The following article focuses on the cubic measure of the volume of the sphere in Arabic mathematics. After a short presentation of the Greek and Chinese ancient legacies on this topic, the article surveys thoroughly the different formulas methods proposed by the mathematicians of the Arabic-Islamic civilization from the 9th to the 17th century to measure the volume of the sphere. The achievements of eminent scholars are thus presented: Banu Musa, Al-Buzgani, Al-Karaji, Ibn Tahir al-Baghdadi, Ibn al-Haytham, Ibn al-Yasamin, Al-Khawam al-Baghdadi, Kamal al-Din al-Farisi, Jamshid al-Kashi, and Baha' al-Din al-'Amili.
By Professor Dr. Mustafa Mawaldi^{1}
Table of contents
1. Introduction
2. Historical survey
2.1. The volume of the sphere in Greeks mathematics: Archimedes
2.2. The volume of the sphere in Chinese mathematics
3. The volume of the sphere in Arabic mathematics
3.1. Banu Musa
3.2. Abu ‘l-Wafa al-Buzgani
3.3. Al-Karaji
3.4. Ibn Tahir al-Baghdadi
3.5. Ibn al-Haytham
3.6. Ibn al-Yasamin
3.7. Ibn al-Khawwam
3.8. Kamal al-Din al-Farisi
3.9. Al-Kashi
3.10. Bahā' al-Dīn al-'āmilī
4. Conclusion
* * *
Abbreviations
- The diameter = d
- The radius = r
- The periphery = p
- The area of sphere surface = s
- The area of the greatest circle in the sphere = S_{1}
- Volume of the sphere = v
- Volume of right-cone = V_{1}
- The area of cone-base = S_{4}
- Cone height = H_{4}
- Volume of a right-cylinder = V_{2}
- The area of the cylinder base = S_{2}
- The cylinder height = H_{2}
- The area of the side surface of a circular right-cylinder = S_{3}
- Volume of the cube = V_{3}
- Volume of vertical ring = V_{4}
Figure 1: A sphere circumscribed in a cylinder: the sphere has two thirds of the volume and surface area of the circumscribing cylinder. |
The mathematicians of the Arabic civilization endeavoured to find a rule through which the sphere volume can be calculated. Some of them had got a cubic measure of it in comparison with the known volume of solids such as the cone, the cylinder, and so on. Likewise, they obtained a figure of the volume by finding out a relationship that links the elements of the sphere such as its surface to its radius. Consequently, the value of π played an important role in the accuracy of the cubic measure. Thus, some of the mathematicians of the Islamic tradition had the right measure, whilst others had the wrong one and proposed erroneous values.
Basically, this research is concerned with the cubic measure of the volume of the sphere in the mathematical tradition of the Arabic civilization. We begin by surveying the ancient contribution with a presentation of Archimedes' results in the Greek tradition on this issue, besides a survey of the development of cubic measure of sphere in Chinese mathematics. The fundamental points of our study discuss the following subjects. After a historical introduction on the volume of the sphere in Greek and Chinese mathematics, we present a thorough survey of the same topic in the mathematics of the Arabic-Islamic civilization from the 9th to the 17th century, especially in the works of Banu Musa, Abu ‘l-Wafa al-Buzgani, Al-Karaji, Ibn Tahir al-Baghdadi, Ibn al-Haytham, Ibn al-Yasamin, Al-Khawam al-Baghdadi, Kamal al-Din al-Farisi, Jamshid al-Kashi, and Baha' al-Din al-‘Amili. Finally, a set of conclusions is deduced.
2.1. The volume of the sphere in Greeks mathematics: Archimedes
Figure 2: Two manuscript pages of the Greek text of Archimedes' Sphere and Cylinder (Source). |
The mathematical problem of the measure of the volume of the sphere is discussed by Archimedes in his known book The Sphere and Cylinder. Archimedes is considered as the best Greek scientist in the fields of mathematics and mechanical engineering. He died in 212 BCE. His scientific legacy consists in a group of influential texts presenting several important theories [2].
His book The Sphere and Cylinder [3] is composed of two parts. In the first one, Archimedes presented a number of definitions and postulates. Then he discussed the surfaces and volumes of some solids, such as the surface area of the sphere as well as its volume. In the second part, he developed some constructions and demonstrations related to the theories that he had mentioned in the first part.
Archimedes gives a rule of the volume of the sphere in comparison with the cone and cylinder. In the words of Nasir al-Din al-Tusi's edition and recension of his book The Sphere and Cylinder, Archimedes' first formula is formulated as follows:
"Each sphere is equal to four times a cone whose base is equal to the greatest circle in that sphere, and the height [of this cone] is equal to the radius of that sphere. [4]"
We may write such a rule as follows: V = 4V_{1}; assuming that S_{1} = S_{4} and r = H_{4}. This formula corresponds to the eighth formula of Al-Kashi, as we will see below.
Afterwards, Archimedes expressed the second formula as follows:
"Each cylinder of which the base is equal to the greatest circle that exists in a sphere, and its height is equal to the diameter of its base, [such a cylinder] is equal to one and a half of the sphere [5]."
That is, in symbolic language: 3/2V = V_{2}; assuming that S_{1} = S_{2} and d = H_{2}. This formula corresponds to the seventh formula of Al-Kashi.
In the Arabic edition of Archimedes' treatise composed by Nasir al-Din al-Tusi, we find the complete demonstration of these two formulas [6]. In addition, the modern mathematical analysis of the Archimedean theorem of the volume of the sphere is developed by Marshall Clagett in his article on Archimedes in the Dictionary of Scientific Biography [7].
2.2. The volume of the sphere in Chinese mathematics
Figure 3: Modern imaginary portrait of the mathematician Muhamad al-Bujzani known as Abu ‘l-Wafa (940-997 CE) (Source). |
The book of The Arithmetic Art in Nine Chapters (jiuzhang suanshu) by an unknown author is considered as an important source of Chinese mathematics. It was probably collected in the 1st century CE and was used by Chinese mathematicians as an essential source until the 13th century CE [8].
The Chinese gave special interest to the cubic measure of the volume of the sphere. Thus, we find in the fourth chapter of The Arithmetic Art in Nine Chapters two problems related to the calculation of the diameter of a sphere that has a definite volume, then, the solution was obtained by finding the following cubic- root:
[(16/9) V]^{1/3}.
We may formulate the above relation as follows: V = (9/16)d^{3}. Certainly, this is a wrong formula of the cubic measure of the volume of the sphere, as it is greater than the real volume by the amount (13/336 d^{3}).
The historians refer the origin of that formula to different sources and analyses such as:
1. The practical method: An unknown interpreter of the book The Arithmetic Art in Nine Chapters reached the above formula by a trial of the cubic measure as follows: a cubic weight of copper, its diameter being one inch/+/16x200g, and the weight of a sphere of copper having a diameter 9x200g; from here the two numbers 16 and 9 were deduced.
