Louis Charles Karpinski

“ It seems strange that such a labor-saving device should have struggled for nearly a thousand years after its system of place value was perfected before it replaced such crude notations as the one that the Roman conqueror made substantially universal in Europe. Such, however, is the case, and there is probably no one who has not at least some slight passing interest in the story of this struggle. To the mathematician and the student of civilization the interest is generally a deep one; to the teacher of the elements of knowledge the interest may be less marked, but nevertheless it is real ”

“ 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. ”

“ The Egyptians, for example, had three systems of writing, with a numerical notation for each; the Greeks had two well-defined sets of numerals, and the Roman symbols for number changed more or less from century to century. Even to-day the number of methods of expressing numerical concepts is much greater than one would believe before making a study of the subject, for the idea that our common numerals are universal is far from being correct. It will be well, then, to think of the numerals that we still commonly call Arabic, as only one of many systems in use just before the Christian era. ”

“ 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. ”

“ It is therefore reasonable to conclude that in the Middle Ages, as in the time of Boethius, it was a simple matter for any inquiring scholar to become acquainted with such numerals of the Orient as merchants may have used for warehouse or price marks. And the fact that Gerbert seems to have known only the forms of the simplest of these, not comprehending their full significance, seems to prove that he picked them up in just this way. ”

“ 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. ”

“ When, therefore, we consider the rise of the numerals in the land of the Sindhu, [51] it must be understood that it is only the large movement that is meant, and that there must further be considered the numerous possible sources outside of India itself and long anterior to the first prominent appearance of the number symbols.
No one attempts to examine any detail in the history of ancient India without being struck with the great dearth of reliable material. ”

“ Other tribes of northern Africa have borrowed the decimal notation from their civilized neighbors.
Without a single exception the ancient civilized peoples of all the world—Egyptians, Babylonians, Hebrews, Chinese, Greeks, Romans, Hindus—all used the decimal systems. Such striking uniformity among all the races of the earth requires a natural origin for the decimal number base. As Herodotus first suggested, man counts by tens because he has ten fingers. While there may be logical grounds for the advocates of a duo-decimal system, the ten system is too deep-rooted to be dislodged. ”

“ As a matter of fact there is abundant reason for believing that Hindu numerals would naturally have been known to the Arabs, and even along every trade route to the remote west, long before the zero entered to make their place-value possible, and that the characters, the methods of calculating, the improvements that took place from time to time, the zero when it appeared, and the customs as to solving business problems, would all have been made known from generation to generation along these same trade routes from the Orient to the Occident. ”

“ 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. ”

“ He was the most celebrated mathematician of his time, either in the East or West, writing treatises on arithmetic, the sundial, the astrolabe, chronology, geometry, and algebra, and giving through the Latin transliteration of his name, algoritmi, the name of algorism to the early arithmetics using the new Hindu numerals. ”

“ 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. ”

“ A human mind cannot readily create simple forms that are absolutely new; what it fashions will naturally resemble what other minds have fashioned, or what it has known through hearsay or through sight. A circle is one of the world's common stock of figures, and that it should mean twenty in Phœnicia and in India is hardly more surprising than that it signified ten at one time in Babylon. [124] It is therefore quite probable that an extraneous origin cannot be found for the very sufficient reason that none exists. ”

“ 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. ”

“ So little sympathy have the people with any save those of their own caste that a general literature is wholly lacking, and it is only in the observations of strangers that any all-round view of scientific progress is to be found. There is evidence that primary schools [15] existed in earliest times, and of the seventy-two recognized sciences writing and arithmetic were the most prized. [53] In the Vedic period, say from 2000 to 1400 B.C., there was the same attention to astronomy that was found in the earlier civilizations of Babylon, China, and Egypt, a fact attested by the Vedas themselves. ”

“ 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. ”

“ The introduction of myriads as in the Greek, and thousands as in Arabic and in modern numeration, is really a step away from a decimal scheme. So in the numbers below one hundred, in English, eleven and twelve are out of harmony with the rest of the -teens, while the naming of all the numbers between ten and twenty is not analogous to the naming of the numbers above twenty. To conform to our written system we should have ten-one, ten-two, ten-three, and so on, as we have twenty-one, twenty-two, and the like. The Sanskrit is consistent, the units, however, preceding the tens and hundreds. ”

“ 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. ”

“ CHAPTER II EARLY HINDU FORMS WITH NO PLACE VALUE
While it is generally conceded that the scientific development of astronomy among the Hindus towards the beginning of the Christian era rested upon Greek [42] or Chinese [43] sources, yet their ancient literature testifies to a high state of civilization, and to a considerable advance in sciences, in philosophy, and along literary lines, long before the golden age of Greece. From the earliest times even up to the present day the Hindu has been wont to put his thought into rhythmic form. ”

“ As to the Arabs, it is a mistake to feel that their activities began with Mohammed. Commerce had always been held in honor by them, and the Qoreish [330] had annually for many generations sent caravans bearing the spices and textiles of Yemen to the shores of the Mediterranean. In the fifth century they traded by sea with India and even with China, and Ḥira was an emporium for the wares of the East, [331] so that any numeral system of any part of the trading world could hardly have remained isolated. ”

“ On the other hand, we must bear in mind the fact that a traveler in Germany in the year 1700 would probably have heard or seen nothing of decimal fractions, although these were perfected a century before that date. The élite of the mathematicians may have known the zero even in Āryabhaṭa's time, while the merchants and the common people may not have grasped the significance of the novelty until a long time after. On the whole, the evidence seems to point to the west coast of India as the region where the complete system was first seen. ”

“ As it then existed the system was no better than many others, it was of late origin, it contained no zero, it was cumbersome and little used, [2] and it had no particular promise. Not until centuries later did the system have any standing in the world of business and science; and had the place value which now characterizes it, and which requires a zero, been worked out in Greece, we might have been using Greek numerals to-day instead of the ones with which we are familiar. ”

“ Only as Western civilization is making its way into the commercial life of the East do the numerals as used by us find place, save as the Sanskrit forms appear in parts of India. It is therefore with surprise that the student of mathematics comes to realize how modern are these forms so common in the West, how limited is their use even at the present time, and how slow the world has been and is in adopting such a simple device as the Hindu-Arabic numerals. ”

“ Foremost among these writers was the great master of the golden age of Bagdad, one of the first of the Arab writers to collect the mathematical classics of both the East and the West, preserving them and finally passing them on to awakening Europe. This man was Moḥammed the Son of Moses, from Khowārezm, or, more after the manner of the Arab, Moḥammed ibn Mūsā al-Khowārazmī, [8] a man of great [5] learning and one to whom the world is much indebted for its present knowledge of algebra [9] and of arithmetic. Of him there will often be occasion to speak ”