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Meet Me On The Equinox - Death Cab for Cubie Chords by Joan's Genius Intro: Gm Eb Gm Eb Verse 1: Gm Eb Bb Gm Meet me on the Equinox, meet me halfway Bb Cm Eb Gm When the sun is perched. Create and get +5 IQ. Play. I don't see myself as having any special status compared to anyone else. .. who had to kill a white person on the equinox to gain magical powers. .. As a man with a genius IQ, do you think you've learned more from what. who had to kill a white person on the equinox to receive magical powers. Do you feel like you are lacking in areas not covered by an IQ score, e.g. social skills ? Hello, they call me Fisher and I'm a genius. This is When it comes to how we deal with this, you'll being to see the difference between us.
This version of Brainiac was obsessed with all things Kryptonian and the Silver Age Superman was significantly more scientifically adept than any version of the character since the Crisis on Infinite Earths. Superman has been shown far less intellectually capable since the Man of Steel series in Future Earth, super-smart general level of intelligence for Humanity on Earth. But this was the collective intellect of the species.
Which probably means individuals were only at 6th or 7th level with the outliers pushing the numbers collectively higher. But this implies any citizen of the 31st century was smarter than ANY average citizen of the 20th century by a wide margin.
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The scientist Jor-El of Krypton was individually considered at the 8th level of intellect. The average citizen of Colu is individually considered to be at the 8th level of intellect. Lex Luthor has been estimated of being a th level intellect and arguably one of the smartest humans on the DC Earth. His greatest weakness is likely a lack of significant exposure to information and education to challenge his intellect to achieve its greatest state.
In the modern DC Universe, Superman has been shown to use his natural aptitutes for skill acquisition and super computation in addition to a genius-level aptitude with using his powers and abilities. One of the greatest standard display of his super-intellect is his Fortress of Solitude. Pre-Crisis, Silver-Age Superman, particularly in the s and early s was the most scientifically inquisitive and skilled version of the character. He built his Fortress of Solitude by himself, no Kryptonian super-crystals as seen in the Donner Superman movies.
He created a shrink-ray so that he could visit the Bottle City of Kandor, shrunken by the evil Brainiac. He would also create a technology for the Kandorians to be free of their bottle prison. He tended a zoo of alien animals he saved from extinction within the Fortress. No idea how he made food for them, but they always looked health and exotic… He created his Superman robots to help tend the Fortress, stand in for him when he needed to protect his secret-identity and to assist him when he needed to assist in more than one place at one time.
While the robots lacked all of his abilities, they were quite capable of standing in for him for most issues. They did possess a degree of superhuman strength, flight and heat vision. He also programmed their artificial intelligence. He maintained a laboratory where he conducted experiments of an unknown naturehe maintained an armory of super-weapons from all over the known galaxy, and occasionally built devices the Supermobile and other exotic equipment.
He also maintained a communication array that he used to talk with aliens all over the galaxy. Post-Crisis Superman Superman, as he has been written from the Post Crisis Era to the last iteration of the DC Universe, was not considered to have as great a superhuman intellect.
However, he would rediscover technologies from Krypton, utilize them as he needed and archive them within the Fortress of Solitude.
He would rebuild his Fortress a number of times to improve security. He would also use hybridized Kryptonian and human technology created by John Henry Irons aka Steel and would also utilize alien technologies over time Thanagarian.
Whether this will remain true in the latest DCnU remains to be seen. Lacking a super-human intellect did not mean he was not intelligent, after all he was the son of the greatest scientists of Krypton, Jor-El and Lara-El.
But he did not get the grounding in science he would have needed to equal his parents capabilities. His current Fortress is still an impregnable stronghold filled with technology from all over the galaxy.
While he may not be considered a super-genius, his ability must be significant enough he is able to rebuild and maintain his Kryptonian stronghold on Earth. He does possess a photographic memory. He is able to remember everything he has read and can draw upon it at will. He has taught himself surgery, at superspeed no less. It is theorized, if he made the effort he could learn and retain any particular skill he had an interest in applying himself to. It is unclear if this is a byproduct of his powers or a natural Kryptonian ability.
