Born : c.
1135, Tus, present-day Iran
Died : c.
1213
Occupation : Mathematician.
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).
Biography
Tusi was probably born in Tus, Iran. Little is known
about his life, except what is found in the biographies of other scientists.
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.
According to Ibn Abi Usaibi'a, Sharaf al-Din was
"outstanding in geometry 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 move to the dynamic function was made 5 centuries
after him, by Gottfried Leibniz. 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. 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). 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 .
Al-Tusi
gave no indication of how he discovered the expressions m for the maxima of the functions f(x). 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. 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 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 , 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 x3 + d = b⋅x2 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. 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.
Sharaf
al-Din al-Tusi's "Treatise on equations" has been described as
inaugurating the beginning of algebraic geometry.
Astronomy
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.
Honours
The main-belt asteroid 7058 Al-Ṭūsī, discovered by
Henry E. Holt at Palomar Observatory in 1990, was named in his honor.
Sumber
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Mathematician
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