Sharaf al-Dīn al-Ṭūsī

Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī (Persian: شرف‌الدین مظفر بن محمد بن مظفر توسی‎; c. 1135 – c. 1213) was an Iranian mathematician and astronomer of the Islamic Golden Age (during the Middle Ages).[1]

Sharaf al-Dīn al-Ṭūsī
Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī

c. 1135
Tus, present-day Iran
Diedc. 1213
EraIslamic Golden Age


Tusi was probably born in Tus, Iran. Little is known about his life, except what is found in the biographies of other scientists.[2]

Around 1165, he moved to Damascus and taught mathematics there. He then lived in Aleppo for three years, before moving to Mosul, where he met his most famous disciple Kamal al-Din ibn Yunus (1156-1242). This Kamal al-Din would later become the teacher of another famous mathematician from Tus, Nasir al-Din al-Tusi.[2]

According to Ibn Abi Usaibi'a, Sharaf al-Din was "outstanding in geometry and the mathematical sciences, having no equal in his time".[3][4]


Sharaf al-Din used what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also developed a novel method for determining the conditions under which certain types of cubic equations would have two, one, or no solutions.[5] The equations in question can be written, using modern notation, in the form  f(x) = c, where  f(x)  is a cubic polynomial in which the coefficient of the cubic term  x3  is  −1, and  c  is positive. The Muslim mathematicians of the time divided the potentially solvable cases of these equations into five different types, determined by the signs of the other coefficients of  f(x).[6] For each of these five types, al-Tusi wrote down an expression  m  for the point where the function  f(x)  attained its maximum, and gave a geometric proof that  f(x) < f(m)  for any positive  x  different from  m. He then concluded that the equation would have two solutions if  c < f(m), one solution if  c = f(m), or none if   f(m) < c .[7]

Al-Tusi gave no indication of how he discovered the expressions  m  for the maxima of the functions  f(x).[8] Some scholars have concluded that al-Tusi obtained his expressions for these maxima by "systematically" taking the derivative of the function  f(x), and setting it equal to zero.[9] This conclusion has been challenged, however, by others, who point out that al-Tusi nowhere wrote down an expression for the derivative, and suggest other plausible methods by which he could have discovered his expressions for the maxima.[10]

The quantities   D = f(m) − c  which can be obtained from al-Tusi's conditions for the numbers of roots of cubic equations by subtracting one side of these conditions from the other is today called the discriminant of the cubic polynomials obtained by subtracting one side of the corresponding cubic equations from the other. Although al-Tusi always writes these conditions in the forms  c < f(m),  c = f(m), or   f(m) < c, rather than the corresponding forms   D > 0 ,   D = 0 , or   D < 0 ,[11] Roshdi Rashed nevertheless considers that his discovery of these conditions demonstrated an understanding of the importance of the discriminant for investigating the solutions of cubic equations.[12]

Sharaf al-Din analyzed the equation x3 + d = bx2 in the form x2 ⋅ (b - x) = d, stating that the left hand side must at least equal the value of d for the equation to have a solution. He then determined the maximum value of this expression. It is arguable that the isolation of this expression is an early approach to the notion of a "function". A value less than d means no positive solution; a value equal to d corresponds to one solution, while a value greater than d corresponds to two solutions. Sharaf al-Din's analysis of this equation was a notable development in Islamic mathematics, but his work was not pursued any further at that time, neither in the Muslim world nor in Europe.[13]

Sharaf al-Din al-Tusi's "Treatise on equations" has been described as inaugurating the beginning of algebraic geometry.[14]


Sharaf al-Din invented a linear astrolabe, sometimes called the "staff of Tusi". While easier to construct and was known in al-Andalus, it did not gain much popularity.[3]


The main-belt asteroid 7058 Al-Ṭūsī, discovered by Henry E. Holt at Palomar Observatory in 1990, was named in his honor.[15]