Besides, it is curious to find that the theory of the practical method to make a cubic measure of the volume of the sphere of Al-Karaji corresponds to the practical method used in the Chinese treatise, although it differs from it in certain details, as the Chinese mathematician made the diameter of the sphere equal to a cube diameter, whilst Al-Karaji made the diameter of the sphere equal to one side of the cube. Consequently the volume of the sphere in the Chinese treatise by the practical method was greater than the real volume –as mentioned above– by the amount of (13/336 d^{3}); whilst the volume of the sphere in Al-Karaji by the practical method was greater than the real volume by the amount of (31/315 d^{3}).
Both approaches led to a greater value than the exact one for the calculation of the volume of the sphere, but the Chinese practical method is closer to the real volume of the sphere. We need to mention here that we don't know the history of the Chinese practical method.
2. The value of π. Some historians of mathematics [9] link the original formula found in the Chinese calculation of the volume of the sphere to the history of π value. Hence, the author of the Arithmetic Art in Nine Chapters would use π with the value 25/8. Actually, the value of π in this book is considered roughly as 3 in general.
Finally, we find that the Chinese author made a thorough study and after several arduous attempts got the formula of the correct volume of the sphere.
3. The volume of the sphere in Arabic mathematics
Figure 4: Figures of the geometrical proof of the Pythagorean theorem by Abu ‘l-Wafa al-Buzgani and its application in ornamental tiles: two equal squares are easily combined into a bigger square; Abu ‘l-Wafa's method works even if the squares are different (Source). |
The three brothers Banu Musa, who flourished in Baghdad in the 3rd century H/9th century CE, studied the volume of the sphere in their well known mathematical treatise Kitab ma'rifat masahat al-ashkal al-basita wa-‘l-kuriya (Book about the knowledge of the area of plain and spherical figures).
The book consists in an introduction and eighteen theorems. In general, the book investigates the rules to calculate the areas of the spherical and plane surfaces with their volumes. It also discusses a set of geometric problems such as the division of angles into three equal portions, placing two quantities between two quantities in order to create series of one proportion. Hence, the book includes a method to find the approximate cube root of any figure required by the calculator [10].
The book is ascribed to the three brothers Banu Musa Muhammed, Ahmed, and Al-Hasan, who were known as the Banu Musa. They excelled in mathematics, astronomy, mechanics, music, and philosophy. They were among the best Muslim scientists during the 3rd century H / the 9th century CE. As a result of our recent study of the book, we mostly relate it to the three brothers in common, but Al-Hasan, the mathematician of the group, had the biggest share in its authorship [11].
Banu Musa gave a rule of the volume of sphere, then they proved it in the 15th theorem of their book. We present hereinafter this theorem:
[Theorem 15]: "For each sphere, the product of multiplying its radius in the third of the area of the sphere surface is equal to its volume [12]."
As a result of this statement, Banu Musa gave the correct rule of the volume of the sphere as follows: V = 1/3 r.(S). Thereafter, this theorem was considered as identical with Abu ‘l-Wafa al-Buzgani's second formula for the volume of the sphere. It was also mentioned by Ibn Tahir al-Baghdadi, and it corresponds to the first formulas of Al-Kashi and Al-‘Amili, as we shall see later. Furthermore, the Banu Musa used the proof by contradiction (also known as burhan al-khulf or reductio ad absurdum) to prove the rightness of the above mentioned formula. Before them, Archimedes gave the volume of sphere with reference to the volume of the cone. He said:
"Each sphere has four times the volume of a cone of which the base is equal to the greatest circle that may be inscribed in that sphere and of which the height is equal to half the radius of that sphere [13]."
Accordingly, Al-Dabbagh confirms the importance of the assumption that Banu Musa defined the volumes like magnitudes and not by comparing them with other volumes as Archimedes did. In other words, they used arithmetic operations to find the geometric magnitudes, and this approach may be considered as an important step to extend the numeric system so that it comprises natural as well as rational numbers [14].
On the other hand, Banu Musa's demonstration differs from that of Archimedes. This feature was remarked by Roshdi Rashed, who referred to it as a feature stressing the importance of the mathematical achievements of Banu Musa [15].
Also, we notice that Kamal al-Din al-Farisi depended on the book of Banu Musa so that he could correctly find the volume of the sphere. He corrected the wrong formula used to measure the volume of the sphere in his time, referring to the 15th theorem in the book of Banu Musa [16].
Figure 5: General view of the western and southern courtyards of the Friday Mosque in Isfahan where Abu ‘l-Wafa's figure is applied (Source). |
Abu ‘l-Wafa al-Buzgani (328 H/940 CE–388 H/998 CE) is an important mathematician of Islam. He is the author of Kitab fi ma-yahtaju ilayhi al-kuttab wa-‘l-‘ummal wa-ghayruhum min ‘ilm al-hisab (Book in what is needed by secretaries, artisans and others in the science of arithmetic). Known as Abu al-Wafa al-Buzgani, his full name is Muhammad b. Muhammad b. Yahya bin Isma'il bin Al-‘Abbas. He was born in Buzgan, in the region of Nishabur in Kuhistan, Iran, in Ramadan 328 H (940 CE), and he moved to Iraq in 348 H/959 CE. He then lived in Baghdad where he wrote the above mentioned mathematical book in addition to numerous other works in mathematics and astronomy. He died in Baghdad in 388 H/998 CE [17].
Al-Buzgani divided his book into seven parts or chapters devoted to the following subjects: the proportion, multiplication and division, surveying, taxation, division of inheritances, several varieties of arithmetic that are needed by secretaries (state employees), and calculations for commercial transactions [18].
Concerning the volume of the sphere, Al-Buzgani gave a rule that he included in the chapter on the surface area of the cone. He states:
"We get the surface area of a sphere by multiplying by four the area of the greatest circle inside it, the resultant is the surface area of the sphere. As for the surface area of the solid, Archimedes used to multiply the diameter of the sphere in itself and he added the periphery of the greatest circle on it, then he takes its sixth; the result is the surface area of the sphere. Another method: we obtain the area of the sphere by multiplying the diameter of the greatest circle that exists on it by the periphery of that circle, the product will be the area of its surface. Furthermore, we get the surface area of the solid by multiplying the third of the surface area by the radius of the sphere, the product will be the surface area of the sphere [19]."
Consequently, Al-Buzgani gives a rule for the volume of a sphere by two correct formulas:
1. The first formula: Before giving the first formula, Al-Buzgani gives the surface area of the sphere as follows: S =4S_{1}. Then, he gives the rule of the volume of the sphere referring to Archimedes: V = 1/6 d^{2}.p. This formula corresponds to the fifth formula of Al-Kashi.
2.The 2nd formula: Then Al-Buzgani gives, first, the surface area of the sphere surface, then its volume. The surface area of the sphere is S = d.P; the rule of the volume of the sphere is V = 1/3 S.r. This formula corresponds to the first one of Al-Kashi; it was also mentioned by Banu Musa who proved it.
Figure 6: Two cases of the theorem of Al-Kashi or the law of cosines applied in the case of an unknown side and unknown angle by the method of "cutting and pasting". In trigonometry, Al-Kashi's law or the ‘cosine formula' is a statement about a general triangle which relates the lengths of its sides to the cosine of one of its angles (Source). |
Abu Baker Muhammed ibn al-Hasan al-Karaji, lived in Baghdad, in the age of the king Fakhr al-Mulk Abu Ghaleb Muhammed bin Khalaf. Most sources indicated that he died around 419 H/1029 CE [20].
