History of gravitational theory

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In physics, theories of gravitation postulate mechanisms of interaction governing the movements of bodies with mass. There have been numerous theories of gravitation since ancient times. The first extant sources discussing such theories are found in ancient Greek philosophy. This work was furthered by ancient Indian and medieval Islamic physicists, before gaining great strides during the Renaissance and Scientific Revolution, culminating in the formulation of Newton's law of gravity. This was superseded by Albert Einstein's theory of relativity in the early 20th century.

Greek philosopher Aristotle (fl. 4th century BC) believed that objects tend toward a point due to their inner gravitas (heaviness). Vitruvius (fl. 1st century BC) understood that objects fall based on their specific gravity. In the 6th century CE, Byzantine Alexandrian scholar John Philoponus modified the Aristotelian concept of gravity with the theory of impetus. In the 7th century, Indian astronomer Brahmagupta spoke of gravity as an attractive force. In the 14th century, and influenced by certain Islamic scholars, European philosophers Jean Buridan and Albert of Saxony linked impetus to the acceleration and mass of objects. Albert also developed a law of proportion regarding the relationship between the speed of an object in free fall and the time elapsed.

In the early 17th century, Galileo Galilei found that all objects tend to accelerate equally in free fall. In 1632, he put forth the basic principle of relativity. The existence of the gravitational constant was explored by various researchers from the mid-17th century, helping Isaac Newton formulate his law of universal gravitation. Newton's classical mechanics were superseded in the early 20th century, when Einstein developed the special and general theory of relativity. The force carrier of gravity remains an outlier in the search for a theory of everything, which various models of quantum gravity are candidates for.

Antiquity[edit]

Greco-Roman world[edit]

The Ionian Greek philosopher Heraclitus (c. 535 – c. 475 BC) used the word logos ('word') to describe a kind of law which keeps the cosmos in harmony, moving all objects, including the stars, winds, and waves.[1]

In the 4th century BC, the Greek philosopher Aristotle taught that there is no effect or motion without a cause. The cause of the downward motion of heavy bodies, such as the element earth, was related to their nature, which caused them to move downward toward the centre of the universe, which was their natural place. Conversely, light bodies such as the element fire, move by their nature upward toward the inner surface of the sphere of the Moon. Thus in Aristotle's system heavy bodies are not attracted to the Earth by an external force, but tend toward the centre of the universe because of an inner gravitas or heaviness.[2][3]

The 3rd-century-BC Greek physicist Archimedes discovered the centre of mass of a triangle.[4] He also postulated that if the centres of gravity of two equal weights was not the same, it would be located in the middle of the line that joins them.[5] Two centuries later, the Roman engineer and architect Vitruvius contended in his De architectura that gravity is not dependent on a substance's weight but rather on its 'nature' (cf. specific gravity):

If the quicksilver is poured into a vessel, and a stone weighing one hundred pounds is laid upon it, the stone swims on the surface, and cannot depress the liquid, nor break through, nor separate it. If we remove the hundred pound weight, and put on a scruple of gold, it will not swim, but will sink to the bottom of its own accord. Hence, it is undeniable that the gravity of a substance depends not on the amount of its weight, but on its nature.[6]

In the 6th century CE, the Byzantine Alexandrian scholar John Philoponus proposed the theory of impetus, which modifies Aristotle's theory that "continuation of motion depends on continued action of a force" by incorporating a causative force which diminishes over time.

Indian subcontinent[edit]

The Shatapatha Brahmana, a Hindu text written by the Vedic Sage Yajnavalkya, states that the "sun strings to himself ... worlds on a thread".[7] The prose is dated to 300 BCE, but some elements stem from earlier sources,[8] perhaps from the 10th–6th centuries BCE.[9]

The Indian mathematician/astronomer Brahmagupta (c. 598 – c. 668 CE) first described gravity as an attractive force, using the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" to describe it within a heliocentric view of the Solar System as had been defined by Aryabhata:[10][11][12][13]

The earth on all its sides is the same; all people on the earth stand upright, and all heavy things fall down to the earth by a law of nature, for it is the nature of the earth to attract and to keep things, as it is the nature of water to flow ... If a thing wants to go deeper down than the earth, let it try. The earth is the only low thing, and seeds always return to it, in whatever direction you may throw them away, and never rise upwards from the earth.[14][15][a]

Islamic world[edit]

