From 0 to Infinity in 26 Centuries

by Chris Waring

  An Education

  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.

  All Part of Nature

  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.

  Geoffrey Chaucer (1343–1400)

  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...

  The Renaissance Onwards

  The Renaissance began in Italy as early as the twelfth century and witnessed great advances in all fields of intellectual endeavour. It sparked a new-found interest in the culture of the classical civilizations, which made an appreciation for – not to mention an investment in – science, culture and philosophy, the done thing among the wealthy patrons of fourteenth-century Europe.

  The initial epicentre of the Renaissance was Florence, Italy, where a wealthy merchant family, the Medicis, became sponsors of art and culture. One artist who benefited from their philanthropy was Leonardo da Vinci.

  LEONARDO DA VINCI (1452–1519)

  Legendary for his talent in almost every field of intellectual pursuit, da Vinci was equally adept in the arts and the sciences. His superior ability and imagination enabled him both to paint the Mona Lisa and to invent flying machines, among other extraordinary feats.

  Perfectly proportioned

  Da Vinci was also a keen anatomist, perhaps out of a desire to bring an element of realism to his art. He was very interested in the relative proportions of the human body, and his famous drawing of the Vitruvian Man demonstrates his unders
tanding.

  The name of the picture harks back to a Roman architect called Vitruvius. He believed the proportions of the human body are naturally pleasing to the eye, which led him to design his own buildings along similar proportions. Vitruvius considered the navel to be the natural centre of a man’s body. He believed a square and a circle drawn over an image of a man with his legs and arms outstretched would represent the natural proportions of the body. Many artists tried to draw human figures that adhered to Vitruvius’ proportions, but all looked somehow misshapen. Da Vinci, however, discovered the correct drawing could be made if the centre of the circle and the centre of the square are positioned differently.

  A Helping Hand

  Although da Vinci himself was not a trained mathematician, he did spend time receiving instruction from Luca Pacioli, a highly regarded maths teacher and accountant. Da Vinci created many drawings of solids for one of Pacioli’s books, and his technical expertise with perspective helped to make the diagrams clear.

  NICOLAUS COPERNICUS (1473–1543)

  A Polish astronomer, Nicolaus Copernicus was one of the first to propose the heliocentric model of the universe: the earth is not the centre of the universe; it does, in fact, orbit the sun. While this was not a mathematical discovery in itself, the way in which Copernicus devised his theory had significant implications for mathematics and science.

  The march of science

  According to the Bible, the earth was the centre of the universe, which was a perfectly reasonable, if slightly self-important, assumption to have made. After all, both the sun and the moon appear to orbit the earth every day; indeed, all other objects in the night sky appear to do the same.

  However, an exploration into the world of astronomy soon revealed problems with this assumption. For example, there are times when the planets appear to move in reverse, which could not be explained if the earth is stationary.

  Scientists at the time worked empirically, which means they made observations of phenomena and then came up with an explanation to fit what had been observed. But Copernicus did something that was considered very backward by scientists at the time – he first came up with a theory about how the solar system might work and then tested it against observations, using mathematics as his main tool.

  While Copernicus’s heliocentric model did not cause much of a stir at the time, his way of working theoretically was one of the first examples of a new way of conducting modern scientific methods.

  JOHN NAPIER (1550–1617)

  A Scottish nobleman, John Napier was responsible for inventing a new kind of abacus called Napier’s bones. He also discovered logarithms.

  Napier’s table

  Logarithms are very important in many fields of mathematics and Napier’s book Description of the Wonderful Rule of Logarithms was quickly adopted by those who had to conduct such calculations on a regular basis. It took Napier an astonishing twenty years to perform the calculations required for the tables of logarithms in the book – that’s some dedication.

  The logarithm of a number is the number we have to raise ten by in order to generate that number. For example:

  The logarithm of 100 is 2 because 100 = 102

  The logarithm of 1000 is 3 because 1000 = 103

  The shorthand for writing this would be log (1000) = 3

  We can find logarithms for numbers that are not whole powers of ten too:

  log (25) = 1.39794 because 25 = 101.39794

  We can also find the logarithm for numbers using any number, not just 10, as a base:

  log5 (25) = 2 because 25 = 52

  Natural logarithms (see here) are logarithms with a base of e, which is a very special number in mathematics that allows many difficult calculus problems to be solved.

  Logarithms today are computed using a calculator or computer, but originally they were worked out either by hand or using Napier’s tables. Before desktop calculators became commonplace people would use logarithms and a slide rule (see here)to perform difficult calculations.

