Instruct: Given a question, retrieve documents that answer the question.
Query: Who is considered the father of geometry?
Euclid (; – "Eukleídēs", ; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His "Elements" is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the "Elements", Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and rigor.
Pythagoras, whose dates are commonly given as 569–475 BC, used algebraic methods to construct Pythagorean triples, according to Proclus's commentary on Euclid. Proclus, however, wrote between 410 and 485 AD. According to Thomas L. Heath (1861–1940), no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived. However, when authors such as Plutarch and Cicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted. "Whether this formula is rightly attributed to Pythagoras personally, [...] one can safely assume that it belongs to the very oldest period of Pythagorean mathematics."
In India, the "Baudhayana Sulba Sutra", the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle.
The "Apastamba Sulba Sutra" (c. 600 BC) contains a numerical proof of the general Pythagorean theorem, using an area computation. Van der Waerden believed that "it was certainly based on earlier traditions". Carl Boyer states that the Pythagorean theorem in "Śulba-sũtram" may have been influenced by ancient Mesopotamian math, but there is no conclusive evidence in favor or opposition of this possibility.
With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, the Chinese text "Zhoubi Suanjing" (周髀算经), ("The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven") gives a reasoning for the Pythagorean theorem for the (3, 4, 5) triangle—in China it is called the "Gougu theorem" (勾股定理). During the Han Dynasty (202 BC to 220 AD), Pythagorean triples appear in "The Nine Chapters on the Mathematical Art", together with a mention of right triangles. Some believe the theorem arose first in China, where it is alternatively known as the "Shang Gao theorem" (商高定理), named after the Duke of Zhou's astronomer and mathematician, whose reasoning composed most of what was in the "Zhoubi Suanjing".
Al-Khwarizmi's popularizing treatise on algebra ("The Compendious Book on Calculation by Completion and Balancing", c. 813–833 CE) presented the first systematic solution of linear and quadratic equations. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because he was the first to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation), he has been described as the father or founder of algebra. The term "algebra" itself comes from the title of his book (specifically the word "al-jabr" meaning "completion" or "rejoining"). His name gave rise to the terms "algorism" and "algorithm". His name is also the origin of (Spanish) "guarismo" and of (Portuguese) "algarismo", both meaning digit.
Algebra was also pioneered by Persian Scientist Muhammad ibn Mūsā al-Khwārizmī during this time in his landmark text, Kitab al-Jabr wa-l-Muqabala, from which the term algebra is derived. He is thus considered to be the father of algebra. The terms algorism and algorithm are also derived from the name of al-Khwarizmi, who was responsible for introducing the Arabic numerals and Hindu-Arabic numeral system beyond the Indian Subcontinent.
Hypatia (born 350–370; died 415 AD) was a Hellenistic Neoplatonist philosopher, astronomer, and mathematician, who lived in Alexandria, Egypt, then part of the Eastern Roman Empire. She was a prominent thinker of the Neoplatonic school in Alexandria, where she taught philosophy and astronomy. She is the first female mathematician whose life is reasonably well recorded. Hypatia was renowned in her own lifetime as a great teacher and a wise counselor. She is known to have written a commentary on Diophantus's thirteen-volume "Arithmetica", which may survive in part, having been interpolated into Diophantus's original text, and another commentary on Apollonius of Perga's treatise on conic sections, which has not survived. Many modern scholars also believe that Hypatia may have edited the surviving text of Ptolemy's "Almagest", based on the title of her father Theon's commentary on Book III of the "Almagest".
Lemoine has been described by Nathan Altshiller Court as a co-founder (along with Henri Brocard and Joseph Neuberg) of modern triangle geometry, a term used by William Gallatly, among others. In this context, "modern" is used to refer to geometry developed from the late 18th century onward. Such geometry relies on the abstraction of figures in the plane rather than analytic methods used earlier involving specific angle measures and distances. The geometry focuses on topics such as collinearity, concurrency, and concyclicity, as they do not involve the measures listed previously.
There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof. According to Joran Friberg, a historian of mathematics, evidence indicates that the Pythagorean theorem "was well-known to the mathematicians of the Old Babylonian period" (20th to 16th centuries BC), which would have been over a thousand years before Pythagoras was born. The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system.
With Arthur Cayley, Menger is considered one of the founders of distance geometry; especially by having formalized definitions to the notions of "angle" and of "curvature" in terms of directly measurable physical quantities, namely ratios of "distance" values.
