Chapter 1, Part 2
Why was quadrature a Holy Grail pursued by mathematicians through the ages? Quadrature is the classical construction of a square equal in measure to another plane figure. Dunham opines that the Greek approach to all knowledge was to build complex ideas, in a systematic and logical way, from simple ones. It became important to them to either discover or impose a simple symmetry or regularity on the workings of the world [1]. Many natural objects can be modeled by near-perfect (or severely distorted) polygons, however, many more objects exhibit curved forms that are more precisely modeled by curves such as circles. The Greeks had accomplished the quadrature of polygons, but they were less successful in knowing the properties of circles and other curvilinear forms. They could relate various curves such as conics to one another, but until a squaring was achieved there would be no unification with the rectilinear objects, and thus no unification of the natural world.
Numbers had proven unreliable, even unfaithful, to the Pythagoreans and other mathematicians who pursued unification. It was discovered that some numbers were incommensurable, or in our terminology, irrational. To be commensurable, numbers either must be rational (e.g., the reason we can measure thirds with the unit fifths, is that there are units such as fifteenths), or at least related by a rational factor, as the square root of two is related to its double, triple, and so on. How could Greek universal principles rest on the unfamiliar, the irrational? One simple application of the Pythagorean theorem, that which calculates the diagonal of the unit square, had proved the existence of irrational numbers.
Perhaps geometry would be a stabler, more concrete foundation for mathematical knowledge. It was known that all types of polygons could be equated, in measure, to the square, and the square seemed an ideal unit of areal measure. The next two figures show, in turn, the quadratures of the triangle and the rectangle. The first task is a simple matter of building a rectangle of equal base and one-half the height of the triangle. To square the rectangle, we transform it through a right triangle which is created by an intersection on a semicircle. The semicircle and the triangle involve both the sum and the difference of the rectangle's dimensions [2].
Eves [3] thinks that the Greeks knew a trick for reducing a convex polygon of any number of sides to a triangle of equal area. In the figure below, we see a side CF extended to meet a line GK from an adjacent vertex, running parallel to the diagonal FH in the polygon. We cut a corner by connecting H with K, while maintaining the same area. So we see a complex figure equated to a simpler one.
What about the quadrature of the sums of areas? Suppose we were able to square two polygons. Is there a way to make a single square of equal area? Absolutely. If we construct a right triangle, letting each square's side measure be a leg, then the Pythagorean theorem allows us to conclude that the square on the hypotenuse meets the specification.
Can the circle be squared? The conclusive answer, in the form of rigorous proof, was unknown until 1880 when Lindemann proved that the number pi was not constructible. This proof logically dashed all hopes because we know that for a rectangle or a triangle to be equal in area to a circle, one or both of its dimensions would have to be proportional to pi [4].
Hippocrates came so close though, that he teased mathematicians, both professional and amateur, into pursuing the circle, in spite of logic, even to this day. What he did was pick three specific lunes, square them, and then show how the squaring of a fourth type would also succeed in squaring the circle.
In the preceding figure, we see on the left side the first lune that Hippocrates squared. The region is bounded by a large circle and a semicircle on one side of the inscribed square. The lune's area is equal to that of the right triangle and is thus squarable. The details of the argument are left for the reader to supply. Note that the lune spans a ninety-degree sector of the large square. Hippocrates then succeeded in squaring two other lunes, one spanning an obtuse sector, and the other an acute one. Does this seem to suggest that all lunes are squarable? That appears to be the conclusion that some wrongly assumed, and that others accused Hippocrates' of jumping to. The reader mustn't be tempted to, however, because Hippocrates picked three squarable lunes rather than three representative types, and it later was shown that there are only five lune types that are squarable by these means [5].
I claim that if all lunes are squarable then the impossible job of squaring the circle is possible. Consider the following figure, in which circle C has one-half the radius of D. If the lune on the hexagon is squarable then circle C can be squared [6].
Many have difficulty in crediting Hippocrates with a rigorous solution of even the first lune. The proof requires knowledge of the angle inscribed in a semicircle, and an application of the Pythagorean theorem, both of which would have been at Hippocrates' disposal. In addition, however, a third vital fact seems to have come along after Hippocrates had departed the scene. I refer to Euclid XII.2, which states that the areas of circles are to one another as are the squares of their diameters. The proof (appendix) of this depends on the exhaustion techniques of Eudoxus. Both Euclid and Archimedes assign the credit to Eudoxus [7.
The one possibility that Hippocrates knew and used the method of exhaustion, informally perhaps, lies in the fact that a mathematician from his own century, namely Antiphon, nearly succeeded in determining the area of a circle 250 years prior to Archimedes, while employing an exhaustion technique similar to that of Eudoxus [8]. We will examine circles and other area measures of conic sections in Chapter 3 of this report.
In summary then, we have seen that the Greeks placed a heavy reliance on geometry as a result of their mistrust of number. Geometry thrives as a discipline to this day and has its stalwart devotees. Unfortunately, circle-squarers are still busily engaged in an impossible task. Even the great Leonardo di Vinci was, for a time, fascinated with lune constructions. Coolidge [9] found sketches of hundreds of them, alone and in combinations with other figures, and he wondered what would foster such devotion to an "unimportant" topic. Of course, Leonardo may have had artistic uses for designs, but we should keep in mind also that the squaring of the circle was not a logically shut case in his time.
Still, geometry, when burdened with the rules of construction, seemed a cumbersome way to solve certain problems, and no one was expected to carry an exhaustion proof to its infinitesimal extreme. Geometry was dealt its most serious blow in the twentieth century: thanks to Kurt Godel, we know that any logical system must be incomplete, that is, have theorems which are unprovable, no matter the choice of axioms.
Other cultures had devised their own ways of dealing with arithmetic, irrationals, and practical problems. What was lacking was the ability to generalize individual examples. The disparate contributions from Egypt, Babylon, Greece, and the Far Eastern cultures needed to find a focus, a center around which they could be synthesized and generalized. For that center, we now turn eastward, even as the might of empire shifted out of Rome and into the kingdom of Islam: a subject of our next installment.
Uncle Bob's Puzzle Corner | Uncle Bob and Aunt Claire's Place | Math Menu | Math History Menu