Glimpses in the History of a Great Number: Pi in Arabic Mathematics

by Moustafa Mawaldi Published on: 22nd September 2008

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The Greek letter pi (symbolized by π) is defined as the ratio of the circumference of the circle to its diameter. It is considered to be a vital element in the calculations of the area and sizes of several mathematical figures: the circle, the cube, the cone and the sphere, from which infinite practical applications have sprung. As a result, mathematicians in many civilizations: Greek, Chinese, Indian, Arabian and European have been highly concerned with calculating π as carefully as possible. This article by Professor Mustapha Mawaldi, the Director of the Institute for the History of Arabic Science in Aleppo, sheds light on the contribution of mathematicians of the Islamic civilisation in refining the value of pi. The works surveyed are those of Al-Khwarizmi, Al-Kashi, Al-Biruni, Al-Quhi, and Al-Kashi.

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Note of Editor: This article was translated from Arabic by Haya Zedan (FSTC). A thorough revision and copy editing was performed by the editorial board of Muslim Heritage.

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Table of contents

  1. Introduction
  2. The meaning of π
  3. Approximating calculations of π through history

3.1. The Greek civilisation
3.2. The Chinese civilisation
3.3. The Indian civilisation
3.4. The Arabo-Islamic Civilisation

  1. The calculation of Pi in Arabic mathematics

4.1. Banu Musa
4.2. Muhammad Al-Khwārizmī
4.3. Wijan b. Rustum Al-Quhi
4.4. Abu ‘l-Rayhan al-Biruni
4.5. Jamshīd Al-Kāshī

  1. Further Reading
  2. References

1. Introduction

The Greek letter pi (symbolized by π) is defined as the ratio of the circumference of the circle to its diameter. It is considered to be a vital element in the calculations of the area and sizes of several mathematical figures: the circle, the cube, the cone and the sphere, from which infinite practical applications have sprung. As a result, mathematicians in Greek, Chinese, Indian, Arabic and European civilizations have been highly concerned with calculating π as carefully as possible.

Fig. 1. Pi, symbol and value. (Source)

Our aim in this article is to survey the history of the mathematical research carried on in different scientific traditions to find approximated values for π. Our focus will be on the discussion of the values of π in the mathematical tradition of the Arabo-Islamic civilisation. The analysis also sheds light on the innovations and contributions brought by Islamic scientists in this issue and stresses on the interest and motivation of Arab scholars to arrive at the nearest possible approximation for the value of π, depending on presenting new and original mathematical proofs.

2. The meaning of π

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Fig. 2: Front cover of Pi: A Source Book by L. Berggren, J.M. Borwein and P.B. Borwein (Springer-Verlag, 2nd edition, 2000). (Source)

The symbol π is used mathematically to describe the function of the proportion of the circumference of a circle to its diameter. It is said[1] that this symbol has been in use since the year 1766, and it is a small Greek letter and the first letter of the word “circumference” in the Greek language.

Under the entry (Pi) in a French dictionary,[2] the author contends that the value π is of Greek origin, only entering into the French language in the 19th century. π is the sixth letter in the Greek alphabet, and its counterpart in European modern languages is the letter p. From an engineering perspective, it is the truncation of the Greek word “peripheria”, and it is a symbol for the number that represents the static value of the fixed ratio of the circumference to the diameter of a circle, equal approximately to 3.1415926.

3. Approximating calculations of π through history

3.1. The Greek civilisation

Archimedes (287-212 BCE) is considered one of the greatest Greek scholars, for he has put forward many innovations in both the fields of mathematics, physics and engineering. Of his many contributions, the most significant is his calculation of the ratio of the circumference to the diameter of a circle (π), as he narrowed the value of π within the following equation:

3 10/71 < π < 3 1/7

Archimedes submitted a proof of the above equation in his treatise Measurement of a Circle[3] which contained three propositions. In the first proposition, he proves that

“…the area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.”

In the second proposition, Archimedes goes on to prove his theory by saying:

“The area of a circle is to the square on its diameter as 11 to 14”

That is the circumference of a circle is greater than three times its diameter, less than one-seventh of the diameter and more than one-tenth of seventy-one parts of the diameter.

