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In the 9th century, Al-Mahani and Al-Nayrizi copied the Byzantine tradition of writing commentaries on Greek works by scholars such as Euclid.

Muslim mathematicians invented the concept of the tangent function of trigonometry.

Several Muslim mathematicians discovered methods of “doubling the cube” (using a ruled straightedge and a compass to construct a line segment whose length is the cube root of 2 times the length of a given line segment) that were different than the 5 different methods discovered by 5 different Greeks over a thousand years earlier.

The astronomer Abū Sahl al-Qūhī (940 – 1030) may have been the first to inscribe a pentagon inside a square.

Alhazen (Ibn al-Haytham) was the first person to have integrated a fourth-degree polynomial. He extended the 3rd century BC integral calculus heuristics of Archimedes into an algorithm for integrating polynomials of any positive degree.

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COMMENTARIES ON EUCLID

University-educated scholars of the [Christian, Greek-speaking] Byzantine Empire [what had been the eastern half of the Roman Empire] had a tradition of writing commentaries on the published works of Greek scholars such as the Greek mathematician Euclid.

In the 8th or 9th century, Baghdad scholar Al-Hajjaj ibn Yusuf translated Euclid’s “Elements” [13 books mostly about geometry written around 300 BC] from Greek into Arabic.

In the 9th century, Al-Mahani and Al-Nayrizi copied the Byzantine tradition of writing commentaries on Euclid.

In the 10th century Abu’l-Hasan translated books written by Euclid from Greek into Arabic.

THEODOSIUS OF BITHYNIA

Theodosius of Bithynia was a late 2nd century BC astronomer who wrote about spherical geometry of the sky, was translated into Arabic in the 9th century.

PLANE GEOMETRY THEOREMS

In the late 9th century, Al-Nayrizi devised a different proof of the Pythagorean theorem (that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the sides).

In the late 11th century, Al-Mu’taman ibn Hud on Muslim Spain discovered a proof of a theorem in plane geometry 600 years before Giovanni Ceva did, but Giovanni Ceva is often falsely given credit for the discovery of the proof in 1678.

http://en.wikipedia.org/wiki/Ceva’s_theorem

EUCLID’S PARALLEL POSTULATE

Al-Abbās ibn Said al-Jawharī (800 – 860) and several other Muslim mathematicians unsuccessfully attempted to prove Euclid’s parallel postulate.

Alhazen thought he proved Euclid’s fifth postulate (the parallel postulate), but later mathematicians discovered Alhazen had used an unproved assumption.

FERMAT’S LAST THEOREM

Abu-Mahmud al-Khujandi (Khojandi) (940 – 1000) attempted to prove the special case of Fermat’s last theorem for n = 3 but his “proof” had an error.

He was a Mongolian and probably an Ismaili (Shia Sevener).

Hundreds of years later. Pierre de Fermat worked on this theorem.

PRIME NUMBER CONJECTURE

Alhazen was the first to state a conjecture (that is of no practical value, not even for determining whether a number is a prime) about prime numbers, now known as Wilson’s Theorem: n is prime if and only if (n−1)! is 1 less than a multiple of n.

The general case is a bit difficult for a non-mathematician to understand, so I will show the particular case of the prime number 11.

10! = (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)

= (1) ((2)(6)) ((3)(4)) ((5)(9)) ((7)(8)) (10)

= (1)(12)(12)(45)(56)(10)

= (1)(11+1)(11+1)(44+1)(55+1)(11-1)

= (11k + 1)(11-1) = which is one less than a multiple of 11.

Alhazen stated (but did not prove) that if n is a prime number then all the numbers (such as 5 and 9) can be paired so that their product is 1 more than a multiple of n, but cannot be paired if n is not a prime. The first person to adequately prove this was Lagrange in 1793.

CENTER OF GRAVITY

Abu Sahl al-Kuhi in the 10th century expanded upon the writings of Archimedes on the center of gravity of geometrical shapes. Abu Sahl al-Kuhi thought that a book by Archimedes had an incorrect value for the center of gravity of a semicircle, but it was the calculation by Abu Sahl al-Kuhi that was wrong.

