192 
REPORTS OF SOCIETIES. 
Aug., 1892. 
OXFORDSHIKE NATURAL HISTORY SOCIETY.—At a 
meeting of the Society on June 9th, in the geological lecture room at the 
Museum, Prof. A. H. Green taking the chair, Mr. G. Claridge Druce 
exhibited some interesting specimens of plants found for the first time 
in Berkshire—an intermediate form of Geum, a hybrid between urbanum 
and rivale\ a Lychnis hybrid between dioica and vespertina, and 
Trifolium subterraneum. Dr. E. B. Tylor gave a lecture on “ Primitive 
Arithmetic.” He began by explaining that all mankind, however 
low in civilisation, can count by the obvious use of fingers and toes. 
Thus certain terms used as numerals have come into existence in 
various barbaric languages, as where the words denoting 5, 10, and 20 
respectively, signify “ one hand,” “two hands,” and “ a whole man. ” 
In languages like our own, where no such meaning can be discerned, 
the fact of our counting by tens and sqores shows that our arithmetic 
began also by counting on the hands and feet, the primitive decimal 
apparatus. The next development in calculation is that of substituting 
small pieces of stick or stones for the fingers, a heap of ten such being 
expressed by a larger piece, and a hundred by one larger still. A 
curious discovery has lately been made regarding the form of the 
Egyptian symbols for one, ten, and a hundred, namely, that they 
represent pieces of cord of varying length, thus showing that the 
written sign was simply a copy of the actual objects formerly used as 
counters. After this, by the simple device of marking parallel 
columns on a board, it became possible to show by the column, in 
which a counter was placed, whether it stood for a unit, or for a group 
of ten, a hundred, Ac. Thus arose the abacus of the ancient Egyptians 
and Babylonians, which passed from them into Greek and Roman 
arithmetic. It was shown how numbers were added, subtracted, 
Ac., by counters on the Roman abacus, and that this instru¬ 
ment, though now unknown to us, is still in large use. The Chinese 
swan-pan, or calculating dish, with its balls strung on wires, is an 
evident copy of the classical abacus which found its way into the far 
East in the Middle Ages, and it still transacts the whole mercantile 
calculations of Chinese counting-houses. A similar instrument 
remains in active use in Russia, and on the retreat from Moscow 
a French officer brought it back to France and introduced it as a 
teaching instrument, the so-called “boullier” adopted in England, and 
it became the familiar ball-frame of our infant schools. The lecturer 
gave some interesting illustrations of the way in which calculations are 
made on the abacus, and worked some sums on the Chinese 
“calculating dish” in a method taught him by a Chinese merchant. 
He then proceeded to prove that the method of our “ciphering” 
is derived from the abacus or counting board. Figures or characters 
for numbers, such as the Hebrew or Greek letters in their alphabetical 
order, had been in use for ages, but the simplest arithmetical rules 
were extremely cumbrous and confusing, as anyone may see who will 
perform even a short multiplication with Greek or Roman numerals. 
When, however, such figures were written down in the columns of the 
counting-board—and it further occurred to the Hindu arithmeticians 
to use a dot or o as the^sign of an empty space—it became possible to 
transfer the quick and exact method of the counting-board to the 
written paper, the order in which figures were set down showing their 
value as units, tens, Ac., and the o corresponding to a vacant column. 
It shows well how the method of numeral figures grew out of the old 
counter-abacus to notice that till the last century a form of abacus 
calculation remained in use in England as the auditors’ method. 
Examples were given from the famous 17th century arithmetic of 
Robert Recorde, where the pupil is taught to reckon “ on the line ” by 
drawing pictures of the counters on the abacus and working with 
them, though he was also instructed in reckoning “with the pen,” 
that is to say, in the modern ciphering with numeral figures. 
