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Some reviewers have been puzzled as to the style of this book, deep mathematics for the unsophisticated reader. This is explained by its origin in the 1960's when many bright high school students were not offered calculus until college. Hence some top colleges experimented with very high level introductions to calculus aimed at gifted and committed students who had never seen calculus. Possibly Spivak took such a course, but certainly his book was used as the text for one at Harvard, and was still used more recently at a few schools still offering this course, such as University of Chicago. Unfortunately today, due to the somewhat misguided AP movement, which is oriented to standardized test performance rather than understanding, almost all mathematically talented high school students take calculus before college, receiving significantly inferior preparation to what they would receive in college. The result is that many top colleges where the Spivak type course originated, no longer see the need to offer it. This means that gifted freshmen at schools such as Harvard and Stanford are now asked to begin with an advanced honors calculus course for which Spivak is the ideal prerecquisite, although those same schools do not offer that prerecquisite. Thus if you are a high school student hoping to become a mathematician and planning to attend many elite colleges, almost the only way to be adequately prepared for an honors level mathematics program is to read this book first. It may be that a book like Stewart or even Calculus Made Easy, is useful as a first introduction to calculus, but it will not get you to the level you need for a course out of Apostol vol. 2, or Loomis and Sternberg. Although some seasoned mathematicians eventually come to prefer other treatments such as those of Apostol or Courant, the appeal for a young student of Spivak's light hearted, clear, and enthusiastic presentation is probably unmatched by any other work. For forty years there has been a cult of fanatical adherents to the books of Spivak, and they still deliver the goods to the audience for which they are intended. There is only one caveat that may even be unimportant for beginners, but it is that Spivak is almost too clear and thorough in his explanations. I.e. a future researcher needs to learn to question, to discover and work things out on his own, whereas Spivak explains everything for you. So every now and then resist reading his explanation and try working out a proof or a generalization on your own, and see if it doesn't become even clearer afterwards. Understanding how a proof was thought of is different from just knowing how it goes.