1352:(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.
22:
1251:
436:. Otherwise, if we want to calculate the attraction between massive bodies, we need to add all the point-point attraction forces vectorially and the net attraction might not be exact inverse square. However, if the separation between the massive bodies is much larger compared to their sizes, then to a good approximation, it is reasonable to treat the masses as a point mass located at the object's
1443:
674:, the effective origin is located far behind the beam aperture. If you are close to the origin, you don't have to go far to double the radius, so the signal drops quickly. When you are far from the origin and still have a strong signal, like with a laser, you have to travel very far to double the radius and reduce the signal. This means you have a stronger signal or have
693:, the inverse-square law is used to determine the “fall off” or the difference in illumination on a subject as it moves closer to or further from the light source. For quick approximations, it is enough to remember that doubling the distance reduces illumination to one quarter; or similarly, to halve the illumination increase the distance by a factor of 1.4 (the
1212:, in his 2020 paper "Non-Euclidean Newtonian Cosmology," elaborates on the behavior of force (F) and potential (Φ) within hyperbolic 3-space (H3). He illustrates that F and Φ obey the formulas F ∝ 1 / R^2 sinh^2(r/R) and Φ ∝ coth(r/R), where R and r represent the curvature radius and the distance from the focal point, respectively.
697:), and to double illumination, reduce the distance to 0.7 (square root of 1/2). When the illuminant is not a point source, the inverse square rule is often still a useful approximation; when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1%.
1312:
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
1215:
The concept of the dimensionality of space, first proposed by
Immanuel Kant, is an ongoing topic of debate in relation to the inverse-square law. Dimitria Electra Gatzia and Rex D. Ramsier, in their 2021 paper, argue that the inverse-square law pertains more to the symmetry in force distribution than
467:
both expounded gravitation in 1666 as an attractive force. Hooke's lecture "On gravity" was at the Royal
Society, in London, on 21 March. Borelli's "Theory of the Planets" was published later in 1666. Hooke's 1670 Gresham lecture explained that gravitation applied to "all celestiall bodys" and added
1478:
Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses: See page 239 in Curtis Wilson (1989), "The
Newtonian achievement in astronomy", ch.13 (pages 233–274) in "Planetary astronomy from the Renaissance to the rise of astrophysics: 2A:
1320:
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,
1278:
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
1896:
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,
1286:
Just as spherical surfaces, for which the source of light is the center, from the wider to the narrower, so the density or fortitude of the rays of light in the narrower , towards the more spacious spherical surfaces, that is, inversely. For according to 6 & 7, there is as much light in the
402:
gets farther from the source, it is spread out over an area that is increasing in proportion to the square of the distance from the source. Hence, the intensity of radiation passing through any unit area (directly facing the point source) is inversely proportional to the square of the distance from
33:
depends on the strength of the light source and is constant with increasing distance, where a greater density of flux lines (lines per unit area) means a stronger energy field. The density of flux lines is inversely proportional to the square of the distance from the source because the surface area
1602:
In a letter to Edmund Halley dated 20 June 1686, Newton wrote: "Bullialdus wrote that all force respecting ye Sun as its center & depending on matter must be reciprocally in a duplicate ratio of ye distance from ye center." See: I. Bernard Cohen and George E. Smith, ed.s,
248:
334:
704:(Φ) for indirectly ionizing radiation with increasing distance from a point source can be calculated using the inverse-square law. Since emissions from a point source have radial directions, they intercept at a perpendicular incidence. The area of such a shell is 4π
838:
would decrease by 6.02 dB per doubling of distance. When referring to measurements of power quantities, a ratio can be expressed as a level in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the reference value.
1219:
Within the realm of non-Euclidean geometries and general relativity, deviations from the inverse-square law might not stem from the law itself but rather from the assumption that the force between bodies depends instantaneously on distance, contradicting
499:
The force of attraction or repulsion between two electrically charged particles, in addition to being directly proportional to the product of the electric charges, is inversely proportional to the square of the distance between them; this is known as
427:
The gravitational attraction force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of their separation distance. The force is always attractive and acts along the line joining
468:
the principles that the gravitating power decreases with distance and that in the absence of any such power bodies move in straight lines. By 1679, Hooke thought gravitation had inverse square dependence and communicated this in a letter to
740:). At large distances from the source (compared to the size of the source), this power is distributed over larger and larger spherical surfaces as the distance from the source increases. Since the surface area of a sphere of radius
160:
167:
1888:
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,
1321:
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 .
