Abu-Abdullah Muhammad ibn Īsa
Māhānī (ابوعبدالله محمد بن عیسی
ماهانی, flourished c. 860 and died c.
880) was a Persian Muslim mathematician and astronomer born in Mahan, (in today
Kermān, Iran) and active in Baghdad, Abbasid Caliphate.
His known mathematical
works included his commentaries on Euclid's Elements, Archimedes' On the Sphere
and Cylinder and Menelaus' Sphaerica, as well as two independent treatises. He
unsuccessfully tried to solve a problem posed by Archimedes of cutting a sphere
into two volumes of a given ratio, which was later solved by 10th century
mathematician Abū Ja'far al-Khāzin.
His only known surviving work on
astronomy was on the calculation of azimuths. He was also known to make
astronomical observations, and claimed his estimates of the start times of
three consecutive lunar eclipses were accurate to within half an hour.
Biography
Historians know little of
Al-Mahani's life due to lack of sources. He was born in Mahan, Persia (hence
the Nisba Al-Mahani). He was active in the 9th century CE or 3rd century AH,
lived in Baghdad c. 860 and died c. 880. From a reference in Ibn Yunus'
Hakimite Tables, he was known to make astronomical observations between 853 and
866, allowing historians to estimate the time of his life and activities.
Mathematics
His works on mathematics covered
the topics of geometry, arithmetic, and algebra. Some of his mathematical work
might have been motivated by problems he encountered in astronomy. The 10th
century catalogue Kitab al-Fihrist mentions al-Mahani's contributions in
mathematics but not those in astronomy.
He also worked on current
mathematical problems at his time. He wrote commentaries on Greek mathematical
works: Euclid's Elements, Archimedes' On the Sphere and Cylinder and Menelaus'
Sphaerica. In his commentaries he added explanations, updated the language to
use "modern" terms of his time, and reworked some of the proofs. He
also wrote a standalone treatise Fi al-Nisba ("On Relationship") and
another on the squaring of parabola.
His commentaries on the Elements
covered Books I, V, X and XII; only those on Book V and parts of those on book
X and XII survive today. In the Book V commentary, he worked on ratio,
proposing a theory on the definition of ratio based on continued fractions that
was later discovered independently by Al-Nayrizi.
In the Book X commentary, he
worked on irrational numbers, including quadratic irrational numbers and cubic
ones. He expanded Euclid's definition of magnitudes—which included only
geometrical lines—by adding integers and fractions as rational magnitudes as
well as square and cubic roots as irrational magnitudes. He called square roots
"plane irrationalities" and cubic roots "solid
irrationalities", and classified the sums or differences of these roots,
as well as the results of the roots' additions or substractions from rational
magnitudes, also as irrational magnitudes. He then explained Book X using those
rational and irrational magnitudes instead of geometric magnitudes like in the
original.
His commentaries of the Sphaerica
covered book I and parts of book II, none of which survive today. His edition
was later updated by Ahmad ibn Abi Said al-Harawi (10th century). Later, Nasir
al-Din al-Tusi (1201–1274) dismissed Al-Mahani and Al-Harawi's edition and
wrote his own treatment of the Sphaerica, based on the works on Abu Nasr
Mansur. Al-Tusi's edition became the most widely known edition of the Sphaerica
in the Arabic-speaking world.
Al-Mahani also attempted to solve
a problem posed by Archimedes in On the Sphere and Cylinder, book II, chapter
4: how to divide a sphere by a plane into two volumes of a given ratio. His
work led him to an equation, known as "Al-Mahani's equation" in the
Muslim world: . However, as documented
later by Omar Khayyam, "after giving it lengthy meditation" he
eventually failed to solve the problem. The problem was then considered
unsolvable until 10th century Persian mathematician Abu Ja'far al-Khazin solved
it using conic sections.
Astronomy
His astronomical observations of conjunctions as well as solar and lunar
eclipses was cited in the zij (astronomical tables) of Ibn Yunus (c. 950 –
1009). Ibn Yunus quoted Al-Mahani as saying that he calculated their timings
with an astrolabe. He claimed his estimates of the start times of three
consecutive lunar eclipses were accurate to within half an hour.
He also wrote a treatise, Maqala fi ma'rifat as-samt li-aiy sa'a aradta wa
fi aiy maudi aradta ("On the Determination of the Azimuth for an Arbitrary
Time and an Arbitrary Place"), his only known surviving work on astronomy.
In it, he provided two graphical methods and an arithmetic one of calculating
the azimuth—the angular measurement of a heavenly object's location. The
arithmetic method corresponds to the cosine rule in spherical trigonometry, and
was later used by Al-Battani (c. 858 – 929).
He wrote another treatise, whose title, On the Latitude of the Stars, is
known but its content is entirely lost. According to later astronomer Ibrahim
ibn Sinan (908–946), Al-Mahani also wrote a treatise on calculating the
ascendant using a solar clock.
Sumber
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