What Historical Math Textbooks Offer Unique Insights?

[sponsoredwalk]

Just last week I relieved myself of a mathematical burden, freeing up some time for myself.
Coincidentally I came across the following passage by Arnol'd around the same time:

To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since
addition is commutative". He did not know what the sum was equal to and could not
even understand what he was asked about! Another French pupil (quite rational, in my
opinion) defined mathematics as follows: "There is a square, but that still has to be
proved". Judging by my teaching experience in France, the university students' idea
of mathematics (even of those taught mathematics at the École Normale Supérieure -
I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as
that of this pupil. Mentally challenged zealots of "abstract mathematics" threw all the
geometry (through which connection with physics and reality most often takes place
in mathematics) out of teaching. Calculus textbooks by Goursat, Hermite, Picard were
recently dumped by the student library of the Universitiés Paris 6 and 7 (Jussieu) as
obsolete and, therefore, harmful (they were only rescued by my intervention).
link

So, having a bit of free time & the ability to understand old textbooks on mathematics,
both of which I'd previously not been fortunate enough to be in possession of, I got
Goursat's books on calculus & differential equations on archive.org. Wow! Phenomenal
stuff so far! This pushed me to look for a historical algebra text to go over some of the
more elementary stuff. I found a book by G. Chrystal on Algebra that gives a proof of
the partial fraction expansion & derives the lagrange interpolation formula from scratch!
I haven't been able to find much on either of these topics tbh, especially not in the way
that is done in that book. They have been very non-intuitive explanations that I have
gotten from more modern books.

So, with this in mind, could people recommend other similar books from the late 19th,
early 20th century era that contain similar gold? I'm speaking particularly of the geometric
aspect that Arnol'd is describing, further elaborated by this passage:

...these students have never seen a paraboloid and a question on the form of the
surface given by the equation xy = z^2 puts the mathematicians studying at ENS into a
stupor. Drawing a curve given by parametric equations (like x = t^3 - 3t, y = t^4 - 2t^2)
on a plane is a totally impossible problem for students (and, probably, even for most
French professors of mathematics).

Beginning with l'Hospital's first textbook on calculus ("calculus for understanding of
curved lines") and roughly until Goursat's textbook, the ability to solve such problems
was considered to be (along with the knowledge of the times table) a necessary part of
the craft of every mathematician.

http://www.geniebusters.org/Riemann_intro.html

Now I don't think Hardy's book is a good example of this, I have it & I don't like it that
much. I'm just hoping people would be aware of a lot of other books that one could check
out, contrast & compare etc... to find something akin to what Arnol'd is talking about,
especially other books that people know about & have a good reputation that are not
the Elements, or the Principia, or Archimedes or something I'm not interested
in newer books as I'm aware of what they contain, we're talking about the seedy
undercurrent of old mathematics textbooks

 

[fourier jr]

sounds like theory of equations fits into that category. Most courses & books on Galois theory do the usual stuff with field extensions & automorphism groups (etc) but no mention of actually solving cubics or quartics (except for Stewart), or it's at least not emphasised. In other words no real mention of how to find out what to adjoin to a field. (btw weird how as theory of equations advanced it got taught in more advanced courses, the opposite of what usually happens) I'm not sure what the best textbooks are but I've got ones by Dickson & Uspensky, and a newer (Moore-method style) one, Polynomials by Barbeau which I like.

zygmund's trig series is another good one. It's got lots of stuff in it that you probably wouldn't find in books on locally compact abelian groups, for example.

 

[Petek]

Take a look at the http://store.doverpublications.com/by-subject-science-and-mathematics-dover-phoenix-editions.html . They include subjects other than math and are more expensive than Dover's standard line of books, but they otherwise meet your criteria.

 

[fourier jr]

burnside's book on finite groups is another one that I thouht of which is also in that series. I don't have a copy but I've read that it all about permutation groups, which I would think is (by the Cayley-Hamlton thm) more or less good enough.

 

[mathwonk]

i recommend reading euclid's elements and eulers elements of aLgebra and eulers analysis of the infinite[ies].

 

[mathwonk]

i recommend reading euclid's elements and eulers elements of aLgebra and eulers analysis of the infinite[ies].

as well as archimedes and Newton. i.e. i suggest you reconsider them. or maybe gauss.