it was gradually accepted that the Earth was spherical (a fact known to the Greeks), but their voyages also offered opportunities for observation of the heavens from the perspective of the southern hemisphere and other regions never before visited. Exploration also expanded trade and supplied increased wealth, and learning grew exponentially. At the same time, mathematical and astronomical techniques were improving, and the traditional notions of Ptolemy, Aristotle and Plato were challenged on an intellectual as well as a practical and observational level.
Mathematics had been refined greatly by the Arabs throughout the period of the Dark Ages in Europe. Based in cities such as Isfahan and Baghdad, Arabic mathematicians merged material gained from Alexandria, India and China. Preoccupied with astrology, they were greatly taken with Ptolemy’s system, preserving the Almagest after the sacking of the library of Alexandria and passing it on to the West around the thirteenth century.
During this period, Arabic mathematicians refined Greek geometry and developed algebra significantly, and when Europe finally emerged from the Dark Ages the Arabic system of numerals was adopted along with the Arabs’ place-valued decimal system – the system gives values to numbers according to their positions on either side of the decimal point, increasing by factors of ten from right to left and decreasing by factors of ten from left to right.
All of these advances helped to re-establish analytical astronomy in the West. Although there remained a strong tradition of interest in astrology, and many of the alchemists and magicians who travelled around Europe well into the seventeenth century also earned money from practising astrology, the Christian Church did not officially recognise the art. Because learning had been sustained by a unification of science and theology, many of those interested in astronomy and mathematics were monks and theologians and they could not be seen to be dabbling in astrology. Astronomy, however, could be justified as a purely intellectual pursuit, as a component of worship. Indeed, the man who led the re-evaluation of the Ptolemic system was a priest: the Polish canon Nicolas Koppernigk, known to posterity as Nicolas Copernicus, who wrote a book called On the Revolutions of the Heavenly Spheres, first published in 1543.
The simple image of Copernicus upturning the established, fourteen-hundred-year-old ideas of Ptolemy almost overnight and revolutionising our thinking about the structure of the universe is quite false. Copernicus’s work was revolutionary, but, except in one respect, it did not offer a completely new system. Copernicus’s model of the solar system was actually more complex than Ptolemy’s and comprised forty-eight circles or wheels within wheels to account for the observed facts (eight more than in Ptolemy’s model). The one vital and controversial aspect of Copernicus’s work that challenged accepted thinking was his belief that the Earth did not lie at the centre of the universe.
Having said that, Copernicus did not place the Sun at the centre of the system either, but accounted for astronomical observations by keeping most of Ptolemy’s epicycles and having the Sun, along with the Earth and the known planets, orbiting a point (close to the Sun) which he claimed to be the true centre of the universe.
Copernicus suppressed his findings for over thirty years, and, through fear of religious persecution, did not allow his book to be published until he was on his deathbed. In fact he began his treatise boldly, by asserting that the Sun lies at the centre of the universe, but then appeared to change his mind. After the first few pages, he went on to complicate his theory more and more with unnecessary refinements, finally placing the Sun slightly off-centre. This prevarication made the entire work almost unreadable and frequently contradictory. Perhaps it was because of this that, despite containing what became one of the most influential theories in the history of science, in commercial terms On the Revolutions of the Heavenly Spheres was one of the least successful books ever written.
Running to 212 sheets in small folio, the heart of Revolutions is contained in the first twenty pages, in which Copernicus outlines the central tenets of his theory, stating, ‘in the midst of all dwells the Sun … Sitting on the royal throne, he rules the family of planets which turn around him … We thus find in this arrangement an admirable harmony of the world.’8
The central tenets of Copernicus’s theory were revolutionary for the time in which he lived, and they bear comparison with the modern view far better than Ptolemy’s geocentric picture. Copernicus states that the Sun lies stationary at the centre of a finite universe bounded by the fixed stars. The Earth and all the planets orbit the Sun in circles; the Moon revolves around the Earth. The apparent daily rotation of the firmament about the Earth is not because it is at the centre of the universe but because the planet revolves on its axis. The apparent annual motion of the Sun around the Earth in Ptolemy’s description is actually due to the passage of the Earth around the Sun. Finally, Copernicus could explain the bug-bear of the Greeks’ system – the apparent retrograde motion of certain planets – by describing how the planets all orbit the Sun at different speeds. He was even able to explain the slight irregularities of the seasons as being due to the Earth ‘wobbling’ on its axis.
It is possible that Copernicus confused the written account of his theory deliberately, to ward off the expected attacks of religious orthodoxy. But it is equally likely that, having come to the irrefutable conclusion that the Earth revolved around the Sun, he could not himself accept the consequences of the simple system he had stated at the beginning of his book. Copernicus was an Aristotelian, rooted in medieval thinking, and had been educated to accept Greek teaching verbatim. In no sense was he the revolutionary figure posterity has painted him as being. Throughout his notebooks there are points where he could have reached far more profound conclusions but missed them because he was strait-jacketed by his religious convictions and his traditional education.
In spite of his fears, Copernicus’s On the Revolutions of the Heavenly Spheres had little immediate impact upon religious or scientific thinking, and was not included in the Index Prohibitorum (the list of books banned by the Catholic Church) until 1616, seventy-three years after its publication. What did change history and posthumously placed Copernicus at the centre of an ecclesiastical and intellectual storm was the work of his successors, who based their ideas upon his discoveries and combined them with meticulous observation and more refined mathematical knowledge.
This mathematical knowledge had been developing in parallel with advancing ideas in astronomy based upon observation. The Babylonians had developed a simple form of algebra five or six thousand years before Copernicus, but their understanding of the subject had been limited to the use of what mathematicians call linear and quadratic equations. These describe equations of different levels of complexity. A linear equation contains just numbers and symbols which may be added, subtracted, divided or multiplied together. A quadratic equation contains terms (such as x or y) which have been squared (raised to the power of 2). Both of these types of equation are simple to solve and fall well within today’s secondary-school curriculum. More difficult, but far more powerful in the hands of the scientist, are solutions to what are called cubic equations – equations which contain terms raised to the power of three (or cubed).
Cubic equations had resisted all attempts at solution until the beginning of the sixteenth century, when two mathematicians could independently lay claim to success: Niccolo Tartaglia and Scipione Ferro. Their methods of solving cubic equations were eventually published in 1545, two years after the death of Copernicus, in a book by Gerolamo Cardano called Ars Magna which paved the way for a succession of new algebraic techniques.
The sixteenth century was also a time when international trade and mercantile enterprise flourished and businessmen and economists employed new and more efficient mathematical techniques. Towards the end of the century, in 1585, Simon Stevinus of Bruges created rules for solving equations of higher powers than three, and three decades later, in 1614, the Scottish mathematician John Napier devised the technique of logarithms – an incredibly powerful algebraic and arithmetic tool that opened the door to further rapid advance.
