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. Bas