Mathematics Newton's work has been said "to distinctly advance every branch of mathematics then studied". His work on
calculus, usually referred to as fluxions, began in 1664, and by 20 May 1665 as seen in a manuscript, Newton "had already developed the calculus to the point where he could compute the tangent and the curvature at any point of a continuous curve". His work by 1665 amounted to a systematic calculus that unified differentiation and integration, which he applied to the dynamic analysis of algebraic and transcendental curves, an approach described by scholar
Tom Whiteside as "radically novel, indeed unprecedented" and which later directly informed the theory of central-force orbits in the
Principia. Another manuscript of October 1666, is now published among Newton's mathematical papers. Newton recorded a definitive tract of calculus in what is called his "Waste Book". Despite this, the notation of Leibniz is recognised as the more convenient notation, being adopted by continental European mathematicians, and after 1820, by British mathematicians. The historian of science
A. Rupert Hall notes that while Leibniz deserves credit for his independent formulation of calculus, Newton was undoubtedly the first to develop it, stating: Hall further notes that in
Principia, Newton was able to "formulate and resolve problems by the integration of differential equations" and "in fact, he anticipated in his book many results that later exponents of the calculus regarded as their own novel achievements." Hall notes Newton's rapid development of calculus in comparison to his contemporaries, stating that Newton "well before 1690 . . . had reached roughly the point in the development of the calculus that Leibniz, the two Bernoullis, L'Hospital, Hermann and others had by joint efforts reached in print by the early 1700s". Despite the convenience of Leibniz's notation, it has been noted that Newton's notation could also have developed multivariate techniques, with his dot notation still widely used in
physics. Some academics have noted the richness and depth of Newton's work, such as the physicist
Roger Penrose, stating "in most cases Newton's geometrical methods are not only more concise and elegant, they reveal deeper principles than would become evident by the use of those formal methods of calculus that nowadays would seem more direct." The mathematician
Vladimir Arnold stated that "Comparing the texts of Newton with the comments of his successors, it is striking how Newton's original presentation is more modern, more understandable and richer in ideas than the translation due to commentators of his geometrical ideas into the formal language of the calculus of Leibniz." His work extensively uses calculus in geometric form based on limiting values of the ratios of vanishingly small quantities: in the
Principia itself, Newton gave demonstration of this under the name of "the method of first and last ratios" and explained why he put his expositions in this form, remarking also that "hereby the same thing is performed as by the method of indivisibles." Because of this, the
Principia has been called "a book dense with the theory and application of the infinitesimal calculus" in modern times and in Newton's time "nearly all of it is of this calculus." His use of methods involving "one or more orders of the infinitesimally small" is present in his
De motu corporum in gyrum of 1684 and in his papers on motion "during the two decades preceding 1684". It has been argued that Newton had an imprecise or limited understanding of
limits. However, the mathematician Bruce Pourciau contends that in his
Principia, Newton actually demonstrated a more sophisticated understanding of limits than he is generally credited with, including being the first to present an epsilon argument. Newton had been reluctant to publish his calculus because he feared controversy and criticism. He was close to the Swiss mathematician
Nicolas Fatio de Duillier. In 1691, Duillier started to write a new version of Newton's
Principia, and corresponded with Leibniz. In 1693, the relationship between Duillier and Newton deteriorated and the book was never completed. Starting in 1699, Duillier accused Leibniz of plagiarism. The mathematician
John Keill accused Leibniz of plagiarism in 1708 in the
Royal Society journal, thereby deteriorating the situation even more. The dispute then broke out in full force in 1711 when the Royal Society proclaimed in a study that it was Newton who was the true discoverer and labelled Leibniz a fraud; it was later found that Newton wrote the study's concluding remarks on Leibniz. Thus began the bitter controversy which marred the lives of both men until Leibniz's death in 1716. Newton's first major mathematical discovery was the
generalised binomial theorem, valid for any exponent, in 1664–65, which has been called "one of the most powerful and significant in the whole of mathematics." He discovered
Newton's identities (probably without knowing of earlier work by
Albert Girard in 1629),
Newton's method, the
Newton polygon, and classified
cubic plane curves (
polynomials of degree three in two
variables). Newton is also a founder of the theory of
Cremona transformations, and he made substantial contributions to the theory of
finite differences, with Newton regarded as "the single most significant contributor to finite difference
interpolation", with many formulas created by Newton. He was the first to state
Bézout's theorem, and was also the first to use fractional indices and to employ
coordinate geometry to derive solutions to
Diophantine equations. He approximated
partial sums of the
