836-901 Thabit Ibn Qurrah ( THEBIT)
[Astronomy ,
Mechanic)
Thabit Ibn Qurra Ibn Marwan al-Sabi al-Harrani was born in
the year 836 C.E. at Harran (present Turkey). As the name indicates he was
basically a member of the Sabian sect, but the great Muslim mathematician
Muhammad Ibn Musa Ibn Shakir, impressed by his knowledge of languages, and
realising his potential for a scientific career, selected him to join the
scientific group at Baghdad that was being patronised by the Abbasid Caliphs.
There, he studied under the famous Banu Musa brothers. It was in this setting
that Thabit contributed to several branches of science, notably mathematics,
astronomy and mechanics, in addition to translating a large number of works
from Greek to Arabic. Later, he was patronised by the Abbasid Caliph
al-M'utadid. After a long career of scholarship, Thabit died at Baghdad in 901
C.E.
Thabit's major contribution lies in mathematics and
astronomy. He was instrumental in extending the concept of traditional geometry
to geometrical algebra and proposed several theories that led to the development
of non-Euclidean geometry, spherical trigonometry, integral calculus and real
numbers. He criticised a number of theorems of Euclid's elements and proposed
important improvements. He applied arithmetical terminology to geometrical
quantities, and studied several aspects of conic sections, notably those of parabola and
ellipse. A number of his computations aimed at determining the surfaces and
volumes of different types of bodies and constitute, in fact, the processes of
integral calculus, as developed later.
Thabit ibn Qurra was a native of Harran and a member of the
Sabian sect. The Sabian religious sect were star worshippers from Harran often
confused with the Mandaeans (as they are in [1]). Of course being worshipers of
the stars meant that there was strong motivation for the study of astronomy and
the sect produced many quality astronomers and mathematicians. The sect, with
strong Greek connections, had in earlier times adopted Greek
culture, and it was common for members to speak Greek although after the
conquest of the Sabians by Islam, they became Arabic
speakers. There was another language spoken in southeastern
Turkey, namely Syriac, which was based on the East Aramaic dialect of Edessa.
This language was Thabit ibn Qurra's native language, but he was fluent in both
Greek and Arabic.
Some accounts say that Thabit was a money changer as a young
man. This is quite possible but some historians do not agree. Certainly he
inherited a large family fortune and must have come from a family of high
standing in the community. Muhammad ibn Musa ibn Shakir, who visited Harran,
was impressed at Thabit's knowledge of languages and, realising the young man's
potential, persuaded him to go to Baghdad and take lessons in mathematics from
him and his brothers (the Banu Musa). In
Baghdad Thabit received mathematical training and also training in medicine,
which was common for scholars of that time. He returned to Harran but his
liberal philosophies led to a religious court appearance when he had to recant
his 'heresies'. To escape further persecution he left Harran and was appointed
court astronomer in Baghdad. There Thabit's patron was the Caliph, al-Mu'tadid,
one of the greatest of the 'Abbasid caliphs.
At this time there were many patrons who employed talented scientists to
translate Greek text into Arabic and Thabit, with his great skills in languages
as well as great mathematical skills, translated and revised many of the
important Greek works. The two earliest translations of Euclid's Elements were
made by alHajjaj. These are lost except for some fragments. There are, however,
numerous manuscript versions of the third translation into Arabic which was
made by Hunayn ibn Ishaq and revised by Thabit. Knowledge today of the complex
story of the Arabic translations of Euclid's Elements indicates that all later
Arabic versions develop from this revision by Thabit.0
In fact many Greek texts survive today only because of this
industry in bringing Greek learning to the Arab world. However we must not
think that the mathematicians such as Thabit were mere preservers of Greek
knowledge. Far from it, Thabit was a brilliant scholar who made many important
mathematical discoveries.
Although Thabit contributed to a number of areas the most
important of his work was in mathematics where he [1]:-
... played an important role in preparing the way for such
important mathematical discoveries as the extension of the concept of number to
(positive) real numbers, integral calculus, theorems in spherical trigonometry,
analytic geometry, and non-euclidean geometry. In astronomy Thabit was one of
the first reformers of the Ptolemaic system, and in mechanics he was a founder
of statics.
We shall examine in more detail Thabit's work in these
areas, in particular his work in number theory on amicable numbers. Suppose
that, in modern notation, S(n)denotes the sum of the aliquot parts of n, that
is the sum of its proper quotients. Perfect
numbers are those numbers nwith S(n) = n while mand n
are amicable if S(n) = m, and S(m) = n.
In Book on the determination of amicable numbers Thabit claims that Pythagoras
began the study of perfect and amicable numbers. This claim, probably first
made by Iamblichus in his biography of Pythagoras written in the third century
AD where he gave the amicable numbers 220 and 284, is almost certainly false.
