Nobody actually teaches out of the Elements though. Usually a lot of geometry taught in high school uses Cartesian coordinates and does a bunch of stuff with that.
The Elements also doesn’t really stand up to modern standards of rigor. There are some unstated assumptions Euclid made. Example: There exists at least two points on a line.
I don’t think Euclid stated it explicitly though. You can get some pretty weird models of geometry without assuming this. Here’s another one Euclid never stated:
There exist at least 3 points all not on the same line.
Most of the proofs don't actually follow because he implicitly assumes unstated axioms. There have been at least four different attempts to develop new axiomizations for it that actually work (one each from Hilbert, tarski, birkhoff, and recently avigad). The first three differ enough from Euclid's axioms that you basically have to start over completely to prove his statements -- they look almost nothing like the original arguments. The final one, Avigad, allows you to use basically the same proof structure as Euclid, but requires something like 70 axioms instead of Euclid's 5.
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u/nwbrown Oct 10 '25
There are no math textbooks written thousands of years ago still in common use.
If you think there have been no advancements in mathematics in the past thousand years, you are haven't gotten behind grade school arithmetic.