Page:Popular Science Monthly Volume 75.djvu/495

Rh involved, but the symbols are given by combination of the symbols for 1 to 9, with the symbol for zero. As our development will show the symbol for nothing was the great stumbling block in the development of a scientific method of writing the numerals. A place system to the base five would require only the addition of a symbol for zero to the symbols for 1, 2, 3 and 4. Leibnitz occupied himself with the binary system, as this required only two characters, one for unity and one for non-entity. To illustrate a binary place system the numbers from 1 to 16 are written, using only 1 and 0.

Three written as 11, means one, two and one unit. Nine written as 1001 represents one cube of two, no squares of two, no first powers of

two, and one unit. The construction of the arithmetic universe out of the single unit afforded Leibnitz some philosophical satisfaction in connection with his system of monads. All the operations of ordinary arithmetic are possible in this system. We catch a glimpse of our slight comprehension of the infinite totality of numbers in noting that any number that can be expressed with our ordinary ten digits can also be expressed with these two digits, and that even though we used a thousand digits we could add no new numbers. Doubtless it would afford Leibnitz some gratification to know that his binary system is used in modern mathematical analysis in certain delicate proofs. The study of these number systems is not wholly foreign to the history of the decimal system, as traces of the binary and quinary systems appear among primitive peoples.

Among the South Australian tribes the binary system of numeration is almost universal. This is undoubtedly due to the fact that the hands and feet and eyes and ears occur in groups of two in each normal individual. These tribes are not advanced enough to have a system of symbols; such a development would imply a degree of intelligence which would proceed to a higher and more convenient number base. The system is seen in their words; three is given as two and one,