The principle of Archimedes’ proof rests on the drawing of a figure that has ninety-six sides that are equal within a circle, and another outside the circle so that the circumference of the outside figure is larger than that of the circle itself, and the circumference of the sides that rest within the circle is smaller than the circumference of the circle. From this, he calculates the relationship between the perimeter of this polygon within the circle and the diameter of the circle.

The third theorem asserts the following:

“The ratio of the circumference of any circle to its diameter is less than 31/7 but greater than 310/71.”

He comments that if the circumference of the circle is three times the diameter plus one-seventh − this is an approximation that has been used by surveyors – then the ratio of the area of the circle to the square of its diameter is equal to the ratio of eleven to fourteen.

Archimedes derived his proof of the value of π from the theory of Euclid (330-290 BCE) that is found in the XIIth book of The Elements.[4] In that theorem, Euclid asserts the existence of a set ratio between the area of a circle and the area of a square of which the angle is equal to half the diameter of the circle. The first estimation is a simplified calculation that is used to solve geometric questions using practical methods.[5]

3.2. The Chinese Civilisation

To study the history of Chinese mathematics one must return to the text Jiuzhang suanshu or The Art of Mathematics in Nine Chapters[6] of which the author is unknown. It is probable that this text was collated in the 1st century CE. It was depended upon as the main text until the 13th century.

The text put forth several theories for the calculation of the area of a circle, and in those we find some that are incorrect:

1) S1 = 3/4d2 (the correct formula is S1 = pd2/4);

2) S1= 1/12p2 (the correct formula is S1p2/4p)

(with p = the circumference of a circle, d = the diameter, and S1 = the area).

In these two equations, π is used with a value that is equal to three, which leads to an error in calculating the area of a circle.

At the end of the 5th century,[7] Zu Chongzhi gave a rounded value of to π, and from that his own son Zu Kengzhi advanced this knowledge with his own work, using the value for (π).

Rosenfeld and Youschkevitch list in the Encyclopedia of the History of Arabic Science[8] the Chinese scientist and astronomer Chang Hêng (139-78 CE) as having suggested the value for π. Chinese mathematicians continued to use the value of 3 for π until the 9th century, as it was a simple value to use in calculations and formulae.

3.3. The Indian Civilisation

Indian civilisation contributed a great deal to the calculations of the value of π. Since the 7th century, the astronomer Brahmagupta (born in 598 CE) gave π the value of. Historians also credit the Indian astronomer Âryabhata (born 476 CE) with several values of (π), notably:  and or 3.1416.[9] However, he was said to have used the value of  or 3 for π. [10]

Nine centuries later, the astronomer and mathematician Mâdhava (fl. in the 5th century CE) arrived at a calculation of the area of the circle by using the approximated value of π that is correct up to 10 digits, namely π = 3.14159265359, as stated by Guy Mazars. However, the last digit (9) is incorrect and must be replaced with the digit (8). [11]

In the 6th century CE, an Indian mathematician used the value  as an approximated value for π, and put forward the following rule for its calculation:

π/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – …

3.4. The Arabo-Islamic Civilisation

Arab scholars contributed to the quest for the calculation of the value of π, and the proportion of the circumference of a circle to its diameter, and their efforts impacted the history of mathematics.

Among the mathematicians whose names are referred to with regard to this issue, we mention the scholars who flourished from the 9th to the 15th centuries. First, the three brothers Banū Mūsā bin Shākir (Baghdad, 3rd-century H/9th century CE), the famous mathematician known as the father of algebra, Muhammad b. Mūsā Al-Khwārizmī (3rd century H / 9th century CE), the Persian scholar author of various mathematical works Abū Sahl Wījan b. Rustam Al-Qūhī (around 390 H/ around 1000 CE), the well-known polymath scholar Abū Al-Rayhān Al-Bīrūnī (died 440 H/ 1048 CE) and finally the author of the widely diffused mathematical text Miftah al-hisab (Keys of arithmetic) Jamshīd b. Mas‛ūd b. Mahmūd Al-Kāshī (died 1429 CE). In this section, we point out some of their achievements in approximating the value of π and the methods used in this sophisticated research.

