Demonstrate knowledge of Euclidean geometry, including the history and basics of Euclid’s Elements and its influence on math as a discipline.

SLO 7 Summary

The study of many mathematical subjects can be illuminated by looking at their histories; this is especially true of geometry. Euclidean geometry is named after Euclid, the Greek mathematician who lived in Alexandria around 300 BCE. Euclid systematized the knowledge of geometry and included it in thirteen books called The Elements. In The Elements, Euclid set out a sequence of Definitions, Postulates (axioms for geometry), Common Notions (axioms common to all mathematical subjects), and Propositions (theorems derived logically from the preceding materials). For most of the 2400 years since it was written, it was considered to be an essential text and the gold standard of mathematical rigor. Students need to know this history to place modern ideas about proof into context and to understand mathematics as a human endeavor.

SLO 7 Narrative

Euclidean geometry is named after Euclid, the Greek mathematician who lived in Alexandria around 300 BCE. Euclid synthesized what was known at the time about Euclidean geometry into the thirteen books of The Elements. In The Elements, Euclid sets out a sequence of definitions, postulates (axioms for geometry), common notions (axioms common to all mathematical subjects), and propositions (theorems derived logically from the preceding materials). It is the “oldest extant large-scale deductive treatment of mathematics” (“Euclid’s Elements,” 2022). For most of the 2400 years since it was written, it was considered to be an essential text that any educated person would have studied; it is only in the last 150 years that this was no longer true. Likewise, for most of that time, it was considered to be the gold standard of mathematical rigor; again, it is only in the last 150 years that its rigor has been surpassed.

Many mathematical disciplines can be illuminated by considering their histories, and this is especially true of geometry. Students need to know this history to place modern ideas about proof into context. A geometry class is likely where students will first encounter the notion of axiomatic proof, and for some of them it may be their only encounter with it. [See SLO 4 on Axiomatic Systems.] Euclid’s first three postulates also set the stage for constructions using straightedge and compass [see SLO 8 on Constructions].

One particularly interesting piece of this history concerns Euclid’s fifth Postulate: “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles” (Euclid, 1956). Over the years, this postulate was considered inferior to the others, and many mathematicians tried to prove it from the others. It was only in the early nineteenth century that it was discovered that there is a geometry in which the other postulates are true, but this one is false. This is how hyperbolic geometry was discovered [see SLO 9 on non-Euclidean geometry].

Most students will not have seen Euclid’s fifth postulate in its original form but would likely know something more like Playfair’s postulate that given a line and a point not on the line that there is a unique line through the point parallel to the given line. Proving the logical equivalence to Euclid’s fifth postulate is a good homework problem.

One question that remains relevant to modern geometry teachers is to what extent a geometry course should cover Euclid’s actual writings and methods, or to what extent they should be replaced by simplified treatments (as is done in many high school geometry books) or more technical rigorous methods (such as those based on Hilbert’s Foundations of Geometry). As discussed above, for most of the time since it was written, almost all educated people studied The Elements. The debate about what treatments of geometry might be better than Euclid for modern students started in the mid-nineteenth century. There was enough debate that in 1879, Charles Dodgson, the mathematician better known as Lewis Carroll, published a book entitled Euclid and his Modern Rivals, which argued that Euclid’s treatment of geometry was superior for teaching students than any other treatment then proposed. This debate continues to this day.

One reason to consider teaching Euclid’s original proofs, besides their historical interest, is that they are generally at the right level of sophistication for students when more modern treatments that are considered more rigorous might also be too complicated for students. One theory of cognitive development espoused by Jean Piaget, among others, states that “ontogeny recapitulates phylogeny,” that is, individual development often follows a similar path to the historical development of a subject (Gould, 1977). Given the outsized role that Euclid’s work has played in the historical development of geometry, it is not surprising that it might be at just the right level of sophistication for many students.

Looking at Euclid also makes it natural to consider where Euclid’s treatment may have gaps. For example, Euclid’s first postulate says that it is possible “to draw a straight line from any point to any point,” but if you look at the way he uses it, he really assumes that the line drawn is unique. Looking at his proofs through the lens of a geometry where any two points can be connected but not necessarily uniquely, such as on the sphere, makes this issue immediately clear.

Euclid also introduces the idea of superposition to prove the Side-Angle-Side congruence condition for triangles. This argument was criticized because of the assumptions his proof makes about transformations without stating them explicitly. This can lead to a discussion of the transformations approach to Euclidean geometry. 

References

Dodgson, C. (1885). Euclid and His Modern Rivals. MacMillan and Co., 2nd edn. 

Euclid’s Elements (2022, January 27). In Wikipedia. https://en.wikipedia.org/wiki/Euclid%27s_Elements.

Euclid (1956). The Elements. Dover, 2nd edn. Translated with introduction and commentary by Thomas L. Heath.

Gould, S. J. (1977). Ontogeny and Phylogeny. Belknap Press of Harvard University Press.

Hilbert, D. (1971). Foundations of Geometry. Open Court Publishing Co., 4th edn. Translated by Leo Unger.