Without doubt, Euclid's Elements is the most influential book in the whole history of mathematics, and has been described as the world's most widely printed book second only to the Bible. Composed some time before 300 BC by Eucleides, a teacher of mathematics and philosophy at Alexandria, this essentially encyclopaedic work draws on Greek mathematical traditions which were a good three centuries old by the time of writing, and preserves a vast wealth of earlier Greek geometrical writing which may not otherwise have survived. Euclid's editor, Proclus, specifically mentions material deriving from Eudoxus (c. 408-355 BC) and Theaetetus (c. 414-369 BC), while Books I and II draw from the school begun by Pythagoras (c. 580-500 BC) and perhaps Thales (fl. 580 BC).
Euclid's Elements was originally composed in thirteen Books, or Sections, though later editors added a fourteenth and a fifteenth, as is the case in the present volume. It was, however, the first six books that laid the systematic foundations of 'plane' or flat geometry, and which for centuries would be known as 'Euclid', especially in English education: the bane, or the delight, of centuries of schoolboys and younger undergraduates. (Dr Robert Hooke, the brilliant Christ Church protégé of Warden John Wilkins of Wadham in the 1650s, and an early Fellow of the Royal Society, when a schoolboy at Westminster, so his friend and early biographer John Aubrey tells us in his Brief Life of Hooke, 'in one weeke's time made himself master of the first VI books of Euclid to the Admiration of Mr. Busby' the Headmaster. A formidable achievement by any standards in any age!)
Books I and II deal with the Pythagorean geometry of straight lines, while III and IV discuss the circle, and also the circle's relation to straight lines and their resulting triangles. Parts of these books have been ascribed to Hippocrates of Chios (c. 470-400 BC), who lived almost 100 years before Euclid, and is not to be confused with Hippocrates the eminent physician of 'Hippocratic Oath' fame. Books V and VI cover the mathematics of proportions, and may derive from Eudoxus. Books VII-IX deal with number theory and prime numbers, and Book X with proportions. Euclid uses geometrical notation and words for what in more modern times would be expressed in algebraic notation. Books XI-XIII discuss the geometry of regular solids, which Hippocrates believed were only five in number.
It is important to remember that Euclid's Elements is one of the great achievements of Greek civilisation, and by extension, of Western civilisation, while during the medieval centuries, when Latin versions of Euclid were being studied in Europe, Arabic translations were being studied in the Arab world. (John Aubrey also tells us how the English philosopher and classicist, Thomas Hobbes - alumnus of Magdalen Hall, Oxford - was converted to mathematical study by coming upon a copy of Euclid in an Italian gentleman's library when he was about forty, which would have been around 1638. Opening it by chance at the 47th proposition, Hobbes 'By G-, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is impossible! . . . This made him in love with Geometry.' He then diligently worked backwards through the preceding propositions to convince himself of the logical truth, or scientia, of geometry. This chance encounter with Euclid was to change Hobbes's whole intellectual life.)
For geometry was foundational to Greek and to later science, in its exploration of that realm of pure and unfalsifiable truths which is mathematics. For while the Greek philosophers were all too well aware that chaos and confusion reigned in the ordinary world, the realm of the human intellect could gain access to something much higher, and apparently perfect. This was the world of geometry and mathematics, with its possible cognate connections to Plato's realm of Forms, to logic, and the four immutable elements of Aristotle. Were these truths related to a greater truth, such as the Logos of Heracleitus? For to the philosophical Greeks, geometry too possessed an important religious dimension.
Of course, for centuries before Thales and Pythagoras, the Egyptians and Babylonians had been masters of practical measurement, in land surveying and architecture in particular, while Babylonian tablets of c. 1700 BC survive describing processes of calculation. But what made Greek mathematics so different from Egyptian and Babylonian was its philosophical dimension, for Euclid and others were not interested in practical counting and measuring, but in exploring the nature of number and form as eternal truths in themselves.
And many others would be inspired to take things further, such as Apollonius of Perga (262-190 BC), whose Conics treatise on the geometry of the cone, and the circle, ellipse, parabola, hyperbola, and such shapes that resulted when angled sections were cut through a cone, would prove invaluable to later generations of astronomers. (And Oxford's own Dr Edmond Halley, 1656-1742, of The Queen's College, would produce a critical edition of Apollonius's Conics in 1706, when he was Savilian Professor of Geometry, and working out how regularly-changing gravitational forces in the solar system made the planets rotate around the sun in Apollonian elliptical, conic section, orbits.)
And what made Greek astronomy so radically different from that of Egypt and Babylonia was its philosophical advancement beyond the simple observing, tabulating, and reckoning of those cultures. For the Greeks infused astronomy with a geometrical and mathematical understanding, quite removed from the gods and spirits which drove the Egyptian and Babylonian cosmos. For by the time of Hipparchus (c.170-120 BC) and then Ptolemy (second century AD) one finds a mature mathematical cosmology, where observed planetary motions or eclipses were expressed and predicted by mathematical and geometrical means. Observations were first made with degree-graduated mathematical instruments, and then related to the geometry of the celestial sphere.
And while Euclid does not discuss astronomy or the heavenly spheres - others, such as Eudoxus of Cnidus (c.400-347 BC), had already done so - the philosophical geometry of the Elements, and its systematic, logical treatment, would bestow an incalculable legacy on the mathematical and scientific thinking of centuries to come.
And one of the great practical applications of classical geometry, later described by writers as diverse as Pliny and Vitruvius, and which is still taught in schools today, was Eratosthenes' measurement of the size of the earth around 230 BC. For Eratosthenes compared the lengths of midsummer noon shadows at Cyrene in upper Egypt and in Alexandria. And by next measuring the distance between the two places, and comparing their respective shadow angles to a segment of a circle, he was able to derive the terrestrial circumference in stadia.
And by way of a footnote, it is interesting to observe that the present Basel 1533 edition of the Elements (Euclid was originally printed in 1482) contains a prefatory reference to 'Doctiss. Viro Cuthberto Tonstallo Pontifici Simon Grynaeus Salutem...'. For in 1533, Cuthbert Tunstall had held the Episcopal Sees of London and Durham, and had been sent by Henry VIII on diplomatic embassies to France and to the Holy Roman Emperor. He was a friend of the great classical scholar Erasmus and a patron of Renaissance classical scholarship. It says something of the high regard in which Euclidean geometry and mathematics were then held to see Tunstall's name and greeting associated with the volume.