The name of this all-important symbol also demands some attention, especially as we are even yet quite undecided as to what to call it. We speak of it to-day aszero, naught, and evencipher; the telephone operator often calls itO, and the illiterate or careless person calls itaught. In view of all this uncertainty we may well inquire what it has been called in the past.[211]
As already stated, the Hindus called itśūnya, "void."[212]This passed over into the Arabic asaṣ-ṣifrorṣifr.[213]When Leonard of Pisa (1202) wrote upon the Hindu numerals he spoke of this character aszephirum.[214]Maximus Planudes (1330), writing under both the Greek and the Arabic influence, called ittziphra.[215]In a treatise on arithmetic written in the Italian language by Jacob of Florence[216](1307) it is calledzeuero,[217]while in an arithmetic of Giovanni di Danti of Arezzo (1370) the word appears asçeuero.[218]Another form iszepiro,[219]which was also a step fromzephirumto zero.[220]
Of course the Englishcipher, Frenchchiffre, is derived from the same Arabic word,aṣ-ṣifr, but in several languages it has come to mean the numeral figures in general. A trace of this appears in our wordciphering, meaning figuring or computing.[221]Johann Huswirt[222]uses the word with both meanings; he gives for the tenth character the four namestheca, circulus, cifra, andfigura nihili. In this statement Huswirt probably follows, as did many writers of that period, theAlgorismusof Johannes de Sacrobosco (c. 1250A.D.), who was also known as John of Halifax or John of Holywood. The commentary ofPetrus de Dacia[223](c. 1291A.D.) on theAlgorismus vulgarisof Sacrobosco was also widely used. The widespread use of this Englishman's work on arithmetic in the universities of that time is attested by the large number[224]of MSS. from the thirteenth to the seventeenth century still extant, twenty in Munich, twelve in Vienna, thirteen in Erfurt, several in England given by Halliwell,[225]ten listed in Coxe'sCatalogue of the Oxford College Library, one in the Plimpton collection,[226]one in the Columbia University Library, and, of course, many others.
Fromaṣ-ṣifrhas comezephyr, cipher,and finally the abridged formzero. The earliest printed work in which is found this final form appears to be Calandri's arithmetic of 1491,[227]while in manuscript it appears at least as early as the middle of the fourteenth century.[228]It also appears in a work,Le Kadran des marchans, by JehanCertain,[229]written in 1485. This word soon became fairly well known in Spain[230]and France.[231]The medieval writers also spoke of it as thesipos,[232]and occasionally as thewheel,[233]circulus[234](in Germandas Ringlein[235]),circularnote,[236]theca,[237]long supposed to be from its resemblance to the Greek theta, but explained by Petrus de Dacia as being derived from the name of the iron[238]used to brand thieves and robbers with a circular mark placed on the forehead or on the cheek. It was also calledomicron[239](the Greeko), being sometimes written õ orφto distinguish it from the lettero. It also went by the namenull[240](in the Latin booksnihil[241]ornulla,[242]and in the Frenchrien[243]), and very commonly by the namecipher.[244]Wallis[245]gives one of the earliest extended discussions of the various forms of the word, giving certain other variations worthy of note, asziphra,zifera,siphra,ciphra,tsiphra,tziphra,and the Greekτζίφρα.[246]
THE QUESTION OF THE INTRODUCTION OF THE NUMERALS INTO EUROPE BY BOETHIUS
Just as we were quite uncertain as to the origin of the numeral forms, so too are we uncertain as to the time and place of their introduction into Europe. There are two general theories as to this introduction. The first is that they were carried by the Moors to Spain in the eighth or ninth century, and thence were transmitted to Christian Europe, a theory which will be considered later. The second, advanced by Woepcke,[247]is that they were not brought to Spain by the Moors, but that they were already in Spain when the Arabs arrived there, having reached the West through the Neo-Pythagoreans. There are two facts to support this second theory: (1) the forms of these numerals are characteristic, differing materially from those which were brought by Leonardo of Pisa from Northern Africa early in the thirteenth century (before 1202A.D.); (2) they are essentially those whichtradition has so persistently assigned to Boethius (c. 500A.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 (711A.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, as they certainly did, from those that came through Bagdad. The theory is that the Hindu system, without the zero, early reached Alexandria (say 450A.D.), and that the Neo-Pythagorean love for the mysterious and especially for the Oriental led to its use as something bizarre and cabalistic; that it was then passed along the Mediterranean, reaching Boethius in Athens or in Rome, and to the schools of Spain, being discovered in Africa and Spain by the Arabs even before they themselves knew the improved system with the place value.
