Extreme and Mean
‘Extreme and Mean Ratio’
The Greek geometers never speak of the ‘golden ratio’ and the first recorded use of the term is as late as 1835 ─ when Ohm referred to it as the goldener Schnitt. Nor does any ancient Greek give a numerical value for what we now know as phi or Φ. What we do find in Euclid and other ancient writers is repeated mention of a certain manner of dividing a line segment in “extreme and mean ratio”. Euclid VI Proposition 30 shows you how to do this. In our terms, this method of division results in “the ratio of the larger to the smaller part of the line segment being equal to the ratio of the whole to the larger part” i.e. a:b = (a + b) : a where a > b .
← (a + b) →
———————————————————————
← a →← b →
Why was this important to the ancient Greeks? Not apparently because of the supposed aesthetic properties of the associated ‘Golden Rectangle’ (formed by making the smaller portion into one of the sides). Although it is sometimes claimed that Phidias used the Golden Section in some of his Parthenon statues this is mere speculation; it was only Renaissance painters and architects who superimposed the proportions of the golden rectangle onto the human figure as in the famous Leonardo da Vinci drawing and claimed there was something especially beautiful about the ‘divine proportion’, as they called it.
Nonetheless, to judge by the number of theorems relating to it in Euclid and numerous references to it in other extant ancient manuscripts, the ‘section’, as Proclus calls it, was famous. So why did the ancient Greek mathematicians consider the division of a line in ‘extreme and mean ratio’ significant? Because it was a prerequisite for the ruler and compass construction of a regular pentagon (five-sided figure with all sides and angles equal) and thus for the construction of the pentacle (regular pentagon within a circle) and the starry pentagram (five-pointed star). The pentacle already had a certain history as a ‘magic symbol’, being originally associated with the ‘morning star’ (Venus), and this esoteric reputation has lasted right up to the present day ─ Dr. Faust uses it and so do some contemporary Wicca groups. In ancient Greek times the pentacle had a more respectable, but still somewhat offbeat, reputation since the Pythagoreans, originally a kind of scientific secret society, used it as a sign of recognition amongst the Fraternity ─ compare the Freemason handshake. They sometimes put letters at each point of the five pointed star and these letters spelled out the Greek word for health (ugieia) ─ so it was a sort of “Good Health to you, fellow Pythagorean” message.
But the pentagon had a more serious meaning still for educated Hellenistic Greeks and Romans. Although he did not invent them, Plato was an ardent propagandist for the importance of the regular solids, still called Platonic solids in his honour. For Plato, shape was more fundamental than substance and the supreme shapes were the perfect forms of geometry such as the circle and the regular polyhedral. These ideal Forms were changeless and harmonious whereas everything on the terrestrial physical plane was erratic and unpredictable. The five Platonic solids, which Plato identified with the four elements, Earth, Air, Fire and Water (plus a subtle fifth element Ether), had much the same status as the elements of the Periodic Table have in our eyes today. Indeed, it would hardly be going too far to say that, for Plato, these ideal Forms were cosmic computer programmes while the entire physical world consisted of the fallible execution of such programmes, software compared to hardware, genotype to phenotype. In consequence, it was very important for Platonists to know how to construct these forms, if only in imagination. The five solids are:
- The Tetrahedron (four triangular faces);
- The Octahedron (eight triangular faces);
- The Cube (six square faces);
- The Dodecahedron (twelve pentagonal faces);
- The Icosahedron (twenty triangular faces).
Euclid concludes his great work with Book XIII which is entirely devoted to the construction of the five Platonic solids. Although Euclid is generally regarded today as the originator, or at any rate greatest early expositor, of the axiomatic method, this gives the modern reader the wrong impression. Today, the axiomatic treatment of a mathematical topic implies complete disregard of practicalities and ‘realistic’ concerns, but Euclid always has his eye on the actual construction of figures inasmuch as this is feasible. The very first Proposition (Heath calls ‘theorems’ Propositions) of Book I is “On a given finite straight line to construct an equilateral triangle”. And the penultimate Proposition of his Elements (Book XIII. 17) tells you how to “construct a dodecahedron and comprehend it in a sphere”. To be sure, this construction is so complicated, likewise that of a icosahedron (20-sided regular polygon), that one is hard put to follow the steps in the argument, let alone produce an actual model in wood or metal. Nonetheless, the mathematical presentation is not abstract in the way that, say, a theorem about Baruch spaces in modern mathematics is.
