HERE'S LOOKING AT EUCLID
Professor Robin Wilson
This is the second of three lectures on the multi-cultural origins of mathematics, shown here on this time-line. In my last lecture I introduced you to the world of papyruses and clay tablets, the mathematics of Ancient Egypt and Mesopotamia. Today I'll move forward more than a thousand years to the world of Greek mathematics, before spending the next lecture on the cultures of China, India and Central America.
The period of Greek mathematics lasted for about a thousand years, starting around 600 BC, and falls into three main sections. The first of these concerns the semi-legendary figures of Thales and Pythagoras; the second deals mainly with Athens and Plato's Academy; and for the third we'll visit Alexandria, starting with Euclid and extending over seven hundred years. We'll also make a brief excursion to the island of Sicily to meet Archimedes.
In which parts of the Greek empire was the story acted out? Thales came from Miletus in modern Turkey, while Pythagoras was based in Crotona, now in Italy. Plato's Academy was in Athens, while Alexandria is now situated in Egypt. Archimedes was based in Syracuse in Sicily.
Before proceeding, I'd like to mention the available sources. Unlike Ancient Egypt, where there are a few well-preserved papyruses, and Mesopotamia where there are many thousands of surviving clay tablets, we have hardly any Greek primary sources. As in Egypt, the Greeks wrote on papyrus which didn't last the centuries, although a few fragments have survived from the later period. There were also disasters, such as the fire in the library at Alexandria, in which many primary sources perished.
So we have to rely entirely on commentaries and later versions. The best-known commentator on Greek mathematics was Proclus from the fifth century AD, who supposedly derived much of his material from earlier commentaries (now lost) by Eudemus of Rhodes, a fourth century BC student of Aristotle. But Proclus lived some 800 years after Euclid – it's rather like us trying to comment on contemporary oral and written accounts of Robin Hood – so we have to treat his commentaries with scepticism, while recognising that they're all we've got.
We also have translations and commentaries by Islamic scholars, which are most useful – but the fact remains that our earliest surviving copy of Euclid's Elements is an Arabic translation in the Bodleian Library in Oxford, dating from the year 888, which is as far removed from Euclid's time as it is from our own.
The first period
Moving on to our first period, that of Thales and Pythagoras, we have to admit that we don't know much about either of them. According to legend, Thales came from Miletus, he brought geometry to Greece from Egypt , he predicted a solar eclipse in 585 BC, and he showed how rubbing feathers with a stone produced electricity.
Further, as Proclus commented while writing about Euclid's Elements:
The famous Thales is said to have been the first to demonstrate that the circle is bisected by the diameter.
If you wish to demonstrate this mathematically, imagine the diameter drawn and one part of the circle fitted upon the other.
If it is not equal to the other, it will fall either inside or outside it, and in either case it will follow that a shorter line is equal to a longer.
For all the lines from the centre to the circumference are equal, and hence the line that extends beyond will be equal to the line that falls short, which is impossible.
You'll notice that the Thales extract is concerned with mathematical proof. Of all the many mathematical contributions by the Greeks – and they include the introduction and study of conics and the study of the three classical problems, as you'll see – the idea of deductive reasoning and mathematical proof is the most fundamental. Starting with some initial assumptions – or axioms – we derive some simple results, and then more complicated ones, and so on, eventually creating a great hierarchy of results, each depending on previous ones.
The Greeks adopted several methods of proof – the Thales extract used a proof by contradiction (or reductio ad absurdum), where we first assume that the result we want to prove is incorrect and then deduce a result that contradicts our assumptions. You'll see other examples of this later on.
These ideas were developed around 550 BC by Pythagoras and the Pythagoreans – the brotherhood that supposedly gathered around him in the Greek seaport of Crotona, now in Italy. There's a representation of Pythagoras in Raphael's fresco The School of Athens, even though this School (Plato's Academy) was two hundred years after Pythagoras.
The Pythagoreans believed that all is number – that everything worthy of study can be quantified – and they then divided the mathematical sciences into four parts. Two of these are arithmetic, which deals with numerical quantities, and music, which deals with intervals regarded as simple ratios between these quantities – for example, an octave corresponds to a ratio of 2 to 1 and a perfect fourth gives a ratio of 3 to 2. The other two are geometry, which deals with magnitudes at rest; andastronomy, which deals with magnitudes in motion. These four mathematical areas were later combined into the quadrivium which, together with the trivium of grammar, rhetoric and logic, formed the seven liberal arts – the subjects studied in academies and universities for the next 2000 years or so.