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 theḥurūf al-ġobār, the ġobār or dust numerals, as distinguished from theḥurūf al-jumalor 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 ġobār 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-Bīrūnī 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 Kairwān, in Tunis, in which the author speaks of one of his works on ġobār calculation;[251]and, much later, the Arab writerAbū Bekr Moḥammed ibn ‛Abdallāh, surnamedal-Ḥaṣṣār(the arithmetician), wrote a work of which the second chapter was "On the dust figures."[252]
The ġobār numerals themselves were first made known to modern scholars by Silvestre de Sacy, who discovered them in an Arabic manuscript from the library of the ancient abbey of St.-Germain-des-Prés.[253]The system has nine characters, but no zero. A dot above a character indicates tens, two dots hundreds, and so on,5 with dotmeaning 50, and5 with 3 dotsmeaning 5000. It has been suggested that possibly these dots, sprinkled like dust above the numerals, gave rise to the wordġobār,[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 ġobār system was ofArabic 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 Hellenic number system. When we consider, however, that the dot is found for zero in theBakhṣālīmanuscript,[257]and that it was used in subscript form in theKitāb 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 ġobār 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 theSēfer Yeṣīrāhby Abū Sahl ibn Tamim (probably composed at Kairwān, c. 950) speaks of "the Indian arithmetic known under the name ofġobāror dust calculation."[261]All this suggests that the Arabs may verylikely have known the ġobār forms before the numerals reached them again in 773.[262]The term "ġobār 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 manuscripts 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 passage in theKitāb al-Fihrist[265](987A.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 ġobār. In this document the dots are placed below the characters, instead of being superposed as described above. The significance was the same.
In form these ġobār numerals resemble our own much more closely than the Arab numerals do. They varied more or less, but were substantially as follows:
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 ġobār 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 fromthis source has no foundation. The first four Egyptian cardinal numerals[272]resemble more the modern Arabic.
Demotic and Hieratic OrdinalsDemotic 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 ġobār, as is seen in the third line on p.69, where the forms are from a manuscript written at Shiraz about 970A.D., and in which some western Arabic forms, e.g.symbolfor 2, are also used. Probably most significant of all is the fact that the ġobār 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 ġobār numerals. We now have to consider the question as to whether Boethius knew these ġobār 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 circumstantial evidence that he did know them? (4) What are the probabilities in the case?
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 wasaccordingly looked upon as a saint,[282]his bones were enshrined,[283]and as a natural consequence his books were among the classics 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 Cassiodorus[285](c. 490-c. 585A.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,distinguishing the trivium and quadrivium,[287]writing the only classics 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, ambassadors, 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.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](980-1037A.D.) in a short biography of himself relates that when his people were living at Bokhāra 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 ġobār numerals, the Boethius question, Gerbert, Leonardo of Pisa, and other names and events, that a digression for its consideration now becomes necessary.[290]
Even in very remote times, before the Hindu numerals were sculptured in the cave of Nānā Ghāt, 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 centuryB.C., Hecatæus,[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 Hecatæus, Scylax, a Persian admiral under Darius, from Caryanda on the coast of Asia Minor, traveled tonorthwest 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 Phœnicians and the Sabæans, and later were taken by the Greeks and Romans.[298]
In the fourth centuryB.C.the West and East came into very close relations. As early as 330, Pytheas of Massilia (Marseilles) 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 measurethe roads of India; and one of the great highways is described by Megasthenes, who in 295B.C., as the ambassador of Seleucus, resided atPātalīpuṭra, 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 Rāmāyana 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 Dicæarchus of Messana (about 320B.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 termsfound their way, doubtless at this time, into the Sanskrit language.[305]Even as late as from the second to the fifth centuriesA.D., Indian coins showed the Hellenic influence. The Hindu astronomical terminology reveals the same relationship to western thought, for Varāha-Mihira (6th centuryA.D.), a contemporary ofĀryabhaṭa, entitled a work of his theBṛhat-Saṃhitā, a literal translation ofμεγάλη σύνταξιςof 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 Phœnicians 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, themusic 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 that about 200B.C.their arms were successful in the far west, and that in 180B.C.an ambassador 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 114B.C.to Hadrian's time were found at Paklī, a part of the Hazāra district, sixteen miles north of Abbottābād,[317]and numerous similar discoveries have been made from time to time.
