ntroduction into Europe. There are two general theories as to this introduction. The first is that they were carried by the Moors

o Boethius (c. 500 A.D.), and which he would naturally have received, if at all, from these same Neo-Pythagoreans or from the sources from which they derived them. Furthermore, Woepcke points out that the Arabs on entering Spain (711 A.D.) would naturally have followed their custom of adopting for the computation of taxes the numerical systems of the countries they conquered,[248] so that the numerals brought from Spain to Italy, not having undergone the same modifications as those of the Eastern Arab empire, would have differed,

ell-known writer. Bubnov holds that the forms first found in Europe are derived from ancient symbols used on the abacus, but

at was the nature of these Spanish numerals, and how we

btless that these numerals were written on the dust abacus, this plan being distinct from the counter method of representing numbers. It is also worthy of note that Al-Bīrūnī states that the Hindus often performed numerical computations in the sand. The term is found as early as c. 950, in the vers

ve a character indicates tens, two dots hundreds, and so on, meaning 50, and meaning 5000. It has been suggested that possibly these dots, sprinkled like dust above the numerals, gave rise to the word ?obār,[254] but this is not at all probable. This system o

ould have adopted such a scheme, since their diacritical marks would suggest it, not to speak of the possible influence of the Greek accents in the Hellenic number system. When we consider, however, that the dot is found for zero in the Bakh?ālī manuscript,[257]

ebrew scholars of Northern Africa in the tenth century knew these numerals as Indian forms, for a commentary on the Sēfer Ye?īrāh by Abū Sahl ibn Tamim (probably composed at Kairwān, c. 950) speaks of "the Indian arithmetic known under the name of ?obār or dust calculation."[261] All this suggests that the Arabs may very likely have known the ?obār forms before the numera

he people of India. Notwithstanding the importance of this reference for the early history of the numerals, it has not been mentioned by previous writers on this subject. The numeral forms given are those

more closely than the Arab numerals do. They vari

[2

[2

[2

[2

[2

[2

forms. They would as appropriately be used in connection with the Hindu forms, and the evidence of a relation of the first three with all these systems is apparent. The only further resemblance is in

nd Hierat

, where the forms are from a manuscript written at Shiraz about 970 A.D., and in which some western Arabic forms, e.g. for 2, are also used. Probably most significant of all is the fact that the ?obār numerals as given by Sacy are all, with the exception o

Could he have known these numerals? (3) Is there any positive or strong circumst

of two such powerful families allowed him to move in the highest circles.[278] Standing strictly for the right, and against all iniquity at court, he became the object of hatred on the part of all the unscrupulous element near the throne, and his bold defense of the ex-consul Albinus, unjustly accused of treason, led to his imprisonment at Pavia[279] and his execution in 524.[280] Not many generations after his death, the period being one in which historical criticism was at i

stronomy of Ptolemy are read by those of Italy, and the arithmetic of Nicomachus and the geometry of Euclid are known to those of the West."[286] Founder of the medieval scholasticism, distingui

. The numerals had existed, without the zero, for several centuries; they had been well known in India; there had been a continued interchange of thought between the East and West; and warriors, ambassadors, scholars, and the restless trader, all

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 Occiden

nnected with the ?obār numerals, the Boethius question, Gerbert, Leonardo of Pisa, and ot

, and the farther Orient.[292] In the sixth century B.C., Hecat?us,[293] the father of geography, was acquainted not only with the Mediterranean lands but with the countries as far as the Indus,[294] and in Biblical times there were regular triennial voyages to India. Indeed, the story of Joseph bears witness to the caravan trade from India, across Arabia, and on to the banks

ow Sesostris[297] fitted out ships to sail to that country, and mentioning the routes to the east. These routes were generall

of the German Sea, while Macedon, in close touch with southern France, was also sending her armies under Alexander[299] through Afghanistan as far east as the Punjab.[300] Pliny tells us that Alexander the G

nt source of historical information.[302] The Rāmāyana speaks of merchants traveling in great caravans and embarking by sea for foreign lands.[303] Ceylon traded with M

