Projective Geometry: An Introduction (Oxford-Warburg Studies)

Projective Geometry: An Introduction (Oxford Warburg Studies)
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Following on from his Doubts on Ptolemy , Alhazen described a new, geometry-based planetary model, describing the motions of the planets in terms of spherical geometry, infinitesimal geometry and trigonometry. He kept a geocentric universe and assumed that celestial motions are uniformly circular, which required the inclusion of epicycles to explain observed motion, but he managed to eliminate Ptolemy's equant.

In general, his model didn't try to provide a causal explanation of the motions, but concentrated on providing a complete, geometric description that could explain observed motions without the contradictions inherent in Ptolemy's model. Alhazen wrote a total of twenty-five astronomical works, some concerning technical issues such as Exact Determination of the Meridian , a second group concerning accurate astronomical observation, a third group concerning various astronomical problems and questions such as the location of the Milky Way ; Alhazen argued for a distant location, based on the fact that it does not move in relation to the fixed stars.

In mathematics , Alhazen built on the mathematical works of Euclid and Thabit ibn Qurra and worked on "the beginnings of the link between algebra and geometry ". He developed a formula for summing the first natural numbers, using a geometric proof to prove the formula. Alhazen explored what is now known as the Euclidean parallel postulate , the fifth postulate in Euclid's Elements , using a proof by contradiction , [] and in effect introducing the concept of motion into geometry.

In elementary geometry, Alhazen attempted to solve the problem of squaring the circle using the area of lunes crescent shapes , but later gave up on the impossible task. Alhazen's contributions to number theory include his work on perfect numbers. Alhazen solved problems involving congruences using what is now called Wilson's theorem.

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In his Opuscula , Alhazen considers the solution of a system of congruences, and gives two general methods of solution. His first method, the canonical method, involved Wilson's theorem, while his second method involved a version of the Chinese remainder theorem. Alhazen discovered the sum formula for the fourth power, using a method that could be generally used to determine the sum for any integral power. He used this to find the volume of a paraboloid. He could find the integral formula for any polynomial without having developed a general formula.

Alhazen also wrote a Treatise on the Influence of Melodies on the Souls of Animals , although no copies have survived. It appears to have been concerned with the question of whether animals could react to music, for example whether a camel would increase or decrease its pace. In engineering , one account of his career as a civil engineer has him summoned to Egypt by the Fatimid Caliph , Al-Hakim bi-Amr Allah , to regulate the flooding of the Nile River.

He carried out a detailed scientific study of the annual inundation of the Nile River, and he drew plans for building a dam , at the site of the modern-day Aswan Dam. His field work, however, later made him aware of the impracticality of this scheme, and he soon feigned madness so he could avoid punishment from the Caliph.

In his Treatise on Place , Alhazen disagreed with Aristotle 's view that nature abhors a void , and he used geometry in an attempt to demonstrate that place al-makan is the imagined three-dimensional void between the inner surfaces of a containing body. Alhazen also discussed space perception and its epistemological implications in his Book of Optics.

Introduction to Projective Geometry - WildTrig: Intro to Rational Trigonometry - N J Wildberger

In "tying the visual perception of space to prior bodily experience, Alhazen unequivocally rejected the intuitiveness of spatial perception and, therefore, the autonomy of vision. Without tangible notions of distance and size for correlation, sight can tell us next to nothing about such things.

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Alhazen was a Muslim; it is not certain to which school of Islam he belonged. As a Sunni, he may have been either a follower of the Ash'ari school, [] or a follower of the Mu'tazili school. Alhazen wrote a work on Islamic theology in which he discussed prophethood and developed a system of philosophical criteria to discern its false claimants in his time. There are occasional references to theology or religious sentiment in his technical works, e. Truth is sought for its own sake Finding the truth is difficult, and the road to it is rough.

For the truths are plunged in obscurity.

