A Short Account of the History of Mathematics · W. W. Rouse Ball
The mathematics of this period has been discussed by Cantor, by S. Günther, Geschichte des mathematischen Unterrichtes im deutschen Mittelalter, Berlin, 1887; and by H. Weissenborn, Gerbert, Beiträge zur Kenntniss der Mathematik des Mittelalters, Berlin, 1888; and Zur Geschichte der Einführung der jetzigen Ziffers, Berlin, 1892.
The first few centuries of this second period of our history are singularly barren of interest; and indeed it would be strange if we found science or mathematics studied by those who lived in a condition of perpetual war. Broadly speaking we may say that from the sixth to the eighth centuries the only places of study in western Europe were the Benedictine monasteries. We may find there some slight attempts at a study of literature; but the science usually taught was confined to the use of the abacus, the method of keeping accounts, and a knowledge of the rule by which the date of Easter could be determined. Nor was this unreasonable, for the monk had renounced the world, and there was no reason why he should learn more science than was required for the services of the Church and his monastery. The traditions of Greek and Alexandrian learning gradually died away. Possibly in Rome and a few favoured places copies of the works of the great Greek mathematicians were obtainable though with difficulty, but there were no students, the books were unvalued, and in time became very scarce.
Three authors of the sixth century--Boethius, Cassiodorus, and Isidorus--may be named whose writings serve as a connecting link between the mathematics of classical and of medieval times. As their works remained standard text-books for some six or seven centuries it is necessary to mention them, but it should be understood that this is the only reason for doing so; they show no special mathematical ability. It will be noticed that these authors were contemporaries of the later Athenian and Alexandrian schools.
Boethius. Anicius Manlius Severinus Boethius, or as the name is sometimes written Boetius, born at Rome about 475 and died in 526, belonged to a family which for the two preceding centuries had been esteemed one of the most illustrious in Rome. It was formerly believed that he was educated at Athens: this is somewhat doubtful, but at any rate he was exceptionally well read in Greek literature and science.
Boethius would seem to have wished to devote his life to literary pursuits; but recognizing "that the world would be happy only when kings became philosophers or philosophers kings," he yielded to the pressure put on him and took an active share in politics. He was celebrated for his extensive charities, and, what in those days was very rare, the care that he took to see that the recipients were worthy of them. He was elected consul at an unusually early age, and took advantage of his position to reform the coinage and to introduce the public use of sun-dials, water-clocks, etc. He reached the height of his prosperity in 522 when his two sons were inaugurated as consuls. His integrity and attempts to protect the provincials from the plunder of the public officials brought on him the hatred of the Court. He was sentenced to death while absent from Rome, seized at Ticinum, and in the baptistery of the church there tortured by drawing a cord round his head till the eyes were forced out of the sockets, and finally beaten to death with clubs on October 23, 526. Such at least is the account that has come down to us. At a later time his merits were recognized, and tombs and statues erected in his honour by the state.
Boethius was the last Roman of note who studied the language and literature of Greece, and his works afforded to medieval Europe some glimpse of the intellectual life of the old world. His importance in the history of literature is thus very great, but it arises merely from the accident of the time at which he lived. After the introduction of Aristotle's works in the thirteenth century his fame died away, and he has now sunk into an obscurity which is as great as was once his reputation. He is best known by his Consolatio, which was translated by Alfred the Great into Anglo-Saxon. For our purpose it is sufficient to note that the teaching of early medieval mathematics was mainly founded on his geometry and arithmetic.
His Geometry consists of the enunciations (only) of the first book of Euclid, and of a few selected propositions in the third and fourth books, but with numerous practical applications to finding areas, etc. He adds an appendix with proofs of the first three propositions to shew that the enunciations may be relied on. His Arithmetic is founded on that of Nicomachus.
Cassiodorus. A few years later another Roman, Magnus Aurelius Cassiodorus, who was born about 490 and died in 566, published two works, De Institutione Divinarum Litterarum and De Artibus ac Disciplinis, in which not only the preliminary trivium of grammar, logic, and rhetoric were discussed, but also the scientific quadrivium of arithmetic, geometry, music, and astronomy. These were considered standard works during the middle ages; the former was printed at Venice in 1598.
Isidorus. Isidorus, bishop of Seville, born in 570 and died in 636, was the author of an encyclopaedic work in twenty volumes called Origines, of which the third volume is given up to the quadrivium. It was published at Leipzig in 1833.
When, in the latter half of the eighth century, Charles the Great had established his empire, he determined to promote learning so far as he was able. He began by commanding that schools should be opened in connection with every cathedral and monastery in his kingdom; an order which was approved and materially assisted by the popes. It is interesting to us to know that this was done at the instance and under the direction of two Englishmen, Alcuin and Clement, who had attached themselves to his court.