However Thabit then states quite correctly that although Euclid and Nicomachus
studied perfect numbers, and Euclid gave a rule for determining them ([6] or
[7]):- ... neither of these authors
either mentioned or showed interest in [amicable numbers].
Thabit continues ([6] or [7]):- Since the matter of [amicable numbers] has
occurred to my mind,and since I have derived a proof for them, I did not wish
to write the rule without proving it perfectly because they have been neglected
by [Euclid and Nicomachus]. I shall therefore prove it after introducing the
necessary lemmas.
After giving nine lemmas Thabit states and proves his
theorem:
for n > 1, let pN = 3.2n -1
and qn = 9.2 2n-1 -1.
If pn-1 , pn , and
qn
are prime numbers, then
a = 2n pn-1pn and b = 2n qn
are amicablenumbers while a is abundant and b is
deficient. Note that an abundant number n satisfies S(n) > n, and a
deficient number n satisfies S(n) < n. More details are given in [9] where
the authors
conjecture how Thabit might have discovered the rule. In
[13] Hogendijk shows that Thabit was
probably the first to discover the pair of amicable numbers 17296, 18416.
Another important aspect of Thabit's work was his book on
the composition of ratios. In this Thabit deals with arithmetical operations
applied to ratios of geometrical quantities. The Greeks had dealt with
geometric quantities but had not thought of them in the same way as numbers to
which the usual rules of arithmetic could be applied. The authors of [22] and
[23] stress that by introducing arithmetical operations on quantities
previously regarded as geometric and non-numerical, Thabit started a trend
which led eventually to the generalisation of the number concept.
Thabit generalised Pythagoras's theorem to an arbitrary
triangle (as did Pappus). He also discussed parabolas, angle trisection and
magic squares. Thabit's work on parabolas and paraboliods is of particular
importance since it is one of the steps taken towards the discovery of the
integral calculus. An important consideration here is whether Thabit was
familiar with the methods of Archimedes. Most authors (see for example [29])
believe
that although Thabit was familiar with Archimedes' results
on the quadrature of the parabola, he did not have either of Archimedes' two treatises on the topic. In
fact Thabit effectively computed the integral of x and
[1]:-
The computation is based essentially on the application of
upper and lower integral sums, and the proof is done by the method of
exhaustion: there, for the first time, the segment of integration is divided
into unequal parts.
Thabit also wrote on astronomy, writing Concerning the
Motion of the Eighth Sphere. He believed (wrongly) that the motion of the
equinoxes oscillates. He also published observations of the Sun. In fact eight
complete treatises by Thabit on astronomy have survived and the article [20]
describes these.
The author of [20] writes:-
When we consider this body of work in the context of the
beginnings of the scientific movement in ninth-century Baghdad, we see that
Thabit played a very important role in the establishment of astronomy as an
exact science (method, topics and program), which developed along three lines:
the theorisation of the relation between observation and theory, the
'mathematisation' of astronomy, and the focus on the conflicting relationship
between 'mathematical' astronomy and 'physical' astronomy.
An important work Kitab fi'l-qarastun (The book on the beam
balance) by Thabit is on mechanics. It was translated into Latin by Gherard of
Cremona and became a popular work on mechanics.
In this work Thabit proves the principle of equilibrium of
levers. He demonstrates that two equal
loads, balancing a third, can be replaced by their sum placed at a point
halfway between the two without destroying the equilibrium. After giving a
generalisation
Thabit then considers the case of equally distributed
continuous loads and finds the conditions for the equilibrium of a heavy beam.
Of course Archimedes considered a theory of centres of gravity, but in [14] the
author argues that Thabit's work is not based on Archimedes' theory.
Finally we should comment on Thabit's work on philosophy and
other topics. Thabit had a student Abu Musa Isa ibn Usayyid who was a Christian
from Iraq. Ibn Usayyid asked various questions of his teacher Thabit and a
manuscript exists of the answers given by Thabit, this manuscript being
discussed in [21]. Thabit's concept of number follows that of Plato and he
argues that numbers exist, whether someone knows them or not, and they are
separate from numerable things. In other respects Thabit is critical of the
ideas of Plato and Aristotle, particularly regarding motion. It would seem that
here his ideas are based on an acceptance of using arguments concerning motion
in his geometrical arguments.
Thabit also wrote on [1]:-
... logic, psychology, ethics, the classification of
sciences, the grammar of the Syriac language, politics, the symbolism of
Plato's Republic ... religion and the customs of the Sabians.
His son, Sinan ibn Thabit, and his grandson Ibrahim ibn
Sinan ibn Thabit, both were eminent scholars who contributed to the development
of mathematics. Neither, however, reached the mathematical heights of Thabit.
Reference:
1. Biography in Dictionary of Scientific Biography (New
York 1970-1990).
2. Biography in Encyclopaedia Britannica.
3. F J Carmody, The Astronomical Works of Thabit b. Qurra
(Berkeley-LosAngeles, 1960).
