With the above exceptions, this study will, it is hoped, be found to be quite complete; and as the subject here investigated has never before been treated in any thorough and comprehensive manner, it is hoped that this book may be found helpful. The collections of numeral systems illustrating the use of the binary, the quinary, and other number systems, are, taken together, believed to be the most extensive now existing in any language. Only the cardinal numerals have been considered. The ordinals present no marked peculiarities which would, in a work of this kind, render a separate discussion necessary. Accordingly they have, though with some reluctance, been omitted entirely. Sincere thanks are due to those who have assisted the author in the preparation of his materials. Especial acknowledgment should be made to Horatio Hale, Dr. D.G. Brinton, Frank Hamilton Cushing, and Dr. A.F. Chamberlain. WORCESTER, MASS., Nov. 12, 1895. CONTENTS. Chapter I. Counting 1 Chapter II. Number System Limits 21 Chapter III. Origin of Number Words 37 Chapter IV. Origin of Number Words (_continued_) 74 Chapter V. Miscellaneous Number Bases 100 Chapter VI. The Quinary System 134 Chapter VII. The Vigesimal System 176 * * * * * Index 211 THE NUMBER CONCEPT: ITS ORIGIN AND DEVELOPMENT. CHAPTER I. COUNTING. Among the speculative questions which arise in connection with the study of arithmetic from a historical standpoint, the origin of number is one that has provoked much lively discussion, and has led to a great amount of learned research among the primitive and savage languages of the human race. A few simple considerations will, however, show that such research must necessarily leave this question entirely unsettled, and will indicate clearly that it is, from the very nature of things, a question to which no definite and final answer can be given.