# Eutocius of Ascalon

### Quick Info

Born
Ascalon (now Ashqelon), Israel
Died

Summary
Eutocius was a Greek mathematician who wrote commentaries on works of Archimedes and Apollonius.

### Biography

Eutocius of Ascalon was for a long time thought to have been born in 530. It is instructive to see how this came about for it shows how many pitfalls there are in the study of history. Eutocius wrote commentaries on three works of Archimedes. His commentary on Book II of On the Sphere and Cylinder ends with the statement:-
... the edition was revised by Isidorus of Miletus, the mechanical engineer, our teacher.
From this it was thought that Eutocius was a pupil of Isidorus and his dates were deduced from this information. However, further investigation showed that this contradicted other information such the dedications that Eutocius makes in some of his other works. It was then realised that the comment at the end of Eutocius's commentary to Archimedes' On the Sphere and Cylinder was inserted by a later editor of the work who was indeed a pupil of Isidorus of Miletus. It is thought that the first of Eutocius's commentaries on Archimedes was written around 510.

Ascalon, where Eutocius was born, had a long history and is mentioned in the Old Testament as Askelon. It received its Greek name after it was conquered by Alexander the Great in 332 BC and the city had fine public buildings built by Herod the Great. It seems likely that Eutocius left Ascalon and went to Alexandria to study.

Paul Tannery has argued convincingly (see [5]) that Eutocius was almost certainly a pupil of Ammonius in Alexandria and it appears that he went on to become head of the Alexandrian School after Ammonius. Ammonius himself was a pupil of Proclus and Eutocius dedicates his commentary on Book I of Archimedes' On the sphere and cylinder to him. Eutocius addresses Ammonius in the preface asking him to [1]:-
... bear with him if he should have erred through youth. He explains that he found no satisfactory commentaries on Archimedes before his own time and promises further elucidation of the master if his work should meet with approval from Ammonius.
Certainly this reads as if Eutocius is addressing his teacher and Paul Tannery's deduction seems secure. One has to assume that indeed Ammonius did approve, for Eutocius went on to write commentaries on other works by Archimedes, namely Measurement of the circle and On plane equilibria. However, Bulmer-Thomas in [1] is convinced that Eutocius did not know of certain other works by Archimedes such as Quadrature of a parabola and On spirals for he claims that he would have referred to them at certain natural places in his commentaries rather than give much less suitable references at these points.

Eutocius also edited and wrote commentaries on the first 4 books of the Conics of Apollonius. Heath writes [2]:-
Eutocius's commentary on Apollonius's "Conics" is extant for the first four Books, and it is probably owing to their having been commented on by Eutocius, as well as to their being more elementary than the rest, that these four Books alone survive in Greek.
One sees immediately that commentators such as Eutocius are very important in the history of mathematics and many important works have only survived due to the work of the commentators.

Eutocius does not appear to have done any original work. However, his commentaries contain much that is invaluable in the nature of historical information which might otherwise have been completely lost. Heath lists some of these important pieces of information [2]:-
1. the account of the solutions of the problem of the duplication of the cube, or the finding of two mean proportionals, by Plato, Heron, Philon, Apollonius, Diocles, Pappus, Sporus, Menaechmus, Archytas, Eratosthenes, Nicomedes;
2. the fragment discovered by Eutocius himself containing the missing solution, promised by Archimedes in On the Sphere and Cylinder Book II. 4, of the auxiliary problem amounting to the solution by means of conics of the cubic equation $(a - x) x^{2} = b c^{2}$.
3. the solutions (a) by Diocles of the original problem of II.4 without bringing in the cubic, (b) by Dionysodorus of the auxiliary cubic equation.
As to contributions to astronomy, Eutocius did write an introduction to Book I of the Almagest but Neugebauer writes [3]:-
Eutocius did not write a "commentary" of the ordinary type that follows a given text chapter by chapter. ... the main part concerns methods of sexagesimal computation: multiplication, division, square roots etc. Another chapter concerns isoperimetric problems, followed by a short section about the shape and size of the earth, based on Ptolemy's norm of 500 stades for the equatorial degree. Obviously nothing of real astronomical interest has come down from Eutocius.

### References (show)

1. I Bulmer-Thomas, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
3. O Neugebauer, A history of ancient mathematical astronomy (New York, 1975).
4. R Lorch, The Arabic transmission of Archimedes' 'Sphere and cylinder' and Eutocius' commentary, Z. Gesch. Arab.-Islam. Wiss. 5 (1989), 94-114.
5. P Tannery, Eutocius et ses contemorains, Bull. des sciences mathématique 7 (1883), 278-291.