  1. ^ Smith (1997a, p.75),"This was invented by Iranian mathematician Sharaf al-Din al-Tusi (d. ca. 1213), and was known as "Al-Tusi's cane""
  2. ^ a b O'Connor & Robertson (1999).
  3. ^ a b Berggren 2008.
  4. ^ Mentioned in the biography of the Damascene architect and physician Abu al-Fadhl al-Harithi (d. 1202-3).
  5. ^ O'Connor & Robertson (1999). To al-Tusi, "solution" meant "positive solution", since the possibility of zero or negative numbers being considered genuine solutions had yet to be recognised at the time (Hogendijk, 1989, p.71; 1997, p.894; Smith, 1997b, p.69).
  6. ^ The five types were:
    • a x2x3 = c
    • b xx3 = c
    • b xa x2x3 = c
    • b x + a x2x3 = c
    • b x + a x2x3 = c
    where  a  and  b  are positive numbers (Hogendijk, 1989, p.71). For any other values of the coefficients of  x  and  x2, the equation  f(x) = c  has no positive solution.
  7. ^ Hogendijk (1989, p.71–2).
  8. ^ Berggren (1990, p.307–8).
  9. ^ Rashed (1994, p.49), Farès (1995).
  10. ^ Berggren (1990), Hogendijk (1989).
  11. ^ Hogendijk (1989).
  12. ^ Rashed (1994, pp.46–47, 342–43).
  13. ^ Katz, Victor; Barton, Bill (October 2007). "Stages in the History of Algebra with Implications for Teaching". Educational Studies in Mathematics. 66: 192.
  14. ^ Rashed (1994, pp.102-3)
  15. ^ "7058 Al-Tusi (1990 SN1)". Minor Planet Center. Retrieved 21 November 2016.


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Greater Khorasan

Khorasan (Middle Persian: Xwarāsān; Persian: خراسان‎ Xorāsān, Persian pronunciation: [xoɾɒːˈsɒːn] listen ), sometimes called Greater Khorasan, is a historical region lying in northeast of Greater Persia, including part of Central Asia and Afghanistan. The name simply means "East, Orient" (literally "sunrise") and loosely includes the territory of the Sasanian Empire north-east of Persia proper. Early Islamic usage often regarded everywhere east of so-called Jibal or what was subsequently termed 'Iraq Ajami' (Persian Iraq), as being included in a vast and loosely-defined region of Khorasan, which might even extend to the Indus Valley and Sindh. During the Islamic period, Khorasan along with Persian Iraq were two important territories. The boundary between these two was the region surrounding the cities of Gurgan and Qumis (modern Damghan). In particular, the Ghaznavids, Seljuqs and Timurids divided their empires into Iraqi and Khorasani regions.

The main cities of Khorasan in the Islamic period were Balkh and Herat (now in Afghanistan), Mashhad and Nishapur (now in northeastern Iran), Merv and Nisa (now in southern Turkmenistan), and Bukhara and Samarkand (now in southern Uzbekistan). The cities of Merv and Nisa have since been abandoned but the other cities remain integral parts of their respective states. The term Khorasan tended to further extend from these urban centers into the rural regions of their respective west, east, north and south. Sources from the 10th-century onwards refer to areas in the south of the Hindu Kush as the Khorasan Marches, forming a frontier region between Khorasan and Hindustan.Greater Khorasan is today sometimes used to distinguish the larger historical region from the modern Khorasan Province of Iran (1906–2004), which roughly encompassed the western half of the historical Greater Khorasan.

Horner's method

In mathematics, the term Horner's rule (or Horner's method, Horner's scheme etc) refers to a polynomial evaluation method named after William George Horner expressed by

This allows evaluation of a polynomial of degree n with only multiplications and additions. This is optimal, since there are polynomials of degree n that cannot be evaluated with fewer arithmetic operations.[citation needed]

This algorithm is much older than Horner. He himself ascribed it to Joseph-Louis Lagrange but it can be traced back many hundreds of years to Chinese and Persian mathematicians.

Horner's root-finding method: Until computers came into general use in about 1970 the term 'Horner's method' was used the refer to a root-finding method for polynomials named after Horner who described a similar method in 1819. This method was widely used and became a standard method for hand calculation. It gave a convenient way for using the Newton–Raphson method for polynomials. It relied on the algorithm for polynomial evaluation now named after Horner. After the introduction of computers this root-finding method went out of use and as a result the term Horner's method (rule etc) has become understood to mean just the polynomial evaluation algorithm.

List of Shia Muslims

The following is a list of notable Shia Muslims.

List of minor planets named after people

This is a list of minor planets named after people, both real and fictional.


Mashhad (Persian: مشهد‎, Mašhad [mæʃˈhæd] (listen)), also spelled Mashad or Meshad, is the second most populous city in Iran and the capital of Razavi Khorasan Province. It is located in the northeast of the country, near the borders with Turkmenistan and Afghanistan. It has a population of 3,001,184 inhabitants (2016 census), which includes the areas of Mashhad Taman and Torqabeh. It was a major oasis along the ancient Silk Road connecting with Merv to the east.