Al-Karaji discussed the problem of the volume of the sphere in his mathematical treatise Al-Kafi fi ‘l-hisab (The sufficient book in arithmetic) [21]. This book is particularly assigned to employees and to the public in general, and to those interested in the calculation of Islamic welfare (al-zakat) and legacies. While being of a prominent scientific level, the book was structured with methods that could be understood by those for whom it was written. Basically, Al-Karaji didn't use numbers such as the hand calculation of his age, but he expressed them by words. Moreover, the book was classified as one of by-hand calculation books. It discusses subjects such as arithmetic, geometry, and algebra.
Al-Karaji gave a rule for the volume of the sphere in the case of solids. He said:
"The third part: the sphere and its surface area. You should multiply its diameter in itself, then the sum is multiplied by the diameter, then you subtract the seventh and a half from the product, and you subtract the seventh and a half from the remainder. If you take a body of wax erected at right angles with three equal dimensions and you weigh to find it equal to thirty dirhams, then you make a solid sphere from it such that its diameter is equal to one side of the solid, so that you find that it weighs roughly less than eighteen and two-thirds, this requires that the diameter of sphere to be cubed and you subtract approximately a third and two fifths of its ninth from it. Consequently, we find a close distinction between the two works, but I believe that the first is the correct one [22]."
Al-Karaji gives the volume of the sphere by two methods: the first is theoretical, and the second is practical.
The theoretical method obtains the formula:
V = [d^{3} - (1/7+1/2.1/7)d^{3}] – (1/7+1/2.1/7)[d^{3} - (1/7+1/2.1/7) d^{3}];
i.e.: V = (11/14)^{2}.d^{3}.
When compared with the correct volume of the sphere (11/21).d^{3}, we find that Al-Karaji's formula produces a value greater than the correct volume by the amount (55/588.d^{3}).
Al-‘Amili [23] gives the same rule as Al-Karaji of the volume of the sphere, but Ibn Muhammed al-Khawam [24] credits the same wrong rule of the volume of the sphere mentioned by Al-Karaji. Furthermore, we find the following statement in the treatise Al-kafi fi ‘l-hisab: "The surface area of the sphere = (1 – 1/7 – 1/2.1/7), (the diameter) = (1 – 1/7 – 1/2.1/7)d^{3}." Baha' al-Din al-‘Amili considers that the surface area of the sphere = 11/14. (the diameter)^{3}[25], that is = (11/14)d^{3} (note that the equation is not meant to be dimensionally correct). This contradicts what was mentioned by Al-Karaji in his book Al-kafi fi ‘l-hisab recently edited, and also what was mentioned by Al-‘Amili who said the following in this concern: "Or you subtract the seventh and a half of its seventh from the cube of the diameter, then do the same with the result [26]."
The practical method: with this method, we obtain:
V = d^{3} - (1/3 + 2/5.1/19)d^{3} = d^{3}(1 – 1/3 – 2/5.1/9) = (28/45)d^{3}. Comparing with the correct volume of the sphere, which is (11/21)d^{3}, we find that the rule of the volume of the sphere of Al-Karaji with the practical method gives a value greater than the correct volume by the amount of (31/315)d^{3}.
By comparison with the distinction of the first method and its amount (825/8820)d^{3} = (55/588)d^{3}, with the distinction of the second method and its amount (868/8820)d^{3} = (31/315)d^{3}, we may conclude that the rule of the practical method is more erroneous than that of the theoretical method. This is confirmed by Al-Karaji himself when he said: "There is a slight discrepancy between the two methods, and I believe that the first is the correct one". Thus, by calculating the distinction between the two methods we find it equal to the following amount: (868/8820)d^{3} - (825/8820)d^{3} = (43/8820)d^{3}.
Finally, we note that Al-Karaji gives two rules to calculate the volume of the sphere, and both are wrong. Saidan mentions an opinion of an interpreter of the book of Al-Karaji about the volume of the sphere as follows [27]: "Al-Shahrzuri [28] justifies the first rule that the square of the circle is (11/14)d^{3}, and thus the volume of the sphere should be 11/14.11/14.d^{3}, and he objects to the amendment mentioned by Al-Karaji as it differs from what was stated by the ancients!"
Figure 7: Portraits of Ibn al-Haytham on two stamps issued by Qatar in 1971 and Pakistan in 1969 (Source). |
Our mathematician Al-Baghdadi is Ibn Mansur Abdul Qahir bin Tahir b. Muhammed b. Abdullah al-Shafi'i al-Baghdadi. He is called a faqih. He died in 429 H/1037 CE. He was a scientist and an important scholar who wrote many books dealing with the science of fiqh (Islamic law), principles of jurisprudence (usul al-fiqh) and arithmetic [29]. He dealt with the volume of the sphere in his book Al-takmila fi ‘l-hisab (The completion of arithmetic).
The book comprises the following seven chapters of arithmetic: in the knowledge of Indian arithmetic figures, calculation on the board, whole numbers and revealing their numeric figures, calculating fractions figures, calculating degrees and minutes figures, hand calculation figures; particular sections of roots and cubes; the different properties of numbers, some specific arithmetic operations, on transactional arithmetic, and finally a chapter on base derivation [30].
Concerning the volume of the sphere, Ibn Tahir al-Baghdadi gives a rule of this mathematical problem in the chapter devoted to the area of solids (i.e., their volumes). He said:
"In order to know the volume of the sphere solid, third of surface area of the greatest circle on it should be multiplied by its radius, thus, all of its volume will be known by calculating the surface area related to this circle, as it is quarter to all of its area of sphere surface. Allah knows best [31]."
Specifically, the volume of the sphere is given as follows: V = (S).r/3, and this is a correct formula. Thus, to calculate the surface area of the sphere, Ibn Tahir al-Baghdadi gives the following connection: S_{1} = S.1/4 (assuming that S_{1} = the area of the greatest circle). Thus, Ibn Tahir al-Baghdadi defined the rule of the volume of the sphere in its correct form, and it is identical with that of Banu Musa for this mathematical problem.
Figure 8: Manuscript view of the opening of the mathematical poem of Abdullah ibn Muhammad Ibn al-Yasamin (d. 1202) al-Urjuzah al-Yâsamîniyya fî l-jabr (Source). |
Risala fi misahat al-kura (treatise on the surface area of the sphere) is another treatise of Arabic mathematics in which the problem of the volume of the sphere is demonstrated. The text is due to the famous scientist Al-Hasan ibn al-Haytham, born in 965 CE and died 1039 CE. Ibn al-Haytham was creative in many fields from optics, mathematics, astronomy, to philosophy and divine sciences.
