The History of the Calculus and Its Conceptual Development (Dover Books on Mathematics) First Edition

This is a truly enlightening history of calculus from Eudoxus to Weierstrass. This 1949 book by Carl Benjamin Boyer, republished by Dover, places the developments of Newton and Leibniz within the long sequence of historical developments of calculus including Pythagoras, Zeno, Eudoxus, Aristotle, Euclid, Archimedes, Oresme, Viète, Stevin, Cavalieri, Torricelli, Kepler, Galileo, Descartes, Fermat, Wallace, Barrow, Leibniz, Newton, Maclaurin, Euler, Lagrange, Lacroix, Bolzano, d'Alembert, Cauchy, Weierstrass, Cantor, and Dedekind, roughly in that order.Within this context, it becomes crystal clear that the old arguments about the relative precedence of Newton and Leibniz are a relatively minor matter. Both of them relied heavily on a very long sequence of earlier developments, and both of them fell very far short of a satisfactory, logically self-consistent, meaningful formalism, which required another 200 years to develop.This book has about a thousand footnotes. It is very thoroughly researched indeed. Boyer describes how various opposing forces were at work in the development of the calculus. Very important, for example, was the weight of tradition, such as the method of exhaustion of the ancient Greeks, which was held in high esteem. There was a strong resistance to abandonment of geometrical intuition as the basis of calculus, although ultimately a satisfactory axiomatization of the real numbers permitted calculus to be liberated from its geometric origins. Unsatisfactory concepts of infinity and the infinitesimal strongly constrained or encouraged many mathematicians to reject some formulations while accepting others.One thing which worried me a lot is how much of modern calculus is still taught in the same way as many centuries ago, using ways of thinking which are obsolete, meaningless or logically circular. In fact, one of the themes of this book is how ancient ill-founded concepts have frequently been "rediscovered" and adopted by mathematicians, long after they had been superseded by better-founded concepts.I ended with the suspicion that our modern-day calculus (or "analysis") is not the end of the road. Even our current calculus is a mixture of intuition, metaphysics, pragmatism and sometimes empty formalism. This book seems to put some of our currently accepted calculus concepts in doubt.

I bought this book for my class course and I found that this book is written and edited in a very properway. Every next chapter has some kind of connectivity with the previous chapter. I will recommend this book to others.

I am still reading it, but am half way through. The book has a lot of interesting information. For the first time I understand how Archimedes found the volume of a sphere by "weighing" it, and that is only one of many great insights. I am not giving it five stars because, as others have said, it is written a style that may have been common at the time, but to the modern reader can be frustrating. I think the same information, with the same nuance where necessary, could be written in a less tedious manner today.

I bought this after taking a physics course and becoming interested in the historical and conceptual development of this level of mathematics that I've never studied (I only went up to Pre-Calculus in high school). I had to read it slowly, as the narrative is very dense with information, but it's great, because it doesn't waste any time in explaining the different problems and solutions that arose historically. Truly fascinating and worthwhile read. I highly recommend it, and I'm NOT a math person.

The first thing I noticed about this book is that it is written with an intellectually arrogant, indecipherable style which (I hope) would today prevent its being published at all. Here is a paragraph, verbatim, from the introduction:"At this point it may not be undesirable to discuss these ideas, with reference both to the intuitions and speculations from which they were derived and to their final rigorous formulation. This may serve to bring vividly to mind the precise character of the contemporary conceptions of the derivative and the integral, and thus to make unambiguously clear the <I>terminus ad quem</I> of the whole development."I admit that back in 1939, when this book was originally written, it was common for academics to express themselves in that sort of haughty, impenetrable prose. But that doesn't make it any easier to read today, and it doesn't really provide those people with an excuse for having written that way. Didn't it occur to them that their writing might be read by real human beings? There are plenty of mathematical writers today who can write in real English without sacrificing rigor or depth.Secondly, I recommend that everyone read the review by the reader from Phoenix (February 7, 2001). In particular, I agree with the criticism that this book takes a backwards approach to the history of Calculus, interpreting each historical idea and contribution in terms of the way we think of those ideas today. As Boyer certainly should have known, the proper way to relate the history of ideas is to place each idea in the context of its own time. Instead, he writes this book as if each ancient mathematician had tried and failed to reach the level of understanding which we superior moderns are now gifted with. I think it is important for a reader to read this book with this defect clearly in mind.Having got those two criticisms off my chest, however, I have to admit that there is a wealth of interesting material in this book, and I don't know of any other place where it is all gathered together in one volume. If you want a detailed, in-depth account of how mathematicians and philosophers (they used to be the same people!) eventually evolved the ideas and methods of calculus, then this book is probably the best place to find it.(I just wish the publisher would hire someone to translate it into real English!)

worth reading

I really like this book. As a math major in undergrad, I never got the full story about calculus, only bits and pieces. This book is very conversational, and is very easy to follow, even for the layperson.

A fascinating and detailed look at the more than 200 year struggle to define and refine 'The Calculus'. Any lover of Mathematics will enjoy this history.