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Fluxion

A fluxion is the infinitesimal rate of change of a varying quantity, known as a fluent, which Isaac Newton introduced as a foundational concept in his development of calculus during the mid-1660s.^[1] Newton conceptualized fluents as continuously flowing or generated quantities, such as distances traveled by moving points, with fluxions representing their instantaneous velocities or derivatives, often denoted by accented variables like \dot{x} or x'.^[2] This approach allowed Newton to solve problems in tangents, areas, and curvatures, building on earlier ideas from mathematicians like Isaac Barrow while establishing the inverse relationship between differentiation (fluxions) and integration (inverse fluxions).^[3] Newton first outlined the method of fluxions in a private tract written in October 1666, during his annus mirabilis at Woolsthorpe, though it remained unpublished for decades and circulated only in manuscript form among contemporaries.^[1] He expanded the work in 1671 as De Methodis Serierum et Fluxionum, incorporating infinite series expansions to handle algebraic equations, but withheld publication due to concerns over priority and philosophical objections to infinitesimals.^[4] The treatise was eventually published posthumously in Latin in 1736 and in English translation shortly after, revealing fluxions as a rigorous tool for geometry and mechanics, including applications to maximum and minimum problems and centers of curvature.^[5] Fluxions played a pivotal role in the calculus priority dispute with Gottfried Wilhelm Leibniz, whose independent notation (differentials) gained wider adoption, but Newton's framework profoundly influenced British mathematics and later unified with Leibnizian methods in the 18th century.^[3] Despite initial criticisms for relying on "ghosts of departed quantities" (infinitesimals), fluxions provided an early statement of the Fundamental Theorem of Calculus, linking fluxions to the quadrature of curves, and remain a historical cornerstone of differential calculus.^[1]

Fundamentals

Definition

In mathematics, fluxions refer to the instantaneous rates of change of quantities that vary continuously over time, a concept developed by Isaac Newton as part of his foundational work on calculus. These varying quantities, termed fluents, represent entities that "flow" or evolve, much like positions in physical motion, while fluxions capture their velocities of change at any given moment. This approach allowed Newton to model dynamic processes mathematically, treating change not as discrete steps but as an unbroken continuum of infinitesimal increments.^[3] The distinction between fluxions and fluents is central: a fluent is the evolving quantity itself, such as the position of a moving body or the area under a curve, whereas the fluxion is the measure of its rate of variation with respect to time or another parameter. Newton conceived fluxions as arising from the "generating motion" that produces the fluent, envisioning them as composed of an infinite series of vanishingly small changes that collectively describe the flow. This perspective emphasized the geometric and kinematic origins of the method, where quantities are generated by the motion of points along curves.^[6]^[7] Newton's motivation for fluxions stemmed from his investigations into planetary motion and the geometry of curves during the 1660s, where he sought tools to quantify how such entities alter instantaneously. In essence, fluxions provide a principle for determining how fluents "flow" or transform at a precise instant, laying the groundwork for analyzing rates in natural phenomena. This framework prefigures the modern concept of derivatives in calculus.^[3]^[7]

Notation

Newton employed a symbolic system in which the fluxion of a quantity x, representing its rate of change, is denoted by a dot placed above the letter, written as \dot{x}. Higher-order fluxions are indicated by additional dots, such as the second fluxion \ddot{x} and the third \dddot{x}.^[5] To handle infinitesimal increments, Newton used the symbol o to denote a small, fixed moment of time, treating higher powers like o^2 as negligible quantities that vanish in the limit.^[8] Fluents, the flowing quantities whose fluxions are taken, were conceptualized as generated by accumulating fluxions over successive moments of time, akin to integration, but without a dedicated symbolic notation for the operation itself.^[7] In various manuscripts, Newton's notation showed inconsistencies before standardization, including the use of corresponding small letters for fluxions of capital-letter fluents, as well as primes or acute accents over letters to mark fluxions in earlier drafts.^[9]^[10]