The city is named after the "shrine" of Imam Reza, the eighth Shia Imam. The Imam was buried in a village in Khorasan, which afterwards gained the name Mashhad, meaning the place of martyrdom. Every year, millions of pilgrims visit the Imam Reza shrine. The Abbasid caliph Harun al-Rashid is also buried within the shrine.

Mashhad has been governed by different ethnic groups over the course of its history. The city enjoyed relative prosperity in the Mongol period.

Mashhad is also known colloquially as the city of Ferdowsi, after the Iranian poet who composed the Shahnameh. The city is the hometown of some of the most significant Iranian literary figures and artists, such as the poet Mehdi Akhavan-Sales, and Mohammad-Reza Shajarian, the traditional Iranian singer and composer. Ferdowsi and Akhavan Sales are both buried in Tus, an ancient city that is considered to be the main origin of the current city of Mashhad.

On 30 October 2009 (the anniversary of the death of Imam Reza), Iran's then-President Mahmoud Ahmadinejad declared Mashhad to be "Iran's spiritual capital".


Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity, structure, space, and change.Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.

Mathematics in medieval Islam

Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra (named for The Compendious Book on Calculation by Completion and Balancing by scholar Al-Khwarizmi), and advances in geometry and trigonometry.Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries.

Muhammad ibn Musa al-Khwarizmi

Muḥammad ibn Mūsā al-Khwārizmī (Persian: محمد بن موسى خوارزمی‎; c. 780 – c. 850), formerly Latinized as Algorithmi, was a Persian scholar who produced works in mathematics, astronomy, and geography under the patronage of the Caliph Al-Ma'mun of the Abbasid Caliphate. Around 820 AD he was appointed as the astronomer and head of the library of the House of Wisdom in Baghdad.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.

In the 12th century, Latin translations of his textbook on arithmetic (Algorithmo de Numero Indorum) which codified the various Indian numerals, introduced the decimal positional number system to the Western world. The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester in 1145, was used until the sixteenth century as the principal mathematical text-book of European universities.In addition to his best-known works, he revised Ptolemy's Geography, listing the longitudes and latitudes of various cities and localities. He further produced a set of astronomical tables and wrote about calendaric works, as well as the astrolabe and the sundial. He also made important contributions to trigonometry, producing accurate sine and cosine tables, and the first table of tangents.

Nasir al-Din al-Tusi

Muhammad ibn Muhammad ibn al-Hasan al-Tūsī (Persian: محمد بن محمد بن حسن طوسی‎‎ 24 February 1201 – 26 June 1274), better known as Nasir al-Din Tusi (Persian: نصیر الدین طوسی‎; or simply Tusi in the West), was a Persian polymath, architect, philosopher, physician, scientist, and theologian.

He established trigonometry as an independent branch of mathematics.

He was a Twelver Shia Muslim. Ibn Khaldun (1332–1406) claimed Tusi was the greatest of the later Persian scholars.

Sharaf al-Din

Sharaf al-Din (Arabic: شرف الدين‎) and Sharif al-Din (Arabic: شریف الدین‎) are two related male Muslim given names. They may refer to:

Abd al-Husayn Sharaf al-Din al-Musawi (1872–1957), Shi'a twelver Islamic scholar

Abdullah-Al-Muti Sharfuddin (1930–1998), Bangladeshi educationalist and popular science writer

Al-Hadi Sharaf ad-Din (1820–1890), claimant for the Zaidi imamate of Yemen

Sharaf ad-Din Ali Yazdi (died 1454), 15th-century Persian historian

Sharaf al-Dīn al-Ṭūsī, in full Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī (1135–1213), Persian mathematician and astronomer of the Islamic Golden Age

Sharafuddin of Selangor (born 1945), sultan of Selangor, Malaysia since November 2001

Sharfuddin Shah Wilayat, early Iraqi Sufi, active in India

Sheikh Sharaf ad-Din ibn al-Hasan (died 1258), head of the ‘Adawiyya Ṣūfī Order

Syed Sharifuddin Pirzada (1923–2017), Pakistani lawyer and politician

Shaikh Sharafuddeen Bu Ali Qalandar Panipati, known as Bu Ali Shah Qalandar, thirteenth century Azerbaijani-Indian Sufi saint

Timeline of science and engineering in the Islamic world

This timeline of science and engineering in the Islamic world covers the time period from the eighth century AD to the introduction of European science to the Islamic world in the nineteenth century. All year dates are given according to the Gregorian calendar except where noted.

Mathematical works
People of Khorasan
Islamic scholars
Poets and artists
Historians and
political scientists

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