In the 11th century CE, Persian polymath Ibn Sina (Avicenna) agreed with Philoponus' theory that "the moved object acquires an inclination from the mover" as an explanation for projectile motion.[16] Ibn Sina then published his own theory of impetus in The Book of Healing (c. 1020). Unlike Philoponus, who believed that it was a temporary virtue that would decline even in a vacuum, Ibn Sina viewed it as a persistent, requiring external forces such as air resistance to dissipate it.[17][18][19] Ibn Sina made distinction between 'force' and 'inclination' (mayl), and argued that an object gained mayl when the object is in opposition to its natural motion. He concluded that continuation of motion is attributed to the inclination that is transferred to the object, and that object will be in motion until the mayl is spent.[20]

Another 11th-century Persian polymath, Al-Biruni, proposed that heavenly bodies have mass, weight, and gravity, just like the Earth. He criticized both Aristotle and Ibn Sina for holding the view that only the Earth has these properties.[21] The 12th-century scholar Al-Khazini suggested that the gravity an object contains varies depending on its distance from the centre of the universe (referring to the centre of the Earth). Al-Biruni and Al-Khazini studied the theory of the centre of gravity, and generalized and applied it to three-dimensional bodies. They also founded the theory of ponderable lever, and created the science of gravity. Fine experimental methods were also developed for determining the specific gravity or specific weight of objects, based the theory of balances and weighing.[22]

In the 12th century, Abu'l-Barakāt al-Baghdādī adopted and modified Ibn Sina's theory on projectile motion. In his Kitab al-Mu'tabar, Abu'l-Barakat stated that the mover imparts a violent inclination (mayl qasri) on the moved and that this diminishes as the moving object distances itself from the mover.[23] According to Shlomo Pines, al-Baghdādī's theory of motion was "the oldest negation of Aristotle's fundamental dynamic law [namely, that a constant force produces a uniform motion], [and is thus an] anticipation in a vague fashion of the fundamental law of classical mechanics [namely, that a force applied continuously produces acceleration]."[24]

European Renaissance[edit]

In the 14th century, both the French philosopher Jean Buridan and the Merton College of Oxford rejected the Aristotelian concept of gravity.[25][b] They attributed the motion of objects to an impetus (akin to momentum), which varies according to velocity and mass;[25] Buridan was influenced in this by Ibn Sina's Book of Healing.[19] Buridan and the philosopher Albert of Saxony (c. 1320–1390) adopted Abu'l-Barakat's theory that the acceleration of a falling body is a result of its increasing impetus.[23] Influenced by Buridan, Albert developed a law of proportion regarding the relationship between the speed of an object in free fall and the time elapsed.[26] He also theorized that mountains and valleys are caused by erosion[c]—displacing the Earth's centre of gravity.[27][d] Also in that century, the Merton College developed the mean speed theorem, which was proved by Nicole Oresme (c. 1323–1382) and would be influential in later gravitational equations.[25]

Leonardo da Vinci (1452–1519) wrote that the "mother and origin of gravity" is energy. He describes two pairs of physical powers which stem from a metaphysical origin and have an effect on everything: abundance of force and motion, and gravity and resistance. He associates gravity with the 'cold' classical elements, water and earth, and calls its energy infinite.[29][e] By 1514, Nicolaus Copernicus had written an outline of his heliocentric model, in which he stated that Earth's centre is the centre of both its rotation and the orbit of the Moon.[31][f] In 1533, German humanist Petrus Apianus described the exertion of gravity:[g]

Since it is apparent that in the descent [along the arc] there is more impediment acquired, it is clear that gravity is diminished on this account. But because this comes about by reason of the position of heavy bodies, let it be called a positional gravity [i.e. gravitas secundum situm][34]

By 1544, according to Benedetto Varchi, the experiments of at least two Italians had dispelled the Aristotelian claim that objects fall proportionately to their weight.[36] In 1551, Domingo de Soto suggested that objects in free fall accelerate uniformly.[36] This idea was subsequently explored in more detail by Galileo Galilei, who derived his kinematics from the 14th-century Merton College and Jean Buridan,[25] and possibly De Soto as well.[36] Galileo successfully applied mathematics to the acceleration of falling objects,[37] correctly hypothesizing in a 1604 letter to Paolo Sarpi that the distance of a falling object is proportional to the square of the time elapsed.[38][i] Galileo suggested in his Two New Sciences (1638) that the slight variance of speed of falling objects of different mass was due to air resistance, and that objects would fall completely uniformly in a vacuum.[39]

