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bers. Quantitative reasoning is the component of human thought that is easiest to simulate with a machine, and so thinking machines began with mathematics machines.


One of the earliest of these devices automatically paired objects with events in order to count them. In Roman times, military chariots had a mechanism mounted on the axle that dropped a stone into a cup each time a certain distance was covered, to keep track of total dis- tance traveled (in Latin, a small stone or pebble is a calculus,and the word remains in the names of two important mathematical techniques, differential and integral calculus).

Later the more sophisticated abacus helped people do arithmetic. With forerunners dating back to 500 BCE, its present forma wire frame on which are strung sliding beadsappeared in China around 1200 CE. The beads do not automatically perform calculations as they are moved (they do not accomplish carriesfrom the unitscolumn to the tenscolumn, and so on, as numbers are added), but only keep track of the operators arithmetic. Still, the device repre- sented a conceptual advance over counting pebbles because it intro- duced symbolic or positional notation; some beads carry a value of one,whereas others are valued at five”—an innovation that speeds up calculations and is echoed in modern computers.

The next step came much later, when seventeenth-century in- ventors (including two eminent mathematicians,the Frenchman Blaise Pascal and the German Gottfried Leibniz) developed automatic or semiautomatic mechanical calculators. One adding machine worked like a modern automobile odometer. Six interlocking rotating wheels, each numbered 0 to 9, represented the units,”“tens,and other col- umns of a six-digit number. Numbers were entered by turning the wheels.As values accumulated, for instance in the unitscolumn, and that wheel rotated through its whole range, it moved the adjoining tenswheel from 0 to 1, and so on. This took proper account of carries from one column to the next. Mechanical calculators contin-

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