Alcuin. Of these the more prominent was Alcuin, who was born in Yorkshire in 735 and died at Tours in 804. He was educated at York under archbishop Egbert, his "beloved master," whom he succeeded as director of the school there. Subsequently he became abbot of Canterbury, and was sent to Rome by Offa to procure the pallium for archbishop Eanbald. On his journey back he met Charles at Parma; the emperor took a great liking to him, and finally induced him to take up his residence at the imperial court, and there teach rhetoric, logic, mathematics, and divinity. Alcuin remained for many years one of the most intimate and influential friends of Charles and was constantly employed as a confidential ambassador; as such he spent the years 791 and 792 in England, and while there reorganized the studies at his old school at York. In 801 he begged permission to retire from the court so as to be able to spend the last years of his life in quiet: with difficulty he obtained leave, and went to the abbey of St. Martin at Tours, of which he had been made head in 796. He established a school in connection with the abbey which became very celebrated, and he remained and taught there till his death on May 19, 804.
Most of the extant writings of Alcuin deal with theology or history, but they include a collection of arithmetical propositions suitable for the instruction of the young. The majority of the propositions are easy problems, either determinate or indeterminate, and are, I presume, founded on works with which he had become acquainted when at Rome. The following is one of the most difficult, and will give an idea of the character of the work. If one hundred bushels of corn be distributed among one hundred people in such a manner that each man receives three bushels, each woman two, and each child half a bushel: how many men, women, and children were there? The general solution is (20 - 3n) men, 5n women, and (80 - 2n) children, where n may have any of the values 1, 2, 3, 4, 5, 6. Alcuin only states the solution for which n = 3 ; that is, he gives as the answer 11 men, 15 women, and 74 children.
This collection however was the work of a man of exceptional genius, and probably we shall be correct in saying that mathematics, if taught at all in a school, was generally confined to the geometry of Boethius, the use of the abacus and multiplication table, and possibly the arithmetic of Boethius; while except in one of these schools or in a Benedictine cloister it was hardly possible to get either instruction or opportunities for study. It was of course natural that the works used should come from Roman sources, for Britain and all the countries included in the empire of Charles had at one time formed part of the western half of the Roman empire, and their inhabitants continued for a long time to regard Rome as the centre of civilization, while the higher clergy kept up a tolerably constant intercourse with Rome.
After the death of Charles many of his schools confined themselves to teaching Latin, music, and theology, some knowledge of which was essential to the worldly success of the higher clergy. Hardly any science or mathematics was taught, but the continued existence of the schools gave an opportunity to any teacher whose learning or zeal exceeded the narrow limits fixed by tradition; and though there were but few who availed themselves of the opportunity, yet the number of those desiring instruction was so large that it would seem as if any one who could teach was sure to attract a considerable audience.
A few schools, where the teachers were of repute, became large and acquired a certain degree of permanence, but even in them the teaching was still usually confined to the trivium and quadrivium. The former comprised the three arts of grammar, logic, and rhetoric, but practically meant the art of reading and writing Latin; nominally the latter included arithmetic and geometry with their applications, especially to music and astronomy, but in fact it rarely meant more than arithmetic sufficient to enable one to keep accounts, music for the church services, geometry for the purpose of land-surveying, and astronomy sufficient to enable one to calculate the feasts and fasts of the church. The seven liberal arts are enumerated in the line, Lingua, tropus, ratio; numerus, tonus, angulus, astra. Any student who got beyond the trivium was looked on as a man of great erudition, Qui tria, qui septem, qui totum scibile novit, as a verse of the eleventh century runs. The special questions which then and long afterwards attracted the best thinkers were logic and certain portions of transcendental theology and philosophy.
We may sum the matter up by saying that during the ninth and tenth centuries the mathematics taught was still usually confined to that comprised in the two works of Boethius together with the practical use of the abacus and the multiplication table, though during the latter part of the time a wider range of reading was undoubtedly accessible.
Gerbert. In the tenth century a man appeared who would in any age have been remarkable and who gave a great stimulus to learning. This was Gerbert, an Aquitanian by birth, who died in 1003 at about the age of fifty. His abilities attracted attention to him even when a boy, and procured his removal from the abbey school at Aurillac to the Spanish march where he received a good education. He was in Rome in 971, where his proficiency in music and astronomy excited considerable interest: but his interests were not confined to these subjects, and he had already mastered all the branches of the trivium and quadrivium, as then taught, except logic; and to learn this he moved to Rheims, which Archbishop Adalbero had made the most famous school in Europe. Here he was at once invited to teach, and so great was his fame that to him Hugh Capet entrusted the education of his son Robert who was afterwards king of France.
