String, Straightedge, and Shadow the Story of Geometry

by Julia E. Diggins

  Elsewhere in the city, the impact of the new geometry took another form. On the narrow streets of Athens walked world-famous philosophers, talking to the people, lecturing on mathematics, geography, rhetoric, how to live the good life. Socrates and others asked, ‘What is beauty? What is virtue?’ – and tried to teach people to think out the answers.

  Their method was borrowed from the geometers. They called it dialectics, and it was based on the deductive reasoning and proofs of geometry. ‘For geometry,’ they said, ‘will lead the soul toward truth and create the spirit of philosophy.’

  Geometry itself made tremendous strides in the Golden Age and the darker time that followed. Even after Athenian democracy collapsed in the war with Sparta, geometry continued to flourish in the Athens of the restored aristocracy.

  Eudoxus of Cnidus

  Plato’s Academy

  In the fourth century BC, study was carried on in schools with grounds and buildings of their own. The first and most famous of these was the Academy, headed by the great philosopher Plato. It was located in an olive grove a half-mile outside of town, and over its gate was this inscription:

  Let none ignorant of geometry enter here

  Plato’s Academy was the earliest institution of higher learning. Its curriculum was frankly inspired by the old programme of the Pythagorean Order. Studies were broader now – the highest branch was moral and political philosophy. But the ideal was still pure wisdom, and the basic training was still in the mathemata.

  When his teacher, Socrates, was put to death by the Athenian government, Plato had fled to Sicily. There he studied mathematics under noted Pythagoreans, picked up mystical ideas and dabbled in aristocratic politics. Finally, he came home to Athens to found his own school and make it the great mathematical centre of the Greek world. Most of the mathematicians of that era were his friends or associated with his Academy.

  Perhaps the most gifted geometer to study at Plato’s Academy was Eudoxus of Cnidus, who finally broke the deadlock of the irrational numbers, and freed geometry for the advances that were to come. How he did this – with his work on the Golden Mean and his new theory of proportion – is an exciting story. And if we add a bit of imagination, it gives us a fascinating glimpse of Athens and the Academy in Plato’s time.

  At the age of twenty-four, Eudoxus came to Athens from his home town of Cnidus on the Black Sea, in order to study at Plato’s Academy. He was so poor that he could not afford lodgings in the city, but lived in the small seaport of Piraeus and walked to school every day. Of course, he had already studied some geometry; it was the entrance requirement. But at the Academy he got particularly interested in the matter of an irrational number on a geometric figure. For in Athens the problem was in plain sight every day, in a concrete (or rather, a marble) form.

  On the high Acropolis, against the shimmering sky, stood the beautiful temple called the Parthenon – the most wonderful monument of the Age of Pericles, the ‘perfect’ building whose ruins enthrall us even today. The Parthenon had been designed by Iktinos and Callicrates according to mathematical principles. Its surrounding pillars were an example of ‘number’ applied: 8 pillars in front, an even number, as Pythagoras had advised, so no central posts would block the view: but 17 pillars on each side, where it was all right to have an odd number. And some of its lines were deliberately curved and slanted to correct optical distortions.

  But above all, the Parthenon was a crowning example of proportion in architecture. Scholars still marvel at the logical and harmonious ratios in the whole building and its various parts. And this beauty was achieved with one of the ‘dynamic rectangles’ then in vogue.

  Like many Greek temples of the time, the Parthenon used the ‘root five rectangle’, a rectangle with an irrational side – the square root of 5. How did this ‘root five rectangle’ come to be used? How was it constructed and shown to be irrational? How did Eudoxus analyse in it the most beautiful of all linear proportions, the Golden Section, or Golden Mean? That is our story.

  Architectural constructions

  Greek builders, we must remember, did not have a minutely graduated measuring rod, in inches or centimetres, like ours. Ground plans were still laid out in the old way, with string (rope), straight-edge, level and carpenter’s right angle or ‘set square’. And some of the older temples, and even a few new ones, were quite carelessly designed.

  But as geometry became popular in Athens, architects took to drawing careful plans with string and straight-edge, for geometric constructions could be enlarged easily and accurately in the building itself.

  Temples remained severely rectangular, but now the favourite rectangle was made by a ‘construction’: a square circumscribed by a semicircle. This figure gave you the shape of the rectangle: it was as long as the semicircle’s diameter, as high as the inscribed square. Calculating its numerical dimensions was easy with the Pythagorean theorem; any builder or Academy student knew how in those days. The rectangle had an irrational dimension. When its width was 1, its length was √5.

  The geometry of irrational numbers

  This ‘root five rectangle’ was enough to discourage any member of the Pythagorean Order – but Eudoxus belonged to a new age. After studying for a while at the Academy, he went to Egypt, where he studied under the learned priests. Afterward, he travelled and established his own school. Then, years later, he returned to Athens to revisit his former master Plato.

  This time, Eudoxus arrived not as a poor student, but as an acknowledged master of geometry. In token of his importance, he now wore his beard and eyebrows shaved in the Egyptian style. He was accompanied by several of his own disciples. A holiday in his honour was declared at the Academy. All the students wanted to see him and they crowded into the famous open-air lecture space shaded by the grove of olive trees. And there – we may imagine – he gave them the geometric solution of the proportion in the ‘root five rectangle’, which had puzzled him as a student.

