The Principia. Mathematical Principles of Natural Philosophy (Concise edition)

By Isaac Newton, Kirill Krasnov and Marika Taylor

Newton's bold masterwork helped shaped the cultural landscape of the world today. Now in a digestible, pocket format for the modern reader.

New concise edition with a new introduction, abridged for the modern reader. The Principia. Mathematical Principles of Natural Philosophy is one of the most important scientific works ever to have been written and has had a profound impact on modern science. Consisting of three separate books, the Principia states Newton’s laws of motion and Newton’s law of universal gravitation. Understanding and acceptance of these theories was not immediate, however by the end of the seventeenth century no one could deny that Newton had far exceeded all previous works and revolutionised scientific thinking.

The FLAME TREE Foundations series features core publications which together have shaped the cultural landscape of the modern world, with cutting-edge research distilled into pocket guides designed to be both accessible and informative.

• Language: English
• Publisher: Flame Tree 451
• Release date: Apr 9, 2024
• ISBN: 9781804175705

Isaac Newton

Sir Isaac Newton (1642–1727) was an English physicist and mathematician who was a leading figure in the scientific revolution. His work throughout the seventeenth century provided the basis for modern science, including his three laws of motion and the law of universal gravitation. Newton’s career was prolific. He was president of the Royal Society and in 1705 he was knighted, becoming the first ever scientist to receive the honour.

Book preview

9781804175705.jpg

The Principia. Mathematical Principles of Natural Philosophy

Isaac Newton

A New Introduction by Kirill Krasnov

Series Foreword by Professor Marika Taylor

flametreepublishing.com

FLAME TREE 451

London & New York

Series Foreword

Science has always been a source of fascination, pushing the frontiers of knowledge. This series of books spans some of the most important scientific developments of all time. From the origins of life to the nature of the universe, these books showcase remarkable achievements of scientists through the ages and the limitless potential of human curiosity.

In this series we can follow the evolution of science from ancient times through to the modern era. Aristotle’s writings cover a wide range of topics related to the natural world, including physics, metaphysics and biology. His work is notable for its depth and breadth as well as for its immense influence on the development of scientific thought in the Western world.

Copernicus and Newton were also great polymaths. Copernicus is best known for his work on the heliocentric nature of the solar system as presented in On the Revolutions of the Heavenly Spheres but he also made substantial contributions to mathematics and medicine. Newton’s Principia is recognized as one of the most important scientific works ever written. His theories of motion and gravity laid the foundations for modern physics and revolutionized our understanding of the ‘Nature’ of the world. Within mathematics, Newton developed calculus, as well as algebra, trigonometry and geometry.

Newton’s Principia was originally read primarily by intellectuals. By contrast, Darwin’s work On the Origin of Species immediately caused a significant stir in both scientific and public circles, particularly through its challenge to traditional beliefs. Darwin’s theory of evolution was debated and refined, gradually becoming a foundational concept of modern biology.

At the beginning of the twentieth century, physics experienced two major paradigm shifts: relativity and quantum theory. Einstein’s Special and General Relativity fundamentally changed our understanding of space, time and gravity. The works of Bohr and Planck explain the early development of quantum theory, starting from Planck’s revolutionary concept of quanta.

Readers of this series can step back in time to explore how the foundations of science are described by its pioneers. The transformative nature of these works and their profound impacts on society can only inspire a sense of awe and wonder.

Professor Marika Taylor

University of Southampton

A New Introduction to Aristotle

An Enduring Legacy

This volume presents an abridged version of Mathematical Principles of Natural Philosophy, or Philosophiae Naturalis Principia Mathematica in Latin, the language in which it was written, by Isaac Newton (1643–1727). The publication of this book in 1687 was a truly singular event in the history of science. This introductory article pays yet another tribute to its author and the role that this book played in so many subsequent developments of science.

While it has become common to associate Newton’s name with the discovery of the law of universal gravitation and the invention of calculus, and also to attribute both of these discoveries to the Principia, we will see that the true story is much more interesting and complex. In fact, we will see that Newton was indeed ‘standing on the shoulders of giants’, to use his own words, in regards to both gravitation and calculus. We will instead argue that Newton’s greatest contribution may be in the very method he used to achieve the remarkable conclusions described in this book. By formulating this method, Newton single-handedly put human scientific thought on the path along which it still moves today. And while his law of universal attraction was later superseded by Einstein’s theory of gravitation, his ‘principles of natural philosophy’ changed the very way in which scientific problems are attacked. This is as relevant now as it was in Newton’s time.