  Napier’s bones

  Napier also devised a faster and more convenient way of performing multiplication, based on a lattice method that Fibonacci had learned from the Arabs.

  Napier’s bones was a useful tool that consisted of a set of sticks engraved with numbers, and each stick had a times table written on it, from 2 times the number up to 9 times the number:

  If you wanted to multiply 567 by 3, for example, you would collect together the three sticks that match the large number and then highlight the third row:

  You then add the diagonal rows shown on the bones in order to calculate the answer.

  Good Point

  Napier was also one of the first people to use the decimal point. Although the Hindu-Arabic numeral system was in common use across Europe by this point, a standard way of writing fractions was yet to be formalized. Because Napier needed a concise way to write them for his log tables, he adopted the Hindu-Arabic decimal fractions we use today.

  WILLIAM OUGHTRED (1574–1660)

  An English mathematician and teacher, William Oughtred continued Napier’s work on logarithms. He is credited with inventing the slide rule, a calculating device that allowed the user to multiply large numbers together using a ruler marked with logarithmic scales (see here), which meant the answer to the multiplication could simply be read off the ruler. Slide rules were used by scientists, engineers and mathematicians up until electronic calculators became commonplace in the 1970s.

  In action

  The slide rule works because of the law of logarithms:

  log (a × b) = log (a) + log (b)

  Let’s test this out using some easy numbers:

  log (10 × 1000) = log (10) + log (1000)

  = 1 + 3

  = 4

  From this we know that 10 × 1000 = 104 = 10000.

  Therefore, if you want to multiply two numbers together you can ‘take the log’ of each number, add them together and then raise ten by this number. Before the slide rule, you would have had to look up this answer in a big, expensive book of tables – so the slide rule was a remarkably convenient development for mathematicians and scientists.

  Jumping ahead

  Rather than spacing out successive markings along an axis by adding on the same value each time, a logarithmic scale spaces them out by a multiple (normally of 10) each time. For example, a graph with a normal linear scale would go 0, 1, 2, 3; a graph with a logarithmic scale would go 0, 1, 10, 100. The graphs of y=10x shown below are with a linear and logarithmic scale respectively.

  RENÉ DESCARTES (1596–1650)

  Born in France, René Descartes was an important philosopher, perhaps best known for coining the statement ‘I think, therefore I am’.

  A different equation

  Descartes was a sickly child and was allowed to sleep in every morning to recuperate, which became a lifelong habit. Apparently, during one such lie-in Descartes was watching a fly walk across the ceiling and wondered how he might be able to describe accurately the position of the fly at any given time. He realized that if he mapped out the ceiling with a square grid he could use what we now call coordinates to record exactly where the fly was positioned.

  These Cartesian graphs proved to be incredibly useful in mathematics because they linked the fields of geometry and algebra. Equations could now be drawn on a set of numbered axes, which allowed them to be investigated more easily by sight, rather than solving them algebraically.

  Cartesian geometry encouraged mathematicians to think about the graphical properties of equations, such as whether or not the lines of equations are parallel. To find out whether lines are parallel you need to work out their gradient. If two lines have the same gradient they must be parallel. Only the simplest linear equations are a straight line, which makes their gradients straightforward to work out. Curves, however, have a changing gradient, but thanks to Descartes’ breakthrough the way was paved for Isaac Newton and Gottfried Wilhelm Leibniz to discover calculus (see here).<
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  The Proof of the Matter

  Descartes’ philosophy also assumed a mathematical bent because he believed the universe was set out according to mathematical rules. For example, in his book Meditations on First Philosophy, Descartes discards any beliefs he holds that are unproven, and then builds his philosophy of the way in which reality works without recourse to anything other than proven fact. This sceptical viewpoint is one of the ideas that underpins modern science and makes mathematics the scientist’s most powerful and productive tool.

  PIERRE DE FERMAT (1601–65)

  A French lawyer, Pierre de Fermat spent his spare time pursuing his love of mathematics. Although he did not publish any of his ideas when he was alive, he did share them in letter form with his mathematical contemporaries. Frustratingly, however, Fermat seldom found it necessary to provide proof for his work.

  Fermat worked in many areas of mathematics, but the area for which he is famed is number theory: the study of integers (whole numbers) and the attempt to find integer solutions to equations.

  Fermat is famous for his ‘last theorem’, which he wrote in his copy of Diophantus’ Arithmetica, and which was discovered by his son after his death (see here). Fermat was also interested in perfect numbers (see box here) and primes.

  In their prime

  Fermat devised a method of testing whether or not a number is prime that relies on an algebraic trick known as the difference of two squares. This says that: