No science teaches the harmonies of nature more clearly than mathematics. In arithmetic we explore a universe of figures by counting; in geometry we discover another universe by drawing lines and laying down definite directions in the conceptual field of the imagination; in algebra we produce magnitudes of a still more abstract nature that are expressed by letters. In all of these cases the first step is to establish the general conditions that lay down the rules which all further steps are subject to. Every one of these “universes” is dominated by a consistency that produces an excellent symmetry.

Certainly the world, the universe, and nature can be reliably understood using mathematics. In other words, we can say that nature is mathematics. For instance, the arrangement of seeds in a sunflower can be understood using Fibonacci numbers. Every number in the Fibonacci sequence equals the sum of the two previous except for the first two numbers. Sunflower heads, like other flowers, contain two families of interlaced spirals—one winding clockwise, the other counterclockwise. The shape assumed by a delicate spider web suspended from fixed points, or the cross section of sails bellying in the wind, is a catenary—a simple curve defined by a simple formula. Seashells, animal horns, and the cochlea of the ear are logarithmic spirals that can be generated using a mathematical constant known as the golden ratio. Mountains and the branching patterns of blood vessels and plants are fractals, a class of shapes that exhibit similar structures at different magnifications. Einstein’s E = mc² defines the fundamental relationship between energy, matter, and the speed of light. Additionally, a few simple constants—the gravitational constant, Planck’s constant, and the speed of light—are profound examples for better understanding our universe as a whole.

Magic squares are conspicuous instances of the intrinsic harmony of numbers. They are also noticeable instances of the inherent harmony of numbers that some consider as an interpreter of the cosmic order that dominates all existence. Can they? Maybe. Though they are a mere intellectual play, they not only illustrate the nature of mathematics but also, incidentally, the nature of reality dominated by mathematical regularity. This is incidental, because its discovery has a very extraordinary story. Magic Squares are square grids with a unique arrangement of numbers in them. These numbers are exceptional because every row, column, and diagonal adds up to the same number. They are understood from every part in their overall square shape, the sum of which gives the same number – the sum of rows, columns, and diagonals. Magic Squares, one of the Sudoku-style question types, may vary in the numbers to be placed in the square according to its size. For example, let’s say there is a 3×3 square and ask: how can you put the numbers 1 to 9 each in their own square so that all of the the rows, columns, and cross sums are always the same? I recommend you stop here for a while and work on the question. You can see that it is not as easy as it sounds. Otherwise, the answer is in Figure 1.

The historical development of magic squares is also very fascinating. The earliest record of magic squares is from China in about 2200 BC and is called Lo-Shu. It tells that the first magic square was incidentally discovered by the Chinese emperor Yu 4,200 years ago. Emperor Yu discovered a tortoise while sauntering along the Yellow River (Lo River). The tortoise’s shell had a magic square shape, and this square divided into grids contained dots instead of numbers. Imagine having the same quantity as spots instead of numbers (Figure 2).

According to the myth, Emperor Yu took the tortoise and kept caring for it as a valuable guest in his palace. Its fame spread all over the world and extends beyond time to today’s modern world. Perhaps, it deserves the title of the world’s most famous mathematical tortoise with an outstanding mark on him. After the tortoise, the magic squares have become a very significant mystical symbol in China and in the Karma philosophy. In Karma, it is said that the world is surrounded by four essential elements and this is symbolized with magic squares. If we consider the number 5 in the center as the world, the followers of this philosophy surrounded the world with numbers that were supposed to be the four elements of Yin Yang. 4 and 9 symbolize metal, 2 and 7 symbolize fire, 1 and 6 symbolize water, and 3 and 8 signify wood, as shown in Figure 3.

We see that magic squares have mystical meanings not only in China but also in many places. For example, in pre-Islamic Arabia, people hung magic squares in their homes as amulets that they believe protected their houses from evil spirits. One explanation to this superstition could be that Arab astrologers used magic squares to cast horoscopes.

Magic squares were introduced to the Western world in the work of Theon of Smyrna. Theon’s most important work is “Expositio rerum mathematicarum ad legendum Platonem utilium.” This work is a handbook for philosophy students to show how prime numbers, geometrical numbers such as squares, progressions, music, and astronomy are interrelated. Its rather curious title means that it was intended as an introduction to a study of the works of Plato, which sounds rather fanciful. In the section on numbers, Theon adopts a Pythagorean approach and writes about all kinds of numbers: odd, even, prime, composite, square, oblong, triangular, polygonal, circular, spherical, solid numbers with three factors, pyramidal, perfect, deficient, and abundant numbers. The work of the Greek mathematician Moschopoulos in 1300 AD helped to spread knowledge about magic squares. So here we are now, more than 700 years later, and teachers are using them in class for problem-solving and practicing addition.

In the 16^th century the Catholic physician, astrologer, and theologian Cornelius Agrippa (1486–1535) constructed squares of orders from 3, 4, 5, 6, 7, 8, and 9 which he associated with the seven known astrological planets: Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon (the Sun and Moon were considered to be planets at that time.) Agrippa had a colorful life that included various dangerous run-ins with the Church and jobs as an occult scholar, lawyer, and military strategist. Agrippa’s De Occulta Philosophia stimulated Renaissance study of magic and got his name into early Faust legends. Agrippa believed that a magic square containing the digit 1—which exhibits the magic constant of 1 in all directions—represented God’s eternal perfection.

I do not think that anyone goes beyond the Indian people in terms of expecting help from magic squares for healing diseases and even finding a spouse. They also took first place in producing magic squares known as the Jaina inscriptions. The first 4×4 square was discovered on a door in Khajuraho, India, around 1100 AD (Figure 4). Jainism is an Indian philosophy that was established in the 6th century BC, which may imply how important magic squares are for Indians who follow this tradition.

India brought the usage of these squares to daily life. One superstition still practiced today is to write a 3×3 magic square to find someone missing and hang it everywhere like a notice. Strangely, the name of the missing person is not in the notice! The magic square’s merits are not limited to this. It is believed that a couple can save their marriage if they can get a copy of a square that is drawn on a Wednesday or Friday.

For mathematicians, it makes sense if the sums of squares across each row, column, or diagonal are equal. But Indians have different expectations from the squares. For example, the square used by young girls to get married is shown in Figure 5. This square is painted on a plate with pastels. The plate is washed in the Ganges River. The girl who is seeking her desired husband drinks the pastel-colored water from the river. This tradition is still being practiced today.

There are also squares used by Muslims in the Islamic period. One of the most beautiful examples of these squares is one designed to express God’s name as shown in Figure 6. The square with 66 letters in total on each side evokes the sum of the Arabic word Allah in Abjad calculation (In Arabic, every letter is assigned a numerical value, and this calculation is called Abjad).

Throughout history, humanity has been influenced by mathematics and numbers and has tried to reveal the unknowns by investigating the mysteries of numbers. Humanity’s long-time fascination with mathematics has arisen because the universe is constructed from a mathematical fabric. In 1623, Galileo Galilei reinforced this belief by stating his credo: “Nature’s great book is written in mathematical symbols.” Plato’s doctrine was that God is a geometer, and Sir James Jeans believed God experimented with arithmetic. Newton supposed that the planets were originally thrown into orbit by God, but even after God decreed the law of gravitation, the planets required continual adjustments to their orbits. Likewise, magic squares have been one such mysterious dimension of mathematics that have influenced human life in various ways.

Figure 1 – 3×3 Magic Square

Figure 2 – Spots on the tortoise

Figure 3 – Five Elements in Magic Square

Figure 4. Magic square at the Parshvanatha temple, Khajuraho

Figure 5 — Magic Square Drawn on a Plate with Pastel Paint

Figure 6- The sum of Allah’s words with Abjad calculation is 66. The grid is formed by the letters in the word “Allah,” whose numerical value is also 66 (nineteenth century, Damascus).

Mathematics, or “the queen of the sciences” as Carl Friedrich Gauss called it, plays an important role in our lives. A world without mathematics is unimaginable. Throughout history, many scholars have made important contributions to this science, among them a great number of Muslims. It is beyond the scope of a short article like this one to mention all the contributions of Muslim scholars to mathematics; therefore, I will concentrate on only four aspects: translations of earlier works, and contributions to algebra, geometry, and trigonometry. In order to understand fully how great were the works of scholars in the past, one needs to look at them with the eye of a person of the same era, since things that are well-known facts today might not have been known at all in the past.

There has never been a conflict between science and Islam. Muslims understand everything in the universe as a letter from God Almighty inviting us to study it to have knowledge of Him. In fact, the first verse of the Qur’an to be revealed was:

Read! In the Name of your Lord, Who created… (Alaq 96:1).

