Louis Charles Karpinski

“ It is entirely within the bounds of possibility that the first development of the Hindu, commonly called Arabic, place system was due to some oriental scholar who was familiar with the writings of these ancient Babylonians. Abundant testimony exists tending to prove the communication between Europe and the east. Having special symbols, such as existed in India for 1, 2, 3, 4, 5, 6, 7, 8 and 9 as early as the second century, acquaintance with this advancement of the Babylonians may have suggested the step to a decimal place system and the innovation of a zero. ”

“ 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. ”

“ Do the Germans take to philosophy more easily than other people because of some peculiarly philosophical bias of their language? These are speculative questions which can never be satisfactorily answered. It may, however, safely be asserted that the literature of a language is immediately dependent upon the written alphabet. ”

“ As literature and general culture are related to the alphabet and written language, so scientific advancement is related to the number system in use and to the system of writing numbers. A slight study of the Roman numerals gives the clue to the reason why the advancement along scientific lines lagged so far behind the general advancement achieved by the Roman peoples. The Greeks had a peculiar genius for arithmetical research,-but with them long division was a difficult operation, on account of the symbols. ”

“ Original work in science and mathematics did not come from Arabic hands, but the debt of civilization is none the less great as they were long the conservors of the learning of the Greeks and Hindus. The revival of Euclid was brought about by translations made from the Arabic; indeed many important Greek works in all the sciences have come to us only from Arabic sources. ”

“ The hypothesis of Aures is justified by the existence of the special symbols and names for 10, 100 and 1,000, and many other curious mixtures of decimal, duo-decimal and sexagesimal systems in the Babylonian measures. There is some comfort to be found in the reflection that ours is not the first civilization to struggle with diverse systems of notation and measurement.
The most striking fact of Babylonian mathematics is that they were in possession of a sixty place system. ”

“ Wide-spread use in India did not occur until towards the ninth century. With nations as with individuals, the complete significance of great idea is not achieved in a moment; even as this idea itself in its unfolding required the labor of master minds of many centuries, so the appreciation and application of this advance required centuries for its completion.
The intellectual awakening of the Arabs beginning about the middle of the eighth century, manifested itself in the appearance of numerous translations of Greek, Syrian and Hindu works. ”

“ Barbarians as they undoubtedly were at the period of their first conquests, the Arabs distinguished themselves by their desire for the further conquests of the science and literature of the subjugated peoples. ”

“ In order to comprehend the essence of our own number system, it is necessary to distinguish between a number system and a place system. A ten system involves having symbols for 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 groups of objects, respectively, and beyond that separate symbols for the successive powers of 10—100, 1,000, 10,000, 100,000.... ”

“ The numerals of the ancient Tamils, who, like the mountaineers of Appalachian America, conserve the traditions of a more remote civilization, show us that the Hindu peoples originally had special symbols not only for the first nine units, but also for the nine tens, the nine hundreds and even the nine thousands. ”

“ The Roman numerals like their prototypes in the Attic system of Greece and the more ancient Semitic systems, left no traces upon our current arithmetic. However, the Roman system of calculating upon a reckoning table was one of the vital factors in the development of the decimal place system. This system was not peculiarly Roman, as ancient Greek reckoning tables are found in several continental museums. The Chinese suan-pan, in popular use in Chinese laundries, is familiar to most readers. A similar instrument is found in Russian elementary schools. ”

“ The use of a special word to indicate absence of a unit is not hypothesis, but is found in verses in the ancient Indian book on astronomy, the Sourya-Siddhanta, and in numerous other ancient Hindu writings. ”

“ This longing for the infinite found expression in the construction of ever increasing numbers. Buddha calculates the number of grains of sand in a mile and shows how to compute the number in a sphere whose radius is the distance to one of the fixed stars. ”