## Summary

Sir Isaac Newton (25 December 1642 – 20 March 1727 according to the Julian calendar, in force in England until 1752, or January 4, 1643 – March 31, 1727 by the Gregorian calendar) – English physicist, mathematician, mechanic and astronomer, one of the founders of classical physics. Author of the fundamental work “Mathematical Beginnings of Natural Philosophy,” in which he outlined the law of universal gravitation and the three laws of mechanics, which became the basis of classical mechanics. He developed differential and integral calculus, color theory, laid the foundations of modern physical optics, and created many other mathematical and physical theories.

Member (1672) and president (1703-1727) of the Royal Society of London.

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## Early Years

Isaac Newton was born in the village of Woolsthorpe, Lincolnshire, on the eve of the Civil War. Newton”s father, a small but prosperous farmer Isaac Newton (1606-1642), did not live to see his son born. The boy was born prematurely, was sickly, so he long hesitated to be baptized. Yet he survived, was baptized (January 1), and named Isaac in memory of his father. The fact that he was born on Christmas Day was considered by Newton as a special sign of fate. Despite his poor health in infancy, he lived 84 years.

Newton sincerely believed that his family descended from fifteenth-century Scottish nobles, but historians discovered that in 1524 his ancestors were poor peasants. By the end of the sixteenth century, the family had grown rich and passed into the category of yeomen (landowners). Newton”s father left a large inheritance for the time of 500 pounds sterling and several hundred acres of fertile land occupied by fields and forests.

In January 1646, Newton”s mother, Hannah Ayscough (1623-1679), remarried. She had three children with her new husband, a 63-year-old widower, and began to pay little attention to Isaac. The boy”s patron was his maternal uncle, William Ayscough. As a child, Newton, according to his contemporaries, was silent, withdrawn and isolated, liked to read and make technical toys: sundial, water clock, mill, etc. All his life he felt lonely.

His stepfather died in 1653, and part of his inheritance went to Newton”s mother, who immediately registered it in Isaac”s name. His mother returned home, but her main focus was on her three youngest children and her extensive household; Isaac was still left to himself.

In 1655, 12-year-old Newton was sent to study at a nearby school in Grantham, where he lived in the house of Clark the apothecary. Soon the boy showed outstanding ability, but in 1659 his mother Anne returned him to the estate and tried to entrust the 16-year-old son part of the management of the economy. The attempt was unsuccessful – Isaac preferred all other activities reading books, poetry, and especially the construction of various mechanisms. At this time Anna was approached by Stokes, Newton”s schoolteacher, and began to persuade her to continue teaching her unusually gifted son; this request was joined by Uncle William and Humphrey Babington, a Grantham acquaintance of Isaac (a relative of Clark the apothecary), a member of Cambridge Trinity College. With their combined efforts, they eventually got their way. In 1661 Newton successfully graduated from school and went to continue his education at Cambridge University.

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## Trinity College (1661-1664)

In June 1661, the 18-year-old Newton arrived in Cambridge. According to the charter, he was given an examination in Latin, after which he was informed that he had been admitted to Trinity College (Holy Trinity College) at Cambridge University. More than 30 years of Newton”s life are associated with this institution.

The college, like the entire university, was going through a difficult time. The monarchy had just been restored in England (1660), King Charles II often delayed the payments due to the university and dismissed a large part of the teaching staff appointed during the Revolution. A total of 400 people lived at Trinity College, including students, servants, and 20 beggars, to whom the charter required the college to give alms. The educational process was in a deplorable state.

Newton was enrolled as a “sizar” student, who was not charged tuition (probably on Babington”s recommendation). According to the norms of the time, a sizer was obliged to pay for his education through various jobs at the University, or through services to wealthier students. Very little documentary evidence or recollection of this period of his life has survived. During these years Newton”s character was finally formed – a desire to get to the bottom of things, intolerance of deceit, slander and oppression, and indifference to public fame. He still had no friends.

In April 1664 Newton, having passed his examinations, was promoted to the higher class of senior students (scholars), which gave him the right to a scholarship and to continue his studies at the college.

Despite Galileo”s discoveries, science and philosophy at Cambridge were still taught according to Aristotle. However, Newton”s extant notebooks already mention Galileo, Copernicus, Cartesianism, Kepler, and Gassendi”s atomistic theory. Judging from these notebooks, he continued to make (mostly scientific instruments), and was passionate about optics, astronomy, mathematics, phonetics, and music theory. According to his roommate”s recollections, Newton devoted himself wholeheartedly to his studies, forgetting food and sleep; it was probably, despite all the difficulties, the way of life he himself desired.

The year 1664 was also rich in other events in Newton”s life. Newton experienced a creative upsurge, began independent scientific activity and compiled an extensive list (of 45 items) of unsolved problems in nature and human life (Questiones quaedam philosophicae, Latin). Such lists subsequently appeared more than once in his workbooks. In March of the same year, in the newly founded (1663) Department of Mathematics at the college, lectures began by a new instructor, 34-year-old Isaac Barrow, a major mathematician, Newton”s future friend and teacher. Newton”s interest in mathematics increased dramatically. He made his first significant mathematical discovery: the binomial expansion for an arbitrary rational exponent (including negative ones), and through it he came to his main mathematical method: the expansion of a function into an infinite series. At the very end of the year, Newton became an undergraduate.

Newton”s scientific support and inspiration for his work were, to a great extent, the physicists Galileo, Descartes, and Kepler. Newton completed their writings by combining them into a universal system of the world. Lesser but significant influences came from other mathematicians and physicists: Euclid, Fermat, Huygens, Wallis, and his immediate teacher Barrow. In Newton”s student notebook there is a program phrase:

There can be no sovereign in philosophy but truth… We should put monuments of gold to Kepler, Galileo, and Descartes and write on each one: “Plato is a friend, Aristotle is a friend, but the chief friend is truth.

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## “The Plague Years.” (1665-1667)

On Christmas Eve, 1664, red crosses began to appear on London houses – the first marks of the Great Plague Epidemic. By the summer, the deadly epidemic had greatly expanded. On August 8, 1665, classes at Trinity College were terminated and the staff disbanded until the end of the epidemic. Newton went home to Woolsthorpe, taking his basic books, notebooks, and instruments with him.

These were disastrous years for England – a devastating plague (a fifth of the population died in London alone), a devastating war with Holland, and the Great Fire of London. But much of Newton”s scientific discoveries were made in the seclusion of the “plague years. The surviving notes show that the 23-year-old Newton was already fluent in the basic methods of differential and integral calculus, including the expansion of functions into series and what was later called the Newton-Leibniz formula. In a series of witty optical experiments, he proved that white is a mixture of the colors of the spectrum. Newton later recalled these years:

At the beginning of 1665 I found the method of approximate series and the rule of converting any degree of a two-manifold into such a series… in November I got the direct method of fluxions; in January of the following year I got the theory of colors, and in May I set about the inverse method of fluxions… At this time I was at the best period of my youth and was more interested in mathematics and philosophy than at any other time afterwards.

But his most significant discovery during these years was the law of universal gravitation. Later, in 1686, Newton wrote to Halley:

In papers written more than 15 years ago (I cannot give the exact date, but at any rate it was before the beginning of my correspondence with Oldenburg), I expressed the inverse quadratic proportionality of the planets” gravitation to the Sun as a function of distance and calculated the correct ratio of the Earth”s gravity and conatus recedendi of the Moon to the center of the Earth, although not quite precisely.

The inaccuracy mentioned by Newton was caused by the fact that Newton took the size of the Earth and the value of the acceleration of free fall from Galileo”s Mechanics, where they were given with a significant error. Later, Newton obtained more accurate data from Picard and was finally convinced of the truth of his theory.

It is a well-known legend that Newton discovered the law of gravitation by observing an apple falling from a tree branch. Newton”s apple was first glimpsed by Newton”s biographer William Stukeley (in his book Memoirs of Newton”s Life, 1752):

After lunch the weather was warm, we went out into the orchard and drank tea in the shade of the apple trees. He told me that the thought of gravity had occurred to him while he was sitting under a tree just like that. He was in a contemplative mood when suddenly an apple fell from a branch. “Why do apples always fall perpendicular to the ground?” – he thought.

The legend became popular thanks to Voltaire. In fact, as can be seen from Newton”s workbooks, his theory of universal gravitation developed gradually. Another biographer, Henry Pemberton, cites Newton”s reasoning (without mentioning the apple) in more detail: “comparing the periods of several planets and their distances from the Sun, he found that … this force must decrease in quadratic proportion with increasing distance. In other words, Newton discovered that from Kepler”s third law, which relates the periods of the planets” orbits to their distance from the Sun, follows precisely the “inverse square formula” for the law of gravitation (in the circular orbit approximation). The final formulation of the law of gravitation, which entered the textbooks, Newton wrote out later, after the laws of mechanics became clear to him.

