Albert Einstein's school-leaving examination in mathematics 1896 (Q5945674)

The majority of this article is a well-commented reproduction of Einstein's school-leaving examination in mathematics at the Aargau Kantonsschule. The author notes that, contrary to a popular legend, Einstein had been a good student. He had hated the militarism of his previous school, the Luitpold-Gymnasium in München and had procured a doctor's certificate of ill-health to enable him to leave and rejoin his family in Milan. (They moved there to open a new business in collaboration with a firm in Padua when the family engineering business failed after losing a competitive bidding for the electrification of the city center in München.) The author notes that the examination papers are still preserved, but are written in the Sütterlin script and as a result are very difficult for the unpracticed to read. NEWLINENEWLINENEWLINEThe first problem in the geometry examination, which took place from 7:00 to 11:00 a.m. on September 19, 1896, was to solve a triangle given the ratios of its altitudes and the radius of the circumscribed circle. In the second problem a family of parallel chords in a circle is given, and the student is to prove that the circles having these chords as diameters are all tangent to an ellipse whose semiaxes are the radius of the circle and the radius multiplied by the square root of 2, then to find the distance from the midpoint of the radius that bisects all these chords to the point at which the tangency ceases to hold. NEWLINENEWLINENEWLINEIn the algebra examination, which took place from 9:30 to 11:30 a.m. on September 21, the first problem was to find the radius of a circle inscribed in a triangle, given that the distances from the center of the circle to the three vertices are 1, 1/2, and 1/3. In this problem, which leads to a cubic equation, Einstein applied the Cardan formula and noted that the discriminant was negative. He commented that the roots would then be ``irrational''. The examiner underlined this word, and Einstein's biographer Albrecht Fölsing commented that the word should have been ``imaginary''. But of course, neither of these is the case. The \textit{formula} may involve complex numbers, but the \textit{roots} can even be integers, as for example is the case with the equation \(x^3 - 7x + 6=0\). It is of interest that the Cardan formula was still taught in Einstein's day, since it disappeared from the standard curriculum, at least in the United States, decades before hand-held calculators came along to provide an easy solution of cubic equations. Of course, the formula was always more useful theoretically than practically, and numerical methods of solution, such as Horner's method, were generally applied on the rare occasions when it was necessary to solve a cubic equation.