255:
1869:
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
602:(energy per unit of area perpendicular to the source) is inversely proportional to the square of the distance from the source, so an object (of the same size) twice as far away receives only one-quarter the
1883:
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.
826:
1879:
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.
1092:
870:, the decrease is still 6.02 dB, since dB represents an intensity ratio. The pressure ratio (as opposed to power ratio) is not inverse-square, but is inverse-proportional (inverse distance law):
573:
459:'s solution for circular motion (motion in a straight line pulled aside by the central force). Indeed, Bullialdus maintained the sun's force was attractive at aphelion and repulsive at perihelion.
1447:
1243:
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
1970:
1304:(1605–1694) refuted Johannes Kepler's suggestion that "gravity" weakens as the inverse of the distance; instead, Bullialdus argued, "gravity" weakens as the inverse square of the distance:
1170:
1003:
910:
1834:
724:
of heat “as the point source is magnification by distances, its radiation is dilute proportional to the sin of the angle, of the increasing circumference arc from the point of origin”.
66:
from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.
346:
which is the resultant of radial inverse-square law fields with respect to one or more sources is proportional to the strength of the local sources, and hence zero outside sources.
1585:
Newton acknowledged Wren, Hooke and Halley in this connection in the
Scholium to Proposition 4 in Book 1 (in all editions): See for example the 1729 English translation of the
107:
In mathematical notation the inverse square law can be expressed as an intensity (I) varying as a function of distance (d) from some centre. The intensity is proportional (see
485:
acknowledged that Hooke, along with Wren and Halley, had separately appreciated the inverse square law in the solar system, as well as giving some credit to
Bullialdus.
114:
1015:
of the particle velocity that is 90° out of phase with the sound pressure and does not contribute to the time-averaged energy or the intensity of the sound. The
961:
936:
720:
treatment planning, though this proportionality does not hold in practical situations unless source dimensions are much smaller than the distance. As stated in
1224:. General relativity instead interprets gravity as a distortion of spacetime, causing freely falling particles to traverse geodesics in this curved spacetime.
432:
If the distribution of matter in each body is spherically symmetric, then the objects can be treated as point masses without approximation, as shown in the
34:
of a sphere increases with the square of the radius. Thus the field intensity is inversely proportional to the square of the distance from the source.
1397:
766:
1940:
1027:
component of the RMS particle velocity, both of which are inverse-proportional. Accordingly, the intensity follows an inverse-square behaviour:
1392:
1032:
509:
481:
1621:
Williams, E.; Faller, J.; Hill, H. (1971), "New
Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass",
243:{\displaystyle {\frac {{\text{intensity}}_{1}}{{\text{intensity}}_{2}}}={\frac {{\text{distance}}_{2}^{2}}{{\text{distance}}_{1}^{2}}}}
1936:
Translation of the Latin quote from
Bullialdus' 'Astronomia Philolaica' … is from: O'Connor, John J. and Roberson, Edmund F. (2006)
55:
1126:
968:
875:
347:
407:
is similarly applicable, and can be used with any physical quantity that acts in accordance with the inverse-square relationship.
1452:
329:{\displaystyle {\text{intensity}}_{1}\times {\text{distance}}_{1}^{2}={\text{intensity}}_{2}\times {\text{distance}}_{2}^{2}}
1247:(ca. 1349), stating that it was not linearly proportional to the distance, but was unable to expose the Inverse-square law.
659:(1 AU)—an approximate threefold increase in distance results in an approximate ninefold decrease in intensity of radiation.
1947:, The MacTutor History of Mathematics Archive, School of Mathematics and Statistics, University of Saint Andrews, Scotland.