harmonic series by
logarithms (a precursor to
Euler's summation formula) and was the first to use
power series with confidence and to revert power series. He introduced the
Puisseux series. He also provided the earliest explicit formulation of the general
Taylor series, which appeared in a 1691-1692 draft of his
De Quadratura Curvarum. He originated the
Newton-Cotes formulas for
numerical integration. Newton's work on infinite series was inspired by
Simon Stevin's decimals. He also initiated the field of
calculus of variations, being the first to formulate and solve a problem in the field, that being
Newton's minimal resistance problem, which he posed and solved in 1685, later publishing it in
Principia in 1687. It is regarded as one of the most difficult problems tackled by variational methods prior to the twentieth century. He then used calculus of variations in his solving of the
brachistochrone curve problem in 1697, which was posed by
Johann Bernoulli in 1696, and which he famously solved in a night, thus pioneering the field with his work on the two problems. He was also a pioneer of
vector analysis, as he demonstrated how to apply the
parallelogram law for adding various physical quantities and realised that these quantities could be broken down into components in any direction. He is credited with introducing the notion of the
vector in his
Principia, by proposing that physical quantities like velocity, acceleration, momentum, and force be treated as directed quantities, thereby making Newton the "true originator of this mathematical object". Newton was probably first to develop a system of
polar coordinates in a strictly analytic sense, with his work in relation to the topic being superior, in both generality and flexibility, to any other during his lifetime. His 1671
Method of Fluxions work preceded the earliest publication on the subject by
Jacob Bernoulli in 1691. He is also credited as the originator of
bipolar coordinates in a strict sense. A private manuscript of Newton's which dates to 1664–66 contains what is the earliest known problem in the field of
geometric probability. The problem dealt with the likelihood of a negligible ball landing in one of two unequal sectors of a circle. In analysing this problem, he proposed substituting the enumeration of occurrences with their quantitative assessment, and replacing the estimation of an area's proportion with a tally of points, which has led to him being credited as founding
stereology. Newton was responsible for the modern origin of
Gaussian elimination in Europe. In 1669 to 1670, Newton wrote that all the algebra books known to him lacked a lesson for solving simultaneous equations, which he then supplied. His notes lay unpublished for decades, but once released, his textbook became the most influential of its kind, establishing the method of substitution and the key terminology of 'extermination' (now known as elimination). In the 1660s and 1670s, Newton found 72 of the 78 "species" of cubic curves and categorised them into four types, systemising his results in later publications. However, a 1690s manuscript later analysed showed that Newton had identified all 78 cubic curves, but chose not to publish the remaining six for unknown reasons. Newton briefly dabbled in
probability. In letters with
Samuel Pepys in 1693, they corresponded over the
Newton–Pepys problem, which was a problem about the probability of throwing sixes from a certain number of dice. For it, outcome A was that six dice are tossed with at least one six appearing, outcome B that twelve dice are tossed with at least two sixes appearing, and outcome C in which eighteen dice are tossed with at least three sixes appearing. Newton solved it correctly, choosing outcome A, Pepys incorrectly chose the wrong outcome of C. However, Newton's intuitive explanation for the problem was flawed.
Optics in 1672 (the first one he made in 1668 was loaned to an instrument maker but there is no further record of what happened to it). In 1666, Newton observed that the spectrum of colours exiting a
prism in the position of
minimum deviation is oblong, even when the light ray entering the prism is circular, which is to say, the prism refracts different colours by different angles. This led him to conclude that colour is a property intrinsic to light – a point which had, until then, been a matter of debate. From 1670 to 1672, Newton lectured on optics. During this period he investigated the
refraction of light, demonstrating that the multicoloured image produced by a prism, which he named a
spectrum, could be recomposed into white light by a
lens and a second prism. Modern scholarship has revealed that Newton's analysis and resynthesis of white light owes a debt to
corpuscular alchemy. In his work on
Newton's rings in 1671, he used a method that was unprecedented in the 17th century, as "he
averaged all of the differences, and he then calculated the difference between the average and the value for the first ring", in effect introducing a now
standard method for reducing noise in measurements, and which does not appear elsewhere at the time. He extended his "error-slaying method" to studies of equinoxes in 1700, which was described as an "altogether unprecedented method" but differed in that here "Newton required good values for each of the original equinoctial times, and so he devised a method that allowed them to, as it were, self-correct." Newton "invented a certain technique known today as
linear regression analysis", as he wrote the first of the two 'normal equations' known from
ordinary least squares, averaged a set of data, 50 years before
