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Brian Clegg
A Brief History of Infinity
The book starts with the early ideas of the Greeks such as Zeno, and goes on to look at Archimedes use of huge numbers in The Sand Reckoner. Moving on to the middle ages we see the links between religion and the infinite, and how some of Galileo's troubles were due to his ideas of infinity. Clegg goes on to discuss the development of the calculus, including the battles between Leibniz and Newton. In the nineteenth century the problems with infinitesimals were dealt with, but a new can of worms was opened with Cantor's transfinite numbers.
Note that, as the title says, this is a book on the history of infinity and so doesn't deal with the most up to date research. Nonstandard analysis is mentioned, but as Clegg tells us, that subject is now over half a century old. Some might also find the book a bit slow going  it is very much aimed at those without previous knowledge. If that's what you think would suit you, or if you want to see how belief gradually moved from potential to actual infinity then you should take a look at this book.