The theory of impetus (from the Latin impetus 'push, impulse') is a natural philosophical theory according to which the cause of the movement of abandoned bodies is some force (impetus), invested in them by an external source. The theory of impetus appeared as a result of criticism of some of the provisions of Aristotle's physics , but in general corresponds to it.
General characteristics of the theory of impetus
Key
The theory of impetus was an attempt to answer the question: what moves a body thrown near the surface of the Earth? The presence of a driving force was considered necessary in connection with the general provisions of the mechanics of Aristotle , according to which movement is possible only if there is a driving force. In the theory of impetus, it was assumed that during a joint movement with an abandoned body (stone, arrow, cannonball), the mover (human hand, bowstring, sling, firearm, etc.) puts some force into the abandoned body (among the thinkers of the East - violent tendency), which makes the body move on [1] . This invested force was named in the fourteenth century. impetus . Impetus was considered a new quality of a moving body, which is absent in a motionless body, similar to how heat is a quality of a hot body, which is absent in a cold body. The process of impetus transmission was thought by analogy with heat transfer. During the movement of the body, the impetus was gradually exhausted, due to which the abandoned body eventually fell to the surface of the Earth.
Controversial Issues
Initially, the theory of impetus developed in the context of commenting on the works of Aristotle or even theological treatises, and only at the end of the XVI - beginning of the XVII centuries. compositions were written containing attempts to build on its basis a consistent physical theory ( Jambatista Benedetti , Galileo Galilei ). However, such a theory was never created [2] .
Great disagreement has always been raised by the question of whether impetus is exhausted during the movement of the body spontaneously or only due to the resistance of external factors (friction against air, the effect of gravity). Philopon , al-Baghdadi , Francesco from March spoke in favor of self-exhaustion of impetus , Nikolai Orem , in favor of the disappearance of impetus due to the resistance of external factors - Avicenna , Jean Buridan , Albert of Saxony .
Further, some thinkers believed that a body moving under the influence of impetus does not experience gravity (Avicenna), while others believed that gravity and impetus act simultaneously (al-Baghdadi), at least on some part of the trajectory (Albert Saxon).
Later (from the 14th century), disputes began on the question of how the initial movement (i.e., the combined movement of an abandoned body and mover) contributes to the emergence of impetus: due to the presence of speed or acceleration? In the first case, the impetus also gives rise to the speed of an abandoned body, in the second - also acceleration. In favor of the first option, Buridan spoke out, the second - Nikolai Orem. Another controversial concept is associated with Buridan - the idea that impetus can cause the rotation of a rigid body around its axis; it was rejected by Jambatista Benedetti in favor of the suggestion that impetus can only cause a rectilinear movement of the body.
The lack of clarity in the general provisions of the theory of impetus affected its application to the solution of specific physical problems. For example, some scientists and philosophers used the theory of impetus to substantiate the hypothesis of the Earth 's rotation around the axis ( Giordano Bruno ), while others, on the contrary, to refute it ( Jean Buridan , Giovanni Battista Riccioli ). Another disagreement was raised by another question: is it necessary to involve the existence of special spiritual entities, the so-called “intelligences” (varieties of angels ), to explain the movements of celestial bodies (as was supposed in the Middle Ages attached to the celestial spheres), or is it sufficient to assume that the movement of celestial bodies occurs due to the impetus invested in them when God created the world. Philopon , Jean Buridan , Albert of Saxony spoke out in favor of the second possibility, while Avicenna , Nikolai Orem thought that it was impossible to do without involving the intelligentsia. There were compromise solutions to this problem ( al-Bitruji , Francesco from March , Riccioli ).
Impetus and Inertia
At the beginning of the 20th century, the opinion was expressed (mainly by Pierre Duhem ) that the theory of impetus is a direct predecessor, a kind of medieval shell of modern ideas about inertia , and impetus itself is an analogue of impulse . Indeed, in some versions of this theory, the impetus communicated to the body was considered to change only due to external influences and was calculated using the same formula as the momentum in classical mechanics (such were the versions of Avicenna and Buridan).