Gerbert was especially famous for his construction of abaci and of terrestrial and celestial globes; he was accustomed to use the latter to illustrate his lectures. These globes excited great admiration; and he utilized this by offering to exchange them for copies of classical Latin works, which seem already to have become very scarce; the better to effect this he appointed agents in the chief towns of Europe. To his efforts it is believed we owe the preservation of several Latin works. In 982 he received the abbey of Bobbio, and the rest of his life was taken up with political affairs; he became Archbishop of Rheims in 991, and of Ravenna in 998; in 999 he was elected Pope, when he took the title of Sylvester II.; as head of the Church, he at once commenced an appeal to Christendom to arm and defend the Holy Land, thus forestalling Peter the Hermit by a century, but he died on May 12, 1003, before he had time to elaborate his plans. His library is, I believe, preserved in the Vatican.
So remarkable a personality left a deep impress on his generation, and all sorts of fables soon began to collect around his memory. It seems certain that he made a clock which was long preserved at Magdeburg, and an organ worked by steam which was still at Rheims two centuries after his death. All this only tended to confirm the suspicions of his contemporaries that he had sold himself to the devil; and the details of his interviews with that gentleman, the powers he purchased, and his effort to escape from his bargain when he was dying, may be read in the pages of William of Malmesbury, Orderic Vitalis, and Platina. To these anecdotes the first named writer adds the story of the statue inscribed with the words "strike here," which having amused our ancestors in the Gesta Romanorum has been recently told again in the Earthly Paradise.
Extensive though his influence was, it must not be supposed that Gerbert's writings shew any great originality. His mathematical works comprise a treatise on arithmetic entitled De Numerorum Divisione, and one on geometry. An improvement in the abacus, attributed by some writers to Boethius, but which is more likely due to Gerbert, is the introduction in every column of beads marked by different characters, called apices, for each of the numbers from 1 to 9, instead of nine exactly similar counters or beads. These apices lead to a representation of numbers essentially the same as the Arabic numerals. There was however no symbol for zero; the step from this concrete system of denoting numbers by a decimal system on an abacus to the system of denoting them by similar symbols in writing seems to us to be a small one, but it would appear that Gerbert did not make it. He found at Mantua a copy of the geometry of Boethius, and introduced it into the medieval schools. Gerbert's own work on geometry is of unequal ability; it includes a few applications to land-surveying and the determination of the heights of inaccessible objects, but much of it seems to be copied from some Pythagorean text-book. In the course of it he however solves one problem which was of remarkable difficulty for that time. The question is to find the sides of a right-angled triangle whose hypotenuse and area are given. He says, in effect, that if these latter be denoted respectively by c and h^2 , then the lengths of the two sides will be 12c^2+4h^2 + c^2-4h^2 and 12c^2+4h^2-c^2-4h^2.
Bernelinus. One of Gerbert's pupils, Bernelinus, published a work on the abacus which is, there is very little doubt, a reproduction of the teaching of Gerbert. It is valuable as indicating that the Arabic system of writing numbers was still unknown in Europe.
At the end of the eleventh century or the beginning of the twelfth a revival of learning took place at several of these cathedral or monastic schools; and in some cases, at the same time, teachers who were not members of the school settled in its vicinity and, with the sanction of the authorities, gave lectures which were in fact always on theology, logic, or civil law. As the students at these centres grew in numbers, it became desirable to act together whenever any interest common to all was concerned. The association thus formed was a sort of guild or trades union, or in the language of the time a universitas magistrorum et scholarium. This was the first stage in the development of the earliest medieval universities. In some cases, as at Paris, the governing body of the university was formed by the teachers alone, in others, as at Bologna, by both teachers and students; but in all cases precise rules for the conduct of business and the regulation of the internal economy of the guild were formulated at an early stage in its history. The municipalities and numerous societies which existed in Italy supplied plenty of models for the construction of such rules, but it is possible that some of the regulations were derived from those in force in the Mohammedan schools at Cordova.
We are, almost inevitably, unable to fix the exact date of the commencement of these voluntary associations, but they existed at Paris, Bologna, Salerno, Oxford, and Cambridge before the end of the twelfth century: these may be considered the earliest universities in Europe. The instruction given at Salerno and Bologna was mainly technical--at Salerno in medicine, and at Bologna in law--and their claim to recognition as universities, as long as they were merely technical schools, has been disputed.