Tobias Mayer, the person originally thought to be the oldest to do so, and he also summed the residuals to zero, forcing the regression line through the average point. He differentiated between two uneven sets of data and may have considered an optimal solution regarding bias, although not in terms of effectiveness. He showed that coloured light does not change its properties by separating out a coloured beam and shining it on various objects, and that regardless of whether reflected, scattered, or transmitted, the light remains the same colour. Thus, he observed that colour is the result of objects interacting with already-coloured light rather than objects generating the colour themselves. This is known as
Newton's theory of colour. His 1672 paper on the nature of white light and colours forms the basis for all work that followed on colour and colour vision. separating white light into the colours of the spectrum, as discovered by Newton From this work, he concluded that the lens of any
refracting telescope would suffer from the
dispersion of light into colours (
chromatic aberration). As a proof of the concept, he constructed a telescope using reflective mirrors instead of lenses as the
objective to bypass that problem. Building the design, the first known functional reflecting telescope, today known as a
Newtonian telescope, involved solving the problem of a suitable mirror material and shaping technique. Newton grounded his own mirrors out of a custom composition of highly reflective
speculum metal, using Newton's rings to judge the
quality of the optics for his telescopes. In late 1668, he was able to produce this first reflecting telescope. It was about eight inches long and it gave a clearer and larger image. Newton reported that he could see the four
Galilean moons of
Jupiter and the
crescent phase of
Venus with his new reflecting telescope. which he later expanded into the work
Opticks. When
Robert Hooke criticised some of Newton's ideas, Newton was so offended that he withdrew from public debate. However, the two had brief exchanges in 1679–80, when Hooke, who had been appointed Secretary of the Royal Society, opened a correspondence intended to elicit contributions from Newton to Royal Society transactions, In astronomy, Newton is further credited with the realisation that
high-altitude sites are superior for observation because they provide the "most serene and quiet Air" above the dense, turbulent atmosphere ("grosser Clouds"), thereby reducing star
twinkling. , commenting on Briggs'
A New Theory of Vision Newton argued that light is composed of particles or corpuscles, which were refracted by accelerating into a denser medium. He verged on soundlike waves to explain the repeated pattern of reflection and transmission by thin films (
Opticks Bk. II, Props. 12), but still retained his theory of 'fits' that disposed corpuscles to be reflected or transmitted (Props.13). Despite his known preference of a particle theory, Newton noted that light had both particle-like and wave-like properties in
Opticks; he believed that corpuscles must interact with waves in a medium to explain
interference patterns and the general phenomenon of
diffraction. In his
Hypothesis of Light of 1675, Newton posited the existence of the
ether to transmit forces between particles. The contact with the
Cambridge Platonist philosopher
Henry More revived his interest in alchemy. Newton contributed to the study of
astigmatism by helping to erect its mathematical foundation through his discovery that when oblique pencils of light undergo refraction, two distinct image points are created. This would later stimulate the work of
Thomas Young. In 1704, Newton published
Opticks, in which he expounded his corpuscular theory of light, and included a set of queries at the end, which were posed as unanswered questions and positive assertions. In line with his corpuscle theory, he thought that normal matter was made of grosser corpuscles and speculated that through a kind of alchemical transmutation, with query 30 stating "Are not gross Bodies and Light convertible into one another, and may not Bodies receive much of their Activity from the Particles of Light which enter their Composition?"
Opticks has been referred to as one of the "earliest exemplars of experimental procedure". His design was probably built as early as 1677. It is notable for being the first quadrant to use two mirrors, which greatly improved the accuracy of measurements since it provided a stable view of both the horizon and the celestial body at the same time. His quadrant was built but appears to have not survived to the present.
John Hadley would later construct his own double-reflecting quadrant that was nearly identical to the one invented by Newton. However, Hadley likely did not know of Newton's original invention, causing confusion regarding originality. In 1704, Newton constructed and presented a
burning mirror to the Royal Society. It consisted of seven concave glass mirrors, each about one foot in diameter. It is estimated that it reached a maximum possible radiant energy of 460 W cm⁻², which has been described as "certainly brighter thermally than a thousand Suns (1,000 × 0.065 W cm⁻²)" based on estimating that the intensity of the
Sun's radiation in London in May of 1704 was 0.065 W cm⁻². As a result of the maximum radiant intensity possibly achieved with his mirror he "may have produced the greatest intensity of radiation brought about by human agency before the arrival of
nuclear weapons in 1945."
David Gregory reported that it caused metals to smoke, boiled
gold and brought about the
vitrification of
slate.
