Born in the hamlet of Woolsthorpe, Newton was the only son of a local yeoman, also Isaac Newton, who had died three months before, and of Hannah Ayscough. That same year, at Arcetri near Florence, Galileo Galilei had died; Newton would eventually pick up his idea of a mathematical science of motion and bring his work to full fruition. A tiny and weak baby, Newton was not expected to survive his first day of life, much less 84 years. Deprived of a father before birth, he soon lost his mother as well, for within two years she married a second time; her husband, the well-to-do minister Barnabas Smith, left young Isaac with his grandmother and moved to a neighbouring village to raise a son and two daughters. For nine years, until the death of Barnabas Smith in 1653, Isaac was effectively separated from his mother, and his pronounced psychotic tendencies have been ascribed to this traumatic event. That he hated his stepfather we may be sure. When he examined the state of his soul in 1662 and compiled a catalog of sins in shorthand, he remembered “Threatning my father and mother Smith to burne them and the house over them.” The acute sense of insecurity that rendered him obsessively anxious when his work was published and irrationally violent when he defended it accompanied Newton throughout his life and can plausibly be traced to his early years.

After his mother was widowed a second time, she determined that her first-born son should manage her now considerable property. It quickly became apparent, however, that this would be a disaster, both for the estate and for Newton. He could not bring himself to concentrate on rural affairs—set to watch the cattle, he would curl up under a tree with a book. Fortunately, the mistake was recognized, and Newton was sent back to the grammar school in Grantham, where he had already studied, to prepare for the university. As with many of the leading scientists of the age, he left behind in Grantham anecdotes about his mechanical ability and his skill in building models of machines, such as clocks and windmills. At the school he apparently gained a firm command of Latin but probably received no more than a smattering of arithmetic. By June 1661, he was ready to matriculate at Trinity College, Cambridge, somewhat older than the other undergraduates because of his interrupted education.

When Newton arrived in Cambridge in 1661, the movement now known as the scientific revolution was well advanced, and many of the works basic to modern science had appeared. Astronomers from Copernicus to Kepler had elaborated the heliocentric system of the universe. Galileo had proposed the foundations of a new mechanics (mechanism) built on the principle of inertia. Led by Descartes, philosophers had begun to formulate a new conception of nature as an intricate, impersonal, and inert machine. Yet as far as the universities of Europe, including Cambridge, were concerned, all this might well have never happened. They continued to be the strongholds of outmoded Aristotelianism, which rested on a geocentric view of the universe and dealt with nature in qualitative rather than quantitative terms.

Like thousands of other undergraduates, Newton began his higher education by immersing himself in Aristotle's work. Even though the new philosophy was not in the curriculum, it was in the air. Some time during his undergraduate career, Newton discovered the works of the French natural philosopher René Descartes (Descartes, René) and the other mechanical philosophers, who, in contrast to Aristotle, viewed physical reality as composed entirely of particles of matter in motion and who held that all the phenomena of nature result from their mechanical interaction. A new set of notes, which he entitled “Quaestiones Quaedam Philosophicae” (“Certain Philosophical Questions”), begun sometime in 1664, usurped the unused pages of a notebook intended for traditional scholastic exercises; under the title he entered the slogan “Amicus Plato amicus Aristoteles magis amica veritas” (“Plato is my friend, Aristotle is my friend, but my best friend is truth”). Newton's scientific career had begun.

The “Quaestiones” reveal that Newton had discovered the new conception of nature that provided the framework of the scientific revolution. He had thoroughly mastered the works of Descartes and had also discovered that the French philosopher Pierre Gassendi (Gassendi, Pierre) had revived atomism, an alternative mechanical system to explain nature. The “Quaestiones” also reveal that Newton already was inclined to find the latter a more attractive philosophy than Cartesian natural philosophy, which rejected the existence of ultimate indivisible particles. The works of the 17th-century chemist Robert Boyle (Boyle, Robert) provided the foundation for Newton's considerable work in chemistry. Significantly, he had read Henry More (More, Henry), the Cambridge Platonist, and was thereby introduced to another intellectual world, the magical Hermetic tradition, which sought to explain natural phenomena in terms of alchemical and magical concepts. The two traditions of natural philosophy, the mechanical and the Hermetic, antithetical though they appear, continued to influence his thought and in their tension supplied the fundamental theme of his scientific career.

Although he did not record it in the “Quaestiones,” Newton had also begun his mathematical studies. He again started with Descartes, from whose La Géometrie he branched out into the other literature of modern analysis with its application of algebraic techniques to problems of geometry. He then reached back for the support of classical geometry. Within little more than a year, he had mastered the literature; and, pursuing his own line of analysis, he began to move into new territory. He discovered the binomial theorem, and he developed the calculus, a more powerful form of analysis that employs infinitesimal considerations in finding the slopes of curves and areas under curves.

By 1669 Newton was ready to write a tract summarizing his progress, De Analysi per Aequationes Numeri Terminorum Infinitas (“On Analysis by Infinite Series”), which circulated in manuscript through a limited circle and made his name known. During the next two years he revised it as De methodis serierum et fluxionum (“On the Methods of Series and Fluxions”). The word fluxions, Newton's private rubric, indicates that the calculus had been born. Despite the fact that only a handful of savants were even aware of Newton's existence, he had arrived at the point where he had become the leading mathematician in Europe.

When Newton received the bachelor's degree in April 1665, the most remarkable undergraduate career in the history of university education had passed unrecognized. On his own, without formal guidance, he had sought out the new philosophy and the new mathematics and made them his own, but he had confined the progress of his studies to his notebooks. Then, in 1665, the plague closed the university, and for most of the following two years he was forced to stay at his home, contemplating at leisure what he had learned. During the plague years Newton laid the foundations of the calculus and extended an earlier insight into an essay, “Of Colours,” which contains most of the ideas elaborated in his Opticks. It was during this time that he examined the elements of circular motion and, applying his analysis to the Moon and the planets, derived the inverse square relation that the radially directed force acting on a planet decreases with the square of its distance from the Sun—which was later crucial to the law of universal gravitation. The world heard nothing of these discoveries.

Newton was elected to a fellowship in Trinity College in 1667, after the university reopened. Two years later, Isaac Barrow, Lucasian professor of mathematics, who had transmitted Newton's De Analysi to John Collins in London, resigned the chair to devote himself to divinity and recommended Newton to succeed him. The professorship exempted Newton from the necessity of tutoring but imposed the duty of delivering an annual course of lectures. He chose the work he had done in optics as the initial topic; during the following three years (1670–72), his lectures developed the essay “Of Colours” into a form which was later revised to become Book One of his Opticks.

Beginning with Kepler's Paralipomena in 1604, the study of optics had been a central activity of the scientific revolution. Descartes's statement of the sine law of refraction, relating the angles of incidence and emergence at interfaces of the media through which light passes, had added a new mathematical regularity to the science of light, supporting the conviction that the universe is constructed according to mathematical regularities. Descartes had also made light central to the mechanical philosophy of nature; the reality of light, he argued, consists of motion transmitted through a material medium. Newton fully accepted the mechanical nature of light, although he chose the atomistic alternative and held that light consists of material corpuscles in motion. The corpuscular conception of light was always a speculative theory on the periphery of his optics, however. The core of Newton's contribution had to do with colours (colour). An ancient theory extending back at least to Aristotle held that a certain class of colour phenomena, such as the rainbow, arises from the modification of light, which appears white in its pristine form. Descartes had generalized this theory for all colours and translated it into mechanical imagery. Through a series of experiments performed in 1665 and 1666, in which the spectrum of a narrow beam was projected onto the wall of a darkened chamber, Newton denied the concept of modification and replaced it with that of analysis. Basically, he denied that light is simple and homogeneous—stating instead that it is complex and heterogeneous and that the phenomena of colours arise from the analysis of the heterogeneous mixture into its simple components. The ultimate source of Newton's conviction that light is corpuscular was his recognition that individual rays of light have immutable properties; in his view, such properties imply immutable particles of matter. He held that individual rays (that is, particles of given size) excite sensations of individual colours when they strike the retina of the eye. He also concluded that rays refract at distinct angles—hence, the prismatic spectrum, a beam of heterogeneous rays, i.e., alike incident on one face of a prism, separated or analyzed by the refraction into its component parts—and that phenomena such as the rainbow are produced by refractive analysis. Because he believed that chromatic aberration could never be eliminated from lenses, Newton turned to reflecting telescopes; he constructed the first ever built. The heterogeneity of light has been the foundation of physical optics since his time.

There is no evidence that the theory of colours, fully described by Newton in his inaugural lectures at Cambridge, made any impression, just as there is no evidence that aspects of his mathematics and the content of the Principia, also pronounced from the podium, made any impression. Rather, the theory of colours, like his later work, was transmitted to the world through the Royal Society of London, which had been organized in 1660. When Newton was appointed Lucasian professor, his name was probably unknown in the Royal Society; in 1671, however, they heard of his reflecting telescope and asked to see it. Pleased by their enthusiastic reception of the telescope and by his election to the society, Newton volunteered a paper on light and colours early in 1672. On the whole, the paper was also well received, although a few questions and some dissent were heard.

Among the most important dissenters to Newton's paper was Robert Hooke (Hooke, Robert), one of the leaders of the Royal Society who considered himself the master in optics and hence he wrote a condescending critique of the unknown parvenu. One can understand how the critique would have annoyed a normal man. The flaming rage it provoked, with the desire publicly to humiliate Hooke, however, bespoke the abnormal. Newton was unable rationally to confront criticism. Less than a year after submitting the paper, he was so unsettled by the give and take of honest discussion that he began to cut his ties, and he withdrew into virtual isolation.

In 1675, during a visit to London, Newton thought he heard Hooke accept his theory of colours. He was emboldened to bring forth a second paper, an examination of the colour phenomena in thin films, which was identical to most of Book Two as it later appeared in the Opticks. The purpose of the paper was to explain the colours of solid bodies by showing how light can be analyzed into its components by reflection as well as refraction. His explanation of the colours of bodies has not survived, but the paper was significant in demonstrating for the first time the existence of periodic optical phenomena. He discovered the concentric coloured rings in the thin film of air between a lens and a flat sheet of glass; the distance between these concentric rings ( Newton's rings) depends on the increasing thickness of the film of air. In 1704 Newton combined a revision of his optical lectures with the paper of 1675 and a small amount of additional material in his Opticks.

A second piece which Newton had sent with the paper of 1675 provoked new controversy. Entitled “An Hypothesis Explaining the Properties of Light,” it was in fact a general system of nature. Hooke apparently claimed that Newton had stolen its content from him, and Newton boiled over again. The issue was quickly controlled, however, by an exchange of formal, excessively polite letters that fail to conceal the complete lack of warmth between the men.

Newton was also engaged in another exchange on his theory of colours with a circle of English Jesuits in Liège, perhaps the most revealing exchange of all. Although their objections were shallow, their contention that his experiments were mistaken lashed him into a fury. The correspondence dragged on until 1678, when a final shriek of rage from Newton, apparently accompanied by a complete nervous breakdown, was followed by silence. The death of his mother the following year completed his isolation. For six years he withdrew from intellectual commerce except when others initiated a correspondence, which he always broke off as quickly as possible.