Besides commanding us to read the Qur’an, by mentioning the creation the verse also draws our attention to the universe. There are many verses which ask Muslims to think, to know, to learn and so on. Moreover, there are various sayings of the Prophet Muhammad, peace be upon him, encouraging Muslims to seek knowledge. One hadith says, “A believer never stops seeking knowledge until they enter Paradise” (al-Tirmidhi).

In another hadith, the Prophet said, “Seeking knowledge is a duty on every Muslim” (Bukhari). Hence it is no surprise to see early Muslim scholars who were dealing with different sciences.

Translations

Prophet Muhammed (pbuh) said, “Knowledge is the lost property of a Muslim; whoever finds it must take it” [1]; hence Muslims started seeking knowledge. One way they did this was to start translating all kinds of knowledge that they thought to be useful. There were two main sources from which Muslim scholars made translations in order to develop the field of science, the Hindus and the Greeks. The Abbasid caliph al-Mamun (804–832) had a university built and ordered its scholars to translate into Arabic many works of Greek scholarship. Between 771 and 773 CE the Hindu numerals were introduced into the Muslim world as a result of the translation of Sithanta from Sanskrit into Arabic by Abu Abdullah Muhammad Ibrahim al-Fazari. Another great mathematician, Thabit ibn Qurra, not only translated works written by Euclid, Archimedes, Apollonius, Ptolemy and Eutocius, but he also founded a school of translation and supervised many other translations of books from Greek into Arabic. While Hajjaj bin Yusuf translated Euclid’s Elements into Arabic, al-Jayyani wrote an important commentary on it which appears in the Fihrist (Index), a work compiled by the bookseller Ibn an-Nadim in 988. A simplified version of Ptolemy’s Almagest appears in Abul-Wafa’s book of Tahir al-Majisty and Kitab al-Kamil. Abu’l Wafa Al-Buzjani commented on and simplified the works of Euclid, Ptolemy and Diophantus. The sons of Musa bin Shakir also organized translations of Greek works.

These translations played an important role in the development of mathematics in the Muslim world. Moreover, the ancient Greek texts have survived thanks to these translations.

Algebra and geometry

The word “algebra” comes from “Al-Jabr”, which is taken from the title of the book Hisab Al-Jabr wal Muqabala by Muhammad ibn Musa al-Khwarizmi (780–850). Al-Khwarizmi, after whom the “algorithm” is named, was one of the great mathematicians of all times. Europe was first introduced to algebra as a result of the translation of Khwarizmi’s book into Latin by Robert Chester in 1143. The book has three parts. The first part deals with six different types of equations:

(ax² = bx) ; (ax² = b) ; (ax = b) ; (ax² + bx = c) ; (ax² + c = bx) ; (bx + c = ax²)

Khwarizmi gives both arithmetic and geometric methods to solve these six types of problems [2]. He also introduces algebraic multiplication and division. The second part of Hisab Al-Jabr deals with mensuration. Here he describes the rules of computing areas and volumes. Since Prophet Muhammad, peace be upon him, said, “Learn the laws of inheritance and teach them to people, for that is half of knowledge,”[3] the last and the largest part of this section concerns legacies, which requires a good understanding of the Islamic laws of inheritance. Khwarizmi develops Hindu numerals and introduces the concept of zero, or “sifr” in Arabic, to Europe. The word “zero” actually comes from Latin “zephirum,” which is derived from the Arabic word “sifr.”

The three sons of Musa bin Shakir (about 800–860) were perhaps the first Muslim mathematicians to study Greek works. They wrote a great book on geometry, Kitab Marifat Masakhat Al-Ashkal (The Book of the Measurement of Plane and Spherical Figures), which was later translated into Latin by Gerard of Cremona. In the book, although they used similar methods to those of Archimedes, they move a step further than the Greeks to consider volumes and areas as numbers, and hence they developed a new approach to mathematics. For example, they described the constant number pi as “the magnitude which, when multiplied by the diameter of a circle, yields the circumference.”[4]

A well-known poet, philosopher and astronomer Omar Khayyam (1048–1122) was at the same time a great mathematician. His most famous book on algebra is Treatise on the Demonstration of Problems of Algebra. In his book besides giving both arithmetic and geometric solutions to second degree equations he also describes geometric solutions to third degree equations by the method of intersecting conic sections. He also discovered binomial expansion [26]. His work later helped develop both algebra and geometry.

