or keeping tally in any manner whatever.  They are no longer necessary, but they are so convenient and so useful that civilization can never dispense with them.  The use of the abacus, in the form of the ordinary numeral frame, has increased greatly within the past few years; and the time may come when the abacus in its proper form will again find in civilized countries a use as common as that of five centuries ago.

In the elaborate calculating machines of the present, such as are used by life insurance actuaries and others having difficult computations to make, we have the extreme of development in the direction of artificial aid to reckoning.  But instead of appearing merely as an extraneous aid to a defective intelligence, it now presents itself as a machine so complex that a high degree of intellectual power is required for the mere grasp of its construction and method of working.

CHAPTER II.

NUMBER SYSTEM LIMITS.

With respect to the limits to which the number systems of the various uncivilized races of the earth extend, recent anthropological research has developed many interesting facts.  In the case of the Chiquitos and a few other native races of Bolivia we found no distinct number sense at all, as far as could be judged from the absence, in their language, of numerals in the proper sense of the word.  How they indicated any number greater than one is a point still requiring investigation.  In all other known instances we find actual number systems, or what may for the sake of uniformity be dignified by that name.  In many cases, however, the numerals existing are so few, and the ability to count is so limited, that the term number system is really an entire misnomer.

Among the rudest tribes, those whose mode of living approaches most nearly to utter savagery, we find a certain uniformity of method.  The entire number system may consist of but two words, one and many; or of three words, one, two, many.  Or, the count may proceed to 3, 4, 5, 10, 20, or 100; passing always, or almost always, from the distinct numeral limit to the indefinite many or several, which serves for the expression of any number not readily grasped by the mind.  As a matter of fact, most races count as high as 10; but to this statement the exceptions are so numerous that they deserve examination in some detail.  In certain parts of the world, notably among the native races of South America, Australia, and many of the islands of Polynesia and Melanesia, a surprising paucity of numeral words has been observed.  The Encabellada of the Rio Napo have but two distinct numerals; tey, 1, and cayapa, 2.[20] The Chaco languages[21] of the Guaycuru stock are also notably poor in this respect.  In the Mbocobi dialect of this language the only native numerals are yna tvak, 1, and yfioaca, 2.  The Puris[22] count omi,