Robert Hooke?s Seminal Contribution to Orbital Dynamics

Nauenberg, Michael

doi:10.1007/s00016-004-0226-y

Cited by 32 publications

(13 citation statements)

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“…The existence of this device may not have been recorded because in the years just prior to 1681, Hooke stopped producing a full account of his astronomical instruments for the Royal Society, including one that described ‘all variety of Ellipses’ (Waller, ). We also know that his formulation of the physical principles for orbital motion relied on mechanical analogues, such as the conical pendulum, a ball rolling inside a half‐sphere and an inverted cone (Nauenberg, ). Although its uses are unclear, as an object in the painting, the armillary sphere provides a mechanical analogue of the diagram.…”

Section: The Armillary Spherementioning

confidence: 99%

“…Hooke was at the time probably aware of De Motu , but his treatment in the 1685 manuscript was different. Hooke demonstrated the converse, using the math ‘in a novel way to obtain the orbit of a body moving in an attractive central field of force that varies linearly with distance’ (Nauenberg, ). This was what both Hooke and Newton had set out to do, but Newton did not specifically state a case for extending his proofs to the converse until he commented on it in the second edition of Principia Mathematica (Whiteside, ).…”

Section: The Disputementioning

confidence: 99%

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The lost portrait of Robert Hooke?

Griffing

2020

Journal of Microscopy

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Summary This letter considers the ‘Portrait of a Mathematician’ attributed to Mary Beale in the 1680s as a likely candidate for a portrait of Robert Hooke made during his lifetime. It closely matches the physical descriptions of Hooke made by his biographers who knew him, Richard Waller (d. 1715) and John Aubrey (1626–1697). The portrait contains a remarkable diagram, as well as its mechanical analogue, demonstrating the elliptical orbit of a body under constant force similar to an unpublished, unfinished 1685 manuscript by Hooke. It also contains a landscape, which is likely to be Lowther Castle and its associated church. Hooke provided designs for the Lowther castle church renovations completed in 1686. The portraitist, Mary Beale, knew Hooke and was very familiar with the Lowther family, who commissioned 30 portraits from her. The diagram and armillary sphere in the portrait establish Hooke's contribution to the theory of gravity. Their presence may have been one of the reasons the portrait was not purchased and kept by the Royal Society. It might have diminished the legacy of Isaac Newton.

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Section: The Armillary Spherementioning

confidence: 99%

Section: The Disputementioning

confidence: 99%

The lost portrait of Robert Hooke?

Griffing

2020

Journal of Microscopy

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“…A extensão da contribuição de Hooke para a mecânica de Newton em geral, e para o entendimento e a enunciação da Lei da Gravitação Universal em particular,é objeto de amplo debate acadêmico ainda hoje [19], debate aliás iniciado pelo próprio Hooke com sua reclamação de prioridade na descoberta da referida Lei. Esse debateé tão amplo que uma exposição mais aprofundada mereceria uma abordagemà parte, que foge aos objetivos deste trabalho.…”

Section: ]unclassified

Breve histórico da dinâmica newtoniana do movimento curvilíneo

Porto¹

2015

Rev. Bras. Ens. Fis.

