446 



THE POPULAR SCIENCE MONTHLY. 



brings us to the top of the column. Take the pawn back to its zero- 

 place, and lift the pawn in the next column, up one place, calling it 

 ten. Then we begin with the right-hand pawn again, two, and count 

 eleven, twelve, etc., to nineteen. Then bring the first pawn back to 



9 

 8 

 1 

 G 



5. 

 5 

 2 

 1 

 



Fig. 9.— The Nkw Universai, Abacus. 



zero, and, lifting the second pawn to another square, call it twenty, to 

 which we may add the units formed by raising the first pawn, as be- 

 fore. So we may go on with all the pawns, giving each successive 

 piece, as we go to the left, ten times the value of the preceding one. 



Our new abacus has the advantages that the value of its places in- 

 crease in the same direction as the written numbers they represent, 

 while the counters increase in arithmetical value as they are raised 

 higher. As arranged in the cut the board represents the number 

 0,369,258,147. The capacity of this table, which is now equal to 

 the expression of a thousand millions, may be indefinitely increased 

 by adding columns to the left. The capacity of the board may also 

 be changed by adding to it or subtracting from it in a vertical direc- 

 tion, whereby, instead of counting by tens, hundreds, and thousands, 

 we may count by dozens, grosses, and so on, or by multiples of eight, 

 six, four, two, or any other number. Every system of numeration is 

 thus founded on the employment of units of different orders, each of 

 which contains the preceding one a certain number of times, or, in 

 other words, upon a geometrical progression, the ratio of which is 

 called the base of the system. Aristotle observed that the number 

 four might take the place of ten, and Weigel, in 1687, published a 