During his time of isolation, Newton was greatly influenced by the Hermetic tradition with which he had been familiar since his undergraduate days. Newton, always somewhat interested in alchemy, now immersed himself in it, copying by hand treatise after treatise and collating them to interpret their arcane imagery. Under the influence of the Hermetic tradition, his conception of nature underwent a decisive change. Until that time, Newton had been a mechanical philosopher in the standard 17th-century style, explaining natural phenomena by the motions of particles of matter. Thus, he held that the physical reality of light is a stream of tiny corpuscles diverted from its course by the presence of denser or rarer media. He felt that the apparent attraction of tiny bits of paper to a piece of glass that has been rubbed with cloth results from an ethereal effluvium that streams out of the glass and carries the bits of paper back with it. This mechanical philosophy denied the possibility of action at a distance; as with static electricity, it explained apparent attractions away by means of invisible ethereal mechanisms. Newton's “Hypothesis of Light” of 1675, with its universal ether, was a standard mechanical system of nature. Some phenomena, such as the capacity of chemicals to react only with certain others, puzzled him, however, and he spoke of a “secret principle” by which substances are “sociable” or “unsociable” with others. About 1679, Newton abandoned the ether and its invisible mechanisms and began to ascribe the puzzling phenomena—chemical affinities, the generation of heat in chemical reactions, surface tension in fluids, capillary action, the cohesion of bodies, and the like—to attractions and repulsions between particles of matter. More than 35 years later, in the second English edition of the Opticks, Newton accepted an ether again, although it was an ether that embodied the concept of action at a distance by positing a repulsion between its particles. The attractions and repulsions of Newton's speculations were direct transpositions of the occult sympathies and antipathies of Hermetic philosophy—as mechanical philosophers never ceased to protest. Newton, however, regarded them as a modification of the mechanical philosophy that rendered it subject to exact mathematical treatment. As he conceived of them, attractions were quantitatively defined, and they offered a bridge to unite the two basic themes of 17th-century science—the mechanical tradition, which had dealt primarily with verbal mechanical imagery, and the Pythagorean tradition, which insisted on the mathematical nature of reality. Newton's reconciliation through the concept of force was his ultimate contribution to science.

Nearly five years later, in August 1684, Newton was visited by the British astronomer Edmond Halley (Halley, Edmond), who was also troubled by the problem of orbital dynamics. Upon learning that Newton had solved the problem, he extracted Newton's promise to send the demonstration. Three months later he received a short tract entitled De Motu (“On Motion”). Already Newton was at work improving and expanding it. In two and a half years, the tract De Motu grew into Philosophiae Naturalis Principia Mathematica, which is not only Newton's masterpiece but also the fundamental work for the whole of modern science.

Significantly, De Motu did not state the law of universal gravitation. For that matter, even though it was a treatise on planetary dynamics, it did not contain any of the three Newtonian laws of motion (Newton's laws of motion). Only when revising De Motu did Newton embrace the principle of inertia (the first law) and arrive at the second law of motion. The second law, the force law, proved to be a precise quantitative statement of the action of the forces between bodies that had become the central members of his system of nature. By quantifying the concept of force, the second law completed the exact quantitative mechanics that has been the paradigm of natural science ever since.

The quantitative mechanics of the Principia is not to be confused with the mechanical philosophy. The latter was a philosophy of nature that attempted to explain natural phenomena by means of imagined mechanisms among invisible particles of matter. The mechanics of the Principia was an exact quantitative description of the motions of visible bodies. It rested on Newton's three laws of motion: (1) that a body remains in its state of rest unless it is compelled to change that state by a force impressed on it; (2) that the change of motion (the change of velocity times the mass of the body) is proportional to the force impressed; (3) that to every action there is an equal and opposite reaction. The analysis of circular motion in terms of these laws yielded a formula of the quantitative measure, in terms of a body's velocity and mass, of the centripetal force necessary to divert a body from its rectilinear path into a given circle. When Newton substituted this formula into Kepler's third law, he found that the centripetal force holding the planets in their given orbits about the Sun must decrease with the square of the planets' distances from the Sun. Because the satellites of Jupiter also obey Kepler's third law, an inverse square centripetal force must also attract them to the centre of their orbits. Newton was able to show that a similar relation holds between the Earth and its Moon. The distance of the Moon is approximately 60 times the radius of the Earth. Newton compared the distance by which the Moon, in its orbit of known size, is diverted from a tangential path in one second with the distance that a body at the surface of the Earth falls from rest in one second. When the latter distance proved to be 3,600 (60 × 60) times as great as the former, he concluded that one and the same force, governed by a single quantitative law, is operative in all three cases, and from the correlation of the Moon's orbit with the measured acceleration of gravity (gravitation) on the surface of the Earth, he applied the ancient Latin word gravitas (literally, “heaviness” or “weight”) to it. The law of universal gravitation (Newton's law of gravitation), which he also confirmed from such further phenomena as the tides and the orbits of comets, states that every particle of matter in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