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Neste trabalho fazemos uma apresentação da dinâmica newtoniana do movimento curvilíneo. Iniciamos fazendo um pequeno retrospecto da questão, a partir da problematização imposta pela concepção cartesiana do movimento inercial, passando quase que imediatamente ao estudo das sucessivas abordagens newtonianas do tema. Apresentamos os tratamentos iniciais de Newton-ainda dominados pela idéia cartesiana de "esforço centrífugo"-nos quais, no entanto, já observamos a sutileza do raciocínio matemático apoiado nas noções de elementos infinitesimais. Destacamos, em seguida, a aparente contribuição do pensamento de Robert Hooke para a concepção newtoniana do movimento curvilíneo, concretizada sobretudo através de diálogos epistolares, ainda que escassos ou reticentes. Enfatizamos como, após esse contato, Newton abandonou a noção da atuação de uma força centrífuga a produzir um equilíbrio dinâmico no movimento, em favor de um entendimento do problema como um sistema mecânico fundamentalmente fora do equilíbrio. Passamos, porúltimo,à concepção newtoniana definitiva do movimento, já expressa em sua essência no tratado De Motu e finalmente apresentada na grande obra dos Princípios Matemáticos da Filosofia Natural. Salientamos, particularmente, a dedução efetuada por Newton das leis do movimento planetário propostas por Kepler, que constituiu uma das mais altas realizações da ciência humana. Palavras-chave: história da mecânica, Newton, força mecânica. In this work we make an exposition on the Newtonian dynamics of curvilinear motion. We begin by making a small retrospect on this question, starting from the moment when Descartes' conception of inertial motion rendered it problematic, and passing almost at once to the study of the different Newtonian approaches to this issue. We present Newton's initial treatments-yet governed by the Cartesian idea of "Centrifugal Endeavour"-where we nonetheless can already observe the subtlety of a mathematical reasoning based on the notions of infinitesimal calculus. Afterwards, we emphasize the seeming contribution of Robert Hooke's thought to the Newtonian conception of curvilinear motion, accomplished most of all by epistolary dialogues, yet rare or reticent. We emphasize how, after this exchange, Newton abandoned the notion of the action of a centrifugal force producing a kind of dynamical equilibrium of motion, in favor of a new understanding of the problem as a basically out-of-equilibrium mechanical system. We end with Newtonian final conception of motion, already framed in its essential aspects in the tract named De Motu, and finally presented in his master work Mathematical Principles of Natural Philosophy. We particularly point Newton's deduction of Kepler's planetary laws of motion, which has become one of the highest achievements of human science.

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“…In 1679, Robert Hooke (Nauenberg 2005) wrote to Isaac Newton asking him …and particularly if you will let me know your thoughts of that of compounding the celestiall motions of the planetts of a direct motion by the tangent & an attractive motion towards the centrall body (Newton 1960) In the Principia, Newton took a great step forward by extending the application of Euclidean geometry to the concept of force impulses which he represented by line segments that in a limit have vanishing small magnitude, 1 leading to the emergence of a finite and continuous force. Such evanescent quantities appeared already in Greek geometry, in the method of exhaustion which, for example, was applied by Archimedes to obtain a rigorous relations between the circumference and the area of a circle, as well as bounds for these quantities that he calculated algebraically.…”

Section: Nauenberg (B)mentioning

confidence: 99%

The early application of the calculus to the inverse square force problem

Nauenberg

2010

Arch. Hist. Exact Sci.

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The translation of Newton's geometrical Propositions in the Principia into the language of the differential calculus in the form developed by Leibniz and his followers has been the subject of many scholarly articles and books. One of the most vexing problems in this translation concerns the transition from the discrete polygonal orbits and force impulses in Prop. 1 to the continuous orbits and forces in Prop. 6. Newton justified this transition by lemma 1 on prime and ultimate ratios which was a concrete formulation of a limit, but it took another century before this concept was established on a rigorous mathematical basis. This difficulty was mirrored in the newly developed calculus which dealt with differentials that vanish in this limit, and therefore were considered to be fictional quantities by some mathematicians. Despite these problems, early practitioners of the differential calculus like Jacob Hermann, Pierre Varignon, and Johann Bernoulli succeeded without apparent difficulties in applying the differential calculus to the solution of the fundamental problem of orbital motion under the action of inverse square central forces. By following their calculations and describing some essential details that have been ignored in the past, I clarify the reason why the lack of rigor in establishing the continuum limit was not a practical problem.

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Robert Hooke?s Seminal Contribution to Orbital Dynamics

Cited by 32 publications

References 25 publications

The lost portrait of Robert Hooke?

The lost portrait of Robert Hooke?

Breve histórico da dinâmica newtoniana do movimento curvilíneo

The early application of the calculus to the inverse square force problem

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