Thabit bin Qurra (836–901) was an important mathematician who made many discoveries in his time. As mentioned in the Dictionary of Scientific Biography [5] he “played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.”

To give an idea of his importance, we will just give here, without details, one of his theorems on amicable numbers. Two natural numbers m and n are called “amicable” if each is equal to the sum of the proper divisors of the other:

for n > 1, let pn=3.2^2n–1 and qn=9.2^2n–1–1. If p_n–1 , p_n and q_n are prime numbers, then a=2n p_n–1 p_n and b=2ⁿq_n are amicable. [6]

Abu Kamil (about 850–930), an Egyptian mathematician, wrote the Book on Algebra which consists of three parts:

(1) Solutions of quadratic equations,

(2) Application of algebra to geometry,

(3) Diophantine equations.[7],[8]

He improved the work of Khwarizmi and applied algebraic methods to geometry. His research was on quadratic equations, multiplication and division of algebraic quantities. His work also includes addition and subtraction of radicals. He found the following formulas:

ax.bx=abx²; a(bx)=(ab)x; (10–x)(10–x)=100+x²–20x

Abu Kamil also wrote the Book On Surveying and Geometry, which was intended for government land surveyors. There, he stated the nontrivial rules for calculating areas, volumes, perimeters, and diagonals of different objects in geometry.[9]

Ibrahim ibn Sinan (908–946), a grandson of Thabit bin Qurra, was both an astronomer and a mathematician. Fuat Sezgin writes, “He was one of the most important mathematicians in the medieval Islamic world.” [10] He studied geometry, and his work on calculation of the area under the graph of a parabola is especially appreciated. Going further than Archimedes, he introduced a more general method of integration. [11]

Abu Bakr ibn Muhammad ibn al-Husayn al-Karaji (953–1029), also known as al-Karkhi, is regarded as the first person to have developed algebraic operations without using geometry. One of his major works was Al-Fakhri fi’l-jabr wa’l-muqabala (Glorious on algebra). Historian Woepcke recognizes Al-Fakhri as the beginning of the theory of algebraic calculus. [12] Here, al-Karkhi introduced the monomials x, x², x³, … and 1/x, 1/x², 1/x³, … and explained product rules among them. Moreover, he was the first to find the solutions of the equations ax²ⁿ+bxⁿ=c. [13] Al-Karkhi proved the sum formula for integral cubes by using the method of proof by induction, and hence became the first to use this method. [14]

Abu’l Hasan ibn Ali al-Qalasadi (1412–1486) was an Andalusian Muslim mathematician. His main contribution was to introduce algebraic symbolism, and he used short Arabic words for his symbols. For example, he used the symbol for the sound “sh” from the Arabic word meaning “thing” to represent what we call x, the unknown. [15]

Trigonometry

Khwarizmi also contributed to trigonometry. He established accurate trigonometric tables for sine and cosine, and he was the first to introduce tangent tables. [16] In 1126, these works were translated into Latin by Adelard of Bath.

Al-Battani or Albetagnius (about 850–929) was a Muslim astronomer and mathematician. In his research on astronomy he used trigonometric methods which were a lot more advanced than the geometric methods used by Ptolemy. [17] He introduced trigonometric ratios. For example, for a right triangle with adjacent sides a and b, he gives the formula b sin(A) = a sin(90⁰ – A), which is equivalent to tan A = a/b. He was the first to introduce the cotangent function. [18]

Muhammad Abu’l Wafa (940–998), born at Buzjan in Khorasan, introduced the use of secant, cosecant and tangent functions. He gave a new method of constructing sine tables. He calculated sin(30^0) with an accuracy of up to eight decimal digits. He improved spherical trigonometry and proved the law of sines for general spherical triangles. [19] In particular, he developed the half/double angle formulas:

2 sin² (x/2)=1–cos x; sin 2x=2sin x cos x

He was the first to introduce the notion of secant and cosecant, and hence completed the list of all six trigonometric functions. [20]

Abu Abd Allah Muhammad ibn Muadh Al-Jayyani (989–1079) was an Arab mathematician from Andalus. He was the author of The Book of Unknown Arcs of a Sphere which was “the first treatise on spherical trigonometry.” [21] Here he mentioned formulas for right handed triangles and law of sines. He also stated the formula for the solution of a spherical triangle in terms of the polar triangle. [21] He had a strong influence on the West.