These discoveries, as well as many of the later ones, were published 20-40 years later than they were made. Newton was not chasing fame. In 1670 he wrote to John Collins: “I see nothing desirable in fame, even if I were able to earn it. It would perhaps increase the number of my acquaintances, but this is precisely what I am most anxious to avoid.” His first scientific work (not found until 300 years later.

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## The Beginning of Scientific Fame (1667-1684)

In March and June of 1666, Newton visited Cambridge. In the summer, however, a new wave of the plague forced him to go home again. Finally, in early 1667 the epidemic subsided, and in April Newton returned to Cambridge. On 1 October he was elected a member of Trinity College, and in 1668 he became a Master. He was allocated a spacious separate room for housing, assigned a salary (£2 a year) and handed a group of students with whom he diligently engaged several hours a week in standard academic subjects. However, neither then nor later did Newton become famous as a teacher; his lectures were poorly attended.

Having strengthened his position, Newton traveled to London, where shortly before, in 1660, the Royal Society of London had been established – an authoritative organization of prominent scientists, one of the first Academies of Sciences. The printed organ of the Royal Society was the journal Philosophical Transactions.

In 1669, mathematical works using expansions into infinite series began to appear in Europe. Although the depth of these discoveries did not compare with Newton”s, Barrow insisted that his student record his priority in this matter. Newton wrote a brief but fairly complete outline of this part of his discoveries, which he called “Analysis by means of equations with an infinite number of terms.” Barrow forwarded this treatise to London. Newton asked Barrow not to reveal the name of the author of the work (but he did let it slip). “Analysis” spread among specialists and gained some fame in England and beyond.

That same year Barrow accepted the King”s invitation to become court chaplain and resigned from teaching. On October 29, 1669, the 26-year-old Newton was chosen to succeed him as “Lucas Professor” of mathematics and optics at Trinity College. In this position Newton received a salary of 100 pounds a year, not including other bonuses and stipends from Trinity. The new post also gave Newton more time for his own research. Barrow left Newton an extensive alchemical laboratory; during this period Newton became seriously interested in alchemy and conducted a host of chemical experiments.

At the same time, Newton continued his experiments in optics and color theory. Newton investigated spherical and chromatic aberration. To minimize them, he built a mixed reflector telescope: a lens and a concave spherical mirror, which he made and polished himself. The project of such a telescope was first proposed by James Gregory (1663), but this idea was never realized. Newton”s first design (1668) was unsuccessful, but the next, with a more carefully polished mirror, despite its small size, gave 40x magnification of excellent quality.

Word of the new instrument quickly reached London, and Newton was invited to show his invention to the scientific public. In late 1671 or early 1672, the reflector was demonstrated in front of the king, and then – at the Royal Society. The apparatus attracted universal rave reviews. Apparently its practical importance also played a role: astronomical observations were used to determine the exact time, which in turn was necessary for navigation at sea. Newton became famous and in January 1672 he was elected a Fellow of the Royal Society. Later, improved reflectors became the basic tools of astronomers, and they were used to discover the planet Uranus, other galaxies, and the red shift.

At first Newton treasured his fellowship with his colleagues in the Royal Society, which included, besides Barrow, James Gregory, John Wallis, Robert Hooke, Robert Boyle, Christopher Wren and other well-known figures of English science. Soon, however, tiresome conflicts ensued, which Newton disliked very much. In particular, a noisy controversy broke out over the nature of light. It began with the fact that in February 1672 Newton published in Philosophical Transactions a detailed description of his classical experiments with prisms and his theory of color. Hooke, who had earlier published his own theory, declared that he was not convinced by Newton”s results; he was supported by Huygens on the grounds that Newton”s theory “contradicted conventional wisdom.” Newton responded to their criticism only six months later, but by this time the number of critics had greatly increased.

Two important events occurred in 1673. First: by royal decree, Newton”s old friend and patron, Isaac Barrow, returned to Trinity, now as head (“master”) of the college. Second: Leibniz, known at the time as a philosopher and inventor, became interested in Newton”s mathematical discoveries. After receiving Newton”s 1669 work on infinite series and studying it in depth, he went on to develop his own version of analysis. In 1676 Newton and Leibniz exchanged letters in which Newton explained a number of his methods, answered Leibniz”s questions, and hinted at the existence of even more general methods, as yet unpublished (meaning general differential and integral calculus). Henry Oldenburg, secretary of the Royal Society, pressed Newton for the glory of England to publish his mathematical discoveries on analysis, but Newton replied that he had been engaged on another subject for five years and did not wish to be distracted. Newton did not respond to Leibniz”s next letter. The first brief publication on Newton”s version of analysis did not appear until 1693, when Leibniz”s version had already spread widely in Europe.

The end of the 1670s was sad for Newton. In May 1677, Barrow, 47, died unexpectedly. In the winter of that year, a massive fire broke out in Newton”s house and part of Newton”s manuscript archives burned down. In September 1677 Oldenburg, Newton”s beneficiary, died as secretary of the Royal Society, and the new secretary was Hooke, who treated Newton with hostility. In 1679 his mother Anne became gravely ill; Newton, leaving everything to do, came to her, took an active part in the care of the sick, but his mother”s condition rapidly deteriorated, and she died. Mother and Barrow were among the few people who brightened Newton”s loneliness.

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## “Mathematical Beginnings of Natural Philosophy” (1684-1686)

The story of this work, one of the most famous in the history of science, began in 1682, when the passage of Halley”s comet sparked a surge of interest in celestial mechanics. Edmond Halley tried to persuade Newton to publish his “general theory of motion,” which had long been rumored in the scientific community. Newton, not wanting to get involved in new scientific disputes and bickering, refused.

In August 1684, Halley came to Cambridge and told Newton that he and Wren and Hooke had discussed how to derive the ellipticity of the orbits of the planets from the formula for the law of gravitation, but did not know how to approach the solution. Newton reported that he already had such a proof, and in November he sent the finished manuscript to Halley. He immediately appreciated the value of the result and the method, immediately visited Newton again and this time managed to persuade him to publish his findings. On December 10, 1684, a historical record appeared in the Minutes of the Royal Society:

Mr. Halley … recently saw Mr. Newton in Cambridge and he showed him an interesting treatise “De motu”. According to the wish of Mr. Halley, Newton promised to send the said treatise to the Society.

Work on the book went on from 1684 to 1686. According to the recollections of Humphrey Newton, a relative of the scientist and his assistant during these years, at first Newton wrote “Elements” in the intervals between alchemical experiments, which he focused on, then he gradually became fascinated and devoted himself with enthusiasm to work on the main book of his life.

The publication was supposed to be financed by the Royal Society, but in early 1686 the Society published a treatise on the history of fishes that was not in demand, and thus depleted its budget. Halley then announced that he would bear the costs of the publication. The Society gratefully accepted this generous offer and, as partial compensation, provided Halley with 50 copies of the treatise on the history of fishes free of charge.

Newton”s work – perhaps by analogy with Descartes” Beginnings of Philosophy (1644) or, according to some historians of science, a challenge to the Cartesians – was called Mathematical Beginnings of Natural Philosophy (Latin Philosophiae Naturalis Principia Mathematica), that is, in modern language, Mathematical Foundations of Physics.

On April 28, 1686, the first volume of “Mathematical Beginnings” was presented to the Royal Society. All three volumes, after some editing by the author, were published in 1687. The circulation (about 300 copies) was sold out in four years – very fast for that time.

Both physically and mathematically, Newton”s work is qualitatively superior to the works of all his predecessors. It lacks Aristotelian or Cartesian metaphysics, with its vague reasoning and vaguely formulated, often far-fetched “prime causes” of natural phenomena. Newton, for example, does not proclaim that the law of gravitation applies in nature, but he strictly proves this fact on the basis of the observed pattern of motion of the planets and their satellites. Newton”s method is to create a model of the phenomenon, “without inventing hypotheses,” and then, if there is enough data, search for its causes. This approach, which began with Galileo, meant the end of the old physics. The qualitative description of nature had given way to the quantitative, with calculations, drawings, and tables occupying a considerable part of the book.

In his book, Newton clearly defined the basic concepts of mechanics, and introduced several new ones, including such important physical quantities as mass, external force and quantity of motion. Three laws of mechanics were formulated. A strict derivation of all three of Kepler”s laws of gravitation is given. Note that hyperbolic and parabolic orbits of celestial bodies unknown to Kepler were also described. Newton does not directly discuss the truth of Copernicus” heliocentric system, but implies it; he even estimates the deviation of the Sun from the center of mass of the solar system. In other words, the Sun in Newton”s system, unlike Kepler”s, does not rest, but obeys the general laws of motion. The general system also includes comets, the type of their orbits caused great controversy at the time.

The weakness of Newton”s theory of gravitation, according to many scientists of the time, was the lack of an explanation of the nature of this force. Newton outlined only the mathematical apparatus, leaving open questions about the cause of gravitation and its material carrier. For a scientific community raised on Descartes” philosophy, this was an unfamiliar and challenging approach, and only the triumphant success of celestial mechanics in the 18th century forced physicists to temporarily come to terms with Newton”s theory. The physical foundations of gravitation only became clearer more than two centuries later, with the advent of the General Theory of Relativity.