A disciple of Galileo, Evangelista Torricelli reiterated Aristotle's model involving a gravitational centre, adding his view that a system can only be in equilibrium when the common centre itself is unable to fall.[33]

European Enlightenment[edit]

The relation of the distance of objects in free fall to the square of the time taken was confirmed by Francesco Maria Grimaldi and Giovanni Battista Riccioli between 1640 and 1650. They also made a calculation of the gravitational constant by recording the oscillations of a pendulum.[40]

Mechanical explanations[edit]

In 1644, René Descartes proposed that no empty space can exist and that a continuum of matter causes every motion to be curvilinear. Thus, centrifugal force thrusts relatively light matter away from the central vortices of celestial bodies, lowering density locally and thereby creating centripetal pressure.[41][42] Utilizing aspects of this theory, between 1669 and 1690, Christiaan Huygens designed a mathematical vortex model. In one of his proofs, he shows that the distance elapsed by an object dropped from a spinning wheel will increase proportionally to the square of the wheel's rotation time.[43] In 1671, Robert Hooke speculated that gravitation is the result of bodies emitting waves in the aether.[44][j] Nicolas Fatio de Duillier (1690) and Georges-Louis Le Sage (1748) proposed a corpuscular model using some sort of screening or shadowing mechanism. In 1784, Le Sage posited that gravity could be a result of the collision of atoms, and in the early 19th century, he expanded Daniel Bernoulli's theory of corpuscular pressure to the universe as a whole.[45] A similar model was later created by Hendrik Lorentz (1853–1928), who used electromagnetic radiation instead of corpuscles.

English mathematician Isaac Newton utilized Descartes' argument that curvilinear motion constrains inertia,[46] and in 1675, argued that aether streams attract all bodies to one another.[k] Newton (1717) and Leonhard Euler (1760) proposed a model in which the aether loses density near mass, leading to a net force acting on bodies.[citation needed] Further mechanical explanations of gravitation (including Le Sage's theory) were created between 1650 and 1900 to explain Newton's theory, but mechanistic models eventually fell out of favor because most of them lead to an unacceptable amount of drag (air resistance), which was not observed. Others violate the energy conservation law and are incompatible with modern thermodynamics.[47]

Portrait of Isaac Newton (1642–1727) by Godfrey Kneller (1689)

Newton's law[edit]

In 1679, Robert Hooke wrote to Isaac Newton of his hypothesis concerning orbital motion, which partly depends on an inverse-square force.[48][l] In 1684, both Hooke and Newton told Edmond Halley that they had proven the inverse-square law of planetary motion, in January and August, respectively.[50] While Hooke refused to produce his proofs, Newton was prompted to compose De motu corporum in gyrum ('On the motion of bodies in an orbit'), in which he mathematically derives Kepler's laws of planetary motion.[50] In 1687, with Halley's support (and to Hooke's dismay), Newton published Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), which hypothesizes the inverse-square law of universal gravitation.[50] In his own words:

I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centres about which they revolve; and thereby compared the force requisite to keep the moon in her orb with the force of gravity at the surface of the earth; and found them to answer pretty nearly.

Newton's original formula was:

where the symbol means "is proportional to". To make this into an equal-sided formula or equation, there needed to be a multiplying factor or constant that would give the correct force of gravity no matter the value of the masses or distance between them (the gravitational constant). Newton would need an accurate measure of this constant to prove his inverse-square law. This was first performed by Henry Cavendish in 1797.[m]

In Newton's theory[54] (rewritten using more modern mathematics) the density of mass generates a scalar field, the gravitational potential in joules per kilogram, by

Using the Nabla operator for the gradient and divergence (partial derivatives), this can be conveniently written as:

This scalar field governs the motion of a free-falling particle by:

At distance r from an isolated mass M, the scalar field is

The Principia sold out quickly, inspiring Newton to publish a second edition in 1713.[55][56] The treatise inspired the French philosopher Voltaire to write his own book explaining aspects of it in 1738, which helped to popularize Newton's theory.[57] In 1755, Prussian philosopher Immanuel Kant published a cosmological manuscript based on Newtonian principles, in which he develops the nebular hypothesis.[58] In 1788, Joseph-Louis Lagrange introduced an improved formulation of classical mechanics. Neither version takes relativistic effects into account, as these had not yet been discovered. Even so, Newton's theory is thought to be exceptionally accurate in the limit of weak gravitational fields and low speeds.[n]