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 embassy from China.[319]Suetonius (first centuryA.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 thenumber 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 centuryB.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 62A.D.) the Chinese emperor sent an ambassador to India, and in 67A.D.a Buddhist monk was invited to China.[326]Then, too, in India itself Aśoka, 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 Firdusī tells us that during the brilliant reign ofKhosrū I,[328]the golden age of Pahlavī literature, the Hindu game of chess was introduced into Persia, at a time when wars with the Greeks were bringing prestige to the Sassanid 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 nameIndicopleustes(the Indian traveler). His map (547A.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 scraps 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 shores of the Mediterranean. In the fifth century they traded by sea with India and even with China, andḤirawas 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 wayalong the southern shores 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 résumé 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 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 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 manuscripts 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 circumstantial evidence that Boethius did know these numerals? The question is not new,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]
The genuineness of the arithmetic and the treatise on music is generally recognized, but the geometry, which contains the Hindu numerals with the zero, is under suspicion.[337]There 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 whofeel that the geometry, or at least the part mentioning the numerals, is spurious.[339]The argument of those who deny the authenticity of the particular passage in question may briefly be stated thus:
1. The falsification of texts has always been the subject of complaint. It was so with the Romans,[340]it was common in the Middle Ages,[341]and it is much more prevalentto-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 unknown editors have mutilated Shakespeare, to see how much more easy it was for medieval scribes to insert or eliminate paragraphs without any protest from critics.[342]
2. If Boethius had known these numerals he would have mentioned them in his arithmetic, but he does not do so.[343]
3. If he had known them, and had mentioned them in any of his works, his contemporaries, disciples, and successors would have known and mentioned them. But neither Capella (c. 475)[344]nor any of the numerous medieval writers who knew the works of Boethius makes any reference to the system.[345]
4. The passage in question has all the appearance of an interpolation by some scribe. Boethius is speaking of angles, in his work on geometry, when the text suddenly changes to a discussion of classes of numbers.[346]This is followed by a chapter in explanation of the abacus,[347]in which are described those numeral forms which are calledapicesorcaracteres.[348]The forms[349]of these characters vary in different manuscripts, but in general are about as shown on page88. 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 Roman numerals for the rest of his discussion, a proceeding so foreign to the method of Boethius as to be inexplicable on the hypothesis of authenticity. Why should such a scholarly writer have given them with no mention of their origin or use? Either he would have mentioned some historical interest attaching to them, or he would have used them in some discussion; he certainly would not have left the passage as it is.
Sir E. Clive Bayley has added[361]a further reason for believing them spurious, namely that the 4 is not of the Nānā Ghāt type, but of the Kabul form which the Arabs did not receive until 776;[362]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 from the chief manuscripts as given by Friedlein,[363]each contains some form of zero, which symbol probably originated in India about this time or later. It could hardly have reached Europe so soon.
As to the fourth question, Did Boethius probably know the numerals? It seems to be a fair conclusion, according to our present evidence, that (1) Boethius might very easily have known these numerals without the zero, but, (2) there is no reliable evidence that he did know them. And just as Boethius might have come in contact with them, so any other inquiring 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 theliterature of the period, unless by chance it should get into the writings of some man like Alcuin. As a matter of fact, it was not until the ninth or tenth century that there is any tangible evidence of their presence in Christendom. They were probably known to merchants here and there, but in their incomplete state they were not of sufficient importance to attract any considerable attention.
As a result of this brief survey of the evidence several conclusions seem reasonable: (1) commerce, and travel for travel's sake, never died out between the East and the West; (2) merchants had every opportunity of knowing, and would have been unreasonably stupid if they had not known, the elementary number systems of the peoples with whom they were trading, but they would not have put this knowledge in permanent written form; (3) wandering 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 ġobār 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 1000A.D., and that they were the subject of an elaborate work in 1202 by Fibonacci, but rather that more extended manuscript evidence of their appearance before that time has not been found. That they were more or less known early in the Middle Ages, certainly to many merchants of Christian Europe, and probably to several scholars, but without the zero, is hardly to be doubted. The lack of documentary evidence is not at all strange, in view of all of the circumstances.
THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS
If the numerals had their origin in India, as seems most probable, when did the Arabs come to know of them? It is customary to say that it was due to the influence of Mohammedanism that learning spread through Persia and Arabia; and so it was, in part. But learning was already respected in these countries long before Mohammed appeared, and commerce flourished all through this region. In Persia, for example, the reign of Khosrū Nuśīrwān,[364]the great contemporary of Justinian the law-maker, was characterized not only by an improvement in social and economic conditions, but by the cultivation of letters. Khosrū fostered learning, inviting to his court scholars from Greece, and encouraging the introduction of culture from the West as well as from the East. At this time Aristotle and Plato were translated, and portions of theHito-padēśa, or Fables of Pilpay, were rendered from the Sanskrit into Persian. All this means that some three centuries before the great intellectual ascendancy of Bagdad a similar fostering of learning was taking place in Persia, and under pre-Mohammedan influences.
The first definite trace that we have of the introduction of the Hindu system into Arabia dates from 773A.D.,[365]when an Indian astronomer visited the court of the caliph, bringing with him astronomical tables which at the caliph's command were translated into Arabic by Al-Fazārī.[366]Al-Khowārazmī andḤabash(Aḥmed ibn ‛Abdallāh, died c. 870) based their well-known tables upon the work of Al-Fāzarī. It may be asserted as highly probable that the numerals came at the same time as the tables. They were certainly known a few decades later, and before 825A.D., about which time the original of theAlgoritmi de numero Indorumwas written, as that work makes no pretense of being the first work to treat of the Hindu numerals.
The three writers mentioned cover the period from the end of the eighth to the end of the ninth century. While the historians Al-Maś‛ūdī and Al-Bīrūnī follow quite closely upon the men mentioned, it is well to note again the Arab writers on Hindu arithmetic, contemporary with Al-Khowārazmī, who were mentioned in chapter I, viz. Al-Kindī, Sened ibn ‛Alī, andAl-Ṣūfī.
For over five hundred years Arabic writers and others continued to apply to works on arithmetic the name "Indian." In the tenth century such writers are‛Abdallāh ibn al-Ḥasan, Abū 'l-Qāsim[367](died 987A.D.) of Antioch, andMoḥammed ibn ‛Abdallāh, Abū Naṣr[368](c. 982), of Kalwādā near Bagdad. Others of the same period orearlier (since they are mentioned in theFihrist,[369]987A.D.), who explicitly use the word "Hindu" or "Indian," areSinān ibn al-Fatḥ[370]ofḤarrān, and Ahmed ibn ‛Omar, al-Karābīsī.[371]In the eleventh century come Al-Bīrūnī[372](973-1048) and‛Ali ibn Aḥmed, Abū 'l-Ḥasan, Al-Nasawī[373](c. 1030). The following century brings similar works byIshāq ibn Yūsuf al-Ṣardafī[374]and Samū'īl ibnYaḥyāibn ‛Abbās al-Maġrebī al-Andalusī[375](c. 1174), and in the thirteenth century are ‛Abdallatīf ibn Yūsuf ibnMoḥammed, Muwaffaq al-Dīn AbūMoḥammedal-Baġdādī[376](c. 1231), and Ibn al-Bannā.[377]
The Greek monk Maximus Planudes, writing in the first half of the fourteenth century, followed the Arabic usage in calling his workIndian Arithmetic.[378]There were numerous other Arabic writers upon arithmetic, as that subject occupied one of the high places among the sciences, but most of them did not feel it necessary to refer to the origin of the symbols, the knowledge of which might well have been taken for granted.
One document, cited by Woepcke,[379]is of special interest since it shows at an early period, 970A.D., the use of the ordinary Arabic forms alongside the ġobār. The title of the work isInteresting and Beautiful Problems on Numberscopied byAḥmed ibn Moḥammed ibn ‛Abdaljalīl, Abū Sa‛īd, al-Sijzī,[380](951-1024) from a work by a priest and physician,Naẓīfibn Yumn,[381]al-Qass (died c. 990). Suter does not mention this work ofNaẓīf.
