Editing Inverse-square law (section)

==History==

[[John Dumbleton]] of the 14th-century [[Oxford Calculators]], was one of the first to express functional relationships in graphical form. He gave a proof of the [[mean speed theorem]] stating that "the latitude of a uniformly difform movement corresponds to the degree of the midpoint" and used this method to study the quantitative decrease in intensity of illumination in his ''Summa logicæ et philosophiæ naturalis'' (ca. 1349), stating that it was not linearly proportional to the distance, but was unable to expose the Inverse-square law.<ref>{{Cite book |last=Freely |first=John |author-link=John Freely |title=Before Galileo: The Birth of Modern Science in Medieval Europe |date=2012 |publisher=Overlook Duckworth |isbn=978-0-71-564536-9}}</ref>
[[File:Kepler_1910.jpg|alt=Kepler 1910|thumb|199x199px|German astronomer [[Johannes Kepler]] discussed the inverse-square law and how it affects the intensity of light.]]
In proposition 9 of Book 1 in his book ''Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur'' (1604), the astronomer [[Johannes Kepler]] argued that the spreading of light from a point source obeys an inverse square law:<ref>{{Cite book |last=Kepler |first=Johannes |author-link=Johannes Kepler |url=https://books.google.com/books?id=-OB9O7FowP4C |title=Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur |date=1604 |publisher=Apud Claudium Marnium & haeredes Ioannis Aubrii |page=10 |language=la}}</ref><ref>Translation of the Latin quote from Kepler's ''Ad Vitellionem paralipomena'' is from: {{Cite journal |last1=Gal |first1=Ofer |last2=Chen-Morris |first2=Raz |date=2005 |title=The Archaeology of the Inverse Square Law: (1) Metaphysical Images and Mathematical Practices |journal=History of Science |volume=43 |issue=4 |pages=391–414 |doi=10.1177/007327530504300402 |bibcode=2005HisSc..43..391G |bibcode-access=free}} ; see especially p. 397.</ref>

{{verse translation|lang=la|
Sicut se habent spharicae superificies, quibus origo lucis pro centro est, amplior ad angustiorem:  ita se habet fortitudo seu densitas lucis radiorum in angustiori, ad illamin in laxiori sphaerica, hoc est, conversim.  Nam per 6. 7. tantundem lucis est in angustiori sphaerica superficie, quantum in fusiore, tanto ergo illie stipatior & densior quam hic.
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Just as [the ratio of] spherical surfaces, for which the source of light is the center, [is] from the wider to the narrower, so the density or fortitude of the rays of light in the narrower [space], towards the more spacious spherical surfaces, that is, inversely.  For according to [propositions] 6 & 7, there is as much light in the narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there.
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In 1645, in his book ''Astronomia Philolaica'' ..., the French astronomer [[Ismaël Bullialdus]] (1605–1694) refuted Johannes Kepler's suggestion that "gravity"<ref>Note:  Both Kepler and William Gilbert had nearly anticipated the modern conception of gravity, lacking only the inverse-square law in their description of "gravitas". On page 4 of chapter 1, Introductio, of ''Astronomia Nova'', Kepler sets out his description as follows:

''"The true theory of gravity is founded on the following axioms:''

''Every corporeal substance, so far forth as it is corporeal, has a natural fitness for resting in every place where it may be situated by itself beyond the sphere of influence of a body cognate with it.''

'''''Gravity is a mutual affection between cognate bodies towards union or conjunction (similar in kind to the magnetic virtue), so that the earth attracts a stone much rather than the stone seeks the earth.'''''

...

''If two stones were placed in any part of the world near each other, and beyond the sphere of influence of a third cognate body, these stones, like two magnetic needles, would come together in the intermediate point, '''each approaching the other by a space proportional to the comparative mass of the other'''.''

''If the moon and earth were not retained in their orbits by their animate force or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance, and the moon fall towards the earth through the other fifty-three parts, and they would there meet, assuming, however, that the substance of both is of the same density."''

Notice that in saying "''the earth attracts a stone much rather than the stone seeks the earth"'' Kepler is breaking away from the Aristotelian tradition that objects ''seek'' to be in their natural place, that a stone ''seeks'' to be with the earth.</ref> weakens as the inverse of the distance; instead, Bullialdus argued, "gravity" weakens as the inverse square of the distance:<ref>{{Cite book |last=Boulliau |first=Ismael |author-link=Ismaël Bullialdus |url=https://books.google.com/books?id=BkZZAAAAcAAJ |title=Astronomia Philolaica |date=1645 |publisher=Simeon Piget |page=23 |language=la |bibcode=1645ibap.book.....B |doi=10.3931/e-rara-549 |bibcode-access=free}}</ref><ref>Translation of the Latin quote from Bullialdus' 'Astronomia Philolaica' … is from: {{Cite web |last1=O'Connor |first1=John J. |last2=Robertson |first2=Edmund F. |date=2006 |title=Ismael Boulliau |url=https://mathshistory.st-andrews.ac.uk/Biographies/Boulliau/ |website=The MacTutor History of Mathematics Archive |publisher=[[University of Saint Andrews]]}}</ref>

{{verse translation|lang=la|
Virtus autem illa, qua Sol prehendit seu harpagat planetas, corporalis quae ipsi pro manibus est, lineis rectis in omnem mundi amplitudinem emissa quasi species solis cum illius corpore rotatur:  cum ergo sit corporalis imminuitur, & extenuatur in maiori spatio & intervallo, ratio autem huius imminutionis eadem est, ac luminus, in ratione nempe dupla intervallorum, sed eversa.
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As for the power by which the Sun seizes or holds the planets, and which, being corporeal, functions in the manner of hands, it is emitted in straight lines throughout the whole extent of the world, and like the species of the Sun, it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances [that is, 1/d²].
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In England, the Anglican bishop [[Seth Ward (bishop of Salisbury)|Seth Ward]] (1617–1689) publicized the ideas of Bullialdus in his critique ''In Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis'' (1653) and publicized the planetary astronomy of Kepler in his book ''Astronomia geometrica'' (1656).

In 1663–1664, the English scientist [[Robert Hooke]] was writing his book ''Micrographia'' (1666) in which he discussed, among other things, the relation between the height of the atmosphere and the barometric pressure at the surface.  Since the atmosphere surrounds the Earth, which itself is a sphere, the volume of atmosphere bearing on any unit area of the Earth's surface is a truncated cone (which extends from the Earth's center to the vacuum of space; obviously only the section of the cone from the Earth's surface to space bears on the Earth's surface).  Although the volume of a cone is proportional to the cube of its height, Hooke argued that the air's pressure at the Earth's surface is instead proportional to the height of the atmosphere because gravity diminishes with altitude.  Although Hooke did not explicitly state so, the relation that he proposed would be true only if gravity decreases as the inverse square of the distance from the Earth's center.<ref>{{harvnb|Gal|Chen-Morris|2005|pp=391–392}}</ref><ref>{{Cite book |last=Hooke |first=Robert |author-link= |url=https://books.google.com/books?id=LsbBada4VVYC |title=Micrographia |date= |publisher=Science Heritage |isbn=978-0-940095-07-6 |pages=227 |quote=I say a ''Cylinder'', not a piece of a ''Cone'', because, as I may elsewhere shew in the Explication of Gravity, that ''triplicate'' proportion of the shels of a Sphere, to their respective diameters, I suppose to be removed in this case by the decrease of the power of Gravity.}}</ref>