4. The Calculation of Pi in Arabic mathematics

4.1. Banu Musa

In their book on Kitab fi ma’rifat misahat al-ashkal al-basita wa al-kuriya (The Measurement of Plane and Spherical Figures, the brothers Banu Musa prove that the proportion of the circumference of a circle to its diameter is greater than the percentage of 3 and 10 parts of 71 to 1, and smaller than the percentage of 3 and 7 to 1.[12] This means: as π is the circumference of the circle, and d is the diameter. Banu Musa also mentions that Archimedes had proved this relationship, which provides the approximation of the value of p:

“than to prove the proportion of the diameter to the circumference along with the method of Archimedes this has not reached us in a form that was in our time, and this method even though it has not reached to the point of measuring one and the other to arrive at the truth, it in fact allows us to derive the value of one from the other closer to that which we desire to arrive.”[13]

However, they considered Archimedes to be incomplete and not arriving at the truth, and Suter[14] agrees that their method was different from that employed by Archimedes.

Sezgin[15] considers the proof of Banu Musa to measure the proportion of the diameter of a circle as an important advancement and more in-depth than Archimedes’ attempt. Roshdi Rashed[16] mentions that Banu Musa had achieved an “explanation to Archimedes method in the approximation of (π), and derived their generalizations from this calculation”. 

4.2. Muhammad Al-Khwārizmī

We find in the chapter on area in his book Al-Jabr wa-‘l-muqabala (that he wrote in the years 813 and 833 CE) rules for the calculation of the circumference of a circle as he states:[17]

“And for every circle (mudawwara) when you multiply the diameter by three-sevenths is the circumference (dawr) that surrounds it, and it is a term among men that is unnecessary, and to those of geometry, there are two other sayings: one is to multiply the diameter by itself, and then by ten, and then to take the root of the result and that is the circumference of the circle. The other is the that used by astronomers is to multiply the diameter by two and sixty thousand and eight hundred and thirty-two then divide that by twenty-thousand and what results is the circumference, and all these are near to each other…”

Al-Khwarizmi gives three values to π and these are:

The editors of his book produce this marginal commentary in which he says:[18]

“It is an approximation not a proof, and no one stands on the truth of this, and no one but Allah knows the true circumference of the circle, as the line is not straight and has no beginning and no end, we merely attempt to approximate and discover the root, but even the root has no definition as no one may know its exact value but Allah, and the best of these approximations that is to multiply the diameter by three-seventh as it is faster and simpler and only Allah might know it true.”

We note also that in his calculation of the area of the circle in his book,[19] he adopts for π the value.

4.3. Wijan b. Rustum Al-Quhi

Abu-Ishaq Al-Sabi had sent a missive to Al-Quhi[20] inquiring of several matters, specifically of Al-Kuhi’s methods to derive the proportion of the diameter to the circumference of the circle, and asks him to forward this to him as he says:[21]

“I desired… to send all that has been derived especially that of the proportion of the diameter to the circumference of the circle. As a ratio of a number to a number then that would be something myself looks for to know and have the benefit of.”

Al-Quhi answers the queries of Al-Sabi and then moves to discuss his text on the centres of gravity and says:[22]

“on the four articles that I have done here we have reached strange things each to prove the greatness and the order of the Creator, such as those matters relating to the Sphere and the Cylinder of Archimedes. Don’t we marvel at how the sphere is equal to two-thirds of the cylinder as he proved and described, and that the paraboloïd is equivalent to its half as it was proved by Thabit ibn Qura, and that the cone is one-third as it was shown by the ancients? We found in the [study of the] centres of gravity much orders to impress us”.

Then he moves on to provide his theory and proof that is found in the text of Ibn Al-Salah, according to which the circumference is three times the diameter and nine parts. He depends in this demonstration on the following three lemmas:

Lemma 1: The centre of gravity of a semi-circle falls on the perpendicular which is drawn from its centre to the circumference on a point of the diameter in the ratio of three-sevenths.

Lemma 2: There are given two portions of two circles which the same centre. If the ratio of the semi-diameter of one to the ratio of the semi-diameter of the other is like three to two, and if they are similar, then the ratio of the centre of gravity of the arc of the smallest arc is equal to the centre of gravity of the largest arc.

Lemma 3: The ratio of each arc to its chord in the circle is like the ratio of the semi-diameter of that circle to the line that is situated between the centre of the circle and the centre of gravity of the chord.