He is also able to speak every language he has encountered or bothered to learn. His brain must process its activity and neural connectivity at a rate far greater than ours. This can be surmised by the fact that he is able to utilize super-speed with both precision and accuracy and has done so pretty much from the beginning of his media career. This was displayed most often during his Pre-Crisis era when he would have a yearly race with the Flash around the world. Their challenge was to keep the race sub-sonic because Superman would case catastrophic damage as he pass areas creating sonic booms.
The Flash does not have this problem, because of his speed aura. He is famous for his prime number Sieve, but more impressive was his work on the cube-doubling problem which he related to the design of siege weapons catapults where a cube-root calculation is needed. Eratosthenes had the nickname Beta; he was a master of several fields, but was only second-best of his time. His better was also his good friend: Archimedes of Syracuse dedicated The Method to Eratosthenes. Euclid, Eudoxus and Archytas are other candidates for this honor.
His writings on conic sections have been studied until modern times; he developed methods for normals and curvature. He is often credited with inventing the names for parabola, hyperbola and ellipse; but these shapes were previously described by Menaechmus, and their names may also predate Apollonius.
Although astronomers eventually concluded it was not physically correct, Apollonius developed the "epicycle and deferent" model of planetary orbits, and proved important theorems in this area. He deliberately emphasized the beauty of pure, rather than applied, mathematics, saying his theorems were "worthy of acceptance for the sake of the demonstrations themselves.
Many of his works have survived only in a fragmentary form, and the proofs were completely lost. Most famous was the Problem of Apollonius, which is to find a circle tangent to three objects, with the objects being points, lines, or circles, in any combination. Constructing the eight circles each tangent to three other circles is especially challenging, but just finding the two circles containing two given points and tangent to a given line is a serious challenge.
Vieta was renowned for discovering methods for all ten cases of this Problem. Other great mathematicians who have enjoyed reconstructing Apollonius' lost theorems include Fermat, Pascal, Newton, Euler, Poncelet and Gauss. In evaluating the genius of the ancient Greeks, it is well to remember that their achievements were made without the convenience of modern notation.
It is clear from his writing that Apollonius almost developed the analytic geometry of Descartes, but failed due to the lack of such elementary concepts as negative numbers. Leibniz wrote "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times. There is some evidence that Chinese writings influenced India and the Islamic Empire, and thus, indirectly, Europe. Although there were great Chinese mathematicians a thousand years before the Han Dynasty as evidenced by the ancient Zhoubi Suanjingand innovations continued for centuries after Han, the textbook Nine Chapters on the Mathematical Art has special importance.
Many of the mathematical concepts of the early Greeks were discovered independently in early China. Chang's book gives methods of arithmetic including cube roots and algebra, uses the decimal system though zero was represented as just a space, rather than a discrete symbolproves the Pythagorean Theorem, and includes a clever geometric proof that the perimeter of a right triangle times the radius of its inscribing circle equals the area of its circumscribing rectangle.
Some of this may have been added after the time of Chang; some additions attributed to Liu Hui are mentioned in his mini-bio; other famous contributors are Jing Fang and Zhang Heng. Nine Chapters was probably based on earlier books, lost during the great book burning of BC, and Chang himself may have been a lord who commissioned others to prepare the book. Moreover, important revisions and commentaries were added after Chang, notably by Liu Hui ca Although Liu Hui mentions Chang's skill, it isn't clear Chang had the mathematical genius to qualify for this list, but he would still be a strong candidate due to his book's immense historical importance: It was the dominant Chinese mathematical text for centuries, and had great influence throughout the Far East.
After Chang, Chinese mathematics continued to flourish, discovering trigonometry, matrix methods, the Binomial Theorem, etc. Some of the teachings made their way to India, and from there to the Islamic world and Europe.
There is some evidence that the Hindus borrowed the decimal system itself from books like Nine Chapters. No one person can be credited with the invention of the decimal system, but key roles were played by early Chinese Chang Tshang and Liu HuiBrahmagupta and earlier Hindus including Aryabhataand Leonardo Fibonacci.
After Fibonacci, Europe still did not embrace the decimal system until the works of Vieta, Stevin, and Napier.
