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Four Areal Views

Introduction

The story of mathematics, and the achievements and biographies of its practitioners, is intriguing in every way. This series of articles will trace one concept, that of the area of planar figures, through four eras in history. We will see those particular problems and applications of area measurement that faced mathematicians in each era. We will examine the strengths and weaknesses of various approaches. We will see evidence of startling creativity in the solutions, and what is more, we will see the "look" and substance of mathematics change forever.

In Part One we visit ancient Greece and witness their best minds struggle with making geometry a logical system. We will see their attempts to create a basis of comparison for all planar areas in a topic known as quadrature. Each achievement gave rise to many new questions. An important one to keep in mind as you read is: could Euclidian geometry ever reach the degree of comprehensiveness and efficiency needed to solve all the quantitative problems of our universe?

In Part Two we will see the achievements of ancient cultures, both Eastern and Western, congregate and synthesize in the Arabian Empire after the fall of Rome. A system to be known as algebra will unify much of Greek geometry, Hindu number theory, and application problems from the earliest civilizations. Area will be a key link in the theory of quadratics.

In Part Three we will take up the subject of area as it has been applied to the conic sections. After a brief look at the methods of Archimedes, we will focus on the seventeenth century and see that the areas in and around ellipses and hyperbolas were keys in aiding astronomers' understanding of the workings of our solar system. Finally, in part Four we will have a modern view, as area measure transforms from a concrete application into an abstract concept, and as the objects and methods of measurement become strange indeed.

In this web-based version of these reports, we have streamlined the presentation by offering links to interactive figures and the appendices. You have the opportunity to click on these and then return to the report. We welcome your comments and questions and include an email link for this purpose.

Hippocrates Squares the Lune -- or Does he?

The Classical Era

If you should happen to read in a book of mathematical history that Hippocrates of Chios, contemporary of Plato in 5th century B.C. Athens, succeeded in squaring the lune, don't believe it! We will show that, if he had squared the lune, that is, constructed a square equal in area to a region enclosed by two circular arcs, then the circle itself would also be squarable, and pi would then be constructible and not a transcendental number. Wouldn't that have caused a few wrinkles in the mathematical tapestry? Don't worry though, we know today that the circle is not squarable, and details of this story follow. We must caution, though, that all of our knowledge of the ancient thinkers comes from editors and commentators working hundreds or even a thousand years later. They deleted, embellished, and corrected the documents they possessed with varying degrees of proficiency, and today, we have no original documents to assess for ourselves.

A Brief Time Line

The Greeks were a people who believed that rational thought was the key to moving their society from a "barbarous past to an undreamed-of future. [They subjected all of] Nature and the human condition to the penetrating light of reason." [1] Thales (640-560 B.C.) is credited with the first few demonstrative proofs in geometry.

Pythagoras (b. 572) followed and established a secretive society which carried the reverence of mathematics to quasi-religious extremes. The society outlasted its founder and eventually was welcomed into the mainstream of the Greek civilization. Due to its secrecy, most of the society's advances in geometry and the theory of numbers were credited to Pythagoras. An exception to that rule would be the many versions of the tale of Hippasus, who discovered the incommensurability of the side and diagonal of a square. Some numbers would thus be irrational, and this fact caused problems for many of the existing theorems which depended on an inadequate theory of proportions. [2] Hippasus then, reaped the credit for an unhappy discovery.

In the fifth century B.C., the great thinker Socrates, and followers Plato and Aristotle, established Athens as a center of learning. Hippocrates of Chios, the lune-squarer, was a contemporary of Plato. He should not be confused with the Hippocrates of Cos, known as the Father of Medicine. This era seems to have been an unsettling one in mathematics for several reasons. First, the philosophers subjected all knowledge to critical thought. There were logical gaps in Pythagorean geometry, mainly in those theorems employing the definition of proportions, which, unfortunately, were valid for rational numbers only. In addition, there was no organizational structure to support the many and varied theorems in geometry. Without structure, mathematicians had difficulty knowing which theorems depended on which of the others. Logical pitfalls, such as circular reasoning, were playing havoc. Hippocrates of Chios was the first known mathematician to begin to assemble the "elements" of geometry: those principles upon which most of the other theorems are based [3].

Eudoxus (408-355 B.C.) repaired the rent in the fabric of
proportional reasoning by creating a definition which does not depend
on commensurable quantities. He also contributed the "method of
exhaustion" later employed by Euclid (c. 300 B.C.) and Archimedes (c.
225) in developing the geometry of the circle. This project will
revisit Archimedes in Part Three, which looks at areas related to the
conics. Some time between Eudoxus and Euclid, Eudemus wrote a
*History of Geometry*. Nothing survives of the work, but it is
cited by later commentators as, one, referring to Hippocrates'
successes with lune quadratures, and also, alluding to the logical
objections of others regarding this work. Later (210 A.D.), Alexander
Aphrodisiensis objected to Hippocrates' claim that in squaring lunes,
he established the squaring of the circle. Most historians doubt that
Hippocrates himself ever made such a claim [4]. It seems
sensible though that the squaring of the circle was a powerful
motivator for Hippocrates' look into lunes [5].

Next we come to Euclid (fl. 300 B.C.) who did set out the basic propositions for a geometry of the plane and 3-space, and for number theory as well. Euclid's collection of theorems is not only admirably comprehensive, but concisely and logically sequenced. The content of Elements has endured as a core of geometric curriculum to the present day. A proposition which Hippocrates would have required for a rigorous proof of his lune quadrature appeared as Euclid's XII.2. Could Hippocrates have proved and utilized it a century-and-a-half before Euclid? Most historians think not. Proposition XII.2 itself depends on the method of exhaustion attributed to Eudoxus [6].

Commentators such as the aforementioned Alexander, along with Simplicius and Proclus, who all wrote in the Christian era, provide us with indirect and wholly undocumented evidence of the facts just set forth, and of those to follow [7].

Endnotes

- Dunham, William Journey Through Genius. Penguin, New York, 1991. p.6.
- Heath, Thomas A History of Greek Mathematics, Vol.1. Oxford, London, 1921. pp. 168&endash;169.
- Heath, T.L. (ed. and translator), Euclid, the Thirteen Books of the Elements. Dover, New York, 1956 (2nd ed.). pp. 116&endash;117.
- Heath, History... pp. 183-200.
- Eves, Howard An Introduction to the History of Mathematics.(3rd ed.) Holt, New York, 1969. p. 88.
- Heath, ed., Elements. Vol. 3. pp. 365-378.
- Heath, History... ibid. and pp. 119-121.