Al-Quhi concludes his missive by saying:[23]

“And when we look at Archimedes as he writes, that the circumference of the circle, is less than three times the diameter and ten parts of seventy, meaning one-seventh, and this agrees with our work and does not depart from this, and one-ninth is less than one-seventh, and he also stated: that it is greater than three times and ten parts of seventy, and this does not agree unless he means: ninety-one parts instead of seventy-one to become in agreement, not more, and we do not presume to think in any manner that is not well of those who preceded us in this work, as he is Archimedes and he is a forerunner in this field.”

Al-Quhi had attempted to find a more accurate value for π but he failed, and it was some time before this was achieved, many others tried and failed as well.

4.4. Abu ‘l-Rayhan al-Biruni

In the fifth chapter[24] (On the Proportion between the Circumference and the Diameter) from the third book of Al-Qanun al-Mas’udi, Abu Al-Rayhan calculates the circumference of an angular figure which has 180 sides within the circle, and also calculated the circumference of an angular figure that has 180 sides surrounding the circle and takes the median of the two. From that, he calculates the value of π and reaches the value:[25] This is a value which is not much more accurate than what was known to the Indian civilisation before.

4.5. Jamshīd Al-Kāshī

Fig. 3: A stamp issued in 1979 in Iran commemorating al-Kāshī. (Source)

Al-Kāshī is considered one of the greatest Islamic scholars that have put forward significant scientific achievements that have propelled modern civilisation forward. Among his works, Al-Risala al-muhitiya (The Letter of Circumference) that sets out a very particular calculation for the value of π. Al-Kāshī measured the circumference of an equal-sided angular figure that is surrounded by a circle, and another that is surrounded by a circle that has 3 x 228 = 805306368 sides,[26] when Archimedes and Banu Musa limited themselves – as we have seen- to a figure with 3 x 25 = 96 sides. Al-Kāshī determined the number (28) as the difference between the circumferences of the two shapes is equal to its diameter 600000 times the diameter of the earth, less than the width of a hair.

After his calculation of the circumferences of the two figures, he presumed that the circumference of a circle is equal to the median of the two results, and he arrived at this conclusion (in the sexagesimal system):

π = 3;8, 29,44,0,47,25,53,7,25

And he then transformed it into the decimal system:

π = 3. 14 159 265 358 979 325

Rosenfeld and Youschkevitch that the last numeral (5) from this result is the only one that is erroneous, and that the correct one is (38), and they point to the fact that in Europe 150 years after Al-Kāshī, a man from Holland by the name of (A.Van Roomen) reached the same approximation of π.

Fig. 4: Manuscript page of Al-Kāshī’s calculation of π: jadwal tadhā’if nisbat al-muhīt wa-‘l-qutr (table of the multiples of the ratio of the circumference to the diameter). (Source).

Al-Kāshī points out in his book The Keys to Calculations that some mathematicians had determined the value of π in which he states:[27]

“Know that the circumference is three times the diameter, and this is less than one-seventh of the diameter, however many have sufficed to use the one-seventh for ease of use”.

Al-Kāshī states that his calculations are more precise than those of Archimedes, and determines a value for π[28], by saying:

“Archimedes has said that it must be less than one-seventh and more than ten-parts of seventy, and from what we have gathered and mentioned in our text on the circumference which is: 44 829 3 thirds, after subtracting the fourths and what follows, if the diameter is one. This is a more precise calculation than that of Archimedes, and closer to the accurate value of π , and Al-Kāshī reiterates that it is the “closest that might be, save for the knowledge of Almighty Allah.”

Fig. 5: Applications of the law of cosines: unknown side and unknown angle. In trigonometry, the law of cosines is known as Al-Kāshī law; it is a statement about a general triangle which relates the lengths of its sides to the cosine of one of its angles. Al-Kāshī was the first to provide an explicit statement of this law.

Al-Kāshī’s exacting calculations might not have been needed in his time, but he was a forward thinker, he was a scholar who searched for accurate results to advance progress.