The mathematical apparatus and the general structure of the book were constructed by Newton as close as possible to the standard of scientific rigor recognized by his contemporaries, Euclid”s Elements. He deliberately avoided using mathematical analysis almost anywhere – the use of new, unfamiliar methods would have jeopardized the credibility of the results. This caution, however, devalued Newton”s method of presentation for future generations of readers. Newton”s book was the first work on the new physics and at the same time one of the last serious works to use the old methods of mathematical investigation. All of Newton”s followers were already using the powerful methods of mathematical analysis he had created. D”Alambert, Euler, Laplace, Clero and Lagrange were the greatest direct successors of Newton”s work.

During the author”s lifetime, the book survived three editions, and with each reprint the author made significant additions and refinements to the book.

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## Administrative Activities (1687-1703)

The year 1687 was marked not only by the publication of the great book, but also by Newton”s conflict with King James II. In February, the king, in his consistent line of restorative Catholicism in England, ordered the University of Cambridge to grant a master”s degree to the Catholic monk Alban Francis. The university authorities hesitated, wanting neither to break the law nor to annoy the king; soon a delegation of scholars, including Newton, was summoned for reprisals before Lord High Justice George Jeffreys, known for his rudeness and cruelty. Newton opposed any compromise that infringed on university autonomy and persuaded the delegation to take a principled stand. In the end the vice-chancellor of the university was dismissed, but the king”s wish was never carried out. In one of his letters of these years, Newton outlined his political principles:

Every honest man is bound by the laws of God and man to obey the lawful commands of the king. But if His Majesty is advised to demand something which cannot be done by law, no one should suffer harm if he neglects such a demand.

In 1689, after the overthrow of King James II, Newton was first elected to parliament from Cambridge University and sat there for just over a year. He was again a member of parliament in 1701-1702. There is a popular anecdote that he took the floor to address the House of Commons only once, asking to close the window to avoid a draught. In fact, Newton carried out his parliamentary duties with the same integrity with which he treated all his affairs.

Around 1691 Newton became seriously ill (most likely poisoned during chemical experiments, although other versions also exist – overwork, shock after a fire that resulted in the loss of important results, and age-related ailments). Relatives feared for his sanity; several of his surviving letters from this period do show evidence of mental distress. It was not until late 1693 that Newton”s health fully recovered.

In 1679 Newton met at Trinity with an 18-year-old aristocrat, a lover of science and alchemy, Charles Montague (1661-1715). Newton probably made the strongest impression on Montague, because in 1696, becoming Lord Halifax, president of the Royal Society and Chancellor of the Exchequer (ie, Minister of Finance of England), Montague offered to the king to appoint Newton as curator of the mint. The king gave his consent, and in 1696 Newton took the post, left Cambridge and moved to London.

To begin with, Newton thoroughly studied the technology of coinage, put the paperwork in order, and redesigned the accounting for the past 30 years. At the same time, Newton energetically and expertly assisted Montague”s monetary reform, restoring confidence in the thoroughly neglected English monetary system of his predecessors. In England of these years there were almost exclusively incomplete, and in considerable quantities, counterfeit coins. Cutting the edges of silver coins was widespread, and coins of new minting disappeared as soon as they came into circulation, since the masses went into overfilling, were exported abroad and hidden in chests. In this situation, Montague concluded that the situation could only be changed by recoining all the coins circulating in England and banning the circulation of cut coins, which required a drastic increase in the productivity of the Royal Mint. This required a competent administrator, and such a man was Newton, who in March 1696 took the post of curator of the Mint.

Thanks to the energetic actions of Newton during 1696 a network of branches of the Mint was established in cities in England, particularly in Chester, where Newton put his friend Halley as director of the branch, which allowed to increase the production of silver coinage by 8 times. Newton introduced the use of inscribed girth into coinage technology, after which criminal grinding of the metal became virtually impossible. The old, incomplete silver coins were completely withdrawn from circulation and recoined in 2 years, the issue of new coins increased in order to keep up with the demand for them, their quality improved. Earlier, during similar reforms, the population had to change the old money by weight, after which the amount of cash used to decrease both at individuals (private and legal) and in the whole country, but the interest and credit liabilities remained the same, which caused the economy to stagnate. Newton proposed to exchange money at face value, which prevented these problems, and the inevitable shortage of funds after this was made up by borrowing from other countries (most of all from the Netherlands), inflation declined sharply, but the external public debt rose to unprecedented sizes in the history of England by mid-century. But during this time there was noticeable economic growth, because of it the tax deductions to the treasury increased (equal in size to the French, despite the fact that France was populated by 2.5 times more people), at the expense of this public debt was gradually repaid.

In 1699 the recoining of coins was completed and, apparently as a reward for his services, in that year Newton was appointed governor (“master”) of the Mint. However, an honest and competent man at the head of the Mint did not suit everyone. From the earliest days of Newton poured complaints and denunciations, constantly appearing inspection commissions. As it turned out, many of the denunciations came from counterfeiters irritated by Newton”s reforms. Newton was generally indifferent to backbiting, but never forgave if it affected his honor and reputation. He was personally involved in dozens of investigations, and more than 100 counterfeiters were tracked down and convicted; in the absence of aggravating circumstances, they were most often exiled to the North American colonies, but several ringleaders were executed. The number of counterfeit coins in England declined considerably. Montague, in his memoirs, praised Newton”s extraordinary administrative abilities, which ensured the success of the reform. Thus, the reforms carried out by the scholar not only prevented the economic crisis, but decades later led to a significant increase in the wealth of the country.

In April 1698 the Russian Tsar Peter I visited the Mint three times during the “Great Embassy”; unfortunately, the details of his visit and communication with Newton have not been preserved. It is known, however, that in 1700 in Russia was carried out a coin reform, similar to the English. And in 1713 the first six printed copies of the 2nd edition of the “Elements” Newton sent to Tsar Peter in Russia.

The symbol of Newton”s scientific triumph became two events in 1699: began teaching Newton”s system of the world in Cambridge (since 1704 – and Oxford), and the Paris Academy of Sciences, a bulwark of his opponents, the Cartesians, elected him its foreign member. All this time Newton was still listed as a member and professor of Trinity College, but in December 1701 he officially resigned from all his posts at Cambridge.

In 1703 the president of the Royal Society, Lord John Somers, died, having attended only twice during the five years of his presidency. In November Newton was elected his successor and ruled the Society for the rest of his life – more than twenty years. Unlike his predecessors, he personally attended all the meetings and did everything to ensure that the British Royal Society took an honorable place in the scientific world. The number of members of the Society grew (among them, apart from Halley, we can single out Denis Papin, Abraham de Moivre, Roger Cotes, Brook Taylor), interesting experiments were conducted and discussed, the quality of journal articles was greatly improved, and financial problems were alleviated. The society acquired paid secretaries and its own residence (in Fleet Street); Newton paid the cost of the move out of his own pocket. During these years Newton was often invited as a consultant to various government commissions, and Princess Caroline, the future Queen of Great Britain (wife of George II), spent hours with him at the palace in conversations on philosophical and religious topics.

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## Recent years

In 1704, the monograph “Optics” was published (first in English), which determined the development of this science until the beginning of the 19th century. It contained an appendix “On the Quadrature of Curves,” the first and fairly complete presentation of Newton”s version of mathematical analysis. It is in fact Newton”s last work on the natural sciences, although he still lived more than 20 years. The catalog of the library he left behind contained books mostly on history and theology, and it was these that Newton devoted the rest of his life to. Newton remained steward of the Mint, for this position, unlike that of overseer, did not require much activity on his part. Twice a week he went to the Mint, once a week to a meeting of the Royal Society. Newton never traveled outside England.

In 1705 Queen Anne knighted Newton. Henceforth he was Sir Isaac Newton. This was the first time in English history that the rank of knight was conferred for scientific merit; the next time it happened more than a century later (1819, with regard to Humphrey Davy). Some biographers, however, believe that the Queen was motivated not by science but by politics. Newton acquired his own coat of arms and a not very reliable pedigree.

In 1707 Newton published a collection of lectures on algebra called “Universal Arithmetic. The numerical methods it contained marked the birth of a new and promising discipline: numerical analysis.

In 1708, an open priority dispute with Leibniz began (see below), in which even royalty were involved. This tussle between the two geniuses cost science dearly – the English mathematical school soon became inactive for a century, and the European school ignored many of Newton”s outstanding ideas, rediscovering them much later. The conflict was not extinguished even by Leibniz”s death (1716).

The first edition of Newton”s “Beginnings” had long ago been sold out. Newton”s many years of work to prepare the 2nd edition, clarified and supplemented, was crowned with success in 1710, when the first volume of the new edition came out (the last, third – in 1713). The initial circulation (700 copies) was clearly insufficient, and in 1714 and 1723 additional copies were printed. In finalizing the second volume, Newton, as an exception, had to return to physics to explain the discrepancy between the theory and experimental data, and he immediately made a major discovery – the hydrodynamic contraction of the jet. The theory was now in good agreement with experiment. Newton added an “Exhortation” at the end of his book with a devastating critique of the “vortex theory” with which his Cartesian opponents were trying to explain the motion of the planets. To the natural question “how is it really?” the book follows the famous and honest answer: “The reason … of the properties of gravity I still could not deduce from the phenomena, I do not invent hypotheses.