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Hipparchus of Nicaea and Rhodes ca BC Greek domain Ptolemy may be the most famous astronomer before Copernicus, but he borrowed heavily from Hipparchus, who should thus be considered along with Galileo and Edwin Hubble to be one of the three greatest astronomers ever.
Careful study of the errors in the catalogs of Ptolemy and Hipparchus reveal both that Ptolemy borrowed his data from Hipparchus, and that Hipparchus used principles of spherical trig to simplify his work.
Classical Hindu astronomers, including the 6th-century genius Aryabhata, borrow much from Ptolemy and Hipparchus. Hipparchus is called the "Father of Trigonometry"; he developed spherical trigonometry, produced trig tables, and more. He produced at least fourteen texts of physics and mathematics nearly all of which have been lost, but which seem to have had great teachings, including much of Newton's Laws of Motion.
He invented the circle-conformal stereographic and orthographic map projections which carry his name. As an astronomer, Hipparchus is credited with the discovery of equinox precession, length of the year, thorough star catalogs, and invention of the armillary sphere and perhaps the astrolabe.
He had great historical influence in Europe, India and Persia, at least if credited also with Ptolemy's influence. Hipparchus himself was influenced by Babylonian astronomers. The Antikythera mechanism is an astronomical clock considered amazing for its time.
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It may have been built about the time of Hipparchus' death, but lost after a few decades remaining at the bottom of the sea for years. The mechanism implemented the complex orbits which Hipparchus had developed to explain irregular planetary motions; it's not unlikely the great genius helped design this intricate analog computer, which may have been built in Rhodes where Hipparchus spent his final decades. Recent studies suggest that the mechanism was designed in Archimedes' time, and that therefore that genius might have been the designer.
Menelaus of Alexandria ca Egypt, Rome Menelaus wrote several books on geometry and trigonometry, mostly lost except for his works on solid geometry. His work was cited by Ptolemy, Pappus, and Thabit; especially the Theorem of Menelaus itself which is a fundamental and difficult theorem very useful in projective geometry. He also contributed much to spherical trigonometry. Disdaining indirect proofs anticipating later-day constructivists Menelaus found new, more fruitful proofs for several of Euclid's results.
This theorem has many useful corollaries; it was frequently applied in Copernicus' work. Ptolemy also wrote on trigonometry, optics, geography, map projections, and astrology; but is most famous for his astronomy, where he perfected the geocentric model of planetary motions.
For this work, Cardano included Ptolemy on his List of 12 Greatest Geniuses, but removed him from the list after learning of Copernicus' discovery. Interestingly, Ptolemy wrote that the fixed point in a model of planetary motion was arbitrary, but rejected the Earth spinning on its axis since he thought this would lead to powerful winds.
Ptolemy discussed and tabulated the 'equation of time,' documenting the irregular apparent motion of the Sun. It took fifteen centuries before this irregularity was correctly attributed to Earth's elliptical orbit. Heliocentrism The mystery of celestial motions directed scientific inquiry for thousands of years. With the notable exception of the Pythagorean Philolaus of Croton, thinkers generally assumed that the Earth was the center of the universe, but this made it very difficult to explain the orbits of the other planets.
This problem had been considered by Eudoxus, Apollonius, and Hipparchus, who developed a very complicated geocentric model involving concentric spheres and epicyles.Death Cab For Cutie Meet me on the Equinox (+ Lyrics)
Ptolemy perfected or, rather, complicated this model even further, introducing 'equants' to further fine-tune the orbital speeds; this model was the standard for 14 centuries. While some Greeks, notably Aristarchus and Seleucus of Seleucia and perhaps also Heraclides of Pontus or ancient Egyptiansproposed heliocentric models, these were rejected because there was no parallax among stars.
Aristarchus guessed that the stars were at an almost unimaginable distance, explaining the lack of parallax. Aristarchus would be almost unknown except that Archimedes mentions, and assumes, Aristarchus' heliocentrism in The Sand Reckoner. I suspect that Archimedes accepted heliocentrism, but thought saying so openly would distract from his work.
