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In man's eternal quest to understand his purpose in the universe, much effort and time was spent looking at complex and mysterious things in mathematics, geometry and music to try and find connections to and reflections of the divine. Consequently patterns and symbols were embedded in art as a constant reminder of these ideas. The Mesopotamians were the first to devise a system of proportion. The oldest pentagram originates in ancient Mesopotamia (Iraq) around 3000 B.C. It was referred to as the heavenly body or the "Star". The development of geometry was also important to the Ancient Egyptians to survey their irrigation system and for division of land, since every year it was flooded and the land markings were lost. In their religion their god Osiris was reflected in the waters and his consort Isis epitomised earth. Therefore the prospect of finding a divine strategy to establish the sacredness of this process became a necessity.
The desire to manifest the sacredness of man and his role in the universe, to show that within him there is the cosmos and that he reflects it, drove him to build temples as sanctuaries for the souls of the dead. The human body of the ruler was to rest in the temple for the sake of its soul. The architectural theme of the Centrality of Creation was primordial for the Mesopotamian. Man was formed at the "navel of the earth" where the bond of Heaven and Earth is located. The calculation is determined on a human figure in a circle of which the navel is the centre. Hence the translation into Greek "Omphalos", meaning navel.The designs of such buildings reflect not only the know-how of a nation but also its religious conviction. The equation of the body with a temple is thus an ancient Near Eastern and Ancient Egyptian concept.
Despite the absence of a proper numerical system, the Ancient Near Eastern and Ancient Egyptians were able to divide lengths into simple fractions using a rope-folding method. Although they were confined to drawing their ground plans with ropes and pegs, they had nevertheless managed to trace squares and rectangles correctly by using a grid of intersecting circles; the circles being the one form that can easily be traced with a rope and two pegs. The accurate dimensions of their temples and monuments provide evidence for this. They had also discovered that there existed certain right-angled triangles whose sides were in the proportion of whole numbers, in particular the 3:4:5 triangle. This appears to have been considered as sacred and was known by the Ancient Egyptians as "The Triangle of Osiris". It symbolized the Tuat (Underworld).
Irrational numbers, particularly 21/2 and 51/2 can be identified in some of their temple designs. It seems that, in the absence of a numeric system, these were obtained by constructing dynamic rectangles. The association with cosmological ideas and themes of creation and fertility gives ample evidence that they were copying the proportions found in nature. This has been evidenced by applying modern computer-aided design techniques to the structures of the Ancient Egyptian temple of Sesotris I at Tod (XIIth Dynasty c.1950 B.C.) and the Tomb of Rameses IV (XXnd Dynasty c.1140 B.C.). Their geometric system appears to be based on square grids, which shows that their monuments were built using a reproducible geometric method based on squares, circles, polygons etc. Furthermore they employed fractions as a means of computing differences. (Issam Es Said/ Ayse Parman 1976). Finally, in many of their monuments, there is clear evidence of proportions governed by the Golden Ratio F.
In its simplest form the Golden Ratio is obtained by a geometric construct based on a square, the base of which (AP) is bisected. The mid-point thus obtained is joined to the opposing corner of the square. Using the line thus produced as radius and the mid-point of AP as centre, an arc is drawn that cuts the extension of the base at B. The Golden Ratio thus obtained is expressed by both AP/PB and AB/AP. Given that AP is unity, the ratio AB/AP can be expressed as ½(1 + 51/2), which gives a value of 1.618.
Ancient Egyptian and Babylonian mathematics were transmitted to the Greeks, (the Golden Rectangle was used for the Parthenon, 400 B.C.). Later Pythagoras, who wrote on the Golden mean, referring to it as "analogia", received his information from the Egyptian priests. Polycleitus, (c. 450/470 B.C.) was a well-known Greek architect and sculptor. His usage of the mathematical proportions of the human body is reflected in his statues: Doryphorus, (Naples Museum) and Diadumenus (Athens Museum) and the Amazon (New York Museum). The Roman architect and engineer Marcus Vitruvius Pollio (c.70-25 B.C.), wrote his Roman Canon "De Architectura" based on earlier Greek treatises including those of Polycleitus. The idea of a circle with the navel of a human figure as its centre is firmly an ancient esoteric and religious concept. Vitruvius relied upon the ancient sexagesimal system of calculation from which is derived our degree of sixty minutes and our minute of sixty seconds within a three hundred and sixty degree circle. This system of mensuration was bequeathed from Ancient Mesopotamia and Ancient Egypt to the Greeks.
Following the collapse of the Roman Empire at the beginning of the 5th century man's concern was primarily focussed upon security and stability, whilst art and science were of necessity neglected. For two hundred years all progress stagnated in the wake of barbarian invasions and the resulting lack of maintenance of public works, such as dams, aqueducts and bridges. With the advent of Islam in the 7th century a new type of society emerged, which quickly established its supremacy and its constructive identity in large sections of the known world. The citizen, whether Muslim or not, soon became confident in the future stability of his environment, so that trade not only reached its previous levels but also began to expand.