The Pythagoreans were concerned with many areas of mathematics. For them, arithmetic meant both 'arithmos' – ordinary arithmetic, calculating with whole numbers, and also what we now call number theory. For example, they knew how to find the triangular numbers by adding consecutive integers (for example, 15 = 1 + 2 + 3 + 4 = 5), and they realised that square numbers are obtained by adding consecutive odd numbers – for example, 16 = 1 + 3 + 5 + 7. They also studied prime numbers, which we'll meet later.
The Pythagoreans were also very interested in commensurable andincommensurable numbers, which came to play an important role in Euclid's Elements. We say that the numbers 8 and 12 are commensurablebecause they can both be 'measured' an exact number of times by a ruler of length 4, and the numbers 3π and 5π are commensurable because they can both be measured a whole number of times by a ruler of length π. Essentially, two numbers are commensurable if their ratio can be written as a fraction (a ratio of whole numbers) – so 5π divided by 3π is just 5/3, which is a fraction. However, as they discovered, the diagonal and side of a square are not commensurable.
The current proof of this last fact is well known and is typical of the Greek approach, except that here I'll use algebra, whereas the Greeks would have couched everything in geometrical terms. The proof is by contradiction.
By Pythagoras's theorem the ratio of the diagonal and side of a square is √2, and we must prove that this number cannot be written as a fraction – as a/b, where a and b are whole numbers. So, to obtain a contradictionwe assume that √2 can be written as a fraction, and we can assume that this fraction is written in its lowest terms, so a and b have no common factor. By squaring, we can rewrite this as a 2 = 2b 2, which means that a2 must be an even number. But if a 2 is even, then a must also be even (because otherwise, a is odd, so a 2 is odd). Since a is even, we can writea = 2k, for some whole number k. So 2b 2 = 4k 2, which gives b 2 = 2k2, so b 2 is even, and b is even. But this gives us a contradiction – a andb are both even, so both are divisible by 2, contradicting the fact that aand b had no common factor. This contradiction arises from our original assumption – that √2 can be written as a fraction – so this assumption is wrong: √2 cannot be written as a fraction, and so the diagonal and side of a square are incommensurable.
For no apparent reason, Pythagoras's name is associated withPythagoras'sTheorem, even though the Mesopotamians knew the result a thousand years earlier, as we saw last time. But it was in Greek times that the theorem was first proved.
The theorem states that in any right-angled triangle, the area of the square on the hypotenuse (the longest side) is equal to the sum of the areas of the squares on the other two sides. So it's a geometrical result about areas – there's no mention of an algebraic equation such as a 2 + b2 = c 2.
A dissection proof of Pythagoras's theorem is typical of the Pythagorean school. We can draw two different dissections of a square of side a + b. Removing the four triangles in each case, and comparing the red squares, we see that the largest area must be the sum of the two smaller ones. It's a far cry from the axiomatic proof that would appear later in Book I of Euclid's Elements.
Charles Dodgson, better known as Lewis Carroll, was a great enthusiast for Pythagoras's theorem:
It is as dazzlingly beautiful now as it was in the day when Pythagoras first discovered it, and celebrated the event, it is said, by sacrificing a hecatomb of oxen [that's a whole oxen] – a method of doing honour to Science that has always seemed to me slightly exaggerated and uncalled-for. One can imagine oneself, even in these degenerate days, marking the epoch of some brilliant scientific discovery by inviting a convivial friend or two, to join one in a beefsteak and a bottle of wine. But a hecatomb of oxen! It would produce a quite inconvenient supply of beef.
The second period
The second great period of Greek mathematics took place in Athens, with the founding of Plato's Academy around 387 BC in a suburb of Athens called 'Academy' – that's where the word Academy comes from. Plato's Academy is featured in Raphael's fresco The School of Athens with Plato and Aristotle at the top of the steps, and with Pythagoras on the left and Euclid on the right.
Plato's Academy soon became the focal point for mathematical study and philosophical research, and it is said that over its entrance were the wordsLet no-one ignorant of geometry enter these doors.
