- COUNTING AROUND THE WORLD 119
_ = = * h U 7 S?
Brahmiι? 4 t Μ
Indian (Gwalior)Sanskrlt-Devanagari (Indian)ß r* y
Wast Arabic (Gobar) East Arabic
º11th Century (Apices), æ i a. (, e «\ 8 9 · 1 * $ 4- ß 7 $9 ·
15th Century 16th Century (Diirer)FIGURE 2. Evolution of the Hindu-Arabic numerals from India to
modern Europe. ©Vandenhoeck & Ruprecht, from the book by
Karl Menninger, Zahlwort und Ziffer, 3rd ed., Gottingen, 1979.Some later Chinese numbering seems to reflect contact with India. Buddhism
entered China during the Han dynasty (202 BCE-220 CE), and Buddhist monks
had a fondness for large numbers. Unmistakable evidence of influence from India
can be found in the Suan Shu Chimeng by Zhu Shijie, who introduced names for
very large powers of 10, including the term "sand of the Ganges" for 10^96.
3.5. Greece and Rome. You are familiar with Roman numerals, since books still
use them to number pages in the front matter, and some clock faces still show the
hours in Roman numerals. Although these numerals were adequate for counting
and recording, you can well imagine that they were rather inefficient for any kind of
calculation. Adding, say MDCLIX to CCCIV, would take noticeably longer than
adding 1659 to 304, and the idea of multiplying or dividing these two numbers
seems almost too horrible to contemplate.^7 The Greek numeral system was hardly
better as far as calculation is concerned. The 24-letter Greek alphabet used today,
together with 3 older letters, provided symbols for 1,..., 9, 10,..., 90, 100,..., 900,
essentially the system used by the Egyptians. These symbols are shown in Fig. 3.
The 3 older letters were à (digamma) for 6 (now usually written as the letter sigma
in the form ò that it assumes at the end of a word),ö (qoppa) for 90, and \ (sampi)
for 900. When letters were used as numbers, they were usually given a prime, so
(^7) Nevertheless, the procedure for doing so can be learned in a fairly short time. Detlefsen and
co-authors (1975) analyzed the procedure and compared it to a "paper-and-pencil abacus."