Despite Europe’s plunge into the Dark Ages – so-called because it was thought that following the fall of the Roman Empire the continent had reverted back to a barbaric state of tribal warfare and religious fundamentalism – there remained a coterie of individuals intent on pushing the boundaries of mathematics even during these difficult times.
BEDE (672–735)
The Venerable Bede is known more perhaps for his contribution as a historian than for the role he played in the development of mathematics. Bede was a monk living in north-eastern England and his translation of a number of scholarly works into the English of the time helped to spread an enormous amount of knowledge.
Bede’s contribution to mathematics began when he attempted to develop a way to calculate accurately when Easter would fall. At the time it was thought to fall on the first Sunday after the first full moon following the spring equinox. Missing Easter mass following the calculation of an incorrect date would have resulted in excommunication, and therefore damnation, so Bede’s was no trivial task.
Dating in the Dark Ages
In order to calculate the date of Easter, it was necessary for Bede to rationalize the date of the spring equinox with the lunar calendar. This was a difficult task in itself because the date of the equinox varied because the Julian calendar in use at the time was unreliable. Because the date of the equinox varied each year, and full moons come at alternate 29- or 30-day intervals, it meant that there was a 19-year cycle of possible dates for Easter. The procedure for calculating the date of Easter has been known as computus (meaning ‘computation’) ever since.
Once Bede had completed the computus, he decided to sort out dating the rest of history as well. Prior to Bede’s endeavours, historians had been dating things in reference to the lifetime of the current emperor or king, for example: ‘the Vikings first attacked in the third year of Aethelred’s reign.’ This method, of course, relied on the reader knowing when Aethelred was around in the first place. Bede decided that it would be far more sensible to date everything occurring either before or after the birth of Jesus Christ. Although not originally Bede’s idea – that responsibility lay with Dionysius Exiguus, a south-eastern European monk active during the sixth century – such was his influence that we have been using AD (Anno Domini, Year of the Lord) and BC (Before Christ) ever since.
Finger Talk
Bede also wrote a book called On Counting and Speaking With the Fingers, which allowed the reader to use hand signals for numbers into the millions – a super-sized version of the systems we saw Stone Age cultures using. Again, such was his influence, people were still referencing Bede’s book 1,000 years later.
ALCUIN OF YORK (730–804)
A gifted poet, scholar, teacher and mathematician, Alcuin of York began his academic life under the instruction of Archbishop Ecgbert of York, who in turn had been tutored by Bede. Alcuin’s main mathematical work was a textbook for students titled Propositiones ad acuendos juvenes (Problems to Sharpen the Young). The book contains many word-based logic puzzles, a few of which have become quite famous, including the following two river-crossing problems.
The first problem relates to a man trying to cross a river with a wolf, a goat and a cabbage. The man’s boat is very small and he can only fit one thing in the boat with him at a time. However, if he leaves the goat and the wolf together, the wolf will eat the goat. If he leaves the goat and the cabbage together, the goat will eat the cabbage. How does he get them all safely across the river?
Answer: this is a good medieval example of lateral thinking. Clearly, on his first run the man can only take the goat across the river. On his second trip he brings the wolf across but takes the goat back with him; he then leaves the goat there and takes the cabbage across, and then makes a final trip for the well-travelled goat.
Family matters
In the second problem, a couple, who are of equal weight, have two children, each of whom weighs half the weight of one of the adults. All four people need to cross a river, but their boat will only hold the weight of one adult. How can they cross in safety?
Answer: the children cross the river in the boat. One child stays on the far bank while the other child returns. Dad crosses to the far bank and the child returns with the boat to be with the mum and the other child. The two children cross again and one remains on the far side with Dad. The other child returns to be with Mum. Mum crosses to the far side, and the child with the dad returns to collect the other child to reunite the family.
In 781 Alcuin joined the court of Charlemagne, the King of the Franks, where his skills as a teacher were in great demand. While there Alcuin introduced the trivium and quadrivium, which he had encountered during his time in York.
During medieval times only seven subjects were taught in schools and universities. The trivium (Latin for ‘three roads’) comprised logic, grammar and rhetoric. Logic was seen as the way to organize one’s thinking, grammar the way to express these thoughts without confusion and rhetoric was the way to persuade others that your correctly expressed thoughts were worth listening to.
After graduating in the trivium, worthy students could attempt the quadrivium (Latin for ‘four ways’): geometry, arithmetic, astronomy and music.
The trivium was the equivalent of an undergraduate course and the quadrivium a Master’s degree. Succeeding in these courses of study gave access to the Doctorates, either of Philosophy or Theology.
GERBERT D’AURILLAC, POPE SYLVESTER II (946–1003)
Born in France, D’Aurillac joined a monastery during his teenage years, from where he was sent to Spain for further education. Under significant Arabic influence, Spain exposed D’Aurillac to the wonderful discoveries of the Islamic mathematicians. D’Aurillac carved a name for himself as an excellent teacher and was taken on as a royal tutor. His political career soon followed, culminating in him becoming the first French Pope in the year 999.
