What motivated Democritus to discover the volume of a cone?

Question

In the works of Archimedes, it is mentioned that Democritus was the first to discover that the volume of a cone is 1/3 that of the cylinder with the same base and same height.

I am curious to know about what motivated Democritus to discover this.

Other than intellectual curiosity, which is a somewhat unsatisfying answer, was there some operational motivation, i.e. in Greek astronomy or mechanics?

I can't think of why someone in Greek times was trying to find the volume of a cone, other than as an intermediate step to calculate the volume of a sphere maybe as Archimedes did, or for some mechanical/architectural reasons which the Greeks had that I am unaware of.

Any known theories or sources?

Answer 1

Democritus was one of the earliest "Atomists" who though that all large things (even mathematical things) could be built up by the most elementary things.

While thinking about the Paradoxes of Zeno & other related things (tiniest matter , tiniest time interval , tiniest length) , Democritus considered the thinnest layers of various Solids , which were termed laminae.

These laminae could be put together to get back the Solid.
Now there was some Contradiction there :
Will the laminae of Cylinder match the laminae of Cone ?
If YES , then how can the Cone differ from Cylinder ?
If NO , then how can the "atomic" laminae differ ?
Paradox !

We need not go into the resolution there , except to take it that the laminae for Cone should be "jagged" when stacked up , whereas the Cylinder will have a "smooth" stack.

Those "Atomism" Concepts were the motivations for Democritus.

Motivation was most likely not some Engineering Activity or Mathematical theorem. Proof of that "formula" was not a concern for Democritus ( Proof was eventually given by Eudoxus )

[[ I am interpolating on what concepts/ideas/thoughts Democritus had & what texts are currently known & what later writers wrote. That is a reasonable approach , given that there is nothing else available , mostly getting lost in the mists of time ]]

Answer 2

The physical fact of equal volumina for different shapes, cylinder and cuboid as a gauge, is known to the Babylonians.

The discussion of the equality of becomes critical with the invention of coins and overseas trade in oil, wine etc during the 6th century BC. By the new economy this question became a topic at courts with 1000 methods of cheating.

Even the fact, that the volume is the same independent of its content was a topic of mathematical discussions. Democrit made the important statement that the empty container has the same volume as the full. It's a giant step in philosophy and geometry, to postulate that the empty space has metric properties.

Its an old topic, discussion going back to Plutarchos, one of the few sources about Democritus.

REINHARD SEIDE: On the problem of geometric atomism in Democritus

Translation thanks to google:

p. 8

As Plutarch tells us (Chapter 39, Moralia 1079 EF), the Stoic Chrysippus believed that with this theory he could solve the dilemma that Democritus had encountered in his stereometric work. According to him, the Abderite was not able to state whether the surfaces that result from the intersection of a right circular cone with planes parallel to its base are equal or unequal; because none of the alternatives preserves the integral shape of the cone. In the case of inequality, the cone's mantle has steps and is therefore not smooth, while in the case of equality, a cylinder results26.

This passage has already been discussed many times, without the factual and historical context being taken into account in every case (see Appendix II). When Otto TOEPLITZ remarks in his lecture on the genesis of infinitesimal calculus on the occasion of the determination of the volume of a cone by Democritus that “from the many things that have been printed on the subject” it is still not clear what Democritus’ method really looked like, then one must say that this situation has not changed to this day.