Euclid and Incommensurables

by Dr. Andrew Seeley, President and Co-Founder of the Boethius Institute

Last week I was reminded of one of the many moments of wonder that the study of Euclid’s Elements provokes. Jeff Lehman and I have been teaching a semester course in Geometry and Arithmetic to a group of very bright Master’s students at the Pascal Institute in the Netherlands. We were discussing the definitions and opening propositions of Book X of the Elements. This book concludes his study of plane ge

ometry, which begins with six books of numberless study of equalities, inequalities, constructions, proportions and similarities to be found among plane figures, followed by three books devoted to the study of what we call “natural numbers” (1, 2, 3, etc.), although Euclid just called them numbers.

In Book X, he turns to one of the foundational problems of ancient mathematics, namely that not all lines and figures can be known through numbers. He begins by defining commensurable and incommensurable magnitudes: Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure. Two lines, for example, are incommensurable which cannot be divided into equal smaller parts in such a way that the same part measures both. He then makes the claim in definition 3 that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively. In our discussion, one of the students expressed the natural response to such a claim: “It is not possible that two lines can be incommensurable! You can divide them into parts as small as you want. You must be able to find a common measure!”

I remember feeling that astonishment myself when I first studied Euclid. Walking through propositions 5-9 can convince you that it must be true, but it doesn’t relieve the astonishment. This is one of the great moments that show why the liberal study of mathematics is such an important part of a liberal education. The problem of incommensurability is not a practical problem; you can always find an adequate common measure in any real world instance. It is simply theoretical, and yet it is deeply disturbing precisely because our mind confronts mystery even in the world of its own thought.

Perhaps for the first time, I wondered why we do not encounter this mystery in the course of ordinary high school mathematics. We are taught about irrational numbers, admit them into the number line, and learn to work with them in equations. Yet we never realize that they are incommensurable with ordinary numbers. The common definitions given do not draw attention to this fact: An irrational number is a real number that cannot be expressed as a ratio of integers. Again, the decimal expansion of an irrational number is neither terminating nor recurring. These definitions introduce them as a problem of calculation. They made me dislike irrational numbers because they were messy and could only be worked with approximately. It took Euclid to give me a glimpse at and an experience of one of the deep mysteries of the world.