Newton's theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted by the actions of the other planets. Calculations by John Couch Adams and Urbain Le Verrier both predicted the general position of the planet. In 1846, Le Verrier sent his position to Johann Gottfried Galle, asking him to verify it. The same night, Galle spotted Neptune near the position Le Verrier had predicted.[60] By the end of the 19th century, Le Verrier showed that the orbit of Mercury could not be accounted for entirely under Newtonian gravity, and all searches for another perturbing body (such as a planet orbiting the Sun even closer than Mercury) were fruitless.[61]

At the end of the 19th century, many tried to combine Newton's force law with the established laws of electrodynamics (like those of Wilhelm Eduard Weber, Carl Friedrich Gauss, and Bernhard Riemann) in order to explain the anomalous perihelion precession of Mercury. In 1890, Maurice Lévy succeeded in doing so by combining the laws of Weber and Riemann, whereby the speed of gravity is equal to the speed of light. In another attempt, Paul Gerber (1898) succeeded in deriving the correct formula for the perihelion shift (which was identical to the formula later used by Albert Einstein). These hypotheses were rejected because of the outdated laws they were based on, being superseded by those of James Clerk Maxwell.[47]

Modern era[edit]

In 1900, Hendrik Lorentz tried to explain gravity on the basis of his ether theory and Maxwell's equations. He assumed, like Ottaviano Fabrizio Mossotti and Johann Karl Friedrich Zöllner, that the attraction of opposite charged particles is stronger than the repulsion of equal charged particles. The resulting net force is exactly what is known as universal gravitation, in which the speed of gravity is that of light. Lorentz calculated that the value for the perihelion advance of Mercury was much too low.[62]

In the late 19th century, Lord Kelvin pondered the possibility of a theory of everything.[63] He proposed that every body pulsates, which might be an explanation of gravitation and electric charges. His ideas were largely mechanistic and required the existence of the aether, which the Michelson–Morley experiment failed to detect in 1887. This, combined with Mach's principle, led to gravitational models which feature action at a distance.

Albert Einstein developed his revolutionary theory of relativity in papers published in 1905 and 1915; these account for the perihelion precession of Mercury.[61] In 1914, Gunnar Nordström attempted to unify gravity and electromagnetism in his theory of five-dimensional gravitation.[o] General relativity was proved in 1919, when Arthur Eddington observed gravitational lensing around a solar eclipse, matching Einstein's equations. This resulted in Einstein's theory superseding Newtonian physics.[64] Thereafter, German mathematician Theodor Kaluza promoted the idea of general relativity with a fifth dimension, which in 1921 Swedish physicist Oskar Klein gave a physical interpretation of in a prototypical string theory, a possible model of quantum gravity and potential theory of everything.

Einstein's field equations include a cosmological constant to account for the alleged staticity of the universe. However, Edwin Hubble observed in 1929 that the universe appears to be expanding. By the 1930s, Paul Dirac developed the hypothesis that gravitation should slowly and steadily decrease over the course of the history of the universe.[65] Alan Guth and Alexei Starobinsky proposed in 1980 that cosmic inflation in the very early universe could have been driven by a negative pressure field, a concept later coined 'dark energy'—found in 2013 to have composed around 68.3% of the early universe.[66]

In 1922, Jacobus Kapteyn proposed the existence of dark matter, an unseen force that moves stars in galaxies at higher velocities than gravity alone accounts for. It was found in 2013 to have comprised 26.8% of the early universe.[66] Along with dark energy, dark matter is an outlier in Einstein's relativity, and an explanation for its apparent effects is a requirement for a successful theory of everything.

In 1957, Hermann Bondi proposed that negative gravitational mass (combined with negative inertial mass) would comply with the strong equivalence principle of general relativity and Newton's laws of motion. Bondi's proof yielded singularity-free solutions for the relativity equations.[67]

Early theories of gravity attempted to explain planetary orbits (Newton) and more complicated orbits (e.g. Lagrange). Then came unsuccessful attempts to combine gravity and either wave or corpuscular theories of gravity. The whole landscape of physics was changed with the discovery of Lorentz transformations, and this led to attempts to reconcile it with gravity. At the same time, experimental physicists started testing the foundations of gravity and relativity – Lorentz invariance, the gravitational deflection of light, the Eötvös experiment. These considerations led to and past the development of general relativity.