Another outstanding mathematician Nasir al-Din al-Tusi (1201–1274) wrote Treatise On The Quadrilateral, considered the best book on trigonometry written in medieval times, [25] later translated into French by Alexandre Carathéodory Pasha in 1891. In his book al-Tusi made enormous advances in plane and spherical trigonometry. The Dictionary of Scientific Biography [22] states, “This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.” The well-known sine law is also stated in this work: a/sin A = b/sin B = c/sin C.

Ghiyath al-Din al-Kashi (1393–1449) produces sine tables of up to eight decimal places. In 1424, he computed 2π to an accuracy of sixteen decimal digits. He wrote a very impressive book on mathematics: Miftah al-Hussab (Key to Arithmetic). His main purpose in this book is to provide sufficient knowledge of mathematics for those who are working on astronomy,surveying, architecture, accounting and trading. [23] He also describes how to find the fifth root of any number. [24]

Unfortunately, the contributions of Muslims often go unrecognized. Muslim scholars contributed to science in many aspects such as mathematics, astronomy, geography, philosophy, medicine, art, architecture and so on. However, today few realize that in that era Islam played an important role in all aspects of life. Europe faced losing the works of major scholars, but as a result of their translations into Arabic most of this scholarship not only survived, but was further developed. Inspired by the Qur’an and hadiths, Muslims sought knowledge for the benefit of humankind. As the Qur’an says, “Are those who know equal to those who know not?”(Zumar 39:9). We should appreciate the scholars of all eras for their contributions to science.

Shirali Kadyrov is a PhD candidate at the Ohio State University, Mathematics Department.

References

1. Tirmidhi, `Ilm, 19.

2. B.L. van der Waerden, A History of Algebra.

3. Ibn Maja, Hadith No: 2719.

4. D. El-Dabbah, The geometrical treatise of the ninth-century Baghdad mathematicians Banu Musa (Russian), in History Methodology Natur. Sci., No. V, Math. Izdat. (Moscow, 1966), 131–139.

5. Y. Dold-Samplonius, A. T. Grigorian, B. A. Rosenfeld, Biography in Dictionary of Scientific Biography (New York 1970–1990).

6. For more, see S. Brentjes and J. P. Hogendijk, Notes on Thabit ibn Qurra and his rule for amicable numbers, Historia Math. 16 (4) (1989), 373–378.

7. R. Lorch, Abu Kamil on the pentagon and decagon, Vestigia mathematica (1993), 215–252.

8. J. Sesiano, La version latine medievale de ‘l’Algebre d’Abu Kamil, in Vestigia mathematica (Amsterdam, 1993), 315–452.

9.J. Sesiano, Le Kitab al-Misaha d’Abu Kamil, Centaurus 38 (1996), 1–21.

10. F. Sezgin, History of Arabic literature (German) Vol. 5 (Leiden, 1974), 292–295.

11. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Ibrahim.html

12. F. Woepcke, Extrait du Fakhri, traite d’Algebre par Abou Bekr Mohammed Ben Alhacan Alkarkhi (Paris, 1853).

13. Boyer, Carl B. (1991). “The Arabic Hegemony”, A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 239. ISBN 0471543977.

14. Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255–259. Addison-Wesley. ISBN 0321016181.

15. J. Samso, Las ciencias de los antiguos en al-Andalus (Madrid, 1992).

16. http://en.wikipedia.org/wiki/History_of_trigonometry

17. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Battani.html

18. http://www.unhas.ac.id/~rhiza/saintis/battani.html

19. http://www.britannica.com/EBchecked/topic/2127/Abul-Wafa

20. http://www.bhatkallys.com/article/article.asp?aid=3442

21. O’Connor, John J. & Robertson, Edmund F., Abu Abd Allah Muhammad ibn Muadh Al-Jayyani.

22. S. H. Nasr, Biography in Dictionary of Scientific Biography (New York 1970–1990).

23. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Kashi.html

24. http://www.bhatkallys.com/article/article.asp?aid=3442

25. http://members.tripod.com/worldupdates/newupdates10/id142.htm

26. Heinrich Dorrie, David Antin (1965). 100 Great Problems of Elementary Mathematics: Their History and Solution, p.34–36. ISBN 0486613488.