However, at present this point of view is outdated [3] . Significant differences are due to the fact that in medieval theory the state of rest was considered to be something primary, and it was necessary to explain its termination, i.e. there is bringing the body into motion, the appearance of speed in the body. In some cases, the cause of movement was, for example, heaviness, in others - impetus. In general, the theory of impetus was consistent with Aristotle's physics , since force was considered the cause of body motion and speed was considered proportional to force. In modern science, peace is just a special case of movement, and the change of state of movement is subject to explanation, i.e. acceleration; according to Newton’s second law , acceleration is proportional to force.
Further, impetus was considered some special quality, which is endowed with a moving body, similar, for example, to heat. In modern physics, in accordance with the principle of relativity , a moving body is not considered to have any special qualities compared to a motionless one.
At the same time, in some respects the theory of impetus contributed to the appearance of classical mechanics , since it criticized some of the provisions of Aristotle's mechanics. Beginning with Galileo, the term “impetus” was increasingly used in the same sense as “impulse”.
Historical Review
Antiquity
The origins of the theory of impetus lie in antiquity - the physics of Aristotle .
Aristotle. According to Aristotle , each type of matter corresponds to its natural place within the Universe: the place of the element of the earth is in the very center of the world, then the natural places of the elements of water, air, fire, ether follow. The sublunar world was characterized by movement in vertical straight lines; such a movement must have a beginning and an end, which corresponds to the perishability of everything earthly. If an element of the sublunar world is removed from its natural place, it will strive to fall into its natural place. So, if you raise a handful of earth, it will be natural for it to move vertically down. Since the elements of earth and water in their natural movement tended down to the center of the world, they were considered absolutely heavy; elements of air and fire rushed up to the border of the sublunar area, so they were considered absolutely light. Aristotle explained the increase in the speed of the falling body as the body approached its final point - the Earth. Upon reaching a natural place, the movement of the elements of the sublunar world ceases.
The movement of the body to its natural place was called the natural movement . Otherwise, the movement was called violent . Aristotle believed that violent movement is possible only if force is applied to the body from the side of another body: “everything that is in motion must move with something else”; movable and mover must be in direct contact [4] . Aristotle considered the body speed to be proportional to the applied force.
In this theory, an elementary fact with difficulty found its explanation: when a person throws a stone, the stone continues to move after the termination of contact with the hand. Indeed, the stone belongs to the category of heavy bodies, its natural place is located below, on Earth. While he is in the hand, he makes a violent movement, but after the thrower takes his hand, the stone, it would seem, should make a natural movement to the center of the world, i.e. fall to the surface of the earth. But the stone moves in a completely different way: it first rises up or moves at an angle to the horizon, and only then falls to the ground. According to Aristotle , the movement of the stone is supported by air, which, in turn, was informed of the movement of a person’s hand [5] [6] .
Hipparchus. Another solution to the problem of abandoned bodies was given by Hipparchus of Nicaea in the book On bodies moving downward under the influence of their gravity . This book itself did not reach us, but we are familiar with its basic ideas in retelling Simplicia :
Hipparchus writes that if you throw a piece of land straight up, the cause of the upward movement will be the casting force, while it exceeds the weight of the abandoned body; at the same time, the greater the force that is thrown, the faster the object moves up. Then, as the force decreases, the upward movement will occur with ever decreasing speed, until, finally, the body begins to move downward under the influence of its own attraction - although to some extent the abandoned force will still be present in it; as it runs dry, the body will move down faster and faster, reaching its maximum speed, when this force finally disappears [7] .
According to the most common interpretation of this passage, Hipparchus's “abandoned power” is the same as impetus. In this case, Hipparchus contains the first statement of the concept of impetus [8] .
Late Antiquity and Early Middle Ages
The idea of the existence of some internal engines in moving bodies was expressed by the Athenian philosopher of the late II – early III centuries Alexander Afrodisiysky [9] . Similar ideas (in a theological context) are found in the fifth-century Christian thinker Sinesia , a student of the legendary Hypatia [10] [11] .
However, the VIth century Alexandrian thinker John Philopon is usually considered the real author of the concept of impetus.
