The Pythagoreans andirrational numbers

Initially, the Pythagoreans believed that all things could be measured by the discrete natural numbers (1, 2, 3, …) and their ratios (ordinary fractions, or the rational numbers). This belief was shaken, however, by the discovery that the diagonal of a unit square (that is, a square whose sides have a length of 1) cannot be expressed as arational number. This discovery was brought about by their ownPythagorean theorem, which established that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides—in modern notation,c² =a² +b². In a unit square, the diagonal is the hypotenuse of a right triangle, with sidesa =b = 1; hence, its measure isSquare root of√2—anirrational number. Against their own intentions, the Pythagoreans had thereby shown that rational numbers did notsuffice for measuring even simple geometric objects. (SeeSidebar: Incommensurables.) Their reaction was to create anarithmetic ofline segments, as found in Book II ofEuclid’sElements (c. 300bce), that included a geometric interpretation of rational numbers. For the Greeks, line segments were more general than numbers, because they included continuous as well as discrete magnitudes.

Indeed,Square root of√2 can be related to the rational numbers only via aninfinite process. This was realized by Euclid, who studied the arithmetic of both rational numbers and line segments. His famousEuclidean algorithm, when applied to a pair of natural numbers, leads in a finite number of steps to their greatest common divisor. However, when applied to a pair of line segments with an irrationalratio, such asSquare root of√2 and 1, it fails to terminate. Euclid even used this nontermination property as acriterion for irrationality. Thus, irrationality challenged the Greek concept of number by forcing them to deal with infinite processes.

Zeno’s paradoxes and the concept of motion

Just asSquare root of√2 was a challenge to the Greeks’ concept of number,Zeno’s paradoxes were a challenge to their concept ofmotion. In hisPhysics (c. 350bce),Aristotle quoted Zeno as saying:

There is no motion because that which is moved must arrive at the middle [of the course] before it arrives at the end.

Zeno’s arguments are known only through Aristotle, who quoted them mainly to refute them. Presumably, Zeno meant that, to get anywhere, one must first go half way and before that one-fourth of the way and before that one-eighth of the way and so on. Because this process of halving distances would go on intoinfinity (aconcept that the Greeks would not accept as possible), Zeno claimed to “prove” that reality consists of changeless being. Still, despite their loathing of infinity, the Greeks found that the concept was indispensable in the mathematics of continuous magnitudes. So they reasoned about infinity as finitely as possible, in a logical framework called thetheory of proportions and using themethod of exhaustion.

The theory of proportions was created byEudoxus about 350bce and preserved in Book V of Euclid’sElements. It established an exact relationship between rational magnitudes and arbitrary magnitudes by defining two magnitudes to be equal if the rational magnitudes less than them were the same. In other words, two magnitudes were different only if there was a rational magnitude strictly between them. This definition served mathematicians for twomillennia and paved the way for thearithmetization of analysis in the 19th century, in which arbitrary numbers were rigorously defined in terms of the rational numbers. The theory of proportions was the first rigorous treatment of the concept of limits, an idea that is at the core of modern analysis. In modern terms, Eudoxus’ theory defined arbitrary magnitudes as limits of rational magnitudes, and basic theorems about the sum, difference, and product of magnitudes were equivalent to theorems about the sum, difference, and product of limits.

The method of exhaustion

Themethod of exhaustion, also due to Eudoxus, was a generalization of the theory of proportions. Eudoxus’s idea was to measurearbitrary objects by defining them as combinations of multiple polygons or polyhedra. In this way, he could compute volumes and areas of many objects with the help of a few shapes, such as triangles and triangular prisms, of known dimensions. For example, by using stacks of prisms, Eudoxus was able to prove that the volume of a pyramid is one-third of the area of its baseB multiplied by its heighth, or in modern notationBh/3. Loosely speaking, the volume of the pyramid is “exhausted” by stacks of prisms as the thickness of the prisms becomes progressively smaller. More precisely, what Eudoxus proved is that any volume less thanBh/3 may beexceeded by a stack of prisms inside the pyramid, and any volume greater thanBh/3 may be undercut by a stack of prisms containing the pyramid. Hence, the volume of the pyramid itself can be onlyBh/3—all other possibilities have been “exhausted.” Similarly, Eudoxus proved that the area of a circular disk is proportional to the square of its radius (seeSidebar: Pi Recipes) and that the volume of a cone (obtained by exhausting it by pyramids) is alsoBh/3, whereB is again the area of the base andh is the height of the cone.

The greatest exponent of the method of exhaustion wasArchimedes (287–212/211bce). Among his discoveries using exhaustion were the area of a parabolic segment, the volume of a paraboloid, thetangent to a spiral, and aproof that the volume of a sphere is two-thirds the volume of the circumscribing cylinder. His calculation of the area of the parabolic segment involved the application ofinfinite series to geometry. In this case, the infinitegeometric series1 +1/4 +1/16 +1/64 +⋯ =4/3 is obtained by successively adding a triangle with unit area, then triangles that total1/4 unit area, then triangles of1/16, and so forth, until the area is exhausted. Archimedes avoided actual contact with infinity, however, by showing that the series obtained by stopping after a finite number of terms could be made to exceed any number less than4/3. In modern terms,4/3 is the limit of the partial sums. For information on how he made his discoveries,seeSidebar: Archimedes’ Lost Method.