If one is to buy into Plato's theory of perfect forms, then I must say that this comes infinitesimally close to being a "perfect introductory Calculus book". I couldn't help but get the impression that this was a book that was crafted to be enjoyed. Even without looking at the content, its physical properties are admirable. It's much smaller than those over-size Calculus textbooks you're used to lugging around in school, yet the print is large enough that it's easilly readable. The organization is quite impressive. The book allows you to delve into the complexities of hyperreals from the get-go, or skip the technicalities and still understand enough of the concepts to apply to the rest of the book. But the most remarkable trait of this book is that it is actually entertaining!!! Not because it consists of a lot of lame jokes that detract from the book's mathematical content as other "friendly Calculus" books sometimes do, but because the authors actually appear to be competent writers as well as mathematicians! Background is intermixed with theory, and in the midst of it, you'll find lots of interesting little anecdotes interwoven in the sidebars that enlighten your perspective of mathematical concepts and the personalities of the matematicians who discovered them. Content-wise, the book is completely rigourous, concise, and very consistent. It's such a tiny book that I was sure that it must have skipped something important, but comparing it to the much longer long-winded Spivak book, I couldn't find anything missing...except epsilons and deltas. That of course is the main goal of the book, to take the traditional introductory material of a first-year Calculus class and apply the techniques of Nonstandard Analysis, which were discovered in the last few decades. The result is that the authors have created a very concrete and rigorous treatment of Calculus that has all of the traditional uglyness removed from it. The authors even provide one epsilon-delta proof in the beginning, just to show how much more cumbersome it is compared to their elegant hyperreal system. The system itself is very abstract, but the authors take us to the point where we can see that abstraction and intuition do converge! Amazing. My only warning about this book is that it may not help you very much with your current curriculum, simply because the approaches it uses are so different than norm. Most of the topics in this book are not even covered during undergraduate studies, much less a first year class. If this book was actually used as a textbook for a real math course, I'd be the first to enroll!

The authors have created an introduction to logic that can be used as a textbook in a logic course but is also filled with enough puzzles, paradoxes, quizzes and ideas to interest any general reader. My only quibble is the somewhat HIGH price (which is probably out of the authors' control.