Some of the Greek and Roman metrical and astronomical terms found their way, doubtless at this time, into the Sanskrit language.[305] Even as late as from the second to the fifth centuries A.D., Indian coins showed the Hellenic influence. The Hindu astronomical terminology reveals the same relationship to western thought, for Varāha-Mihira (6th century A.D.), a contemporary of āryabha?a, entitled a work of his the B?hat-Sa?hitā, a literal translation of μεγ?λη σ?νταξι? of Ptolemy

water clock, the music system, the use of the myriad,[312] the calendars, and in many other ways.[313] In passing through the suburbs of Peking to-day, on the way to the Great Bell temple, one is constantly reminded of the semi-Greek architecture of Pompeii, so closely does modern China touch the old classical civilization of the Mediterranean. The Chinese historians tell us t

ndia, frankincense from Persia, and silks from China, being more in demand than the exports from the Mediterranean lands, the balance of trade was against the West, and thus Roman coin found its way eastward. In 1898, for exam

aks in his history of these relations,[320] as do several of his contemporaries,[321] and Vergil[322] tells of Augustus doing battle in Persia. In Pliny's time the trade of the Roman Empire with A

oof may sometime come to light to show that the Greeks and Romans knew somet

ched eastern Turkestan, and had probably gone thence to China. Some centuries later (in 62 A.D.) the Chinese emperor sent an ambassador to India, and in 67 A.D. a Buddhist monk was invited to China.[326] Then, too, in India itself A?oka,

at Jondi-Sapur admitted both the Hindu and the Greek doctrines, and Firdusī tells us that during the brilliant reign of Khosrū I,[328] the golden age of P

the name Indicopleustes (the Indian traveler). His map (547 A.D.) shows some knowledge of the earth from the Atlantic to India. Such a man would, with hardly a doubt, have observed every nu

or 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 e

from India. Others went north to Damascus, while still others made their way along the southern shores of the Mediterranean. Ships sailed from the isthmus of Suez to all the commerci

ven to those of any other seaport of the Mediterranean, in the time of Boethius. The Brāhmī numerals would not have attracted the attention of scholars, for they had no zero so far as we know, and therefore they were no better and no worse than those of dozens of ot

he slightest doubt, that he could easily have known them, and that it would have been strange if a man of his inquirin

it much nearer being answered than it was over two centuries ago when Wallis (1693) expressed his doubts about it[333] soon after Vossius (1658) had called attention t

here are plenty of supporters of the idea that Boethius knew the numerals and included them in this book,[338] and on the other hand there are as many who feel that the geometry,

alent to-day than we commonly think. We have but to see how every hymn-book compiler feels himself authorized to change at will the classics of our language, and how u

s he would have mentioned them in his

and successors would have known and mentioned them. But neither Capella (c. 475)[344] nor any of t

erent manuscripts, but in general are about as shown on page 88. They are commonly written with the 9 at the left, decreasing to the unit at the right, numerous writers stating that this was because they were derived from Semitic sources in which the direction of writing is the opposite of our own. This practice continued until the sixteenth century.[350] The writer then leaves the subject entirely, using the Roma

s, Largely from Work

[3

[3

[3

[3

[3

[3

[3

[3

[3

so that it is not likely, even if the characters were known in Europe in the time of Boethius, that this particular form was recognized. It is worthy of mention, also, that in the six abacus forms

quiring mind might have done so either in his time or at any time before they definitely appeared in the tenth century. These centuries, five in number, represented the darkest of the Dark Ages, and even if these numerals were occasionally met and studied, no trace of them would be likely to show itself in the literature of the period, unless by chanc

ering scholars would have known many and strange things about the peoples they met, but they too were not, as a class, writers; (4) there is every reason a priori for believing that the ?obār numerals would have been known to merchants, and probably to some of the wandering scholars, long before the Arabs conquered northern Africa; (5) the wonder is not that the Hindu-Arabic numerals were known about 1000 A.D., and that they were the s