believing that the visible sky was a dome or half of a complete sphere. But it was not until the fourth century BC, when Greek explorers began to travel further afield, that this idea began to be widely accepted. An indication of the rapid progress that was made during this period is that by the third century BC, around 300 years after Anaximander, Greek astronomy had progressed to the point where Eratosthenes, a contemporary of Archimedes, was able to estimate the circumference of the Earth, putting it at 24,000 miles (only 800 miles short of the modern measurement). He was also able to calculate the distance between the Earth and the Sun, assigning it a figure of 92 million miles (a little over 1 per cent out from the modern value of 93 million miles).
This progress in astronomical knowledge was due largely to the development of geometry between the lifetimes of Anaximander and Eratosthenes. Many advances derived from a strong need for practical mathematical tools for use by land surveyors and farmers – ‘rules of thumb’ and practical guidelines. Such developments helped philosophers and mathematicians to derive axioms and general principles that led to further discoveries. The first great mathematician to work in this way was Pythagoras, a man most people remember from school maths lessons as the creator of a theorem concerning rightangled triangles.
In fact Pythagoras, who was born at Samos shortly after Anaximander’s death, derived much more than a single geometric relationship: he was the most important figure in formulating the whole basis of pure mathematics. His school was pseudo-mystical, in that he and his followers believed that the universe had been designed around hidden numeric relations and that its entire structure and the complex interplay of the four elements (later popularised by Aristotle) were governed by mathematical patterns. He and his followers discovered the mathematical relationship between sounds, using vibrating strings, and originated the concept of the ‘music of the spheres’ – the idea that the ratios observed between notes on the musical scale could be mirrored in the distances of the planets from the Earth.*
Fortunately, many of Pythagoras’s ideas were preserved by another great mathematician, who lived two centuries later, Euclid of Alexandria – the man most commonly perceived as the father of modern geometry. Although Euclid was an original thinker and added much to the knowledge of geometry, his greatest contribution was to collect earlier work, especially that of the Pythagorean school, and to rationalise it into a collection of books he produced around 300 BC. These have survived to the present day and formed the basis of all geometry until the middle of the last century. So fundamental is this work to our understanding of mathematics that the three-dimensional space in which we perceive the universe is known as ‘Euclidean’ space, and it was only during the nineteenth century that mathematicians began to speculate about the possibility of non-Euclidean space – geometry which did not adhere to Euclidean rules.†
Astronomy and mathematics developed little between the waning of Greek culture and the domination of the Arabic intellectual system which began to emerge during the second half of the first millennium AD. The exceptional name from this era is the Alexandrian Ptolemy (c. AD 100–170), who codified the geocentric theory, a concept that remained at the heart of astronomical thinking until the sixteenth century.
Little is known of Ptolemy’s life, but he made astronomical observations from Alexandria during the years 127–41 and probably spent most of his life there. Principally a geographer, he wrote a treatise entitled Almagest which contained many of his own observations and theories as well as summaries of Graeco-Roman thinkers. He also produced geometric models which he used to predict the positions of the planets, imagining all heavenly bodies to travel in a complex set of circles known as epicycles, within the framework of a geocentric system supplied by many of the Greek astronomers.
To the modern mind, the concept of the Earth lying at the centre of the universe is an absurdity, but there were very good reasons why this concept held sway for so long and became so thoroughly ingrained in Western intellectual systems. It was certainly not born out of ignorance on the part of the Greek philosophers and astronomers who created it. These same philosophers could, after all, measure the distance between the Earth and the Sun with an accuracy of 1 per cent. It was more to do with deliberate obfuscation of the facts in order to comply with the Greeks’ anthropocentric vision.
This historical interpretation has become fashionable only in this century and has been championed by a number of historians of science, including the eminent writer Arthur Koestler, who popularised it in his influential work The Sleepwalkers.4 The Greeks, like the scholars of Europe in the Middle Ages, were obsessed with the idea that man was the centre of Creation and that consequently the Earth must be at the centre of the universe. To accommodate this dogma they created an incredibly elaborate mechanical system that would account for their observations of the heavens. If there had been no philosophical imperative for the Sun, the Moon and the five known planets to orbit the Earth in perfect circles, then it would have been quite within the powers of the late Greek and Alexandrian astronomers to show that the Earth, along with Mercury, Venus, Mars, Jupiter and Saturn, orbited the Sun, and that the Moon orbited the Earth. They might even have been able to deduce that the orbits were elliptical rather than circular. Instead, in order to account for the observed movements of the known heavens and to satisfy the prevailing philosophy, Ptolemy needed to create a system of forty different ‘wheels within wheels’ – a crazy pattern of gears or Ferris wheels.
The most difficult problem he faced was how to explain what is called the retrograde movement of some planets: that at certain times of the year planets appear to move backwards against the backdrop of stars from one night to the next. Today we know that this is because planets follow elliptical orbits around the Sun and move at different speeds, so there will be times when the Earth appears to ‘overtake’ the slower-orbiting outer planets and they appear to travel backwards.
To picture this clearly, imagine the solar system as a motor-racing track with the planets represented by the cars in different lanes. If the cars move at very different speeds, from the viewpoint of a car moving quickly on the inside track a slower car in an outside lane would appear to be moving backwards.
Because it complied with the anthropocentric world-view, Ptolemy’s complex, but quite incorrect, system was adopted as the only valid description of the universe and was later sanctioned by Christian theology. But even then there were some who doubted. When he learned of Ptolemy’s thousand-year-old system, the thirteenth-century Spanish King Alphonso X, known as Alphonso the Wise, declared, ‘If the Almighty had consulted me before embarking upon the Creation, I should have recommended something simpler.’5
Koestler believed that ‘There is something profoundly distasteful about Ptolemy’s universe.’6 What he meant by this was that, as we learn more about the universe, we find that the underlying rules by which it operates are fundamentally elegant and simple. This is what scientists mean when they talk about the ‘beauty’ of the mathematics representing universal laws such as those of gravitation or radioactive decay. Although science appears to the uninitiated to be incredibly complex, the laws that govern the behaviour of matter and energy are remarkably simple. In retrospect we can see that any system like Ptolemy’s had to be wrong. He and others of his time were trying to squeeze the facts into a false theory in order to fit a belief. They were starting with a rigid conviction and attempting to make the universe suit their dogma. Ptolemy himself wrote, ‘We believe that the object which the astronomer must strive to achieve is this: to demonstrate that all the phenomena in the sky are produced by uniform and circular motions.’7
Like Aristotelian dogma, Ptolemy’s system survived the Renaissance, but it was gradually eroded by a growing awareness that came from exploration of our own world. As European explorers circumnavigated