Philopon. In his comments on Aristotle's Physics , Philopon criticized Aristotle's solution to the problem of abandoned bodies and proposed another solution to this problem. In his opinion, the “throwing agent” (for example, a hand or bowstring) informs the abandoned body of the driving force (later called impetus), which moves the body after the termination of contact; here again the influence of Aristotle's physics manifested itself, in which the speed of the body was considered proportional to the force. The surrounding air does not help the movement, as Aristotle believed, but prevents it [12] . However, even in the void, the impetus of the body would have to spontaneously decrease (be exhausted).
Philopon also applied the theory of impetus to the movement of celestial bodies. He denied the notions that existed at that time (expressed, for example, by Theodore of Mopsuestian and Cosma Indikoplevst ) that heavenly bodies are transported in space by angels . In his opinion, the movement of celestial bodies is due to the driving force invested in them when God created the world [13] .
Islamic East
Avicenna. The theory of the driving force of Philopon became famous among scholars of Muslims. So, it was mentioned by one of the founders of the Arabic philosophy of al-Farabi (IX-X centuries). A significant contribution to its development was made by the outstanding philosopher and scientist of the 11th century Avicenna (Ibn Sina) ( Book of Healing , c. 1020). In his opinion, the “engine” tells the moving body some “desire”, similar to the way fire transfers heat to water. The role of the engine can be not only a hand or a bowstring, but also heaviness.
There are three types of “aspiration”: psychic (among living beings), natural and violent. "Natural aspiration" is the result of the action of gravity and is manifested in the fall of the body, that is, in the natural movement of the body, in agreement with Aristotle . In this case, “aspiration” can exist even in a motionless body, manifesting itself in resistance to immobility. "Violent desire" is an analogue of the Philoponian driving force - it is communicated to the abandoned body by its "engine". As the body moves, “violent desire” decreases due to environmental resistance; as a result, the speed of the body tends to zero. In the void, “violent desire” would not change and the body could make an eternal movement. One could see in this an anticipation of the concept of inertia, but Avicenna did not believe in the existence of a void.
According to Avicenna , “natural” and “violent desire” cannot coexist in one body. The abandoned body will move under the influence of “violent desire” until it is exhausted under the influence of the external environment (the portion of the trajectory AB in the figure on the left). Immediately after this, for a moment, the body will stop and begin to move under the influence of "natural aspiration", that is, to fall vertically downward (the portion of the aircraft trajectory in the left figure). Thus, in Avicenna’s theory, gravity does not affect it on a certain part of the trajectory of an abandoned body.
Avicenna tried to quantify the "violent desire": in his opinion, it is proportional to the weight and speed of the body [14] .
Al-Baghdadi. The further development of the theory of impetus is connected with the Baghdad philosopher Abu l Barakat al-Baghdadi (XII century). Unlike Avicenna , al-Baghdadi believed that the “violent tendency” in the body is exhausted even in the absence of environmental resistance, in empty space, the existence of which he did not deny. In addition, al-Baghdadi considered coexistence in one body both “natural” and “violent inclination” possible. As the abandoned body moves, its “violent tendency” gradually decreases, while the “natural tendency” remains constant, and eventually the body begins to move down.
Significant merit of al-Baghdadi was the inclusion in the picture of the movement of the falling body of acceleration. In his opinion, as the body moves, its severity informs the body of more and more portions of “violent tendency,” thanks to which the movement of the body is accelerated.
A follower of al-Baghdadi in this matter was the next generation philosopher Fahr al-Din ar-Razi [15] . On the contrary, an outstanding Persian scholar of the 13th century. Nasir ad-Din at-Tusi , sharing the idea of the existence of “violent addiction” in abandoned bodies, was inclined to the version of Avicenna [16] .
Al-Bitruji. Another 12th-century scientist, Nur al-Din al-Bitroji , used the theory of impetus to explain the causes of planetary motion. If most scientists of that time were sure that the planets move under the influence of spiritual incorporeal engines (“intelligentsia”, or angels ), then al-Bitruji gave a mechanical explanation: the higher celestial sphere receives a driving force from the Prime Mover and transfers it to the lower spheres to which are attached planets; as it moves toward Earth, this force weakens [16] [17] . Al-Bitruji cited the fall of an abandoned stone as an analogy: the driving force put into the stone by the hand weakens with time, as a result of which heaviness begins to dominate in the stone and the stone falls to the ground.
