Ever since we discussed Mark Kac's Statistical Independence in Probability, Analysis and Number Theory, I've thought about other truly great non-recreational* math books. I have not necessarily read all of these books!!! But I have at least spent serious time dipping into each of them extensively, and have found that with most of these, with adequate background, you can dip in almost anywhere and have a very enjoyable read. They are all pellucid, at least most of the time. Here are a few among many: Tom Apostol, Calculus vol. 1 Tom Apostol, Calculus vol. 2 William Feller, An Introduction to Probability Theory and Its Applications, vol. I discrete probability William Feller, An Introduction to Probability Theory and Its Applications, vol. II continuous probability. Covers many subjects a second time in a more advanced way. G.H. Hardy & E.M. Wright, An Introduction to the Theory of Numbers algebraic but not analytic John Conway & Neil Sloane, Sphere Packings, Lattices and Groups Walter Rudin, Principles of Mathematical Analysis George Pólya & Gordon Latta, Complex Variables Tom Apostol, Modular Functions and Dirichlet Series Anthony Knapp, Elliptic Curves pre-FLT, largely based on a lecture series by Don Zagier, but filled with fascinating math James Munkres, Topology ultra-clear William Massey, Algebraic Topology does homotopy, and not homology or cohomology, but has many fascinating nonstandard topics John Milnor, Morse Theory* best to have a background in differential geometry. Explains the amazing Bott Periodicity theorem via geometry. John Milnor & James Stasheff, Characteristic Classes* exquisite introduction to many profound and beautiful results about manifolds that are available with algebraic and differential topology. Based on mimeographed notes Stasheff took on 1957 lectures by Milnor, the book did not appear until 1974, which is probably one reason the writing is so very polished. Although everything is done rigorously, Milnor makes it appear almost effortless. Alexandru Scorpan, The Wild World of 4-Manifolds* Utterly fascinating tour through the topology of dimension 4, which is unlike any other dimension. (For instance, each Euclidean space R^n for n <> 4 has exactly one smooth structure, up to equivalence (diffeomorphism). R^4, on the other hand, has infinitely many inequivalent smooth structures. And the infinity is not aleph_0. It is 2^aleph_0, the continuum.) The author is extremely generous to the reader, explaining virtually everything in detail, often more than one place in the book, with ample cross-references (that save the reader from having to flip back and forth to the index), and with voluminous explanatory footnotes. 0 Dimension 4 is also the only unresolved case of the smooth Poincaré conjecture PC4: PC4: If M is a compact 4-manifold such that every map of a circle S^1 or a sphere S^2 into it can be continuously shrunk to a point, then M is diffeomorphic to the standard 4-sphere S^4. --Dan ________________________________________________________________________________________ * The last three require a solid background in topology, and at least an introduction to homotopy, homology, and cohomology. The 4-manifold book is best read with some background in differential topology. _________________________________________________________________________ * I almost used the word "medical" instead.