In April 1714, Newton summarized his experience with financial regulation and submitted his article “Observations Concerning the Value of Gold and Silver” to the Treasury. The article contained specific proposals for adjusting the value of precious metals. These proposals were partially accepted, and this had a favorable effect on the English economy.

Shortly before his death, Newton was one of the victims of a financial scam by the government-backed South Seas Trading Company. He purchased the company”s securities for a large sum and also insisted that they be purchased by the Royal Society. On September 24, 1720, the company”s bank declared itself bankrupt. His niece Catherine recalled in her notes that Newton lost more than 20,000 pounds, after which he declared that he could calculate the motion of celestial bodies, but not the degree of madness of the crowd. However, many biographers believe that Catherine did not mean the actual loss, but the failure to receive the expected profit. After the company went bankrupt, Newton offered to compensate the Royal Society out of his own pocket, but his offer was rejected.

Newton devoted the last years of his life to writing The Chronology of the Ancient Kingdoms, which he was engaged in for about 40 years, as well as to preparing the third edition of the Elements, which came out in 1726. In contrast to the second, the changes in the third edition were minor – mainly the results of new astronomical observations, including a fairly complete guide to the comets observed since the fourteenth century. Among others the calculated orbit of Halley”s comet was presented, whose new appearance at that time (1758) clearly confirmed the theoretical calculations of (by that time already deceased) Newton and Halley. The circulation of the book for a scientific publication of those years could be considered huge: 1250 copies.

In 1725 Newton”s health began to deteriorate noticeably and he moved to Kensington near London, where he died at night, in his sleep, March 20 (31), 1727.He did not leave a written will, but he gave a large part of his large fortune shortly before his death to his closest relatives. He is buried in Westminster Abbey. Fernando Savater, from Voltaire”s letters, describes Newton”s funeral in this way:

The whole of London took part. First the body was displayed for all to see in a magnificent hearse, flanked by huge lamps, then it was carried to Westminster Abbey, where Newton was buried among kings and prominent statesmen. At the head of the funeral procession was the Lord Chancellor, followed by all the royal ministers.

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## Character traits

It is difficult to draw up a psychological portrait of Newton, since even his sympathizers often attribute different qualities to Newton. We have to take into account the cult of Newton in England, which forced the authors of memoirs to endow the great scientist with every conceivable virtue, ignoring the real contradictions in his nature. In addition, by the end of his life, Newton”s character was characterized by such traits as good-naturedness, condescension, and sociability that were not previously characteristic of him.

On the outside, Newton was short, of sturdy build, with wavy hair. He hardly ever got sick, retained thick hair (already quite gray from the age of 40) and all but one of his teeth until old age. He never (according to other reports, almost never) used eyeglasses, though he was somewhat nearsighted. Almost never laughed or became irritated, there is no mention of his jokes or other displays of humor. He was careful and frugal with money, but not stingy. He was never married. Usually he was in deep inner concentration, which often made him absent-minded: for instance, once when he invited guests, he went to the pantry for wine, but then some scientific idea struck him and he rushed to his study and did not return to the guests. He was indifferent to sports, music, art, theater, travel. His assistant recalled: “He did not allow himself any rest and respite … considered lost every hour that is not devoted to studies … I think he saddened the need to spend time on food and sleep. With all this said, Newton was able to combine worldly practicality and common sense, evident in his successful management of the Mint and the Royal Society.

Raised in the Puritan tradition, Newton set for himself a series of rigid principles and self-restrictions. And he was not inclined to forgive others what he would not forgive himself; this is the root of many of his conflicts (see below). He was warm to relatives and many colleagues, but had no close friends, did not seek the company of others, kept aloof. At the same time Newton was not heartless and indifferent to the fate of others. When, after the death of his half-sister Anne”s children were left without a livelihood, Newton appointed minor children allowance, and later daughter Anne, Catherine, took over the education. Constantly helped other relatives as well. “Being thrifty and calculating, he was at the same time very free with money and was always ready to help a friend in need, without being obtrusive. He was especially generous to young people. Many famous English scientists – Stirling, McLaren, the astronomer James Pound and others – remembered with deep gratitude the help Newton provided at the beginning of their scientific careers.

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## Conflicts

In 1675 Newton sent the Society his treatise with new studies and speculations on the nature of light. Robert Hooke stated at the meeting that everything of value in the treatise was already available in Hooke”s previously published book, Micrography. In private conversations he accused Newton of plagiarism: “I have shown that Mr. Newton used my hypotheses about impulses and waves” (from Hooke”s diary). Hooke challenged the priority of all Newton”s discoveries in the field of optics, except those with which he disagreed. Oldenburg immediately notified Newton of these accusations, and he regarded them as insinuations. This time the conflict was extinguished and the scientists exchanged letters of conciliation (1676). However, from that moment until Hooke”s death (1703), Newton did not publish any works on optics, although he had accumulated a huge material, systematized by him in his classic monograph “Optics” (1704).

Another priority controversy was the discovery of the law of gravitation. As early as 1666, Hooke concluded that the motion of the planets is a superposition of falling on the Sun due to the force of gravity on the Sun, and motion by inertia on a tangent to the trajectory of the planet. In his opinion, this superposition of motion and causes the elliptical shape of the trajectory of the planet around the Sun. However, he could not prove it mathematically, and he sent a letter to Newton in 1679, where he offered cooperation in solving the problem. This letter also outlined the assumption that the force of attraction to the Sun decreases in inverse proportion to the square of the distance. In response, Newton remarked that he had previously dealt with the problem of planetary motion, but had abandoned these studies. Indeed, as the documents found later show, Newton was dealing with the problem of planetary motion back in 1665-1669, when he on the basis of Kepler”s III law found that “the tendency of the planets to move away from the Sun will be inversely proportional to the squares of their distances from the Sun”. However, the idea of the orbit of the planet as the result of the equality of the forces of attraction to the Sun and the centrifugal force was not yet fully developed by him in those years.

Subsequently, the correspondence between Hooke and Newton broke off. Hooke returned to trying to construct the trajectory of the planet under the force decreasing under the law of inverse squares. However, these attempts also proved unsuccessful. Meanwhile, Newton returned to the study of planetary motion and solved the problem.

When Newton was preparing his Elements for publication, Hooke demanded that Newton stipulate in the preface Hooke”s priority of the law of gravitation. Newton objected that Bullwald, Christopher Wren and Newton himself had arrived at the same formula independently and before Hooke. A conflict erupted that poisoned the lives of both scientists.

Modern authors pay tribute to both Newton and Hooke. Hooke”s priority is the formulation of the problem of constructing the trajectory of the planet due to the superposition of its fall to the Sun according to the law of inverse squares and motion by inertia. It is also possible that it was Hooke”s letter that directly prompted Newton to complete the solution of this problem. However, Hooke himself did not solve the problem, nor did he guess the universality of gravity,

If we combine all of Hooke”s assumptions and thoughts about the motion of the planets and gravitation, expressed by him for almost 20 years, we meet almost all the main conclusions of Newton”s “Elements”, only expressed in an uncertain and little provable form. Without solving the problem, Hooke found the answer. However, we do not have before us a randomly thrown idea, but undoubtedly the fruit of long years of work. Hooke had the genius of a physicist-experimenter who discerns the true relations and laws of nature in a labyrinth of facts. With a similar rare intuition of the experimenter we meet in the history of science with Faraday, but Hooke and Faraday were not mathematicians. Their work was completed by Newton and Maxwell.The aimless struggle with Newton for priority cast a shadow on the glorious name of Hooke, but history is time, after almost three centuries, to pay tribute to each. Hooke could not follow the straight, immaculate path of Newton”s Mathematical Beginnings, but by his roundabout paths, the traces of which we can no longer find, he came to the same place.

Later, Newton”s relationship with Hooke remained strained. For example, when Newton presented the Society with a new construction of the sextant that he had invented, Hooke immediately stated that he had invented such a device more than 30 years ago (although he had never built a sextant). Nevertheless, Newton was aware of the scientific value of Hooke”s discoveries and in his “Optics” mentioned his, already deceased, opponent several times.

In addition to Newton, Hooke had priority disputes with many other English and Continental scientists, including Robert Boyle, whom he accused of appropriating an improvement of the air pump, and the secretary of the Royal Society, Oldenburg, claiming that with Oldenburg”s help, Huygens had stolen from Hooke the idea of the spiral spring clock.

The myth that Newton allegedly ordered the destruction of Hooke”s only portrait is discussed below.

