Medieval mathematics

Mathematics Through the Middle Ages (320-1660AD)

An idosyncratic essay by Paul Dickson for

History of Mathematics 07305
(University of South Australia, 1996)

In the history of mathematics as a science there existed a so called 'Golden Age' centred in ancient Greece and the surrounding Mediterranean from about 600BC to 300AD, many advances were made and recorded in this time.
Then there was the decline of the Dark (or early Middle) Ages that started with the sacking of Rome and the destruction of most of the knowledge contained therein. During this time much of the remaining knowledge of the ancient world was preserved by Byzantium, the rest lay scattered in small monasteries spread throughout Mediterranean Europe. In the period from 300AD to 1600AD there existed two major sub-divisions, the early Middle Ages, or Dark Ages, and the late Middle Ages, just before the Renaissance. In the early Middle Ages mathematics made no progress, but in the late Middle Ages there were a few advances and much of what had been forgotten from the ancient world was rediscovered and re-evaluated. In the late Middle Ages education was introduced in earnest by the Catholic Church and knowledge of these rediscovered techniques was spread to the common man.

The early Middle Ages of Europe spanned nearly a millennia, from 300AD to 1100AD, and saw little advance made in the field of mathematics. The two major contributions of this time were firstly the translation of many Greek works into the language of the time, and the second was the implementation of formal schooling. Major advances in mathematics were a thing of the past, even the errors in ancient texts were taken as truths in mathematics of the time. Kline (1977) suggests that the reason no advances were made in the Middle Ages is because those in power were too concerned with the civitas dei and the preparation for the latter world. Whereas the reason for the stagnation of the Roman Empire was for the exact opposite reason, they were too concerned with the civitas mundi or practical results. Through this and other historical evidence it becomes apparent that mathematics cannot bloom under either of these social climates.
The collapse of the Western Roman Empire did not see the complete distruction of the western civilised world, there existed two surviving splinter groups. One with sufficient power to resist the initial invasion of the Vandals, Goths and Visigoths and the other so effectively spread throughout the civilised and barbarian world by the declining Roman Empire that it's destruction would have been a task much beyond the scope of the barbarian mind. The first of these groups was a remanent of the glory that was once Golden Age Greece, known as Byzantium or the Eastern Roman Empire. The other was the Catholic Church that spread throughout western Europe during the twilight reign of the Western Roman Empire.
Byzantium remained a safe haven for the collected works of ancient Greece and maintained its civilisation from Constantinople until the late Middle Ages when it was overrun by the Arabs and Turks in 1453 AD. It's only achievement of worth in the field of mathematics was preservation of knowledge, no major discoveries or advances were made by Byzantine mathematicians. However it's application of Greek mathematical techniques in architecture and the arts was one of the finest in the ancient world.
The Catholic Church unlike the static Byzantine Empire expanded and flourished, converting many of the barbarians to civilised, god-fearing citizens. It's major contributions were the education of the common person and the translation of ancient works from Arabic and Greek into Latin. Although at a glance this would seem to place the contributions of the Church in a much rosier light than those of Byzantium. However, this is not really the case as the Church also instilled a fear of knowledge in the common populace. The Church guarded it's meagre, corrupted secrets with such fervour that it held inquisitions and witch hunts when it felt threatened in it's monopoly of knowledge. The Church taught what it deemed right as described in the Bible. A major achievement for the Church was the conversion of Philosophy into 'black magic'. St Augustine is accredited with saying "whatever knowledge man has acquired outside of Holy Writ, if it be harmful it is there condemned; if it be wholesome it is there contained" during his time (354 to 430 AD) and seems to have forecast the attitude of the Church toward mathematics up until about 1100 AD. Education was the first beneficial act accomplished by the Church, a major task considering the scientific level of the northern European area, as Kline (1977) states 'An account of the learning, arts, and sciences of the Germanic tribes is readily given. There was nothing.' The Church's education program consisted of schools which taught what was dictated by the Bible and the Pope, they were attached to churches, operated by monks and taught from the geometric, musical, and arithmetic compilations of Anicius Manlius Severinus Boethius (480-524 AD). Later, in Charlemaine's time Alcuin of York (730-804 AD) travelled from his home in England to France on invitation of Charlemaine where he was given the task of organising the schools and implemented a curriculum of Christian theology and music. The first universities arose soon afterwards from large monastery schools with Dominican and Franciscan monks as teachers, the first was the University of Salerno, near Naples in Italy, founded in the 9th century for the further study of medicine.

