Article: On the argument of simplicity in Elements and schoolbooks of Geometry

Evelyne Barbin: On the argument of simplicity in Elements and schoolbooks of Geometry, Educational studies in mathematics (2007) 66: 225-242.

We all want to present the mathematics in a simple way to our students, don't we? Sure we do, but what do we mean by "simple" exactly? In this article, the author very interestingly traces the concept of "simplicity" through several editions of Euclid's Elements as well as in some recent French textbooks. And it turns out that "simplicity" is not as simple as we thought...

Proclus, in his commentary to Euclid, proposed two "orders of simplicity": simplicity of a figure and simplicity of theorems. "A figure is more complex than another if it is obtained from the former with additions of lines or circles." A theorem is simpler than another if it comes earlier in the order of logical deduction.

Peletier preferred clearness of proofs rather than simplicity of figures.

Arnauld sought to follow the "natural order". "It is contrary to this natural order to prove, as Euclid does, propositions on perpendicular straight lines or on parallel straight lines (simple things) using triangles (compound things)." Therefore, the whole book was ordered according to the natural order. A consequence of this was that "perpendicular lines [were] studied without using angles and angles [were] studied without using triangles."

Hoüel wanted simplicity of "principles and proofs". He defined "the straight line by the motion of a point, and the plane by the motion of a straight line. Angles are defined by the ideas of motion and direction." (I find this last point particularly interesting, as it has been a problem in Norwegian textbooks that angles have been defined as static things only, without any dynamic dimension.)

Then, of course, in the 20th century, the notion of sets was regarded as "the simplest of all the notions." This had absurd consequences; "In the Mathématiques, classe de sixième, directed by Mauguin (1977), for eleven-year old pupils, the angle is defined as an equivalence class."

Finally, in a textbook from 1996, the angle is introduced with drawings of fans which are opened to different degrees. This quite good idea is however partly destroyed by a confusing use of colors.

I have just given a tiny idea of the contents of the article, of course. The author shows how these different ideas of simplicity is determining the structure of the books. Choose your notion of simplicity, and the structure will have to change. She ends with a reference to Descartes:

"As Descartes writes, arithmetic and geometry are more certain than the other sciences because their object is "so pure and so simple". [...] History of mathematics invites us to come back and to work with the notion of simplicity to construct the teaching of geometry."