The treatise of Ibn al-Haytham on misahat al-kura is still unedited. It is extant in several manuscripts. One is a manuscript copy held at the National Algerian Library in Algiers under no.1446 (pp. 113r and 119v). The text consists in 14 manuscript pages, each page contains 21 lines. Another manuscript source of this treatise is preserved at Istanbul, Atif Library, under no. 1714/20 (pp. 211v-218r; 14 pages). Each page consists of 25 lines. The treatise comprises a proof of the formula for the volume of the sphere. Professor Ali al-Ayeb presented recently the proof of Ibn al-Haytham in two congress articles [32].
Ibn al-Haytham gives the formula of the volume of the sphere as follows:
"Each sphere is two thirds of a rounded cylinder of which the base is the greatest circle located in the sphere and its height is such as the diameter of the sphere [33]."
This relationship can be expressed in symbolic language as follows: (the volume of the sphere) V = 2/3 V_{2} (volume of the cylinder), the diameter of the sphere d being = height of cylinder H_{2}, the area of the greatest circle in the sphere s_{1} = area of the base of the cylinder s_{2}; thus, the volume of the sphere is:
V = 2/3 (V_{2}) = 2/3 (πr^{2}.H_{2}) = 2/3 (S_{1}.d).
By this transformation, we find that the above formula corresponds to the second formula of Al-Kashi.
Professor Ali al-Ayeb mentioned in the introduction to his study of the treatise of Ibn al-Haytham that his work "is related to one feature of the Arabic mathematical activities in infinitesimal quantities… This subject became subsequently a basic introduction of the chapter of integral calculus which provided easy solutions to many difficult problems and complicated operations, furnishing thus a great help to the [mathematical] discovery and invention [34]."
Figure 9: The first two pages of Al-Urjuzah al-Yasaminiyah by Ibn al-Yasamin. © The British Library, Oriental and India Office Collections, MS Or 3130 (Source). |
Abu Muhammed Abdullah bin Muhammed bin Hajjaj al-Adrini known as Ibn al-Yasamin was a citizen of Fez and Marrakech, in Morocco. He left written works in mathematics and poetry. He died in Morocco in 601 H/1204 CE [35]. One of his mathematical texts is Talkih al-afkar fi al-'amal bi rushum al-ghubar. It is a book of arithmetic that comprises five parts in forty chapters dealing with the most important operations needed in arithmetic, algebra, and geometry [36].
Ibn al-Yasamin gave a rule for the volume of a sphere in the chapter dedicated to the measurement of solids. He said:
"The third part: the sphere. You get its area by multiplying its diameter by itself, then we multiply the resultant by the diameter, then you subtract the seventh and a half of its seventh from the resultant. Al-Haj Abu Bakr said: If you weigh a perpendicular equilateral solid of wax to find it equal to thirty dirhams, then you make a sphere of it as plain as you can, making its diameter similar to one side of the solid and you find it weighing less than eighteen and roughly two-thirds, thus this involves the diameter of the sphere to be cubed, then approximately a third and the two-fifths of its ninth should be subtracted. There is a slight difference between the two methods, but I believe that the first is the correct one [37]."
Ibn al-Yasamin gives the volume of the sphere through two methods - as Al-Karaji did:
The first method: V = d^{3} - (1/7 + 1/2.1/7)d^{3}; thus, V = 11/14 d^{3}. As we know that Al-Karaji subtracted the following amount from the above one, then: (1/7 + 1/2.1/7)[d^{3} - (1/7+1/2.1/7)d^{3}]. As we mentioned before, still Al-Karaji's formula gives the volume of the sphere greater than its correct volume by the amount (55/588 ^{3}).
In comparison, between the volume of the sphere formula of Ibn al-Yasamin (11/14 - d^{3}) with the correct formula (11/21 d^{3}), we find that the error in the formula of Ibn al-Yasamin (11/42 d^{3}) is greater than that of Al-Karaji (55/588 d^{3}).
The second method: Ibn al-Yasamin mentioned the practical second method to calculate the volume of the sphere which was mentioned by Al-Karaji in his book Al-kafi, and he related it to Al-Haj Abu Bakr, i.e. Al-Karaji; it is: V = d^{3} – (1/3 + 2/5.1/9)d^{3}.
In conclusion, Ibn al-Yasamin depends on what was mentioned by Al-Karaji concerning calculating the volume of the sphere; hence he adds some errors to the first formula, and keeps on the error of the second one.
Figure 10: Incipit of Khulasat fi 'ilm al-hisab wa-'l-jabr wa-'l-muqabala (Summa of arithmetic and algebra) of Baha' al-Dīn al-Amilī, a mathematical treatise in ten sections (Source). |
Abdullah b. Muhammad Al-Khawam al-Baghdadi received his teachings by Nasir al-Din al-Tusi (597-672 H/1201-1274 CE). His presence at Isfahan city was mentioned around the year 675 H/1277 CE, as he was teaching there the sons of princes and ministers at the beginning of the 8th Hijra century. He also taught the fiqh at Dar al-Zahab in Baghdad, and he lived in it until he died in 724 H/1324 CE [38].
The book in which he hit upon the calculation of the volume of the sphere is Al-fawa'id al-baha'iya fi al-qawa'id al-hisabiya (The Baha'i uses in the arithmetical rules). The treatise deals with arithmetic, algebra, and geometry. Al-Khawam arranged it in an introduction, five chapters and a conclusion.
Ibn Muhammad al-Khawwam gives the rule for the volume of the sphere as follows:
"The sphere is a solid circumscribed by one surface inside which there is a point, and all the outgoing straight lines towards the circumscribed surface are equal. Its area is a cube of the diameter after subtracting its seventh and a half of its seventh, then subtracting the seventh and a half of its seventh from the resultant [39]."
Thus, Ibn Muhammad al-Khawam gives a rule for the volume of the sphere as follows:
V = [d^{3} - (1/7 + 1/2.1/7)d^{3}] - (1/7 + 1/2.1/7) [d^{3} - (1/7 + 1/2.1/7)d^{3}].
In other words: V = (11/14)^{2}.d^{3}.
Evidently, this is an errenous rule about the volume of the sphere. It is identical to its above mentioned counterpart in Al-Karaji. On the other hand, Kamal al-Din al-Farisi revealed this error and its amount; we shall mention this in the next section.
Figure 11: Front cover of Les mathématiques arabes: VIIIe-XVe siècles by the Russian historian of science Adolf P. Youschkevitch (French translation, Paris: Vrin, 1976). |
Al-Hasan bin Ali bin al-Hasan al-Farisi called Kamal al-Din was born in Iran in 665 H/1266-67 CE. He was an excellent scholar in mathematics and optics, fields in which he left many original writings. He died in Tabriz in 718 H/1319 CE [40].
One of his mathematical texts is Asas al-qawa'id fi usul al-fawa'id (The base of the rules in the principles of uses). It comprises an introduction and five chapters dealing with arithmetic, notarial and sales rules, the areas of surfaces and solids, and the last two essays are on algebra. The book is a commentary on the treatise of Al-Baha'i uses in the arithmetic rules of Al-Khawam al-Baghdadi [41].
