Continuity: Eleven Sketches from the Past of Mathematics

This book offers a scattershot overview of some themes in the history of geometry and analysis. The author proceeds chronologically from classical geometry, to medieval kinematics, to infinitesimals and calculus, to foundations of analysis. Much of the material discussed can be said to pertain to foundational questions regarding continuity, although the coverage is hardly systematic. The first half of the book includes a fair amount of mathematical detail and complete proofs, but then the author abandons this and instead opts for a more purely historical narrative format.

The book has the flavor of a professor telling stories off the cuff: no coherent argument and not very reliable as historical scholarship, but occasional nuggets of value acquired over a long career. Although most of the material is standard and well known, there are a few twists. New to me were for example the idea of proving that parallelograms with the same base and height are finitely equidecomposable by induction from the case where they have overlapping top sides (the case that requires the least number of cuts), and the suggestion, in the context of Fourier analysis, that we should go to the beach and experience the nature of wave motion by holding our foot just above the surface as a wave approaches: “most people are surprised to find that the water hits their feet from below” (172).

Also refreshing are the author’s occasional historical hot takes: “We don’t love the 18th century”; it had “a certain sterility” (173). Mathematicians were perhaps driven to develop non-Euclidean geometry purely in order to spite Kant since they found his account of the inviability of Euclidean geometry too dogmatic (88). “[A medieval] scholastic is like a student who, when solving problems looks up the answers [which in their case means in Aristotle] and regards them as the main criterion of truth” (96, via Boris Kuznetzov).

The book leaves much to be desired as a work of historical scholarship, however. It gives few references, except some to obscure sources in Polish. Let us illustrate the problems that arise from this. After seeing a standard infinitesimal proof of the area of a circle, we are told that “This ‘proof’ is so persuasive that the 16th-century Hindu mathematician Gamesha was presumably satisfied with dividing the semicircles into six parts” (124). This sentence comes completely out of the blue: there is nothing else in the book about “Gamesha” or the Hindu geometrical tradition. I decided to try to find out more in other sources. I could not find anything about any Hindu mathematician called “Gamesha.” After some more thorough research I realized that “Gamesha” is probably a typo for “Ganesha,” a possible transliteration of the name usually rendered as Gaṇeśa, who was indeed a 16th-century Hindu mathematician. There is a translation of the relevant text in the book Ramasubramanian et al. (eds.), Bhaskara-prabha (Springer, 2019). I am still puzzled, however, by the remark that “Gamesha” was “presumably” satisfied with a small number of slices. In the translation in the Bhaskara-prabha book, Ganesa speaks of “cutting [the area] into as many as desired number of needle-shaped parts” (110), which seems to directly contradict Mioduszewski’s account. So while I admittedly learned something new from reading Mioduszewski on this occasion, it that was weirdly and misleadingly presented in a way that left me having to do quite a bit of work to find out more.

There are other historical issues as well. The book claims that Newton’s term “fluxion” was taken from “the traditional terminology of the Calculators of Merton College” (142). I am quite sure this is wrong. I have never head this claim before and I have seen many scholars attribute this terminology to Newton himself. Again, no reference is given by Mioduszewski. To give another example, there were not twenty years between Galileo’s Dialogo and his Discorsi, as Mioduszewski erroneously claims (117). Elsewhere we are told that “in his proof of Euclid’s [parallel] postulate Al Jawhari made use of the obvious fact that if one has an angle and a point in the interior of that angle, then one can lead through that point a straight line intersecting both arms of that angle” (86). In Mioduszewski’s hasty treatment, it is implied that Al Jawhari thought this was “obvious” and didn’t realize the need for a proof. But this sells Al Jawhari short. He did in fact state this as a proposition, and gave a proof. This proof was flawed because of mistaken assumptions several propositions earlier in his chain of reasoning, but that is still a very different thing than merely assuming it as an “obvious fact.”

Unfortunately the English is very poor throughout, and made more difficult to read by an abundance of typos. The book was supposedly translated by Abe Shenitzer. It is difficult to understand how this can be reconciled with the usually excellent quality of Shenitzer’s other translations and writings. Let me illustrate with a few examples. To understand Cavalieri’s Principle, we should “imagine a puck of cards” (130). Archimedes thought infinitesimal arguments were “ultramathematical” (125, 142) and therefore avoided them in favor of more rigorous proofs by the method of exhaustion, such as in his theorem on the area of a “cirrcle” (124). Some mathematical traditions emphasize the visual, others the algorithmic: “uphanging may play a role” (111) in shaping such preferences. A famous quote from Proclus on the origins of geometry is rendered completely nonsensical: “[Egyptians] needed [geometry] because the inhabitants of the Nile washed out balks” (10). This is a butchered version of the sentence that in the standard Morrow translation reads: “[Geometry] was necessary for them because the Nile overflows and obliterates the boundary lines between their properties.”

Despite occasional charm and a few interesting tidbits, this book is unfortunately unlikely to be very useful to any audience. Almost all of the material can be found in readily available books that are more reliable and written in much better English.

Viktor Blåsjö is an assistant professor at the Mathematical Institute of Utrecht University. You can follow him on Twitter @viktorblasjo and listen to his Opinionated History of Mathematics podcast.