Historical Context

Newton's Development

During his annus mirabilis in 1665–1666, while isolated at his family estate in Woolsthorpe due to the Great Plague, Isaac Newton began developing the concept of fluxions as part of his foundational work in mathematics.^[11] This period marked the inception of his ideas on rates of change, building on his studies of infinite series and influenced by John Wallis's Arithmetica Infinitorum (1656), which provided methods for interpolating series expansions.^[11] Newton's early notations and approaches to instantaneous motion emerged from this time, though he did not yet formalize them under the term "fluxions."^[11] Newton kept his mathematical innovations largely private, recording them in personal manuscripts rather than seeking immediate publication. In 1669, he composed De Analysi per Aequationes Numero Terminorum Infinitas, a tract demonstrating his methods for solving equations through infinite series, which Isaac Barrow, his mentor, shared with the mathematical correspondent John Collins.^[11] This manuscript, circulated among a small circle of scholars, included preliminary applications of fluxional ideas to problems like finding tangents and areas under curves, but Newton withheld broader dissemination.^[12] By 1671, he expanded these into the comprehensive treatise Methodus Fluxionum et Serierum Infinitarum, outlining the systematic use of fluxions for quadratures (integration) and tangents (differentiation), yet it remained unpublished during his lifetime.^[4] Newton applied fluxions in his early scientific investigations, integrating them into unpublished notes and preliminary papers on optics and mechanics from the late 1660s. In optics, he employed series expansions derived from fluxional methods to analyze light refraction and prism experiments conducted around 1666, as referenced in his correspondence and later works.^[11] For mechanics, fluxions informed his initial formulations of motion and force in 1666 waste-book entries, enabling calculations of trajectories and gravitational effects that foreshadowed his later Principia.^[11] These applications demonstrated fluxions' utility beyond pure mathematics, though Newton presented them sparingly in public forums like his 1669 meeting with Collins in London.^[11] The delay in publishing Methodus Fluxionum until 1736 stemmed from Newton's perfectionism and aversion to controversy; he continually revised the manuscript, fearing incomplete presentation or criticism, and prioritized other pursuits like his Lucasian professorship duties.^[4] John Colson's English translation, prepared from Newton's notes, finally appeared posthumously, ensuring the work's preservation despite the decades-long concealment.^[4]

The Calculus Priority Dispute

The priority dispute over the invention of calculus arose between Isaac Newton and Gottfried Wilhelm Leibniz, centering on their independent developments of infinitesimal methods, with fluxions representing Newton's approach. In 1676, during their correspondence mediated by Henry Oldenburg, Newton included an anagram in his letter dated October 24 to obscure the details of his fluxional method while establishing priority: "6accdæ13eff7i3l12mfkwprx," which unscrambles to the Latin phrase describing the process of finding fluxions from fluents and vice versa in any equation involving fluent quantities.^[13] This cryptic device, common in the era for protecting intellectual property, hinted at Newton's earlier work on fluxions from the 1660s without revealing specifics.^[13] Leibniz publicly introduced his differential notation in 1684 with the paper "Nova methodus pro maximis et minimis, itemque tangentibus," published in Acta Eruditorum, marking the first formal presentation of calculus using differentials (dx, dy) for rates of change. In contrast, Newton delayed full publication of his fluxion method; although privately circulated earlier, it appeared publicly in 1711 with the release of "Analysis per quantitatum series, fluxiones ac differentias," a compilation including his 1669 tract "De Analysi," which detailed infinite series and fluxions. This timing fueled suspicions, as Leibniz's earlier publication contrasted with Newton's reticence. The controversy intensified in the early 18th century amid accusations of plagiarism, with Newton's supporters, including John Keill, claiming in 1711 that Leibniz had borrowed ideas from the 1676 letters.^[14] In response, the Royal Society formed an ad hoc committee in 1711, chaired by Newton himself and composed largely of his allies, to investigate the matter; the biased panel declared Newton the original inventor, issuing a report titled "Commercium epistolicum" that dismissed Leibniz's independent contributions.^[15] Leibniz countered by appealing to the broader European community, but mutual plagiarism charges escalated, exacerbating nationalistic divides between English mathematicians favoring fluxions and Continental scholars adopting Leibniz's notation.^[14]^[16]

