I will research the reasons why Isaac Newton did not use algebraic notation in his Principia Mathematica, incorporating both the historical and mathematical context. The analysis will be in-depth and will rely exclusively on primary sources. I will let you know when the report is ready for review.Introduction
Isaac Newton’s Philosophiæ Naturalis Principia Mathematica (1687) is famously written in the style of classical geometry rather than the emerging algebraic calculus notation. Modern readers are often surprised to find no formulas with $x$ or $y$, no $\dfrac{d}{dt}$ differentials, and none of the “fluxional” notation that Newton himself invented. Instead, Newton presents his revolutionary laws of motion and gravitation through Euclidean geometric propositions, ratios of line segments, and classical lemmas. To understand why Newton eschewed algebraic notation in the Principia, one must consider the late 17th-century context: the influence of Greek geometry, Newton’s own methodological philosophy, the status of algebra and calculus at the time, differences between Newton’s private work and public presentation, and comparisons with contemporaries like Gottfried W. Leibniz (who embraced algebraic methods). This report examines these factors using Newton’s own writings and correspondence and contemporary documents. The evidence reveals a deliberate choice by Newton to align with ancient geometric rigor, to ensure clarity and acceptance of his physical arguments, and to avoid introducing a novel calculus in a foundational work of natural philosophy.1. Classical Geometric Tradition and Newton’s Ideals
Newton was deeply influenced by classical Greek mathematics and the ideal of rigorous geometric proof. In the Principia’s preface “To the Reader,” he explicitly invokes the ancient approach. Citing the geometer Pappus, Newton notes that the ancients greatly esteemed the “science of mechanics in the investigation of natural things” and distinguished between rational (demonstrative) and practical mechanicsen.wikisource.org. He then pointedly says “I have in this treatise cultivated mathematics so far as it regards philosophy”en.wikisource.org– meaning he will apply mathematical reasoning (primarily geometry) to natural philosophy. Newton goes on to praise geometry’s deductive power, remarking that “it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things”en.wikisource.org. This reverence shows Newton’s mindset: geometry, the method of Euclid and Archimedes, was the gold standard for certainty in mathematics. A classical geometric proof, built from first principles, was seen as more rigorous and comprehensible than the newer algebraic methods. Newton, who had been schooled in Euclid’s Elements and admired the works of ancient geometers, strove to put his Principia on the same footing – “laying its foundation in the most solid proofs of geometry,” as Roger Cotes later described Newton’s approachmathshistory.st-andrews.ac.uk.
This classical orientation meant that Newton composed the Principia in the form of propositions, theorems, and corollaries, much like an ancient text. He avoided the “analysis” (analytic/algebraic derivation) in the published proofs, instead giving synthetic demonstrations. It was common in the 17th century to distinguish the method of analysis (finding a solution through exploration, often algebraically) from synthesis (presenting the solution with a formal proof). Newton adhered to this ancient paradigm. As Cotes explained in 1713, Newton’s method was twofold: “from some select phenomena they deduce by analysis the forces of nature…and from thence by synthesis show the constitution of the rest. This is that incomparably best way of philosophizing, which our renowned author most justly embraced”mathshistory.st-andrews.ac.uk. In other words, Newton may have discovered truths by analytic calculation, but he chose to demonstrate them synthetically. By casting the Principia in the form of classical geometry, Newton was consciously placing his work in the respected tradition of Plato’s geometry “friend” and avoiding the appearance of mere algebraic trickery. Geometry was not just a tool for Newton – it was a statement of philosophical rigor. As the 1846 English edition of the Principia commented (reflecting Newton’s own view), “Newton, himself the inventor of fluxions, censured the handling of geometrical subjects by algebraical calculations; and the maturest opinions which he expressed were in favour of the geometrical method”en.wikisource.org. While this comment comes long after Newton, it encapsulates the historical reality that Newton esteemed ancient geometry highly and regarded the algebraic approach as less elegant for presenting truths of nature.
