MVT: A Most Valuable Theorem

The book is about the mean value theorem, mostly its past. There is much material on related topics. The author begins by giving a plausibility argument for the truth of the theorem and says

The argument given is not mathematically rigorous, but it can be made so first by offering precise definitions of the terms involved (“curve”, “tangent”) and isolating the conditions under which the Theorem [sic — the author is terrifically fond of capital letters] is to hold, and then appealing to more advanced theory to justify the individual steps. This will be done in the course of this book.

He begins with curves, starting with the ancient Greeks. Most teachers of calculus, for whom the book is most appropriate, may not be aware that they had very few. After lines and circles came the quadratrix of Hippias (around 400 BC) that, as its name implies, can be used to square the circle. It can also be used to trisect the angle, and to heptasect, nonasect, and divide it into any number of parts. As the author does throughout, he gives history and follows with mathematics. Readers armed with a quadratrix will, after getting to page 12, be able to square any circle that they encounter.

The conic sections arrived later, and they (two parabolas, actually) can be used to take care of the third classical Greek problem, duplicating the cube. Though the book is not a textbook, after explaining some of the properties of conic sections, the author includes some exercises. He does this throughout the book; no answers are provided, but the exercises could be useful.

There follows material on the conchoid, the spiral of Archimedes, and the cycloid; all good things to know even if not directly related to the mean value theorem. The author always ranges widely.

Next come continuous curves, including the struggle to define continuity (not done “correctly” — i.e., as we would do it — until 1817). When the author goes into Peano’s continuous space-filling curve, he gives a translation of Peano’s 1890 paper in full, all four pages of it. This is an admirable characteristic of the book: it has many long excerpts of original sources.

The next forty pages are devoted to smooth curves, and the difficulties of defining what a tangent is. There is a consideration of Isaac Barrow, who has been too much neglected. Speaking of tangents, the is a tangential discussion of, and excerpts from, Bishop Berkeley’s objection to Newton’s fluxions.

Though we haven’t even got to the mean value theorem yet, I won’t continue to go through the book’s content. A look at the table of contents shows what it contains. It continues as it begins. It is split between history and mathematics, both well done. It is a rich, rich book. I won’t say that all teachers of calculus should buy it because I have known some who are indifferent to history, but I will say that all right-thinking teachers of calculus should have a copy, or at least know where to locate it in the library. It will enlarge their mathematical horizons, even for those like me who thought that they were vast already.

There are many references and the book is well produced.

As I indicated, the author is inordinately fond of capital letters. The names of subjects — calculus, analytic geometry, topology, and so on — are invariably capitalized, and we have passages like (page 328)

… apply the Higher Order L’Hôpital’s Rule to conclude Taylor’s Theorem with the Lagrange Form of the Remainder, i.e., the Higher Order Mean Value Theorem.

More than half of the words capitalized! If this annoys you, as it does me, you will constantly annoyed. Well, no book is perfect.

Woody Dudley graded his first calculus test in 1957. He has not kept track of how many times he asked students to find \(c\) so that \(f'(c) = (f(b)-f(a))/(b-a)\). His compilation Readings for Calculus (1993) was, the last time he looked, languishing in 2,392,151st place in Amazon’s list of best sellers.