1. IntroductionThe Greek letter

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

Figure 1:Pi, symbol and value.Our aim in this article is to survey the history of the mathematical research carried on in different scientific traditions to find approximated values for p. Our focus will be on the discussion of the values of p in the mathematical tradition of the Arabo-Islamic civilisation. The analysis also sheds light on the innovations and contributions brought by Islamic scientists in this issue, and stresses on the interest and motivation of Arab scholars to arrive at the nearest possible approximation for the value of p, depending on presenting new and original mathematical proofs.

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

Figure 2:Front cover ofPi: A Source Bookby L. Berggren, J.M. Borwein and P.B. Borwein (Springer-Verlag, 2^{nd}edition, 2000).Under the entry (Pi) in a French dictionary [2], the author contends that the value p is of Greek origin, only entering into the French language in the 19

^{th}century. p is the sixth letter in the Greek alphabet, and its counterpart in European modern languages is the letterp. From a geometrical perspective, it is the truncation of the Greek word “peripheria”, and it is a symbol for the number that represents the static value of the fixed ratio of the circumference to the diameter of a circle, equal approximately to 3.1415926.

3. Approximating calculations of p through history3.1. The Greek civilisation

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

3 10/71 < p < 3 1/7.

Archimedes submitted a proof of the above equation in his treatise

Measurement of a Circle[3] that contained three propositions. In the first proposition, he proves that “the area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.”In the second proposition, Archimedes goes on to prove his theory by saying: “the circumference of a circle is greater than three times its diameter, less than one-seventh of the diameter and more than one-tenth of seventy-one parts of the diameter.”

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

The third proposition asserts the following: “The ratio of the circumference of any circle to its diameter is less than 3 1/7 but greater than 3 10/71.” He comments that if the circumference of the circle is three times the diameter plus one-seventh – this is an approximation that has been used by surveyors – then the ratio of the area of the circle to the square of its diameter is equal to the ratio of eleven to fourteen.

Archimedes derived his proof of the value of p from the theory of Euclid (330-290 BCE) that is found in the XII

^{th}book ofThe Elements[4]. In that theorem, Euclid asserts the existence of a set of ratios between the area of a circle and the area of a square of which the side is equal to half the diameter of the circle. The first estimationp˜ 3 1/7 = 22/7 is a simplified calculation that is used to solve geometric questions using practical methods [5].

3.2. The Chinese civilisationTo study the history of Chinese mathematics one must return to the text

Jiuzhang suanshuorThe Art of Mathematics in Nine Chapters[6] of which the author is unknown. It is probable that this text was collated in the 1^{st}century CE. It was depended upon as a main text until the 13^{th}century.The text put forth several theories for the calculation of the area of a circle, and in those we find some that are incorrect:

1)

S_{1}= 3/4d^{2}(the correct formula isS_{1}=pd^{2}/4);2)

S_{1}= 1/12p^{2}(the correct formula isS_{1}=p^{2}/4p)(with p = the circumference of a circle, d = the diameter, and

Sthe area)._{1}=In these two equations p is used with a value that is equal to three, which leads to an error in calculating the area of a circle.

At the end of the 5

^{th}century [7] Zu Chongzhi gave a rounded value of 355/113 to p, and from that his own son Zu Kengzhi advanced this knowledge with his own work, using the value 22/7 for (p).Rosenfeld and Youschkevitch list in their article in the

Encyclopedia of the History of Arabic Science[8] the Chinese scientist and astronomer Chang Hêng (78-139 CE) as having suggested the value Ö10 for p. Chinese mathematicians continued to use the value of 3 for p until the 19^{th}century, as it was a simple value to use in calculations and formulae.

3.3. The Indian civilisationIndian mathematicians contributed a great deal to the calculations of the value of p. Since the 7

^{th}century, the astronomer Brahmagupta (born in 598 CE) gave p the value of Ö10. Historians also credit the Indian astronomer Âryabhata (born 476 CE) with several values of (p), notably: 62832/20000 and 3 177/1250 or 3.1416 [9]. However he was said to have used the value of Ö10 or 3 for p [10].Nine centuries later, the astronomer and mathematician Mâdhava (fl. in the 15

^{th}century CE) arrived at a calculation of the circumference of the circle by using the approximated value of p that is correct up to 11 digits, namely p = 3.14159265359, as stated by Guy Mazars. However the last digit (9) is incorrect, and must be replaced with the digit (8) [11].In the 16

^{th}century, an Indian mathematician used the value 355/113 as an approximated value for p, and put forward the following rule for its calculation:

p/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – …

3.4. The Arabo-Islamic CivilisationArab scholars contributed to the quest for the calculation of the value of p, and the proportion of the circumference of a circle to its diameter, and their efforts impacted the history of mathematics.

