This is a follow-up to some of my comments to the OP and to the answer that @Conifold gave. A few days ago I purchased a copy of the book that I had mentioned, Measure and the Integral by Lebesgue (1966). I was correct that this book is where, many years ago (1980s, possibly even late 1970s), I had read about Lebesgue's fascination with this “paradox”. However, contrary to what I thought, Lebesgue's fascination is not discussed in Kenneth O. May's biographical essay, but instead it appears in Section 66 on pp. 97-98. Those wishing to read more about these issues involving arc length and surface area will find Chapter V: Lengths of Curves. Areas of Surfaces (Sections 62-83; pp. 92-124) especially instructive. Below is all of Section 66, followed by some bibliographic information about this 1966 book.
66 A similar paradox for lengths. $\;\;\;\;$ If mathematicians had not been hypnotized by the word “inscribed,” if they had not forgotten that inscribing had been chosen only as one way of approximating, they would have seen that the difficulty encountered for areas existed equally for curves. Now it was just this difference between curves and surfaces that was most shocking. Allow me to refer to my own recollections.
When I was a schoolboy, it was agreed in France, as I have said [previously], that one could evaluate lengths, areas, and volumes by passing to the limit. Soon doubts began to appear in the textbooks. The students who had heard Schwarz’ objections in Hermite’s analysis course had now in their turn become teachers. Besides, everything then predisposed us to a critical analysis of concepts: researches on functions of a real variable and on sets, which people were beginning to consider, Tannery’s teaching, which had aroused in many of his students the desire for complete comprehension or at least verbal precision. People began to doubt, sometimes without knowing what they doubted. For example, the determination of the area of a circle by means of the areas of the polygons that it contained or that contained it (see section 42) was confused with an argument about limits.
Formerly, when I was a schoolboy, the teachers and pupils had been satisfied with this reasoning by passage to the limit. However, it ceased to satisfy me when some of my schoolmates showed me, along about my fifteenth year, that one side of a triangle is equal to the sum of the other two and that $\pi = 2.$ Suppose that $ABC$ is an equilateral triangle and that $D,$ $E,$ and $F$ are the midpoints of $BA,$ $BC,$ and $CA.$ The length of the broken line [= polygonal path] $BDEFC$ is $AB + AC.$ If we repeat this procedure with the triangles $DBE$ and $FEC,$ we get a broken line of the same length made up of eight segments, etc. Now these broken lines have $BC$ as their limit, and hence the limit of their lengths, that is, their common length $AB + AC,$ is equal to $BC.$ The reasoning with regard to $\pi$ is analogous.
Nothing, absolutely nothing, distinguishes this reasoning from what we used to evaluate the circumference and area of a circle, the surface and volume of a cylinder, a cone, and a sphere. This result has been extremely instructive to me.
Besides, every paradox is highly instructive. In my opinion, the critical examination of paradoxes and the correction of erroneous reasoning should be standard exercises, frequently repeated at the secondary level.
The preceding example shows that passing to the limit for lengths, areas, or volumes requires justification, and, like Schwarz’ example, it is enough to arouse all one’s suspicions.
Henri Léon Lebesgue (1875-1941), Measure and the Integral, edited with a biographical essay by Kenneth Ownsworth May (1915-1977), The Mathesis Series, Holden-Day, 1966, xii + 194 pages.
This book is an English translation of two works by Lebesgue. The first work is on pp. 12-175 and the second work is on pp. 178-194.
The first work was originally published in L’Enseignement Mathématique with the title Sur la mesure des grandeurs and consists of an Introduction and 8 sections published in 6 parts: (i) L’E. M. (1) 31 #2 (1932), pp. 173-206 [Introduction (pp. 173-174); I. Comparaison des Collections; Nombres Entiers (pp. 175-181); II. Longueurs; Nombres (pp. 182-206)]. (ii) L’E. M. (1) 32 #1 (1933), pp. 23-51 [III. Aires (pp. 23-51)]. (iii) L’E. M. (1) 33 #1 (1934), pp. 22-48 [IV. Volumes (pp. 22-48)]. (iv) L’E. M. (1) 33 #2 (1934), pp. 177-213 [V. Longueurs des Courbes. Aires des Surfaces (pp. 177-213)]. (v) L’E. M. (1) 33 #3 (1934), pp. 270-284 [VI. Grandeurs Mesurables (pp. 270-284)]. (vi) L’E. M. (1) 34 #2 (1935), pp. 176-219 [VII. Intégration et Dérivation (pp. 176-212); VIII. Conclusions (pp. 212-219)]. [[ Note: “(1) 33 #3” means “series 1, volume 33, issue 3”. I do not know the precise dates of the issues, or even whether such more precise dates exist, so the years are for the volumes. ]]
The first work was published as a book with the title Sur la Mesure des Grandeurs by Gauthier-Villars (Paris) and L’Enseignement Mathématique (Geneva) in 1956 (iv + 184 pages), which was reprinted with the title La Mesure des Grandeurs by Albert Blanchard (Paris) in 1975 (iv + 184 pages).
The second work is the published version of a conference talk that Lebesgue gave in Copenhagen on 8 May 1926 and it was originally published with the title Sur le développement de la notion d’intégrale in Matematisk Tidsskrift B [after 1952: Mathematica Scandinavica] 1926 (1926), pp. 54-74, and reprinted with the same title in Revue de Métaphysique et de Morale 34 #2 (April-June 1927), pp. 149-167, and translated into Spanish and published in Revista Matemática Hispano-Americana with the title Evolución de la noción de integral and published in 2 parts: (i) R. M. H.-M. (2) 2 #3 (March 1927), pp. 65-74. (ii) R. M. H.-M. (2) 2 #4 (April 1927), pp. 97-106.
Book Reviews I know of: Truman Arthur Botts, Science (N.S.) 155 #3765 (24 February 1967), p. 992; A. S. G., Current Science 36 #7 (5 April 1967), p. 194; Thomas William Hawkins, American Mathematical Monthly 75 #6 (June-July 1968), pp. 696-697; Roger Philip Rigelhof, Canadian Mathematical Bulletin 11 #5 (December 1968), pp. 753-754; Mark Edward Noble, Mathematical Gazette 52 #382 (December 1968), 412-413; André Reix, Revue Philosophique de la France et de l'Étranger 166 #4 (October-December 1976), 437-438 (in French).