{65}

A recent theory set forth by Bubnov[249] also deserves mention, chiefly because of the seriousness of purpose shown by this well-known writer.

Bubnov holds that the forms first found in Europe are derived from ancient symbols used on the abacus, but that the zero is of Hindu origin. This theory does not seem tenable, however, in the light of the evidence already set forth.

Two questions are presented by Woepcke's theory: (1) What was the nature of these Spanish numerals, and how were they made known to Italy? (2) Did Boethius know them?

The Spanish forms of the numerals were called the _[h.]ur[=u]f al-[.g]ob[=a]r_, the [.g]ob[=a]r or dust numerals, as distinguished from the _[h.]ur[=u]f al-jumal_ or alphabetic numerals. Probably the latter, under the influence of the Syrians or Jews,[250] were also used by the Arabs. The significance of the term [.g]ob[=a]r is doubtless 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-B[=i]r[=u]n[=i] states that the Hindus often performed numerical computations in the sand. The term is found as early as c. 950, in the verses of an anonymous writer of Kairw[=a]n, in Tunis, in which the author speaks of one of his works on [.g]ob[=a]r calculation;[251] and, much later, the Arab writer Ab[=u] Bekr Mo[h.]ammed ibn 'Abdall[=a]h, surnamed al-[H.]a[s.][s.][=a]r {66} (the arithmetician), wrote a work of which the second chapter was "On the dust figures."[252]

The [.g]ob[=a]r numerals themselves were first made known to modern scholars by Silvestre de Sacy, who discovered them in an Arabic ma.n.u.script from the library of the ancient abbey of St.-Germain-des-Pres.[253] The system has nine characters, but no zero. A dot above a character indicates tens, two dots hundreds, and so on, [5 with dot] meaning 50, and [5 with 3 dots] meaning 5000. It has been suggested that possibly these dots, sprinkled like dust above the numerals, gave rise to the word _[.g]ob[=a]r_,[254] but this is not at all probable. This system of dots is found in Persia at a much later date with numerals quite like the modern Arabic;[255] but that it was used at all is significant, for it is hardly likely that the western system would go back to Persia, when the perfected Hindu one was near at hand.

At first sight there would seem to be some reason for believing that this feature of the [.g]ob[=a]r system was of {67} Arabic origin, and that the present zero of these people,[256] the dot, was derived from it. It was entirely natural that the Semitic people generally should 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 h.e.l.lenic number system.

When we consider, however, that the dot is found for zero in the Bakh[s.][=a]l[=i] ma.n.u.script,[257] and that it was used in subscript form in the _Kit[=a]b al-Fihrist_[258] in the tenth century, and as late as the sixteenth century,[259] although in this case probably under Arabic influence, we are forced to believe that this form may also have been of Hindu origin.

The fact seems to be that, as already stated,[260] the Arabs did not immediately adopt the Hindu zero, because it resembled their 5; they used the superscript dot as serving their purposes fairly well; they may, indeed, have carried this to the west and have added it to the [.g]ob[=a]r forms already there, just as they transmitted it to the Persians.

Furthermore, the Arab and Hebrew scholars of Northern Africa in the tenth century knew these numerals as Indian forms, for a commentary on the _S[=e]fer Ye[s.][=i]r[=a]h_ by Ab[=u] Sahl ibn Tamim (probably composed at Kairw[=a]n, c. 950) speaks of "the Indian arithmetic known under the name of _[.g]ob[=a]r_ or dust calculation."[261] All this suggests that the Arabs may very {68} likely have known the [.g]ob[=a]r forms before the numerals reached them again in 773.[262] The term "[.g]ob[=a]r numerals"

was also used without any reference to the peculiar use of dots.[263] In this connection it is worthy of mention that the Algerians employed two different forms of numerals in ma.n.u.scripts even of the fourteenth century,[264] and that the Moroccans of to-day employ the European forms instead of the present Arabic.

The Indian use of subscript dots to indicate the tens, hundreds, thousands, etc., is established by a pa.s.sage in the _Kit[=a]b al-Fihrist_[265] (987 A.D.) in which the writer discusses the written language of the 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 which have usually been called Indian,[266] in opposition to [.g]ob[=a]r. In this doc.u.ment the dots are placed below the characters, instead of being superposed as described above. The significance was the same.

