*THE POPULAR SCIENCE MONTHLY.*

Denary. | Tonal. | |

12 | .5 | .8 |

14 | .25 | .4 |

18 | .125 | .2 |

116 | .0625 | .1 |

132 | .03125 | .08 |

164 | .015625 | .04 |

1128 | .0078125 | .02 |

1256 | .00390625 | .01 |

The disadvantages also of this scale are many. It requires a multiplication-table for all numbers from one to sixteen. The mental labor of calculating would, therefore, be increased. The number of symbols required would not be quite so large, but the advantage from this source would be slight. It may be noted, in passing, that in the Hindoo notation, the smaller the radix the greater is the number of symbols required to express any number, but the easier the mental work of calculating. The binary scale, which has the smallest possible radix, is an extreme example under this rule. For instance, the lady who would be called 23 under the tonal system would have to confess to no less than 100011 summers were she living among people who counted with a binary scale! On the other hand, the larger the radix the less the number of symbols required, but the greater the difficulty of computation. Thus the tonal system expresses numbers more compendiously than the decimal, but the difficulty of its many tables would make the use of it a continual and severe strain upon the mind.

Its author proposed also a tonal unit of linear, superficial, and cubical measurement, a tonal watch, a tonal compass, tonal wet and dry measures, a tonal currency for the world, a tonal division of time, tonal thermometers and barometers, and tonal postage-stamps. There is not opportunity in this paper to describe these schemes.

But other numbers might be used as radices, though most of them will be found to be ill adapted to the purpose. The number three would furnish a system which would possess no merits whatever. Its scale would present only two digits, and the first ten numbers would be 1, 2, 10, 11, 12, 20, 21, 22, 100, 101. But three is an odd number, and the first bisection would result in an endless fraction. The same is true of all systems in which odd numbers form the radix.

The number four, however, would furnish a practicable scale. It is a square, and can be bisected indefinitely without producing an odd number except at unity. The notation would be simple, and the tables of combinations easy to learn. Theoretically, the scale would be an excellent one, but calculations in it would require much manual labor, and consequently be more tedious than similar computations in our system. There would be three digits and the first ten numbers would be 1, 2, 3, 10, 11, 12, 13, 20, 21, 22.

The five scale, which is in use to a very limited extent among savage