In his elevated position, D’Aurillac introduced the Hindu-Arabic number system to Europe (see here), although it did not immediately become widely accepted. He was also responsible for re-introducing the abacus, which had not been used since Roman times but which soon became commonplace.
LEONARDO OF PISA (FIBONACCI) (c. 1170–1250)
The son of an Italian trader, Fibonacci lived near Algiers in North Africa, where he gained his first taste for Arabic mathematics. He travelled widely around the Islamic world to further his learning and published a seminal book on his findings called Liber Abaci (Book of the Abacus). Fibonacci’s approach to writing the book showed a keen head for business – not only did he expound the advantages of the Hindu-Arabic number system, he also applied it directly to banking and accounting. Fibonacci’s book became very popular among medieval European scholars and businessmen, and his success earned him the patronage of the Holy Roman Emperor. A triumph, Fibonacci was then able to continue his mathematical work in the fields of geometry and trigonometry.
Fibonacci’s name is well known for the sequence of numbers named in his honour. The sequence derived from, of all things, a problem about rabbits that he posed in his Liber Abaci.
At it like rabbits
Fibonacci numbers were known to Hindu mathematicians long before Fibonacci encountered them, but, much like Blaise Pascal (see here), Fibonacci became eponymous with the sequence after it appeared in his book. In his problem, Fibonacci considers the growth of a rabbit population in a field. Fibonacci conjectured rabbits could start mating after they’d reached the age of one month, and could reproduce every month thereafter. Therefore, if you start with one pair of newborn rabbits (one male and one female) in a field, how many pairs will you have in one year (if each female rabbit continues to breed one male and one female)?
You can see the pattern emerging in the right column: 1, 1, 2, 3, 5. You can calculate the next number in the sequence by adding the previous two numbers in the sequence together. So, next month there will be 3 + 5 = 8 pairs in the field. If you continue with this pattern until there are 13 terms in the sequence (which takes you to the end of the twelfth month of the year), you get the following sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233. Therefore there will be 233 pairs of rabbits at the end of a highly sexed, not to mention very incestuous, year.
Although the rabbit example is biologically inaccurate, the Fibonacci numbers do crop up in all manner of natural settings:
The number of petals on some flowers form part of the Fibonacci number sequence.
Plant shoots often split in such a way that the number of stems follows a Fibonacci
pattern.
The scales on a pineapple make three spirals, each of which contains a Fibonacci number of
scales.
A SYMBOLIC SHIFT
Today, even the least mathematically minded schoolchild understands the four symbols we use for arithmetic: + - × ÷, and the sign we use to show our answers, =. Before the invention of these shorthand ways of writing, the words were written out in full, which made following a mathematical treatise even more cumbersome than it is now.
One of the great poets of the Middle Ages, Geoffrey Chaucer is not normally associated with the disciplines of science and mathematics. Chaucer, however, also led a sideline in astronomy and alchemy, the latter of which sought to discover the philosopher’s stone, the means by which base metals could be turned into gold.
As part of these activities Chaucer became an expert at using a device called an astrolabe, a circular disc that allows you to find certain celestial bodies in the night sky for any given latitude. Having detected his son Lewis’ interest in science from an early age, Chaucer wrote A Treatise on the Astrolabe in his honour. Understandably, perhaps, it is a rather dry text, despite Chaucer’s attempts to enliven the subject by writing the book in verse.
The words plus and minus are, respectively, Latin for ‘more’ and ‘less’. In medieval times the letters ‘p’ and ‘m’ were used to denote these two actions, until German mathematician Johannes Widmann first used the + and - symbols in his 1489 work Nimble and Neat Calculation in All Trades.
Next came the equals sign: =, first used in Welsh physician and mathematician Robert Recorde’s catchily named book The Whetstone of Witte (1557) – one of the first books on algebra to be published in Britain. In The Whetstone of Witte, Recorde states his intention to use symbols ‘to avoide the tediouse repetition of these words’.
The multiplication symbol, ×, came later on in Englishman William Oughtred’s book The Key to Mathematics, which was published in 1631. John Wallis, chief cryptographer for Parliament, first used the ouroboros symbol, ∞, to mean infinity, in his 1665 book De sectionibus conicis, in which he considers cones and planes intersecting to form curves (which today is referred to as conic sections).
The division sign, ÷, is technically called an obelus, and it was first used in Swiss mathematician Johann Rahn’s book Teutsche Algebra in 1659. Ever concise, Rahn was also the first to use ‘.·.’ to mean ‘therefore’.
The Dark Ages, it seems, weren’t quite so barren after all. The slow diffusion of mathematical knowledge from the East allowed Europeans to catch up gradually with their Islamic counterparts. And what happened next allowed the Europeans to take the lead...
