Arab Contributions to Trigonometry

Overview

Trigonometry is one of the most practical branches of mathematics, finding uses in engineering, physics, chemistry, surveying, and virtually every other science and applied science. It is also one of the oldest branches of applied mathematics; practical problems in crude trigonometry have been dated to Egypt in about 1850 b.c., and the ancient Greeks developed more sophisticated trigonometry about 2,000 years later. Since that time, trigonometry has played a crucial role in many branches of mathematics and science, and is indispensable to our understanding of science and technical disciplines today.

Background

The earliest mention of a problem relating to trigonometry is in an Egyptian papyrus dated to about 1850 b.c. Although the concepts used are not stated in conventional trigonometric terms, it is obvious from the context that a form of "proto-trigonometry" existed at this time and was used to help ensure the pyramids were constructed according to the architect's specifications. However, it is almost certain the Egyptians did not place their calculations in a mathematical context that would allow them to draw any other conclusions from their results—the math involved was only applied to construction projects.

The next milestone in the development of trigonometry as a true mathematical discipline was reached by the Babylonians when they divided the circle into 360 equal divisions, or degrees. They did this because a year in their calendar had 360 days, so each day represented a degree. Since the Babylonians used a base-60 number system (as opposed to our base-10 system), 360 degrees was a tidy "fit" into their existing mathematics. The Babylonians also invented the gnomon, a device for measuring the angular distance of stars or planets above the horizon, which was similar to the protractor.

It is interesting to note how deeply ingrained the Babylonian numbering system is today: our hours have 60 minutes of 60 seconds each, we continue to use circles with 360 degrees, and our maps use 60 minutes of arc to a degree and 60 arc seconds to an arc minute. Clocks, maps, and protractors throughout the world are based on this system, even though a decimal (base-10) system would be easier to use.

The Greeks were first to elevate trigonometry to the level of an independent branch of mathematics. Greek trigonometers such as Pythagorus, Euclid, and Aristarchus advanced trigonometric theory and also championed new practical uses. Perhaps the most ambitious of these uses were Erastosthenes's calculation of the circumference of Earth and Hipparchus's determination of the distance of the Moon from Earth. In both cases, the final results were surprisingly close to current accepted values, in spite of the crude instruments used at the time.

In India, the Hindus made further advances during and after the fifth century. These advances included the construction of some early trigonometric tables and, more important, the invention of a new numbering system that made calculating much simpler. Hindu mathematicians based their version of trigonometry on variants of the sine function. The Hindu system led not only to the sine function, but to the cosine, tangent, and other familiar trigonometric functions we use today.

During their centuries of contact with the Greeks and Hindus, Arabic mathematicians adopted many of their mathematical discoveries. Among prominent Arabic mathematicians who helped translate Hindu mathematical texts or introduced Hindu mathematics to the Arabs were al-Battani (c. 850-929), Abu al-Wafa (940-998), and al-Biruni (973-?). Al-Battani adapted Greek trigonometry and astronomical observations to make them more useful. Al-Biruni was among the first to use the sine function in astronomy and geography, and Abu al-Wafa helped apply spherical trigonometry to astronomy, among other important contributions.

Impact

Arab mathematicians and scientists of the Middle Ages did more than translate Greek texts into Arabic, they translated specific Greek texts to use as reference materials for their own research in these areas. The Arab world lay between two other intellectual powerhouses—India and Greece. Arab scientists were exposed to the rich mathematical tradition of their own culture and, to this, they added the best of both Greek and Hindu mathematics and science. They were then able to synthesize these elements into a new way of looking at mathematics, as well as putting their mathematics to work on practical problems.

Abu al-Wafa made several significant contributions to the mathematics of the day. He made the first recorded mention of negative numbers in a book he wrote in the latter half of the tenth century. Today we take negative numbers for granted, but a thousand years ago, negative numbers were not widely accepted because they did not make intuitive sense to the people of that time. For example, we can all visualize having an apple, but how do you visualize having a "negative" apple? What does it look like? How do you count it? People of Abu al-Wafa's day were not used to thinking in these terms, and many simply refused to. Abu al-Wafa described negative numbers in monetary terms, referring to them as a "debt." This description of negative numbers could be grasped intuitively and was instrumental in bringing negative numbers into mainstream mathematics.

Abu al-Wafa's construction of tables of sines was also important. Having tables of sines may seem mundane because today we have calculators that instantly calculate all the trigonometric functions. To use trigonometric functions in calculations 1,000 years ago, one had to know their values, and these came either from hand calculation or from tables that had been laboriously calculated by hand and distributed. When he decided to calculate the value of the sine function for all angles at 15' (¼-degree) increments, Abu al-Wafa committed himself to an arduous and mind-numbingly repetitive task that required not only a great deal of commitment but also an almost unimaginable attention to detail. However, his work made these tables available for future generations of mathematicians who used his tables or their derivatives for centuries.

Abu al-Wafa was also first to introduce the concept of the tangent, the secant, and the cosecant to Arab mathematics. These functions, all derivatives of the sine function, are extremely useful in many areas of study, including physics, engineering, architecture, and surveying. The tangent had been described by Hindu mathematicians, but Abu al-Wafa showed how all the concepts could be used in mathematical calculations. By introducing these functions, Abu al-Wafa helped to increase the value of trigonometry by creating concepts that expanded its range.

If Abu al-Wafa had only translated some obscure texts into Arabic and generated some interesting functions, he might have passed into history without further notice. However, Abu al-Wafa and other Arab scholars helped to blend mathematical concepts from two distinct mathematical traditions into a synthesis which was much more important than either of its parts. Arab mathematicians took the geometric trigonometry (trigonometric identities derived from geometric drawings) of the Greeks, and added the mathematical sophistication and superior numbering system of Hindu mathematics, to create a trigonometry that very much resembles that of today. By doing this, they helped to create one of the most useful branches of mathematics.

Since the time of Abu al-Wafa, trigonometry has become essential to virtually all the sciences and applied sciences. Consider these examples:

1) The motion of rotating objects is usually described in terms using trigonometric functions. Engineering designs that involve rotating pieces (including wheels, camshafts, gears, motors, and fans), depend on trigonometry.

2) The motion of objects moving cyclically, such as pendulums, bridges swaying in a strong wind, and buildings oscillating after an earthquake is described using trigonometric functions. In fact, any periodic or oscillatory action can be described using trigonometry, including electrical equipment that uses oscillating electric fields, electronics, and the orbits of spacecraft.

3) Trigonometric functions also underlie much of physics, including quantum physics. These functions help describe phenomena ranging from probability functions that describe electron orbitals around an atom to the rotation of galaxies.

The technology, sciences, and mathematics upon which industrialized societies depend are based on trigonometry. Abu al-Wafa and his fellow Arab mathematicians and scientists could not have imagined how their work would someday be applied, but their discoveries are an indispensable part of our modern society.

P. ANDREW KARAM