The second reason for not ascribing too much credit to the purely Arab influence is that the Arab by himself never showed any intellectual strength. What took place afterMoḥammedhad lighted the fire in the hearts of his people was just what always takes place when different types of strong races blend,—a great renaissance in divers lines. It was seen in the blending of such types at Miletus in the time of Thales, at Rome in the days of the early invaders, at Alexandria when the Greek set firm foot on Egyptian soil, and we see it now when all the nations mingle their vitality in the New World. So when the Arab culture joined with the Persian, a new civilization rose and flourished.[382]The Arab influence came not from its purity, but from its intermingling with an influence more cultured if less virile.
As a result of this interactivity among peoples of diverse interests and powers, Mohammedanism was to the world from the eighth to the thirteenth century what Rome and Athens and the Italo-Hellenic influence generally hadbeen to the ancient civilization. "If they did not possess the spirit of invention which distinguished the Greeks and the Hindus, if they did not show the perseverance in their observations that characterized the Chinese astronomers, they at least possessed the virility of a new and victorious people, with a desire to understand what others had accomplished, and a taste which led them with equal ardor to the study of algebra and of poetry, of philosophy and of language."[383]
It was in 622A.D.thatMoḥammedfled from Mecca, and within a century from that time the crescent had replaced the cross in Christian Asia, in Northern Africa, and in a goodly portion of Spain. The Arab empire was an ellipse of learning with its foci at Bagdad and Cordova, and its rulers not infrequently took pride in demanding intellectual rather than commercial treasure as the result of conquest.[384]
It was under these influences, either pre-Mohammedan or later, that the Hindu numerals found their way to the North. If they were known beforeMoḥammed's time, the proof of this fact is now lost. This much, however, is known, that in the eighth century they were taken to Bagdad. It was early in that century that the Mohammedans obtained their first foothold in northern India, thus foreshadowing an epoch of supremacy that endured with varied fortunes until after the golden age of Akbar the Great (1542-1605) and Shah Jehan. They also conquered Khorassan and Afghanistan, so that the learning and the commercial customs of India at once found easyaccess to the newly-established schools and the bazaars of Mesopotamia and western Asia. The particular paths of conquest and of commerce were either by way of the Khyber Pass and through Kabul, Herat and Khorassan, or by sea through the strait of Ormuz to Basra (Busra) at the head of the Persian Gulf, and thence to Bagdad. As a matter of fact, one form of Arabic numerals, the one now in use by the Arabs, is attributed to the influence of Kabul, while the other, which eventually became our numerals, may very likely have reached Arabia by the other route. It is in Bagdad,[385]Dār al-Salām—"the Abode of Peace," that our special interest in the introduction of the numerals centers. Built upon the ruins of an ancient town byAl-Manṣūr[386]in the second half of the eighth century, it lies in one of those regions where the converging routes of trade give rise to large cities.[387]Quite as well of Bagdad as of Athens might Cardinal Newman have said:[388]
"What it lost in conveniences of approach, it gained in its neighborhood to the traditions of the mysterious East, and in the loveliness of the region in which it lay. Hither, then, as to a sort of ideal land, where all archetypes of the great and the fair were found in substantial being, and all departments of truth explored, and all diversities of intellectual power exhibited, where taste and philosophy were majestically enthroned as in a royal court, where there was no sovereignty but that of mind, and no nobility but that of genius, where professors wererulers, and princes did homage, thither flocked continually from the very corners of theorbis terrarumthe many-tongued generation, just rising, or just risen into manhood, in order to gain wisdom." For here it was thatAl-Manṣūrand Al-Māmūn and Hārūn al-Rashīd (Aaron the Just) made for a time the world's center of intellectual activity in general and in the domain of mathematics in particular.[389]It was just after theSindhindwas brought to Bagdad thatMoḥammedibn Mūsā al-Khowārazmī, whose name has already been mentioned,[390]was called to that city. 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.[391]Appreciating at once the value of the position system so recently brought from India, he wrote an arithmetic based upon these numerals, and this was translated into Latin in the time of Adelhard of Bath (c. 1180), although possibly by his contemporary countryman Robert Cestrensis.[392]This translation was found in Cambridge and was published by Boncompagni in 1857.[393]