John Flemsteed, the eminent English astronomer, met Newton at Cambridge (1670), when Flemsteed was still a student and Newton a master. However, almost simultaneously with Newton, Flemsteed also became famous in 1673 when he published astronomical tables of outstanding quality, for which the king granted him a personal audience and the title of “Royal Astronomer”. Moreover, the king had an observatory built at Greenwich near London and placed at Flemstead”s disposal. However, the king considered the money to equip the observatory an unnecessary expenditure, and almost all of Flemsteed”s income was spent on building instruments and the economic needs of the observatory.

At first, the relationship between Newton and Flemsteed was good-natured. Newton was preparing a second edition of the Elements and badly needed accurate observations of the moon to construct and (in the first edition, the theory of the motion of the moon and comets was unsatisfactory. It was also important for the validation of Newton”s theory of gravitation, which had been sharply criticized by Cartesians on the Continent. Flemsteed willingly gave him the requested data, and in 1694 Newton proudly informed Flemsteed that a comparison of calculated and experimental data showed a practical match. In some letters, Flemsteed urged Newton to stipulate his, Flemsteed”s, priority if observations were used; this primarily applied to Halley, whom Flemsteed disliked and suspected of scientific dishonesty, but could also mean a lack of confidence in Newton himself. In Flemsteed”s letters, resentment began to appear:

I agree: the wire is more valuable than the gold of which it is made. I have, however, collected this gold, cleaned and washed it, and I dare not think that you appreciate my help so little just because you received it so easily.

The open conflict began with a letter from Flemsteed in which he apologized for finding a number of systematic errors in some of the data given to Newton. This jeopardized Newton”s theory of the moon and forced him to redo the calculations, and the credibility of the other data was also shaken. Newton, who could not tolerate dishonesty, was extremely irritated and even suspected that Flemsteed had deliberately made the mistakes.

In 1704, Newton visited Flemstead, who had by this time obtained new, extremely accurate observational data, and asked him to pass these data on; in return, Newton promised to help Flemstead in publishing his major work, the Great Star Catalog. Flemsteed, however, began to procrastinate for two reasons: the catalog was not quite ready yet, and he no longer trusted Newton and was afraid of stealing his invaluable observations. Flemstead used the experienced calculators given to him to complete his work to calculate the positions of the stars, while Newton was primarily interested in the moon, planets and comets. Finally, in 1706, printing of the book began, but Flemstead, who suffered from excruciating gout and was becoming increasingly suspicious, demanded that Newton not open the sealed printer”s copy until printing was complete; Newton, who needed the data urgently, disregarded this prohibition and wrote out the necessary quantities. The tension was growing. Flemsteed scandalized Newton for attempting to personally make minor corrections of errors. The printing of the book was extremely slow.

Due to financial difficulties, Flemsteed failed to pay his membership fee and was expelled from the Royal Society; a new blow came from the Queen, who, apparently at Newton”s request, transferred control of the observatory to the Society. Newton gave Flemsteed an ultimatum:

You have submitted an imperfect catalog in which many things are missing, you have not given positions of stars which were desirable, and I heard that the printing has now stopped because of their non-provision. Thus, the following is expected of you: either you will send the end of your catalog to Dr. Arbetnott, or at least you will send him the observational data necessary to finish it, so that printing may proceed.

Newton also threatened that further delays would be considered insubordination to Her Majesty”s orders. In March 1710 Flemsteed, after heated complaints about the injustice and intrigues of his enemies, nevertheless handed over the final sheets of his catalog, and in early 1712 the first volume, entitled “Celestial History,” was published. It had all the data Newton needed, and a year later a revised edition of the Iniquities, with a much more accurate theory of the moon, was not slow to appear either. A vindictive Newton did not include his thanks to Flemstead and crossed out all the references to him that were present in the first edition. In response, Flemsteed burned all of the unsold 300 copies of the catalog in his fireplace and began to prepare a second edition, already to his own taste. He died in 1719, but through the efforts of his wife and friends this remarkable edition, the pride of English astronomy, was published in 1725.

Flemsteed”s successor at the Royal Observatory was Halley, who also immediately classified all the results of his observations to prevent his rivals from stealing the data. There was no conflict with Halley, but at meetings of the Society, Newton repeatedly reprimanded Halley for his unwillingness to share the data that Newton needed.

From extant documents, historians of science have found that Newton created differential and integral calculus as early as 1665-1666, but did not publish it until 1704. Leibniz developed his version of analysis independently (from 1675), although the initial impetus for his thought probably came from rumors that Newton already had such an calculus, as well as from scientific conversations in England and correspondence with Newton. Unlike Newton, Leibniz immediately published his version, and later, along with Jacob and Johann Bernoulli, widely promoted this epochal discovery throughout Europe. Most scientists on the continent had no doubt that Leibniz had discovered analysis.

Responding to the entreaties of his friends, who appealed to his patriotism, Newton reported in Book 2 of his Beginnings (1687):

In letters, which about ten years ago I exchanged with the very skilful mathematician Mr. Leibniz, I informed him that I possessed a method for determining maxima and minima, tangents and solutions of similar questions, equally applicable to both rational and irrational terms, and I concealed the method by rearranging the letters of the following sentence: “when an equation containing any number of current quantities is given, find the fluids and back. The most famous husband answered me that he also attacked such a method and informed me of his method, which turned out to be hardly different from mine, and that only in terms and in the outlining of formulas.

In 1693, when Newton finally published the first summary of his version of analysis, he exchanged friendly letters with Leibniz. Newton reported:

Our Wallis has attached to his “Algebra,” which has just appeared, some of the letters I wrote to you in my time. In doing so, he demanded of me that I set forth openly the method which I had at that time concealed from you by rearranging the letters; I did this as briefly as I could. I hope that I did not write anything that would be unpleasant to you, but if I did, I beg you to let me know, because friends are dearer to me than mathematical discoveries.

After the first detailed publication of Newton”s analysis (mathematical appendix to Optics, 1704), an anonymous review appeared in Leibniz”s journal Acta eruditorum with insulting allusions to Newton. The review made it clear that Leibniz was the author of the new calculus. Leibniz himself strongly denied that the review was written by him, but historians have managed to find a draft written in his handwriting. Newton ignored Leibniz”s article, but his students responded indignantly, after which an all-European priority war erupted, “the most shameful squabble in the entire history of mathematics.”

On January 31, 1713, the Royal Society received a letter from Leibniz containing conciliatory wording: he agreed that Newton had arrived at the analysis independently, “on general principles similar to our own. An angry Newton demanded an international commission to clarify the priority. It did not take long: a month and a half later, after studying Newton”s correspondence with Oldenburg and other documents, the commission unanimously recognized Newton”s priority, and in a formulation, this time offensive to Leibniz. The decision of the commission was printed in the Proceedings of the Society with all the supporting documents attached. Stephen Hawking and Leonard Mlodinow argue in A Brief History of Time that the commission consisted only of scientists loyal to Newton, and that most of the articles in Newton”s defense were written in his own hand and then published on behalf of friends.

In response, beginning in the summer of 1713, anonymous pamphlets flooded Europe, defending Leibniz”s priority and claiming that “Newton is appropriating for himself the honor that belongs to another. The pamphlets also accused Newton of stealing the results of Hooke and Flemsteed. Newton”s friends, for their part, accused Leibniz himself of plagiarism; according to their version, while in London (1676), Leibniz had read Newton”s unpublished papers and letters at the Royal Society, after which Leibniz published the ideas there and passed them off as his own.

The war continued unabated until December 1716, when Antonio Schinella Conti, Abbot of Conti, informed Newton, “Leibniz is dead – the dispute is over.

Newton”s works are associated with a new era in physics and mathematics. He completed the creation, begun by Galileo, of theoretical physics based, on the one hand, on experimental data and, on the other hand, on quantitative and mathematical descriptions of nature. Powerful analytical methods appeared in mathematics. In physics, the main method of investigation of nature becomes the construction of adequate mathematical models of natural processes and intensive investigation of these models with systematic involvement of all the power of the new mathematical apparatus. Subsequent centuries proved the exceptional fruitfulness of such an approach.

## Philosophy and the Scientific Method

Newton firmly rejected the popular approach of Descartes and his Cartesian followers at the end of the 17th century, which prescribed that in constructing a scientific theory one must first, by the “discernment of the mind”, find the “root causes” of the phenomenon under study. In practice, this approach often led to far-fetched hypotheses about “substances” and “hidden properties” that could not be verified by experience. Newton believed that in “natural philosophy” (i.e. physics) only such assumptions (“principles”, now prefer the name “laws of nature”), which directly follow from reliable experiments, generalize their results, are allowed; he called assumptions, insufficiently substantiated by experiments, hypotheses. “Everything … that is not deduced from phenomena must be called a hypothesis; hypotheses of metaphysical, physical, mechanical, hidden properties have no place in experimental philosophy.” Examples of principles are the law of gravitation and the 3 laws of mechanics in the “Elements”; the word “principles” (Principia Mathematica, traditionally translated as “mathematical principles”) is also contained in the title of his main book.

