So who here has heard of Sonya Kovalevsky? Not me....
[Sonya] continued to study by candlelight, and when her father confiscated her candles, she memorized texts during the day and worked out problems in her head at night. The family of French mathematician Sophie Germain -- inventor of "Germain primes," i.e., double a Germain prime and add 1 to get another prime number -- used a similar tactic to dissuade their equally precocious daughter from studying geometry, algebra and calculus... to no avail. Sonya also studied under the covers at night, borrowing an algebra textbook from one of her tutors.
Then a neighbor, who taught science, gave the family a copy of a basic physics book he'd written. Sonya turned to the section on optics, and discovered trigonometry. Even though she'd never encountered it before, she managed to make sense of the derivations for small angles by substituting "a chord for the mysterious sine." In short, she independently rediscovered the same method by which the whole concept of a sine had been developed historically. Impressed, the neighbor convinced Sonya's father to let her study analytic geometry and calculus privately in St. Petersburg. She mastered both subjects in a single winter. Her astonished tutor noted that it was almost as if she'd known the concepts in advance.
...
Weierstrass wasn't a familiar name to me, but at the time he was the most renowned German mathematician, a professor at the University of Berlin. Sonya came to him bearing glowing recommendations from her Heidelberg professors, yet even then, he was skeptical, and far from enthusiastic about taking her on. To discourage the young woman, he gave her a set of problems he'd prepared for his most advanced students, assuming she'd never make sense of them. Instead, she solved them in record time; not only that, her solutions were clear and original, demonstrating a grasp of the material lacking in most of his male students (Mittag-Leffler being one notable exception). So he agreed to teach her privately, and came to consider her among the most brilliant and promising of all his students.
Sonya didn't disappoint her mentor. By the age of 25, she had produced three original papers, each of which was deemed worthy of a PhD degree: one on the shape of Saturn's rings, another on elliptical integrals, and a third on partial differential equations. Not that Berlin would ever award a woman a PhD, especially one that had never been officially matriculated. Anywhere. (And how could she possibly matriculate when they wouldn't allow it? Yes. Exactly.) To his credit, Weierstrass fought for her, eventually convincing the University of Gottingen to award her a PhD in mathematics, summa cum laude.