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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.

      As well as assisting the pure mathematician, all of these methods had a beneficial impact upon the development of astronomy. Application of these techniques was most ably exploited by a man who was himself a mathematician as well as a practising astronomer, Johannes Kepler. Kepler, born in Württemberg, first appeared on the scene in 1596, when, at the age of twenty-four, he published a book called Cosmographic Mystery, in which he postulated his earliest model of the solar system and defended Copernicus’s ideas.

      Although Cosmographic Mystery was a promising work for someone so young, it gave no better mechanism for how the planets moved or how the mechanical structure of the solar system was maintained than did Copernicus’s complex model. But, a short time after its publication, Kepler was offered an opportunity that was to transform his career and lead to an advancement in astronomy almost as profound as Copernicus’s own.

      Impressed by the Cosmographic Mystery, in 1600 the ageing Tycho Brahe – mathematician to the court of Emperor Rudolph II in Prague – invited Kepler to come to Prague as his assistant. A year later Brahe died, leaving the twenty-nine-year-old Kepler to take over his observatory and inherit his vast collection of astronomical data.

      At heart Kepler was a Pythagorean, in that he believed that the universe was an harmonic entity, that number ruled every aspect of Creation, and that regular simple patterns lay behind all facets of the observable realm. Combined with meticulous observation and mathematical rigour, it was these convictions that led him to his great discoveries.

      Using the vast body of data that Tycho Brahe had collected during twenty years of observations, Kepler discovered that there was a minor discrepancy between the observed position of the planet Mars and that calculated using Copernicus’s model. He could trust the observational data because Brahe had used a set of newly invented sextants and quadrants that were accurate to between one and four minutes of arc. (A minute of arc is one-sixtieth of an angular degree, or one five-thousand-four-hundredth of a right angle.) The difference between the calculated value (based upon Copernicus’s theory) and the observed value for the position of Mars was eight minutes of arc.

      Starting from this discrepancy, Kepler came to the conclusion that the orbits of the planets around the Sun were not circular, as Copernicus had proposed, but elliptical, and he went on to prove it by matching precisely his calculations for planetary positions based upon elliptical orbits with accurate observations from Brahe’s data. This offered conclusive support for the heliocentric, or sun-centred, model of the solar system, because accurate observation matched values derived from independent calculation. Kepler was then able to formulate three laws (since known as Kepler’s Laws) first described in his two great books, New Astronomy, published in 1609, and Harmony of the World, which appeared in 1619.

      The first law is a simple statement of Kepler’s discovery – that all the planets travel in paths which are ellipses with the Sun at one focus. The second law states that the area swept out in any orbit by the straight line joining the centres of the Sun and a planet is proportional to the time taken for the orbit. In other words, the area of space an orbit borders is proportional to the length of time the planet takes to orbit the Sun: the further a planet is from the Sun, the larger the area and the longer an orbit will take.

      Kepler’s third law came some time later, with publication of his Harmony of the World. This law describes the mathematical relationship between the distances of the planets from the Sun and the times they take to complete their orbits. Kepler found that the square of the periodic time which a planet takes to describe its orbit is proportional to the cube of the planet’s mean distance from the Sun.

      Just as Kepler was devising these laws, the telescope was being turned into a powerful tool by the Italian natural philosopher Galileo. The instrument was actually invented by a Dutchman, Hans Lippershey, in 1608, but Galileo’s device, designed and built within two years of Lippershey’s, was far superior and could magnify up to thirty times – making it powerful enough to distinguish craters on the Moon’s surface and to observe a set of moons orbiting the planet Jupiter.

      So revolutionary was this invention that (perhaps through fear of the inevitable consequences of such a discovery) many could not contemplate its uses, and several leading political and military figures of the time had to be teased into trying out the instrument.

      What Galileo observed immediately confirmed Kepler’s laws. The system of moons around Jupiter could be visualised as a model of the solar system, and the moons’ revolutions could be measured and compared to calculated values, showing their paths to be elliptical. Furthermore, Aristotle had believed that the heavenly sphere (anything outside of the Earthly realm) was perfect and featureless, yet the telescope showed clearly visible craters on the surface of the Moon.

      Gradually, the edifice of Aristotelian ideas and the ancient astronomy of Ptolemy and the Greeks was crumbling and being replaced with accurate observation supported by the clinical precision of mathematics. Together these would eventually become an irresistible force for a major change in the way the universe was perceived.

      But the road to empiricism was bumpy and dangerous. The sixteenth and early seventeenth centuries were a period of huge ecclesiastical upheaval throughout Europe that included the worse excesses of the witch-hunts and the terror of the Inquisition. Kepler’s own mother, who lived in the small town of Leonberg that was sympathetic to Catholic activists, was accused of being a witch in 1615 and suffered over a year in jail and faced torture several times before being acquitted.

      Kepler and Galileo corresponded, and the German astronomer sought the Italian’s public support for his theory, asking him openly to support what he had accepted in private. But Galileo was unable to do this. Living as he did in the most volatile religious environment in Europe, and keeping only one step ahead of the ecclesiastical authorities, who were fearful that this new science would undermine their authority, Galileo could not risk such an endorsement. Kepler was lucky to be living and working in northern Europe, where, for the most part, he experienced far greater religious tolerance.

      Soon after the invention of the telescope, the powerful cleric Cardinal Bellarmine began to undermine Galileo’s work. In 1616 he pronounced that the Copernican system was ‘false and altogether opposed to Holy Scripture’.⁹ Galileo was then famously

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