2. Newton’s Methodological Choices and Philosophy
Newton’s decision to avoid algebraic calculus in the Principia was not due to any lack of ability – as we shall see, he was a master of analytic methods – but stemmed from his methodological convictions. Newton believed that natural philosophy (science) should be demonstrated with certainty from first principles and phenomena, an ideal he associated with geometry. He was wary of speculative or unproven methods. In the Principia (Book 1, Section 1), Newton introduces a set of foundational Lemmae on the limits of ratios – essentially building a rigorous basis for what we would call calculus, but doing so in geometric form. In an instructive scholium following those lemmas, Newton explicitly justifies his approach. He acknowledges that using “the method of indivisibles” (the informal infinitesimal calculus as developed by e.g. Cavalieri) can lead to shorter proofs, “for demonstrations are more contracted by the method of indivisibles.” However, he immediately adds that “because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations…to the first and last sums and ratios of nascent and evanescent quantities, that is, to the limits of those sums and ratios”en.wikisource.org. Here in Newton’s own words is a key reason for eschewing standard calculus: the usual infinitesimal methods were seen as “less geometrical” and founded on a dubious “hypothesis.” To Newton, it was “harsh” to assume infinitesimal quantities without a firm foundation. So he reformulated calculus ideas into the language of limits (his “first and last ratios”) within a classical Euclidean framework. By doing so, “the same thing is performed as by the method of indivisibles,” Newton says – but now “we may use them with more safety”en.wikisource.org. This scholium shows Newton’s priority was rigorous justification: he wanted the benefits of calculus (determining areas, tangents, etc.) but via a method that could not be accused of algebraic looseness.
Newton then cautions the reader that if he ever speaks as if a curve is composed of infinitesimal line segments or a quantity composed of “particles,” it is only a manner of speaking; in truth, “I would not be understood to mean indivisibles, but evanescent divisible quantities…not the sums and ratios of determinate parts, but always the limits”en.wikisource.org. This clarification was effectively Newton’s assurance that the Principia contained nothing as un-rigorous as unproved infinitesimals; everything was backed by the geometric limit process. Newton’s philosophical stance was that in mathematics (especially in a foundational scientific work) one must prove every step or base it on accepted principles. He famously declared “hypotheses non fingo” (I frame no hypotheses) in the Principia’s General Scholium (2nd ed. 1713), emphasizing that he would not introduce unsupported assumptions. In the mathematical methods, this philosophy translated to avoiding a “new hypothesis” of calculus in favor of classical demonstrations.
It is also important to note Newton’s intended audience and purpose. The Principia was not written as a mathematics textbook but as a work of natural philosophy establishing the laws governing the heavens and earth. Newton’s aim was to convince the scientific community of the truth of universal gravitation and the validity of his three laws of motion. To do this, he provided mathematical proofs for phenomena like planetary orbits, the motion of projectiles, tides, and the precession of the equinoxes. He likely judged that introducing a completely new formalism (his fluxional calculus) would distract from the physical arguments or, worse, invite criticism that his results relied on untested mathematics. Newton was acutely aware of potential criticisms – ever since his early papers, he had faced challenges (for example, Robert Hooke and others had attacked his 1672 paper on optics). By couching his Principia in familiar geometric terms, Newton made it harder for critics to dismiss his work on methodological grounds. Indeed, the proofs in the Principia are crafted so meticulously that even without modern notation, a skilled contemporary reader could follow the logic. Newton even premised those initial lemmas on limits specifically “to avoid the tediousness of deducing perplexed demonstrations ad absurdum, according to the method of the ancient geometers”en.wikisource.org. In other words, he gave himself a slightly more modern toolkit (the concept of a limit, framed geometrically) to streamline proofs, but he stayed within what he considered the bounds of proper geometry. This choice reflects both a respect for the classical forms and a pragmatic understanding that his readership would more readily accept geometric proofs. Huygens, Halley, and other leading scientists of Newton’s day were well-versed in Euclidean geometry; many were less familiar with (or even unaware of) the “fluxions” Newton had invented in private. Newton’s philosophical commitment to clear, demonstrative reasoning thus dovetailed with a practical strategy: to present the most important scientific work of his life in a form that would appear irreproachable and “transparent” (to 17th-century eyes) rather than in an experimental new notation.