1376:
452:
939:
1109:
is zero outside the source. This can be generalized to higher dimensions. Generally, for an irrotational vector field in
1674:
474:
my supposition is that the attraction always is in duplicate proportion to the distance from the center reciprocall
1458:
1330:
1850:
2006:
1404:
76:
return, so the inverse square for both paths means that the radar will receive energy according to the inverse
1287:
narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there.
444:
628:
varies inversely with the square of the distance from the source (assuming there are no losses caused by
2011:
479:
Hooke remained bitter about Newton claiming the invention of this principle, even though Newton's 1686
404:
1925:
1937:
1239:, was one of the first to express functional relationships in graphical form. He gave a proof of the
1102:
1755:
1194:. The inherent curvature in these spaces impacts physical laws, underpinning various fields such as
1729:
464:
1623:
1429:
383:
91:, which acts like a canal does for water, or how a gun barrel restricts hot gas expansion to one
1851:"The Archaeology of the Inverse Square Law: (1) Metaphysical Images and Mathematical Practices,"
1409:
1187:
1117:, the intensity "I" of the vector field falls off with the distance "r" following the inverse (
25:
S represents the light source, while r represents the measured points. The lines represent the
1608:
1590:
1493:
1506:
1713:
1632:
73:
1991:
1301:
448:
8:
1424:
1191:
945:
920:
753:
629:
583:
1717:
1636:
1105:
in three-dimensional space, the inverse-square law corresponds to the property that the
736: be the total power radiated from a point source (for example, an omnidirectional
712:
is the radial distance from the center. The law is particularly important in diagnostic
1986:
1891:
each approaching the other by a space proportional to the comparative mass of the other
1703:
1559:
1543:
1386:
1240:
1236:
1221:
1199:
1183:
1008:
737:
679:
663:
456:
1792:
1521:
1313:
imminutionis eadem est, ac luminus, in ratione nempe dupla intervallorum, sed eversa.
1670:
1551:
915:
667:
652:
51:
1563:
1270:
argued that the spreading of light from a point source obeys an inverse square law:
678:
in the direction of the narrow beam relative to a wide beam in all directions of an
1767:
1721:
1640:
1535:
1462:
1020:
1012:
694:
648:
618:
501:
59:
155:{\displaystyle {\text{intensity}}\ \propto \ {\frac {1}{{\text{distance}}^{2}}}\,}
1944:
1511:
of the
Medicean planets deduced from physical causes] (Florence, (Italy): 1666)
1366:
1267:
1255:
1114:
1016:
614:
1725:
1644:
1526:
1414:
1381:
1371:
1232:
1209:
852:
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690:
494:
437:
47:
2000:
1419:
1203:
433:
1771:
1555:
1345:
1258:
discussed the inverse-square law and how it affects the intensity of light.
717:
675:
599:
469:
460:
399:
387:
379:
343:
77:
1335:
In
Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis
713:
686:
587:
351:
1833:(Frankfurt, (Germany): Claude de Marne & heir Jean Aubry, 1604),
1106:
633:
591:
423:
is the attraction between objects that have mass. Newton's law states:
339:
1901:
the earth attracts a stone much rather than the stone seeks the earth"
1547:
830:
The energy or intensity decreases (divided by 4) as the distance
1903:
Kepler is breaking away from the Aristotelian tradition that objects
1195:
859:
848:
625:
374:
The inverse-square law generally applies when some force, energy, or
363:
92:
88:
30:
21:
1831:
Ad Vitellionem Paralipomena, quibus astronomiae pars optica traditur
1337:(1653) and publicized the planetary astronomy of Kepler in his book
1264:
Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur
504:. The deviation of the exponent from 2 is less than one part in 10.
1708:
1691:
1539:
63:
87:
while propagating a signal, certain methods can be used such as a
1524:(1952). "An Unpublished Letter of Robert Hooke to Isaac Newton".
867:
835:
701:
420:
39:
1876:"The true theory of gravity is founded on the following axioms:
1333:(1617–1689) publicized the ideas of Bullialdus in his critique
862:
radiating from a point source decreases by 50% as the distance
856:
622:
603:
391:
96:
84:
1818:
Before Galileo: The Birth of Modern Science in Medieval Europe
1250:
1174:
given that the space outside the source is divergence free.
