The Origins of the Infinitesimal Calculus (Dover Classics of Science and Mathematics)

This book is marginally useful at best. It consists almost entirely of convoluted and muddled exposition of sample theorems and proofs of one mathematician after another without much cohesion. Baron's tendency to obscure or even severely distort the point of an argument may be illustrated by the following example, where she is in addition promoting the modern propaganda myth that 17th century mathematicians committed numerous mistakes and were guided by "a happy instinct" (p. 109) rather than reason."When [Kepler] argued by analogy he sometimes made mistakes. Unable to determine theoretically the length of an elliptic arc he argues that, since the area of an ellipse is equal to that of a circle, the radius of which is the geometric mean of the major and minor axes (area = pi ab = pi r^2, where a/r = r/b), then the circumference of the ellipse should correspondingly be equal to the circumference of a circle the radius of which is the arithmetic mean of the semi-axes, i.e. pi (a+b)." (p. 109)The only one making a "mistake" here is Baron. What Baron portrays as a crackpot "analogy" is in fact a perfectly sound and intelligent argument. The argument is this: since the circumference of the circle with the same area as the ellipse is 2 pi sqrt(ab), the circumference of the circle must be somewhat greater (since the circle has the least possible perimeter for a given area). Kepler then proposes to use the *approximation* pi (a+b) for the circumference, being fully aware and completely explicit that it is an approximation only ("elliptica circumferentia est proxime...", we read on the very page of the Opera Omnia referred to by Baron)---and indeed a very good and very convenient approximation at that for ellipses of small eccentricity (such as the orbit of Mars, which is Kepler's interest).