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I believe their is a pre- and post Weierstrass era of mathematics (loosely speaking). Afterwards there was epsilon-delta, before 'infinitesimals' (with certain rules, ideas and theorems, of course not holding up to todays standards).

When I read the books of Landau and Lifshitz I get the impression they are firmly grounded in the infinitesimal approach. So I would like to understand this better. So I am looking for material which would be considered a prerequesite (not looking for a modern approach). So a question would be:

  • What math books would students lets say in Russia have studied before reading LL at that time?
  • What math books actually follow a similiar approach?

I am not looking for modern texts (I have a firm grounding in modern mathematics), and I also do not want to translate the expressions to modern math, I just would like to know the historical math text which would have/could have been used before studying LL.

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    $\begingroup$ I think you are overestimating the level of mathematical rigor that Landau and Lifshitz is using here. LL was written around a century after Weierstrass' work and I am sure the the writers were well aware they were being a little sloppy. It is simply that the added rigor of the epsilon-delta approach is not really needed in the simple cases that turn up in most practical application, whilst the formal machinery of analysis can be cumbersome and obscure what is really going on. $\endgroup$
    – By Symmetry
    Commented Apr 21, 2021 at 16:05
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    $\begingroup$ I would simply bare in mind that whenever LL talk about infinitesimals they are always ultimately looking to construct a Riemann sum or a derivative. Read their argument to the end and then try to construct a rigorous version yourself if you really want $\endgroup$
    – By Symmetry
    Commented Apr 21, 2021 at 16:06
  • $\begingroup$ @BySymmetry I am not saying they are rigerous, I just want to understand more where they are actually coming from. Thanks for the advise. Sometimes they use this to argue some things. physics.stackexchange.com/questions/48310/… I am actually looking for an introduction to these informal ideas. $\endgroup$
    – lalala
    Commented Apr 21, 2021 at 16:10
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    $\begingroup$ I think "Calculus Made Easy", by Silvanus Thompson, could be of some use, but I am not sure whether it was the text Landau learned Calculus from or anything like that (i.e., it covers calculus using infinitesimals in a sloppy way, so to speak, but I've got no clue whether it is a historical reference for your specific purposes) $\endgroup$
    – Níckolas Alves
    Commented Aug 10, 2021 at 3:01

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