beyond which
they do not count. The Damaras of South Africa only count to three; and
Mr. Galton gives a curious description of how one of them was hopelessly
puzzled when he had sold two sheep for two sticks of tobacco each, and
received four sticks in payment. He could only find out that he was
correctly paid by taking two sticks and then giving one sheep, then
receiving two sticks more and giving the other sheep. Even the
comparatively intellectual Zulus can only count up to ten by using the
hands and fingers. The Ahts of North-West America count in nearly the
same manner, and most of the tribes of South America are no further
advanced.[230] The Kaffirs have great herds of cattle, and if one is
lost they miss it immediately, but this is not by counting, but by
noticing the absence of one they know; just as in a large family or a
school a boy is missed without going through the process of counting.
Somewhat higher races, as the Esquimaux, can count up to twenty by using
the hands and the feet; and other races get even further than this by
saying "one man" for twenty, "two men" for forty, and so on, equivalent
to our rural mode of reckoning by scores. From the fact that so many of
the existing savage races can only count to four or five, Sir John
Lubbock thinks it improbable that our earliest ancestors could have
counted as high as ten.[231]

When we turn to the more civilised races, we find the use of numbers
and the art of counting greatly extended. Even the Tongas of the South
Sea islands are said to have been able to count as high as 100,000. But
mere counting does not imply either the possession or the use of
anything that can be really called the mathematical faculty, the
exercise of which in any broad sense has only been possible since the
introduction of the decimal notation. The Greeks, the Romans, the
Egyptians, the Jews, and the Chinese had all such cumbrous systems, that
anything like a science of arithmetic, beyond very simple operations,
was impossible; and the Roman system, by which the year 1888 would be
written MDCCCLXXXVIII, was that in common use in Europe down to the
fourteenth or fifteenth centuries, and even much later in some places.
Algebra, which was invented by the Hindoos, from whom also came the
decimal notation, was not introduced into Europe till the thirteenth
century, although the Greeks had some acquaintance with it; and it
reached Western Europe from Italy only in the sixteenth century.