In a letter to Pardis, Newton formulated the “golden rule of science:

The best and safest method of philosophizing, it seems to me, should be first to diligently investigate the properties of things and to establish these properties by experiment, and then to progress gradually to hypotheses explaining these properties. Hypotheses can only be useful in explaining the properties of things, but there is no need to burden them with the responsibility of defining these properties beyond the limits revealed by experiment…for one can invent many hypotheses to explain any new difficulties.

Such an approach not only put speculative fantasies outside science (for example, the Cartesians” reasoning about the properties of “thin matter” that seemed to explain electromagnetic phenomena), but was more flexible and fruitful, because it allowed mathematical modeling of phenomena for which no root causes had yet been discovered. This happened with gravitation and the theory of light – their nature became clear much later, which did not interfere with the successful application of Newtonian models for centuries.

The famous phrase “Hypotheses non fingo” does not mean, of course, that Newton underestimated the importance of finding “root causes” if they are unequivocally confirmed by experience. The general principles derived from experiment and their corollaries must also undergo experimental verification, which may lead to a correction or even a change of principles. “The whole difficulty of physics … is to recognize the forces of nature from the phenomena of motion, and then from these forces to explain the other phenomena.”

Newton, like Galileo, believed that all natural processes were based on mechanical motion:

It would be desirable to deduce from the principles of mechanics the rest of the phenomena of nature … for many things lead me to assume that all these phenomena are caused by some forces, with which the particles of bodies, due to reasons still unknown, either tend to each other and join together into regular figures, or mutually repel and move away from each other. Since these forces are unknown, so far the attempts of philosophers to explain the phenomena of nature have remained fruitless.

Newton formulated his scientific method in his book “Optics:

As in mathematics, so in the testing of nature, in the investigation of difficult questions, the analytical method must precede the synthetic method. This analysis consists in drawing general conclusions from experiments and observations by induction, and in allowing no objections against them that would not come from experiments or other reliable truths. For hypotheses are not considered in experimental philosophy. Although the results obtained by induction from experiments and observations cannot yet serve as proof of general conclusions, this is still the best way to draw conclusions, which the nature of things allows.

Rule I. No other causes should be accepted in nature than those which are true and sufficient to explain phenomena… nature does nothing in vain, and it would be vain to accomplish by many what can be done by less. Nature is simple and does not luxuriate in superfluous causes of things…

Newton”s mechanistic views turned out to be wrong – not all natural phenomena are derived from mechanical motion. However, his scientific method has established itself in science. Modern physics successfully investigates and applies phenomena whose nature has not yet been clarified (e.g., elementary particles). Since Newton, natural science has evolved with the firm belief that the world is knowable because nature is organized according to simple mathematical principles. This certainty became the philosophical basis for the tremendous progress of science and technology.

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## Mathematics

Newton made his first mathematical discoveries as a student: the classification of algebraic curves of order 3 (curves of order 2 were studied by Fermat) and the binomial expansion of arbitrary (not necessarily integer) degree, which begins Newton”s theory of infinite series, a new and most powerful tool of analysis. Newton regarded series expansion as the basic and general method of analyzing functions, and he reached the pinnacle of excellence in it. He used series to calculate tables, solve equations (including differential equations), and study the behavior of functions. Newton was able to obtain decompositions for all the functions that were standard at the time.

Newton developed differential and integral calculus simultaneously with G. Leibniz (a little earlier) and independently of him. Before Newton, operations with infinitesimals were not integrated into a unified theory and were of the nature of scattered witticisms (see Method of Indivisibles). The creation of systematic mathematical analysis reduced the solution of the corresponding problems, to a large extent, to the technical level. A set of concepts, operations, and symbols appeared, which became the starting point for the further development of mathematics. The next, the 18th century, was a century of rapid and extremely successful development of analytical methods.

Probably, Newton came to the idea of analysis through difference methods, which he dealt with extensively and deeply. However, in his “Elements” Newton almost did not use infinitesimals, sticking to ancient (geometric) methods of proof, but in other works he used them freely.The starting point for differential and integral calculus were the works of Cavalieri and especially Fermat, who was already able (for algebraic curves) to draw tangents, find extremes, inflection points and curvature of the curve, calculate the area of its segment. Of other predecessors, Newton himself named Wallis, Barrow, and the Scottish scientist James Gregory. The concept of function did not yet exist; he treated all curves kinematically as trajectories of a moving point.

Already as a student, Newton realized that differentiation and integration are reciprocal operations. This basic theorem of analysis already loomed more or less clearly in the works of Torricelli, Gregory and Barrow, but only Newton realized that on this basis one could obtain not only individual discoveries, but a powerful systematic calculus, similar to algebra, with clear rules and gigantic possibilities.

Newton did not care about publishing his version of the analysis for almost 30 years, although in letters (particularly to Leibniz) he willingly shared much of what he had achieved. In the meantime, Leibniz”s version has been widely and openly circulating in Europe since 1676. It is not until 1693 that the first presentation of Newton”s version appears – as an appendix to Wallis”s Treatise on Algebra. We have to admit that Newton”s terminology and symbolism are rather clumsy in comparison with Leibniz”s: fluxia (derivative), fluenta (first form), moment of magnitude (differential), etc. Only the Newtonian notation “o” for infinitesimal dt has survived in mathematics (however, this letter was used earlier by Gregory in the same sense), and the point above the letter as a symbol of the time derivative.

Newton published a sufficiently complete statement of the principles of analysis only in “On the Quadrature of Curves” (1704), attached to his monograph “Optics. Almost all of the material set forth was ready as early as the 1670s-1680s, but only now did Gregory and Halley persuade Newton to publish the work, which, 40 years late, became Newton”s first printed work on analysis. Here Newton appeared derivatives of higher orders, found values of integrals of a variety of rational and irrational functions, and gave examples of solutions of differential equations of order 1.

In 1707 the book “Universal Arithmetic” was published. It contains a variety of numerical methods. Newton always paid great attention to approximate solutions of equations. Newton”s famous method made it possible to find the roots of equations with previously unimaginable speed and accuracy (published in Wallis” Algebra, 1685). Joseph Raphson (1690) gave Newton”s iterative method a modern form.

In 1711, “Analysis by means of equations with an infinite number of terms” was finally printed, 40 years later. In this work, Newton explores both algebraic and “mechanical” curves (cycloid, quadratrix) with equal ease. Partial derivatives appear. In the same year, “Method of Differences” is published, where Newton proposed an interpolation formula for carrying through (n + 1) given points with equally spaced or unequally spaced abscissas of the nth-order polynomial. This is the difference analog of the Taylor formula.

In 1736 he posthumously published his final work “Method of fluctuations and infinite series”, considerably advanced in comparison with “Analysis by means of equations”. It contains numerous examples of finding extrema, tangents and normals, calculating radii and centers of curvature in Cartesian and polar coordinates, finding inflection points, etc. In the same work, quadrature and straightening of various curves are produced.

Newton not only sufficiently developed analysis, but also made an attempt to rigorously substantiate its principles. While Leibniz was inclined to the idea of actual infinitesimals, Newton proposed (in “Elements”) a general theory of limiting transitions, which he somewhat floridly called “the method of first and last relations. It is the modern term “limit” (lat. limes) that is used, although there is no clear description of the essence of this term, implying an intuitive understanding. The theory of limits is set forth in 11 lemmas of Book I of the Beginnings; one lemma is also in Book II. The arithmetic of limits is absent, there is no proof of the uniqueness of the limit, and its relationship to infinitesimals is not revealed. However, Newton rightly points out the greater rigor of this approach compared to the “crude” method of indivisibles. Nevertheless, in Book II, by introducing “moments” (differentials), Newton again confuses the matter, in fact treating them as actual infinitesimals.

It is noteworthy that Newton was not at all interested in number theory. Apparently, physics was much closer to him than mathematics.

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## Mechanics

Newton is credited with solving two fundamental problems.

In addition, Newton definitively buried the notion, ingrained since antiquity, that the laws of motion of terrestrial and celestial bodies are completely different. In his model of the world, the entire universe is subject to a single law that allows mathematical formulation.

Newton”s axiomatics consisted of three laws, which he himself formulated as follows.

1. 1. Every body continues to be held in a state of rest or uniform and rectilinear motion as long as and as long as it is not compelled by applied forces to change this state. 2. The change in the quantity of motion is proportional to the applied force and occurs in the direction of the straight line along which the force acts. 3. There is always an equal and opposite counteraction to the action; otherwise, the interactions of two bodies with each other are equal and directed in opposite directions.

The first law (the law of inertia), in a less clear form, was published by Galileo. Galileo allowed free motion not only in a straight line, but also in a circle (apparently, for astronomical reasons). Galileo also formulated the most important principle of relativity, which Newton did not include in his axiomatics, because for mechanical processes, this principle is a direct consequence of the equations of dynamics (consequence V in the Elements). In addition, Newton considered space and time as absolute concepts, unified for the entire universe, and explicitly pointed this out in his “Elements”.