The late Middle Ages saw the establishment of many new Universities, starting at Bologna and Padua in Italy and Oxford and Cambridge in England in the 13th century. This expansion of knowledge centres continued into the 14th and 15th century with the establishment of Sorbonne in Paris, and Prague, Heidelburg and Louvain in central and northern Europe. Of course many of these institutions were only possible due to the wealth of nobles, church officials and merchants. In this period the Church's monopoly on knowledge was overturned.
Unfortunately the stable society reached in the late Middle Ages was based on the teachings of the Church and the legacy of Rome, since neither was particularly inclined towards mathematics it is doubtful that Western European mathematics would have progressed much at all if it hadn't been for the Crusades and their precursor wars. Sicily was retaken in 1091 AD making the works held there freely accessible. The First Crusade was ordered by Pope Urban II in 1095AD, because of the occupation of the Turks of the pilgrim route to Jerusalem and their subsequent maltreatment of the pilgrims, the Crusaders set out in 1097 AD and retook Jerusalem in 1099 AD. The contact with the Arabs brought old texts back into the possession of the western Europeans and started a hunger burning within the European scholastic community, with a renewed vigour they sought out old texts held by the Arabians in their centers of learning in Africa, Spain, southern France, Sicily and the Near East. Many a European mathematician became a student of Greek thought in this time, disregarding and discrediting their own discoveries in favour of the wisdom of the ancient Greeks. Translation followed quickly afterwards as missions to relearn and recover ancient knowledge were made by many scholars. Adelard of Bath (1090-1150 AD) and Leonardo of Pisa (1175-1250AD), later known as Fibonacci, were two of the most well recognised of these knowledge seekers and travelled far afield to gather information for the European mathematical community (although their motives were probably less nobly based). Leonardo of Pisa wrote Liber Abaci, a free rendition of Greek and Arabic works in Latin which taught the Hindu methods of calculation with integers and fractions, square roots and cube roots, this book made available the masses the number systems heretofore sequestered in monasteries throughout Europe. Unfortunately due to the poor knowledge of Greek as a language the translations made weren't very accurate, although far better than the Arabic translations of Greek, and new, more accurate translations continued until the late 17th century.
Due to the late Medieval scholars, the Scholastics (a group who questioned doctrines based on the authority of the Church Fathers and of Aristotle) in particular, an atmosphere of rationalism was born. The mathematicians of the Renaissance were infused with the idea that 'nature was the creation of God and that God's ways could be understood.' (Kline, 1977) Adelard of Bath said that he would not listen to those who are 'led in a halter;.... Wherefore if you want to hear anything from me, give and take reason.' This was indicative of the attitude of the times as individuals began to pit their own reason against the authority of the Church. One of the most eloquent of the protesters against authority was Roger Bacon (1214-1294AD), who declared his willingness to destroy the inherited works of Aristotle on the basis that they were full of errors and encouraged ignorance. Bacon was one of the most knowledgable of all scholars of his time and had even mastered Arabic, he knew how to obtain reliable knowledge from application of experimentation (previous to this most experimentation was conducted in a search for magical powers) and mastery of texts and he foresaw the inventions of automobiles, aircraft and submarines. Bacon was both a product of his times and a victim, he believed in magic and astrology and maintained that the goal of all learning was theology, yet he was imprisoned for his stance on the priority and independence of human reason. In the mid 14th century Nicole Oresme (1323-1382 AD), the Bishop of Lisieux, wrote a book named the Algorismis Proportionum (c. 1360 AD) which was never published but dealt with notation and computation of fractional exponents. Oresme's work continued in the study of change; uniform motion (motion with constant velocity), difform motion (motion with varying velocity), and uniformly difform motion (motion with constant acceleration). A colleague of Oresme named Jean Buridan (c1300-1360 AD) expanded on the concept of motion, he put forth his theory of impetus. This theory defined impetus as the quantity of matter multiplied by the velocity; hence, in modern terms, it is momentum. Jerome Cardan (1501-1576 AD) was responsible for a major treatise on probability, but this was mostly a by-product of his gambling addiction. The mathematical sections of Cardan's work were without equal, mixed as they were with astronomy and the fading mysticism of the Middle Ages. Cardan's life was like one big probability experiment, it fluctuated wildly but ended with a pension from the Pope. He is variously described as 'a genius, a fool and a charlatan who embraced and amplified all the superstition of his age, and all its learning.' (Cameron, 1983) The last two contributors to mathematics in the late Middle Ages were Rene Descartes (1596-1650 AD) and Pierre Fermat (1601-1665 AD). Descartes is responsible for advances in optics and analytical geometry. Fermat developed formulae for parabolas, hyperbolas, ellipses and spirals of Fermat, all this was done in his spare time as he was a lawyer by trade. These two men made great contributions to mathematics but it is doubtful they would have had the impact they did if it weren't for their friend Father Marin Mersenne (1588-1648 AD) who although not a mathematician himself was responsible for the transmission of all modern advancements in mathematics. Not since the ancient Greeks had there been such an extensive communications network for mathematics, what one mathematician knew soon after a great circle of peers knew too.

In conclusion it seems probable that the Middle Age mathematicians devoted themselves to the tried and tested methods of the Greeks which they gained as a polluted version from the Catholic Church who in turn got a polluted version from the Romans. Later in the period the new Arabic transcriptions gained around the time of the Crusades caused expansion in the merchant world and this new number theory branched quickly into geometry, arithmetic and astronomy. The whole of the period was unproductive for new thought and there seem to have been two contributing factors, one minor the other major. The narrow mindedness of the Catholic Church was the major reason. The friction between the Church and the people grew to such an extent that many openly defied the Church despite the penalties, thus progress was made through defiance of established custom. The minor reason was the government of the time, independent dukedoms, principalities, more or less oligarchic city-states and the Papal States, all warred upon each other and their resources were spent in surviving rather than advancing.

Bibliography


Cameron, Malcolm, 1983. "Heritage Mathematics", 
    Hargreen Publishing Company.

Collins, 1963. "New Age Encyclopedia", 
    Collins Clear-Type Press.

Kline, Morris, 1977. "Mathematics in Western Culture", 
    Pelican Books.

Myers, A. R., 1965. "Late Middle Ages", 
    Pelican Books. 

Stahl, William H., 1962. "Roman Science: Origins, Development 
    and Influence to the later Middle Ages", 
    The University of Wisconsin Press.

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