I am curious as to what mathematicians such as Leibnitz and Gauss and the Bernoulli's studied when they were students in university. I find it fascinating how we are taught calculus and abstract algebra in a few semesters, usually as late teenagers or early 20s, yet in the 17th and 18th century these were brand new concepts that certainly took a long time to develop by mature mathematicians. By the time we get to college, we already know elementary algebra and many major ideas that were developed before calculus. So, before abstract algebra and analysis and topology and PDEs..., what were our great mathematical ancestors studying in school?
What did mathematicians study as an undergraduate/graduate before modern mathematics such as modern algebra and analysis?
2$\begingroup$ I bet Gauss just rediscovered all the important theorems of number theory by himself during his years at university. $\endgroup$– KrijnOct 20, 2014 at 23:23
3$\begingroup$ Surely they studied Euclidean geometry. $\endgroup$– Tony PiccoloOct 20, 2014 at 23:37
$\begingroup$ @Krijn: I bet he did that when he was in pre-school. :-P $\endgroup$– Asaf Karagila ♦Oct 21, 2014 at 0:47
6$\begingroup$ "By the time we get to college, we already know elementary algebra and pretty much everything that was developed before calculus." - This is an entirely naive view of mathematics. $\endgroup$– Tac-TicsOct 21, 2014 at 1:07
9$\begingroup$ @Johann: That is not the major problem with the claim. For example, a great deal of combinatorics, matrix theory, and solid geometry was known before Newton. Almost none of that makes it to elementary education (at least in the American system) $\endgroup$– Eric StuckyOct 21, 2014 at 1:56
To give one example: Cauchy. I choose to discuss him because he straddled the threshold of "modern analysis."
He entered the lycée ("high school") École Centrale du Panthéon in 1802, studying humanities. Then, in 1805 he entered the École Polytechnique at age 15.
The "core" mathematics curriculum at the École Polytechnique was:
Analysis instruction: the Cours d'Analyse Algébrique by Garnier and the Traité Élémentaire de Calcul Différentiel et Intégral by Lacroix;
Mechanics instruction: the Traité de Mécanique, using the methods of Prony and edited by Francoeur and the Plan Raisonne du Cours de Prony;
Descriptive geometry instruction: the Géométrie Descriptive by Monge;
Applied analysis instruction: the Feuilles d'Analyse Appliquée a la Géométrie by Monge and the Application de l’Algèbre a la Géométrie by Monge and Hachette.
(Belhoste, Bruno. Augustin-Louis Cauchy: A Biography. New York: Springer-Verlag, 1991. pp. 7-10. Appendix II is Cauchy's outlines of his analysis courses he taught at the École Polytechnique from 1816-1819.
See the distribution of courses at the École Polytechnique when Cauchy was a student there.)
Also, according to the Oxford English Dictionary, "analysis" originally meant
the proving of a proposition by resolution into simpler propositions already proved or admitted (obs.); (later) algebra (now hist.). Now: the branch of mathematics concerned with the rigorous treatment of functions and the use of limits, continuity, and the operations of calculus.