Elements

Ah, a literalist. I see your literality, and raise: enabled a pivotal moment in my life Like that time you beat Archimedes to the punch and moved the world with a cantilever? To not just 'standup philosopher'-post: Gödel, Escher, Bach (already mentioned by another I see); Euclid's Elements; Polya's How To Solve It; The Rise and Fall of the Third Reich; Old Possum's Book of Practical Cats - clearly non-fiction, if you know cats. :)

From what I've heard, Euclid is fairly accessible and was for centuries the standard geometry textbook for children; the Principia is incredibly daunting, and Newton even admitted that he made it extra confusing on purpose to deter readers who weren't already experts.

That would be a hard one. A book of appetizers without a main course. I read it when I was an undergrad wanting to get a broad sense of different areas in Math I could pursue. If I had to pick one math book it would have to be a classic with some depth to it: Euclid’s Elements, Newton’s Principia, or Gauss’ Disquisitiones.

Spinoza's Ethics Euclid's Elements Aristotle's Nicomachean Ethics

Euclid's _Elements_.

What you say makes a lot of sense. However, notation actually does affect the ideas, and deeply. Euclid did not consider 1 a number, for perfectly sound reasons. Multiplication is a very different animal in Euclid's math, and his proportions are not just funny-looking fractions. Overall, you are right. If someone is looking to get a broad survey of mathematics, Euclid is not a good source. But to start from scratch and learn rigorous math-by-proof, he has no equal.

Invalidates? Absolutely none. Good mathematics is eternal and the Elements has some beautiful mathematics. What I meant to say is that in the fullness of time they have been expanded on and surpassed. You can still learn good and valid mathematics from Euclid, but you can learn it better form more modern sources. As one example amoung many, good modern geometry books will let the reader know about the existence of non-Euclidean geometry and its differences, even if the focus is on traditional Euclidean geometry. Also, modern books will use modern notation. Notation of course does not affect the ideas which are what is important, but someone trying to learn math today will need to learn the notation so that they can read other more advanced books that will assume the reader knows that stuff. Not to mention, notation can make things easier or harder (try doing multiplication in Roman Numerals). Finally, many things that were proven tediously before have easier proofs developped over time, often with the ability to take recourse in Algebra. For the same reason, many open questions in Euclid's time now have answers.

I heartily endorse reading the classic works of geometry as a way to both the subject as well as a way of thinking about math, proof, and argumentation. Start with Euclid's _Elements_, and then move onto Archimedes' short books on levers and floating bodies, Apollonius's wonderful treatise on conic sections, and Ptolemy's _Almagest_. They are excellent for self-education, providing both geometrical knowledge in itself, as well as extensive training in sound reasoning. Don't be fooled by the antiquity of their origin: they teach more clearly than most modern day texts, and their content is timeless.