However, al-Bitruji still has to turn to the idea of animated spheres to explain the unevenness of the apparent motion of the planets (in particular, backward movements): each of the spheres has a certain desire to “imitate” the motion of the sphere of fixed stars, driven directly by the Prime Mover. This "imitation" and leads to the appearance of unevenness [17] .
Medieval Europe
In Catholic Europe, the idea of invested power became known back in the 12th century. It seems likely that European authors borrowed elements of the theory of driving forces from scientists from the East [18] .
About the "power of the cast" mentions the French natural philosopher of the XII century. Thierry of Chartres [19] . The theory of impetus was briefly mentioned by the great scholastics of the 13th century, Roger Bacon , Albert the Great and Thomas Aquinas , but rejected in favor of the theory of Aristotle . A rather detailed exposition of the theory of impetus is contained in the philosopher of the second half of the 13th century. Peter John Olivi, who, however, also rejected it [20] . William Ockham was critical of the theory of impetus, claiming that it explains the unknown through the even more unknown; impetus was interpreted as an additional quality of moving bodies, similar to heat, while Occam believed that a moving body is basically no different from a motionless one (an example of using Occam's razor ). Aristotle's interpretation of the problem of abandoned bodies, however, he also rejected.
Francesco from March. The first European philosopher to agree with the theory of impetus was the Italian theologian Francesco of March ( Commentary on the "Sentii" by Peter Lombardsky , c. 1320) His motives lay in the field of theology: according to Francesco, the adoption of the sacrament of communion can promote the believer to God, instill in him divine mercy. Francesco considered the communication of a thrown stone with a hand of some strength due to which he continues to move after the cessation of contact with the hand to be an analogy of the sacrament of communion in the material world [21] .
According to Francesco, the driving force should be exhausted as the body moves, even if the movement occurs in a void, as in Philopon and al-Baghdadi [22] [23] [24] . A little later, he was also supported by the Parisian philosopher Nikolai Bonetus, who paid great attention to the problem of movement in the void [25] .
Francesco of March applied the theory of impetus to the movement of celestial bodies. In the Middle Ages, the prevailing idea was that luminaries were attached to the celestial spheres, which move under the influence of “intelligentsia” - special spiritual beings, usually identified with angels [26] . According to Francesco, angels rotate the celestial spheres through the transmission of impetus to them [27] . Since the impetus does not persist, but decreases spontaneously, the angels are forced to do this continuously [28] .
Buridan. The theory of impetus owes its greatest development to the outstanding scholasticist of the mid-14th century, professor of the University of Paris Jean Buridan , who also owns the term “impetus”:
A person throwing a stone moves his hand along with the stone, and when shooting from a bow the bowstring moves for some time with the arrow, pushing the arrow; and the same is true with respect to a mantle that disperses a stone, or with respect to machines throwing huge stones. And as long as the thrower pushes the abandoned body, being in contact with it, the movement is slower at the beginning, for then only the external engine moves the stone or arrow; but when moving, an impetus is continuously acquired, which, together with the above-mentioned external engine, moves a stone or arrow, due to which their movement becomes more and more rapid. But after separation from the thrower, he no longer moves the abandoned body, but only the acquired impetus moves it, and this impetus, due to environmental resistance, is constantly weakening, and therefore the movement becomes more and more slow [29] .
Buridan believed that impetus does not decrease spontaneously, but because of the resistance of the external environment, and also because of gravity, which (according to Aristotle ) acts on all earthly bodies and is fundamentally unremovable factor [30] . He considered the measure of impetus to be the product of the speed of the body and the amount of matter . It is possible that these ideas can be borrowed from Avicenna [31] .
Buridan considered gravity as an analogue of a hand when moving abandoned bodies: gravity reports impetus to falling bodies. However, unlike the hand, gravity acts constantly. From here followed his explanation of the acceleration of bodies during a fall (very similar to the theory of al-Baghdadi ): the movement of a falling body is accelerated due to the fact that as the body moves, its heaviness gives the body more and more portions of impetus. Thus, the cause of the acceleration of the falling bodies is not gravity (which only indicates the direction of movement), but the impetus acquired by the body due to gravity and the movement that has already begun [32] . Perhaps Buridan meant that the speed is acquired by the body not continuously, but in discrete portions [33] [34] .
