaking
during World War II.
Meanwhile, fixed-purpose machines that could perform only a limited number
of set arithmetic operations, continued to be built and improved upon. In the late
days of World War II the "latest thing" in automated calculation was a machine
with many fragile moving parts.
Hand-held calculators, at first very limited, began to supplant the slide rule
during the 1970s. They were easier to use, infinitely faster, and much more accu-
rate than slide rules, and they soon became much cheaper. The only disadvantage
they had in comparison with slide rules was in durability.^6 A popular American
textbook of college algebra published in 1980 weighed the advantages and disad-
vantages of slide rules, calculators, and tables of logarithms, and summed up for
the jury in favor of using tables of logarithms, which had the cheapness of slide
rules but more accuracy. Even that recently it was an extremely refined and ex-
pensive hand calculator that had more than a few dozen memory cells. It was not
possible to foresee the explosion of computing power that was to result from the
development of methods of producing huge quantities of memory on tiny chips at
extremely low prices.
For several decades following World War II there were two types of calculat-
ing devices: Slide rules and cheap adding machines served the individual; more
expensive calculators and the early gigantic computers such as ENIAC (Electronic
Numerical Integrator And Computer) were used by large corporations. The over-
whelming penetration of modern computers into nearly every human activity, es-
pecially their use for word processing and graphics, is due to the vision of people
such as Charles Babbage, who realized that they must be able to use Boolean logic
in addition to their calculating capacity.
Two mathematicians figure prominently in the development of this vision of
the computer. One was Alan Turing (1912 1954), a British mathematician whose
1937 paper "On computable numbers" contained the idea of a universal computer
(^6) One might think that there would be no further market for slide rules. To the contrary, an entire
website is devoted to buying and selling them. The webmaster points out to the site visitor that
the computer on which the purchase is being made will be in a landfill 50 years hence, whereas
the slide rule will be only well broken in.