Page:Popular Science Monthly Volume 26.djvu/462

446 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



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 before. 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 increase 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 direction, 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