In form these [.g]ob[=a]r numerals resemble our own much more closely than the Arab numerals do. They varied more or less, but were substantially as follows:

{69}

1[267][Ill.u.s.tration]

2[268][Ill.u.s.tration]

3[269][Ill.u.s.tration]

4[270][Ill.u.s.tration]

5[271][Ill.u.s.tration]

6[271][Ill.u.s.tration]

The question of the possible influence of the Egyptian demotic and hieratic ordinal forms has been so often suggested that it seems well to introduce them at this point, for comparison with the [.g]ob[=a]r 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 the Demotic 4 and in the 9, so that the statement that the Hindu forms in general came from {70} this source has no foundation. The first four Egyptian cardinal numerals[272]

resemble more the modern Arabic.

[Ill.u.s.tration: DEMOTIC AND HIERATIC ORDINALS]

This theory of the very early introduction of the numerals into Europe fails in several points. In the first place the early Western forms are not known; in the second place some early Eastern forms are like the [.g]ob[=a]r, as is seen in the third line on p. 69, where the forms are from a ma.n.u.script written at Shiraz about 970 A.D., and in which some western Arabic forms, e.g. [symbol] for 2, are also used. Probably most significant of all is the fact that the [.g]ob[=a]r numerals as given by Sacy are all, with the exception of the symbol for eight, either single Arabic letters or combinations of letters. So much for the Woepcke theory and the meaning of the [.g]ob[=a]r numerals. We now have to consider the question as to whether Boethius knew these [.g]ob[=a]r forms, or forms akin to them.

This large question[273] suggests several minor ones: (1) Who was Boethius?

(2) Could he have known these numerals? (3) Is there any positive or strong circ.u.mstantial evidence that he did know them? (4) What are the probabilities in the case?

{71}

First, who was Boethius,--Divus[274] Boethius as he was called in the Middle Ages? Anicius Manlius Severinus Boethius[275] was born at Rome c.

475. He was a member of the distinguished family of the Anicii,[276] which had for some time before his birth been Christian. Early left an orphan, the tradition is that he was taken to Athens at about the age of ten, and that he remained there eighteen years.[277] He married Rusticiana, daughter of the senator Symmachus, and this union 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 its lowest ebb, the church found it profitable to look upon his execution as a martyrdom.[281] He was {72} accordingly looked upon as a saint,[282] his bones were enshrined,[283] and as a natural consequence his books were among the cla.s.sics in the church schools for a thousand years.[284] It is pathetic, however, to think of the medieval student trying to extract mental nourishment from a work so abstract, so meaningless, so unnecessarily complicated, as the arithmetic of Boethius.

He was looked upon by his contemporaries and immediate successors as a master, for Ca.s.siodorus[285] (c. 490-c. 585 A.D.) says to him: "Through your translations the music of Pythagoras and the astronomy 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, {73} distinguishing the trivium and quadrivium,[287] writing the only cla.s.sics of his time, Gibbon well called him "the last of the Romans whom Cato or Tully could have acknowledged for their countryman."[288]

The second question relating to Boethius is this: Could he possibly have known the Hindu numerals? In view of the relations that will be shown to have existed between the East and the West, there can only be an affirmative answer to this question. 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, amba.s.sadors, scholars, and the restless trader, all had gone back and forth, by land or more frequently by sea, between the Mediterranean lands and the centers of Indian commerce and culture. Boethius could very well have learned one or more forms of Hindu numerals from some traveler or merchant.

To justify this statement it is necessary to speak more fully of these relations between the Far East and Europe. It is true that we have no records of the interchange of learning, in any large way, between eastern Asia and central Europe in the century preceding the time of Boethius. But it is one of the mistakes of scholars to believe that they are the sole transmitters of knowledge. {74} 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.

It must always be kept in mind that it was to the tradesman and the wandering scholar that the spread of such learning was due, rather than to the school man. Indeed, Avicenna[289] (980-1037 A.D.) in a short biography of himself relates that when his people were living at Bokh[=a]ra his father sent him to the house of a grocer to learn the Hindu art of reckoning, in which this grocer (oil dealer, possibly) was expert. Leonardo of Pisa, too, had a similar training.

