mistakes before they reached 7, and of Humboldt,[48] who says that a Chayma might be made to say that his age was either 18 or 60, has been duplicated by all investigators who have had actual experience among savage races.  Nor, on the other hand, is the development of a numeral system an infallible index of mental power, or of any real approach toward civilization.  A continued use of the trading and bargaining faculties must and does result in a familiarity with numbers sufficient to enable savages to perform unexpected feats in reckoning.  Among some of the West African tribes this has actually been found to be the case; and among the Yorubas of Abeokuta[49] the extraordinary saying, “You may seem very clever, but you can’t tell nine times nine,” shows how surprisingly this faculty has been developed, considering the general condition of savagery in which the tribe lived.  There can be no doubt that, in general, the growth of the number sense keeps pace with the growth of the intelligence in other respects.  But when it is remembered that the Tonga Islanders have numerals up to 100,000, and the Tembus, the Fingoes, the Pondos, and a dozen other South African tribes go as high as 1,000,000; and that Leigh Hunt never could learn the multiplication table, one must confess that this law occasionally presents to our consideration remarkable exceptions.

While considering the extent of the savage’s arithmetical knowledge, of his ability to count and to grasp the meaning of number, it may not be amiss to ask ourselves the question, what is the extent of the development of our own number sense?  To what limit can we absorb the idea of number, with a complete appreciation of the idea of the number of units involved in any written or spoken quantity?  Our perfect system of numeration enables us to express without difficulty any desired number, no matter how great or how small it be.  But how much of actually clear comprehension does the number thus expressed convey to the mind?  We say that one place is 100 miles from another; that A paid B 1000 dollars for a certain piece of property; that a given city contains 10,000 inhabitants; that 100,000 bushels of wheat were shipped from Duluth or Odessa on such a day; that 1,000,000 feet of lumber were destroyed by the fire of yesterday,—­and as we pass from the smallest to the largest of the numbers thus instanced, and from the largest on to those still larger, we repeat the question just asked; and we repeat it with a new sense of our own mental limitation.  The number 100 unquestionably stands for a distinct conception.  Perhaps the same may be said for 1000, though this could not be postulated with equal certainty.  But what of 10,000?  If that number of persons were gathered together into a single hall or amphitheatre, could an estimate be made by the average onlooker which would approximate with any degree of accuracy the size of the assembly?  Or if an observer were stationed at a certain point,