Taisbak has already written two other handbooks for Euclid: "Division and Logos: A Theory of Equivalent Couples and Sets of Integers, Propounded by Euclid in the Arithmetical Books of the Elements", a commentary on Elements VII-IX which sorts out what it means to say that one number is "parts" of another, and
Coloured Quadrangles (Opuscula Graecolatina)
, on Elements X.
The extant works attributed to Euclid are the Elements, the Data, the "On Divisions"
Euclid's Book on Division of Figures: With a Restoration Based on Woepcke's Text and on the Practica Geometriae of Leonardo Pisano
, the Optics, the Phaenomena
Euclid's Phaenomena: A Translation and Study of a Hellenistic Treatise in Spherical Astronomy (History of Mathematics)
, the "Sectio Canonis"
Greek Musical Writings: Volume 2, Harmonic and Acoustic Theory (Cambridge Readings in the Literature of Music)
.
In this book Taisbak translates Euclid's Data, along with the commentary of Marinus of Neapolis, and gives substantial explanation. The Data is a book of theorems showing that if some things are given in some sense then some other things are given in some sense (data=dedomena=givens): "That an object is given to us means that it is, in some relevant sense and scope, put at our disposal. In mathematics, however, the term given is used in an idiomatic way about conditions at the outset of a problem. Often the Given is, at the same time, thrust upon you, not to get rid of, so that you must use that object and obey that relation and no others."
From the start the Data assumes the notion of ratio of magnitudes, which is worked out in Elements V, although Taisbak says that the notion of magnitude used here may be earlier than the Eudoxan theory in Elements V. The right audience for the Data are people who have mastered Elements I-VI and who want a deeper understanding of ancient Greek mathematics, especially how mathematical objects were thought about.
Jumping around the translation reading the statements and proofs, it all feels weird and opaque; but since this was written by the author of the Elements or at least taken at the time to be written by him, the weirdness and opacity show that I do not have as much intimacy with Greek mathematics as I would like to have. It would be a worthy project meticulously to read the translation and Taisbak's commentary and to write an exposition of the Data. Probably more than any other work in Greek mathematics, the Data would benefit from an exposition by someone with deep knowledge of Greek philosophy and logic like Jonathan Barnes.
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