730 POPULAR SCIENCE MONTHLY. 



plation of tlie fact that decimal notation grew out of the posses- 

 sion of a bundle of ten fingers, and the distinguished author might 

 have declared, in harmony with what he has said before concerning 

 time and circular measure, that it was in a very large degree 

 "dictated by Nature." But while aflBrming that time and the 

 compass have been so riveted upon us as to defy any attempt to 

 change, he leads gradually up to the conclusion that counting by 

 tens is not the only way of counting, and that Nature's group of 

 ten units doesn't mean anything in particular after all. Attention 

 is then called to the greater divisibility of the number twelve, 

 furnishing aliquot parts " which in sundry cases Nature insists 

 upon"; to the fact that we have twelve ounces in a pound 

 (we have also sixteen), twelve lines to the inch, twelve sacks to 

 the last (whatever that may be, De Morgan fails me here), 

 twelve things in a dozen, and that our multiplication table goes 

 up to twelve times twelve! While admitting that "these par- 

 ticular twelve divisions are undesirable, as being most of them 

 arbitrary and unrelated to one another," he maintains that they 

 " make it clear that a general system of twelfths is called for by 

 trading needs and industrial needs." No time for breath- draw- 

 ing is allowed after this astounding bit of logic before one is 

 confronted with the following : " It needs only a small alteration 

 in our method of numbering to make calculation by groups of 

 twelve exactly similar to calculation by groups of ten ; yielding 

 just the same facilities as those now supposed to belong only to 

 decimals. This seems a surprising statement, but I leave you to 

 think about it, and if you can not make out how it will be I will 

 explain presently." 



In the original chronology of these letters as they were pub- 

 lished in the Times it appears that Mr. Spencer's readers were 

 allowed two days in which to think over this " surprising state- 

 ment," and to recover from any condition into which they might 

 have been thrown by its announcement. I can not refrain from 

 saying just at this point that thousands of American schoolboys 

 would have needed only two minutes in which to explain how 

 it might be done, for it has been common knowledge among 

 arithmeticians from the earliest days of the study of the prop- 

 erties of numbers. In the third letter, however, the thing is gone 

 into at great length, and Mr. Spencer generously shares with the 

 reader the knowledge that it is only needed to invent two char- 

 acters to stand for ten and eleven, and then we should have a 

 system suited to a twelve-fingered race and greatly superior to 

 the decimal system now in use. It is useless to repeat that all 

 this is old, very old, and it is but justice to Mr. Spencer to repeat 

 that he prepared it from memoranda of his own roade more than 

 fifty years ago. No one denies that much advantage might come 