When the Royal Society received the completed manuscript of Book I in 1686, Hooke raised the cry of plagiarism, a charge that cannot be sustained in any meaningful sense. On the other hand, Newton's response to it reveals much about him. Hooke would have been satisfied with a generous acknowledgment; it would have been a graceful gesture to a sick man already well into his decline, and it would have cost Newton nothing. Newton, instead, went through his manuscript and eliminated nearly every reference to Hooke. Such was his fury that he refused either to publish his Opticks or to accept the presidency of the Royal Society until Hooke was dead.

The Principia immediately raised Newton to international prominence. In their continuing loyalty to the mechanical ideal, Continental scientists rejected the idea of action at a distance for a generation, but even in their rejection they could not withhold their admiration for the technical expertise revealed by the work. Young British scientists spontaneously recognized him as their model. Within a generation the limited number of salaried positions for scientists in England, such as the chairs at Oxford, Cambridge, and Gresham College, were monopolized by the young Newtonians of the next generation. Newton, whose only close contacts with women were his unfulfilled relationship with his mother, who had seemed to abandon him, and his later guardianship of a niece, found satisfaction in the role of patron to the circle of young scientists. His friendship with Fatio de Duillier, a Swiss-born mathematician resident in London who shared Newton's interests, was the most profound experience of his adult life.

Almost immediately following the Principia's publication, Newton, a fervent if unorthodox Protestant, helped to lead the resistance of Cambridge to James II's attempt to Catholicize it. As a consequence, he was elected to represent the university in the convention that arranged the revolutionary settlement. In this capacity, he made the acquaintance of a broader group, including the philosopher John Locke (Locke, John). Newton tasted the excitement of London life in the aftermath of the Principia. The great bulk of his creative work had been completed. He was never again satisfied with the academic cloister, and his desire to change was whetted by Fatio's suggestion that he find a position in London. Seek a place he did, especially through the agency of his friend, the rising politician Charles Montague, later Lord Halifax. Finally, in 1696, he was appointed warden of the mint. Although he did not resign his Cambridge appointments until 1701, he moved to London and henceforth centred his life there.

In the meantime, Newton's relations with Fatio had undergone a crisis. Fatio was taken seriously ill; then family and financial problems threatened to call him home to Switzerland. Newton's distress knew no limits. In 1693 he suggested that Fatio move to Cambridge, where Newton would support him, but nothing came of the proposal. Through early 1693 the intensity of Newton's letters built almost palpably, and then, without surviving explanation, both the close relationship and the correspondence broke off. Four months later, without prior notice, Samuel Pepys and John Locke, both personal friends of Newton, received wild, accusatory letters. Pepys was informed that Newton would see him no more; Locke was charged with trying to entangle him with women. Both men were alarmed for Newton's sanity; and, in fact, Newton had suffered at least his second nervous breakdown. The crisis passed, and Newton recovered his stability. Only briefly did he ever return to sustained scientific work, however, and the move to London was the effective conclusion of his creative activity.

As warden and then master of the mint, Newton drew a large income, as much as £2,000 per annum. Added to his personal estate, the income left him a rich man at his death. The position, regarded as a sinecure, was treated otherwise by Newton. During the great recoinage, there was need for him to be actively in command; even afterward, however, he chose to exercise himself in the office. Above all, he was interested in counterfeiting. He became the terror of London counterfeiters, sending a goodly number to the gallows and finding in them a socially acceptable target on which to vent the rage that continued to well up within him.

Newton found time now to explore other interests, such as religion and theology. In the early 1690s he had sent Locke a copy of a manuscript attempting to prove that Trinitarian (Trinity) passages in the Bible were latter-day corruptions of the original text. When Locke made moves to publish it, Newton withdrew in fear that his anti-Trinitarian views would become known. In his later years, he devoted much time to the interpretation of the prophecies of Daniel and St. John, and to a closely related study of ancient chronology. Both works were published after his death.