Plato wrote a short dialogue called Meno in which Socrates asks a slave boy how to double the size of a square. The boy first suggests doubling the side of the square, but that gives four times the area. Eventually he settles on the square based on the diagonal of the original square. It's a wonderful example of teaching by experiment and it's far removed from anything we saw in Egypt and Mesopotamia.
Plato believed that the study of mathematics and philosophy provided the finest training for those who were to hold positions of responsibility in the state, and in his Republic he discussed at length the importance of each of the mathematical arts for the so-called 'philosopher-ruler': arithmetic, geometry (which he divided into plane geometry and solid geometry), astronomy and harmonics.
His book Timaeus includes a discussion of the five regular, or 'Platonic', solids – the tetrahedron, cube, octahedron, dodecahedron and icosahedron – in which the faces are all regular polygons of the same type and the arrangement of polygons at each corner is the same: for example, the cube has six square faces, with three meeting at each corner. He also linked four of the polyhedra with the Greek elements of earth, air, fire and water, and assigned the cosmos to the dodecahedron, which had only recently become known. One hundred years later, Archimedes would find all the semi-regular solids, in which the faces are all regular polygons, but they're not all the same – for example, the truncated cube, obtained from a cube by chopping off the corners, is made up of triangles and octagons.
One of the students at the Academy was Aristotle, who remained there for some twenty years. He was fascinated by logical questions and systematised the study of logic and deductive reasoning. In particular, he studied the nature of mathematical proof, and considered syllogisms such as All men are mortal, Socrates in a man; therefore Socrates is mortal. Another early student there was the mathematician and astronomer Eudoxus of Cnidus, who advanced the hypothesis that the sun, moon and planets move around the earth on rotating concentric spheres, a hypothesis later adopted in modified form by Aristotle. Eudoxus is often credited with developing the theory behind two of the books in Euclid'sElements – Book V on proportion and Book XII on the so-called method of exhaustion.
It was around this time that the Greeks adopted a new counting system. This was a decimal system in which separate Greek letters were used for 1, 2, 3, … up to 9; then new letters for 10, 20, … up to 90; and then nine further letters taking them to 900.
It was also around this time that the three classical problems emerged. Each asked for a construction that uses only a straight edge and a pair of compasses – no measuring was to be allowed. The first problem was that of doubling thecube, said to be based on the need to double the size of an altar at Delos to appease the Gods: given a cube, construct another cube with twice the volume – this effectively involved constructing the cube root of 2. The second is trisecting the angle: given any angle, construct two lines that divide it into three equal parts. The third is the best known –squaring the circle: given a circle, construct a square with the same area. These problems fascinated the Greeks, but in the event it would take two thousand years until all three constructions were proved to be impossible.
The third period
Around 300 BC, with the rise to power of Ptolemy I, mathematical activity moved to the Egyptian part of the Greek empire. In Alexandria Ptolemy founded a university that became the intellectual centre for Greek scholarship for the next 800 years – our third period of Greek mathematics. Ptolemy also started its famous library, which held over half-a-million manuscripts before eventually being destroyed by fire.
A number of important mathematicians were associated with Alexandria – Apollonius (who wrote the standard work on conics), the great astronomer Ptolemy (after whom the Ptolemaic system of planetary motion is named), and the Neo-platonists such as the geometers Pappus and Hypatia (one of the most important women mathematicians of all time). But the greatest of all, and the first important mathematician to be associated with Alexandria , was Euclid, who lived and taught there around 300 BC.
We know virtually nothing about Euclid's life. As well as the Elements, he's credited with writing many other books, including several in geometry (the Data and On Divisions of Figures), the Porisms (a three-book work on problems which hasn't survived), a four-book work on Conics (which also hasn't survived), and books on Astronomy and Optics.
Euclid's Elements was a compilation of results known at the time, organised in a systematic way. It wasn't the earliest such work – Hippocrates of Chios and others had written Elements earlier, though these haven't survived. However, Euclid's was the most important. As the commentator Proclus observed:
It is a difficult task in any science to select and arrange properly the elements out of which all other matters are produced and into which they can be resolved.
Such a treatise ought to be free of everything superfluous, for that is a hindrance to learning; the selections chosen must all be coherent and conducive to the end proposed, in order to be of the greatest usefulness for knowledge; it must devote great attention both to clarity and to conciseness, for what lacks these qualities confuses our understanding.
