- On Ptolemy. It is incorrect to say that Ptolemy's description of the motion of the planets with epicycles was "wrong". In the modern times, when we want to compute the positions of Sun, Moon and planets, with respect to Earth (and to do astronomical observations from Earth you need exactly this!) we use long trigonometric series. Mathematically, trigonometric series is the same as epicycles. Some coefficients of these series are obtained by solving differential equations, others are empirical. In Ptolemy, all coefficients were empirical. But the FORM of the description is the same as in Ptolemy.

Ref.: Jean Meeus, Astronomical algorithms, Willmann-Bell, VA, 1998.

- On mathematical theories which became obsolete because better theories were invented.
An example is "vector calculus" which is still taught to all undergraduates for the reasons that escape me. It is superceded by much simpler formalism of differential forms.
(Poor students still have to memorize the complicated expression in Stokes' theorem in the
Stokes original formulation!)

Many proofs with elementary geometry methods are replaced nowadays with analytic geometry
and calculus. Remember, Newton wrote his Principia without explicit use of calculus. No one uses his obsolete arguments in this form anymore.

EDIT. To see that epicycle is the same as a trigonometric series, use complex numbers.
The uniform motion of a point about the center is $Ae^{i\omega_1t}$. In the case of one epicycle, we have a deferent and epicycle; the motion has the form is expressed in the form $Ae^{i\omega_1t}+Be^{i\omega_2t}$,
the center of the epicycle is $Ae^{i\omega_1t}$, it is moving with uniform speed on the deferent. The planet moves around this point with angular speed $\omega_2$.
By the way, in this notation the famous theorem of Apollonius "on the equivalence of excentric and epicycle description" says exactly that the addition of complex numbers is commutative:-)
In more complicated models, more than two terms is required. In the modern theory of the Moon, we have several hundred such summands. The summands other than the first one are traditionally called "inequalities". The first two are due to Ptolemy. The third and fourth to Brahe, and so on. You can see the whole series in the book by Meeus cited above.

EDIT2. Undergraduate textbooks using differential forms:

P.Bamberg, S.Sternberg, A course in mathematics for students of physics, 2 volumes, Cambridge University Press, Cambridge, 1991. (Used to be the standard text for science students in Harvard)

H. Cartan, Formes différentielles. Applications élémentaires au calcul des variations et à la théorie des courbes et des surfaces, Hermann, Paris 1967 (This is the second part. First part is called Calcul diffenentielle).

G. Grauert, I. Lieb, V. Fisher, Differential- und Integralrechnung, 3 volumes. Springer 1967-1968.

All these are general, complete calculus textbooks. I am not mentioning textbooks which contain differential forms only, like Spivak and Flanders.

Sometheorywithinmath, singular...) $\endgroup$ – Hauke Reddmann Mar 31 '15 at 11:18