*THE POPULAR SCIENCE MONTHLY.*

systems need no description. The Hindoo notation now in use, which superseded the Roman, differs from those which preceded it in many respects, all of which are to its advantage. It requires only nine symbols, together with the dot or zero. Its chief excellence, however, arises from its principle of "local values." Each symbol has two values: one intrinsic, and the other local. The intrinsic value is that which the symbol has when it occupies the unit place. Thus, the nine significant digits express the numbers from one to nine. The local value is that which a digit derives from its position in the number to which it belongs. Thus thirty is expressed by 30, the 3 by being thrown into the second place obtaining a positional value which is ten times greater than its absolute value. Since this increase is tenfold, the system is called decimal. If the figures are removed one place farther to the left, their value is again increased tenfold, and a like increase obtains for each removal. If removed to the right, their value is decreased ten times for each place of removal. The number by which the positional value changes is termed the root or radix of the system. It is one of the advantages of this notation that it enables us to express numbers with great ease, but its principal advantage appears in the simplicity which it gives to computations of all kinds. Another peculiar merit appears when fractions are involved, in the facility with which "decimal" fractions may be used.

But the merit of the Hindoo notation does not arise from the fact that it is decimal, but from its system of "local values." Ten was used as its radix simply because that number happened to be the basis of numeration universally in use when the notation was invented. Any other radix might have been used, since the principle of local values may be applied to all numbers. It has not, however, been popularly applied except to the number ten. Discussion, however, has arisen from time to time as to the merits of the number ten in comparison with other numbers. It appears to be admitted by all who have considered the matter that ten is far from being the best number for the purpose. It would be remarkable if it were. It came into use not on account of any intrinsic excellence, but because the number of the fingers is ten. For no other reason, ten was the number of objects placed in each group when the device of grouping came into use; then, naturally, it became the basis of the early systems of notation, and when the Hindoo notation was invented, it was taken for the radix of that system. It evidently was not selected on account of its fitness for the position. Were we eight-fingered, we should without doubt perform all our calculations with a scale of eight, to our great advantage in all arithmetical work.

Ten is, theoretically, ill suited for the radix of a system of notation, because it permits of only one bisection. The half of it is five, an odd number. It also is incapable of any other division. On account of these defects the system is ill adapted to the operations of the shop