Archimedes Is Reticent to Publish Proofs Using Indivisibles
This post follows on from these three posts on Archimedes (287 BCE-212 BCE):
The Rediscovery of The Archimedes Palimpsest and His Use of the Infinite
André Koch Torres Assis And Ceno Pietro Magnaghi Write About Archimedes's Method
In the first post, I wrote about the remarkable discovery made by Reviel Netz and Ken Saito, related in the book The Archimedes Codex1, that the text of Archimedes’s Method is peppered with the words isos plethei [equal in magnitude], and that in fact Archimedes is comparing infinite sets in the same manner as is done by today’s mathematicians:
He was saying that, with the infinitely many slices produced in the cube, after all the random slices were made and the cube was entirely cut, the triangles, produced in the cube by all the possible random cuts, were “equal in multitude” to the lines in the rectangle. You see? In each random slice there was a triangle in the cube, standing on top of a line in the rectangle. And Archimedes pointed out that the number of triangles of which the prism was made was the same as the number of lines of which the rectangle was made. Surely he meant this to be verified by the fact that there was a one-to-one relationship. Each triangle stood on an individually separate line and each line was at the bottom of an individually separate triangle. [The Archimedes Codex, p.201, my emphasis]
One of Netz’s conclusions is as follows:
we find that Archimedes calculated with actual infinities in direct opposition to everything historians of mathematics have always believed about their discipline. Actual infinities were known already to the ancient Greeks. [The Archimedes Codex, p.203, my emphasis]
Although the exact comparison of the size of infinite sets may not have been previously known to historians of science, several did write about the use of indivisibles, and in so doing, necessarily the use of infinite sets. In particular, the Netz-Noel book states that “The best general book on Archimedes’ scientific achievement is likely to remain for many years to come” Eduard Jan Dijksterhuis’s (1892-1965) Archimedes2.
Here is what Dijksterhuis writes:
in Greek mathematics there existed, side by side with the strict and official method of the indirect passage to a limit, also the less strict, but heuristically more fertile method of indivisibles, and that Archimedes himself diligently used it as a method of investigation. In this method a solid is regarded as the sum of plane sections, a surface as the sum of lines, while it also includes the view of a curve as being generated by juxtaposition of points; this might easily suggest the idea that thus the difference between two solids might be a surface, that between two surfaces a length. [Archimedes, p.148, my emphasis]
The indivisibles are not to be confused with Leibniz’s infinitesimals, which I will write about in future posts.
Dijksterhuis writes in depth about problems perceived with the use of indivisibles after presenting the proof of the first proposition of the Method. See my third post on Archimedes, Archimedes Uses a Lever to Prove Geometry Theorems, for more details about that proof.
The method which Archimedes wishes to explain emerges so clearly from the proposition dealt with that we can proceed to discuss it already here.
We may note first of all that it is characterized by the application of two different principles: in the first place it makes use of considerations taken from mechanics in that it conceives geometrical figures to be attached to a lever in such a way that the latter remains in equilibrium, and then draws up conditions for such equilibrium; and it is further based on the view that the area of a plane figure is to be looked upon as the sum of the lengths of all the line segments drawn therein in a given direction and of which the figure is imagined to be made up; this view will be extended to space in the following propositions in the sense that a solid, too, is conceived to be made up of all the intersections determined therein by a plane of fixed inclination that is displaced, and that subsequently also the volume of the solid is looked upon as the sum of the areas of those intersections. We shall designate these two methodic principles by the references: “barycentric method” and “method of indivisibles”. [Archimedes, pp.318-319, my emphasis]
A barycenter is the center of mass of a set of bodies. So where Dijksterhuis writes about the “barycentric method”, he is referring to the balancing on a lever two geometric objects, necessarily implying the materialization of these objects. He continues:
We further saw that Archimedes is not prepared to recognize the results obtained with this twofold method as actually proved conclusions. It might now be asked where in his view resides the lack of exactness, in the barycentric character of the arguments, in the application of indivisibles, or in both.
The answer to this question may be given without much doubt: the mathematical deficiency is exclusively a consequence of the use of indivisibles; there is not the least objection from the mathematical point of view against properly founded barycentric considerations, such as we already found applied in Prop. 1.
That this is actually the view of Archimedes is particularly evident from the fact that in his treatise Quadrature of the Parabola, which constitutes an official publication satisfying all requirements of exactness, he proves the insight gained in Prop. 1 on the area of any segment of an orthotome [parabola] once more by means of statical considerations, but this time without indivisibles. [Archimedes, p.319, my emphasis]
So, although Archimedes used the method of indivisibles to prove theorems to himself, his published proofs were all of a geometrical nature, as the latter were considered to be better.
But where did this reticence to public use of the indivisibles come from? For this, we will need to look at Enrico Rufini’s (1890-1924) Il “Metodo” di Archimede e le origini del calcolo infinitesimale nell’antichità3 [Archimedes’s “Method” and the Origins of the Infinitesimal Calculus in Antiquity], which is referred to by Dijksterhuis. Therein, Rufini explains the contradictions that Parmenides and Zeno of Elea point out in the Pythagorean understanding of matter and number, and the implications that this will have on the perception of indivisibles.
These questions will be dealt with in detail in the coming posts.
Reviel Netz and William Noel. The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity’s Greatest Scientist. Philadelphia: Da Capo Press, 2007. Chapter 8: “Archimedes’ Method, 2001, or Infinity Unveiled”.
E.J. Dijksterhuis. Archimedes. Translated by C. Dikshoorn. With a new bibliographic essay by Wilbur R. Knorr. Princeton University Press, 1987. Chapters I-V of this work were formerly published in Dutch (Archimedes, I, P. Noordhoff, Groningen, 1938). The subsequent chapters appeared in the Dutch periodical Euclides (XV-XVII, XX; 1938-1944).
Enrico Rufini. Il “Metodo” di Archimede e le origini del calcolo infinitesimale nell’antichità. Milano: Giangiacomo Feltrinelli, 1961. First published in 1926 by Nicola Zanichelli Editore, Bologna.