Among the mathematicians whose names are referred to with regard to this issue, we mention the scholars who flourished from the 9

^{th}to the 15^{th}centuries. First, the three brothers Banu Musa bin Shakir (Baghdad, 3^{rd}century H/9^{th}century CE), the famous mathematician known as the father of algebra, Muhammad b. Musa Al-Khwarizmi (3^{rd}century H / 9^{th}century CE), the scholar author of various mathematical works Abu Sahl Wijan b. Rustam Al-Quhi (d. 390 H/ ca. 1000 CE), the well known polymath scholar Abu Al-Rayhan Al-Biruni (died 440 H/ 1048 CE) and finally the author of the widely diffused mathematical textMiftah al-hisab(Keys of arithmetic) Jamshid b. Mas‘ud b. Mahmud Al-Kashi (died 1429 CE). In this section we point out some of their achievements in approximating the value of p and the methods used in this sophisticated research.

4. The calculation of Pi in Arabic mathematics

4.1. Banu MusaIn their book on

Kitab fi ma’rifat misahat al-ashkal al-basita wa al-kuriya(The Measurement of Plane and Spherical Figures, the brothers Banu Musa prove that the proportion of the circumference of a circle to its diameter is greater than the percentage of 3 and 10 parts of 71 to 1, and smaller than the percentage of 3 and 1/7 to 1 [12]. This means: 3 10/71 <p/d< 3 1/7, as p is the circumference of the circle, and d is the diameter. Banu Musa also mention that Archimedes had proved this relationship, that provides the approximation of the value of p: “then to prove the proportion of the diameter to the circumference along the method of Archimedes this has not reached us in a form that was in our time, and this method even though it has not reached to the point of measuring one and the other to arrive at the truth, it in fact allows us to derive the value of one from the other closer to that which we desire to arrive [13].” However, they considered Archimedes’ method to be incomplete and not arriving at the truth, and Suter [14] agrees that their method was different than that employed by Archimedes.Sezgin [15] considers the proof of Banu Musa to measure the proportion of the diameter of the circumference of a circle as an important advancement and more in depth than Archimedes’ attempt. Roshdi Rashed [16] mentions that Banu Musa had achieved an “explanation to Archimedes method in the approximation of (p), and derived their generalizations from this calculation”.

4.2. Muhammad Al-KhwarizmiWe find in the chapter on area in his book

Al-Jabr wa-‘l-muqabala(that he wrote in the years 813 and 833 CE) rules for the calculation of the circumference of a circle as he states:“And for every circle (

mudawwara) when you multiply the diameter by three and sevenths is the circumference (dawr) that surrounds it, and it is a term among men that is unnecessary, and to those of geometry there are two other sayings: one is to multiply the diameter by itself, and then by ten, and then to take the root of the result and that is the circumference of the circle. The other is that used by astronomers is to multiply the diameter by sixty two thousand and eight hundred and thirty-two then divide that by twenty-thousand and what results is the circumference, and all these are near to each other…” [17].Al-Khwarizmi gives three values to p and these are: 62832/20000, Ö10, 3 1/7. The editors of his book produce this marginal commentary in which he says: “It is an approximation not a proof, and no one stands on the truth of this, and no one but Allah knows the true circumference of the circle, as the line is not straight and has no beginning and no end, we merely attempt to approximate and discover the root, but even the root has no definition as no one may know its exact value but Allah, and the best of these approximations that is to multiply the diameter by three and seventh as it is faster and simpler and only Allah might know it true” [18].

We note also that in his calculation of the area of the circle in his book he adopts for p the value 22/7 [19].

4.3. Wijan b. Rustum Al-QuhiAbu-Ishaq Al-Sabi had sent a missive to Al-Quhi [20] inquiring of several matters, specifically of Al-Kuhi’s method to derive the proportion of the diameter to the circumference of the circle, and asks him to forward this to him as he says [21]: “I desired… to send all that has been derived especially that of the proportion of the diameter to the circumference of the circle. As a ratio of a number to a number then that would be something myself looks for to know and have benefit of.”