William Derham thought it be to the most powerful burning mirror in Europe at the time. Newton also made early studies into electricity, as he constructed a primitive form of a frictional
electrostatic generator using a
glass globe, the first to do so with glass instead of
sulfur, which had previously been used by scientists such as
Otto von Guericke to construct their globes. He detailed an experiment in 1675 that showed when one side of a glass sheet is rubbed to create an electric charge, it attracts "light bodies" to the opposite side. He interpreted this as evidence that electric forces could pass through glass. Newton also reported to the Royal Society that glass was effective for generating static electricity, classifying it as a "good electric" decades before this property was widely known. His idea in
Opticks that optical
reflection and
refraction arise from interactions across the entire surface is seen as a precursor to the field theory of the electric force. His theory of nervous transmission had an immense influence on the work of
Luigi Galvani, as Newton's theory focused on electricity as a possible mediator of nervous transmission, which went against the prevailing Cartesian hydraulic theory of the time. He was also the first to present a clear and balanced theory for how both electrical and chemical mechanisms could work together in the nervous system. Newton's mass-dispersion model, ancestral to the successful use of the
least action principle, provided a credible framework for understanding refraction, particularly in its approach to refraction in terms of momentum. In
Opticks, Newton introduced prisms as
beam expanders and multiple-prism arrays, prismatic configurations that nearly 278 years later were incorporated into
tunable lasers, where
multiple-prism beam expanders became central to the development of
narrow-linewidth systems. The use of these prismatic beam expanders led to the
multiple-prism dispersion theory. Newton was the first to theorise the
Goos–Hänchen effect, an
optical phenomenon in which
linearly polarised light undergoes a small lateral shift when
totally internally reflected. He provided both experimental and theoretical explanations for it using a mechanical model. Science came to realise the difference between perception of colour and mathematisable optics. The German poet and scientist
Johann Wolfgang von Goethe could not shake the Newtonian foundation but "one hole Goethe did find in Newton's armour, ... Newton had committed himself to the doctrine that refraction without colour was impossible. He, therefore, thought that the object-glasses of telescopes must forever remain imperfect, achromatism and refraction being incompatible. This inference was proved by
Dollond to be wrong."
Philosophiæ Naturalis Principia Mathematica '' with Newton's hand-written corrections for the second edition, now housed in the
Wren Library at
Trinity College, Cambridge Newton had been developing his theory of gravitation as far back as 1665. In 1679, he returned to his work on
celestial mechanics by considering gravitation and its effect on the orbits of planets with reference to
Kepler's laws of planetary motion. Newton's reawakening interest in astronomical matters received further stimulus by the appearance of a comet in the winter of 1680–1681, on which he corresponded with
John Flamsteed. After his exchanges with
Robert Hooke, Newton worked out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector. He shared his results with
Edmond Halley and the Royal Society in , a tract written on about nine sheets which was copied into the Royal Society's Register Book in December 1684. As part of this work, Newton also coined the term
centripetal force. This tract contained the nucleus that Newton would develop and expand to form the
Principia. The was published on 5 July 1687 with encouragement and financial help from Halley. In this work, Newton stated the
three universal laws of motion. Together, these laws describe the relationship between any object, the forces acting upon it and the resulting motion, laying the foundation for
classical mechanics. His work achieved the
first great unification in physics. In the same work, Newton presented a calculus-like method of geometrical analysis using 'first and last ratios', gave the first analytical determination (based on
Boyle's law) of the speed of sound in air, inferred the oblateness of Earth's spheroidal figure, accounted for the precession of the equinoxes as a result of the Moon's gravitational attraction on the Earth's oblateness, initiated the gravitational study of the
irregularities in the motion of the Moon, provided a theory for the determination of the orbits of comets, and much more. He provided the first calculation of the
age of Earth by experiment, and also described a precursor to the modern
wind tunnel. Newton identified two "principal cases of attraction"—the
inverse-square law and a central force proportional to distance—showing that both yield stable conic-section orbits and that spherically symmetric bodies behave as if their mass were concentrated at a point; in modern terms, this linear force law is mathematically equivalent to the force associated with the
cosmological constant. Through Book II of the
Principia, Newton was an important pioneer of
fluid mechanics, and later analysis has shown that of its 53 propositions almost all are correct, with only two or three open to question. Propositions 1–18 of the book are the first comprehensive treatment of motion under resistance proportional to velocity or its square, leading the scholar
Richard S. Westfall to remark that 'almost without precedent, Newton created the scientific treatment of motion under conditions of resistance, that is, of motion as it is found in the world'. In Section IX of Book II, he formulated the
