4 22 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 superseded the Roman, differs from those which preceded it in many 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 sym- 

 bols, 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 be- 

 longs. 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 " deci- 

 mal " 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 con- 

 sidered 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, natural- 

 ly, 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 cal- 

 culations with a scale of eight, to our great advantage in all arithmeti- 

 cal 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 