CHORDS

In 1027 in his book “The Book on the Determination of Chords in a Circle” al-Biruni presents 4 proofs he invented (all based on a theorem of Ptolemy) and 5 proofs of other mathematicians of the following theorem about the length of chords of a circle:

If A and G are 2 points on a circle, and if D is the midpoint of arc AG, then for any point B on the arc ADG, (the length of the line AD) squared = (the length of the line BD) squared + (the length of line AB multiplied by the length of line BG).

http://books.google.com/books?id=nFMvx4ZR_30C&pg=PA130&lpg=PA130&dq=biruni+proof&source=bl&ots=qvlTrBSvbb&sig=Hj4OQgdm11krkadW8hykp2gJSY&hl=en&sa=X&ei=i9veUoqBM6T4yQGCmYD4BA&ved=0CCoQ6AEwAAo

LAW OF SINES

Būzjānī proved that the sines of the angles in a spherical triangle are proportional to the lengths of the opposite sides, but “Theorems in Euclid on lengths of chords are essentially the same ideas we now call the law of sines.“ http://www.pballew.net/lawofsin.html

PROOF OF IRRATIONAL NUMBERS

Al-Khwarizmi wrote that the hypotenuse of a right triangle is often an irrational number, but the Greek mathematician Hippasus had produced a formal proof of this in the 5th century BC.

RAISING TO POWERS PROOF

Al-Khwarizmi invented words for squared and cubed, but in the 3rd century BC the Greek mathematician Archimedes proved (n to power a) multiplied by (n to power b) equals n to power (a+b).

MATHEMATICAL INDUCTION

Euclid in around the year 300 BC used mathematical induction (proving something to be true for a particular positive integer, and then proving that this implies it is also true for higher integers) in his proof that there is no largest prime number.

About 1300 years later, al-Karaji (953 – 1029) a Persian living in Baghdad, gave a proof by mathematical induction that the sum of the first n natural numbers was ½ n(n + 1).

Ibn Sa’id al-Maghribi (1213 -1286) (Samaw’al al-Maghribi) was the son of a Jewish rabbi. Prior to his conversion to Islam at age 33, he used mathematical induction in a proof of the coefficients of the binomial theorem.

OTHER THEOREMS

Alhazen (Ibn al-Haytham)

(965 – 1040) attempted to prove several little known theorems of geometry and mathematics.

Other Muslim mathematicians also worked on proving geometry theorems.

COMPASS AND RULED STRAIGHTEDGE CONSTRUCTIONS

Several Muslim mathematicians discovered methods of trisecting an angle with a ruled straightedge and compass that were different from the methods written about by the Greeks over a thousand years earlier.

Al-Sijzi (945 – 1020) devised an angle trisection method that used the intersection of a circle and an equilateral hyperbola.

Several Muslim mathematicians discovered methods of “doubling the cube” (using a ruled straightedge and a compass to construct a line segment whose length is the cube root of 2 times the length of a given line segment) that were different than the 5 different methods discovered by 5 different Greeks over a thousand years earlier.

The astronomer Abū Sahl al-Qūhī (940 – 1030) may have been the first to inscribe a pentagon into a square.

INTEGRAL CALCULUS AND BINOMIAL THEOREM

Alhazen (Ibn al-Haytham) (965 – 1040)

Alhazen was the first person to have integrated a fourth-degree polynomial. He extended the 3rd century BC integral calculus heuristics of Archimedes into an algorithm for integrating polynomials of any positive degree, but without using modern mathematical notation.

http://apcentral.collegeboard.com/apc/members/features/2015.html

For an explanation of integration, see http://www.mathtutordvd.com/public/Integrating-Polynomials-in-Calculus.cfm

Alhazen borrowed calculus (integration) from Archimedes and Thabit ibn Qurra. In the 3rd century BC, Archimedes, in his book “On Conoids and Spheroids”, calculated the exact volume of a paraboloid using a method similar to modern integral calculus. http://www.math.ubc.ca/~cass/courses/m308-02b/projects/sheppet/308Project.html

A book containing an identical method of computing the volume of a paraboloid was written by Thabit ibn Qurra (a 9th century Sabian star-worshiper living in Baghdad, who could read Greek and is known to have translated at least one work attributed to Archimedes from Greek into Arabic).http://books.google.com/books?id=nLBUitNR8B0C&pg=RA2-PT258&lpg=RA2-PT258&ots=DilzJ3LuQg&focus=viewport&dq=archimedes+translated+arabic&output=html_text

Thabit ibn Qurra was later forced to convert to Islam.

