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

Tus, Iran Tus () was an ancient city in Khorasan near the modern city of Mashhad, Razavi Khorasan province, Iran. To the ancient Greeks, it was known as Susia (). It was also known as Tusa. The area now known as Tus was divided into four cities, Ta ...

–

Iran Iran, officially the Islamic Republic of Iran (IRI) and also known as Persia, is a country in West Asia. It borders Iraq to the west, Turkey, Azerbaijan, and Armenia to the northwest, the Caspian Sea to the north, Turkmenistan to the nort ...

) known more often as Sharaf al-Dīn al-Ṭūsī or Sharaf ad-Dīn aṭ-Ṭūsī, was an

Iranian Iranian () may refer to: * Something of, from, or related to Iran ** Iranian diaspora, Iranians living outside Iran ** Iranian architecture, architecture of Iran and parts of the rest of West Asia ** Iranian cuisine, cooking traditions and practic ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

and

astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...

of the

Islamic Golden Age The Islamic Golden Age was a period of scientific, economic, and cultural flourishing in the history of Islam, traditionally dated from the 8th century to the 13th century. This period is traditionally understood to have begun during the reign o ...

(during the

Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th to the late 15th centuries, similarly to the post-classical period of global history. It began with the fall of the Western Roman Empire and ...

Biography

Al-Tusi was probably born in

. Little is known about his life, except what is found in the biographies of other scientists and that most mathematicians today can trace their lineage back to him. Around 1165, he moved to

Damascus Damascus ( , ; ) is the capital and List of largest cities in the Levant region by population, largest city of Syria. It is the oldest capital in the world and, according to some, the fourth Holiest sites in Islam, holiest city in Islam. Kno ...

and taught mathematics there. He then lived in

Aleppo Aleppo is a city in Syria, which serves as the capital of the Aleppo Governorate, the most populous Governorates of Syria, governorate of Syria. With an estimated population of 2,098,000 residents it is Syria's largest city by urban area, and ...

for three years, before moving to

Mosul Mosul ( ; , , ; ; ; ) is a major city in northern Iraq, serving as the capital of Nineveh Governorate. It is the second largest city in Iraq overall after the capital Baghdad. Situated on the banks of Tigris, the city encloses the ruins of the ...

, where he met his most famous disciple Kamal al-Din ibn Yunus (1156-1242). Kamal al-Din would later become the teacher of another famous mathematician from Tus,

Nasir al-Din al-Tusi Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī (1201 – 1274), also known as Naṣīr al-Dīn al-Ṭūsī (; ) or simply as (al-)Tusi, was a Persians, Persian polymath, architect, Early Islamic philosophy, philosopher, Islamic medicine, phy ...

. According to Ibn Abi Usaibi'a, Sharaf al-Din was "outstanding in

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

and the mathematical sciences, having no equal in his time".

Mathematics

Al-Tusi has been credited with proposing the idea of a function, however his approach being not very explicit, algebra's decisive move to the dynamic function was made 5 centuries after him, by German polymath Gottfried Leibniz. Sharaf al-Din used what would later be known as the " Ruffini- Horner method" to numerically approximate the

root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...

of a

cubic equation In algebra, a cubic equation in one variable is an equation of the form ax^3+bx^2+cx+d=0 in which is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...

. He also developed a novel method for determining the conditions under which certain types of cubic equations would have two, one, or no solutions. 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. The equations in question can be written, using modern notation, in the form , where is a cubic polynomial in which the

coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...

of the cubic term is , and 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 . For each of these five types, al-Tusi wrote down an expression for the point where the function attained its

maximum In mathematical analysis, the maximum and minimum of a function (mathematics), function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given Interval (ma ...

, and gave a geometric proof that for any positive different from . He then concluded that the equation would have two solutions if , one solution if , or none if . Al-Tusi gave no indication of how he discovered the expressions for the maxima of the functions . Some scholars have concluded that al-Tusi obtained his expressions for these maxima by "systematically" taking the derivative of the function , and setting it equal to zero. 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. The quantities 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 In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...

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 , , or , rather than the corresponding forms , , or , 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. Sharaf al-Din analyzed the equation ''x''³ + ''d'' = ''b''⋅''x''² in the form ''x''² ⋅ (''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. 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 or European world. Sharaf al-Din al-Tusi's "Treatise on equations" has been described by Roshdi Rashed as inaugurating the beginning of

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

. This was criticized by Jeffrey Oaks who claims that Al-Tusi did not study curves by means of equations, but rather equations by means of curves (just as al-Khayyam had done before him) and that the study of curves by means of equations originated with Descartes in the seventeenth century.

Astronomy

Sharaf al-Din invented a linear

astrolabe An astrolabe (; ; ) is an astronomy, astronomical list of astronomical instruments, instrument dating to ancient times. It serves as a star chart and Model#Physical model, physical model of the visible celestial sphere, half-dome of the sky. It ...

, sometimes called the "Staff of Tusi". While it was easier to construct and was known in

al-Andalus Al-Andalus () was the Muslim-ruled area of the Iberian Peninsula. The name refers to the different Muslim states that controlled these territories at various times between 711 and 1492. At its greatest geographical extent, it occupied most o ...

, it did not gain much popularity.

Honours

The main-belt asteroid 7058 Al-Ṭūsī, discovered by Henry E. Holt at

Palomar Observatory The Palomar Observatory is an astronomical research observatory in the Palomar Mountains of San Diego County, California, United States. It is owned and operated by the California Institute of Technology (Caltech). Research time at the observat ...

in 1990, was named in his honor.

Notes

References