671:
656:
595:
359:
355:
108:
69:
29:
emanating from the sources and fluxes. The total number of
1576:
Hooke's letter to Newton of 6 January 1680 (Koyré 1952:332).
1279:
fusiore, tanto ergo illie stipatior & densior quam hic.
577:
1508:
Theoricae Mediceorum Planetarum ex Causis Physicis Deductae
1361:
821:{\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.\,}
655:); but only 1367 watts per square meter at the distance of
644:
375:
72:
energy expands during both the signal transmission and the
26:
1607:(Cambridge, England: Cambridge University Press, 2002),
1087:{\displaystyle I\ =\ pv\ \propto \ {\frac {1}{r^{2}}}.\,}
640:
568:{\displaystyle F=k_{\text{e}}{\frac {q_{1}q_{2}}{r^{2}}}}
111:) to the reciprocal of the square of the distance thus:
398:) is proportional to the square of the radius, as the
1756:"Dimensionality, symmetry and the Inverse Square Law"
1129:
1035:
971:
948:
923:
878:
769:
512:
258:
170:
117:
50:
stating that the observed "intensity" of a specified
1749:
1747:
350:
follows an inverse-square law, as do the effects of
1897:
that the substance of both is of the same density."
1793:"Introduction to Non-Euclidean General Relativity"
1754:Gatzia, Dimitria Electra; Ramsier, Rex D. (2021).
1620:
1164:
1086:
997:
955:
930:
904:
820:
567:
328:
242:
164:It can also be mathematically expressed as :
154:
1992:Sound pressure p and the inverse distance law 1/r
1744:
1669:Ryer,A. (1997) “The Light Measurement Handbook”,
639:For example, the intensity of radiation from the
252:or as the formulation of a constant quantity:
95:in order to prevent loss of energy transfer to a
1998:
1520:
1860: : 391–414 ; see especially p. 397.
759:(power per unit area) of radiation at distance
1874:, Kepler sets out his description as follows:
1849:is from: Gal, O. & Chen-Morris, R.(2005)
1658:Lighting for Film and Television – 3rd Edition
1165:{\displaystyle I\propto {\frac {1}{r^{n-1}}},}
1096:
1845:Translation of the Latin quote from Kepler's
1753:
1685:
1683:
1177:
998:{\displaystyle v\ \propto {\frac {1}{r}}\ \,}
905:{\displaystyle p\ \propto \ {\frac {1}{r}}\,}
700:The fractional reduction in electromagnetic
1956:(Gal & Chen-Morris, 2005), pp. 391–392.
1907:to be in their natural place, that a stone
440:while calculating the gravitational force.
1680:
1490:The History of the Royal Society of London
1969:… (London, England: John Martyn, 1667),
1707:
1083:
994:
952:
927:
901:
817:
578:Light and other electromagnetic radiation
151:
1514:
1249:
20:
1262:In proposition 9 of Book 1 in his book
1182:The inverse-square law, fundamental in
598:or other linear waves radiating from a
1999:
1689:
1492:, … (London, England: 1756), vol. 2,
942:with the instantaneous sound pressure
914:The same is true for the component of
1245:Summa logicæ et philosophiæ naturalis
348:Newton's law of universal gravitation
1987:Damping of sound level with distance
1344:In 1663–1664, the English scientist
1306:
1272:
647:per square meter at the distance of
1692:"Non-Euclidean Newtonian Cosmology"
13:
617:per unit area in the direction of
378:is evenly radiated outward from a
14:
2023:
1980:
1924:… (Paris, France: Piget, 1645),
1605:The Cambridge Companion to Newton
1479:Tycho Brahe to Newton", CUP 1989.
1393:William Thomson, 1st Baron Kelvin
1377:Kepler's laws of planetary motion
842:
488:
1446: This article incorporates
1441:
1329:In England, the Anglican bishop
1216:to the dimensionality of space.