Earth's circumference lines are the foundation of the coordinate system of latitude and longitude, introduced in the mid-second century BC by Hipparchus.
Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues. George Peacock was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovered relation algebra in his "Syllabus of a Proposed System of Logic". Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra).
In 1988, "The Theorem of Pythagoras" was the first video produced by the series and reviews the Pythagorean theorem. For all right triangles, the square of the hypotenuse is equal to the sum of the squares of the other two sides ( a + b = c ). The theorem is named after Pythagoras of ancient Greece. Pythagorean triples occur when all three sides of a right triangle are integers such as a = 3, b = 4 and c = 5. A clay tablet shows that the Babylonians knew of Pythagorean triples 1200 years before Pythagoras, but nobody knows if they knew the more-general Pythagorean theorem. The Chinese proof uses four similar triangles to prove the theorem.
Johann Carl Friedrich Gauss (; ; ; (30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and sciences. Sometimes referred to as the "Princeps mathematicorum" () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history's most influential mathematicians.
Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, he confidently solved an arithmetic series problem faster than anyone else in his class of 100 students. Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100. There are many other anecdotes about his precocity while a toddler, and he made his first groundbreaking mathematical discoveries while still a teenager. He completed his magnum opus, "Disquisitiones Arithmeticae", in 1798, at the age of 21—though it was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
Al-Karkhi (953–1029), also known as Al-Karaji, was the successor of Abū al-Wafā' al-Būzjānī (940–998) and he discovered the first numerical solution to equations of the form ax + bx = c. Al-Karkhi only considered positive roots. Al-Karkhi is also regarded as the first person to free algebra from geometrical operations and replace them with the type of arithmetic operations which are at the core of algebra today. His work on algebra and polynomials, gave the rules for arithmetic operations to manipulate polynomials. The historian of mathematics F. Woepcke, in "Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi" (Paris, 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic calculus". Stemming from this, Al-Karaji investigated binomial coefficients and Pascal's triangle.
According to Cleomedes' "On the Circular Motions of the Celestial Bodies", around 240 BC, Eratosthenes, the librarian of the Library of Alexandria, calculated the circumference of the Earth in Ptolemaic Egypt. Using a scaphe, he knew that at local noon on the summer solstice in Syene (modern Aswan, Egypt), the Sun was directly overhead. (Syene is at latitude 24°05′ North, near to the Tropic of Cancer, which was 23°42′ North in 100 BC) He knew this because the shadow of someone looking down a deep well at that time in Syene blocked the reflection of the Sun on the water. He then measured the Sun's angle of elevation at noon in Alexandria by using a vertical rod, known as a gnomon, and measuring the length of its shadow on the ground. Using the length of the rod, and the length of the shadow, as the legs of a triangle, he calculated the angle of the sun's rays. This angle was about 7°, or 1/50th the circumference of a circle; taking the Earth as perfectly spherical, he concluded that the Earth's circumference was 50 times the known distance from Alexandra to Syene (5,000 stadia, a figure that was checked yearly), i.e. 250,000 stadia. Depending on whether he used the "Olympic stade" (176.4m) or the Italian stade (184.8m), this would imply a circumference of 44,100 km (an error of 10%) or 46,100 km, an error of 15%. In 2012, Anthony Abreu Mora repeated Eratosthenes's calculation with more accurate data; the result was 40,074 km, which is 66 km different (0.16%) from the currently accepted polar circumference.
In at least Plato's time, the school did not have any particular doctrine to teach; rather, Plato (and probably other associates of his) posed problems to be studied and solved by the others. There is evidence of lectures given, most notably Plato's lecture "On the Good"; but probably the use of dialectic was more common. According to an unverifiable story, dated of some 700 years after the founding of the school, above the entrance to the Academy was inscribed the phrase "Let None But Geometers Enter Here."
Although Greek mathematician and engineer Heron of Alexandria is noted as the first to have conceived these numbers, Rafael Bombelli first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time imaginary numbers, as well as negative numbers, were poorly understood and regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his "La Géométrie", where the term "imaginary" was used and meant to be derogatory. The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818).
Stefan of Pise translated into Latin around 1127 an Arab manual of medical theory. The method of algorism for performing arithmetic with Indian-Arabic numerals was developed by the Persian al-Khwarizmi in the 9th century, and introduced in Europe by Leonardo Fibonacci (1170–1250). A translation by Robert of Chester of the "Algebra" by al-Kharizmi is known as early as 1145. Ibn al-Haytham (Alhazen, 980–1037) compiled treatises on optical sciences, which were used as references by Newton and Descartes. Medical sciences were also highly developed in Islam as testified by the Crusaders, who relied on Arab doctors on numerous occasions. Joinville reports he was saved in 1250 by a “Saracen” doctor.
The Ptolemaic Kingdom was founded in 305 BC by Ptolemy I Soter, a "diadochus" originally from Macedon in northern Greece who declared himself pharaoh of Egypt and created a powerful Macedonian Greek dynasty that ruled an area stretching from southern Syria to Cyrene and south to Nubia. Scholars also argue that the kingdom was founded in 304 BC because of different use of calendars: Ptolemy crowned himself in 304 BC on the ancient Egyptian calendar, but in 305 BC on the ancient Macedonian calendar; to resolve the issue, the year 305/4 was counted as the first year of Ptolemaic Kingdom in Demotic papyri.
Hipparchus is credited with the invention or improvement of several astronomical instruments, which were used for a long time for naked-eye observations. According to Synesius of Ptolemais (4th century) he made the first "astrolabion": this may have been an armillary sphere (which Ptolemy however says he constructed, in "Almagest" V.1); or the predecessor of the planar instrument called astrolabe (also mentioned by Theon of Alexandria). With an astrolabe Hipparchus was the first to be able to measure the geographical latitude and time by observing fixed stars. Previously this was done at daytime by measuring the shadow cast by a gnomon, by recording the length of the longest day of the year or with the portable instrument known as a "scaphe".
Descartes, like Galileo, was convinced of the importance of mathematical explanation, and he and his followers were key figures in the development of mathematics and geometry in the 17th century. Cartesian mathematical descriptions of motion held that all mathematical formulations had to be justifiable in terms of direct physical action, a position held by Huygens and the German philosopher Gottfried Leibniz, who, while following in the Cartesian tradition, developed his own philosophical alternative to Scholasticism, which he outlined in his 1714 work, "The Monadology". Descartes has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day. In particular, his "Meditations on First Philosophy" continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system — allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system — was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry, important to the discovery of calculus and analysis.
Godfried T. Toussaint is a Canadian Computer Scientist, a Professor of Computer Science, and the Head of the Computer Science Program at New York University Abu Dhabi (NYUAD) in Abu Dhabi, United Arab Emirates. He is considered to be the father of computational geometry in Canada. He does research on various aspects of computational geometry, discrete geometry, and their applications: pattern recognition (k-nearest neighbor algorithm, cluster analysis), motion planning, visualization (computer graphics), knot theory (stuck unknot problem), linkage (mechanical) reconfiguration, the art gallery problem, polygon triangulation, the largest empty circle problem, unimodality (unimodal function), and others. Other interests include meander (art), compass and straightedge constructions, instance-based learning, music information retrieval, and computational music theory.
Apparently the first to do so was G. S. Klügel (1739–1812), a doctoral student at the University of Gottingen, with the support of his teacher A. G. Kästner, in the former's 1763 dissertation "Conatuum praecipuorum theoriam parallelarum demonstrandi recensio" (Review of the Most Celebrated Attempts at Demonstrating the Theory of Parallels). In this work Klügel examined 28 attempts to prove Postulate 5 (including Saccheri's), found them all deficient, and offered the opinion that Postulate 5 is unprovable and is supported solely by the judgment of our senses.
The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Circa 1813, Carl Friedrich Gauss and independently around 1818, the German professor of law Ferdinand Karl Schweikart had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Then, around 1830, the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky separately published treatises on what we today call hyperbolic geometry. Consequently, hyperbolic geometry has been called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter "k". Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences. 
The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868.
Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art. The theoretical basis for descriptive geometry is provided by planar geometric projections. The earliest known publication on the technique was "Underweysung der Messung mit dem Zirckel und Richtscheyt", published in Linien, Nuremberg: 1525, by Albrecht Dürer. Gaspard Monge is usually considered the "father of descriptive geometry" due to his developments in geometric problem solving. His first discoveries were in 1765 while he was working as a draftsman for military fortifications, although his findings were published later on.