3. The State of Algebraic Notation and Newton’s Mastery of Algebra
It must be emphasized that Newton’s avoidance of algebraic calculus notation in the Principia was not due to any inability to use it. On the contrary, Newton was one of the inventors of calculus and a leading algebraist of his era. By the late 17th century, algebraic notation had advanced considerably from its medieval beginnings: François Viète had introduced using letters for known and unknown quantities, René Descartes (in La Géométrie, 1637) had popularized the use of x, y, z and exponential notation (like ), and infinite series and analytic methods were being explored by mathematicians like James Gregory and John Wallis. Newton was fully fluent in all these developments. In the mid-1660s, during the plague years when Cambridge was closed, the young Newton made groundbreaking discoveries in mathematics. He later recounted that he “found the method of fluxions” (his term for calculus) “in the years 1665 and 1666”www.newtonproject.ox.ac.uk. This is confirmed by Newton’s manuscripts and letters. For example, in 1669 he wrote an essay De analysi per æquationes numero terminorum infinitas (“On Analysis by Equations with an infinite number of terms”) which he communicated privately via his mentor Isaac Barrow to John Collins. In this tract, Newton used infinite series to solve problems of quadrature (area under curves) and extraction of roots – work that immediately demonstrated his algebraic prowess. He generalized Wallis’s results and extended Descartes’ analytic geometrywww.britannica.comwww.britannica.com. In short, Newton had a command of “modern” algebraic methods well before he wrote the Principia.
Perhaps the most striking evidence of Newton’s private mastery of calculus is a famous 1676 letter that he wrote (in Latin) to Henry Oldenburg (the secretary of the Royal Society) for transmission to Leibniz. In that letter Newton included a cryptogram – an anagram – to secretly affirm his general method without revealing it outright. Decoded, Newton’s concealed statement reads: “Data æquatione quotcunque fluentes quantitates involvente, fluxiones invenire, et vice versa”www.math.purdue.edu. Translated, this means: “Given any equation involving flowing (changing) quantities, to find the fluxions [derivatives], and vice versa.” This one sentence, hidden in code, essentially declares Newton’s general calculus algorithm – how to differentiate and integrate arbitrary functions. The fact that Newton provided this statement shows that by 1676 he had a fully general calculus in hand. (Indeed, he describes variables as “fluents” and their rates of change as “fluxions,” a terminology he used privately.) Newton also discovered the binomial series formula around this time and communicated it in the same letter, using algebraic notation for infinite series. So, Newton was thoroughly adept in algebraic notation and analysis, using it freely in private papers and letters.
Why, then, did Newton not publish the Principia in this analytic language that he knew so well? The above discussions of Newton’s philosophical leanings provide part of the answer: rigor and reception. It was not due to a lack of confidence in algebra per se – after all, Newton did eventually publish some of his analytic work. In 1693, at the request of the mathematician John Wallis, Newton sent two letters explaining fluxions, which were published (with Newton’s permission) in Wallis’s Algebra. In these letters Newton introduced his “dotted” notation (˙x for the time-derivative of x) for fluxionshsm.stackexchange.com. He also later published De quadratura curvarum (On the Quadrature of Curves) in 1704, which was an exposition of his calculus (using a form of infinite series and fluxional arguments) appended to the Opticks. But notably, when Newton did publish these analytic works, it was after the Principia had established his reputation and the truth of his physical laws. In the 1680s, when composing the Principia, Newton evidently decided that introducing his fluxional calculus in that work would be unwise. He preferred to keep the Principia’s derivations on solid classical ground and perhaps save the calculus for a more controlled introduction. It is telling that in 1672 Newton had suffered vicious criticism from Robert Hooke and others upon the publication of his first paper (on light and colors). That experience made Newton somewhat reluctant to publish at all for many years. When Edmund Halley urged him to write the Principia in 1685–86, Newton was drawn out of his shell to some degree, but he remained cautious. It appears that Newton separated his “private” mathematics from his “public” science in order to avoid controversy and to make the Principia as impregnable as possible.