5. Further Reading

  • Al-Daffa, Ali Abdullah, The Muslim contribution to mathematics. London: Croom Helm, 1977.
  • Anbouba, Adil, “L’algèbre arabe aux IXe et Xe siècles : apercu général”, Journal for the History of Arabic Science, vol. 2 (1), 1978, pp. 66-100.
  • Berggren, J. L., Episodes in the mathematics of Medieval Islam. Berlin: Springer-Verlag, 1986.
  • Berggren, L., Borwein J.M., and Borwein, P.B. Pi: A Source Book. Berlin: Springer-Verlag, 1997; second edition, 2000.
  • Beckmann, Petr, A History of Pi. St. Martin’s Griffin, 1976, (Paperback).
  • Borwein, Jonathan M., The Life of Pi History and Computation. Prepared for Australian Colloquia (June 21-July 17, 2003).
  • Gourdon, Xavier, and Sebah, Pascal, π and its computation through the ages.
  • Ibn Lablan, Kushyar. Principles of Hindu Reckoning. Wisconsin Press, Madison, 1966.
  • Luckey, P., Die Rechenkunst bei Ğamšīd b. Mas’ūd al-Kāšī. Wiesbaden: Steiner, 1951.
  • O’Connor, John J. & Robertson, Edmund F., A History of Pi. MacTutor History of Mathematics archive.
  • O’Connor, John J. & Robertson, Edmund F., Ghiyath al-Din Jamshid Mas’ud al-Kashi. MacTutor History of Mathematics archive.
  • O’Connor, John J. & Robertson, Edmund F., Arabic mathematics: forgotten brilliance? MacTutor Math History Archives, November 1999.
  • Rashed, Roshdi, Entre arithmétique et algèbre: Recherches sur l’histoire des mathématiques arabes. Paris: Les Belles Lettres, 1984.
  • Rashed, Roshdi, The Development of Arabic Mathematics: Between Arithmetic and Algebra. London, 1994.
  • Rosen, F., (ed. and trans.), The Algebra of Mohammed ben Musa. London, 1831, reprinted 1986.
  • Saidan, Ahmad Sa’id, (ed. and trans.), The Arithmetic of al-Uqlidisi. Dordrecht: Reidel, 1978.
  • Siddykov, H., “The role of the scientists of ancient Khorezm in the development of the exact sciences” (in Russian), Vestnik Karakalpak. Filiala Akad. Nauk USSR 2, 1967, pp. 3-14.
  • Struik, D.J., A Source Book in Mathematics, 1200-1800. Cambridge, Mass.: Harvard University Press, 1969. Reprinted at Princeton University Press, 1986.
  • Van Brummelen, Glen, The entity from which ERIC acquires the content, including journal, organization, and conference names, or by means of online submission from the author., “Jamshid al-Kashi: Calculating Genius”, Mathematics in School, vol. 27, N.° 4, 1998, pp. 40-44.
  • Wilson, David, The History of Pi. History of Mathematics, Rutgers University, Spring 2000. 

