Isaac Newton: The Last Sorcerer. Michael White

its simplest form geometry deals with the shapes of things, in either two or three dimensions (although modern mathematicians also deal with multidimensional space – a study still called geometry). This area of mathematics found ready use with the architect and the builder. For the astronomer it was an invaluable tool in the search for patterns in the stars, which in turn fuelled the development of astrology.

      Algebra, which was only scantily formulated before the early seventeenth century, is a language in which symbols are assigned to properties of objects. It enables mathematicians to construct equations that describe a situation or the interplay between properties (either real or imaginary) using strict rules that govern what may be done with representative symbols. A simple example would be the equation s = d/t. In words this would be ‘Speed equals distance travelled divided by time taken’. Further examples would include equations used to find the rate at which water flows through a pipe, how quickly a rocket accelerates from the launch pad, or how efficiently a muscle uses energy from glucose.

      Arithmetic and geometry may be considered more everyday than algebra, in that they represent the world and the things we observe directly. Algebra is one level of abstraction away from reality, because symbols are used to represent properties, rather than being actual measurements of things. This distinction could account for the fact that arithmetic and geometry were developed into sophisticated tools and used widely very much earlier than algebra.

      The Greeks viewed mathematics in a different way to the civilisations that predated them, in that they appear to have been the first to consider pure mathematics – to contemplate mathematical abstractions, rather than using mathematics solely as a tool for constructing religious structures or to develop the mystical arts. Using mathematical skills, the Greeks were able to develop elaborate theories to describe the structure of the observable universe and to postulate ways in which the planets, the Sun and the stars could be arranged in the heavens.

      According to most accounts, Anaximander, who lived between 610 and 545 BC, is thought to be responsible for the first development of what became the geocentric view of the universe – the concept that the Earth lies at the centre of the universe. Before then, the Earth was believed to be a floor with a solid base of limitless depth.³

      Anaximander reached his conclusions by astronomical observation, 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