CALCULATING-MACHINES. 449 



be a matter of 18,446,744,073,709,551,615 removals, and will occupy five 

 million centuries. This prodigious number comes up again wben we 

 calculate the theory of the ring-puzzle of sixty-four rings. Accord- 

 ing 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 move- 

 ments was invented by Pascal in 1642. It is illustrated in Diderot's 

 " Encyclopoedia," and in some editions of Pascal's works. 



Every arithmetical machine is composed of four organs : the gen- 

 erator, the reproducer, the reverser, and the effacer. In Pascal's ap- 

 paratus 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 wath 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 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 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, per- 

 form any additions we wish. Multiplication is performed by successive 

 additions, but the process is slow and tedious, on account of the inefii- 

 ciency 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 count- 

 er 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 to 9. Each groove has corresponding to it a button 



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