Einstein (1905, 1908, 1912)[edit]

In 1905, Albert Einstein published a series of papers in which he established the special theory of relativity and the fact that mass and energy are equivalent. In 1907, in what he described as "the happiest thought of my life", Einstein realized that someone who is in free fall experiences no gravitational field. In other words, gravitation is exactly equivalent to acceleration.

Einstein's two-part publication in 1912[68][69] (and before in 1908) is really only important for historical reasons. By then he knew of the gravitational redshift and the deflection of light. He had realized that Lorentz transformations are not generally applicable, but retained them. The theory states that the speed of light is constant in free space but varies in the presence of matter. The theory was only expected to hold when the source of the gravitational field is stationary. It includes the principle of least action:

where is the Minkowski metric, and there is a summation from 1 to 4 over indices and .

Einstein and Grossmann[70] includes Riemannian geometry and tensor calculus.

The equations of electrodynamics exactly match those of general relativity. The equation

is not in general relativity. It expresses the stress–energy tensor as a function of the matter density.

Lorentz-invariant models (1905–1910)[edit]

Based on the principle of relativity, Henri Poincaré (1905, 1906), Hermann Minkowski (1908), and Arnold Sommerfeld (1910) tried to modify Newton's theory and to establish a Lorentz invariant gravitational law, in which the speed of gravity is that of light. As in Lorentz's model, the value for the perihelion advance of Mercury was much too low.[71]

Abraham (1912)[edit]

Meanwhile, Max Abraham developed an alternative model of gravity in which the speed of light depends on the gravitational field strength and so is variable almost everywhere. Abraham's 1914 review of gravitation models is said to be excellent, but his own model was poor.

Nordström (1912)[edit]

The first approach of Nordström (1912)[72] was to retain the Minkowski metric and a constant value of but to let mass depend on the gravitational field strength . Allowing this field strength to satisfy

where is rest mass energy and is the d'Alembertian,

and

where is the four-velocity and the dot is a differential with respect to time.

The second approach of Nordström (1913)[73] is remembered as the first logically consistent relativistic field theory of gravitation ever formulated. (notation from Pais[74] not Nordström):

where is a scalar field,

This theory is Lorentz invariant, satisfies the conservation laws, correctly reduces to the Newtonian limit and satisfies the weak equivalence principle.

Einstein and Fokker (1914)[edit]

This theory[75] is Einstein's first treatment of gravitation in which general covariance is strictly obeyed. Writing:

they relate Einstein–Grossmann[70] to Nordström.[73] They also state:

That is, the trace of the stress energy tensor is proportional to the curvature of space.

Between 1911 and 1915, Einstein developed the idea that gravitation is equivalent to acceleration, initially stated as the equivalence principle, into his general theory of relativity, which fuses the three dimensions of space and the one dimension of time into the four-dimensional fabric of spacetime. However, it does not unify gravity with quanta—individual particles of energy, which Einstein himself had postulated the existence of in 1905.

General relativity[edit]

Illustration explaining the relevance of the total solar eclipse of 29 May 1919, from the 22 November 1919 edition of The Illustrated London News

In general relativity, the effects of gravitation are ascribed to spacetime curvature instead of to a force. The starting point for general relativity is the equivalence principle, which equates free fall with inertial motion. The issue that this creates is that free-falling objects can accelerate with respect to each other. To deal with this difficulty, Einstein proposed that spacetime is curved by matter, and that free-falling objects are moving along locally straight paths in curved spacetime. More specifically, Einstein and David Hilbert discovered the field equations of general relativity, which relate the presence of matter and the curvature of spacetime. These field equations are a set of 10 simultaneous, non-linear, differential equations. The solutions of the field equations are the components of the metric tensor of spacetime, which describes its geometry. The geodesic paths of spacetime are calculated from the metric tensor.

Notable solutions of the Einstein field equations include:

General relativity has enjoyed much success because its predictions (not called for by older theories of gravity) have been regularly confirmed. For example:

It is believed that neutron star mergers (since detected in 2017)[77] and black hole formation may also create detectable amounts of gravitational radiation.

