In modern textbooks, students are greeted with plenty of exercises. Usually they are also organized in such a way that you have examples and pointers to what concepts are most important. The elements on the other hand (and for that matter a lot of older books of this sort) greet us instead with a series of propositions followed by proofs. Is there any idea on how these were studied?

  • $\begingroup$ Welcome on the site! I am not very sure, how to understand your text. I suggest to give more details by extending your post with some sentences. We have also a sister site matheducators.stackexchange.com . $\endgroup$
    – peterh
    Apr 28, 2019 at 3:08
  • $\begingroup$ I apologize for being unclear. I am asking how students studied mathematics before modern times. What motivated this question was how Euclid's Elements doesn't contain any exercises left to the reader (aside from a few one off things where he says "you can confirm the other cases"). $\endgroup$
    – Darkwisp
    Apr 28, 2019 at 10:38
  • $\begingroup$ Can you explain what you mean by "the elements," and further provide evicend for your claim that some book was "the prime textbook" ? $\endgroup$ Apr 29, 2019 at 13:07
  • $\begingroup$ Sorry. This is what I mean by the elements: en.wikipedia.org/wiki/Euclid%27s_Elements. I don't have any evidence of my own other than from light reading I have done on the subject. $\endgroup$
    – Darkwisp
    Apr 30, 2019 at 9:24

1 Answer 1


Students learned theorems, propositions and their proofs. A teacher would call a student to the blackboard and ask to reproduce a proof. A shorter test would be just to state the theorem. Many students memorized theorems and proofs by heart. A good (but not popular) teacher would make his own picture on the blackboard, with his own notation, different from the book, and ask a student to reproduce the proof.

Exercises are given by the teacher, for homework or in class. Later, separate books of exercises were used. (Including exercises in the main text of the textbook is a British-American custom. In other countries they had separate books of exercises).

I was taught by this system in secondary school in Soviet Union in 1960s. They did not use Euclid, but a book very close to it (Kiselev), with slightly modernized language. And a separate exercise book (Rybkin). Euclid was used everywhere until 19th century.

  • $\begingroup$ I don't think this addresses the question, but then again I don't understand what the OP intends to ask. $\endgroup$ Apr 29, 2019 at 13:08
  • $\begingroup$ @Carl Witthoft: don't you see that two parts of your sentence contradict each other? $\endgroup$ Apr 29, 2019 at 20:58
  • $\begingroup$ yep :-) . Ternary logic FTW $\endgroup$ Apr 30, 2019 at 12:31

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