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!