Newton also gave strict definitions of such physical concepts as quantity of motion (not clearly used by Descartes) and force. He introduced into physics the concept of mass as a measure of inertia and, at the same time, gravitational properties. Previously, physicists had used the concept of weight, but the weight of a body depends not only on the body itself, but also on its surroundings (e.g., the distance from the center of the Earth), so a new, invariant characteristic was needed.

Euler and Lagrange completed the mathematization of mechanics.

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## Universal gravitation and astronomy

Aristotle and his supporters considered gravity as the tendency of bodies of the “sublunar world” to their natural places. Some other ancient philosophers (among them Empedocles and Plato) believed that gravity was the tendency of related bodies to join together. In the 16th century this view was supported by Nicolaus Copernicus, in whose heliocentric system the Earth was considered only one of the planets. Similar views were held by Giordano Bruno and Galileo Galilei. Johannes Kepler believed that the reason bodies fall is not their internal drives, but the force of gravity on the part of the Earth, and not only the Earth attracts the stone, but the stone also attracts the Earth. In his opinion, the force of gravity extends at least as far as the Moon. In his later writings, he suggested that gravity decreases with distance and that all bodies in the solar system are subject to mutual attraction. The physical nature of gravity was attempted by René Descartes, Gilles Roberval, Christiaan Huygens and other seventeenth-century scientists.

Kepler was the first to suggest that the motion of the planets is controlled by forces coming from the Sun. In his theory, there were three such forces: one, circular, pushes the planet along the orbit, acting on a tangent to the trajectory (due to this force the planet moves), the other attracts and pushes the planet from the Sun (due to it the orbit of the planet is elliptical) and the third acts across the ecliptic plane (so the orbit of the planet lies in one plane). He considered the circular force decreasing in inverse proportion to the distance from the Sun. None of these three forces was identified with gravity. Kepler”s theory was rejected by the leading theoretical astronomer of the mid-17th century, Ismael Bulliald, who believed, first, that the planets move around the Sun not under the action of forces emanating from it, but by internal motion, and second, that if a circular force did exist, it would decrease inversely to the second degree of distance, not the first, as Kepler had believed. Descartes believed that the planets were carried around the Sun by giant vortices.

Jeremy Horrocks suggested the existence of a force from the Sun that governs the movement of the planets. According to Giovanni Alfonso Borelli, three forces come from the Sun: one promotes the planet”s orbit, another attracts the planet to the Sun, and the third (centrifugal), on the contrary, repels the planet. The elliptical orbit of a planet is the result of the opposition of the latter two. In 1666 Robert Hooke suggested that the force of attraction to the Sun alone is sufficient to explain the motion of the planets, we just need to assume that the planetary orbit is the result of a combination (superposition) of falling on the Sun (due to the force of attraction) and motion by inertia (tangential to the trajectory of the planet). In his opinion, this superposition of motions and causes the elliptical shape of the trajectory of the planet around the Sun. Similar views, but in a rather uncertain form, were expressed by Christopher Wren. Hooke and Wren guessed that the gravitational force decreases inversely proportional to the square of the distance from the Sun.

However, no one before Newton had been able to clearly and mathematically prove the connection between the law of gravitation (the force inversely proportional to the square of the distance) and the laws of planetary motion (Kepler”s laws). Moreover, it was Newton who first guessed that gravity acts between any two bodies in the universe; the motion of a falling apple and the rotation of the Moon around the Earth are governed by the same force. Finally, Newton not only published the supposed formula for the law of universal gravitation, but actually proposed a complete mathematical model:

Taken together, this triad is sufficient for a complete study of the most complex motions of celestial bodies, thereby creating the foundations of celestial mechanics. Thus, it is only with Newton”s writings that the science of dynamics begins, including its application to the motion of celestial bodies. Prior to the creation of the theory of relativity and quantum mechanics, no fundamental amendments to the above model were needed, although the mathematical apparatus proved necessary to develop considerably.

The first argument in favor of the Newtonian model was the strict derivation of Kepler”s empirical laws on its basis. The next step was the theory of the motion of comets and the moon, set forth in “The Beginnings”. Later, with the help of Newtonian gravitation, all the observed motions of celestial bodies were explained with high accuracy; great credit is due to Euler, Clero and Laplace, who developed for this purpose the theory of perturbations. The foundation of this theory was laid by Newton, who analyzed the motion of the Moon using his usual method of series expansion; in this way he discovered the reasons for the then known irregularities (inequalities) in the motion of the Moon.

The law of gravitation solved not only the problems of celestial mechanics, but also a number of physical and astrophysical problems. Newton pointed out a method for determining the masses of the Sun and planets. He discovered the cause of tides: the attraction of the moon (even Galileo considered tides as a centrifugal effect). Moreover, by processing years of data on the height of the tides, he calculated the mass of the Moon with good accuracy. Another consequence of gravity was the precession of the Earth”s axis. Newton found that because the Earth is flattened near the poles, the Earth”s axis under the influence of the attraction of the Moon and the Sun makes a constant slow shift with a period of 26000 years. Thus, the ancient problem of the “preceding equinoxes” (first noted by Hipparchus) found a scientific explanation.

Newton”s theory of gravitation caused many years of debate and criticism of the concept of long-range action adopted in it. However, the outstanding successes of celestial mechanics in the 18th century confirmed the view that the Newtonian model was adequate. The first observable deviations from Newton”s theory in astronomy (perihelion displacement of Mercury) were discovered only 200 years later. Soon these deviations were explained by the general theory of relativity (the Newtonian theory turned out to be its approximated version. GR also filled the theory of gravitation with physical content, indicating a material carrier of the gravitational force – the space-time metric, and allowed to get rid of long-range action.

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## Optics and Light Theory

Newton made fundamental discoveries in optics. He built the first mirror telescope (reflector) in which, unlike purely lens telescopes, there was no chromatic aberration. He also studied in detail the dispersion of light, showed that the passage of white light through a transparent prism, it decays into a continuous series of rays of different colors due to the different refraction of rays of different colors, thus Newton laid the foundation for the correct theory of color. Newton created the mathematical theory of the interference rings discovered by Hooke, which have since been called “Newton”s rings”. In a letter to Flemsteed, he laid out a detailed theory of astronomical refraction. But his main achievement is the creation of the foundations of physical (not only geometric) optics as a science and the development of its mathematical basis, transforming the theory of light from a haphazard set of facts into a science with rich qualitative and quantitative content, experimentally well substantiated. Newton”s optical experiments became for decades a model of profound physical investigation.

During this period there were many speculative theories of light and chromaticity; mainly the views of Aristotle (“different colors are a mixture of light and darkness in different proportions”) and Descartes (“different colors are created by the rotation of light particles with different speed”) struggled. Hooke, in his “Micrography” (1665) offered a variant of the Aristotelian view. Many believed that color is not an attribute of light, but of the illuminated object. General discord was exacerbated by a cascade of 17th century discoveries: diffraction (1665, Grimaldi), interference (1665, Hooke), double ray refraction (1670, Erasmus Bartolin, studied by Huygens), estimate of light speed (1675, Römer). There was no theory of light compatible with all these facts.

In his speech to the Royal Society, Newton disproved both Aristotle and Descartes and convincingly proved that white light is not primary, but consists of colored components with different “degrees of refractivity. These components are primary – no tricks Newton could change their color. Thus, the subjective sense of color received a solid objective basis – in modern terminology, the wavelength of light, which could be judged by the degree of refraction.

In 1689, Newton stopped publishing in the field of optics (although he continued his research) – according to popular legend, he vowed not to publish anything in this field during Hooke”s lifetime. In any case, in 1704, the year after Hooke”s death, his monograph Optics was published (in English). The preface to it contains a clear hint of conflict with Hooke: “Not wishing to be drawn into disputes on various questions, I delayed this publication and would have delayed it further if not for the insistence of my friends. During the author”s lifetime the Optics, like the Elements, survived three editions (1704, 1717, 1721) and many translations, including three in Latin.

Historians distinguish two groups of hypotheses about the nature of light at that time.

Newton is often considered a supporter of the corpuscular theory of light; in fact, as was his custom, he did not “hypothesize” and willingly admitted that light could also be related to waves in the ether. In a treatise submitted to the Royal Society in 1675, he writes that light cannot simply be vibrations of the ether, since then it could, for example, propagate along a curved tube, as sound does. But, on the other hand, he suggests that the propagation of light excites vibrations in the ether, which gives rise to diffraction and other wave effects. In essence, Newton, clearly aware of the advantages and disadvantages of both approaches, puts forward a compromise, corpuscular-wave theory of light. In his works, Newton described in detail a mathematical model of light phenomena, leaving aside the question of the physical medium of light: “My doctrine of light refraction and colors is only in the establishment of some properties of light without any hypotheses about its origin. Wave optics, when it appeared, did not reject Newton”s models, but absorbed them and expanded them on a new basis.

