plation of the fact that decimal notation grew out of the possession 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 affirming 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 particular 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-drawing 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 published in the Times it appears that Mr. Spencer's readers were allowed two days in which to think over this "surprising statement," 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 properties 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 characters 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 made more than fifty years ago. No one denies that much advantage might come