609:More generally, the irradiance,
451:. But Bullialdus did not accept
443:As the law of gravitation, this
369:
1959:
1950:
1930:
1914:
1863:
1839:
1823:
1810:
1785:
1663:
1459:General Services Administration
1650:
1614:
1596:
1579:
1570:
1499:
1482:
1472:
453:Kepler's second and third laws
415:
410:
1:
1696:Classical and Quantum Gravity
1496:; see especially pages 70–72.
1435:
1398:Power-aware routing protocols
7:
1847:Ad Vitellionem paralipomena
1355:
1300:..., the French astronomer
1097:Field theory interpretation
834:is doubled; if measured in
606:(in the same time period).
10:
2028:
1798:. MIT OpenCourseWare. 2018
1645:10.1103/PhysRevLett.26.721
1505:Giovanni Alfonso Borelli,
1227:
1178:Non-Euclidean implications
1121: − 1) power law
727:
492:
102:
1103:irrotational vector field
447:was suggested in 1645 by
1943:30 November 2016 at the
1726:10.1088/1361-6382/ab8437
1690:Barrow, John D. (2020).
1310:
1276:
1188:non-Euclidean geometries
1186:spaces, also applies to
866:is doubled; measured in
465:Giovanni Alfonso Borelli
455:, nor did he appreciate
376:other conserved quantity
1899:Notice that in saying "
1624:Physical Review Letters
1430:Principle of similitude
1405:Inverse proportionality
1266:(1604), the astronomer
1023:sound pressure and the
405:Gauss's law for gravity
384:three-dimensional space
83:To prevent dilution of
1772:10.1098/rsnr.2019.0044
1454:Federal Standard 1037C
1448:public domain material
1410:Multiplicative inverse
1318:
1284:
1259:
1166:
1088:
1019:is the product of the
999:
957:
932:
906:
822:
569:
430:
330:
244:
156:
56:inversely proportional
35:
2007:Philosophy of physics
1922:Astronomia Philolaica
1911:to be with the earth.
1348:was writing his book
1339:Astronomia geometrica
1298:Astronomia Philolaica
1296:In 1645, in his book
1253:
1167:
1089:
1000:
958:
933:
907:
823:
570:
425:
331:
245:
157:
24:
1656:Millerson,G. (1991)
1235:of the 14th-century
1127:
1033:
1013:quadrature component
969:
946:
921:
876:
767:
510:
256:
168:
115:
1920:Ismail Bullialdus,
1732:on 25 February 2020
1718:2020CQGra..37l5007B
1637:1971PhRvL..26..721W
1465:on 22 January 2022.
956:{\displaystyle p\,}
931:{\displaystyle v\,}
664:isotropic radiators
325:
290:
237:
220:
1854:History of Science
1387:Telecommunications
1260:
1254:German astronomer
1241:mean speed theorem
1237:Oxford Calculators
1222:special relativity
1200:general relativity
1162:
1084:
995:
953:
928:
902:
818:
738:isotropic radiator
670:, headlights, and
668:parabolic antennas
613:the intensity (or
565:
457:Christiaan Huygens
403:the point source.