In his commentary on the book of Al-Khawwam, Al-Farisi reveals the error of the volume of the sphere in the book and explains the reason lying behind this error; then, he gives the correct rule of the volume of the sphere. Here, we quote Ibn al-Khawam's passage with the expression "he said" and Al-Farisi's commentary with the expression "I say [42]":
"He said. Chapter. We mentioned above that the volume of the solid is the quantity of what it has of a cube doubles to the permitted amount, and a cube is an equilateral solid, thus, the sphere is a solid circumscribed by one solid that has a point inside it and all the outgoing straight lines from it towards the circumscribed surface are equal. Its volume is the cube of the diameter after subtracting its seventh and a half of its seventh; also from we substract from the resultant the seventh and a half of its seventh.
"I say: It is a point of view, because Archimedes showed in in the Proposition 36 from the first treatise of his book The Sphere and the Cylinder, that each sphere is four doubles of a cone of which the base is equal to the greatest circle located inside this sphere, and its height is equal to to the radius of the sphere. We will mention to you later on that the volume of each right-angled cone is the product of the third of its height by its base. Hence, the area of each cone of the four is the result of the multiplication of the third of the sphere radius; I mean, a multiplication of the sixth of the sphere diameter by its base; that is, the greatest of its circle is the sphere. Thus, the area of the four – I mean the area of the sphere – is the multiplication of two-thirds of the diameter by the greatest circle, or, the multiplication of the diameter by the two-thirds of the greatest circle. That this is exact and undoubtful. It is supported by the fifteenth theorem of the book of Banu Musa where they claim that for each sphere the product of its radius by the third of the circumscribed surface is equal to its greatest circle. But the circumscribed surface, as mentioned before, is four doubles of its greatest circle, its third is a circle and a third, and a radius in a circle with its third is such the diameter in the two-thirds of the circle, this is the pretence. Therefore, as the diameter in its square is a cube of the diameter and in its circle exists a cylinder whose base is the greatest and its height is the diameter as revealed thereafter, and the circle is from the square is such 11 to 14, then, the cylinder is from the cube is such 11 to 14 too. Thus, as the sphere is such the multiplication of the diameter by the two thirds of its greatest circle, then, it is two thirds of the cylinder, and if seventh and a half of its seventh is subtracted from cube of the diameter, and third of the resultant then the result is the sphere, but the seventh and its half is less than the third with five sixths of the seventh of one. Thus, with this amount, the result becomes more than it should be. Yes, if he said: if we subtract the seventh and a half of its seventh from the cube, and two sevenths and a third of its seventh from the resultant, it would have been correct".
Furthermore, we mentioned above that Al-Khawam al-Baghdadi gave an erroneous formula for the volume of the sphere as follows:
V = [d^{3} - (1/7 + 1/2.1/7)d^{3}] - (1/7 + 1/2.1/7) [d^{3} - (1/7 + 1/2.1/7)d^{3}] = (11/14)^{2}.d^{3}.
Kamal al-Din al-Farisi objects to above mentioned formula, and he reveals the correct formula and how to find it depending on the theorems of Archimedes and Banu Musa, and here are details of his correct formula for the volume of the sphere [43].
Al-Farisi – to start with – depends on the theorem of Archimedes provided in Proposition 36 of Book I of The Sphere and Cylinder which states that "each sphere is four times of a cone whose base equals to the greatest circle located in that sphere, and its height equals to the radius of the sphere". That means: V (volume of sphere) = 4 V_{1}(volume of right cone) as: S_{1}(area of the greatest circle in sphere) = S_{4} (area of the base of cone), and r (radius of sphere) = H_{4} (height of the cone). Assuming that:
V_{1} = 1/3 H_{4}.S_{4} = 1/3 S_{4} = (d/6) S_{4} = (d/6) S_{1}; therefore
V = 4 V_{1} = 4 (d/6).S_{1} = (2/3 d).S_{1} = d.(2/3 S_{1}).
Al-Farisi confirms the rightness of this relationship to account for the volume of the sphere and he supports it by the fifteenth theorem in the book of Banu Musa, in which the three brothers claim: "The product of multiplying the radius by the third of the circumscribed surface of each sphere is equal to its magnitude/size".
Thus, the rule of the volume of the sphere according to the theory of Banu Musa is: V (volume of sphere) = (d/2).(1/3 S)S. But: S = 4.S_{1}; therefore substituting in the rule of the volume of the sphere, we can write:
V (volume of sphere) = (d/2).1/3(4S_{1})=(d/2)(4/3 S_{1}) = d.(2/3 S_{1}).
The result according to the theory of Banu Musa corresponds to that of the theory of Archimedes. Thus, Al-Farisi justifies the construction of the volume of the sphere as follows:
S_{1}/d^{2} = 11/14 ⇒ (S_{1}.d)/(d^{2}.d) = V_{2}/d^{3} = 11/14 ⇒ V_{2} = 11/14 d^{3}.
Assuming that d (the diameter of the sphere) = H_{2} (height of the cylinder), and we have:
V (volume of sphere) = d.2/3.S_{1} = 2/3.V_{2} = V_{2} - 1/3 V_{2} = 11/14 d^{3} - 1/3 (11/14 d^{3}).
Therefore, we can express the volume of sphere as follows:
V = [d^{3} - (1/7 + 1/2.1/7)d^{3}] – 1/3[d^{3} - (1/7 + 1/2.1/7)d^{3}].
That means that the amount which should be subtracted is:
1/3 [d^{3} - (1/7 + 1/2.1/7)d^{3}]
But Al-Khawam subtracts the following amount: (1/7 + 1/2.1/7)[d^{3} - (1/7 + 1/2.1/7)].
Therefore, the volume of the sphere according to Al-Khawam is greater than the correct volume, and the difference equals:
1/3[d^{3} - (1/7 + 1/2.1/7)d^{3}] - (1/7 + 1/2.1/7)[d^{3} - (1/7 + 1/2.1/7)d^{3}] = 5/6.1/7 [d^{3}3]
Al-Farisi says that the result would have been correct had Ibn Muhammad al-Khawam given the following formula:
V (volume of sphere) = [d^{3} - (1/7 + 1/2.1/7)d^{3}] – (2/7 + 1/3.1/7)[d^{3} - (1/7 + 1/2.1/7)d^{3}].
At last, Kamal al-Din al-Farisi correctly defines the volume of the sphere, and he defines the theorems of Archimedes and Banu Musa related to the volume of the sphere. Furthermore, he corrects the formula given by his teacher Al-Khawam al-Baghdadi.
Ghiyath al-Din Jamshid bin Mas'ud bin Mahmud al-Kashi was one of the astronomers of the king-scientist Ulugh Beg. His scientific career was very brilliant in the fields of astronomy and mathematics. Having lived in the 15th century CE, he was one of the last very original scientists of the Islamic tradition. He died in 1429 CE.
Among his works, one of the most known is Miftah al-hisab (The Key of Arithmetic) [44]. The book comprises an introduction and five chapters. The introduction discussed the definition of arithmetic and the numbers and their parts. The first chapter discussed calculating whole numbers; the second one is devoted to the fractions and other different subjects; the third one to the calculations of astrologers; the fourth to areas; and the fifth to evaluating the known quantities by algebra, to identities, to the method of double error and other arithmetic rules. In sum, this book is a real encyclopaedia of mathematics of the 15th century CE.