Newton’s method was to ‘infer particular propositions from the phenomena, and afterwards render them general by induction’, to quote and slightly rephrase his ‘General Scholium’ from the end of Book III of Principia. This means that the basic principles of science should follow directly from the experiment or observation. This methodology is still valid today; indeed it guides much of modern scientific development. It may well be that this methodology will remain with humanity for as long as it seeks to decipher Nature’s puzzles.

Newton’s Life and Times

Newton lived through one of the most turbulent periods of British history. When he was born, in 1643, the English Civil War was still raging, with battles in Yorkshire, the Midlands, Gainsborough in Lincolnshire and the west of England. The execution of the Charles I took place in 1649, when Newton was a young boy, and he grew up in the years of the Commonwealth. He went on to live through the restoration of the monarchy in 1660, followed by the Glorious Revolution of 1688.

In contrast, Newton’s personal life was largely devoid of drama. His life revolved around three cities, slightly more than 100 miles apart. He never left England, nor did he marry. It may well be this absence of external drama in his life that allowed him to concentrate so deeply on science, and so to achieve the remarkable breakthroughs now associated with his name.

One dramatic event did take place before Newton was born, however. His father, also named Isaac Newton, died three months before his birth, which probably had an important impact on the development of his personality. The young Isaac was thus born during a very difficult time for his mother. His birthday was recorded as Christmas Day (25 December) 1642, according to the (Julian) calendar used in England at the time.¹ He was born prematurely in Woolsthorpe, near Grantham, in Lincolnshire – a very small and weak child who was not expected to survive. Nevertheless, Newton went on to have a long life, dying at the age of 85; he was strong and had good health throughout his life.

Isaac’s mother, Hanna Ayscough, remarried when he was three. She went to live with her new husband, the priest Barnabas Smith, leaving young Isaac in Woolsthorpe in the care of his maternal grandmother. Isaac’s mother had three more children in her second marriage.

At age of 12 Newton was sent to study in Grantham, at the King’s School – a decision probably influenced by Newton’s uncle, the priest William Ayscough. In Grantham Newton stayed with the family of the local pharmacist. It is possible that Isaac’s interest in alchemy began here, stimulated by what he saw every day in the chemist’s laboratory. It was an interest that he developed fully in later life.

Isaac was a thoughtful child, excelling at school. But by 1656 Isaac’s mother had become a widow for the second time and returned to live to Woolsthorpe. In 1658, at age of 15, Isaac was removed from school and sent to live back on the farm with his mother and his half-siblings, probably with the intention of making him the man of the farm. It is not known for sure who convinced his mother to overturn her decision and send him back to school. Both Isaac’s uncle and the school headmaster, who saw in Isaac a very promising pupil, may have played a role. Certainly by the autumn of 1660 Newton had returned to the King’s School, and was preparing for entry to Cambridge.

In 1661 Isaac was admitted to Trinity College, Cambridge, on recommendation of his maternal uncle William Ayscough, who had also studied there. Soon after obtaining his BA in 1665, the University was temporarily closed as a precautionary measure against the plague, and Isaac spent two years in his family home in Woolsthorpe. It is believed that the seeds of all his most important later discoveries – in optics, calculus and gravitation – were sown during this two-year period. Indeed, his work on calculus was largely completed during this ‘plague’ period.

In 1667 Newton returned to Cambridge, where he was elected a Fellow of Trinity. By 1669 he was already a recognized scientist, and Isaac Barrow, the first Lucasian professor, ensured when vacating his chair that Newton succeeded him. Newton duly became the second Lucasian professor, a position that he occupied for 33 years.

Newton’s initial studies mostly concerned optics as well as calculus, which was in particular a tool for his geometric optics constructions. Only from the late 1670s to early 1680s did Newton return to the problems related to gravitation. The first edition of Principia was published in 1687. In 1696 Newton moved to London to take up the post of the Master of the Mint. He continued to live in the capital until his death in 1727.