Mathematical Formulation

Fluxions and Fluents

In Newton's calculus, fluents are variable quantities conceived as flowing or generated continuously with respect to time t, such as the position of a moving point denoted x(t).^[17] Fluxions represent the instantaneous rates of change of these fluents, denoted \dot{x} and understood in the limit sense as \dot{x} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}, where \Delta t is the finite difference in time.^[18] This formulation emphasizes the velocity-like nature of fluxions as the "celerity of flowing" of the fluent.^[19] The change in a fluent over an infinitesimal time interval o is expressed through an infinite series expansion derived from the binomial theorem or Taylor-like approximation:

x(t + o) - x(t) = \dot{x} o + \frac{1}{2} \ddot{x} o^2 + \frac{1}{6} \dddot{x} o^3 + \cdots,

where higher-order fluxions like \ddot{x} (the fluxion of \dot{x}) appear, and terms beyond the first are truncated as o becomes vanishingly small.^[18] This series captures the nascent or momentary state of the fluent's variation.^[17] Momentary increments, or "moments," of a fluent are thus dx = \dot{x} o, representing the infinitesimal change in x during the moment o of time, with analogous forms for other fluents like dy = \dot{y} o.^[19] In forming ratios such as \frac{dx}{dy} = \frac{\dot{x}}{\dot{y}}, the infinitesimal o cancels out and vanishes entirely, yielding the ultimate ratio of the fluxions without reliance on finite quantities.^[18] Unlike finite differences, which involve measurable increments over discrete intervals, fluxions and their moments pertain strictly to instantaneous changes, where o is not a fixed small quantity but one that "vanishes" in the limit, ensuring the ratios reflect true momentary velocities rather than approximations from finite steps.^[17] Newton's approach thus prioritizes the continuous, flowing generation of quantities over discrete partitioning.^[18]

Solving Fluxional Equations

Newton employed a direct method to compute fluxions, the instantaneous rates of change of fluents, by expanding quantities using the binomial theorem and considering infinitesimal increments. For a power of a fluent, such as x^n, the fluxion is derived by treating (x + \dot{x} o)^n, where o denotes an infinitesimal moment of time, and expanding via the binomial theorem to yield the leading term n x^{n-1} \dot{x}, with higher-order infinitesimals neglected.^[20] This approach established the general rule \dot{(x^n)} = n x^{n-1} \dot{x} for positive integer exponents, extendable to fractional powers through series expansions. To handle composite expressions, Newton formulated rules analogous to modern differentiation laws. The fluxion of a product uv satisfies \dot{(uv)} = u \dot{v} + v \dot{u}, obtained by considering the infinitesimal change in the rectangle formed by u and v.^[20] For quotients, the rule \dot{\left( \frac{u}{v} \right)} = \frac{ v \dot{u} - u \dot{v} }{v^2} follows from applying the product rule to u \cdot v^{-1}, with the fluxion of the inverse derived similarly./02%3A_Calculus_in_the_17th_and_18th_Centuries/2.01%3A_Newton_and_Leibniz_Get_Started) Compositions, such as powers or roots, were addressed through chained applications of these rules or direct binomial expansions. Higher-order fluxions extended the method to describe accelerations and curvatures in motion. The second fluxion \ddot{x} represents the rate of change of the first fluxion \dot{x}, applicable in equations governing centripetal forces or orbital paths, where successive applications of the direct rules yield expressions like \ddot{x} = \frac{d}{dt} (\dot{x}).^[21] Further orders, such as third or fourth fluxions, were invoked for more complex dynamics, maintaining the infinitesimal framework without limits.^[22] The inverse problem of solving fluxional equations—finding fluents from given fluxions, known as quadratures—involved reversing these operations, often through infinite series expansions or geometric constructions. Newton tabulated common quadratures, such as integrating powers to recover \frac{x^{n+1}}{n+1}, and used series for non-elementary cases, like logarithmic or trigonometric fluents.^[23] Geometric rules, drawing from Cavalieri's method of indivisibles, approximated areas under curves by summing infinitesimal elements, providing a practical means to invert fluxions without algebraic closure.^[18]

Examples and Applications

Algebraic Example

To illustrate the computation of fluxions in an algebraic context, consider the fluent y = x^3, where the independent fluent x = t^2. The goal is to find the fluxion \dot{y} evaluated at t = 1.^[24] The fluxion of x with respect to time is first determined as \dot{x} = 2t, following the general rule for powers derived from infinitesimal increments. Substituting into the fluxion rule for y = x^3, which yields \dot{y} = 3x^2 \dot{x}, gives \dot{y} = 3(t^2)^2 (2t) = 3t^4 \cdot 2t = 6t^5. At t = 1, this evaluates to \dot{y} = 6. This step-by-step process aligns with Newton's method for composing fluxions in polynomial relations.^[24] Newton employed infinitesimals to justify such computations, denoting the moment (infinitesimal change) of y as oy, where o is a vanishing quantity. The expansion is

oy = \dot{y} o + \frac{1}{2} \ddot{y} o^2 + \cdots,

with higher-order terms neglected to isolate the first fluxion \dot{y}. For the given fluents, this confirms \dot{y} = 6t^5 by retaining the linear term in o after expansion and substitution.^[24] Verification of the result can be achieved through series expansion of the fluents. Expanding y = (t^2)^3 = t^6 directly yields the series y = t^6, whose fluxion is \dot{y} = 6t^5, matching the computed value at t = 1 and demonstrating the accuracy of the infinitesimal approach without higher-order discrepancies.^[24]