Newton’s contemporaries were aware that he possessed powerful analytic methods even if he chose not to display them in the Principia. For instance, Edmond Halley, in his dedicatory ode at the beginning of the Principia, hints at Newton’s almost superhuman mathematical ability (without detailing fluxions). And the mathematician David Gregory, after visiting Newton in 1694, wrote in his journal that Newton had shown him a method of solving problems by series and fluxions – indicating that Newton still had not published many of his advanceswww.newtonproject.ox.ac.ukwww.newtonproject.ox.ac.uk. There was even some frustration in the mathematical community that Newton “never published aught of it [his calculus] to this day (which is worse than nonumque prematur in annum)” as one correspondent lamented in 1693www.newtonproject.ox.ac.uk. Leibniz, who had independently developed his version of calculus and published it starting in 1684, was obviously aware by the 1690s that Newton also had this tool. In 1697 Leibniz wrote to Wallis urging Newton to publish: “I have also informed others… It is right that [the calculus] go by the common name, as I have judged suitable to my candour no less than is deserved by [Newton]’s merit. And so I have warned others as well. For it befits my sense of honesty no less than his merit”, adding that he hoped Newton would “print the works which he suppresses”www.newtonproject.ox.ac.ukwww.newtonproject.ox.ac.uk. This gentle prodding from Leibniz shows that even the continental mathematicians knew Newton was holding back on sharing his analytic methods. Newton’s reluctance to publish his calculus until decades later underscores that he had made a conscious choice with the Principia: to prioritize a clear, universally accepted presentation over the introduction of a new notation or method.
4. Private Analysis vs Public Synthesis in the Principia
Historical evidence suggests that Newton often used calculus behind the scenes to derive results, even as he presented them geometrically in the Principia. For example, Newton’s discovery that an inverse-square force law yields elliptical orbits (Kepler’s first law) was likely first obtained by him around 1679–1680 using an early form of calculation. In a 1679 letter to Robert Hooke, Newton hinted that he had derived the elliptical orbit under certain assumptions (though the details were not given)www.newtonproject.ox.ac.uk. When Halley visited Newton in 1684 and asked what orbit a planet would follow under an inverse-square attraction, Newton famously replied, “an ellipse,” and that he had calculated it alreadyhsm.stackexchange.com. Newton had to dig up his earlier calculations and then composed a short tract De Motu Corporum (1684) for Halley, which was written in more or less geometrical form. Draft manuscripts of the Principia (some of which survive in the Cambridge University Library) show geometrical reasoning, but scholars like D.T. Whiteside have noted that Newton’s scratch work sometimes uses analytic techniques which he then “back-translated” into classical geometryhsm.stackexchange.com. Newton himself, later in life, claimed that Principia’s propositions were invented “by geometry” rather than first by fluxionshsm.stackexchange.com– perhaps an overstatement intended to emphasize the geometrical soundness of his work. Whether or not he initially derived every result by calculus, it is clear that Newton saw the geometrical presentation as the final, perfected form of his discoveries. In one case, Newton even says he chose a longer synthetic proof in the Principia to avoid a messy analytic one: he mentions in the preface that his treatment of lunar motion was kept brief and inserted as corollaries “to avoid being obliged to propose and distinctly demonstrate the several things… in a method more prolix than the subject deserved, and interrupt the series of propositions”en.wikisource.org. This hints that a full treatment might have required tools or digressions (perhaps analytic) that Newton preferred to omit.
Thus, there was a clear divide between Newton’s private working method and his public presentation. In private, he was an “analyst” capable of innovative shortcuts; in public, he was a “synthesist” providing rigorous demonstrations. The two modes complemented each other. As Roger Cotes observed, Newton “proceeded in a twofold method, synthetical and analytical”, deducing forces by analysis (i.e. in discovery) and then “from thence by synthesis” constructing the system of the worldmathshistory.st-andrews.ac.uk. This approach was in line with Newton’s philosophy of science. He believed that one should derive general principles from phenomena (analysis) and then demonstrate other phenomena from those principles (synthesis)mathshistory.st-andrews.ac.uk. The analytic part – which in Newton’s case could be done with calculus or other algebra – was a means to an end; the synthetic part was what one shared to substantiate the results. It is interesting that only after securing his scientific legacy with the Principia did Newton allow the publication of his mathematical papers (like the 1693 letters on fluxions and the 1704 Quadrature). By then, the priority of calculus was becoming an issue and Newton wanted to assert his role, but earlier he had deliberately kept these methods in the background.