Barrow, Calculus and the Reflecting Telescope

Before moving to the main subject of this article, Newton’s Principia, we should pay tribute to his work on optics. Although this is perhaps less known to us today, understanding this work is essential for understanding the author of the Principia.

Work on optics occupied Newton for nearly two decades at the start of his scientific career. Such studies were the main motivation for his development of calculus, and it was this work that made him well-known during his day. His career progression was swift, as his optics work propelled him from a junior fellowship at Trinity in 1667 to a senior fellowship in 1668; he then became magister in the same year, and the second Lucasian professor just one year later. We will not be able to form an accurate image of Newton’s personality without appreciating what guided him during this period, and what his most important achievements were.

It appears certain that Newton’s interest in optics and the calculus of infinitesimals were shaped by interactions with Isaac Barrow, the first Lucasian professor. Isaac Barrow (1630–77) was a person of many interests. He first professorial position in Cambridge, which he obtained in 1660, after being ordained and thus becoming a priest, was a professor of Greek language. He translated the ancient Greek geometers, such as Aristotle and Euclid. In 1662 he became a professor of geometry at Gresham College in London, and from 1663 was a member of the Royal Society. He was also selected as the first Lucasian professor of mathematics in Cambridge when the chair was created, and it was there that he first met the young Newton. After leaving this post in 1669, leaving Isaac Newton to be his successor, Barrow devoted the rest of his life to the study of divinity. He became Master of Trinity College in 1672, occupying the post for five years until his death.

Barrow’s most influential works were his lectures on optics and geometry, written during his brief tenure as the first Lucasian professor. It is in the introduction to these works, already published when he left the Lucasian chair, that the name of a young scientist, Isaac Newton, is first mentioned in print:

Isaac Newton (a man of exceptional ability and remarkable skill) has revised the copy, warning me of many things to be corrected, and adding some things from his own work, which you will see annexed with praise here and there.²

It is surprising how much of what we now know as calculus is already developed in these lectures. In particular, what is now known as the fundamental theorem of calculus, namely the statement that integration and differentiation are each other’s inverses, was already known to Barrow and was described by him. The frequently encountered statement that it was Newton who invented calculus is therefore, to an extent, an exaggeration. Isaac Newton did indeed ‘stand on the shoulders of giants’, to quote his famous words, uttered in the later period of his life.

The original contribution of Newton was the invention of the method of power series, conceived very early on in his career. He was already working upon this method in 1664–65, when he was listening to Barrow’s lectures, and he developed it further in the quietness and concentration of his two ‘plague’ years.

Newton realized that many of the practical problems in which he was interested could be solved by decomposing functions in questions in what we now call power series (or Taylor series), and comparing the coefficients. Every function can be better and better approximated by a polynomial of some given degree, as the degree of this polynomial becomes larger and larger. One thus gets an infinite series representation of a function, as a sum of powers of the argument of the function with some coefficients. It is this method that Newton invented and used with great ingenuity for practical calculations. In later years, during his priority dispute with the German mathematician Gottfried Wilhelm Leibnitz (1646–1716) over the invention of calculus, Newton claimed to have invented his method of power series in 1664–65. He applied it, for example, to compute the area of a hyperbola to 52 digits accuracy. It is very probable that the invention of this method was indeed Newton’s original idea. But for the rest of what we now call calculus, he was to a large extent an improver on what his brilliant teacher Barrow had taught him in geometric lectures. Additionally, it was Gottfried Wilhelm Leibnitz (1646–1716) who invented the notations that every student now learns when studying calculus.

The motivations that guided Newton in his mathematical investigations were very practical, and driven by his interest in optics. The first telescope was invented in Europe in 1608; shortly afterwards Galileo Galilei (1564–1642)³ used his version of the instrument to observe celestial objects. Early telescopes used two lenses to magnify and refocus light. Improved versions of such a telescope were soon developed, but still using two lenses; one was used from 1611 by the German astronomer Johannes Kepler (1571–1630). Newton was the first person to build a functioning reflecting telescope that used a mirror rather than a lens as a light-gathering device. His reflecting telescope had much better properties, and became the standard astronomical device for several centuries.

