I was thinking about Abel's advice to read the masters, when I got an idea. Many people claim that reading the old masters (Euclid, Archimedes, Newton, Euler, Gauss, Lagrange, and so on) is not a good idea today, because modern mathematics has vastly simplified their ideas and moved on so much that it is useless to read old books for actually gaining knowledge in the relevent branches. However, this isn't necessarily true, because (or so I think) once a branch of mathematics is cast into its final form (including notations, etc.), it doesn't usually change very much.

There are many examples.

Euclid's Elements - Euclidean geometry is still the same.

Euler's Calculus Trilogy (His Introductio, Differential, and Integral calculus books) - calculus as taught in schools is still very much the same, including notations. In fact, Andre Weil even recommended that students would be better off studying precalculus using Euler's Introductio (which Euler intended as a precalculus book, i.e. intro to his two calculus volumes).

Gauss's Disquisitiones Arithmeticae - Still the best introduction to elementary number theory in my opinion. Recommended by Alan Baker (fields medal winner) in his "concise introduction to number theory" (and also recommended in his expanded version: "comprehensive course in number theory", showing that his views hadn't changed on this point) as being the best introduction to elementary number theory.

Lagrange's Mechanique Analytique - still the best exposition of Lagrangian mechanics.

The list can go on, the point being that old books which give an exposition of a topic in modern notation are still the best works to consult to learn that branch of mathematics. Why go to Burton when you can go straight to the master, Gauss, who was there in the battlefield himself, and knows the landscape of elementary number theory (I will not say modern number theory, since most of it was either not invented or in the process of being invented then) better than probably anyone else who ever lived. Why go to Stewart's Calculus, when you can go straight to the analysis incarnate himself, Euler?

To avoid people getting the wrong idea, I must emphasize that books where the subject in question was either in its infancy, or still being developed (e.g. Newton's Treatise on Fluxions - definitely not possible to use for learning "school calculus"), can only be consulted out of curiosity, and should not be recommended for learning the subject.

tl;dr: Once a branch of mathematics is cast into its final form (including notations, etc.), it doesn't change very much. With this assumption, it should be possible to learn at least the earlier branches of mathematics by going straight to the masters. Gauss for elementary number theory, Euler for calculus, Lagrange for lagrangian mechanics, Euclid for euclidean geometry, Newton for the elementary mechanics (or at least "elementary mechanics for the ambitious student who wants to learn elementary mechanics straight from the horse's mouth"). What do you think of this idea?

I look forward to hearing your thoughts!


closed as primarily opinion-based by Aloizio Macedo, Rory Daulton, user186473, hardmath, drhab Jan 2 '16 at 16:34

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 4
    $\begingroup$ This question seems to broad and opinion-based for this site. Could you narrow it down and make it more straightforward? $\endgroup$ – Rory Daulton Jan 2 '16 at 16:07

The notion that Euclidean geometry is "still the same" is a bit naive. A pass through Moise's book "Elementary Geometry from an Advanced Standpoint" will convince you of this. In particular, modern Euclidean geometry relies pretty solidly on the notion of measure -- the length of a line, the measure of an angle -- rather than on ideas of proportion as an abstract way of comparing things. This modern approach allows one to use algebraic methods to prove many things more easily.

In much the same way, learning projective geometry from Hartshorne's little book shows you the connections between various geometric things (Fano's Axiom, for instance, or Pappus' Theorem) and the underlying algebraic properties of an algebraic entity $F$ that's associated to a (rich enough) projective geometry (e.g., for geometries that satisfy Pappus' Theorem, the associated $F$ might have a commutative multiplication operation, although I may be misremembering this -- it might be that the operation is associative). You can read a lot of early but well-developed projective geometry texts that will not tell you about this because the formal notions of abstract algebra hadn't yet been developed.

I'm definitely keen on reading older texts, but there's a lot to be said for modern "revisionist" views as well. I'd prefer to read Milnor and Stasheff's Characteristic Classes to anything Whitney wrote, despite my huge admiration for Witney's work on sphere bundles (and almost anything else, too!).

  • 1
    $\begingroup$ I stand by my claim that I would prefer to read Milnor and Stasheff rather than Whitney. But I bow to your superior knowledge. I probably don't know which one led me to greater understanding. $\endgroup$ – John Hughes Jan 3 '16 at 21:57

Not the answer you're looking for? Browse other questions tagged or ask your own question.