4. F J Carmody, Thabit b. Qurra, Four Astronomical Tracts
in Latin (Berkeley,Calif., 1941).
5. E A Moody and M Clagett (eds.), The medieval science
of weights, Treatises ascribed to Euclid, Archimedes, Thabit ibn Qurra,
Jordanus de Nemore, and Blasius of Parma (Madison, Wis., 1952).
6. R Rashed, The development of Arabic mathematics :
between arithmeticand algebra (London, 1994).
7. R Rashed, Entre arithmétique et algèbre: Recherches
sur l'histoire des mathématiques arabes (Paris, 1984).
8. C B Boyer, Clairaut le Cadet and a theorem of Thabit
ibn Qurra, Isis 55(1964), 68-70.
9. S Brentjes and J P Hogendijk, Notes on Thabit ibn
Qurra and his rule for
amicable numbers, Historia Math. 16 (4) (1989), 373-378.
10. F J Carmody, Notes on the astronomical works of
Thabit b. Qurra, Isis 46
(1955), 235-242.
11. Y Dold-Samplonius, The 'Book of assumptions', by
Thabit ibn Qurra (836-901), in History of mathematics (San Diego, CA, 1996),
207-222.
12. H Hadifi, Thabit ibn Qurra's 'al-Mafrudat' (Arabic),
Deuxième Colloque
Maghrebin sur l'Histoire des Mathématiques Arabes (Tunis,
1990), A163-A164, 197-198.
13. J P Hogendijk, Thabit ibn Qurra and the pair of
amicable numbers 17296,
18416, Historia Math. 12 (3) (1985), 269-273.
14. K Jaouiche, Le livre du qarastun de Tabit ibn Qurra.
étude sur l'origine de la notion de travail et du calcul du moment statique
d'une barre homogène, Arch. History Exact Sci. 13 (1974), 325-347.
15. L M Karpova and B A Rosenfeld, The treatise of Thabit
ibn Qurra on sections of a cylinder, and on its surface, Arch. Internat. Hist.
Sci. 24 (94) (1974), 66-72.
16. L M Karpova and B A Rozenfel'd, A treatise of Thabit
ibn Qurra on composite ratios (Russian), in History Methodology Natur. Sci. V
(Moscow, 1966), 126-130.
17. G E Kurtik, The theory of accession and recession of
Thabit ibn Qurra
(Russian), Istor.-Astronom. Issled. 18 (1986), 111-150.
18. G E Kurtik and B A Rozenfel'd, Astronomical
manuscripts of Thabit ibnQurra in the library of the USSR Academy of Sciences
(Russian), VoprosyIstor. Estestvoznan. i Tekhn. (4) (1983), 79-80.
19. K P Moesgaard, Thabit ibn Qurra between Ptolemy and
Copernicus : an
analysis of Thabit's solar theory, Arch. History Exact
Sci. 12 (1974), 199-216.
20. R Morelon, Tabit b. Qurra and Arab astronomy in the
9th century, Arabic
Sci. Philos. 4 (1) (1994), 6; 111-139.
21. S Pines, Thabit Qurra's conception of number and
theory of the mathematical infinite, in 1968 Actes du Onzième Congrès
International d'Histoire des Sciences Sect. III : Histoire des Sciences Exactes
(Astronomie, Mathématiques, Physique) (Wroclaw, 1963), 160-166.
22. B A Rozenfel'd and L M Karpova, Remarks on the
treatise of Thabit ibn
Qurra (Russian), in Phys. Math. Sci. in the East 'Nauka'
(Moscow, 1966), 40-41.
23. B A Rozenfel'd and L M Karpova, A treatise of Thabit
ibn Qurra on composite ratios (Russian), in Phys. Math. Sci. in the East
'Nauka' (Moscow, 1966),
24. A I Sabra, Thabit ibn Qurra on the infinite and other
puzzles : edition and
translation of his discussions with Ibn Usayyid, Z.
Gesch. Arab.-Islam.
Wiss. 11 (1997), 1-33
25. A Sayili, Thabit ibn Qurra's generalization of the
Pythagorean theorem, Isis
51 (1960), 35-37.
26. J Sesiano, Un complément de Tabit ibn Qurra au 'Per“
diairéseon' d'Euclide,
Z. Gesch. Arab.-Islam. Wiss. 4 (1987/88), 149-159.
27. K Taleb and R Bebouchi, Les infiniment grands de
Thabit Ibn Qurra, in
Histoire des mathématiques arabes (Algiers, 1988),
125-131.
28. A P Yushkevich, Note sur les déterminations
infinitésimales chez Thabit ibn
Qurra, Arch. Internat. Histoire Sci. 17 (66) (1964),
37-45.
29. A P Yushkevich, Quadrature of the parabola of ibn
Qurra (Russian), in History Methodology Natur. Sci. V (Moscow, 1966), 118-125.
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