Judged by all these criteria, you will find Euclid's introduction superior to others.
As a result, Euclid's Elements is the best-selling mathematics book of all time, over more than 2000 years – indeed, it's possibly the most printed book ever, apart than the Bible. Much has been written about it: the philosopher Immanuel Kant observed that:
There is no book in metaphysics such as we have in mathematics. If you want to know what mathematics is, just look at Euclid's Elements.
and the Victorian mathematician Augustus De Morgan observed:
The thirteen books of Euclid must have been a tremendous advance, probably even greater than that contained in the 'Principia' of Newton. The sacred writings excepted, no Greek has been so much read and so variously translated as Euclid.
Euclid's Elements consists of thirteen books, traditionally divided into three main parts – on plane geometry, arithmetic and solid geometry. Here's a quick overview.
Books I and II deal with the foundations of plane geometry and the geometry of rectangles; Books III and IV then proceed to the geometry of circles. Book V is on proportion, which is then applied to the geometry of similar figures in Book VI. Books VII, VIII and IX are on arithmetic, and include basic properties such as the divisibility of integers and the so-called Euclidean algorithm, as well as a discussion of prime numbers and perfect numbers. BookX, the longest and most difficult book amounting to a quarter of the whole work, is on incommensurable line segments. The final three books are on solid geometry, and conclude with the construction and classification of the five Platonic solids. I won't go through all these systematically, but I've chosen a few topics that interest me.
Book I starts with twenty-three definitions of basic terms such as point, line and circle. There are then five geometrical Postulates, beginning with three allowable constructions:
- To draw a straight line from any point to any point.
- To produce a finite straight line continuously in a straight line.
- To describe a circle with any centre and distance.
He then observes:
- That all right angles are equal to one other.
The last postulate is much longer:
- That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.
This fifth postulate seems to be of a different style from the others – in fact, as we'll see in a later lecture, it caused no end of problems over the next 2000 years.
Finally, we get down to our first proposition:
Proposition 1. On a given straight line to construct an equilateral triangle.
To do this, we draw the line AB and trace out circles with centre A and radius AB and with centre B and radius AB. These meet at a point C, and the triangle ABC is then the required equilateral triangle. Euclid then proceeded to prove that the construction actually works – that the resulting triangle is equilateral. At each stage there is a reference to a definition or a postulate, and in later propositions there are frequent references to earlier propositions.
Here are some other Propositions from Book I. Proposition 5 is the famous Pons Asinorum, the bridge of asses, that in an isosceles triangle, the angles at the base are equal to each other. This result is credited to Thales, and in medieval universities it was often as far as students of Euclid ever reached: if you could cross the bridge of asses, you could then go on to all the treasures that lay ahead! Propositions 1 to 26 are all basic results and constructions in plane geometry, such as the bisection of an angle and congruence theorems for triangles, and these are followed by nineteen propositions on parallel lines and parallelograms. These include the results that the angles of a triangle add up to two right angles, and that given any triangle, we can construct a rectangle with the same area. Since any polygon, such as a pentagon, can be split into triangles, we can construct a rectangle with the same area as any given polygon – when combined with other results, this shows that we can square any polygon, even though we can't square the circle.
Book III introduces the properties of circles – I'll mention just a couple of results here:
Proposition 20. In a circle the angle at the centre is double the angle at the circumference when the angles have the same arc as base.
A special case is:
Proposition 31. In a circle the angle in a semicircle is right.
and a related result is:
Proposition 22. The opposite angles of quadrilaterals in circles are equal to two right angles.
Books VII to IX take us into arithmetic – but the descriptions are still given in geometrical terms – using lengths of lines to represent numbers, rather than the numbers themselves. A good example of this is Euclid's proof in Book IX that there are infinitely many prime numbers – one of the most famous proofs in the whole of mathematics.
Here's the modern way of writing out the proof: recall that a prime number is a number whose only factors are itself and 1 – for example, 11, 13, 17 and 19 are all prime numbers, whereas 15 isn't.
Suppose, for a contradiction, that the only prime numbers are p 1, p 2, … , p n and form the number N = p 1 p 2 … p n + 1. Since each of the primes p 1, p 2, … , p n divides their product p 1p 2 … p n, none of them can divide N. So N is itself a prime number, or else is divisible by some prime number different from p 1, p 2, … , p n. In either case, there is a prime number different from the given ones, giving the required contradiction. So there are infinitely many primes.