An important innovation of Buridan was the extension of the concept of impetus to the case of rotating solids (the concept of rotational impetus). In his opinion, if you untwist a body planted on an axis, it will be given circular impetus, which will cause it to rotate until the body stops due to environmental resistance. Buridan applied the notion of circular impetus to the explanation of the motion of the celestial spheres. Buridan believed that the existence of intelligentsia (special spiritual entities that carry out the movement of the celestial spheres) does not follow from the Bible and that another explanation of the movement of heaven is also possible:
God at the time of creation told the heavens the same number of movements that exist now, and, setting them in motion, imprinted impetuses in them, thanks to which they then move uniformly, since these impetuses, without encountering resistance, are never destroyed and never are decreasing [35]
(a similar opinion was expressed by John Philopon ). It should be noted that, like other medieval scholastics, in explaining specific astronomical phenomena, Buridan continued to resort to the idea of intelligentsia. So, he believed that the reason for the equality of the periods of motion of the Sun, Mercury and Venus according to the zodiac (manifested in the fact that Mercury and Venus are always in the sky near the Sun) is “the same attitude of moving intelligentsia to moving spheres”, although he knew about the hypothesis according to which these planets revolve around the Sun [36] . Thus, Buridan did not completely abandon the idea of celestial intelligentsia, simply noting that it does not necessarily follow from the Bible , which also corresponds to the idea of “initial impetus” [37] .
Buridan also used the theory of impetus to refute the hypothesis that the Earth rotates around its axis. The traditional argument against this hypothesis was that on a rotating Earth, bodies thrown vertically upwards could not fall to the point from which their movement began: the surface of the Earth would move under the abandoned body. Proponents of the hypothesis of the Earth's rotation responded to this argument that air and all earthly objects (including those thrown upwards) make a movement together with the Earth. Buridan objected to this: the impetus acquired by throwing would resist horizontal movement. He cites the following example: “If a strong wind blew, an arrow fired vertically up would not be able to move as far horizontally as air, but only in part” [38] [39] .
Other representatives of the Paris school. Significant contribution to the development of the theory of impetus was made by other scientists of the University of Paris - younger contemporaries of Buridan.
Albert of Saxony shared the opinion of Buridan that impetus does not decrease spontaneously, but because of the resistance of the environment and gravity, as well as the acceleration of the movement of the falling body due to the fact that as the body moves, its heaviness informs the body of more and more portions of impetus. He even tried to give a mathematical expression for changing the speed of a falling body (speed is proportional to the distance traveled from a state of rest). Albert agreed with the Buridanian theory of "initial impetus" on the causes of the movements of the celestial spheres.
Considering the trajectory of a body launched in a horizontal direction, Albert came to the conclusion that it should consist of three sections. For some time, the body should move under the action of the impetus along a horizontal line, then along a curved path, when gravity gradually begins to act on it, and the impetus decreases, and finally, vertically down, when it moves only under the influence of gravity. From the standpoint of the theory of impetus, he considered a thought experiment: how would a stone move through the Earth if the Earth were drilled through:
When the center of gravity of this falling body would coincide with the center of the world, this body would continue to move in the direction of another part of the sky because of impetus, in it not yet destroyed; and when this impetus is completely consumed during the ascent, this body will begin to descend again, and during the descent it will again acquire a certain small impetus, due to which the center of the Earth will again pass; and when this impetus is also destroyed, it will begin to descend again, and so it will move back and forth around the center of the Earth, hesitating until the impetus remains in it, and finally stops [40] .
This example was cited by the ancient Greek writer Plutarch in the dialogue On the Face, visible on the disk of the Moon , and after Albert of Saxony by other European scholars, including Tartaglia and Galileo .
