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What did the great mathematician, like Cauchy, Lagrange, Euler and Gauss, learn in order to know what they knew?

It seems that they were extremely good in the most basic rules/structures/issues of math:
(infinite) series, limits, trigonometry, (geometry?) etc.

  1. What books are recommended to read in order to know what they knew?
  2. What are the math subjects that they were familiar with?
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    $\begingroup$ Laplace was said to have instructed his students: Lisez Euler, lisez Euler, c'est notre maître à tous, "Read Euler, Read Euler, he is our master in everything." $\endgroup$ – DanielV Sep 1 '15 at 11:06
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    $\begingroup$ There is a story about Euler when he was a young schoolboy. His classmates were being unruly, and to punish them, his teacher said none can be dismissed until they got the correct answer to the problem of adding up all numbers from 1 to 100. In seconds, Euler correctly said "5050". He did this not by actually adding up all the numbers conventionally, but by recognizing that the sum was 100+0 + 99+1 + 98+2 ... + 51+49 + 50. There were 50 pairs of numbers that summed to 100 and 50 left over, making 5050. $\endgroup$ – Michael Tiemann Sep 1 '15 at 11:17
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    $\begingroup$ @MichaelTiemann, Hello, that was Gauss... $\endgroup$ – Benicio Sep 1 '15 at 11:18
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    $\begingroup$ @GudsonChou, Hello, that was actually nobody... $\endgroup$ – Asaf Karagila Sep 1 '15 at 19:37
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    $\begingroup$ Dor, the simplest answer would be to read physics books and engineering books. Find a way to build a time machine, then build it, then go back and ask them in person. Who knows, maybe Gauss got a lot of his inspiration from that one person who went back in time to see what Gauss knew when he was 20-something, and that time traveler told Gauss about all this great discoveries, thus avoiding a paradox. $\endgroup$ – Asaf Karagila Sep 1 '15 at 19:41
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They knew most of the cutting-edge mathematics of their times, by studying with the people who invented it, or by voraciously reading all they could put their hands on. They maintained extensive contact with colleagues, be it by meeting in some Academy or other or just by mail. Not that far from how today's prominent mathematicians came to know what they know. Technology has made it all much easier, and there are more people involved, in ever more specialized branches.

We remember a few names from Euler's day (some almost only because they threw a intriguing conjecture around, which still hasn't been resolved), but there were a lot more mathematicians around then. Just like today, in any field there are a few superstars, a lot of known people, and many, many "also ran"s.

Note: What they did read is mostly moot, e.g. Euler studied under one of the Bernoullis; l'Hôpital had recently finished the first calculus book, and Euler himself wrote precalculus/calculus textbooks that became classics.

Besides, the techniques and notation (pre-Euler, at least) are completely alien (much of our modern notation was introduced by Euler). For some ideas on how they worked, check e.g. William Dunham's "Euler: The master of us all" (AMS, 1999), "Journey through Genius" (Penguin, 1990) or "The Mathematical Universe" (Wiley, 1997), particularly interesting is Leibnitz' proof of what we call integration by parts. Ed Sandifer's "How Euler did it" was a monthly column discussing some of Euler's work, somewhat updating notation (sometimes).

I've no such resources handy for Lagrange or Gauß, sorry.

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    $\begingroup$ I appreciate the answer (+1), though two of my questions weren't resolved. Could you please refer to them? $\endgroup$ – Dor Sep 1 '15 at 20:52

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