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The chapters and authors say it all:
1. A General View of Mathematics (A.D Aleksandrov)
2. Analysis (M. A. Lavrentev & S. M. Nikolski)
3. Analytic Geometry (B. N. Delone)
4. Algebra: The Theory of Algebraic Equations (B. N. Delone)
5. Ordinary Differential Equations (I. M. Petrovski)
6. Partial Differential Equations (S. L Sobolev & O. A. Ladyzhenskaya)
7. Curves and Surfaces (A. D. Alexandrov)
8. The Calculus of Variations (V. I. Krylov)
9. Functions of a Complex Variable (M. V. Keldys)
10. Prime Numbers (K. K. Mardzanisvili & A. B. Postnikov)
11. The Theory of Probability (A. N. Kolmogorov)
12. Approximation of Functions (S. M. Nikolski)
13. Approximation Methods and Computing Techniques (V. I. Krylov)
14. Electronic Computing Machines (S. A. Lebedev & L. V. Kantorovich)
15. Theory of Functions of Real Variable (S. B. Steckin)
16. Linear Algebra (D. K. Faddeev)
17. Non-Euclidean Geometry (A. D. Aleksandrov)
18. Topology (P. S. Aleksandrov)
19. Functional Analysis (I. M. Gelfand)
20. Groups and Other Algebraic Systems (A. I Malcev)
Lest the stature of the authors put you off, one can only say that they uphold the one of the better ideas from the Soviet era, namely that mathematics should not be taught in isolation from its history or its applications. The result is a supremely accessible survey that IMO should be read by any maths undergraduate if not before she begins her course, then at least during the first year.
To those who follow that piece of advice, I give another: Some of your university teachers may tell you that the book is too dated and that you should discard it in favour of one that adopts purely axiomatic approaches to each branch of mathematics. If so, you should lie in wait for them, ambush them and beat the idiots about the head with all 1000+ pages until they recant their heresy.
Dover has done Western mathematicians proud by making this wonderful book available!
This 1999 paperback edition has all three volumes in one binding. The paper is thinner so the book is less bulky. Page number resets at the beginning of volume 2 and 3. (not renumbered.) Thus the book content is not altered from the 2nd edition (three volumes set) from MIT Press in 1969.
One major change is that there is one index at the end of the book which covers all three volumes. The volume is indicated together with the page number. This improves the convenience.
(1) Popular math and science has become quite popular lately. I'm sure that there are many pop-math/pop-sci readers who would like a more 'meatier' treatment of math that still has an accessible style. If you're in that group, then this book is for you.
It basically requires recollection of high school algebra and a willingness to wade through and learn some challenging material. I should add that the book looks intimidating in size but this is mostly due to the fact that Dover has bound 3 volumes as 1 (which is actually a great deal for readers).
(2) I can't think of too may books that can, with effort, take a novice through the bulk of a college/university level math curriculum.
(3) This book contains material by 2 of the greatest mathematicians/scientists of all time: Andrei Kolmogorov (famous for his work on probability and information theory as well as Kolmogorov Complexity/Algorithmic Information Theory) and I. M. Gelfand (for his work on mathematical physics). The book is worth buying for that reason alone.
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