Quantum gravity[edit]

Several decades after the discovery of general relativity, it was realized that it cannot be the complete theory of gravity because it is incompatible with quantum mechanics.[78] Later it was understood that it is possible to describe gravity in the framework of quantum field theory like the other fundamental forces. In this framework, the attractive force of gravity arises due to exchange of virtual gravitons, in the same way as the electromagnetic force arises from exchange of virtual photons.[79][80] This reproduces general relativity in the classical limit, but only at the linearized level and postulating that the conditions for the applicability of Ehrenfest theorem holds, which is not always the case. Moreover, this approach fails at short distances of the order of the Planck length.[78]

Theoretical models such as string theory and loop quantum gravity are current candidates for a possible 'theory of everything'.

See also[edit]

References[edit]

Footnotes

  1. ^ The source of this quote is Al-Biruni's India (c. 1030).[14]
  2. ^ This was interpreted as deriving the weight of objects from the pressure of the air below them.[25]
  3. ^ Leonardo da Vinci tested this theory by observing trace fossils,[27] which he used to argue against the myth of a universal flood.[28]
  4. ^ Furthermore, he hypothesized that the planet is in equilibrium when its centre of gravity coincides with that of its mass.[27]
  5. ^ Leonardo did not publish his manuscripts and they had no direct influence on subsequent science.[30]
  6. ^ He accounted for these movements by explaining, "Rotation is natural to a sphere, and by that very act is its shape expressed."[32]
  7. ^ Physicist Pierre Duhem erroneously attributes this to Jordanus Nemorarius, whom he calls the "precursor of Leonardo". Leonardo alludes to Jordanus in his notebooks, but not to any of his theories.[33]
  8. ^ Some historians consider this to have been a thought experiment rather than a physical test as there is little evidence that it actually took place.[35] Further, the premise had been proven at least two decades before Galileo's birth.[36]
  9. ^ The distance traversed in successive equal intervals of time is calculated with a triangular model whose width (representing maximum velocity) increases by two for every equal section of height (representing time elapsed). This is in part anticipated by the Merton rule.[38]
  10. ^ James Challis repeated this assumption in 1869.
  11. ^ Bernhard Riemann made a similar argument in 1853.
  12. ^ Newton was almost certainly influenced by this correspondence to do his subsequent work on gravitation,[48] although he denied that Hooke had told him of the inverse-square force.[49]
  13. ^ Many sources incorrectly state that this was the first measurement of G (or the Earth's density).[51] There were previous measurements, chiefly by Bouguer (1740) and Maskelyne (1774), but they were very inaccurate.[52][53]
  14. ^ Another imperfection of the theory is that it fails to define a medium in which it interacts, seeming to require action at a distance.[59]
  15. ^ In string theory, dimensions exceeding four allow for the existence of parallel realities—which along with the anthropic principle, help to explain the statistical near-impossibility of our fine-tuned universe.