Newton was able to build the first working reflecting telescope in 1668. The second version of his telescope, built three years later, was sent to London, to be presented to the King and to the newly created (in 1662) Royal Society. Its demonstration was so impressive in early 1672 Newton was elected a fellow of the Royal Society.

Today Newton’s public image does not emphasize the fact that he was more than a mathematician. He was also a chemist, continually looking for better alloys for his mirrors, and thus working very hard in a chemistry laboratory that he had built for himself. He was himself a skilful craftsman, polishing lenses for his own telescopes. He was also, of course, an ingenious conductor of experiments, designing and carefully undertaking physics experiments that guided his theories. We will return to Newton’s experiment-guided method of establishing ‘principles’ in more detail below.

New Theory of Light and Colour

The task of constructing a better telescope was not the sole motivation of Newton’s experiments in optics. Since his Woolsthorpe years he had been preoccupied with the nature of light and colour, and this later became a question of much controversy for him. In his lectures on optics, his teacher and friend Isaac Barrow had described the mathematics needed for the geometric optics in great detail. To a very large extent this mathematics is that of differential calculus, as one of the most important tasks here is to find the tangent and normal to a curve. At the same time, Barrow mentions in passing the theories on the nature of light and colour that existed at the time. For example, according to one of such ‘theories’, colours derive from mixing light and darkness in different degrees. Barrow’s lectures were sceptical about any of these theories; he refers to them for completeness, rather than indicating any preference for one over another.

This was the scientific environment in which Newton presented his ‘New theory of light and colour’ to the Royal Society in 1673. What is particularly remarkable about this work is the method that Newton used to substantiate his conclusions. Already in this work we see essentially the same method that later forms the basis of his Principia.

The main idea of Newton’s method is to use carefully designed experiments to substantiate theoretical conclusions, or ‘principles’. The underlying concept is that the most important principles of a theory must follow unavoidably from experiments. Newton did not want to ‘create hypotheses’: he wanted Nature itself guide him in his search for the truth.

For example, one of the most important postulates of his new theory of colour was that colour is the property of light itself. It thus does not come from light interacting with this or that physical environment. For example, Newton declared that some rays can only produce red colour and no other. To substantiate this, he described a series of experiments designed to change the colour of light of a certain colour. His conclusion is that this is impossible. He then concludes that there are some rays that are composed of basic colours (which we would now call monochromatic), along with other rays composed of several or many different colours, for example, white light. Newton’s main conclusion is that all experiments known to him could be explained if one accepts that different materials reflect the lights of different colours differently.

All this led Newton to conclude that light is a substance. At the same time he was careful to admit that it is hard to reach a definite conclusion about what is light, how it is reflected and refracted, and how light creates an image of colour in the soul. But he emphasized that he did not want to mix speculations with facts.

It is in Newton’s work on the nature of colour that we see the power of his method for the first time. He used experiments to obtain postulates from which all other facts follow (supposedly) unambiguously, but was also careful to distinguish facts from assumptions seeking to build his theory as far as possible on experimental truths rather than theoretical speculations. This same method in application to mechanics was later employed in his Principia.

Hooke and Controversies Surrounding Newton’s Theory of Colour

It is impossible not to mention the philosopher and polymath Robert Hooke (1635–1703) in relation to Newton, his theory of colour and his work on Principia. Both men had very different personalities and their scientific relationship became embittered by a priority controversy.

Robert Hooke was born on the Isle of Wight into a family of the Church of England priest. He was the youngest of four children and the family was rather poor, but Robert nevertheless received a small amount of money from them and managed to acquire an apprenticeship in London. He completed school there, and in 1653 moved to Oxford, where he secured a chorister’s place at Christ Church. He was first employed as a chemical assistant to Dr Thomas Willis, but from about 1655 to 1662 worked as an assistant to Robert Boyle – now remembered for his Boyle’s law on relation between the pressure of a gas and its volume. It is widely suspected that Hooke made the actual observations and did the mathematics necessary to formulate this law. Yet Hooke was only awarded a Master of Art degree in 1662 or 1663.

From 1662 Robert Hooke became Curator of the newly formed Royal Society, a position he held for over 40 years. His role was to demonstrate to the Society experiments confirming newly discovered physics laws, either of his own selection or on the suggestions of members. Society meetings