Question: Newton's law of universal gravitation states that a particle attracts every other particle in the universe using a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers, when Newton's book was presented in 1686 to the Royal Society, which English natural philosopher, architect and polymath, made a claim that Newton had obtained the inverse square law from him?

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Penrose method: The Penrose method (or square-root method) is a method devised in 1946 by Professor Lionel Penrose for allocating the voting weights of delegations (possibly a single representative) in decision-making bodies proportional to the square root of the population represented by this delegation.  This is justified by the fact, that due to the square root law of Penrose, the "a priori" voting power (as defined by the Penrose–Banzhaf index) of a member of a voting body is inversely proportional to the square root of its size.  Under certain conditions, this allocation achieves equal voting powers for all people represented, independent of the size of their constituency.  Proportional allocation would result in excessive voting powers for the electorates of larger constituencies.

Fast inverse square root: Fast inverse square root, sometimes referred to as Fast InvSqrt() or by the hexadecimal constant 0x5F3759DF, is an algorithm that estimates 1/ , the reciprocal (or multiplicative inverse) of the square root of a 32-bit floating-point number "x" in .  This operation is used in digital signal processing to normalize a vector, i.e., scale it to length 1.  For example, computer graphics programs use inverse square roots to compute angles of incidence and reflection for lighting and shading.  The algorithm is best known for its implementation in 1999 in the source code of "Quake III Arena", a first-person shooter video game that made heavy use of 3D graphics.  The algorithm only started appearing on public forums such as Usenet in 2002 or 2003.  At the time, it was generally computationally expensive to compute the reciprocal of a floating-point number, especially on a large scale; the fast inverse square root bypassed this step.

History of the philosophy of field theory: Field theory, the study of dynamical fields in physics, was originally a powerful mathematical formulation of Newtonian mechanics.  The success of Newtonian physics since the publication of the "Principia" provided a strong framework with which to investigate the motion and forces associated with electricity and magnetism.  Charles-Augustin de Coulomb showed in 1785 that the repulsive force between two electrically charged spheres obeys the same (up to a sign) force law as Newton's law of universal gravitation: the force between two bodies is directed along the line separating the bodies and its magnitude is proportional to the product of their charges (for gravitation, their masses) divided by the square of their distance apart.  André-Marie Ampère then showed in 1823 that the force between infinitesimal lengths of current-carrying wires similarly obeys an inverse-square law such that the force is directed along the line of separation between the wire elements.

Gravity model of migration: The gravity model of migration is a model in urban geography derived from Newton's law of gravity, and used to predict the degree of interaction between two places.  Newton's law states that: "Any two bodies attract one another with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them."

Poppy-seed bagel theorem: In physics, the poppy-seed bagel theorem concerns interacting particles (e.g., electrons) confined to a bounded surface (or body) formula_1 when the particles repel each other pairwise with a magnitude that is proportional to the inverse distance between them raised to some positive power formula_2.  In particular, this includes the Coulomb law observed in Electrostatics and Riesz potentials extensively studied in Potential theory.  For formula_3 such particles, an equilibrium (stable) state, which depends on the parameter formula_2, is attained when the associated energy of the system is minimal (the so-called generalized Thomson problem).  For large numbers of points, these equilibrium configurations provide a discretization of formula_1 which may or may not be nearly uniform with respect to the surface area (or volume) of formula_1.  The "Poppy-seed bagel theorem" asserts that for a large class of sets formula_1, the uniformity property holds when the parameter formula_2 is larger than or equal to the dimension of the set formula_1.  For example, when the points ("poppy-seeds") are confined to a torus embedded in 3-dimensions (or "surface of a bagel"), one can create a large number of points that are nearly uniformly spread on the surface by imposing a repulsion proportional to the inverse square distance between the points, or any stronger repulsion (formula_10).  From a culinary perspective, to create the nearly perfect poppy-seed bagel where bites of equal size anywhere on the bagel would contain essentially the same number of poppy seeds, impose at least an inverse square distance repelling force on the seeds.

Curie's law: In a paramagnetic material the magnetization of the material is (approximately) directly proportional to an applied magnetic field.  However, if the material is heated, this proportionality is reduced: for a fixed value of the field, the magnetization is (approximately) inversely proportional to temperature.  This fact is encapsulated by Curie's law:

Robert Hooke: Robert Hooke FRS ( ; 28 July [O.S. 18 July] 1635  – 3 March 1703) was an English natural philosopher, architect and polymath.

Newton's law of universal gravitation: Newton's law of universal gravitation states that a particle attracts every other particle in the universe using a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.  This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning.  It is a part of classical mechanics and was formulated in Newton's work "Philosophiæ Naturalis Principia Mathematica" ("the "Principia""), first published on 5 July 1687.  (When Newton's book was presented in 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him; see the History section below.)

Combined gas law: The combined gas law is a gas law that combines Charles's law, Boyle's law, and Gay-Lussac's law.  There is no official founder for this law; it is merely an amalgamation of the three previously discovered laws.  These laws each relate one thermodynamic variable to another mathematically while holding everything else constant.  Charles's law states that volume and temperature are directly proportional to each other as long as pressure is held constant.  Boyle's law asserts that pressure and volume are inversely proportional to each other at fixed temperature.  Finally, Gay-Lussac's law introduces a direct proportionality between temperature and pressure as long as it is at a constant volume.  The inter-dependence of these variables is shown in the combined gas law, which clearly states that:

Modified Newtonian dynamics: In physics, modified Newtonian dynamics (MOND) is a theory that proposes a modification of Newton's laws to account for observed properties of galaxies.  Created in 1983 by Israeli physicist Mordehai Milgrom, the theory's original motivation was to explain that the velocities of stars in galaxies were observed to be larger than expected based on Newtonian mechanics.  Milgrom noted that this discrepancy could be resolved if the gravitational force experienced by a star in the outer regions of a galaxy was proportional to the square of its centripetal acceleration (as opposed to the centripetal acceleration itself, as in Newton's second law), or alternatively if gravitational force came to vary inversely with radius (as opposed to the inverse square of the radius, as in Newton's law of gravity).  In MOND, violation of Newton's laws occurs at extremely small accelerations, characteristic of galaxies yet far below anything typically encountered in the Solar System or on Earth.

Supporting paragraphs: Newton's law of universal gravitation Robert Hooke
Answer: Robert Hooke