Another factor was Newton’s personality and need for control over his work. The Principia was meticulously polished; introducing the notation of fluxions would have opened the door to queries and misunderstandings that he might not have welcomed. Indeed, even with his geometric approach, some readers found the Principia extremely difficult – John Locke, the philosopher, struggled to read it and had to ask Huygens for help. If Newton had written it in fluxions, very few in 1687 could have read it at all. Newton wanted his work to be authoritative. The geometric style provided an aura of timeless truth, linking Newton’s new physics to the certainties of Euclid. This was arguably a strategic move to bolster the acceptance of the theory of gravitation, which was itself conceptually bold (action at a distance, a universal force acting through void, etc., were controversial ideas). Presenting these novel physical ideas with an unassailable mathematical demonstration was Newton’s way of saying: “Like it or not, the conclusions are true, because here is the proof.” He did not want debates over notation or complaints that his methods were ungrounded. In summary, Newton the mathematician had a powerful new calculus at his disposal, but Newton the natural philosopher chose to “speak the language” of classical geometry in public.5. Algebraic Notation in the 17th Century: Newton vs. Leibniz
The contrast between Newton’s Principia and the works of his great contemporary Gottfried Wilhelm Leibniz highlights why Newton’s approach was distinctive. Leibniz, who independently discovered calculus in the late 1670s, was an ardent proponent of algebraic notation and the potential of analysis. He first published his results in 1684 in the Acta Eruditorum (a scholarly journal in Leipzig), in an article entitled “Nova Methodus pro Maximis et Minimis… et Singulare pro Illa Calculi Genus” – “A New Method for the Maxima and Minima, as well as Tangents, which is not obstructed by Fractional or Irrational Quantities, and a Remarkable Type of Calculus for this”www.britannica.com. In this paper Leibniz introduced the now-familiar d notation for differentials and laid out rules for differentiation (e.g. )www.britannica.com. Leibniz’s approach was purely analytic and symbolic; he demonstrated the power of his “calculus” by solving problems that had stumped earlier geometers (such as finding the tangents of complicated curves). In 1686, Leibniz published another article introducing the symbol for integralswww.britannica.com. He emphasized how this new calculus could penetrate problems that classical geometry found difficult or impossible – for instance, he derived an algebraic formula for the cycloid’s arc length, a challenge that Descartes had deemed unsolvable by standard algebrawww.britannica.com. Leibniz was also a philosophical advocate of symbolic reasoning; he wrote about his dream of a “characteristica universalis” – a universal formal language for reasoning. For Leibniz, the calculus was a shining example of such a language, turning geometry problems into algebra that any trained mathematician could manipulate.
The reception of Leibniz’s methods was notably different from that of Newton’s Principia. By the 1690s and early 1700s, on the European continent, a school of mathematicians (the Bernoulli brothers Johann and Jakob, l’Hôpital in France, Varignon, etc.) had rallied around Leibniz’s calculus. They adopted the , , notation and rapidly applied it to mechanical and geometric problems. They were publishing papers using calculus to solve brachistochrone curves, catenaries, and so forth. In contrast, British mathematicians initially stuck more with Newton’s geometric and fluxional style (and a bit later, unfortunately, fell behind as the continent surged ahead with analysis). But even on the continent, there was great admiration for the Principia’s results, if not its form. Many scholars set about translating Newton’s geometrical proofs into the new calculus idiom. For example, within a few decades, one finds Alexis Clairaut in France and others reproducing Newton’s propositions using Leibnizian calculus. A contemporary report in the 1690s observed that Newton’s methods in the Principia “pass in Europe by the name of Leibniz’s calculus”, simply because continental readers rewrote Newton’s work in that formwww.newtonproject.ox.ac.uk. This somewhat irked Newton’s supporters. In 1708 John Keill, a staunch defender of Newton, accused (in print) that Leibniz “published the same calculus [as Newton’s] afterwards, changing the name and the notation”www.newtonproject.ox.ac.uk, which of course escalated into the famous Newton-Leibniz calculus priority dispute.
For our purposes, the key point is that Leibniz and Newton chose opposite modes of publication. Leibniz presented his new mathematics immediately and unapologetically, with new notation that he believed made the concepts clear. Newton held back his new mathematics and presented his new physics in an old mathematical dress, because he believed that was the surest way to establish truth. Leibniz even gently criticized Newton’s geometric bent. After Newton’s Quadrature of Curves (1704) appeared, Leibniz reviewed it anonymously and noted that Newton “uses fluxions in place of the differences of Leibniz” in the Principia, comparing Newton to an earlier Jesuit geometer (Honoré Fabri) who had tried to avoid indivisibles by using motion conceptswww.newtonproject.ox.ac.ukwww.newtonproject.ox.ac.uk. The implication was that Newton’s approach was essentially calculus in disguise – a point with which Newton would not have disagreed. Leibniz’s 1705 review phrased it as praise that Newton “elegantly made use of [fluxions] in his Mathematical Principles of Natural Philosophy”www.newtonproject.ox.ac.uk. This remark from the Continent underscores that by then everyone understood Newton could have used calculus openly in the Principia – he simply chose not to.