Al-Kashi assigned the fifth chapter of the seventh part (in the volume of solids) of the fourth chapter to the volume of the sphere. He said:
"The volume of the sphere: we multiply its radius by the third of its circumscribed surface to produce the volume.
Another way: we multiply the two thirds of its diameter by the area of the greatest circle in it.
Another way: we write the diameter then we take eleven proportions of twenty-one of it by the famous calculation, meanwhile, by our calculation, we multiply a cube of the diameter by 0 31' 24" 57"'20"", and it is the sixth proportion of the circumference (periphery) to the diameter to give the volume.
Another way: we multiply the sixth cube of the diameter by proportion of circumference to the diameter.
Another way: we multiply two thirds of a cube diameter by the proportion of the area of the circle to the square diameter, which is 0 47^{'} 7^{"} 26"', as mentioned above.
You must know that the volume of the sphere is equal to a cylinder whose base is equal to the greatest circle located in the sphere and its height is equal to two thirds of the diameter of the sphere. Also, it is equal to four cones the base of each being equal to the greatest circle in that sphere, and its height being equal to the radius of that sphere [45]."
As we know that the volume of the sphere is directly related to the value of π (the proportion of the circumference to the diameter), thus Al-Kashi made a calculation of the value of π in his treatise entitled Al-risala al-muhitia and found it equal to (3 8 29 44 third) after subtracting the fourths and thereafter, provided that the diameter is one, and this is what he wrote in The Key of Arithmetic. He said:
"I know that the circumference equals to three times of the diameter with a fraction, and it is less than the seventh of the diameter, but people take it as seventh to facilitate calculation. Archimedes said that fraction is less than the seventh and more than ten proportions of seventy one and of all we get and mentioned in our letter called the peripheral it is: [ 3 8 29 44 third], after subtracting the fourths and thereafter, with the diameter is one.
This calculation is too much more precise than that of Archimedes according to what we revealed and it is closer to the rightness, but in fact nobody knows it except Allah, the most Blessed and High. Thus, when a diameter of a circle is known but its circumference is unknown then we multiply the diameter by that number to produce the periphery. Reflexively, we divide the periphery on that number then the quotient is the diameter [46]."
Thus, the circumference of the circle in Al-Kashi is equal to: P = d . π = d. (3^{o} 8^{'} 29^{"} 44^{"'}) and the diameter of the circle according to him is equal to P / (3^{o} 8^{'} 29^{"} 44^{"'}).
For certainty of the calculation of the volume of the sphere, we need to know the rule of Al-Kashi related to the calculation of surface of the sphere, which he mentioned by stating [47]:
"The fourth chapter: On the surface area of the sphere and the extraction of its diameter. For calculating the area, we multiply the diameter by the periphery of the greatest circle located inside.. Another way, we multiply the square of the diameter by the proportion of the periphery to the diameter , and it is four times of the greatest circle located inside, equal to the surface of a circular right cylinder except the two bases, then, each one of its thickness and the diameter of its base equal to its diameter"
Which means that the surface area of the sphere in Al-Kashi is equal to:
S = d.P = d^{2}.P/d = 4.S_{1} (as: S_{1} = πr^{2})
(assuming that the diameter of the right circular cylinder = the diameter of the base of the right circular cylinder, i.e. S = S_{3}, considering that the area of the side surface of the right circular cylinder as proved by Al-Kashi [48] is equal to S_{3} = (2πr).(H_{2}) = 2πr (2r) = 4πr_{2}; hence we find that the rule of the area of the sphere to Al-Kashi corresponds to the current rule and it is correct.
At present we go back again to Al-Kashi's formulas forth rule of the volume of the sphere:
The first formula V = r . 1/3(S). It is correct and it is the same formula of volume of sphere of Banu Musa, and it corresponds to the second formula to Al-Buzgani.
The second formula V = 2/3. d .S_{1}; it is also correct and it corresponds to the rule formula of Banu Musa, and also to one of the formulas of Al-Farisi.
The third formula: 11/21 . d^{3}; it is correct accounting that π = 22/7; Al-Kashi called it "the famous calculation [49]."
The fourth formula: V = (31^{'} 24^{"} 57^{"'} 20^{"' '}).d^{3} = 1/6.(P/d).d^{3}; it is correct:
(π = 3^{o} 8^{'} 29^{"} 44^{"'}) ⇒ (1/6.(P/d) = π/6 = 31^{'} 24^{"} 57^{"'} 20^{"' '})
The fifth formula: V = d^{3}/6 . P/d is correct, and it corresponds to the first formula of Al-Buzgani.
The sixth formula: V = 2/3 . d^{3} . S_{1}/d^{2} = 2/3 . d^{3} . (47' 7" 26"').
We know that: S_{1}/d^{2} = π/4 and by referring to the above amount as: π = 3^{o} 8^{'} 29^{"} 44^{"'} we find: π/4 = (3^{o} 8^{'} 29^{"} 44^{"'})/4 = 47^{'} 7^{"} 26^{"'}.
The seventh formula: V = V_{2} = S_{2}.H_{2} = S_{1}.(2/3)d (as: S_{1} = S_{2}) 2/3 d = H_{2}.
This formula is similar to the formula mentioned by Archimedes. Knowing that Al-Kashi defined volume of the cylinder by the following relation [50]: V_{2}= S_{2} . H_{2}.
The eighth formula: V = 4V_{1} = 4(1/3 S_{4}).H_{4} = 4.(1/3).S_{1}.r (as: S_{1} = S_{4}, H_{4} = r). This is the formula that Archimedes and Al-Farisi mentioned, knowing that Al-Kashi defined volume of the right-cone by the following relation [51]: V_{1} = 1/3 . S_{4} . H_{4}.
At last, we find that Al-Kashi gave a rule of volume of sphere with its correct form and with many formulas, but without giving a mathematical proof to the rule. Youschkevitch [52] confirmed the rightness of Al-Kashi calculation to the amount π until the seventeenth decimal number, and he summed up his method in calculation and praised it.
Muhammad bin Husayn bin Abdul Samad, called Baha' al-Ddin al-Harithi al-'Amili was born in Ba'labak in 953 H/1547CE. He died in Isfahan in 1031 H/1622 CE. He wrote in many subjects such as mathematics, astronomy, the sciences of religion, the arts, and language [53].
Among his scientific left books we find Kitab khlusat al-hisab (The conclusion of arithmetic). The book comprises an introduction and ten parts. It deals with the basic arithmetical operations such as addition, subtraction, multiplication, and division, and also with the properties of numbers, addition in mathematical series, algebra and equations, the areas of geometrical plane figures, the volumes of regular solids and other subjects [54].
Al-'Amili gives a rule to volume of sphere. He said in the third chapter (in the area of solids) of his book:
"For the sphere, you multiply its radius by the third of its surface, or subtract the seventh and a half of its seventh from the cube of the diameter, then, from the reminder also the same [55]."
Al-'Amili gives the following two formulas of the volume of the sphere:
1. The first formula: V = r. 1/3 . (S). This is the correct formula of the volume of the sphere, and it is the formula of Banu Musa's volume of the sphere.
