“A straight line is a line which lies evenly with the points on itself,” says Definition 4 of Euclid’s Elements.
This definition is clearly meaningless drivel. How is it possible for such a masterful work, which is clearly written by a top-quality mathematician, to open with such junk?
Russo proposed a compelling answer to this conundrum. It goes as follows.
Euclid didn’t define “straight line” at all. His focus was on the overall deductive structure of geometry, and for this purpose the definition of “straight line” is essentially irrelevant, as indeed shown by the fact that the utterly useless Definition 4 is never actually used anywhere in the Elements.
Archimedes agreed with Euclid as far as the Elements were concerned, but in the course of his further researches he found himself needing the assumption that among all lines or curves with the same endpoints the straight line has the minimum length. He therefore stated this property of a straight line as a postulate in the context where this was needed.
The Hellenistic era, which included Euclid and Archimedes, was one of superb intellectual quality. Unfortunately it did not last forever.
Heron of Alexandria lived about 300 years later. This was a much dumber time. Fewer people were capable of appreciating the great accomplishments of the Hellenistic era. But Heron was one of the best of his generation. He could glimpse some of the greatness of the past and tried his best to revive it.
To this end Heron tried to make Euclid’s Elements more accessible to a less sophisticated audience who didn’t have the background knowledge and understanding that Euclid’s original readers would have had. He therefore wrote commentaries on Euclid, trying to explain the meaning of the text.
To these new, dumber readers of geometry, it was necessary to explain for example what a straight line is. Heron realised that Archimedes’s postulate captured well the essence of a straight line. However, it is not itself suitable as a definition, because a line should have the property of being the shortest distance between any two of its points, not be phrased in terms of only two fixed points, as in Archimedes’s postulate.
Heron therefore explained that “a straight line is [a line] which, uniformly in respect to [all] its points, lies upright and stretched to the utmost towards the ends, such that, given two points, it is the shortest of the lines having them as ends.” The phrase “uniformly …” obviously refers to the universality of the shortest-distance property applying to any two points on the line.
Now fast-forward another 300 years. Intellectual quality has now plunged deeper still. Geometry is in the hands of rank fools. Euclid’s Elements, which was once written for connoisseurs of mathematical subtlety, is now used by schoolboys who rarely get past Book I and “learn” that only by mindless rote and memorisation. A time-tested way (still in widespread use today) to “teach” advanced material to students who do not have the capacity to actually understand anything is to have them blindly memorise a bunch of definitions of terms.
In this context, therefore, there is a need for an edition of the Elements which includes many definitions of basic terms, which must be short and memorisable, and which don’t need to make mathematical sense.
In this era of third-rate minds, some compiler set out to put together an edition of the Elements that would satisfy these conditions. Heron’s commentary on the Elements is appealing in this context since it affords opportunities to focus on trivial verbiage instead of hard mathematics. But Heron’s description of a straight line is still too complicated. It’s too long to memorise as a “soundbite” and the mathematical point it makes is moderately sophisticated.
The compiler therefore makes the decision to simply cut Heron’s description off after the bit about “uniformly in respect to [all] its points.” This solves all his problems in one fell swoop. The only drawback is that the “definition” becomes utter and complete nonsense, but since the whole purpose of it is nothing but blind memorisation anyway this doesn’t matter anymore.
This is how the ridiculous Definition 4 — a mutilated vestige of what was once a very good definition — ended up in “Euclid’s” Elements.
I have included excerpts from Russo’s paper in my History of Mathematics Reader. There you can find the above argument in his own words.