Although the organization of these early universities was independent of the neighbouring church and monastic schools they seem in general to have been, at any rate originally, associated with such schools, and perhaps indebted to them for the use of rooms, etc. The universities or guilds (self-governing and formed by teachers and students), and the adjacent schools (under the direct control of church or monastic authorities), continued to exist side by side, but in course of time the latter diminished in importance, and often ended by becoming subject to the rule of the university authorities. Nearly all the medieval universities grew up under the protection of a bishop (or abbot), and were in some matters subject to his authority or to that of his chancellor, from the latter of whom the head of the university subsequently took his title. The universities, however, were not ecclesiastical organizations, and, though the bulk of their members were ordained, their direct connection with the Church arose chiefly from the fact that clerks were then the only class of the community who were left free by the state to pursue intellectual studies.
A universitas magistrorum et scholarium, if successful in attracting students and acquiring permanency, always sought special legal privileges, such as the right to fix the price of provisions and the power to try legal actions in which its members were concerned. These privileges generally led to a recognition of its power to grant degrees which conferred a right of teaching anywhere within the kingdom. The university was frequently incorporated at or about the same time. Paris received its charter in 1200, and probably was the earliest university in Europe thus officially recognized. Legal privileges were conferred on Oxford in 1214, and on Cambridge in 1231: the development of Oxford and Cambridge followed closely the precedent of Paris on which their organization was modelled. In the course of the thirteenth century universities were founded at (among other places) Naples, Orleans, Padua, and Prague; and in the course of the fourteenth century at Pavia and Vienna. The title of university was generally accredited to any teaching body as soon as it was recognized as a studium generale.
The most famous medieval universities aspired to a still wider recognition, and the final step in their evolution was an acknowledgment by the pope or emperor of their degrees as a title to teach throughout Christendom--such universities were closely related one with the other. Paris was thus recognized in 1283, Oxford in 1296, and Cambridge in 1318.
The standard of education in mathematics has been largely fixed by the universities, and most of the mathematicians of subsequent times have been closely connected with one or more of them; and therefore I may be pardoned for adding a few words on the general course of studies in a university in medieval times.
The students entered when quite young, sometimes not being more than eleven or twelve years old when first coming into residence. It is misleading to describe them as undergraduates, for their age, their studies, the discipline to which they were subjected, and their position in the university shew that they should be regarded as schoolboys. The first four years of their residence were supposed to be spent in the study of the trivium, that is, Latin grammar, logic, and rhetoric. In quite early times, a considerable number of the students did not progress beyond the study of Latin grammar--they formed an inferior faculty and were eligible only for the degree of master of grammar or master of rhetoric--but the more advanced students (and in later times all students) spent these years in the study of the trivium.
The title of bachelor of arts was conferred at the end of this course, and signified that the student was no longer a schoolboy and therefore in pupilage. The average age of a commencing bachelor may be taken as having been about seventeen or eighteen. Thus at Cambridge in the presentation for a degree the technical term still used for an undergraduate is juvenis, while that for a bachelor is vir. A bachelor could not take pupils, could teach only under special restrictions, and probably occupied a position closely analogous to that of an undergraduate nowadays. Some few bachelors proceeded to the study of civil or canon law, but it was assumed in theory that they next studied the quadrivium, the course for which took three years, and which included about as much science as was to be found in the pages of Boethius and Isidorus.
The degree of master of arts was given at the end of this course. In the twelfth and thirteenth centuries it was merely a license to teach: no one sought it who did not intend to use it for that purpose and to reside in the university, and only those who had a natural aptitude for such work were likely to enter a profession so ill-paid as that of a teacher. The degree was obtainable by any student who had gone through the recognized course of study, and shewn that he was of good moral character. Outsiders were also admitted, but not as a matter of course. I may here add that towards the end of the fourteenth century students began to find that a degree had a pecuniary value, and most universities subsequently conferred it only on condition that the new master should reside and teach for at least a year. Somewhat later the universities took a further step and began to refuse degrees to those who were not intellectually qualified. This power was assumed on the precedent of a case which arose in Paris in 1426, when the university declined to confer a degree on a student--a Slavonian, one Paul Nicholas--who had performed the necessary exercises in a very indifferent manner: he took legal proceedings to compel the university to grant the degree, but their right to withhold it was established. Nicholas accordingly has the distinction of being the first student who under modern conditions was "plucked."
Although science and mathematics were recognised as the standard subjects of study for a bachelor, it is probable that, until the renaissance, the majority of the students devoted most of their time to logic, philosophy, and theology. The subtleties of scholastic philosophy were dreary and barren, but it is only just to say that they provided a severe intellectual training.
We have now arrived at a time when the results of Arab and Greek science became known in Europe. The history of Greek mathematics has been already discussed; I must now temporarily leave the subject of medieval mathematics, and trace the development of the Arabian schools to the same date; and I must then explain how the schoolmen became acquainted with the Arab and Greek text-books, and how their introduction affected the progress of European mathematics.
· · ·
End of part 11 of 28
Continue readingChapter 9: The Mathematics of the Arabs
Starvard