Such an approach is absolutely in line with the Platonic philosophy. For Plato was not so much an Idealist as a Transcendental Realist: his Ideal Forms were more, not less, real than actual artifacts while not being absolutely divorced from material things either. As certain Sophists in Plato’s own time observed, the figures of geometry, when drawn, did not have all the properties accorded to them by geometers: points on an actual circumference were not always exactly equidistant from the supposed centre, tangents cut a circumference in more than one point &c. &c. “Yes,” Plato might have replied, “but the drawn circle is not the circle of geometry, only a tolerable imitation of it. The true circle and true tangent, of which our human imitations are derivatives, really do have all the properties we ascribe to them, such a tangent really does touch the circumference at a single point only.”
It is interesting to note that Book XIII concludes with the dodecahedron rather than the icosahedron (whose construction is even more complicated) ─ the final Proposition 18 deals with the relations between the entire five Platonic solids and proves as a sort of coda that they are the only possible regular solids. The reason for terminating with the dodecahedron is most likely because the dodecahedron was traditionally associated, not with the four earthly elements, but with starry matter which was considered to be different from, and superior to, earthly matter. (Tradition has it that the Pythagoreans were especially delighted with their discovery of the dodecahedron and sacrificed a hundred oxen to celebrate the occasion.) And, as stated earlier, the division of a line ‘in extreme and mean ratio’ is essential for the construction of the regular pentagon which is itself essential for the construction of the dodecahedron (since all the faces are regular pentagons).
This may go some way to explaining why the ancients had a particular veneration for the ‘section’. Moreover, Allman makes the interesting suggestion that what we call phi, the golden section, was the very first irrational (the Greeks would have said ‘incommensurable’) to be discovered, rather than √2 as is today usually assumed. This would explain the mystery and slightly sinister glamour attached to figures incorporating the golden section such as the pentacle; for the discovery of incommensurables was, as we know, extremely disturbing for Greek mathematicians and philosophers alike. The Pythagoreans seem to have shifted from an attitude of hostility towards irrationals/incommensurables to one of veneration, at least as far as Phi was concerned since they eventually adopted the pentacle as a sort of logo.
Did Euclid have what we might call a philosophical, almost a quasi-religious, aim in giving the ancient world such a detailed exposition of the Elements of geometry? This was certainly the view of Proclus who wrote a commentary on Euclid in which he claimed that Euclid was himself a faithful follower of Plato and that “it was for this reason he set before himself, as the end of the whole Elements, the construction of the so-called Platonic figures”. Heath rejects this out of hand, arguing that Proclus was a biased source since he was himself the leading Neo-Platonist philosopher of his time and keen to claim Euclid as one of his own. Nonetheless, there can be no doubt that philosophical Platonism was inextricably mixed up with late Greek higher mathematics and Heath himself admits that “it is most probable that Euclid received his mathematical training in Athens from the pupils of Plato”. Whether Euclid was himself a Platonist is unknown but he seems to have faithfully transmitted to posterity not only the discoveries of Platonist (or Pythagorean) mathematicians but their overall ‘view of the world’. We do not today consider Book XIII to be the most important part of the Elements and usually single out the ingenious treatment of the problem of incommensurables in earlier books because this treatment anticipates the 19th century approach to irrational numbers as pioneered by Weierstrass and Cantor. But the Elements was not just an exercise in pure mathematics; at any rate for many later Greek mathematicians, it was a sort of technical preamble to Platonic cosmology as laid out in the Timaeus. Kepler, to whom the Alexandrian cultural ambiance of Euclid’s day would have been most congenial, made a persistent attempt to match the orbits of the planets to the outlines of the Platonic solids and, incidentally, singled out the ‘division in extreme and mean ratio’ as the ‘chief jewel of Greek geometry’, on a par with the Pythagorean theorem itself. Although for a long time it was fashionable in scientific circles to look down on interest in the Golden Section as the affair of aesthetes and mystics, it is now known that one version of it, the Golden Angle, does have some importance as a ‘close packing constant’ as Irving Adler relates in his latest book on Phyllotaxis, or Leaf Arrangement. SH 25/09/17
Origins of Mathematics 