I'm looking for books that are written in an old style but are also concise and rigorous. Some examples of what I mean:
All of these books are written very concisely (and as a plus, they're written in a very nice old-style font!). Any other recommendations people would have of math books that resemble these? Thanks for the help!
On
Euclid's Elements are written in an old style (if not very old style) but also concise and rigorous.
On
Two classic and influential ones on abstract algebra which are still missing in the other answers:
There are so many it's impossible to even know where to begin ! For one thing, one has to define what one considers "old". I'll say, somewhat arbitrarily, pre-1970, but really in my case with an emphasis on 1600 - early 20th century.
Since this is a long post I think I'll say at the outset some of my favorite older but not that much older mathematical writers are Courant (I think his 1934 Calculus book is better than any modern Calculus book I have seen), Coxeter, Burnside, and Casey. Going back further, Euler (Gauß as well but I quickly get lost, would love a tour guide), Serrè, and Möbius, and further still Euclid's «Elements», and Appolonius of Perga, «Conics».
I can try to note some of the titles that I think are better in terms of being concise and rigourous, but for myself, I would say that going to the limit of concise and rigourous is not always primary. An extreme example that comes to mind immediately is one of Abel's original articles, translated into German in 1889 by H. Maser, «Abhandlung über die algebraischen Gleichungen, in welcher die Unmöglichkeit der Auflösung der allgemeinen Gleichung fünften Grades bewiesen wird» (1824) on the impossibility of solving fifth degree equations which is literally 4 pages long. Historically, in this case, Abel is said to have been so poor that he could not afford the paper to write as much as he would have liked to. It qualifies as concise and rigourous, but speaking for myself, I have found it almost impossible to make headway because it is so much so.
I'm still glad I own it, which leads into a variety of other reasons one can be interested in older works. I would list them roughly, unordered, as historical perspective (not least, for me, seeing all the amazing calculation techniques that were used before computers which have now been relegated to the dust-bin), connection with the thoughts of brilliant minds, appreciation of old publishing (which contrary to a comment on the original question is sometimes beautifully typeset and illustrated on very high quality paper), study of other languages besides English, accessiblility (I often find myself understanding older authors better than modern ones, especially in what were at the time textbooks, because often nothing was assumed about the reader's preparation as well as having self-learners in view as target audience) discovering some hidden, lost, or forgotten gems (for example, who has read or even heard of Euler's essay on the theory of music? or books such as Praktisches Zahlenrechnen, which might be thought of as an idiot savant's playbook, or Gauß' review of Babbage's Logarithm Tables where he glows over the typesetting and ink, or for that matter his writings on calculating the date of Easter).
A very incomplete listing is below, sort of a semi-random walk, early printings when I've been able to buy them. Although I've spent a fair amount I usually note to myself that on the average I pay less for a vintage volume in good condition than a university student pays for a brand new required undergraduate or graduate coursebook (eg hundreds of dollars). I should note I cannot claim to have read a high proportion of what I own, although there are certainly a few I have read cover to cover. In some cases just reading slivers here and there has been enough, in others it becomes almost like stamp collecting - but a lot more interesting than stamps.
Anyway, here is a very partial unordered list of some older works I own. In some cases the date is unclear; in a few, date is the date of my volume, not the original work. Several of these are translations from the original, but are themselves still very old - so don't go too hard on me if a date is after the author was dead or the book was written!
Some of these I freely admit are too difficult for me to really understand well, but I have read and understood enough of each of them to get something, and to appreciate them. Others are surprisingly accessible. Others are really meant to be on the easy side by those who wrote them, but are not less worthy because of that. There are a few I am not mentioning, because they are almost too obvious, such as whatever you can understand of Gauß, the works of Hardy and his associates that have been receiving a lot of completely deserved attention, etcetera etcetera. These I'm hoping are more obscure. I'm omitting a lot - works aren't great just because they are old - but I'm hoping that all of these have some merit.
«Differential and Integral Calculus», Courant, 1934. (translation into English)
«Theory of Equations», Burnside and Patton, 1918.
Books by Coxeter - many.
«A Treatise on Conic Sections», Casey, 1885 (several other books by Casey also).
«Praktisches Zahlenrechnen», Werkmeister (gotta love the name with the title), 1945.
«Darstellende Geometrie», Haußner, 1908.
«Astronomie», Möbius, 1910.
«Number Theory», Mathews 1890?.
«Algèbra Supérieure», Serrèt, 1910.
«Geometry and the Imagination (Anschaulich Geometrie)», Hilbert, 1946.
«Essai d'une Nouvelle Théorie de la Musique», Euler, 1839.
«Darstellende Geometrie», Fischer, 1921. (and several other books by Fischer).
«Projective Geometry», Veblen and Young, 1910.
«Theory of Groups», Kurosh, 1956.
«Élemens D'Algebre», Euler, 1774 (Translated by one of the Bernoulli's, plus La Grange).
«Induction and Analogy in Mathematics», Polya, 1954.
«Lineare Algebra», Lorenz, 1989 (so not completely 'old' by my definition).
«Gesammelte Mathematische Abhandlungen», Schläfli, 1956.
«Generatingfunctionology», Wilf, 1994. (again breaking my 'old' definition).
«Introduction to the Theory of Groups», Alexandroff, 1959.
«Theory of Equations», Thomas, 1938.
«Introduction to Inequalities», Beckenbach and Bellman, 1961.
«Gesammelte Werke», Steiner, 1881 (incredible fold-out illustrations).
«Die Ausdehnungslehre», Grassmann, 1862.
«Geometric Transformations», Yaglom, 1962.
«Euclidean and Non-Euclidean Geometries», Greenburg, 1972.
«Irrational Numbers», Niven, 1956. (and other books by Niven - he's a good writer).
«The Theory of Algebraic Numbers», Pollard, 1950.
«The Geometry of the Zeroes», Marden, 1949.