Geometric Application

In the geometric application of fluxions, the slope of the tangent line to a curve y = f(x) is determined by the ratio of the fluxions \frac{\dot{y}}{\dot{x}}, representing the instantaneous rate of change in the spatial direction of the curve.^[24] This ratio arises from considering the curve as generated by the motion of a point, where the fluxions capture the velocities along the ordinate and abscissa, allowing for the construction of the tangent as the limiting position of a secant line in the geometric plane.^[24] Such an approach facilitates the drawing of tangents to both mechanical curves, like cycloids, and geometrical curves, such as conic sections, by analyzing the proportional changes in coordinates without relying on algebraic expansion.^[25] A representative example is the parabola y = x^2, where the fluxion of the ordinate yields \dot{y} = 2x \dot{x}, giving the tangent slope \frac{\dot{y}}{\dot{x}} = 2x. The normal line, perpendicular to the tangent, has slope -\frac{\dot{x}}{\dot{y}} = -\frac{1}{2x}.^[24] In this construction, as evanescent increments approach zero, the ratio of fluxions establishes the directions of the tangent and normal, enabling the erection of perpendiculars that define the curve's local orientation and aiding in problems of reflection or incidence on parabolic paths.^[25] This method highlights the spatial interplay between the curve's ordinate and its rate of change, providing a tool for geometric synthesis in figures like projectiles tracing parabolic trajectories. Fluxions also apply to the quadrature of areas under curves, where the area is treated as a fluent generated by the continuous motion of the ordinate along the abscissa, computed via infinite series that approximate the bounded region.^[24] The fluxion of such an area equals the product of the ordinate and the fluxion of the abscissa, \dot{z} = y \dot{x}, offering a geometric measure of the space enclosed by the curve, axis, and a vertical line, often resolved into series for conic sections like hyperbolas or ellipses.^[24] This technique underscores the cumulative spatial extent without exhaustive computation, prioritizing the curve's bounding properties. The connection to kinematics further emphasizes fluxions as velocities driving the generation of conic sections, where a point's motion along varying directions traces parabolas, ellipses, or hyperbolas through proportional fluxions of coordinates.^[24] For instance, uniform motion along the abscissa combined with accelerated motion along the ordinate produces a parabolic path, with fluxions quantifying the instantaneous velocities that define the curve's spatial evolution.^[25] This kinematic perspective integrates motion and geometry, illustrating how fluxions model the dynamic formation of conics in orbital or projectile contexts.^[25]

Criticisms and Responses

Berkeley's Critique

In 1734, George Berkeley, the Irish philosopher and bishop, published the pamphlet The Analyst; or, A Discourse Addressed to an Infidel Mathematician, in which he mounted a sharp philosophical attack on the method of fluxions, arguing that its reliance on infinitesimals undermined its claim to mathematical rigor. Berkeley targeted the core concepts of fluxions, which Newton had introduced as velocities of evanescent increments to capture instantaneous rates of change. He contended that these "increments generated in the least equal particles of time" were ill-defined and led to logical absurdities, as they were invoked without a coherent ontological status.^[26] Central to Berkeley's critique was his dismissal of infinitesimals as "the ghosts of departed quantities," entities that are neither finite quantities, nor quantities infinitely small, nor yet nothing, but mere phantoms invoked to justify calculations. He highlighted inconsistencies in the procedure for finding fluxions, such as the rule for the fluxion of a product, where increments are treated as non-zero to perform division and form ratios, only to be subsequently neglected or set to zero, creating an apparent contradiction in their handling. This, Berkeley argued, rendered the method inconsistent, as it presupposed the existence of vanishing quantities while simultaneously denying their substantiality in the final steps.^[26] Berkeley further accused the fluxional method of circularity, asserting that it depended on the very notion of limits or ultimate ratios that it purported to define and demonstrate, thereby begging the question rather than providing a foundational proof. He challenged mathematicians to explain how such ratios could be determined without already assuming the principles under scrutiny, emphasizing that the approach evaded rather than resolved foundational issues. These arguments were framed not as opposition to the practical utility of fluxions but as a demand for logical clarity comparable to that expected in philosophical or theological reasoning.^[26] The publication of The Analyst profoundly influenced English mathematicians, sparking numerous responses and pamphlets in the ensuing years and setting the agenda for debates on calculus foundations throughout the 1730s and 1740s. It prompted a broader reevaluation of Newtonian methods, contributing to a gradual shift away from fluxions toward alternative formulations that prioritized rigor, such as those based on limits, and diminished the dominance of fluxional notation in British mathematical practice.^[27]^[28]