  ‘I will ask you,’ he said, ‘to disregard numbers entirely and forget all about the numerical dimensions of the ‘root five rectangle’. We will try instead to find a proportion among the geometric quantities. So now, look at the construction itself, the square inscribed in the semicircle.’ Using string and straight-edge, Eudoxus drew it on some sand.

  ‘Look at the straight lines in the whole construction. You will see that there are only two geometric quantities throughout. What are these? One is b, the width of the square, wherever it occurs. Now study the diameter of the semicircle. On that line there are three segments. The long segment is simply b, the base of the square. The two short segments, a and a, on either side, are equal – for each equals the radius minus ½ the base of the square.

  ‘Reduced to its simplest terms, therefore, the problem is to find a proportion between the geometric quantities a and b, irrespective of any numerical dimensions. So here is the figure once more, simplified to show the problem in this simplest form.

  ‘Consider only the line a+b that is, only that portion of the diameter where our two quantities a and b can be defined as a short and a long segment of one line.

  ‘Now here is my question: What is the proportion that links a and b, the short and long segments of this line? Can anybody see how to find out?’

  A ripple of excitement rose from the students gathered in the grove of the Academy, as they peered at the diagram and discussed the problem in whispers. Plato himself stood by, smiling. Finally, when no one volunteered, Eudoxus raised his hand for attention and continued.

  ‘Nothing could be simpler than the answer. It involves a very easy construction that you all know already. From the upper right corner of the square, I will just draw two lines to the ends of the diameter. What does that give you?’ He pointed to an eager student in the front row.

  ‘A right-angled triangle, of course,’ the boy almost shouted. ‘Lines drawn from any point on the circumference to the ends of the diameter make a right angle.’

  ‘What else do y
ou see?’

  Several students answered at once: ‘Inside this large right-angled triangle are two other right-angled triangles. They are formed by a side of the square – but it is now a perpendicular line dropped from the vertex of the large right-angled triangle to its hypotenuse.’

  ‘Absolutely right!’ said Eudoxus, pointing them out. ‘We will call one S for Small, and the other M for Medium; and the large right-angled triangle, of course, can be L for Large. Now, do you see any relationship between these three right-angled triangles?’

  There was a pause, while all the students stared intently at the diagram. Suddenly a boy called out from the back row, ‘The three right-angled triangles are similar, aren’t they?’

  ‘How do you prove that?’ Eudoxus was nodding his approval.

  ‘Sir, they are similar because their angles are equal. If you will kindly spin the three right-angled triangles around and draw them side by side and upright, then everyone else can see the proof.’

  Eudoxus gladly obliged and, using his pointer, he explained for the benefit of the slower students. ‘Notice on the figure that each of the smaller triangles has an angle in common with the large triangle. But we know that in any right-angled triangle the sum of the other two angles is 90°. So each of the remaining angles must be equal respectively. The three right-angled triangles are therefore similar, just as – what is your name, lad? – just as Meno here has said, because their angles are equal. Meno has solved the problem!’

  ‘But sir,’ protested Meno in amazement, ‘of what use is it for us to know that the right-angled triangles are similar?’

  ‘Of what use?’ repeated Eudoxus, laughing. ‘Look again, all of you, and you will see the beautiful proportion that links the geometric quantities a and b.’ He pointed in quick succession to all the drawings on the sand.

  ‘Just take the dimensions from the final figure, and put them on the easy-to-see similar right-angled triangles, just the Small and Medium ones. You know that when right-angled triangles are similar, their corresponding sides are in proportion. Therefore,

  Short Side of Small is to Long Side of Small as

  Short Side of Medium is to Long Side of Medium

  or in other works: a is to b as b is to a + b

  ‘That is your proportion! Just read it off on the line, and you will see how beautiful it is:a:

  a : b = b : (a + b)

  The Short Segment is to the Long Segment as the Long Segment is to the Whole Line.

  ‘High-rete! High-rete! High-rete!’ cried all the students in unison – the Greek equivalent of three cheers.

  The Golden Section

  Plato himself joined the chorus of praise and made a short speech. ‘You have just seen a beautiful demonstration and proof – one of the most ingenious in geometry. This proportion is far more significant than the problem that led to it. So I will ask you all to review the construction for your next assignment.

  ‘By inscribing a square in a semicircle, you can do something truly marvellous with a straight line. You can divide that line into two unequal sections, in such a way that the short section is to the long section as the long section is to the whole line. Do you appreciate this proportion? This section, or cutting, of a line is so important that from now on we will call it the Section.’ Plato drew and wrote on the sand.

  Plato gave a banquet in Eudoxus’ honour that night – history records the event – and heard the rest of the discovery from Eudoxus’ own lips. Before we join them at dinner, let us pause (like the students at the Academy) to appreciate the importance of the Section. Plato himself, in his writings, always called it that. But later writers named it the Golden Section or the Golden Mean.

