96 Transactions. — Miscellaneous. 



decimal radix of weights and measures, universally applied. 

 That certain advantages accrue from this unity of method is 

 too obvious to admit of dispute ; but they are far from making 

 the system the perfect one its advocates represent it to be. 

 Its perfection we are expected to take for granted. But, in 

 view of what we are required to abandon in its favour, it is 

 not only reasonable but necessary to inquire, Is it not 

 possible to overrate the advantages of a universal decimal 

 system ? 



This question brings us to first principles, to the base of 

 the whole fabric — the system of notation and numeration in 

 world-wide use. Is this system itself perfect, either practi- 

 cally or scientifically? It is as much an "accidental in- 

 heritance" — to quote Mr. Mattieu Williams's contemptuous 

 designation of our national standards — as the standards them- 

 selves ; but the reformers did not go to the root of the matter 

 — they made no attempt to reform nor even, apparently, to 

 investigate it. The fact that our numeral signs are called 

 " the ten digits," and the further fact that in many languages 

 the words "five" and "hand" are related, or even identical, 

 associate our system with the primitive method of finger-tally 

 as unmistakably as the verb " calculate " embodies the fact 

 that pebbles were used in bygone days as instruments of 

 reckoning. But, widespread, almost universal, as decimal 

 calculation is and has been, ten has never been the sole radix 

 in common use. In practical concerns it is the geometrical or 

 tangible qualities of numbers with which we have to deal, and 

 the geometrical defects of ten as a radix are a standing dis- 

 qualification, with all its factitious advantages as the root of 

 our notation. Were it possible for the world to start afresh — 

 as the Frenchmen a hundred years ago dreamed of doing — 

 mathematicians would certainly discard ten as a root-number 

 and seek another, and that number would almost certainly be 

 twelve. Had such a proposition come from the French 

 scientists when the change was undertaken it would, even if 

 ultimately found impracticable, have had good scientific rea- 

 sons in its favour; in fact, the difficulties, great though they 

 are, might not then have proved insurmountable. 



Our arithmetical notation expresses actual numbers only 

 from 1 to 9. Thenceforth, both in speech and writing, all 

 numbers are indicated not as they are in themselves, but 

 in their relation to the radix ten. So with fractions. Only 

 those with a single numerator and denominator express 

 actual relations to unity ; beyond these the artificial element 

 makes its appearance. Mr. R. T. Barbour, writing last year 

 in an Australian review, remarked that, instead of vulgar 

 and decimal fractions, we should say vulgar and natural. 

 This is a good instance of a notational convention obscuring 