Newton's Revisions

In the second edition of his Philosophiæ Naturalis Principia Mathematica, published in 1713, Isaac Newton revised his presentation of fluxion theory by shifting from explicit references to infinitesimals—such as the notation "o" for evanescent increments—to the concept of "ultimate ratios" of nascent quantities.^[29] This change allowed Newton to describe the behavior of varying quantities as they converge continually to equality over finite intervals of time, thereby avoiding the philosophical pitfalls associated with infinitely small elements.^[18] Under this redefinition, fluxions represented the limits of ratios of the increments of fluents generated in equal but very small particles of time approaching zero, ensuring that all descriptions remained grounded in finite, assignable magnitudes.^[29] Newton articulated this in the lemmas of Book I, Section I, stating that "quantities, and the ratios of quantities, which in any finite time converge continually to equality... become ultimately equal," thus framing fluxions as precise limits rather than instantaneous velocities tied to vanishing moments.^[18] These adjustments were further emphasized in the 1736 posthumous publication of Newton's The Method of Fluxions and Infinite Series, which integrated extensive discussions of prime ratios (initial tendencies of nascent quantities) and ultimate ratios (final limiting values) to demonstrate fluxional operations without invoking infinitesimals.^[30] The treatise highlighted how such ratios provided a rigorous foundation for solving problems in curve geometry and series expansion, aligning the method with geometric certainty.^[6] Newton's refinements addressed foundational concerns similar to those later raised by Berkeley, emphasizing geometric principles and ultimate ratios to enhance the theory's logical rigor.^[3]

Legacy and Modern Perspective

Influence on Calculus Development

Fluxions played a pivotal role in the adoption of calculus among English mathematicians during the early 18th century, particularly in the development of power series expansions and applications to mechanics. Brook Taylor, in his 1715 work Methodus Incrementorum Directa et Inversa, employed Newtonian fluxion notation to derive infinite series representations of functions, bridging finite differences with infinitesimal changes and laying groundwork for what became known as Taylor series.^[31] Similarly, Colin Maclaurin advanced fluxional methods in his 1742 Treatise of Fluxions, where he applied them to geometric problems and mechanical systems, defending and extending Newton's kinematic approach against contemporary critiques while integrating series expansions for curve analysis.^[32] These efforts solidified fluxions as a core tool in British mathematical practice, influencing subsequent works in approximation techniques and dynamical systems. The framework of fluxions significantly contributed to the formulation of differential equations in physics, most notably in expressing Newton's laws of motion. While the Principia Mathematica (1687) primarily used geometric methods to derive the laws of motion, Newton employed fluxional notation in his private calculations and other works to model the rates of change in velocities and positions, deriving equations that describe accelerated motion under forces, such as the second law relating force to the fluxion of momentum.^[3] Fluxions provided the analytical foundation for representing physical phenomena like planetary orbits and pendular motion as solvable equations, establishing calculus as indispensable for mechanics and inspiring later English treatises on variational problems.^[18] Following the 1736 publication of Newton's The Method of Fluxions and Infinite Series in English, translations facilitated its dissemination across Europe, notably influencing Leonhard Euler's early mathematical endeavors despite his preference for Leibnizian differentials. A French edition appeared in 1740, translated by Georges-Louis Leclerc, Comte de Buffon, which exposed continental scholars to fluxional techniques for infinite series and quadrature.^[33] Euler, in his initial works on mechanics during the 1720s and 1730s, integrated elements of fluxional reasoning with differential methods to analyze rigid body motion and fluid dynamics, as seen in his reformulation of Newtonian laws in Mechanica (1736).^[34] This synthesis helped Euler develop analytical tools for physics, even as he favored symbolic differentials for broader algebraic applications.^[35] Fluxions also factored into resolving the priority dispute between Newton and Gottfried Wilhelm Leibniz over calculus invention, underscoring Newton's independent development. The 1736 publication revealed manuscripts dating to 1671, predating Leibniz's public notations and demonstrating fluxions as an original system conceived in the 1660s, which bolstered Newton's claim amid accusations of plagiarism leveled by his supporters.^[14] This evidence, disseminated through the printed work and translations, affirmed the parallel yet distinct inventions, shifting focus from controversy to the complementary strengths of fluxional and differential approaches in advancing mathematical analysis.^[36]

