The Volume of the Sphere in Arabic Mathematics: Historical and Analytical Survey - Continued

3.6. Ibn al-Yasamin

 Large image 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 = d3 - (1/7 + 1/2.1/7)d3; thus, V = 11/14 d3. As we know that Al-Karaji subtracted the following amount from the above one, then: (1/7 + 1/2.1/7)[d3 - (1/7+1/2.1/7)d3]. 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 - d3) with the correct formula (11/21 d3), we find that the error in the formula of Ibn al-Yasamin (11/42 d3) is greater than that of Al-Karaji (55/588 d3).

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 = d3 – (1/3 + 2/5.1/9)d3.

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.

3.7. Ibn al-Khawwam

 Large image 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 = [d3 - (1/7 + 1/2.1/7)d3] - (1/7 + 1/2.1/7) [d3 - (1/7 + 1/2.1/7)d3].

In other words: V = (11/14)2.d3.

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.

3.8. Kamal al-Din al-Farisi

 Large image 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 = [d3 - (1/7 + 1/2.1/7)d3] - (1/7 + 1/2.1/7) [d3 - (1/7 + 1/2.1/7)d3] = (11/14)2.d3.

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 V1(volume of right cone) as: S1(area of the greatest circle in sphere) = S4 (area of the base of cone), and r (radius of sphere) = H4 (height of the cone). Assuming that:

V1 = 1/3 H4.S4 = 1/3 S4 = (d/6) S4 = (d/6) S1; therefore

V = 4 V1 = 4 (d/6).S1 = (2/3 d).S1 = d.(2/3 S1).

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.S1; therefore substituting in the rule of the volume of the sphere, we can write:

V (volume of sphere) = (d/2).1/3(4S1)=(d/2)(4/3 S1) = d.(2/3 S1).

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:

S1/d2 = 11/14 ⇒ (S1.d)/(d2.d) = V2/d3 = 11/14 ⇒ V2 = 11/14 d3.

Assuming that d (the diameter of the sphere) = H2 (height of the cylinder), and we have:

V (volume of sphere) = d.2/3.S1 = 2/3.V2 = V2 - 1/3 V2 = 11/14 d3 - 1/3 (11/14 d3).

Therefore, we can express the volume of sphere as follows:

V = [d3 - (1/7 + 1/2.1/7)d3] – 1/3[d3 - (1/7 + 1/2.1/7)d3].

That means that the amount which should be subtracted is:

1/3 [d3 - (1/7 + 1/2.1/7)d3]

But Al-Khawam subtracts the following amount: (1/7 + 1/2.1/7)[d3 - (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[d3 - (1/7 + 1/2.1/7)d3] - (1/7 + 1/2.1/7)[d3 - (1/7 + 1/2.1/7)d3] = 5/6.1/7 [d33]

Al-Farisi says that the result would have been correct had Ibn Muhammad al-Khawam given the following formula:

V (volume of sphere) = [d3 - (1/7 + 1/2.1/7)d3] – (2/7 + 1/3.1/7)[d3 - (1/7 + 1/2.1/7)d3].

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.

3.9. Al-Kashi

 Large image Figure 12: Front cover of Riyadhiyyat Baha' al-Dīn al-'Amilī (Mathematical Works of Baha' al-Dīn al-Amilī), edited by Galal Shawki (Cairo: Dar al-Shurūq, 1401/1981, 2nd edition). The book provides an edition of al-'Amili's al-Khulasa fi 'ilm al-hisab and a study of a related passage in Al-'Amili's Kashku¯l.
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. (3o 8' 29" 44"') and the diameter of the circle according to him is equal to P / (3o 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 = d2.P/d = 4.S1 (as: S1 = πr2)

(assuming that the diameter of the right circular cylinder = the diameter of the base of the right circular cylinder, i.e. S = S3, considering that the area of the side surface of the right circular cylinder as proved by Al-Kashi [48] is equal to S3 = (2πr).(H2) = 2πr (2r) = 4πr2; 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 .S1; 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 . d3; it is correct accounting that π = 22/7; Al-Kashi called it "the famous calculation [49]."

The fourth formula: V = (31' 24" 57"' 20"' ').d3 = 1/6.(P/d).d3; it is correct:

(π = 3o 8' 29" 44"') ⇒ (1/6.(P/d) = π/6 = 31' 24" 57"' 20"' ')

The fifth formula: V = d3/6 . P/d is correct, and it corresponds to the first formula of Al-Buzgani.

The sixth formula: V = 2/3 . d3 . S1/d2 = 2/3 . d3 . (47' 7" 26"').

We know that: S1/d2 = π/4 and by referring to the above amount as: π = 3o 8' 29" 44"' we find: π/4 = (3o 8' 29" 44"')/4 = 47' 7" 26"'.

The seventh formula: V = V2 = S2.H2 = S1.(2/3)d (as: S1 = S2) 2/3 d = H2.

This formula is similar to the formula mentioned by Archimedes. Knowing that Al-Kashi defined volume of the cylinder by the following relation [50]: V2= S2 . H2.

The eighth formula: V = 4V1 = 4(1/3 S4).H4 = 4.(1/3).S1.r (as: S1 = S4, H4 = 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]: V1 = 1/3 . S4 . H4.

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.

3.10. Baha' al-Dīn al-'Amilī

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 = [d3 - (1/7 + 1/2.1/7)d3] - (1/7 + 1/2.1/7) [d3 - (1/7 + 1/2.1/7)d3] ⇒ V = (11/14)2 . d3.

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 = d3 {(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 . d3) 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 . d3). 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 = 4d2 - (1/7 + 1/2.1/7)(4d2) = 22/7 d2

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?

4. Conclusion

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

[35] [Ibn al-Yasamin], Al-a'mal al-riyadiyya li ibn al-Yasmin (The mathematical works to ibn Al-Yasmin). A dissertation thesis prepared by Al-Tihami Zamouli for MPhil degree in the history of mathematics from the Highest School of Professorship, Algiers, ENS Al-Qubbah, 1993, pp. 7-10.

[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.

[56] A. Youschkevitch, Les Mathématiques Arabes, op. cit, pp. 107-108.

[57] Al-'Amili, Al-a'mal al-riyadiyya, op. cit., p. 94.

[58] Al-'Amili, Al-a'mal al-riyadiyya, op. cit., pp. 91-92.

by: By Professor Dr. Mustafa Mawaldi, Mon 06 April, 2009