Dedekind in 1872 wrote that equations like $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ ``to the best of my knowledge have never been established before'' (Continuity and Irrational Numbers, p. 22). This was not a crazy thing to say but it makes me wonder what were the standard textbook treatments at the time. A book like Rouse Ball History of the Study of Mathematics at Cambridge would be ideal for this question, except that it's account of instruction ends with the 18th century and it covers the 19th century only in connection with the Tripos.
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$\begingroup$ I'd think that there was no notion of "algebra" in the mid 19th century in Europe. That is, although Vandermonde and Lagrange (pre-1800) knew how to solve many equations in radicals (when they were so-solvable), it seems to have been by seat-of-the-pants algebra, not formal/axiomatic/formal. It is my impression that Euler effectively operated this way, as well. $\endgroup$– paul garrettJan 17, 2018 at 1:04
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$\begingroup$ @paulgarrett In the 1820s Bewick Bridge published several editons of a textbook titled A treatise on the elements of algebra. George Peacock, George has a famous 1830 Treatise on Algebra. It is true that so far as can think of right off hand continental authors were more likely to put this kind of algebra in books with titles like Analysis. But it is there. Or did you only mean they did not have 20th century style formally axiomatized algebra? $\endgroup$– Colin McLartyJan 17, 2018 at 9:03
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$\begingroup$ It is my impression that "rules of algebra" were considered descriptive of palpable realities. $\endgroup$– paul garrettJan 17, 2018 at 13:39
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Here's an example of early 19th century textbooks (from my answer here):
Cauchy 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 à 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.)
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$\begingroup$ A good list. These books all take the algebra of roots entirely for granted. $\endgroup$ Aug 18, 2016 at 17:58
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$\begingroup$ @Colin What do you mean by "the algebra of roots"? And why do they take it "entirely for granted"? $\endgroup$– GeremiaAug 3, 2017 at 16:26
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$\begingroup$ These texts all note things like $\sqrt[n]{a}\times\sqrt[n]{b}=\sqrt[n]{ab}$, and $\sqrt[n]{\sqrt[m]{a}}=\sqrt[nm]{a}$. But they rarely comment in any way on the idea of proving such equations. The equations are simply taken for granted. As to why they do that, I have some ideas but do not really know. They seem to think it is just a consequence of the notation. $\endgroup$ Aug 3, 2017 at 16:55
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$\begingroup$ @ColinMcLarty Do they mention it being an axiom? $\endgroup$– GeremiaAug 3, 2017 at 18:29
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$\begingroup$ No, that language was seriously not current at that time. Axioms were for geometry. And no one that I have found makes any effort to pick out a group of equations to assume, while deriving others from them. Some say that such calculations are "convenient." Basically, I think Dedekind was right, but I'm not certain he was. $\endgroup$ Aug 3, 2017 at 18:55