linear relation between viscous resistance and velocity gradient that now defines a
Newtonian fluid, despite his experiments giving little direct insight into viscosity. Newton also discussed the circular motion of fluids and was the first to analyse
Couette flow, initially in Proposition 51 for a single rotating cylinder and extended in Corollary 2 to the flow between two concentric cylinders. Further, he was the first to analyse the resistance of axisymmetric bodies moving through a
rarefied medium. He further determined the masses and densities of
Jupiter and
Saturn, putting all four celestial bodies (Sun, Earth, Jupiter, and Saturn) on the same comparative scale. This achievement by Newton has been called "a supreme expression of the doctrine that one set of physical concepts and principles applies to all bodies on earth, the earth itself, and bodies anywhere throughout the universe". For Newton, it was not precisely the centre of the Sun or any other body that could be considered at rest, but rather "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World", and this centre of gravity "either is at rest or moves uniformly forward in a right line". (Newton adopted the "at rest" alternative in view of common consent that the centre, wherever it was, was at rest.) Newton was criticised for introducing "
occult agencies" into science because of his postulate of an invisible
force able to act over vast distances. Later, in the second edition of the
Principia (1713), Newton firmly rejected such criticisms in a concluding
General Scholium, writing that it was enough that the phenomenon implied a gravitational attraction, as they did; but they did not so far indicate its cause, and it was both unnecessary and improper to frame hypotheses of things that were not implied by the phenomenon. (Here he used what became his famous expression .) With the , Newton became internationally recognised. He acquired a circle of admirers, including the Swiss-born mathematician
Nicolas Fatio de Duillier.
Other significant work Newton studied heat and energy flow, formulating an
empirical law of cooling which states that the rate at which an object cools is proportional to the temperature difference between the object and its surrounding environment. It was first formulated in 1701, being the first heat transfer formulation and serves as the formal basis of
convective heat transfer, later being incorporated by
Joseph Fourier into his work.
Philosophy of science Newton's role as a philosopher was deeply influential, and understanding the philosophical landscape of the late seventeenth and early eighteenth centuries requires recognising his central contributions. Historically, Newton was widely regarded as a core figure in modern philosophy. For example,
Johann Jakob Brucker's
Historia Critica Philosophiae (1744), considered the first comprehensive modern history of philosophy, prominently positioned Newton as a central philosophical figure. This portrayal notably shaped the perception of modern philosophy among leading Enlightenment intellectuals, including figures such as
Denis Diderot,
Jean le Rond d'Alembert, and
Immanuel Kant. Starting with the second edition of his
Principia, Newton included a final section on science philosophy or method. It was here that he wrote his famous line, in Latin, "hypotheses non fingo", which can be translated as "I don't make hypotheses," (the direct translation of "fingo" is "frame", but in context he was advocating against the use of hypotheses in science). Newton's rejection of hypotheses ("hypotheses non fingo") emphasised that he refused to speculate on causes not directly supported by phenomena. Harper explains that Newton's experimental philosophy involves clearly distinguishing hypotheses—unverified conjectures—from propositions established through phenomena and generalised by induction. According to Newton, true scientific inquiry requires grounding explanations strictly on observable data rather than speculative reasoning. Thus, for Newton, proposing hypotheses without empirical backing undermines the integrity of experimental philosophy, as hypotheses should serve merely as tentative suggestions subordinate to observational evidence. Newton contributed to and refined the
scientific method. In his work on the properties of light in the 1670s, he showed his rigorous method, which was conducting experiments, taking detailed notes, making measurements, conducting more experiments that grew out of the initial ones, he formulated a theory, created more experiments to test it, and finally described the entire process so other scientists could replicate every step. In his 1687
Principia, he outlined four rules, which together form the basis of modern science: • "Admit no more causes of natural things than are both true and sufficient to explain their appearances" • "To the same natural effect, assign the same causes" • "Qualities of bodies, which are found to belong to all bodies within experiments, are to be esteemed universal" • "Propositions collected from observation of phenomena should be viewed as accurate or very nearly true until contradicted by other phenomena" Newton's scientific method went beyond simple prediction in three critical ways, thereby enriching the basic
hypothetico-deductive model. First, it established a richer ideal of empirical success, requiring phenomena to accurately measure theoretical parameters. Second, it transformed theoretical questions into ones empirically solvable by measurement. Third, it used provisionally accepted propositions to guide research, enabling the method of successive approximations where deviations drive the creation of more accurate models. This robust method of theory-mediated measurements was adopted by his successors for extensions of his theory to
astronomy and remains a foundational element in modern physics. == Later life ==