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Four Areal Views

Chapter 1 of this project reported that the Greeks attempted to find a basis with which to compare, at least in two dimensions, the various forms found in nature, through the use of geometric constructions. Chapter 2 will show how some of the findings of the Greeks were synthesized with those of cultures whose mathematics was more focused on numerical generalizations. Area was a key concept in this synthesis.

Chapter 2

al-Khwarizmi Completes the Square


This second installment examines the mathematical contributions of people from the Near East. We will see that the advances of many ancient cultures were collected and nurtured by the Moslems at the height of their territorial conquests. It was this stewardship that was in part responsible for the preservation of what we now know to be the works of ancient Greek and Hindu mathematicians.

Our focus for this chapter will be on one mathematician, al-Khwarizmi, and one topic, his synthesis of the algebra and the geometry which solve the roots of quadratic equations. First we will look at the historical background that led to this achievement. Next, two cases of quadratic solutions will be presented in detail. The conclusion will contain a brief discussion of both the immediate and long-term benefits to the advancement of knowledge bestowed through Islamic efforts.

In Chapter One, we left the Greeks, at about the time of Euclid, struggling with cumbersome constructions, negative and irrational numbers, and the new method called "exhaustion," which seemed infinitely complex. If we had stayed with their history, we would have seen new advances in geometry as well as algebra. Archimedes established the area of the circle by employing the method of exhaustion (and presaging the calculus that was to come 2000 years later). Appolonius unified the knowledge of the conic sections. More importantly I think, these two, together with Diophantus and others, began to move away from the scratches in the sand, toward a systematic way to solve practical problems with numbers. They made the first steps toward the numerization which is evident in their thinking, if not in their figures [1]. A complete system of algebraic theory, together with a system of notation and algorithms, would not be achieved until after the western Renaissance, but it was the Arab cultures who kept the momentum going until then [2].

A Brief Timeline

There exists evidence of mathematical creations that predate the Greeks, as illuminated in the cuneiform tablets of Asia Minor and the papyri of Egypt. The Rhind papyrus, for one example, dates back to approximately 1700 B.C. These and other artifacts show evidence of working number systems, arithmetic, problem solving, measurement, geometry, and astronomy. The sheer number of what we today would call "word problems" is so great that many historians give the Babylonians credit for initiating algebraic thought, merely by the repetitious use of a solving method [3]. These were practical problems of surveying, inheritance, and commerce. Other experts insist that, there being only emphasis on "how" and not on a generalized "why," we give these cultures too much credit by inferring the use of algebra [4]. The Babylonians were comfortable working with irrational numbers though, having an efficient method of extracting roots [5]. By 150 A.D. Ptolemy's Almagest had a fairly accurate system of predicting astronomical events, albeit based on an earth-centered solar system.

In 212 B.C. when a Roman soldier killed Archimedes during the siege of Syracuse, the event was to have multi-faceted symbolism. Greece eventually fell to Rome. To say that the Romans did not value abstract rational thought as much as the Greeks would be to make an understatement. This new empire, in filling the vacuum left by Alexander, made widespread conquests, and at the same time, neglected to mind the store. The libraries fell into disuse or worse. A fire in 391 A.D. destroyed the holdings of a principle library in Alexandria. Many achievements of the best minds of antiquity were lost forever [6]. Those that survived the Roman mismanagement were to be further threatened when Rome fell. That the works were scattered and fragmented is no surprise, and actually may be a reason some escaped annihilation.

The power center of the Christian empire shifted to Constantinople, and here were caretakers of some of the classics. In 532 A.D. Isadorus was ordered by Emperor Justinian to restore St. Sophia and establish a school. Here the works of Euclid and Aristotle were taught and valued [7]. The Byzantine Empire, then, was to be a prime source of mathematics for the Arabs who would come to give algebra a logical foundation. Before they began, however, there was much in the way of sabre swinging.


  1. Dijksterhuis, E.J., Archimedes. Munksgaard, Copenhagen, 1956. p.216
  2. Eves, Howard, An Introduction to the History of Mathematics.(3rd ed.) Holt, New York, 1969. pp.190-193
  3. Aaboe, Asger Episodes from the Early History of Mathematics Random House, New York, 1964. p25
  4. Boyer, Carl B. and Merzbach, Uta A History of Mathematics, 2nd ed. Wiley, New York, 1989. pp.36-48
  5. Wilson, Alistair, The Infinite in the finite. Oxford, 1995. p.65
  6. Dijksterhuis, p.34
  7. ibid., pp.35-36


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