The whole question of this spread of mercantile knowledge along the trade routes is so connected with the [.g]ob[=a]r numerals, the Boethius question, Gerbert, Leonardo of Pisa, and other names and events, that a digression for its consideration now becomes necessary.[290]

{75}

Even in very remote times, before the Hindu numerals were sculptured in the cave of N[=a]n[=a] Gh[=a]t, there were trade relations between Arabia and India. Indeed, long before the Aryans went to India the great Turanian race had spread its civilization from the Mediterranean to the Indus.[291] At a much later period the Arabs were the intermediaries between Egypt and Syria on the west, and the farther Orient.[292] In the sixth century B.C., Hecataeus,[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 of the Nile. About the same time as Hecataeus, Scylax, a Persian admiral under Darius, from Caryanda on the coast of Asia Minor, traveled to {76} northwest India and wrote upon his ventures.[295]

He induced the nations along the Indus to acknowledge the Persian supremacy, and such number systems as there were in these lands would naturally have been known to a man of his attainments.

A century after Scylax, Herodotus showed considerable knowledge of India, speaking of its cotton and its gold,[296] telling how Sesostris[297] fitted out ships to sail to that country, and mentioning the routes to the east.

These routes were generally by the Red Sea, and had been followed by the Phoenicians and the Sabaeans, and later were taken by the Greeks and Romans.[298]

In the fourth century B.C. the West and East came into very close relations. As early as 330, Pytheas of Ma.s.silia (Ma.r.s.eilles) had explored as far north as the northern end of the British Isles and the coasts 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 Great employed surveyors to measure {77} the roads of India; and one of the great highways is described by Megasthenes, who in 295 B.C., as the amba.s.sador of Seleucus, resided at P[=a]tal[=i]pu[t.]ra, the present Patna.[301]

The Hindus also learned the art of coining from the Greeks, or possibly from the Chinese, and the stores of Greco-Hindu coins still found in northern India are a constant source of historical information.[302] The R[=a]m[=a]yana speaks of merchants traveling in great caravans and embarking by sea for foreign lands.[303] Ceylon traded with Malacca and Siam, and Java was colonized by Hindu traders, so that mercantile knowledge was being spread about the Indies during all the formative period of the numerals.

Moreover the results of the early Greek invasion were embodied by Dicaearchus of Messana (about 320 B.C.) in a map that long remained a standard. Furthermore, Alexander did not allow his influence on the East to cease. He divided India into three satrapies,[304] placing Greek governors over two of them and leaving a Hindu ruler in charge of the third, and in Bactriana, a part of Ariana or ancient Persia, he left governors; and in these the western civilization was long in evidence. Some of the Greek and Roman metrical and astronomical terms {78} 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 h.e.l.lenic influence.

The Hindu astronomical terminology reveals the same relationship to western thought, for Var[=a]ha-Mihira (6th century A.D.), a contemporary of [=A]ryabha[t.]a, ent.i.tled a work of his the _B[r.]hat-Sa[m.]hit[=a]_, a literal translation of [Greek: megale suntaxis] of Ptolemy;[306] and in various ways is this interchange of ideas apparent.[307] It could not have been at all unusual for the ancient Greeks to go to India, for Strabo lays down the route, saying that all who make the journey start from Ephesus and traverse Phrygia and Cappadocia before taking the direct road.[308] The products of the East were always finding their way to the West, the Greeks getting their ginger[309] from Malabar, as the Phoenicians had long before brought gold from Malacca.

Greece must also have had early relations with China, for there is a notable similarity between the Greek and Chinese life, as is shown in their houses, their domestic customs, their marriage ceremonies, the public story-tellers, the puppet shows which Herodotus says were introduced from Egypt, the street jugglers, the games of dice,[310] the game of finger-guessing,[311] the water clock, the {79} music system, the use of the myriad,[312] the calendars, and in many other ways.[313] In pa.s.sing 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 cla.s.sical civilization of the Mediterranean. The Chinese historians tell us that about 200 B.C. their arms were successful in the far west, and that in 180 B.C. an amba.s.sador went to Bactria, then a Greek city, and reported that Chinese products were on sale in the markets there.[314] There is also a noteworthy resemblance between certain Greek and Chinese words,[315] showing that in remote times there must have been more or less interchange of thought.