Despite his dislike of hypotheses, Newton placed at the end of “Optics” a list of unsolved problems and possible answers to them. However, in those years he could afford it – the authority of Newton after the “Elements” became unquestionable, and few people dared to pester him with objections. A number of his hypotheses turned out to be prophetic. In particular, Newton predicted:

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## Other works in physics

Newton was the first to deduce the speed of sound in gas, based on the Boyle-Mariotte law. He suggested the existence of the law of viscous friction and described the hydrodynamic compression of the jet. He proposed a formula for the law of resistance of the body in a dilute medium (Newton”s formula) and on its basis he considered one of the first problems concerning the most favorable form of the streamlined body (Newton”s aerodynamic problem). In the “Elements” he expressed and argued the correct assumption that the comet has a solid nucleus whose evaporation under the influence of solar heat forms a vast tail, always directed opposite to the side opposite to the Sun. Newton also dealt with heat transfer, one of the results being called Newton”s law of Richman.

Newton predicted the flatness of the Earth at the poles, estimating it to be about 1:230. At the same time, Newton used the model of a homogeneous fluid to describe the Earth, applied the law of universal gravitation and took into account the centrifugal force. At the same time, similar calculations were made by Huygens, who did not believe in the long-range force of gravity and approached the problem purely kinematically. Accordingly, Huygens predicted more than half as much compression as Newton, 1:576. Moreover, Cassini and other Cartesians proved that the Earth was not compressed, but stretched out at the poles like a lemon. Subsequently, though not immediately (the real compression is 1:298. The reason for the difference between this value and the Huygens value proposed by Newton is that the homogeneous fluid model is still not quite accurate (the density increases markedly with depth). A more accurate theory explicitly taking into account the dependence of density on depth was developed only in the XIX century.

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## Students

Strictly speaking, Newton had no direct disciples. However, a whole generation of English scientists grew up on his books and in communication with him, so they considered themselves Newton”s disciples. Among them the most famous are:

## Chemistry and Alchemy

Along with the research that laid the foundation for today”s scientific (physical and mathematical) tradition, Newton spent a great deal of time in alchemy as well as theology. Books on alchemy constituted one-tenth of his library. He published no works on chemistry or alchemy, and the only known result of this long-standing fascination was Newton”s serious poisoning in 1691. When Newton”s body was exhumed, a dangerous amount of mercury was found in his body.

Stukeley recalls that Newton wrote a treatise on chemistry “explaining the principles of this mysterious art on the basis of experimental and mathematical evidence,” but the manuscript unfortunately burned in a fire, and Newton made no attempt to recover it. The surviving letters and notes suggest that Newton was contemplating the possibility of some unification of the laws of physics and chemistry into a unified system of the world; he placed several hypotheses on this subject at the end of Optics.

Б. Kuznetsov believes that Newton”s alchemical studies were attempts to uncover the atomistic structure of matter and other forms of matter (e.g., light, heat, magnetism). Newton”s interest in alchemy was disinterested and rather theoretical:

His atomistics is based on the idea of a hierarchy of corpuscles formed by less and less intense forces of mutual attraction of parts. This idea of an infinite hierarchy of discrete particles of matter is related to the idea of the unity of matter. Newton did not believe in the existence of elements incapable of transforming into each other. On the contrary, he assumed that the idea of the indecomposability of particles and, consequently, of qualitative differences between the elements is related to the historically limited possibilities of experimental technology.

This assumption is confirmed by Newton”s own statement: “Alchemy does not deal with metals, as ignorant people think. This philosophy is not the kind that serves vanity and deceit, but rather serves to benefit and edify, and the main thing here is the knowledge of God.

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## Theology

As a deeply religious man, Newton viewed the Bible (as well as everything else) from a rationalistic perspective. Newton”s rejection of the Trinity of God seems to be related to this approach. Most historians believe that Newton, who worked for many years at the College of the Holy Trinity, did not believe in the Trinity himself. Researchers of his theological writings have found that Newton”s religious views were close to heretical Arianism (see Newton”s article “A Historical Tracing of Two Notable Distortions of Holy Scripture”).

The degree of closeness of Newton”s views to the various heresies condemned by the church has been variously assessed. The German historian Fiesenmayer suggested that Newton accepted the Trinity, but closer to the Eastern, Orthodox understanding of it. The American historian Stephen Snobelin, citing a number of documentary evidence, firmly rejected this view and classified Newton as a Socinian.

Outwardly, however, Newton remained loyal to the state Anglican Church. There was a good reason for this: the statute of 1697 on the suppression of blasphemy and impiety for denying any of the persons of the Trinity provided for the loss of civil rights and, if the crime was repeated, for imprisonment. For example, Newton”s friend William Whiston was stripped of his professorial rank and expelled from Cambridge University in 1710 for his claims that the creed of the early Church was Arianism. However, in letters to like-minded people (Locke, Halley, etc.) Newton was quite frank.

In addition to anti-Trinitarianism, Newton”s religious worldview contains elements of deism. Newton believed in the material presence of God at every point in the universe and called space “the sensorium of God” (Latin sensorium Dei). This pantheistic idea unites into a single whole the scientific, philosophical and theological views of Newton, “all areas of Newton”s interests, from natural philosophy to alchemy, are different projections and at the same time different contexts of this central idea which undividedly possessed him.

Newton published (in part) the results of his theological studies late in life, but they began much earlier, no later than 1673. Newton proposed his own version of biblical chronology, left works on biblical hermeneutics, and wrote a commentary on the Apocalypse. He studied the Hebrew language, researched the Bible using scientific methodology, bringing in astronomical calculations related to solar eclipses, linguistic analysis, etc. to substantiate his viewpoint. According to his calculations, the end of the world will not come before 2060.

Newton”s theological manuscripts are now preserved in Jerusalem at the National Library.

The inscription on Newton”s grave reads:

Here rests Sir Isaac Newton, who, with an almost divine power of reason, was the first to explain with his mathematical method the movements and shapes of the planets, the paths of comets, and the tides of the oceans.

A statue erected to Newton in 1755 at Trinity College bears verses from Lucretius:

Newton himself estimated his achievements more modestly:

I don”t know how the world perceives me, but I think of myself as just a boy playing on the seashore, having fun from time to time looking for a more colorful pebble or a beautiful seashell, while the great ocean of truth stretches out before me unexplored.

Lagrange said, “Newton was the happiest of mortals, for there is only one universe, and Newton discovered its laws.

The Old Russian pronunciation of Newton”s surname is “Nevton. He, together with Plato, is respectfully mentioned by M. V. Lomonosov in his poems:

According to A. Einstein, “Newton was the first who attempted to formulate elementary laws that determine the temporal course of a wide class of processes in nature with a high degree of completeness and accuracy” and “… had a profound and powerful influence on the worldview as a whole through his writings.

At the turn of 1942-1943, in the most dramatic days of the Battle of Stalingrad, the USSR widely celebrated Newton”s 300th birthday. A collection of articles and a biographical book by S. I. Vavilov were published. In gratitude to the Soviet people, the Royal Society of Great Britain presented to the Academy of Sciences of the USSR a rare copy of the first edition of Newton”s “Mathematical Principles” (1687) and a draft (one of three) of Newton”s letter to Alexander Menshikov, which informed the latter of his election to the Royal Society of London:

The Royal Society has long known that your Emperor has advanced the arts and sciences in his Empire. And now we have learned with great joy from the English merchants that Your Excellency, showing the greatest courtesy, outstanding respect for the sciences and love of our country, intends to become a member of our Society.

Newton is named after him:

Several common legends have already been cited above: “Newton”s apple,” his only parliamentary appearance.

There is a legend that Newton made two holes in his door, one larger and one smaller, so that his two cats, a big one and a small one, could enter the house on their own. In reality, Newton never kept cats or other pets.

Another myth accuses Newton of destroying the only portrait of Hooke once kept at the Royal Society. In fact, there is not a single piece of evidence to support such an accusation. Allan Chapman, Hooke”s biographer, proves that no portrait of Hooke existed at all (not surprising, given the high cost of portraits and Hooke”s constant financial difficulties). The only source for the suggestion that such a portrait existed is a reference to a portrait by a German scholar, Zacharias von Uffenbach, who visited the Royal Society in 1710, but Uffenbach did not speak English and probably had in mind a portrait of another member of the Society, Theodore Haak. The portrait of Haak did exist and has survived to this day. An additional argument in favor of the view that there never was a portrait of Hooke is the fact that Hooke”s friend and secretary Richard Waller published in 1705 a posthumous collection of Hooke”s works with excellent quality illustrations and a detailed biography, but no portrait of Hooke; all of Hooke”s other works do not contain a portrait of the scholar either.

Newton is sometimes credited with an interest in astrology. If he did, it was quickly replaced by disillusionment.

From the fact of Newton”s unexpected appointment as manager of the Mint, some biographers conclude that Newton was a member of a Masonic lodge or other secret society. However, no documentary evidence in favor of this hypothesis has been found.

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## Canonical Editions

A classic complete edition of Newton”s writings in 5 volumes in the original language:

Selected correspondence in seven volumes:

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## Translations into Russian

**Sources**