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.
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 π=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 π 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.