The physical sciences, notably physics and astronomy, have come in the past few decades to depend heavily upon the various devices intended to make the solving of mathematical problems less arduous and more rapid than in the days, and with the paper and pencil methods, of the ancients. The methods, of course, have themselves seen little change; yet machines capable of more and more speed in numerical computing make it possible to calculate problems that before would have been solved only approximately if at all.

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