Relation to Contemporary Mathematics

In modern calculus, Newton's fluxion of a variable x, denoted as \dot{x}, is equivalent to the derivative \frac{dx}{dt} in Leibniz notation, representing the instantaneous rate of change of x with respect to time t.^[37]^[19] This equivalence underscores the fluxion's role as a foundational concept for differentiation, though its explicit dependence on time as the independent variable distinguishes it from the more general modern treatment where the derivative can be with respect to any parameter.^[37] Newton's reliance on infinitesimals in the method of fluxions—treating small increments or "moments" as vanishing quantities—anticipated the rigorous framework of non-standard analysis developed by Abraham Robinson in the 1960s. Robinson's construction of hyperreal numbers, incorporating genuine infinitesimals, provides a logical foundation for the intuitive infinitesimal arguments Newton employed to compute tangents and rates, thereby validating and extending early calculus techniques without the need for limits. A key limitation of the fluxional method was its lack of general rules for finding antiderivatives, or fluents, which Newton often computed via case-specific infinite series expansions or geometric quadratures rather than a unified inverse operation to differentiation./02%3A_Calculus_in_the_17th_and_18th_Centuries/2.01%3A_Newton_and_Leibniz_Get_Started) This approach restricted systematic integration to algebraic or trigonometric forms, leaving more complex functions without elementary antiderivatives. Modern integration theory, particularly the Riemann integral and fundamental theorem of calculus in its rigorous form, resolves these issues by establishing antiderivatives for a broad class of continuous functions through limit-based definitions, enabling general evaluation of definite integrals independently of finding primitives./02%3A_Calculus_in_the_17th_and_18th_Centuries/2.01%3A_Newton_and_Leibniz_Get_Started)^[18] Today, fluxions hold significant educational value in mathematics courses on the history of calculus, where they illustrate the evolution from intuitive motion-based ideas to epsilon-delta rigor, helping students appreciate the conceptual origins of limits and the importance of foundational precision.^[38] By contrasting fluxions with contemporary notation and proofs, instructors demonstrate how early ambiguities, such as those in infinitesimal reasoning, were addressed through 19th-century developments like Weierstrass's limit definitions.^[38] This historical perspective fosters deeper understanding of calculus as a dynamic field, bridging intuitive discovery with formal verification.^[39]

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[34]
La methode des fluxions translated by Georges Louis Le Clerc
Originally written in 1671, in Latin, this was Newton's first comprehensive presentation of his method of fluxions which, according to Hall 'might have effected ...
[35]
(PDF) Newton and Euler - ResearchGate
In this article, we account for Euler's results on the foundation of mechanics and algebraic analysis, in the frame of the development of the Newtonian ...
[36]
[PDF] Euler, Reader of Newton: Mechanics and Algebraic Analysis
A Newtonian way, based on a classic conception of geometry, conceives fluxions as ratios of vanishing quantities. A Leibnitian way, based on the ...
[37]
The Newton–Leibniz Calculus Controversy, 1708–1730
The dispute began in 1708, when John Keill accused Leibniz of having plagiarized Newton's method of fluxions. ... The spark that set fire to the priority dispute ...
[38]
[PDF] Infinite series II - Purdue Math
x = acost, y = bsint, 0 ≤ t ≤ 2π. The formula for the length of a curve from Calculus gives the length as. Z 2π. 0 q. (x0)2 + (y0)2dt = Z 2π. 0 q a2 sin2 t + ...
[39]
[PDF] An Introduction to a Rigorous Definition of Derivative
Nov 30, 2022 · Newton used the term fluxion for a special case of what we now call a derivative.Missing: equivalent | Show results with:equivalent
[40]
Differential Calculus: The Use of Newton's Methodus Fluxionum et ...
History of mathematics enhances understanding of differential ... Newton's exploration of fluxions laid foundational ideas for modern calculus education.