An Introduction to Projective Geometry and Its Applications: An Analytic and Synthetic Treatment

God, however, has not preserved the scientist from error and has not safeguarded science from shortcomings and faults. If this had been the case, scientists would not have disagreed upon any point of science From the statements made by the noble Shaykh, it is clear that he believes in Ptolemy's words in everything he says, without relying on a demonstration or calling on a proof, but by pure imitation taqlid ; that is how experts in the prophetic tradition have faith in Prophets, may the blessing of God be upon them. But it is not the way that mathematicians have faith in specialists in the demonstrative sciences.

I constantly sought knowledge and truth, and it became my belief that for gaining access to the effulgence and closeness to God, there is no better way than that of searching for truth and knowledge. Alhazen made significant contributions to optics, number theory, geometry, astronomy and natural philosophy.

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Alhazen's work on optics is credited with contributing a new emphasis on experiment. A Latin translation of the Kitab al-Manazir was made probably in the late twelfth or early thirteenth century. He made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the problem known as " Alhazen's problem ". Some of his treatises on optics survived only through Latin translation.

During the Middle Ages his books on cosmology were translated into Latin, Hebrew and other languages. The impact crater Alhazen on the Moon is named in his honour, [] as was the asteroid Alhazen. He was voiced by Alfred Molina in the episode. Over forty years previously, Jacob Bronowski presented Alhazen's work in a similar television documentary and the corresponding book , The Ascent of Man. In episode 5 The Music of the Spheres , Bronowski remarked that in his view, Alhazen was "the one really original scientific mind that Arab culture produced", whose theory of optics was not improved on till the time of Newton and Leibniz.

Winter, a British historian of science, summing up the importance of Ibn al-Haytham in the history of physics wrote:.

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After the death of Archimedes no really great physicist appeared until Ibn al-Haytham. If, therefore, we confine our interest only to the history of physics, there is a long period of over twelve hundred years during which the Golden Age of Greece gave way to the era of Muslim Scholasticism, and the experimental spirit of the noblest physicist of Antiquity lived again in the Arab Scholar from Basra.

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An international campaign, created by the Inventions organisation, titled Inventions and the World of Ibn Al-Haytham featuring a series of interactive exhibits, workshops and live shows about his work, partnering with science centers, science festivals, museums, and educational institutions, as well as digital and social media platforms.

Smith has noted that Alhazen's treatment of refraction describes an experimental setup without publication of data. According to medieval biographers, Alhazen wrote more than works on a wide range of subjects, of which at least 96 of his scientific works are known. Most of his works are now lost, but more than 50 of them have survived to some extent. Nearly half of his surviving works are on mathematics, 23 of them are on astronomy, and 14 of them are on optics, with a few on other subjects.

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In seventeenth century Europe the problems formulated by Ibn al-Haytham — became known as 'Alhazen's problem'. Al-Haytham's contributions to geometry and number theory went well beyond the Archimedean tradition. Al-Haytham also worked on analytical geometry and the beginnings of the link between algebra and geometry. Subsequently, this work led in pure mathematics to the harmonious fusion of algebra and geometry that was epitomised by Descartes in geometric analysis and by Newton in the calculus.

Al-Haytham was a scientist who made major contributions to the fields of mathematics, physics and astronomy during the latter half of the tenth century. From Wikipedia, the free encyclopedia. This is the latest accepted revision , reviewed on 22 September For other uses, see Alhazen disambiguation and Ibn al-Haytham disambiguation. Basra , Iraq. Cairo , Egypt. Basra Cairo. Optics Astronomy Mathematics.

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Main article: Book of Optics. See also: Horopter. Further information: Scientific method. Main article: Alhazen's problem. This section contains information of unclear or questionable importance or relevance to the article's subject matter. Please help improve this section by clarifying or removing indiscriminate details. If importance cannot be established, the section is likely to be moved to another article, pseudo-redirected , or removed.

Mark Smith has determined that there were at least two translators, based on their facility with Arabic; the first, more experienced scholar began the translation at the beginning of Book One, and handed it off in the middle of Chapter Three of Book Three. Smith 91 Volume 1: Commentary and Latin text pp.