Al-Quhi answers the queries of Al-Sabi then moves to discuss his text on the centers of gravity and says [22]: “on the four articles that I have done here we have reached strange things each to prove the greatness and the order of the Creator, such as those matters relating to the Sphere and the Cylinder of Archimedes. Don’t we marvel at how the sphere is equal to two-thirds of the cylinder as he proved and described, and that the paraboloïd is equivalent to its half as it was proved by Thabit ibn Qura, and that the cone is one-third as it was shown by the ancients? We found in the [study of the] centres of gravity much orders to impress us”.

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

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

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

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

Al-Quhi concludes his missive by saying: “And when we look at Archimedes as he writes, that the circumference of the circle, is less than three-times the diameter and ten parts of seventy, meaning one-seventh, and this agrees with our work and does not depart from this, and one-ninth is less than one-seventh, and he also stated: that it is greater than three times and ten parts of seventy-one, and this does not agree, unless he means: ninety-one parts instead of seventy-one to become in agreement, not more, and we do not presume to think in any manner that is not well of those who preceded us in this work, as he is Archimedes and he is a forerunner in this field [23].”

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

4.4. Abu ‘l-Rayhan al-BiruniIn the fifth chapter [24] (On the Proportion between the Circumference and the Diameter) from the third book of

Al-Qanun al-Mas’udi, Abu Al-Rayhan calculates the circumference of an angular figure which has 180 sides within the circle, and also calculated the circumference of an angular figure that has 180 sides surrounding the circle, and takes the median of the two. From that, he calculates the value of p and reaches the value :p= 3.1417 [25]. This is a value which is not much more accurate thanp= 3.1416 which was known to the Indian civilisation before.

4.5. Jamshid Al-Kashi

Figure 3:A stamp issued 1979 in Iran commemorating al-Kashi.Al-Kashi is considered as one of the greatest Islamic scholars that have put forward significant scientific achievements that have propelled modern civilisation forward. Among his works,

Al-Risala al-muhitiya(The Letter of Circumference) that sets out a very particular calculation for the value of p. Al-Kashi measured the circumference of an equal sided angular figure that is surrounded by a circle, and another that is surrounded by a circle that has 3 x 2^{28}= 805306368 sides [26], when Archimedes and Banu Musa limited themselves – as we have seen- to a figure with 3 x 2^{5}= 96 sides. Al-Kashi determined the number (28) as the difference between the circumferences of the two shapes is equal to its diameter 600000 times the diameter of the earth, less than the width of a hair.”After his calculation of the circumferences of the two figures, he presumed that the circumference of a circle is equal to the median of the two results, and he arrived at this conclusion (in the sexagesimal system):

p = 3;8, 29, 44, 0, 47, 25, 53, 7, 25

And he then transformed it into the decimal system:

p = 3. 14 159 265 358 979 325

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

Figure 4:Manuscript page of Al-Kashi’s calculation of p:jadwal tadha’if nisbat al-muhit wa-‘l-qutr(table of the multiples of the ratio of the circumference to the diameter). (Source).Al-Kashi points out in his book

The Keys to Calculationsthat some mathematicians had determined the value of 3 1/7 for p in which he states [27] : “Know that the circumference is three times the diameter and fraction, and this is less than one-seventh of the diameter, however many have sufficed to use the one-seventh for ease of use”.Al-Kashi states that his calculations are more precise than those of Archimedes, and determines a value for p [28], in saying that: “Archimedes has said that it must be less than one-seventh and more than ten-parts of seventy, and from what we have gathered and mentioned in our text on circumference which is: 82944,3 thirds, after subtracting the fourths and what follows, if the diameter is one. This is a more precise calculation than that of Archimedes, and closer to the accurate value of p, and Al-Kashi reiterates that it is the “closest that might be, save for the knowledge of Almighty Allah.”

Figure 5:Applications of the law of cosines: unknown side and unknown angle. In trigonometry, the law of cosines is known as Al-Kashi law; it is a statement about a general triangle which relates the lengths of its sides to the cosine of one of its angles. Al-Kashi was the first to provide an explicit statement of this law.Al-Kashi’s exacting calculations might not have been needed in his time, but he was a forward thinker, he was a scholar who searched for accurate results to advance progress.