Why Beauty is Truth by Ian Stewart Read Free Book Online Page A

(“squaring the circle”).
    â€¢Â  Constructing a cube whose volume is exactly twice that of a given cube (“duplicating the cube”).
    It is sometimes said that the Greeks themselves saw these omissions as flaws in Euclid’s monumental work and devoted a great deal of effort torepairing them. Historians of mathematics have found very little evidence to back up these claims. In fact, the Greeks could solve all of the above problems, but they had to use methods that were not available within the Euclidean framework. All of Euclid’s constructions were done with an unmarked straightedge and compass. Greek geometers could trisect angles using special curves called conic sections; they could square the circle using another special curve called a quadratrix. On the other hand, they do not seem to have realized that if you can trisect angles, you can construct a regular 7-gon. (I do mean 7-gon. There is an easy construction for a 9-gon, but there is also a very clever one for a 7-gon.) In fact, they apparently did not follow up the consequences of trisection at all. Their hearts seem not to have been in it.
    Later mathematicians viewed Euclid’s omissions in a rather different light. Instead of seeking new tools to solve these problems, they began to wonder what could be achieved with the limited tools Euclid used: straightedge and compass. (And no cheating with marks on the ruler: the Greeks knew that “neusis constructions” with sliding rulers and alignment of marks could trisect the angle effectively and accurately. One such method was devised by Archimedes.) Finding out what could or could not be done, and proving it, took a long time. By the late 1800s we finally knew that none of the above problems can be solved using straightedge and compass alone.

    How Archimedes trisected an angle.
    This was a remarkable development. Instead of proving that a particular method solved a particular problem, mathematicians were learning to prove the opposite, in a very strong form: no method of this-and-that kind is capable of solving such-and-such problem. Mathematicians beganto learn the inherent limitations of their subject. With the fascinating twist that even as they were stating these limitations, they could prove that they genuinely were limitations.

    In the hope of avoiding misconceptions, I want to point out some important aspects of the trisection question.
    What is required is an exact construction. This is a very strict condition within the idealized Greek formulation of geometry, where lines are infinitely thin and points have zero size. It requires cutting the angle into three exactly equal parts. Not just the same to ten decimal places, or a hundred or a billion—the construction must be infinitely precise. In the same spirit, however, we are allowed to place the compass point with infinite precision on any point that is given to us or is later constructed; we can set the radius of the compass, with infinite precision, to equal the distance between any two such points; and we can draw a straight line that passes exactly through any two such points.
    This is not what happens in messy reality. So is Euclid’s geometry useless in the real world? No. For instance, if you do what Euclid prescribes in Proposition 9, with a real compass on real paper, you get a pretty good bisector. In the days before computer graphics, this is how draftsmen bisected angles in technical drawings. Idealization is not a flaw: it is the main reason mathematics works at all. Within the idealized model, it is possible to reason logically, because we know exactly what properties our objects have. The messy real world isn’t like that.
    But idealizations also have limitations that sometimes make the model inappropriate. Infinitely thin lines do not, for example, work well as painted lane markers on roads. The model has to be tailored to an appropriate context. Euclid’s model was tailored to help us