Alhazen (Ibn al-Haytham), however, sliced the paraboloid into infinitesimally thin slices perpendicular to a different axis.

For Alhazen’s calculation of the volume of a paraboloid, he needed to discover the algorithm for computing the sum of the 4th power of consecutive integers. (The algorithms for computing the sum of consecutive squares, and the sum of consecutive cubes were known in ancient Greece and in India. The Greek mathematician Euclid (325 BC–265 BC) around 300 BC, and the 3rd century BC Indian mathematician Pingala knew about the binomial theorem.) Alhazen used the binomial theorem and some arithmetic calculations to compute (1 + 16 + 81 … + (n to the fourth power)) by adding 6 times n to the 5th power, 15 times n to the 4th power and 10 times n cubed, then subtracting n and dividing by 30.

Al-Samawal al-Maghribi (1130 – 1180) (Al Samaw’ al Ben Yahya al Maghribi) was a Jew who converted to Islam at age 33.

He made the simple calculation that the binomial theorem result for n=12 is (1 12 66 220 495 792 924 792 495 220 66 12 1)

VOLUME OF A SPHERE

Archimedes (287 BC – 212 BC) knew how to calculate the volume of a cylinder, and he did a formal proof (using integration concepts of integral calculus) that the volume of a sphere is 2/3 the volume of a cylinder it is inscribed in. Muslim mathematicians of the Islamic Golden Age rephrased these calculations in the notation invented in the 3rd century AD by Diophantus of Alexandria.

QUADRATIC EQUATIONS AND DIFFERENTIAL CALCULUS?

The Persian mathematician Sharaf al-Dīn al-Ṭūsī (not the same person as Nasir al-Dīn al-Ṭūsī) in around the year 1213 used geometry diagrams to examine cubic polynomial equations.

[I am using modern notation and modern vocabulary to explain the concepts.]

If b is a positive number, and you plot the equation “y = bx – (x cubed)” starting at x = zero, the value of y begins at zero, rises for a while, and then falls faster and faster.

He discovered that the maximum value for y for a positive x value occurs when b = x squared. multiplied by 3. He never mentioned how he obtained that result. Probably he noticed the trend, but had no theoretical understanding of why that is true. Some speculate he knew that the first derivative of x cubed is 3 times the square of x. But he never wrote about derivatives. and there is no evidence he knew about differential calculus.

In a related topic of 3rd degree polynomial equations he combined that curve with the horizontal line produced by plotting y = a (where a is a positive number), He noticed that the line intersects the curve twice, once or not at all.

CHORDS (SIMILAR TO SINES)

A table of chords was compiled by Hipparchus of Nicaea in the 2nd century BC. [To convert a chord to a sine, take the chord of twice the angle and divide by 2.

SINES AND COSINES

Hindus in India produced the first tables of sine values and versine [1 − cosine] values.

http://en.wikipedia.org/wiki/History_of_trigonometric_function

These trigonometric functions were explained around the year 400 in the Hindu book “Surya Siddhanta”. Hindus were probably the first to write about the cosecant.

Al-Biruni (973 – 1048) published a table of sines.

PROOFS ABOUT SINES

A Greek named Euclid (325 BC–265 BC) proved the law of sines (the sines of the angles in a triangle are proportional to the lengths of the opposite sides)

Būzjānī (940 – 998) and Abu Nasr Mansur (970 – 1036) wrote their own proofs of it.

OTHER TRIGONOMETRIC FUNCTIONS

The Muslim mathematician Būzjānī created tables for the other 4 trigonometric functions.

Tangent is the ratio of sine divided by cosine. The Hindus and Greeks did not use it in their astronomy equations, but Muslims did.

Muslims never used logarithmic tables. but someone at a university in Lahore Pakistan http://www.islam4theworld.net/contribution_muslims/muslim_contributions/muslim_contributions.htm falsely claims that “Muslims had been use logarithmic tables, centuries before John Napier”.

TANGENTS

Al-Khwarizmi (780 – 850) produced the first table of tangents. His tables of sines and cosines were more precise than those published earlier by other mathematicians.

Habash al-Hasib al-Marwazi wrote about the “shadow” trigonometric function (similar to a tangent).

Al-Nayrizi in the late 10th century repeated what al-Marwazi had written about tangents.