Another Parisian philosopher, Nicholas Orem , returned to the idea of diminishing impetus even in emptiness. Unlike Buridan, Orem believed that the hand tells the impetus to the thrown stone not only because of its movement (together with the stone), but due to the acceleration of this movement: first, the hand with the stone is motionless, then it accelerates to a certain speed when the palm opens and the stone comes off by hand. Accordingly, impetus causes not only speed, but also the acceleration of bodies [1] .
Among the supporters of the theory of impetus was another well-known Parisian philosopher - Marsilius Ingensky .
Although the number of supporters of the theory of impetus was initially small, the authority and arguments of Parisian philosophers led to its wide distribution in the late Middle Ages.
Renaissance
The popularity of the theory of impetus continued to grow during the Renaissance . In the XV century, it was used to explain various phenomena by Nikolai Kuzansky [35] [41] and Leonardo da Vinci [42] , in the XVI century the Spanish scholastic Domingo de Soto [43] [44] . The famous mathematician and mechanic Niccolo Tartaglia applied the theory of impetus to explain the movement of the cannonball ( New Science , 1537). In his opinion, the trajectory of the nucleus consists of the same three sections as in the theory of Albert Saxon, only the initial portion of the trajectory was not assumed horizontal [45] .
Giordano Bruno in the dialogue Pier on Ashes (1584) uses the theory of impetus to protect the Copernican heliocentric system - explaining the unobservable rotation of the Earth for observers located on its surface. At the same time, he gives an example of a moving ship, as Nikolay Orem did before, but develops the topic deeper:
One of the two people is on a sailing ship, and the other is outside it; at each of them the hand is almost at the same point in the air, and from this place at the same time the first one launches a stone, and the second - another stone, without any shock; the stone of the first, without losing a moment and not deviating from its line, will fall to the appointed place on the ship, and the stone of the second will be left behind. And this hit will occur for the reason that a stone that falls from an outstretched hand on the ship and, therefore, moves, following its movement, has a power communicated to it, which does not have another stone falling out of a hand outside the ship; and all this happens, despite the fact that the stones have the same weight and the same intermediate space that they move (assuming this is possible) from the same point and experience the same impulse.
Here, the “imparted power to the stone” and the “push” are, of course, nothing more than impetus, although the term itself is not used [46] [47] .
An attempt to systematically develop mechanics based on the theory of impetus was made by the outstanding mathematician and physicist of the late Renaissance, Jambatista Benedetti ( Book of various mathematical and physical reflections , 1585).
Scientific Revolution
In one of his works, the theory of impetus was used by Johannes Kepler [48] .
In his treatise On Motion (1590), Galileo Galilei made an attempt to use the theory of impetus in constructing the mechanics of falling bodies. However, he considered impetus self-exhausting. The treatise, however, was never published.
In the work of the Letter on Sunspots (1613), Galileo concluded that the body is at rest until there is some external cause leading it out of this state. Similarly, the body is in a state of inertial motion until there is an external cause leading it out of this state. Thus, to maintain the movement of the body does not require any force, external or internal. While in Aristotle’s physics and in the theory of impetus, motion was considered a process, while peace was considered a state [49] , then for the first time Galileo was called both state [50] . This was a crucial step towards the concept of inertia .
But even in his Dialogue on the two most important systems of the world (1632), Galileo repeatedly used the terms “power input” and “impetus” when describing an abandoned body. As Alexander Koyre showed, he simply had in mind speed or impulse, but he did not say clearly about the non-existence of impetus as a special quality of an abandoned body [51] .
Throughout the 17th century, the terms “embedded power” and “impetus” continued to be used by physicists, mainly in the sense of momentum [52] , but sometimes in the same sense of the additional quality of a moving body, as these terms were used in the Middle Ages . The French Jesuit scientist Honore Fabry tried to give the theory of impetus a mathematical form and build on it the theory of free fall [53] . The Italian Jesuit scientist Giovanni Battista Riccioli ( New Almagest , 1651) tried to use the theory of impetus to refute the rotation of the Earth around the axis [54] , as well as to explain the motion of the planets, joining the opinion of Francesco from March that the angels move the planets through the transmission of impetus (but without the mediation of the celestial spheres) [55] .