Hipparchus was another ancient Greek who considered heliocentrism but, because he never guessed that orbits were ellipses rather than cascaded circles, was unable to come up with a heliocentric model that fit his data. The great skill demonstrated by Ptolemy and his predecessors in developing their complex geocentric cosmology may have set back science since in fact the Earth rotates around the Sun.
The geocentric models couldn't explain the observed changes in the brightness of Mars or Venus, but it was the phases of Venus, discovered by Galileo after the invention of the telescope, that finally led to general acceptance of heliocentrism. Ptolemy's model predicted phases, but timed quite differently from Galileo's observations. Since the planets move without friction, their motions offer a pure view of the Laws of Motion; this is one reason that the heliocentric breakthroughs of Copernicus, Kepler and Newton triggered the advances in mathematical physics which led to the Scientific Revolution.
Heliocentrism offered an even more key understanding that lead to massive change in scientific thought. For Ptolemy and other geocentrists, the "fixed" stars were just lights on a sphere rotating around the earth, but after the Copernican Revolution the fixed stars were understood to be immensely far away; this made it possible to imagine that they were themselves suns, perhaps with planets of their own. Nicole Oresme and Nicholas of Cusa were pre-Copernican thinkers who wrote on both the geocentric question and the possibility of other worlds.
The Copernican perspective led Giordano Bruno and Galileo to posit a single common set of physical laws which ruled both on Earth and in the Heavens. It was this, rather than just the happenstance of planetary orbits, that eventually most outraged the Roman Church And we're getting ahead of our story: Copernicus, Bruno, Galileo and Kepler lived 14 centuries after Ptolemy. Liu Hui ca China Liu Hui made major improvements to Chang's influential textbook Nine Chapters, making him among the most important of Chinese mathematicians ever.
He seems to have been a much better mathematician than Chang, but just as Newton might have gotten nowhere without Kepler, Vieta, Huygens, Fermat, Wallis, Cavalieri, etc.
Among Liu's achievements are an emphasis on generalizations and proofs, incorporation of negative numbers into arithmetic, an early recognition of the notions of infinitesimals and limits, the Gaussian elimination method of solving simultaneous linear equations, calculations of solid volumes including the use of Cavalieri's Principleanticipation of Horner's Method, and a new method to calculate square roots.
Like Archimedes, Liu discovered the formula for a circle's area; however he failed to calculate a sphere's volume, writing "Let us leave this problem to whoever can tell the truth. It seems fitting that Liu Hui did join that select company of record setters: He also devised an interpolation formula to simplify that calculation; this yielded the "good-enough" value 3.
Diophantus of Alexandria ca Greece, Egypt Diophantus was one of the most influential mathematicians of antiquity; he wrote several books on arithmetic and algebra, and explored number theory further than anyone earlier.
He advanced a rudimentary arithmetic and algebraic notation, allowed rational-number solutions to his problems rather than just integers, and was aware of results like the Brahmagupta-Fibonacci Identity; for these reasons he is often called the "Father of Algebra.
His notation, clumsy as it was, was used for many centuries. The shorthand x3 for "x cubed" was not invented until Descartes.
Very little is known about Diophantus he might even have come from Babylonia, whose algebraic ideas he borrowed. Many of his works have been lost, including proofs for lemmas cited in the surviving work, some of which are so difficult it would almost stagger the imagination to believe Diophantus really had proofs. Among these are Fermat's conjecture Lagrange's theorem that every integer is the sum of four squares, and the following: It seems unlikely that Diophantus actually had proofs for such "lemmas.
He wrote about arithmetic methods, plane and solid geometry, the axiomatic method, celestial motions and mechanics. In addition to his own original research, his texts are noteworthy for preserving works of earlier mathematicians that would otherwise have been lost. Pappus' best and most original result, and the one which gave him most pride, may be the Pappus Centroid theorems fundamental, difficult and powerful theorems of solid geometry later rediscovered by Paul Guldin. His other ingenious geometric theorems include Desargues' Homology Theorem which Pappus attributes to Euclidan early form of Pascal's Hexagram Theorem, called Pappus' Hexagon Theorem and related to a fundamental theorem: Two projective pencils can always be brought into a perspective position.
