I looked at a brief sample of the first part of the book and found it beautifully written, interesting and informative. Given the review quoted above, this was rather a surprise. Then I perused the reviews on amazon and found that although over 75% of them are 5 stars, the one quoted is among the 3% or so which are quite negative. This is perhaps evidence that every book has its own audience, and opinions differ based on the goals and background of readers. It may be, from the reviews I saw, that this book appeals more to somewhat more knowledgable readers and less to absolute beginners, since some who liked it mentioned wishing they had seen some of the material in grad school in years past. I myself am a retired mathematics professor, with an appreciation for historical sources like Euclid, Apollonius and Newton, and while I liked the historical section that I read, I was not allowed access to any more advanced parts. I hope you can either find a physical copy to preview, or find a copy so cheap that you will not mind if it is not to your taste. good luck. (Have you considered a more traditional book, like Williamson, Crowell, and Trotter?
https://www.abebooks.com/servlet/SearchResults?an=Williamson, Crowell, trotter&cm_sp=SearchF-_-home-_-Results&ref_=search_f_hp&sts=t)
well I have found one odd statement in the few pages I could preview on the springer site, concerning the fundamental theorem of calculus. it is dangerous to make any criticism of a book without full access to the author's definitions, but it appears that the first statement of the fundamental theorem in this book is false. I.e. the claim that if f is an integrable (defined to mean Riemann integrable in an earlier chapter) function, then there is another function F such that dF/dx = f. This is just false, since any increasing function with only jump discontinuities is Riemann integrable (on a closed bounded interval), but no such function has an antiderivative unless it is actually continuous.
this puzzles me. but it is not a deal breaker for me personally. I have found isolated cases of false statements in many excellent books, and have certainly made more than a few howlers myself, even in articles on topics I am considered "expert" in. It is much easier to find mistakes in other people's books than to write a book not having any. but I suggest the author intended to say in part 1, that f is assumed not just integrable, but continuous. it also suggests to me that this book is more intended to convey useful insight than to belabor the complete mathematical rigor of every argument. that to me makes it only more valuable. but be aware, never accept any mathematical statement as definitely correct, without proof.