326:
309:
274:
240:
221:
204:
152:
44:inverse-square law
36:
2012:Scientific method
1938:"Ismael Boulliau"
1829:Johannes Kepler,
1760:Notes and Records
1327:
1326:
1302:Ismaël Bullialdus
1293:
1292:
1157:
1078:
1062:
1056:
1047:
1041:
993:
989:
977:
916:particle velocity
899:
890:
884:
812:
784:
680:isotropic antenna
563:
526:
449:Ismaël Bullialdus
400:emitted radiation
394:(which is 4π
313:
298:
278:
263:
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225:
208:
197:
189:
177:
149:
141:
131:
125:
121:
52:physical quantity
2019:
1974:
1963:
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1934:
1928:
1918:
1912:
1867:
1861:
1843:
1837:
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1808:
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1803:
1797:
1789:
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1742:
1741:
1739:
1737:
1728:. Archived from
1711:
1687:
1678:
1667:
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1654:
1648:
1647:
1618:
1612:
1600:
1594:
1583:
1577:
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1568:
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1522:Koyré, Alexandre
1518:
1512:
1503:
1497:
1486:
1480:
1476:
1467:
1466:
1461:. Archived from
1445:
1444:
1389:, particularly:
1307:
1273:
1192:hyperbolic space
1171:
1169:
1168:
1163:
1158:
1156:
1155:
1137:
1093:
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1079:
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695:square root of 2
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566:
564:
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561:
552:
551:
550:
541:
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530:
528:
527:
524:
335:
333:
332:
327:
324:
319:
314:
311:
305:
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289:
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119:
2027:
2026:
2022:
2021:
2020:
2018:
2017:
2016:
1997:
1996:
1983:
1978:
1977:
1964:
1960:
1955:
1951:
1945:Wayback Machine
1935:
1931:
1919:
1915:
1872:Astronomia Nova
1868:
1864:
1844:
1840:
1828:
1824:
1815:
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1774:
1752:
1745:
1735:
1733:
1688:
1681:
1668:
1664:
1655:
1651:
1631:(12): 721–724,
1619:
1615:
1601:
1597:
1584:
1580:
1575:
1571:
1519:
1515:
1504:
1500:
1487:
1483:
1477:
1473:
1451:
1442:
1440:
1438:
1425:Square–cube law
1367:Antenna (radio)
1358:
1323:
1315:
1295:
1289:
1281:
1268:Johannes Kepler
1256:Johannes Kepler
1230:
1180:
1145:
1141:
1136:
1128:
1125:
1124:
1115:Euclidean space
1099:
1072:
1068:
1063:
1034:
1031:
1030:
1017:sound intensity
981:
970:
967:
966:
947:
944:
943:
922:
919:
918:
891:
877:
874:
873:
845:
805:
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794:
789:
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580:
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511:
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418:
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143:
138:
137:
132:
118:
116:
113:
112:
105:
17:
12:
11:
5:
2025:
2015:
2014:
2009:
1995:
1994:
1989:
1982:
1981:External links
1979:
1976:
1975:
1965:Robert Hooke,
1958:
1949:
1929:
1913:
1862:
1838:
1822:
1809:
1784:
1766:(2): 333–347.
1743:
1702:(12): 125007.
1679:
1662:
1649:
1613:
1595:
1578:
1569:
1540:10.1086/348155
1534:(4): 312–337.
1513:
1498:
1488:Thomas Birch,
1481:
1470:
1469:
1437:
1434:
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1427:
1422:
1417:
1415:Distance decay
1412:
1407:
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1395:
1384:
1382:Kepler problem
1379:
1374:
1369:
1364:
1357:
1354:
1325:
1324:
1316:
1291:
1290:
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1233:John Dumbleton
1229:
1226:
1210:John D. Barrow
1179:
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1038:
988:
985:
980:
974:
951:
926:
898:
895:
887:
881:
853:sound pressure
844:
843:Sound in a gas
841:
816:
808:
804:
800:
797:
793:
788:
783:
780:
775:
772:
748: = 4
729:
726:
722:Fourier theory
691:stage lighting
579:
576:
560:
556:
549:
545:
539:
535:
522:
518:
515:
495:Electrostatics
493:Main article:
490:
489:Electrostatics
487:
438:center of mass
417:
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371:
368:
323:
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128:
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101:
80:of the range.
