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Domninus was a Syrian, and by religion a Jew, who was born in the town of Larissa (often identified with Laodicea but probably a separate town) on the Orontes River. He went to Athens where he became a pupil of Syrianus who was the head of Plato's Academy there. Proclus, although slightly older than Domninus, was also a pupil of Syrianus at the Academy at the same time. Marinus, who was later a pupil of Proclus and eventually took over as head of the Academy following Proclus, writes about a rivalry between Domninus and Proclus [1]:
[Syrianus] offered to discourse to them on either the Orphic theories or the oracles; but Domninus wanted Orphism, Proclus the oracles, and they had not agreed when Syrianus died...
If at first Domninus and Proclus were merely student rivals, certainly it grew into a more serious disagreement centred on how Plato's philosophy should be interpreted. This serious disagreement saw Proclus come out as the victor in the sense that the Academy preferred his views. Proclus succeeded Syrianus as head of the Academy in Athens while, a short while later, Domninus left Athens and returned to his home town of Larissa.
The mathematical work of Domninus only came to light after the publication of his Manual of Introductory Arithmetic in 1832, and its importance was not realised until Paul Tannery began publishing a number of works on Domninus in 1884. Although Domninus wrote a number of books the only other known in detail is How to take a ratio out of a ratio which was not published until 1883 in [3].
The Manual of Introductory Arithmetic studies numbers, means, and proportion. The book is in five parts [1]:
... an examination of numbers in themselves, an examination of numbers in relation to other numbers, the theory of numbers both in themselves and in relation to others, the theory of means and proportions, and the theory of numbers as figures.
At the end of the book Domninus says that he intends to treat some of the subjects more fully in Elements of Arithmetic but it is not known if he ever wrote it! Certainly he would not be the first or last mathematician to refer to a future work which never materialised. Heath [2] writes of the Manual of Introductory Arithmetic :
It is a sketch of the elements of the theory of numbers, very concise and well arranged, and is interesting because it indicates a serious attempt at a reaction against the Introductio arithmetica of Nicomachus and a return to the doctrine of Euclid.
The second book, How to take a ratio out of a ratio, published in translation in 1883, studies manipulation of ratios into other forms. Heath casts some doubt as to whether this book is actually by Domninus. He writes [2]:
... if it not by Domninus, it probably belongs to the same period.
BulmerThomas in [1] is more certain that it is by Domninus and conjectures that the work was, at least in part, work by Domninus towards the Elements of Arithmetic which he had promised to write.
We do have some indications of the character of Domninus, but these may be very unfair since they are related to us by Damascius, the last head of the Academy. Since Domninus's philosophy was considered oldfashioned and out of favour by the Academy the claims made by Damascius may have been aimed at discrediting him. Damascius wrote that when Domninus was an old man he [1]:
...loved only the conversation of those who praised his superiority and that he would not admit to his company a young man who argued with him about a point in arithmetic.
Article by: J J O'Connor and E F Robertson
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School of Mathematics and Statistics University of St Andrews, Scotland  
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