In October 1828, Évariste Galois joined Louis-Paul Émile Richard's (1795–1849) advanced mathematics class at the lycée Louis-Le-Grand with one goal alone: admission to the École polytechnique. Twelve papers from this period have been identified and published by Robert Bourgne and Jean-Pierre Azra. What followed—the tragedies and setbacks—is well known: his father's suicide on July 2, 1829, and his failed attempts to pass the École polytechnique entrance examination.
On August 12, 1829, the inspector general responsible for administering the University of Paris appealed to his minister to allow Galois to sit the École préparatoire entrance examination. The young man took the tests from August 20 to 27, 1829. All the candidates' papers from the École préparatoire entrance examination are available at the Archives nationales. We therefore have Galois's papers in mathematics, philosophy, physics, Latin and French composition.
The mathematics paper ----------------------
Galois's mathematics paper was examined by Caroline Ehrhardt in a study published in 2008. The historian of science also provides information about how the entrance examination was organized, including the length of the tests, the identities of those who set the questions and the way the papers were marked; she also marked the candidates' mathematics papers herself. The test lasted six hours, and the young men had to answer two questions. The second asked them to *"[s]et out, using the general equation of a conic, the method used to find the asymptotes directly, without having to perform a change of coordinates. Then apply this method to the curve: y2 + 3x2 - 4xy + x + 2y – 1 = 0, also identifying its key lines and points, such as the center, diameters, axes, etc."*
Will you admit defeat—or leave the answer to Galois or one of his classmates? Galois ranked first in the mathematics test. He and most of the other candidates gave the standard answer to this question, which appeared in most school textbooks of the time. About half the candidates proved that a curve has an asymptote if it is possible to write y = cx + d + V, where V tends to 0 as x tends to infinity. They then explained how to obtain c and d, before applying these results to the particular case of conic sections. In doing so, they followed Louis Étienne Lefébure de Fourcy's (1787–1869) Leçons de géométrie almost word for word.