Citations

  1. ^ Smith, Homer W. (1952). Man and His Gods. New York: Grosset & Dunlap. p. 144.
  2. ^ Edward Grant, The Foundations of Modern Science in the Middle Ages, (Cambridge: Cambridge Univ. Pr., 1996), pp. 60-1.
  3. ^ Olaf Pedersen, Early Physics and Astronomy, (Cambridge: Cambridge Univ. Pr., 1993), p. 130
  4. ^ Reviel Neitz; William Noel (13 October 2011). The Archimedes Codex: Revealing The Secrets Of The World's Greatest Palimpsest. Hachette UK. ISBN 9781780221984.CS1 maint: multiple names: authors list (link)
  5. ^ CJ Tuplin, Lewis Wolpert (2002). Science and Mathematics in Ancient Greek Culture. Hachette UK. p. xi. ISBN 9780198152484.
  6. ^ Vitruvius, Marcus Pollio (1914). "7". In Alfred A. Howard (ed.). De Architectura libri decem [Ten Books on Architecture]. VII. Herbert Langford Warren, Nelson Robinson (illus), Morris Hicky Morgan. Harvard University, Cambridge: Harvard University Press. p. 215.
  7. ^ Yajnavalkya, translated by Julius Eggeling, "Satapatha Brahmana Part IV (SBE43): Eighth Kânda: VIII, 7, 3. Third Brâhmana", Internet Sacred Text Archive, retrieved 15 June 2021
  8. ^ The Satapatha Brahmana. Sacred Books of the East, Vols. 12, 26, 24, 37, 47, translated by Julius Eggeling (1882–1900).
  9. ^ Witzel, Michael (1995). Erdosy, George (ed.). The Indo-Aryans of Ancient South Asia: Language, Material Culture and Ethnicity. Berlin. ISBN 978-3-11-081643-3. OCLC 857490772.
  10. ^ Pickover, Clifford (2008). Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. p. 105. ISBN 978-0-19-979268-9.
  11. ^ Bose, Mainak Kumar (1988). Late classical India. A. Mukherjee & Co.[page needed]
  12. ^ Sen, Amartya (2005). The Argumentative Indian. Allen Lane. p. 29. ISBN 978-0-7139-9687-6.
  13. ^ Thurston, Hugh (1993). Early Astronomy. New York: Springer-Verlag. ISBN 978-0-387-94107-3.[page needed]
  14. ^ a b Alberuni's India. London: Kegan Paul, Trench, Trübner & Co., 1910. Electronic reproduction. Vol. 1 and 2. New York: Columbia University Libraries, 2006. p. 272. Retrieved 3 June 2014.
  15. ^ Kitāb al-Jawharatayn al-'atīqatayn al-mā'i'atayn min al-ṣafrā' wa-al-bayḍā' : al-dhahab wa-al-fiḍḍah. Cairo : Maṭba'at Dār al-Kutub wa-al-Wathā'iq al-Qawmīyah bi-al-Qāhirah (Arabic: كتاب الجوهرتين العتيقتين المائعتين من الصفراء والبيضاء : الذهب والفضة) 2004. pp. 43–44, 87. OCLC 607846741.
  16. ^ McGinnis, Jon; Reisman, David C. (2007). Classical Arabic philosophy: an anthology of sources. Hackett Publishing. p. 174. ISBN 978-0-87220-871-1. Retrieved 16 June 2010.
  17. ^ Espinoza, Fernando (2005). "An analysis of the historical development of ideas about motion and its implications for teaching". Physics Education. 40 (2): 141. Bibcode:2005PhyEd..40..139E. doi:10.1088/0031-9120/40/2/002.
  18. ^ Seyyed Hossein Nasr & Mehdi Amin Razavi (1996). The Islamic intellectual tradition in Persia. Routledge. p. 72. ISBN 978-0-7007-0314-2.
  19. ^ a b Aydin Sayili (1987). "Ibn Sīnā and Buridan on the Motion of the Projectile". Annals of the New York Academy of Sciences. 500 (1): 477–482. Bibcode:1987NYASA.500..477S. doi:10.1111/j.1749-6632.1987.tb37219.x. S2CID 84784804.
  20. ^ Espinoza, Fernando. "An Analysis of the Historical Development of Ideas About Motion and its Implications for Teaching". Physics Education. Vol. 40(2).
  21. ^ Starr, S. Frederick (2015). Lost Enlightenment: Central Asia's Golden Age from the Arab Conquest to Tamerlane. Princeton University Press. p. 260. ISBN 9780691165851.
  22. ^ Rozhanskaya, Mariam; Levinova, I. S. (1996). "Statics". In Rushdī, Rāshid (ed.). Encyclopedia of the History of Arabic Science. 2. Psychology Press. pp. 614–642. ISBN 9780415124119. Using a whole body of mathematical methods (not only those inherited from the antique theory of ratios and infinitesimal techniques, but also the methods of the contemporary algebra and fine calculation techniques), Muslim scientists raised statics to a new, higher level. The classical results of Archimedes in the theory of the centre of gravity were generalized and applied to three-dimensional bodies, the theory of ponderable lever was founded and the 'science of gravity' was created and later further developed in medieval Europe. The phenomena of statics were studied by using the dynamic approach so that two trends – statics and dynamics – turned out to be inter-related within a single science, mechanics. The combination of the dynamic approach with Archimedean hydrostatics gave birth to a direction in science which may be called medieval hydrodynamics. ... Numerous fine experimental methods were developed for determining the specific weight, which were based, in particular, on the theory of balances and weighing. The classical works of al-Biruni and al-Khazini can by right be considered as the beginning of the application of experimental methods in medieval science.
  23. ^ a b Gutman, Oliver (2003). Pseudo-Avicenna, Liber Celi Et Mundi: A Critical Edition. Brill Publishers. p. 193. ISBN 90-04-13228-7.
  24. ^ Pines, Shlomo (1970). "Abu'l-Barakāt al-Baghdādī , Hibat Allah". Dictionary of Scientific Biography. 1. New York: Charles Scribner's Sons. pp. 26–28. ISBN 0-684-10114-9.
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