2. The second formula:
V = [d^{3} - (1/7 + 1/2.1/7)d^{3}] - (1/7 + 1/2.1/7) [d^{3} - (1/7 + 1/2.1/7)d^{3}] ⇒ V = (11/14)^{2} . d^{3}.
It is an erroneous formula of the volume of sphere. Here we have to mention that Youschkevitch [56] mistakably mentioned that Al-'Amili gave the volume of the sphere by the following relation:
V = d^{3} {(1 – 3/14) – 3/14(1 – 3/14) – 3/14[(1-3/14) – 3/14(1 – 3/14)]} = (11/14 d)^{3}.
Compared to the correct volume of sphere (11/21 . d^{3}) we find that the rule of the volume of sphere of Al-'Amili – in the second formula – is greater than the correct volume by the amount (55/588 . d^{3}). Thus, the error formula of volume of the sphere of Al-'Amili corresponds to the theoretical formula of volume of sphere of Al-Karaji.
At the end of the third chapter on the measurement of solids, Al-'Amili said:
"The proofs of these works are detailed in our great book called Bahr al-hisab (Sea of arithmetic) [57]."
Which means that he doesn't give proofs to the above formulas in the book of our concer, Kitab khlusat al-hisab.
Actually, in that bookAl-'Amili gives two correct formulas of the surface of the sphere which he expressed as follows:
"As for the surface of the sphere, you multiply its diameter by the periphery of its greatest, or the square of its diameter by four, then, subtract from the resultant its seventh and a half of its seventh [58]."
The first formula of the surface of the sphere could be expressed as follows: S = d . P, and this is a correct formula)
Meanwhile, we find Al-'Amili in the second formula defined the value of the amount π by 22/7 as follows:
S = 4d^{2} - (1/7 + 1/2.1/7)(4d^{2}) = 22/7 d^{2}
At last, we are amazingly wondering to the utmost that a great mathematician such as Al-'Amili gives two formulas of the volume of the sphere, one of them is wrong and the other is correct. Did he compare them?; did he know about the results of his predecessors in Arabic mathematics?
From the above study, we deduce the following conclusions:
1. We may classify the situation of the rule of the volume of sphere of the Arab scientists as follows:
- A category correctly mentioned the rule with an evident proof to it: such as Banu Musa, Ibn al-Haytham, and Kamal al-Din al-Farisi (who only justified the rule).
- A category correctly mentioned the rule without an evident proof to it: such as Al-Buzgani, Ibn Tahir al-Baghdadi, and Jamshid al-Kashi.
- A category mentioned the rule in its error figure only: Al-Karaji, Al-Shahrzuri, Ibn al-Yasamin, and Al-Khawam al-Baghdadi.
- A category mentioned the rule into its two figures, both the correct and the error in their writings: such as Baha' al-Din al-'Amili.
- There might be other proofs of the rule of the volume of sphere of some above mentioned scientists, but we could not get them because of their unavailability.
2. The scientists of the Arabic tradition were influenced by the Greeks, but, they had submitted other evident proofs to the volume of sphere distinguished with their essentiality and independency.
3. The mathematicians of the Chinese and the Arabic traditions submitted a practically similar view of the volume of sphere, but we could not define how the idea transferred between the two civilizations.
4. The historical study of the rule of the volume of sphere in the works of some Arabic mathematicians revealed weak scientific communication devices among the scientists; thus, the scientist that showed his acknowledgement on the works of others submitted correct rule of the volume of sphere, except those scientists whom they got acknowledged with and got impressed by what Al-Karaji wrote.
5. It is strange that a great mathematician such as Al-Karaji submitted an error rule of the volume of sphere, in which he was followed by more than one mathematician such as Al-Shahrzuri, Ibn al-Yasamin, Ibn Muhammad al-Khawam and Al-'Amili, knowing that more than one mathematician lived before Al-Karaji, and knew the correct rule of the volume of sphere. Furthermore, we find another well-known scientist, Baha' al-Din al-'Amili submitted two rules concerning the volume of sphere: one of them correct and the other is an error one.
6. Finally, by the repetition of the rule of the volume of sphere of Al-Karaji in the writings of some of Arab mathematicians, such as Al-'Amili, we conclude that the influence of Al-Karaji on the Arab mathematicians is clear and was strong for long time.
Footnotes
[1] Dean of the Institute for the History of Arabic Science, Aleppo University, Aleppo, Syria.
[2] Marshall Clagett, "Archimedes", Dictionary of scientific Biography, Charles Scribner's Sons, New York, 1970, vol. 1, pp. 213-232.
[3] Archimedes, Kitab al-kura wa-‘l-ustuwana (The book of the sphere and the cylinder), edited by Nasir al-Din al-Tusi, Haydarbad: Da'irat al-ma'arif al-‘Uthmaniya, 1359H. See also James Gow, A Short History of Greek Mathematics, Chelsea Publishing Company, N.Y., 1968, pp. 227-229; Marshall Clagett, "Archimedes", Dictionary of Scientific Biography, edited by C. Gillispie, New York: Charles Scribner's Sons, vol. 1, 1970, pp. 213-231.
[4] Archimedes, Kitab al-kura wa-‘l-ustuwana, op. cit., p. 65.
[5] Archimedes, Kitab al-kura wa-‘l-ustuwana, op. cit., p. 67.
[6] Archimedes, Kitab al-kura wa-‘l-ustuwana, op. cit., pp. 65-67.
[7] M. Clagett, "Archimedes", op. cit., pp. 220-221.
[8] Karine Chemla, "Theoretical Aspects of the Chinese Algorithmic Tradition (First to Third Century)", Historia Scientiarum, No . 42, (1991), p. 75.
[9] Martzloff, Histoire, op.cit, pp. 269-270.
[10] M. Mawaldi, "Ilm al-handasa inda abna' Musa bin Shaker" (Geometry of the sons of Musa Ibn Shaker), in The special book for the celebration of scientists Muhammad, Ahmad, and Al-Hasan, the sons of Musa Ibn Shaker, The 36th Science Week, Aleppo University, 2-7 Nov. 1996; Ministry of High Education, High Committee of Sciences, Damascus, 1998, pp. 99-125; p. 104.
[11] M. Mawaldi, "Ilm al-handasa inda abna' Musa bin Shaker", op. cit., pp. 100, 103.
[12] Banu Musa, Kitab ma'rifat masahat al-ashkal al-basita wa-‘l-kuriya (Book on the areas of of plane and spherical figures), edited by Nasir al-Din al-Tusi, published in Haydearabd by Da'irat al-ma'arif al-uthmaniya, 1359 H, p. 19; see also Banu Musa, Kitab ma'rifat masahat al-ashkal al-basita wa al-kuriya, University of Tehran, Heritage Institute Collection, 720, p. 133.
[13] Archimedes, Kitab al-kura wa-‘l-ustuwana, edited by Nasir al-Din al-Tusi, 1st edition, incl. messages of Al-Tusi, Haydarabad, 1359 H, p. 65.
[14] Dabbagh, "Banu Musa", Dictionary of scientific Biography, Charles Scribner's Sons, New York, 1970, vol. 1, pp. 444-445.
