Toolbox

August 17, 2013

Toolbox is a term which I (Ken Saito) invented in 1994, to indicate the set of propositions and techniques in Greek mathematics which was memorized by Greek mathematicians and used in their investigations.

Reading Greek mathematical texts, one often comes across some apparently fundamental propositions which seem very useful for demonstration of later propositions (e.g., Elements VI.23), but they are never used again, while other propositions whose purpose is not clear for our modern mind are repeatedly used later (e.g., Elements VI.14-17). Then, the Greek mathematicians must have classified the propositions in a different way, and it is interesting to know the set of propositions used by Greek mathematicians for their research. This set must be different what we would gather reading through the Elements (and other elementary works).

So much was my idea in the beginning (see below for further arguments about toolbox by my colleagues), and I began by making an index of the propositions and assumptions used in every step of the demonstration in the first four books of Apollonius' Conics, and book 7 of Pappus' Collection. Here is the result (of quite a few years ago). I still hope to add indices for other major works.

Index of Propositions used in the Conics

Index of the Propositions Used in Book 7 of Pappus' Collection

Further arguments about the "Toolbox"

• Reviel Netz, The Shaping of Deduction in Greek Mathematics. Cambridge University Press, 1999. p. 216ff.
• Karine Chemla, "Mathematical proof: a research programme." p. 30ff., in Chemla ed.The History of Mathematical Proof in Ancient Traditions. Cambridge University Press, 2012. pp.1-68.