Euclid's proof involves the lengths of lines, and starts with only three lines of prime length, representing the general case. Apart from this, the proof is essentially the same.
The final three books deal with aspects of three-dimensional geometry. Of these, Book XIII is the most remarkable. It introduces the five regular solids that we saw earlier, and then shows how they can be constructed. For example, to construct a dodecahedron, we take a cube and add to each face a 'roof' whose proportions are such that the faces all become pentagons; these proportions involve the so-called 'golden ratio', whose geometrical properties are worked out in detail at the beginning of Book XIII.
In this book, Euclid also proved such remarkable results as the following:
If an equilateral pentagon is inscribed in a circle, the side of the pentagon is equal in square to that of the hexagon and that of the decagon inscribed in the same circle.
In other words, if we calculate the lengths of the sides of a pentagon, a hexagon and a decagon inscribed in a circle, then these turn out to be the lengths of the sides of a right-angled triangle. He also presented this complicated diagram in his construction of an icosahedron.
Euclid concluded the Elements by proving that the only possible regular solids are the tetrahedron, cube, octahedron, dodecahedron and icosahedron – there can be no more. This is the first ever 'classification theorem' in mathematics, and forms a fitting climax to this great work.
Euclid's Elements was warmly received, and quickly replaced all its predecessors and competitors. After the invention of printing, an enormous number of printed versions appeared. Two I particularly like – the first English edition, produced by Henry Billingsley in 1570, and Oliver Byrne's colourful edition of the first six books in 1847 in which all the symbols are replaced by coloured diagrams.
One major figure who seems not to have been associated in any substantial way with Alexandria was Archimedes, a native of Syracuse on the island of Sicily. Here's looking at Archimedes – notice how different artists and sculptors had different ideas on what he might have looked like – this is Ribera's portrait of him in the Prado Museum in Madrid.
One of the greatest mathematicians of all time, Archimedes is mainly remembered for running naked through the street shouting Eureka (I have found it). The reason for this outburst is apparently that his friend King Hiero wanted to ascertain whether his crown was of pure gold or whether it was partially made of silver. To solve this problem, Archimedes immersed the crown in water and observed that the weight was reduced by an amount equal to the weight of water displaced, from which he could make the necessary calculations.
Another result of his was the law of the lever – that if unequal weights are placed at opposite ends of a scale, they balance at distances that are inversely proportional to the weights. This is sometimes called the law ofmoments: W 1a = W 2b, or a/b = 1/W 1 divided by 1/W 2.
But Archimedes didn't only work in applied mathematics. A list of his works includes studies on the measurement of a circle, on the so-calledArchimedeanspiral, on properties of the sphere and cylinder and other solids, and The sandreckoner on enormous numbers in which he estimates the number of grains of sand in the universe.
Among his geometrical results are calculations of centres of gravity for a triangle, hemisphere and parallelogram, his impressive calculations of volumes and surface areas such as the sphere, and his celebrated result (which he apparently wanted engraved on his tomb) that the surface area of any horizontal section of a sphere is the same as that of the surrounding cylinder. And although we can't square the circle, Archimedes showed that parabolas can be squared – the area of any parabolic segment is 4/3 times the area of the enclosed triangle, and the corresponding square can then be obtained.
Some of his most profound results can be found in TheMethod, and there was much excitement recently when a palimpsest turned up (and sold for two million dollars) on which a monk had overwritten what was already known to be an Arabic version of The Method. This is still being worked on and interpreted, and is due to go on exhibit around the world in 2007.
One of Archimedes' most well-known results is his invention of a method for estimating π – by drawing hexagons inside and outside a circle and comparing their perimeters with the circumference of the circle. This wasn't very accurate, so he replaced the hexagon by a 12-sided polygon, then 24, 48 and 96 sides, obtaining successively better estimates. He found that π is a little bit less than 22/ 7, the value we learned at school, and a little bit more than 3 10/ 71 – this gives a value of about 3.14, correct to two decimal places.
Back in Alexandria, Apollonius was writing his celebrated treatise onconics. These curves can be traced back to Menaechmus in the fourth century BC, and arise from slicing a cone in various ways. There are three different types – the ellipse (with the circle as a special case), theparabola, and the hyperbola.