In London, Newton assumed the role of patriarch of English science. In 1703 he was elected President of the Royal Society. Four years earlier, the French Académie des Sciences (Academy of Sciences) had named him one of eight foreign associates. In 1705 Queen Anne knighted him, the first occasion on which a scientist was so honoured. Newton ruled the Royal Society magisterially. John Flamsteed (Flamsteed, John), the Astronomer Royal, had occasion to feel that he ruled it tyrannically. In his years at the Royal Observatory at Greenwich, Flamsteed, who was a difficult man in his own right, had collected an unrivalled body of data. Newton had received needed information from him for the Principia, and in the 1690s, as he worked on the lunar theory, he again required Flamsteed's data. Annoyed when he could not get all the information he wanted as quickly as he wanted it, Newton assumed a domineering and condescending attitude toward Flamsteed. As president of the Royal Society, he used his influence with the government to be named as chairman of a body of “visitors” responsible for the Royal Observatory; then he tried to force the immediate publication of Flamsteed's catalog of stars. The disgraceful episode continued for nearly 10 years. Newton would brook no objections. He broke agreements that he had made with Flamsteed. Flamsteed's observations, the fruit of a lifetime of work, were, in effect, seized despite his protests and prepared for the press by his mortal enemy, Edmond Halley. Flamsteed finally won his point and by court order had the printed catalog returned to him before it was generally distributed. He burned the printed sheets, and his assistants brought out an authorized version after his death. In this respect, and at considerable cost to himself, Flamsteed was one of the few men to best Newton. Newton sought his revenge by systematically eliminating references to Flamsteed's help in later editions of the Principia.

In Gottfried Wilhelm Leibniz (Leibniz, Gottfried Wilhelm), the German philosopher and mathematician, Newton met a contestant more of his own calibre. It is now well established that Newton developed the calculus before Leibniz seriously pursued mathematics. It is almost universally agreed that Leibniz later arrived at the calculus independently. There has never been any question that Newton did not publish his method of fluxions; thus, it was Leibniz's paper in 1684 that first made the calculus a matter of public knowledge. In the Principia Newton hinted at his method, but he did not really publish it until he appended two papers to the Opticks in 1704. By then the priority controversy was already smouldering. If, indeed, it mattered, it would be impossible finally to assess responsibility for the ensuing fracas. What began as mild innuendoes rapidly escalated into blunt charges of plagiarism on both sides. Egged on by followers anxious to win a reputation under his auspices, Newton allowed himself to be drawn into the centre of the fray; and, once his temper was aroused by accusations of dishonesty, his anger was beyond constraint. Leibniz's conduct of the controversy was not pleasant, and yet it paled beside that of Newton. Although he never appeared in public, Newton wrote most of the pieces that appeared in his defense, publishing them under the names of his young men, who never demurred. As president of the Royal Society, he appointed an “impartial” committee to investigate the issue, secretly wrote the report officially published by the society, and reviewed it anonymously in the Philosophical Transactions. Even Leibniz's death could not allay Newton's wrath, and he continued to pursue the enemy beyond the grave. The battle with Leibniz, the irrepressible need to efface the charge of dishonesty, dominated the final 25 years of Newton's life. It obtruded itself continually upon his consciousness. Almost any paper on any subject from those years is apt to be interrupted by a furious paragraph against the German philosopher, as he honed the instruments of his fury ever more keenly. In the end, only Newton's death ended his wrath.

During his final years Newton brought out further editions of his central works. After the first edition of the Opticks in 1704, which merely published work done 30 years before, he published a Latin edition in 1706 and a second English edition in 1717–18. In both, the central text was scarcely touched, but he did expand the “Queries” at the end into the final statement of his speculations on the nature of the universe. The second edition of the Principia, edited by Roger Cotes in 1713, introduced extensive alterations. A third edition, edited by Henry Pemberton in 1726, added little more. Until nearly the end, Newton presided at the Royal Society (frequently dozing through the meetings) and supervised the mint. During his last years, his niece, Catherine Barton Conduitt, and her husband lived with him.