You have now had some samples of the problems of beginning calculus and some indications of the questions that cause calculus to develop beyond these. With the background given here, there are many texts on calculus that you should be able to read for yourself and understand without difficulty. There will be others that do not seem to make sense to you at al!. And some books will lie halfway between these. You will be reading quite happily and then you will come to a page that seems to you entirely unnecessary. Perhaps you will not understand what it is saying at all; again, you may find that long arguments are used to reach a conclusion that seems perfectly obvious.

To understand this, you need to know something of the history of mathematics. During the years 1600–1800 a.d., calculus was concerned with very much the kind of problems, and used very much the kind of thinking, that you have seen in this book. Then, gradually, 8 crisis developed. As mathematicians explored deeper and deeper into the subject and studied more and more complicated situations, they began to get answers that were evidently wrong. Their way of thinking, which had been perfectly satisfactory for dealing with simpler situations, was now proving unreliable; they found it necessary to examine very carefully things which before they had taken for granted.