be a matter of 18,446,744,073,709,551,615 removals, and will occupy five million centuries. This prodigious number comes up again when we calculate the theory of the ring-puzzle of sixty-four rings. According to an ancient Indian legend, the Brahmans took their turns day and night on the steps of the altar in the temple at Benares, to execute the readjustment of the sacred tower of Brahma, of sixty-four stories, of fine gold set with diamonds. When they had done, the tower and the Brahmans would fall together, and then would be the end of the world. The principle of this game corresponds with that which is the basis of the binary system. By increasing the number of pins, and slightly modifying the rules, we can make it represent other systems.
The first machine for executing calculations by mechanical movements was invented by Pascal in 1642. It is illustrated in Diderot's "Encyclopædia," and in some editions of Pascal's works.
Every arithmetical machine is composed of four organs: the generator, the reproducer, the reverser, and the effacer. In Pascal's apparatus and in Roth's most recent modification of it the generator is very rudimentary, being nothing but a rod held in the hand. The reproducer is composed of wheels with ten or twelve cogs, mounted on parallel axes, the first wheel on the right representing units, the second tens, the third hundreds, and so on. Each of the wheels bears one or more sets of figures from 0 to 9, and has in front of it a sheet of metal pierced with an opening through which a single figure can be seen at a time. The mechanism is so adjusted that each wheel after the first one advances by one division or tooth as the wheel to the right of it advances from 0 to 9. Over the circumference of each wheel a notch in the covering-plate allows the generator-rod to be applied to the teeth of the wheel to move it as many numbers as may be desired. We can thus, by successive pushings and readings, perform any additions we wish. Multiplication is performed by successive additions, but the process is slow and tedious, on account of the inefficiency of the generator. The object of the third organ, the reverser, is to change addition into subtraction, and multiplication into division. In Pascal's machine, each of the figure-bearing cylinders of the counter carries two scales, the reverse of each other, on parallel circles, the sum of the corresponding figures on which is always 9; so that the addition of four units of any order on one of the scales effects a subtraction of four units on the other scale. The object of the fourth organ, the effacer, is to bring all the numbers back to zero. In Roth's machine, 9 is made, by turning a button, to appear in the addition scale at all the openings; then the wheel is pushed around by the generator so as to add one, and appears in the place of the 9.
The Thomas arithmometer is a much more perfect and practicable machine. The generating apparatus is composed of a horizontal metallic plate, having parallel grooves, along which are written the figures from 0 to 9. Each groove has corresponding to it a button
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