In an empire that stretched from the Pyrenees to India, security of communications was vital. The resultant priority given to safety of travel provided a stimulus to trade. There followed a rapid expansion of commerce in which the economic strengths of the Sassanid, Byzantine, Syrian and western Mediterranean areas were united. The establishment of an efficient fiscal system meant that the state could now invest in large public works projects: mosques, madrasas, public baths, palaces, markets and hospitals. Princes and merchants became patrons of intellectual and scientific development. Waqf (trusts) were created to provide better education. This sponsorship engendered a creative enthusiasm and a flowering of scientific works and scholarly research. The world in effect became greater as mathematicians, geographers, astronomers and philosophers all contributed to a gradual but definite extension of the horizons of man's existence. The dividend of all this expenditure on learning made an immense contribution to the sum of the increase in man's scientific knowledge that occurred between the 9th and the 16th centuries.
Foremost in the achievements of Muslim scholars was the treatment of numbers. It is impossible to conceive how science could have advanced without a sensible logical numeric system to replace the clumsy numerals of the Roman Empire. Fortunately, by the 9th century the Muslim world was using the Arabic system of numerals, an adaptation of the Hindu system, but with the essential addition of the zero. Without the latter, it was impossible to know what power of ten accompanied each digit. Hence 2 3 might mean 23, 230 or 203. The introduction of this numeric system with its zero was thus the ‘sesame' of scientific advancement.
The new numeric system did not only affect science. Its value was manifest in many aspects of daily life, from the calculation of customs dues, taxes, zakat (almsgiving) and transport charges, to the complexity of divisions of inheritance. A further useful innovation was the mine of separation in fractions, which eliminated many frustrating confusions.
Islamic civilization produced from roughly 750 CE to 1450 CE a succession of scientists, astronomers, geographers and mathematicians from the inventor of Algebra to the discoverer of the solution of quadratic equations[1] . The list is far reaching, some are well known whilst others remain anonymous. One of the major advances was contained in the work of Al Khawarizmi, who wrote a mathematical work called Al Jabr wa Al-Muqabala (820 CE), from whose title is derived the name "algebra". Amongst the achievements that Al Khawarizmi left to posterity were:
- Solutions to first and second-degree equations with a single unknown, using both algebraic and geometric methods.
- A method of algebraic multiplication and division.
Al Khawarizmi defined three kinds of quantities:
- Simple numbers, such as 5, 17 and 131.
- The root which is the unknown quantity shay' in Arabic meaning "a thing". However, in translations made in Toledo, (the centre for translation of Arabic books), the absence of a "sh" sound in the Spanish language meant that a suitable letter had to be chosen. The choice fell upon "x", which may well explain why Don Quixote is often pronounced as "Don Quishote".
- "Wealth" (mal) the square of the root (x²).
The algebraic equation expressing the Golden Ratio could therefore be written as
x:y = (x + y)/x
Another virtuoso of algebra was Abu Kamil, a 10th century mathematician nicknamed the "Egyptian calculator". He was capable of rationalizing denominators in expressions that involved dealing with powers of x (the unknown) as high as the eighth and solving quadratic equations with irrational numbers as coefficients. Al Biruni (9th/10th centuries) mathematician and physicist, worked out that the earth rotates on its own axis and succeeded in calculating its circumference. Abu Bakr Al Karaji (10th century) is known for his arithmetization of algebra[2] .. He also drew the attention of the Muslim world to the intriguing properties of triangular arrays of numbers (Berggren 1983). Al Nasawi (10th century) and Kushyar Ibn Labban worked on problems of the multiplication of two decimals. Subsequently Kushyar explained the arithmetic of decimal addition, subtraction and multiplication and also how to calculate square roots. Abu Al Hassan al Uqlidisi (Damascus 10th century) invented decimal fractions, which proved useful for qadis (judges) in inheritance decisions. Al Karkhi (d.1019) found rational solutions to certain equations of a degree higher than two.
Mohamed Al Battani (Baghdad 10th century), mathematician and astronomer, computed sine, tangent and cotangent tables from 0° to 90° with great accuracy. One of his works: Al Zij (Astronomical Treatise and Tables), corrected Ptolemy's observations on the motion of the planets. Al Samaw'al Ben Yahya al Maghribi (1171) drew up charts of computations of long division of polynomials; one of the best contributions to the history of mathematics. Ibn Shatir Al Muwaqqit (Damascus 1375 CE) was an astronomer and the timekeeper of the Damascus mosque. His treatise on making astronomical devices and their usage and his book on celestial motions bear great resemblance to the works of Copernicus (1473-1543 CE). Ghiyat al Din al Kashi (1427 CE) raised computational mathematics to new heights with the extraction of fifth roots. He also showed how to express the ratio of the circumference of a circle to its radius as 6.2831853071795865, identical to the modern formula 2pr.
[1] J.L.Berggren 1986
[2] Roshdie Rashed
