POLYNESIAN AND MALAYAN. 161 
because I feel encouraged to hope that from the result of these consider- 
ations we may be able at some later period to return to the considera- 
tion of the basic /ua with more success and thereby interpolate its 
rational explanation. 
In these studies of the arithmetic of primitive man we have marked 
two well-defined stages. The Polynesian has attained to the decimal 
system, he has firmly grasped the whole initial requirement of the 
science of mathematics, he has the material equipment for all those 
specialized operations of number which are to be acquired from the 
decimal base by the gradual growth of knowledge stimulated by the 
advancing needs of life. ‘The only obstacle which withholds from the 
savage Polynesian the facility of the table of logarithms is that the need 
has not yet appeared in his life-condition to stimulate his mind to higher 
mathematical activity than the operations of addition and subtraction. 
In the introduction to this study of the numerals I have pointed 
out the considerable number of races of savage men within the oceanic 
district of my province who have not yet attained to the facility of the 
decimal system, whose numeration is quinary, whose finger count is 
limited to the digits of a single hand. In the dissection of the quinary 
system we are brought face to face with a yet more primitive concept 
of number and notation. Because it is primitive, because it is a work- 
ing of the dawning intelligence struggling with the comprehension of 
dimly perceived needs of life, where effort to comprehend involves some 
effort to reproduce and to confirm by speech such comprehension, we 
shall not look to find this early numeration restricted by the boundaries 
of any one family of speech. The vocables employed in such expression 
may vary widely between family and family; the principle remains con- 
stant. Weare engaged at first with psychology rather than linguistics. 
Long culture-ages anterior to the development of the decimal base 
which we possess in its completely acquired form in our Polynesian 
speech the quinary system is found as a halting stage of progress toward 
a system of notation. ‘The highest development of the quinary system 
rests in the possession of names for the units from one to five. ‘This is 
exhibited in the language of Aneityum where the numerals are: 1, e thi; 
2,€70; 3,€S€1]; 4,@ manowan; 5,1kman; and no words exist for num- 
ber beyond zkman, which I have shown to be a derivative from the 
eommon lima as hand and five. 
A stage yet more primitive is represented by a system which we 
may continue to call quinary because it is in possession of a word which 
expresses at its minimum connotation the sum of the digits of one hand. 
It differs from the perfect quinary as illustrated in Aneityum in the 
fact that instead of possessing four names for specific number below 
fiveithasbuttwo. We thus have an imperfect quinary of three terms. 
We may go yet lower in the scale and find a numerical system cf but 
two terms. The imperfect quinary of three terms becomes effectively 