48:scientific law
15:
9:
6:
4:
3:
2:
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2013:
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1816:John Freely,
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1420:Fermi paradox
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1234:
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1204:string theory
1201:
1197:
1193:
1189:
1185:
1175:
1172:
1159:
1152:
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1122:
1120:
1116:
1113:-dimensional
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1005:
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802:
798:
795:
791:
786:
781:
778:
773:
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762:
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739:
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683:
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631:
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612:
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589:
585:
575:
558:
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547:
543:
537:
533:
520:
516:
513:
505:
503:
502:Coulomb's law
496:
486:
484:
483:
477:
475:
471:
466:
462:
458:
454:
450:
446:
441:
439:
435:
434:shell theorem
429:
424:
422:
408:
406:
401:
397:
393:
389:
386:. Since the
385:
381:
377:
370:Justification
367:
365:
361:
357:
353:
349:
345:
341:
336:
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316:
306:
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286:
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41:
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23:
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1967:Micrographia
1966:
1961:
1952:
1932:
1921:
1916:
1908:
1904:
1900:
1895:
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1887:
1882:
1881:
1878:
1875:
1871:
1865:
1857:
1853:
1846:
1841:
1830:
1825:
1817:
1812:
1800:. Retrieved
1787:
1775:. Retrieved
1763:
1759:
1734:. Retrieved
1730:the original
1699:
1695:
1665:
1657:
1652:
1628:
1622:
1616:
1604:
1598:
1586:
1581:
1572:
1531:
1525:
1516:
1507:
1501:
1489:
1484:
1474:
1463:the original
1453:
1439:
1350:Micrographia
1349:
1346:Robert Hooke
1343:
1338:
1334:
1328:
1319:
1311:
1297:
1294:
1285:
1277:
1263:
1261:
1244:
1231:
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1214:
1208:
1190:, including
1181:
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1100:
1029:
1024:
1006:
965:
913:
872:
863:
846:
831:
829:
760:
756:
749:
745:
741:
733:
731:
718:radiotherapy
709:
705:
699:
684:
676:antenna gain
661:
638:
610:
608:
600:point source
581:
506:
498:
480:
478:
473:
470:Isaac Newton
461:Robert Hooke
442:
431:
426:
419:
395:
388:surface area
380:point source
373:
344:vector field
337:
251:
163:
106:
82:
78:fourth power
68:
43:
37:
18:
16:Physical law
1494:pages 68–73
1372:Gauss's law
714:radiography
687:photography
619:propagation
588:illuminance
421:Gravitation
416:Gravitation
411:Occurrences
366:phenomena.
2001:Categories
1709:2002.10155
1591:at page 66
1436:References
1107:divergence
1009:near field
634:scattering
630:absorption
592:irradiance
340:divergence
31:flux lines
1971:page 227:
1587:Principia
1331:Seth Ward
1196:cosmology
1184:Euclidean
1150:−
1134:∝
1058:∝
979:∝
886:∝
860:wavefront
857:spherical
849:acoustics
799:π
754:intensity
626:wavefront
623:spherical
584:intensity
482:Principia
364:radiation
307:×
297:intensity
272:×
262:intensity
188:intensity
176:intensity
127:∝
120:intensity
93:dimension
89:waveguide
74:reflected
1941:Archived
1926:page 23.
1835:page 10.
1609:page 204
1564:41626961
1556:13010921
1356:See also
1341:(1656).
1025:in-phase
940:in-phase
938:that is
666:such as
662:For non-
643:is 9126
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352:electric
312:distance
277:distance
224:distance
207:distance
140:distance
64:distance
1802:30 July
1777:30 July
1736:30 July
1714:Bibcode
1633:Bibcode
1228:History
1101:For an
1007:In the
728:Example
702:fluence
651:(0.387
649:Mercury
103:Formula
62:of the
58:to the
46:is any
40:science
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604:energy
392:sphere
362:, and
130:
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85:energy
60:square
1909:seeks
1796:(PDF)
1704:arXiv
1560:S2CID
1544:JSTOR
1450:from
1011:is a
855:of a
657:Earth
645:watts
615:power
611:i.e.,
596:light
594:) of
428:them.
390:of a
360:sound
356:light
342:of a
70:Radar
42:, an
1905:seek
1886:...
1804:2023
1779:2023
1738:2023
1677:p.26
1671:ISBN
1660:p.27
1552:PMID
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1362:Flux
732:Let
716:and
689:and
586:(or
582:The
463:and
338:The
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1768:doi
1722:doi
1641:doi
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1021:RMS
847:In
763:is
744:is
685:In
641:Sun
636:).
632:or
590:or
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445:law
382:in
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1973:""
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1746:^
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836:dB
750:πr
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653:AU
476:.
358:,
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1111:n
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807:2
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771:I
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521:k
517:=
514:F
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292:=
287:2
282:1
267:1
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200:=
193:2
181:1
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109:∝
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