The Romans also exchanged products with the East. Horace says, "A busy trader, you hasten to the farthest Indies, flying from poverty over sea, over crags, over fires."[316] The products of the Orient, spices and jewels from India, 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 example, a number of Roman coins dating from 114 B.C. to Hadrian's time were found at Pakl[=i], a part of the Haz[=a]ra district, sixteen miles north of Abbott[=a]b[=a]d,[317] and numerous similar discoveries have been made from time to time.

{80}

Augustus speaks of envoys received by him from India, a thing never before known,[318] and it is not improbable that he also received an emba.s.sy from China.[319] Suetonius (first century A.D.) speaks 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 Asia amounted to a million and a quarter dollars a year, a sum far greater relatively then than now,[323] while by the time of Constantine Europe was in direct communication with the Far East.[324]

In view of these relations it is not beyond the range of possibility that proof may sometime come to light to show that the Greeks and Romans knew something of the {81} number system of India, as several writers have maintained.[325]

Returning to the East, there are many evidences of the spread of knowledge in and about India itself. In the third century B.C. Buddhism began to be a connecting medium of thought. It had already permeated the Himalaya territory, had reached eastern Turkestan, and had probably gone thence to China. Some centuries later (in 62 A.D.) the Chinese emperor sent an amba.s.sador to India, and in 67 A.D. a Buddhist monk was invited to China.[326] Then, too, in India itself A['s]oka, whose name has already been mentioned in this work, extended the boundaries of his domains even into Afghanistan, so that it was entirely possible for the numerals of the Punjab to have worked their way north even at that early date.[327]

Furthermore, the influence of Persia must not be forgotten in considering this transmission of knowledge. In the fifth century the Persian medical school at Jondi-Sapur admitted both the Hindu and the Greek doctrines, and Firdus[=i] tells us that during the brilliant reign of {82} Khosr[=u]

I,[328] the golden age of Pahlav[=i] literature, the Hindu game of chess was introduced into Persia, at a time when wars with the Greeks were bringing prestige to the Sa.s.sanid dynasty.

Again, not far from the time of Boethius, in the sixth century, the Egyptian monk Cosmas, in his earlier years as a trader, made journeys to Abyssinia and even to India and Ceylon, receiving 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 numeral system used by the people with whom he sojourned,[329] and whether or not he recorded his studies in permanent form he would have transmitted such sc.r.a.ps of knowledge by word of mouth.

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 sh.o.r.es of the Mediterranean. In the fifth century they traded by sea with India and even with China, and [H.]ira 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.

Long before the warlike activity of the Arabs, Alexandria had become the great market-place of the world. From this center caravans traversed Arabia to Hadramaut, where they met ships from India. Others went north to Damascus, while still others made their way {83} along the southern sh.o.r.es of the Mediterranean. Ships sailed from the isthmus of Suez to all the commercial ports of Southern Europe and up into the Black Sea. Hindus were found among the merchants[332] who frequented the bazaars of Alexandria, and Brahmins were reported even in Byzantium.

Such is a very brief resume of the evidence showing that the numerals of the Punjab and of other parts of India as well, and indeed those of China and farther Persia, of Ceylon and the Malay peninsula, might well have been known to the merchants of Alexandria, and even to those of any other seaport of the Mediterranean, in the time of Boethius. The Br[=a]hm[=i]

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 other systems. If Boethius was attracted to them it was probably exactly as any one is naturally attracted to the bizarre or the mystic, and he would have mentioned them in his works only incidentally, as indeed they are mentioned in the ma.n.u.scripts in which they occur.

In answer therefore to the second question, Could Boethius have known the Hindu numerals? the reply must be, without the slightest doubt, that he could easily have known them, and that it would have been strange if a man of his inquiring mind did not pick up many curious bits of information of this kind even though he never thought of making use of them.

Let us now consider the third question, Is there any positive or strong circ.u.mstantial evidence that Boethius did know these numerals? The question is not new, {84} nor is 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 to the matter.[334] Stated briefly, there are three works on mathematics attributed to Boethius:[335]

(1) the arithmetic, (2) a work on music, and (3) the geometry.[336]