6. References

[1] Marcel Boll, Histoire des Mathématiques (Que sais- je? Nº 42). Paris: Presses Universitaires de France, 13e édition, 1979, p. 43.
[2] See Paul Robert, Le Petit Robert, Dictionnaire de la Langue Française, Le Robert, Paris, 1984, pp. 1429-1430, entry “Pi”.
[3] Archimedes, Taksir al-da’ira, maqala mulhaqa bi-kitab “Al-Kura wa-‘l-ustuwana” li-Arkhimidis, tahrir Nasir al-Din al-Tusi, in Rasa’il al-Tusi. Haydarabd: Da’irat al-ma’arif al-‘uthmaniya, 1359 H, pp. 127-133; Heath, T.L. (ed.), The Works of Archimedes (Dover Edition, 1953), 93-98. Originally published in 1897, Cambridge University Press; Archimède, La Mesure du Cercle, Texte établi et traduit par Charles Mugler, Les Belles Lettres, Paris, 1970, vol. 1 , pp. 135-143.
[4] Euclid, The Elements, with Introduction and Commentary by Thomas Heath, 2e Edition, Dover Publications, New York, 1956, vol. 3, Book XII, pp. 365-437.
[5] Emile Noël, Le Matin des Mathématiciens, Entretiens sur l’histoire des mathématiques. Édition Belin – Radio France, 1985, p. 60.
[6] Karine Chemla, “Theoretical aspect of the Chinese algorithmic tradition (first to the third century)”, Historia Scientiarum, No 42, (1991), p. 75.
[7] J.-C. Martzloff, Histoire des Mathématiques Chinoises, Masson, Paris, 1987, pp. 265, 270.
[8] Boris A. Rosenfeld and Adolph P. Youschkevitch, Geometry, in Encyclopedia of the History of Arabic Science, edited by R. Rashed, vol. 2, Routledge, 1996. Arabic translation: Mawsu’at tarikh al-‘ulum al-arabiya. Beirut: Markaz dirasat al-wahda al-‘arabiya, 1997, p. 577.
[9] Rosenfeld and Youschkevitch, op. cit., p. 577.
[10] Qadri Hafez Tuqan, Turath al-‘arab al-‘ilmi fi ‘l-riyadhiyat wa ‘l-falak, Cairo, 1941, p. 19.
[11] E. Noël, Le Matin des Mathématiciens, op. cit., p. 132.
[12] Banu Musa, Mohammad, al-Hassan and Ahmad, Kitab fi ma’rifat misahat al-ashkal al-basita wa al-kuriya, recension (tahrir) by Nasīr al-Dīn al-Tūsī, Hayderabad: Da’irat al-ma’arfial-‘uthmaniya, 1359 H, p. 9. See also Moustafa Mawaldi, “Geometry in Banū Mūsā Ibn Shākir”, in The 36th science week in homage to the Banu Musa (2-7 november 1996) (in Arabic). Damascus: The Supreme Council of Sciences, 1998, p. 107.
[13] Banū Mūsā, Kitab fi ma’rifat misahat al-ashkal al-basita wa al-kuriya, op. cit., p. 6.
[14] Abdel Magīd Nusair, “Mathematics in the Islamic Civilisation”, Proceedings of the Conference on the Arabic Heritage in Exact Sciences, Tripoli (Lybia), 1990, p. 88.
[15] Fuat Sezgin, Conferences in the History of Arabo-Islamic Sciences (in Arabic), Frankfurt, 1984, p. 71.
[16] Roshdi Rashed, “Infinitesimal Determinations, Quadrature of Lunules and Isoperimetric Problems”, in Encyclopedia of the History of Arabic Science, edited by R. Rashed, Arabic translation, Beirut, 1997, vol. 2, p. 542.
[17] Muhammad ibn Musa al-Khwarizmi, Kitab al-jabr wa-‘l-muqabala, edition and commentary Mustafa Musharrafa and Muhammad Musa Ahmad. Cairo: Publications of the Faculty of Sciences, 1939, pp. 55-56.
[18] Al-Khwārizmī, Kitab al-jabr wa-‘l-muqabala, op. cit, pp. 55-56.
[19] Al-Khwārizmī, Kitab al-jabr wa-‘l-muqabala, op. cit, p. 64.
[20]On the mathematician, al-Quhi’s life and works, see Khayr al-Din al-Zirikli, Al-A’lam, 10th edition, Beirut: Dar al-‘ilm li-‘l-mlayin, 1992, vol. 8, p. 127.
[21]Abu Ishaq al-Sabi, Risalat Abi Ishaq al-Sabi ila abi Sahl al-Quhi wa jawabuha, Al-Zahiriya Library, MS 5648 General; Library of the Institute for the History of Arabic Science in Aleppo, microfilm number 1698, folio 1969.
[22] Al-Sabi, Risala…, op. cit., folio 197v.
[23] Al-Sabi, Risala…, op. cit., folio 199v.
[24] Al-Biruni, Al-Qanun al-mas’udi, Haydarabad: Da’irat al-ma’arif al-‘uthmaniya, 1373 H [1954], vol. 1, pp. 303-304.
[25] Adolph P. Youschkevitch, Les Mathématiques Arabes, French translation. Paris: Vrin, 1976, pp. 150-151.
[26] Boris A. Rosenfeld and Adolph P. Youschkevitch, Geometry, in Encyclopedia of the History of Arabic Science, op. cit., pp. 582-584.
[27]Jamshid al-Kashi, Miftah al-hisab, edited by Nadir al-Nablusi, Damascus: Publications of the Ministery of Higher Education, 1977, p. 247.
[28] Al-Kashi, Miftah al-hisab, op. cit., p. 247.

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