The frontispiece of Edmond Halley's 1710 edition of Apollonius's celebrated treatise on conics shows the philosopher Aristippus, shipwrecked on the island of Rhodes, noticing some conics that had been drawn in the sand and claiming that the inhabitants must thus surely be civilised.
In astronomy, too, there was much activity. Eudoxus, whom I mentioned earlier in connection with the theory of proportion in Euclid's Elements, advanced the hypothesis that the sun, moon and planets move around the earth on rotating concentric spheres, a hypothesis later adopted in modified form by Aristotle.
Aristarchus advanced an alternative hypothesis – that the fixed stars and the sun remain unmoved while the earth revolves about the sun in a circle – anticipating by 1700 years the revolutionary work of Copernicus, but his hypothesis found few supporters at the time.
Trigonometry made its first appearance around 150 BC by Hipparchus, possibly the greatest astronomical observer of antiquity. Sometimes called 'the father of trigonometry', he constructed a 'table of chords' giving essentially the sines of certain angles. He also constructed a star catalogue using a coordinate system for the stars.
The earth-centred hypothesis was developed by Ptolemy of Alexandria around 150 AD, and it eventually became known as the Ptolemaic system. Ptolemy wrote an important 13-volume work, usually called by its Arabic name Almagest, containing a mathematical description of the motion of the sun, moon and planets, and including a table of chords listing the sines of angles from 0° to 180° in steps of ½°. He also published a work on map-making called Geographia, discussing various types of map projection and containing the latitude and longitude of 8000 places in the known world: his maps were used by navigators for many centuries.
I'd like to conclude with three mathematicians from later times. Diophantus of Alexandria wrote an important Arithmetic, in which he solved a number of problems whose answers were to be given as whole numbers, or as fractions. A typical example is: find three numbers such that the product of any two added to the third is a square. In the seventeenth-century French translation of Diophantus's Arithmetic by Bachet, Fermat annotated the margin claiming to have a proof of what is now known as 'Fermat's last theorem'. We don't know when Diophantus lived, but it may have been around 250 AD. All we have is a puzzle from the Greek anthology which states that Diophantus spent 1/ 6 of his life in childhood, 1/ 12 in youth, and 1/ 7 more as a bachelor. Five years after his marriage there was a son who died four years before his father, at ½ his father's final age. You can work out from this that Diophantus lived to be 84.
Another later Greek mathematicians is Pappus of Alexandria, from the early fourth century AD, and I'd like to mention two contrasting results of his. In his treatise On the sagacity of bees, he credited bees with a certain geometrical forethought in planning their honeycombs. After showing that there can only be three regular arrangements – with triangles, squares and hexagons – he noted that the bees in their wisdom chose that which has the most angles, perceiving that it would hold more honey than the other two.
Pappus's other result is one of the great theorems of mathematics, and you might like to try it out for yourself. Draw two lines, and mark pointsA, B, C on one and a, b, c on the other. Then join A with b and c, B with aand c, C with a and b – this provides three new points of intersection. Amazingly, whatever points you originally chose, these new points always lie on a straight line.
The last person in today's story was the first great woman mathematician. Women scholars had always been acceptable to the Greeks – for example, they were welcomed at Plato's Academy – and around 400 AD we meet Hypatia, daughter and pupil of the geometer Theon of Alexandria. A fine geometer herself, she became Head of the Neoplatonic school in Alexandria, and was such a renowned expositor and lecturer that people came many miles to hear her. She is credited with impressive commentaries on many classic texts, such as Apollonius's Conics, Ptolemy's Almagest, and Diophantus's Arithmetic, and she showed how to construct various astronomical and navigational instruments, such as the astrolabe. Tragically, her life was cruelly and savagely cut short when she was murdered by a mob of fanatical Christians in 415 AD. After her death, little further was achieved in Alexandria, and the city fell to the Arabs in 641.
This brings to an end one thousand years of Greek mathematics, and I hope that you found the journey worthwhile. If not, may I conclude by recalling that a student in Alexandria who had begun to read geometry with Euclid asked him 'What advantage shall I get by learning these things?' Euclid called his slave and said 'Give him threepence, for he must needs make profit out of what he learns.' I do hope that you don't all want to claim your threepences from me.
Robin Wilson, 27 October 2004