For these theorems, Pappus is sometimes called the "Father of Projective Geometry. He stated but didn't prove the Isoperimetric Theorem, also writing "Bees know this fact which is useful to them, that the hexagon Pappus stated, but did not fully solve, the Problem of Pappus which, given an arbitrary collection of lines in the plane, asks for the locus of points whose distances to the lines have a certain relationship.
This problem was a major inspiration for Descartes and was finally fully solved by Newton. For preserving the teachings of Euclid and Apollonius, as well as his own theorems of geometry, Pappus certainly belongs on a list of great ancient mathematicians. But these teachings lay dormant during Europe's Dark Ages, diminishing Pappus' historical significance.
Greece was eventually absorbed into the Roman Empire with Archimedes himself famously killed by a Roman soldier. Rome did not pursue pure science as Greece had as we've seen, the important mathematicians of the Roman era were based in the Hellenic East and eventually Europe fell into a Dark Age.
The Greek emphasis on pure mathematics and proofs was key to the future of mathematics, but they were missing an even more important catalyst: Top Decimal system -- from India? Laplace called the decimal system "a profound and important idea [given by India] which appears so simple to us now that we ignore its true merit Ancient Greeks, by the way, did not use the unwieldy Roman numerals, but rather used 27 symbols, denoting 1 to 9, 10 to 90, and to Unlike our system, with ten digits separate from the alphabet, the 27 Greek number symbols were the same as their alphabet's letters; this might have hindered the development of "syncopated" notation.
The most ancient Hindu records did not use the ten digits of Aryabhata, but rather a system similar to that of the ancient Greeks, suggesting that China, and not India, may indeed be the "ultimate" source of the modern decimal system.
The Chinese used a form of decimal abacus as early as BC; if it doesn't qualify, by itself, as a "decimal system" then pictorial depictions of its numbers would. Yet for thousands of years after its abacus, China had no zero symbol other than plain space; and apparently didn't have one until after the Hindus.
Ancient Persians and Mayans did have place-value notation with zero symbols, but neither qualify as inventing a base decimal system: Persia used the base Babylonian system; Mayans used base Another difference is that the Hindus had nine distinct digit symbols to go with their zero, while earlier place-value systems built up from just two symbols: The decimal place-value system with zero symbol seems to be an obvious invention that in fact was very hard to invent.
If you insist on a single winner then India might be it. Among the Hindu mathematicians, Aryabhata called Arjehir by Arabs may be most famous. While Europe was in its early "Dark Age," Aryabhata advanced arithmetic, algebra, elementary analysis, and especially plane and spherical trigonometry, using the decimal system. His most famous accomplishment in mathematics was the Aryabhata Algorithm connected to continued fractions for solving Diophantine equations.
Aryabhata made several important discoveries in astronomy, e. He was among the very few ancient scholars who realized the Earth rotated daily on an axis; claims that he also espoused heliocentric orbits are controversial, but may be confirmed by the writings of al-Biruni.
Aryabhata is said to have introduced the constant e. Others claim these were first seen years earlier in Chang Tshang's Chinese text and were implicit in what survives of earlier Hindu works, but Brahmagupta's text discussed them lucidly. Along with Diophantus, Brahmagupta was also among the first to express equations with symbols rather than words. Several theorems bear his name, including the formula for the area of a cyclic quadrilateral: Proving Brahmagupta's theorems are good challenges even today.
In addition to his famous writings on practical mathematics and his ingenious theorems of geometry, Brahmagupta solved the general quadratic equation, and worked on number theory problems. He was first to find a general solution to the simplest Diophantine form.
His work on Pell's equations has been called "brilliant" and "marvelous. He applied mathematics to astronomy, predicting eclipses, etc. He preserved some of the teachings of Aryabhata which would otherwise have been lost; these include a famous formula giving an excellent approximation to the sin function, as well as, probably, the zero symbol itself.
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The "only if" is easy but the difficult "if" part was finally proved by Lagrange in He introduced the Hindu decimal system to the Islamic world and Europe; invented the horary quadrant; improved the sundial; developed trigonometry tables; and improved on Ptolemy's astronomy and geography.
