Читать книгу Isaac Newton: The Last Sorcerer - Michael White - Страница 9

In every piece there is a number – maybe several numbers, but if so there is also a base-number, and that is the true one. That is something that affects us all, and links us all together.

ARVO PÄART (composer)¹

Number and pattern have always held a fascination, and the true origins of mathematics and astronomy are certainly ancient. The earliest form of organised mathematics, in which numbers were meaningfully manipulated and patterns recorded, is credited to the Babylonians of around 4000 BC, who recorded star patterns and named constellations. They had also developed a surprisingly advanced set of mathematical rules, including a sophisticated method of counting – a skill employed by the record-keeper, the farmer and the architect. It is thought that the last of these professions may also have employed simple forms of algebra and geometry.

Modern research, such as John North’s work on ancient stone circles, has demonstrated that the ancient Britons must also have possessed some knowledge of geometry in order to build such structures as Stonehenge, started about 3500 BC,² and the ancient Egyptians had highly developed mathematical and engineering skills which they employed in the building of the Great Pyramid at Giza some 1,000 years later. In these ancient civilisations, mathematics and astronomy were blended together intimately and had rich associations with mysticism and the occult. Astronomy and astrology were viewed as one and the same, and mathematics gained an almost spiritual status as a tool for the astrologer/astronomer. It was not until Greek times that mathematics and, to a lesser extent, astronomy were separated from religion and considered worthy of academic attention. While maintaining their spiritual associations, they then gradually became subjects for pure analysis and reasoning.

All mathematics may be viewed as composed of three central subjects: arithmetic, geometry and algebra. As the most immediately useful to a wide range of crafts and professions, arithmetic was the earliest form of mathematics to be developed, and grew to include all forms of number manipulation.

In 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 ‘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 the globe and discovered America and Australia, 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.

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 persecuted for his support of the heliocentric model of the universe and spent his final years under house arrest. He managed to avoid the punishment of the heretic only by denouncing his convictions and what he knew to be fundamental truth before the Inquisition in 1633, just nine years before Newton’s birth.

But, once started, nothing could stop the flow of progress. By the early part of the seventeenth century the stage was set for another series of revolutionary advances in mathematics that would pave the way for the work of Newton and the triumph of change from a geocentric to a mechanistic and heliocentric viewpoint – from Aristotelian guesswork to empiricism, observational precision and mathematical rigour. The man responsible for this last development before Newton was to take up the baton and reach the law of gravity and the development of the calculus was the French philosopher René Descartes.

Descartes’s great mathematical breakthrough was the realisation that the equation was not the only way in which mathematical terms could be related. During the 1630s he devised the idea of constructing coordinates to represent pairs of numbers relating to algebraic terms (usually x and y). These came to be known as Cartesian coordinates and opened up the vast range of possibilities offered by the drawing of graphs – lines and curves bordered by axes. The technical name for this branch of mathematics is analytical geometry, and it first appeared in an appendix called ‘Geometry’ tacked on to the end of Descartes’s Discourse on the Method, which was first published in 1637.

Descartes’s technique galvanised the world of mathematics, and within a few years of publication the Discourse had influenced the work of mathematicians and astronomers throughout Europe. Such men as the English mathematicians William Wallace and Isaac Barrow, as well as natural philosophers and mathematicians on the Continent, led by Pierre de Fermat and Christiaan Huygens, used Descartes’s findings as a springboard for their own efforts, which began to focus on the properties of the curves that could be drawn using Cartesian coordinates.

A simple example is the graph produced by plotting the distance travelled by a ball dropped from a high tower against the time for which it has fallen. Galileo had shown that the speed of a ball increases with time. If after one second the ball has fallen 16 feet, after two seconds 64 feet and after three seconds 144 feet, clearly it is accelerating. If these values are plotted on a graph with speed on the y-axis and time on the x-axis a curve is produced.

Figure 1. The curve produced by plotting distance against time.

Now it is comparatively easy to calculate properties for straight-line graphs. For example, the area under a straight line can be calculated by simple geometry known to the Babylonians, and the gradient of a straight line (or its steepness) can be found by dividing the change in the values along the y-axis by the corresponding values along the x-axis. So, if the distance-time graph had been a straight line, the gradient would have given us the speed of the ball (the change in

Figure 2. Calculating the gradient and the area under a straight line.

distance with time). But how can the properties of a curve be calculated?

It was soon realised that one way to determine properties of curves, such as the one in our problem, was to imagine them as constantly shifting straight lines: if a straight line was drawn next to a curve and touched it at a particular point, this line could approximate the curve at that point. Mathematicians called this straight line a tangent, and found that they could treat a tangent like any other straight line – they could, for example, find its gradient and therefore work out a value for the speed of the ball at that particular point. But this was still an approximation – and a very limited one at that.

Simple problems concerning objects travelling in circular motion had been studied by earlier generations of philosophers, especially Galileo, but by the 1660s astronomers weaned on Kepler’s work were becoming interested in mathematical models to describe the

Figure 3. The tangent to the curve.

new celestial mechanics – the mathematics of how the planets maintain their orbits around the Sun. They of course realised that the mathematics of curves could lead to a fuller understanding of planetary motion, but limited solutions such as those offered by drawing tangents were not accurate enough to correlate with increasingly sophisticated methods of gathering observational data. Although the mathematicians and astronomers of Europe were exploring methods of working with curves and some, such as Fermat and the great English polymath Christopher Wren, came to very limited solutions that worked in specific cases, there was a need for general solutions, or methods that could be applied to all situations. Newton gradually became aware of this as he studied the work of his predecessors while an undergraduate student at Cambridge during the early 1660s. By the middle of the decade all the elements were in place for a mathematician of genius to produce the required new mathematics. And, thanks to a series of unpredictable events, Newton was able to find the time and inspiration to do just that.