he selection of authorities which have been consulted in the
preparation of this work, and to which reference is made in the following
pages, great care has been taken. Original sources have been drawn upon in
the majority of cases, and nearly all of these are the most recent
attainable. Whenever it has not been possible to cite original and recent
works, the author has quoted only such as are most standard and
trustworthy. In the choice of orthography of proper names and numeral
words, the forms have, in almost all cases, been written as they were
found, with no attempt to reduce them to a systematic English basis. In
many instances this would have been quite impossible; and, even if
possible, it would have been altogether unimportant. Hence the forms,
whether German, French, Italian, Spanish, or Danish in their transcription,
are left unchanged. Diacritical marks are omitted, however, since the
proper key could hardly be furnished in a work of this kind.

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