I want to bring together in this paper two figures from the history of mathematics—one from the ancient world and one from the modern. The first is the Greek, Euclid, and the second, the 19th century Swiss, Jacob Steiner. I have no intention of giving a complete picture of either man’s work (although I do hope readers will end up with their curiosity piqued, especially about the lesser known Steiner). What I do want to consider is a small instance in which both, in some sense, looked at the same geometrical facts. The latter are the theorems concerning the product of the segments of intersecting chords of a circle.
I say “in some sense, looked at the same geometrical facts” because, coming from different worlds, Euclid and Steiner brought to mathematics different perspectives, and that makes speaking about their looking at “the same geometrical fact” at least problematic. Indeed, Euclid’s treatment of these theorems in the Elements and Steiner’s transformation of them in his ‘power of a point’ bring into relief ancient and modern points of view. This brings me to the second part of my title and my second, but really principal, goal in this paper.
For it often happens that in the attempt to combine mathematics education and history of mathematics, the main lesson of the history of mathematics is lost, namely, that mathematics itself is an historical entity (see Fried, 2001 ). When teachers bring problems and mathematical ideas from the past into the classroom, they tend to speak about Roberval’s solution to this or Apollonius’ approach to that, as if the problems and ideas are eternal and only the solutions and approaches change. But to say that mathematics is historical is to say not only that its problems and ideas change but also what mathematics is and what it means to be mathematical.
It is in view of this, I want to consider the often-heard definition of mathematics as the “science of patterns.” Specifically, I want to show, by comparing Euclid and Steiner, that while this is presented to students as a timeless—that is, non-historical—definition, in fact, it represents a modern view of mathematics. I shall show that Greek mathematics, for example, is not a search for patterns but for concrete properties of concrete mathematical objects; and I shall show, conversely, that it is when mathematics becomes symbolic that patterns, as such, are suggested to mathematicians and become objects of their thought.
The example of Euclid and Steiner for this purpose is actually a rather subtle one, and so, before we begin, I should say a word as to why I chose it. There are two main reasons. First, in thinking about how modern mathematics is a science of patterns high school teachers do well to think about mathematics at the level they teach; in this way, an example from elementary geometry is better than one from, say, group theory, which in other respects would be ideal. Second, a subtle example shows how pattern-thinking lurks even where one does not expect.