2010
DOI: 10.1007/s00407-009-0056-z
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The early application of the calculus to the inverse square force problem

Abstract: 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 formu… Show more

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Cited by 11 publications
(11 citation statements)
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References 13 publications
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“…But the assignable triangle T BC consists of the axis, the ordinate, and the tangent, and therefore contains the angle between the direction of the curve (or its tangent) and the axis or base, that is, the inclination of the curve at a given point C. (Struik 1969, 283) The tangent line to the curve C(C) at C is T C, and GL and GC are sides of its "characteristic triangle," GLC, satisfying a "given mutual relation" specified below. But this triangle which Leibniz called inassignable 26 , does not appear in the proof of Leibniz's theorem, while another characteristic triangle, CE C, where C (not indicated in Leibniz's diagram, Fig. 2, is the intersection of the tangent line T C with the extension of (H)(F ), turns out to be relevant to Leibniz's proof.…”
Section: Barrow's Geometrical Proof Of the Fundamental Theorem Of The...mentioning
confidence: 99%
See 1 more Smart Citation
“…But the assignable triangle T BC consists of the axis, the ordinate, and the tangent, and therefore contains the angle between the direction of the curve (or its tangent) and the axis or base, that is, the inclination of the curve at a given point C. (Struik 1969, 283) The tangent line to the curve C(C) at C is T C, and GL and GC are sides of its "characteristic triangle," GLC, satisfying a "given mutual relation" specified below. But this triangle which Leibniz called inassignable 26 , does not appear in the proof of Leibniz's theorem, while another characteristic triangle, CE C, where C (not indicated in Leibniz's diagram, Fig. 2, is the intersection of the tangent line T C with the extension of (H)(F ), turns out to be relevant to Leibniz's proof.…”
Section: Barrow's Geometrical Proof Of the Fundamental Theorem Of The...mentioning
confidence: 99%
“…1, with Newton's diagram for his earliest geometrical proof of the fundamental theorem of the calculus (Guicciardini, 2009, 184). 26 Leibniz chose the Latin word inassignabilis for the characteristic triangle, because its sides are differentials which do not have an assignable magnitude. this law of tangency which he formulated, instead, in terms of the subtangent T F , where T (not shown in Fig.…”
Section: Barrow's Geometrical Proof Of the Fundamental Theorem Of The...mentioning
confidence: 99%
“…The quadrature of the circle that we have just reviewed is of great importance for the treatment of inverse-cube orbits, as is apparent if we consider the integrals (16). Further, Newton could extend the quadrature of the circle to the unit hyperbola…”
Section: The Arccoshmentioning
confidence: 99%
“…C Maxwell [5] who in turn attributed it to Sir William Hamilton. At the start of 18th century, however, able mathematicians like Jacob Hermann, Pierre Varignon and Johann Bernoulli were able to express Newton's relation for central force in Proposition 6 as a differential equation for the orbit in Cartesian coordinates, while Gotfried Leibniz obtained such an equation in polar coordinates [6].…”
Section: Introductionmentioning
confidence: 99%
“…Substituting cosθ = (r/a)cosθ + , and r/a = (1 − 2 )/(1 + cosθ) for the equation of the ellipse in polar coordinates, Eq. 4, one finds (6) δθ = λa r δθ…”
Section: Introductionmentioning
confidence: 99%