  The lasting fame of the Golden Mean rests not only on the sheer beauty of the proportion itself, but on its use in architecture and art. The ‘root five’ and Golden Section rectangles were used frequently in Greek buildings. Scholars have since found that many of the loveliest classical vases and statues cherished today, on the hills of Greece and in museums throughout the world, are based on this same section. And sculptors and painters down the ages have continued to make use of it.

  The Golden Mean correlates surprisingly with the proportions of the human body. Just look at the different lengths in your own hand and fingers and forearm, and you can see this yourself. The length of the first finger joint is to the length of the next two joints as those two are to the length of the whole finger! The length of the middle finger is to the length of the palm as the length of the palm is to the length of the whole hand! The length of the hand is to the length of the forearm as the length of the forearm is to the whole length from fingertip to elbow.

  Experts have made many more measurements, and have found that this proportion runs through the whole human skeleton – not exactly, of course, but as a kind of ‘ideal’ proportion or standard of beauty. That is why the Golden Mean has fascinated some of the greatest artists through the centuries. Leonardo da Vinci, for instance, called it the Divine Section.

  ‘But the most important thing about the Section,’ said Eudoxus to Plato at dinner, ‘is the kind of thinking it stimulates. In the Section, the length is irrational, yet it presents no difficulty because it is handled geometrically. So I have been working on a new definition of proportion – extending the idea of number to include the irrationals, and the idea of length so theorems will be correct for all lines.’

  Of course, this conversation is imaginary, but Eudoxus of Cnidus actually was the greatest mathematician of his age, and the Golden Section theorems were his most striking achievement. Another great geometer, an Athenian friend of his and Plato’s, named Theaetetus, probably worked on the theorems first, and Plato himself doubtless taught the subject at the Academy. But it was Eudoxus who finally broke the tyranny of number, with his magnificent new theory of proportion, so we have made him the hero of our story.

  The Golden Section in art and architecture

  The façade of the Parthenon was apparently designed around the proportions of two large and four small Golden Section, or √5, rectangles, placed above four squares.

  This Greek vase, known as a kylix, was designed to be contained within a Golden Section. The bowl of the vase follows the proportions of four squares placed together horizontally.

  Constructing a pentagram

  Taking the line AB as radius and using A and C as centres, draw arcs intersecting at D. Using AC as radius, draw arcs that cut the long arcs at E and F. Then AC, CF, FD, DE, EA form a pentagon, A five-pointed star can be formed by drawing AF, EC, DA and DC.

  The Section was actually the key to the geometric construction of the pentagon and of the fifth regular solid, the dodecahedron, with its twelve pentagonal faces – not their mere freehand drawing or building up with tiles as before, but their perfect construction with string and straight-edge. These and other beautiful shapes can be drawn easily if you just use the Golden Section.

  16. Applied Geometry

  During the fourth century BC, Greek geometry burst its bounds and went on to the tremendous discoveries of the ‘age of giants’. And Greek culture, too, burst from the mainland of Hellas and spread to most of the eastern Mediterranean.

  Both developments were connected with the romantic figure of Alexander the Great. After Plato’s time, teachers and alumni from the Academy had gone on to found schools of their own. In particular, Plato’s most famous associate, the great philosopher Aristotle, had set up the Lyceum in Athens, and started the systematic classification of human knowledge. And Aristotle’s most renowned pupil was the warrior king Alexander of Macedon, who tried to conquer the world.

  In just thirteen years, Alexander extended his rule over Greece itself, as well as Ionia, Phoenicia, Egypt and the vast Persian domains as far as India. After he died, his empire broke up. But throughout those far-flung lands, he had founded Greek cities and planted the seeds of Greek civilisation – the Greek language, Greek art and, of course, Greek mathematics.

  The roya
l road to geometry

  Mathematicians travelled with his armies. And there is even a story of how hard Alexander himself worked at the study of geometry.

  Menaechmus, who had studied with Plato and Eudoxus, was trying to teach Alexander some geometric proofs. The lesson went badly.

  ‘Master,’ exclaimed Alexander, ‘in my kingdom there are royal roads built smoothly, as shortcuts for the king. Can you not make this task easier for me?’

  Menaechmus made the now famous reply, ‘Sire, there is no royal road to geometry.’

  Practical geometry in the world

  During the fifth and fourth centuries BC, these geometers held the Greek belief that abstract thought, not mechanical work, was the proper occupation for a gentleman. So they kept aloof from practical applications and devoted themselves to smoothing and polishing existing proofs and finding more and more new ones. These thinkers were of various kinds – dreamers, adventurers, discoverers, organisers. Some fitted into one or another of these categories, some belonged to all of them. Among the earlier ones were Archytas of Tarentum, Plato’s geometry teacher, Hippocrates of Chios, who tried to fit together all the rules, and Theodorus of Cyrene, who discovered many of the irrationals. In Plato’s own time, the two greatest were Theaetetus of Athens and Eudoxus of Cnidus. At the Lyceum and at the Academy after Plato, many more worked to improve the definitions and assumptions and make new abstract discoveries – including Menaechmus, Alexander’s taskmaster.