It is instructive to compare a simple result in both styles. For instance, to derive Kepler’s equal-area law, Leibniz or the Bernoullis would use calculus: expressing area as an integral and differentiating. Newton, in the Principia (Book 1, Prop.1), gives a beautiful geometric proof using polygons inscribed in orbits and taking a limit as the time intervals shrink. His proof invokes no symbols, yet it captures exactly the same limiting process a calculus proof would. It convinced even the initially skeptical Huygens. This demonstrates Newton’s success in replacing analytical notation with a geometric equivalent. Even Leibniz acknowledged that ultimately both approaches were getting at the same truths. The difference was largely stylistic and methodological – but those differences mattered to Newton.6. Conclusion
Isaac Newton’s avoidance of algebraic and calculus notation in the Principia was a deliberate, context-driven choice. Historian of mathematics Florian Cajori aptly summarized that the Principia is “written in the geometric style chiefly, no doubt, because Newton wished to place his work in the great classical tradition” – though we rely on Newton’s own words and those of his contemporaries to make the case. We have seen that Newton aligned himself with the rigor of Greek geometry, both out of genuine admiration and out of a strategic desire to make his momentous discoveries in dynamics and gravitation as convincing as possible. Newton’s philosophical commitment to clear first principles led him to develop a geometrical limit approach (his “prime and ultimate ratios”) instead of publishing using infinitesimals or fluxion notation that might have appeared ungrounded. He did have at his disposal a fully developed algebraic calculus – evidenced by his letters and manuscripts – but he kept this “secret weapon” in reserve during the composition of the Principia. In private, Newton could freely employ power series or fluxions to explore a problem, but in public he transformed the solution into a classical demonstration. This allowed the work to be judged on physical and geometric grounds, not on the novelty of its mathematics. It also meant that the Principia’s readership could, in principle, follow the proofs with background only in Euclidean geometry and classical conic sections (a background common to the educated scientific community of the time).By contrast, Leibniz and his followers showcased the power of the new algebraic notation, solving problems with unprecedented ease via calculus. Newton was aware of this – indeed, later in life he had to assert his priority in the invention of calculus. Yet, tellingly, even as the calculus controversy raged, Newton held fast that the Principia had little to gain from being rewritten in fluxions. When asked to prepare a new edition of the Principia, Newton did not seize the opportunity to insert calculus or dot notation into it; the 2nd (1713) and 3rd (1726) editions remain geometrical. Roger Cotes’s preface to the 2nd edition defends Newton’s geometrical method as “the best and safest method of philosophizing” and subtly critiques the Cartesians for favoring algebraic hypotheses over solid demonstrationmathshistory.st-andrews.ac.ukmathshistory.st-andrews.ac.uk. The success of Newton’s approach is evident: the Principia convinced the scientific world of the law of gravitation within a few decades, despite its form. Later mathematicians did translate its contents into calculus, but by then the results were established.
In conclusion, Newton did not use algebraic notation in the Principia because he consciously chose the traditional geometric avenue to ensure that his physics was accepted as sound and exact. The classical geometric style provided a time-tested framework to express new ideas without inviting doubts about the mathematical rigor. Newton’s methodological credo – to derive from phenomena and demonstrate with certainty – meshed with the geometry of the ancients, not the infant calculus. Additionally, the state of mathematical communication in the 1680s favored geometry for a work aiming to reach the broad “natural philosopher” audience. Newton’s private analytical brilliance was thus funneled into a public masterpiece written in Euclid’s language. As a contemporary later remarked, Newton “made the [Mathematical] Principles of Philosophy geometrical so that they might be true*en.wikisource.orgen.wikisource.org. This marriage of ancient form with modern content in the Principia was a masterstroke that secured Newton’s results a place beyond reproach – “geometry triumphant”, carrying the new physics into the annals of science.