The first person who explicitly abandoned the theory of impetus and stated that the movement does not require any force, including internal force, was the Dutch physicist Isaac Beckmann [56] . However, he did not publish this conclusion, formulating it only in his private diary. For the first time, the law of inertia in the correct form was formulated by Rene Descartes in the composition Peace, or a treatise on light (1630) and published in a treatise on the Beginning of Philosophy (1644). The law of inertia was called the first law of motion in Newton in the Mathematical Principles of Natural Philosophy (1687)
See also
- Oxford calculators
Notes
- ↑ 1 2 Damerow et al., 1992 , p. 22-24.
- ↑ Sarnowsky, 2007 .
- ↑ Gaidenko and Smirnov, 1989 , p. 274-277.
- ↑ Sarnowsky, 2007 , p. 123.
- ↑ Rozhanskaya, 1976 , p. thirty.
- ↑ Sarnowsky, 2007 , p. 124.
- ↑ Rozhansky, 1988 , p. 437.
- ↑ See, for example, works: Rozhansky, 1988, p. 438; Crombie, 1996, p. 254. However, there is a slightly different interpretation of the passage cited on the dynamic views of Hipparchus (Wolff, 1989)
- ↑ Pines, 1961 .
- ↑ John Philoponus (The Stanford Encyclopedia of Philosophy)
- ↑ "Bishop Sinesis, who lived in the 5th century, compares the continuous movement of God's will with the art of the same puppet, which" continues to move even when the hand of its controller ceases to pull the threads. "" (S. Neretina, Trails and concepts )
- ↑ Rozhansky, 1988 , p. 439.
- ↑ Sarnowsky, 2007 , p. 125.
- ↑ Rozhanskaya, 1976 , p. 154-155.
- ↑ Rozhanskaya, 1976 , p. 157.
- ↑ 1 2 Rozhanskaya, 1976 , p. 158.
- ↑ 1 2 Samsu, 2007 .
- ↑ Rozhanskaya, 1976 , p. 162-163.
- ↑ Gilson, 2010 , p. 205.
- ↑ Sarnowsky, 2007 , p. 131-132.
- ↑ Funkenstein, 1986 , p. 168.
- ↑ Sarnowsky, 2007 , p. 132-133.
- ↑ Moody, 1951 , p. 392.
- ↑ Hooper, 1998 , p. 161.
- ↑ Grant, 1971 , p. 48.
- ↑ Dales, 1980; Grant, 2009.
- ↑ Grant, 2009 , p. 553.
- ↑ Sarnowsky, 2007 , p. 133.
- ↑ Lupandin I., Cosmology of Jean Buridan
- ↑ Funkenstein, 1972 , p. 342.
- ↑ Sayili, 1987 .
- ↑ Wolff, 1987 , p. 233.
- ↑ Drake, 1975 .
- ↑ The discussion on this topic is contained in Franklin, 1977, Drake, 1977.
- ↑ 1 2 Grigoryan, 1974 , p. 85.
- ↑ Grant, 2009 , p. 314.
- ↑ Dales, 1980 , p. 547-548.
- ↑ Lanskoy, 1999 , p. 91.
- ↑ Grant, 1971 , p. c. 66.
- ↑ Lupandin I. Cosmology of Albert of Saxony
- ↑ Sarnowsky, 2007 , p. 137.
- ↑ Lupandin I. From geocentrism to heliocentrism: Leonardo da Vinci and Copernicus
- ↑ Sarnowsky, 2007 , p. 138.
- ↑ Lupandin I. Development of cosmological representations in the writings of Domingo de Soto and Giovanni Batista Benedetti
- ↑ Sarnowsky, 2007 , p. 141.
- ↑ Koyre, 1943 , p. 342.
- ↑ Sarnowsky, 2007 , p. 139.
- ↑ Rosen, 1966 , p. 613.
- ↑ Coir, 1985 , p. 134-135, 139.
- ↑ Coir, 1985 , p. 141, 212.
- ↑ Hooper, 1998 , p. 162.
- ↑ Sarnowsky, 2007 , p. 142-143.
- ↑ Elazar, 2011 .
- ↑ Grant, 2009 , p. 652-653.
- ↑ Grant, 2009 , p. 553-555.
- ↑ Hooper, 1998 , p. 164.
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