He wrote the book Al-Jabr, which demonstrated simple algebra and geometry, and several other influential books. He also coined the word cipher, which became English zero although this was just a translation from the Sanskrit word for zero introduced by Aryabhata. He was an essential pioneer for Islamic science, and for the many Arab and Persian mathematicians who followed; and hence also for Europe's eventual Renaissance which was heavily dependent on Islamic teachings.
He invented pharmaceutical methods, perfumes, and distilling of alcohol. In mathematics, he popularized the use of the decimal system, developed spherical geometry, wrote on many other topics and was a pioneer of cryptography code-breaking. His work with code-breakig also made him a pioneer in basic concepts of probability. Al-Kindi, called The Arab Philosopher, can not be considered among the greatest of mathematicians, but was one of the most influential general scientists between Aristotle and da Vinci.
As well as being an original thinker, Thabit was a key translator of ancient Greek writings; he translated Archimedes' otherwise-lost Book of Lemmas and applied one of its methods to construct a regular heptagon. He developed an important new cosmology superior to Ptolemy's and which, though it was not heliocentric, may have inspired Copernicus.
He was perhaps the first great mathematician to take the important step of emphasizing real numbers rather than either rational numbers or geometric sizes.
He worked in plane and spherical trigonometry, and with cubic equations. Like Archimedes, he was able to calculate the area of an ellipse, and to calculate the volume of a paraboloid. He produced an elegant generalization of the Pythagorean Theorem: Thabit also worked in number theory where he is especially famous for his theorem about amicable numbers. While many of his discoveries in geometry, plane and spherical trigonometry, and analysis parabola quadrature, trigonometric law, principle of lever duplicated work by Archimedes and Pappus, Thabit's list of novel achievements is impressive.
Among the several great and famous Baghdad geometers, Thabit may have had the greatest genius. He was an early pioneer of analytic geometry, advancing the theory of integration, applying algebra to synthetic geometry, and writing on the construction of conic sections.
He produced a new proof of Archimedes' famous formula for the area of a parabolic section. He worked on the theory of area-preserving transformations, with applications to map-making.
He also advanced astronomical theory, and wrote a treatise on sundials. He's been called the best scientist of the Middle Ages; his Book of Optics has been called the most important physics text prior to Newton; his writings in physics anticipate the Principle of Least Action, Newton's First Law of Motion, and the notion that white light is composed of the color spectrum.
Like Newton, he favored a particle theory of light over the wave theory of Aristotle. His other achievements in optics include improved lens design, an analysis of the camera obscura, Snell's Law, an early explanation for the rainbow, a correct deduction from refraction of atmospheric thickness, and experiments on visual perception.
He studied optical illusions and was first to explain psychologically why the Moon appears to be larger when near the horizon. He also did work in human anatomy and medicine. In a famous leap of over-confidence he claimed he could control the Nile River; when the Caliph ordered him to do so, he then had to feign madness! Alhazen has been called the "Father of Modern Optics," the "Founder of Experimental Psychology" mainly for his work with optical illusionsand, because he emphasized hypotheses and experiments, "The First Scientist.
His best mathematical work was with plane and solid geometry, especially conic sections; he calculated the areas of lunes, volumes of paraboloids, and constructed a heptagon using intersecting parabolas. He solved Alhazen's Billiard Problem originally posed as a problem in mirror designa difficult construction which continued to intrigue several great mathematicians including Huygens.
Alhazen's attempts to prove the Parallel Postulate make him along with Thabit ibn Qurra one of the earliest mathematicians to investigate non-Euclidean geometry.
He is less famous in part because he lived in a remote part of the Islamic empire. He was a great linguist; studied the original works of Greeks and Hindus; is famous for debates with his contemporary Avicenna; studied history, biology, mineralogy, philosophy, sociology, medicine and more; is called the Father of Geodesy and the Father of Arabic Pharmacy; and was one of the greatest astronomers.