[15] R. Rashed, "Les autres disciplines mathématiques", Le Matin des Mathématiciens, Paris: Belin - Radio France, 1985, pp. 163-164.
[16] M. Mawaldi, L'Algèbre de Kamal al-Din al-Farisi. Édition Critique, Analyse mathématique et Étude historique, en 3 Tomes. Thèse du Nouveau Doctorat. Université Paris III, 1989, vol. 1, pp. 573-574.
[17] Saidan, Tarikh ‘ilm al-hisab al-‘arabi [History of the Arabic arithmetic]. Vol. 1: "Hisab al-yad" (hand calculation), a review to the book of "Al-manazil al-sab'a" (the seven arrangements) of Abu ‘l-Wafa al-Buzgani, with an introduction and a comparative study with Al-Kafi book in arithmetic of Al-Karji, Amman, 1971, pp. 58, 268.
[18] Saidan, Tarikh ‘ilm al-hisab al-‘arabi, op. cit., p. 65.
[19] Saidan, Tarikh ‘ilm al-hisab al-‘arabi, op. cit., p. 268.
[20] Al-Karaji, Al-Kafi fi al-hisab, op. cit., pp.18-19.
[21] Al-Karji, Al-Kafi fi al-hisab, study and editing by Sami Shalhoob, published by the Institute for the History of Arabic Science, Aleppo University, 1406 H/1986, p. 7.
[22] Al-Karaji, Al-Kafi fi al-hisab, op. cit., p. 149.
[23] Baha' al-Din Al-‘Amili, Al-a'mal al-riyadiyya li Baha' al-Din al-‘Amili [The Mathematical works of Baha' al-Din al-‘Amili], editin, explanation and analysis by Jalal Shawki. Beirut and Cairo: Dar al-Shuruq, 1981, p. 93.
[24] M. Mawaldi, L'Algèbre de Kamal al-Din al-Farisi, op. cit, vol. 1, p. 573.
[25] Al-Karji, Al-Kafi fi al-hisab, op. cit., p. 281.
[26] Al-‘Amili, Al-a'mal al-riyadiyya li Baha' al-Din al-‘Amili", op. cit., p. 93.
[27] Saidan, Tarikh ‘ilm al-hisab al-‘arabi, op. cit., p. 445.
[28] The title of his book is Al-sharh al-shafi li-kitab al-kafi fi al-hisab" (The sufficient explanation to the book of al-Kafi in calculation) by Muhammed bin Abdullah bin Abi al-Hasan bin Ahmed bin Abdullah al-Shahrzuri (Saidan, op. cit., p. 56).
[29] Abdul Qahir ibn Tahir al-Baghdadi, At-takmila fi ‘l-hisab [The Completion of arithmetics], edited by Ahmed Salim Saidan. Kuwait: the Institute of Arabic Manuscripts, 1985, pp. 10-11.
[30] Ibn Tahir al-Baghdadi, At-takmila fi ‘l-hisab, op. cit., p. 31.[31] Ibn Tahir al-Baghdadi, At-takmila fi ‘l-hisab, op. cit., p. 364.[32] Ali al-Ayeb, Taqdim wa tahlil risalat Ibn al-Haytham fi masahat (hajm) al-kura [Presenting and analysing the letter of ibn Al-Haytham in the survey of (volume) of the sphere], The Proceedings of the 1st National Meeting About the History of Arabic Mathematics, Gherdayeh, April 1993, The Algerian Association for the History of Mathematics, 1996, p. 178.
[33] A. al-Ayeb, Taqdim wa tahlil risalat Ibn al-Haytham fi masahat (hajm) al-kura, op. cit., p. 187.
[34] A. al-Ayeb, "Al-ta'yinat al-lamutanahiya fi al-sighar min khilal risalat Ibn al-Haytham fi masahat al-kura" [The infinitely small determinations in the treatise of Ibn al-Haytham on the surface area of the sphere]. The Second Maghribi Meeting about the History of Arabic mathematics, Tunisia 1-3 December 1988. Tunis: University of Tunisia, The Higher Institution of Education and Continuous Formation, 1990, p. 68.
[36] Ibn al-Yasamin, Al-a'mal al-riyadiyya li-Ibn al-Yasmin, op. cit., p. 14.
[37] Ibn al-Yasamin, Al-a'mal al-riyadiyya li-ibn al-Yasamin, op. cit., pp. 292-293.
[38] Kamal al-Din al-Farisi, Asas al-qawa'id fi usul al-fawa'id [The base of the rules in the principles of uses], edited by M. Mawaldi, Cairo: The Institute of Arabic Manuscripts, 1994, pp. 12.
[39] Abdullah Ibn Mohammed al-Khawam al-Baghdadi, Al-Fawa'id al-bahaiyya fi al-qawa'id al-hisabiyya" [Al-Bahai uses in the arithmetic rules], The British Library, MS N°. or 5615, folio 25v.
[40] Al-Farisi, Asas al-qawa'id fi usul al-fawa'id, op. cit., pp. 9-21.
[41] Al-Farisi, Asas al-qawa'id fi usul al-fawa'id, op. cit., p. 5.
[42] M. Mawaldi, L'Algèbre de Kamal al-Din Al-Farisi, op. cit., vol. 1, pp. 573-574.
[43] M. Mawaldi, L'Algèbre de Kamal al-Din Al-Farisi, op. cit., vol. 3, pp. 1221-1224.
[44] Jamshid al-Kashi, Miftah al-hisab [Key of arithmetic], edited by Nader al-Nabulsi. Damascus: Ministry of High Education, 1977.
[45] Al-Kashi, Miftah al-hisab, op. cit., pp. 315-316.
[46] Al-Kashi, Miftah al-hisab, op. cit., p. 296.
[47] Al-Kashi, Miftah al-hisab, op. cit., p. 296.
[48] Al-Kashi, Miftah al-hisab, op. cit., p. 290.
[49] Al-Kashi, Miftah al-hisab, op. cit., p. 248.
[50] Al-Kashi, Miftah al-hisab, op. cit., p. 302.
[51] Al-Kashi, Miftah al-hisab, op. cit., p. 303.
[52] Adolf Youschkevitch, Les Mathématiques Arabes (VIII-XV siècles), Traduction française par M. Cazenaze et K. Jaouiche. Vrin, Paris, 1976, p.151.
[53] Baha' al-Din al-'Amili, Al-a'mal al-riyadiyya li-Baha' al-Din al-'Amili [The Mathematical Works of Baha' al-Din al-'Amili], edited by Jalal Shawqi. Beirut and Cairo: Dar Al-Shuruq, 1981, pp. 11-14.
[54] Al-'Amili, Al-a'mal al-riyadiyya, op. cit., pp. 11-14.
[55] Al-'Amili, Al-a'mal al-riyadiyya, op. cit., p. 93.
Al-a'mal al-riyadiyya li Baha' al-Din al-‘Amili [The Mathematical works of Baha' al-Din al-‘Amili], editin, explanation and analysis by Jalal Shawki. Beirut and Cairo: Dar al-Shuruq, 1981, p. 93.