Abū Rayhān al-Bīrūnī (973 – 1048)

Al-Biruni wrote that the tangent (which Muslims called the shadow) of an angle greater than 45° can be computed by using the reciprocal of the tangent of (90° minus the angle).

COTANGENTS

Tables of cotangents are not necessary, because you can simply subtract the angle from 90 degrees and instead use the table of tangents. Habash al-Hasib al-Marwazi (805? – 875?) in the year 832 produced the first table of cotangents.

COSECANTS

Cosecants and secants are not of much use, and are almost never used.

Muhammad ibn Jābir al-Harrānī al-Battānī (Albategnius) produced the first table of cosecants, and discovered several formulas about trigonometric functions.

SPHERICAL GEOMETRY

In spherical geometry the sum of the 3 angles of a triangles is not 180 degrees.

Spherical geometry was invented by a Greek named Theodosius of Bithynia in the 2nd century BC.

Al-Jayyani (989 – 1079) copied from Euclid’s “Elements” and from the writings of al-Khwarizmi.

Al-Jayyani’s greatest achievement was replacing the word “sky” with the word “sphere” in his writing about spherical trigonometry. Some Muslim websites are confused and mistakenly claim that he wrote formulas about right handed triangles. What he actually wrote about was formulas about right angled triangles in plane geometry (one angle is 90 degrees) like Pythagoras did.

SPHERICAL TRIGONOMETRY

Spherical trigonometry was an expansion of geometry, and was not considered to be a separate branch of mathematics.

Greeks made early contributions to spherical trigonometry.

Spherical trigonometry was studied by early Greek mathematicians such as Theodosius of Bithynia, a Greek astronomer and mathematician who wrote the Sphaerics, a book on the geometry of the sphere, and Menelaus of Alexandria, who wrote a book on spherical trigonometry called Sphaerica and developed Menelaus’ theorem.

Abu Nasr Mansur (960 – 1036) was a Persian Muslim mathematician who wrote about the law of sines in spherical trigonometry.

Muslim websites falsely claim Al-Biruni (973 – 1048) invented spherical trigonometry.

The Hindu mathematician Bhaskara II wrote about spherical trigonometry in the Siddhānta Śiromaṇ in 1140,

Greek contributions are explained at https://www.math.rutgers.edu/~cherlin/History/Papers2000/hunt.html which is where Ptolemy obtained his information.

The article at https://en.wikipedia.org/wiki/Indian_mathematics describes the Hindu contributions, and claims Hindus first invented many things Muslims caim to have invented.

Encyclopedia Britannica http://www.britannica.com/topic/trigonometry mentions contributions of the Byzantines, Hindus and Muslims to trigonometry.

SPHERICAL LAW OF TANGENTS

The 13th century Persian scholar Nasir al-Din al-Tusi (1201 – 1274) may have written that the law of tangents in plane trigonometry was also true in spherical trigonometry. The law of tangents in plane geometry is explained at https://en.wikipedia.org/wiki/Law_of_tangents

COMPUTING SINES USING POWER SERIES

Qāḍī Zāda al-Rūmī (1364 – 1436) accurately computed the sine of 1° by using the power series for trigonometric functions invented by the Hindu mathematician Madhava of Sangamagrama (1340 – 1425).

Madhava’s mathematics was later rephrased into modern notation by Taylor and Maclaurin http://www.youtube.com/watch?v=dp2ovDuWhro

COORDINATE SYSTEMS

Al-Kashi in the early 15th century produced tables for transforming between coordinate systems on the celestial sphere, such as the transformation from the ecliptic coordinate system to the equatorial coordinate system.

AMICABLE NUMBERS

Muhammad Baqir Yazdi in the 16th century discovered 9,363,584 and 9,437,056 to be a pair of amicable numbers (two different numbers so related that the sum of the proper divisors of each is equal to the other number). Amicable numbers were known to the ancient Greeks, and a general formula for discovering some of them was invented by a Sabian living in Baghdad named Thabit ibn Qurra (826–901).

FEMALE MATHEMATICIAN

Labana of Córdoba was a female Muslim mathematician in 10th century Muslim Spain who was taught (probably by her father) how to build astrolabes and how to solve difficult problems in geometry and algebra.

Image by Michael Hardy via Wikimedia Commons.

image credit https://commons.wikimedia.org/wiki/File:Pythagorean.svg#mw-jump-to-license