He was an early advocate of the Scientific Method. He was also noted for his poetry. He invented but didn't build a geared-astrolabe clock, and worked with springs and hydrostatics. He wrote prodigiously on all scientific topics his writings are estimated to total 13, folios ; he was especially noted for his comprehensive encyclopedia about India, and Shadows, which starts from notions about shadows but develops much astronomy and mathematics.
He anticipated future advances including Darwin's natural selection, Newton's Second Law, the immutability of elements, the nature of the Milky Way, and much modern geology. Among several novel achievements in astronomy, he used observations of lunar eclipse to deduce relative longitude, estimated Earth's radius most accurately, believed the Earth rotated on its axis and may have accepted heliocentrism as a possibility.
In mathematics, he was first to apply the Law of Sines to astronomy, geodesy, and cartography; anticipated the notion of polar coordinates; invented the azimuthal equidistant map projection in common use today, as well as a polyconic method now called the Nicolosi Globular Projection; found trigonometric solutions to polynomial equations; did geometric constructions including angle trisection; and wrote on arithmetic, algebra, and combinatorics as well as plane and spherical trigonometry and geometry.
Al-Biruni's contemporary Avicenna was not particularly a mathematician but deserves mention as an advancing scientist, as does Avicenna's disciple Abu'l-Barakat al-Baghdada, who lived about a century later.
Although he himself attributed the theorem to Archimedes, Al-Biruni provided several novel proofs for, and useful corollaries of, this famous geometric gem. While Al-Biruni may lack the influence and mathematical brilliance to qualify for the Tophe deserves recognition as one of the greatest applied mathematicians before the modern era. He did clever work with geometry, developing an alternate to Euclid's Parallel Postulate and then deriving the parallel result using theorems based on the Khayyam-Saccheri quadrilateral.
He derived solutions to cubic equations using the intersection of conic sections with circles. Remarkably, he stated that the cubic solution could not be achieved with straightedge and compass, a fact that wouldn't be proved until the 19th century.
He was a polymath: He was noted for deriving his theories from science rather than religion. He made achievements in several fields of mathematics including some Europe wouldn't learn until the time of Euler. His textbooks dealt with many matters, including solid geometry, combinations, and advanced arithmetic methods. He was also an astronomer. It is sometimes claimed that his equations for planetary motions anticipated the Laws of Motion discovered by Kepler and Newton, but this claim is doubtful.
In algebra, he solved various equations including 2nd-order Diophantine, quartic, Brouncker's and Pell's equations. His Chakravala method, an early application of mathematical induction to solve 2nd-order equations, has been called "the finest thing achieved in the theory of numbers before Lagrange" although a similar statement was made about one of Fibonacci's theorems.
Earlier Hindus, including Brahmagupta, contributed to this method. In several ways he anticipated calculus: Others, especially Gherard of Cremona, had translated Islamic mathematics, e. Two centuries earlier, the mathematician-Pope, Gerbert of Aurillac, had tried unsuccessfully to introduce the decimal system to Europe. Leonardo also re-introduced older Greek ideas like Mersenne numbers and Diophantine equations. His writings cover a very broad range including new theorems of geometry, methods to construct and convert Egyptian fractions which were still in wide useirrational numbers, the Chinese Remainder Theorem, theorems about Pythagorean triplets, and the series 1, 1, 2, 3, 5, 8, 13, He defined congruums and proved theorems about them, including a theorem establishing the conditions for three square numbers to be in consecutive arithmetic series; this has been called the finest work in number theory prior to Fermat although a similar statement was made about one of Bhaskara II's theorems.
Leonardo's proof of FLT4 is widely ignored or considered incomplete. I'm preparing a page to consider that question. Al-Farisi was another ancient mathematician who noted FLT4, although attempting no proof.
Another of Leonardo's noteworthy achievements was proving that the roots of a certain cubic equation could not have any of the constructible forms Euclid had outlined in Book 10 of his Elements. He also wrote on, but didn't prove, Wilson's Theorem.
Leonardo provided Europe with the decimal system, algebra and the 'lattice' method of multiplication, all far superior to the methods then in use. It seems hard to believe but before the decimal system, mathematicians had no notation for zero. Referring to this system, Gauss was later to exclaim "To what heights would science now be raised if Archimedes had made that discovery!
But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus.