Ammonius himself was a pupil of Proclus and Eutocius dedicated his commentary on Book I of Archimedes' On the sphere and cylinder to him. Ammonius's main work throughout his life was writing critical works on Aristotle and this clearly influenced Simplicius who was himself to write extensive commentaries on Aristotle.
After studying under Ammonius in Alexandria, Simplicius went to Athens where he studied with the Neoplatonist philosopher Damascius. Damascius had written Problems and Solutions about the First Principles which develops the Neoplatonist philosophy as expounded by Proclus. Again Simplicius was exposed to similar views to those he had learnt in Alexandria and his philosophy was built up in a consistent way.
Damascius had been made head of Plato's Academy in about 520 and he was still head when the Christian emperor Justinian closed it in 529. At the same time Justinian closed all the other pagan schools. When Justinian had become emperor, his troops were fighting on the Euphrates River against the armies of the Persian king. It was therefore natural that Damascius, Simplicius and five other members of the Academy, when forced out of Athens, went to Persia to serve at the court of the Persian king Khosrow I. Khosrow was a great patron of culture and Simplicius was well received by the ruler. However, Khosrow and Justinian signed the Treaty of Eternal Peace which was ratified in 532 which led to Simplicius being able to return to Athens.
It is not entirely clear what the terms of the treaty were in regard to Simplicius and the other philosophers who had gone to Persia. Agathias, the Byzantine poet and author of a history of his own times, wrote of these events after the death of Justinian in 565. He wrote that (see ):-
... the terms of the treaty would have guaranteed to the philosophers full security in their own environment: they were not to be compelled to accept anything against their personal conviction, and they were never to be prevented from living according to their own philosophical doctrine.The accuracy of this view by Agathias has been challenged by Cameron in , however, and things may not have been as easy for Simplicius after his return to Athens as Agathias suggested.
There is some evidence that Simplicius did remain in Athens for the rest of his days, writing and undertaking research but certainly not being allowed to lecture. This is reflected by the nature of his writings which are not of the lecture course style, but instead are carefully constructed interpretations of the writings of Aristotle and in particular they attempt to harmonise the views of Plato and Aristotle.
Of the writings of Simplicius that have survived, it is thought that the earliest is his commentary on Epictetus's Enchiridion which some historians believe was written by Simplicius while he was still in Alexandria. This, however, seems highly unlikely and the argument by Cameron in  that it was written during Simplicius's time in Persia seems much more convincing. Certainly this seems to have been written before the commentaries on Aristotle, the first of which is the commentary on De Caelo. This is followed by Simplicius's commentary on Aristotle's Physics and both these works are particularly important for the history of mathematics.
In the commentary on De Caelo Simplicius gives a detailed account of the concentric spheres of Eudoxus and he also relates the modifications to the theory made later by Callippus. Simplicius is quoting from Eudemus's History of Astronomy in giving these details, but he does not quote directly from that work, rather quoting from Sosigenes (who wrote in the second century AD) who in turn quotes from Eudemus.
In his commentary on Aristotle's Physics Simplicius quotes at length from Eudemus's History of Geometry which is now lost. In particular Simplicius quotes the writing on Eudemus on Antiphon's attempts to square the circle and also the attempts of Hippocrates when he squared certain lunes. Also in this commentary on Aristotle's Physics Simplicius gives important quotations from Geminus's summary of Posidonius's Meteorologica.
Simplicius wrote a commentary on Euclid's Elements which survives in an Arabic translation. This is discussed in  where the author discusses the fact that the commentary does not contain an attempt at a proof of the parallel postulate by Simplicius himself, despite the evidence that indeed Simplicius did attempt such a proof. In  there is a discussion of how Simplicius's attempted proof of the parallel postulate entered Arabic mathematics and was first criticised, then incorporated into a new 'proof' designed to take the criticism into account.
The importance of Simplicius as a commentator is described in :-
[Simplicius] did not overestimate his own contributions but was quite aware of his debt to other philosophers, especially to Alexander, Iamblichus, and Porphyry. He did not hesitate to call his own commentaries a mere introduction to the writings of these famous masters, nor did he cling fanatically to his own interpretations; he was happy to exchange them for better explanations. On the other hand, the work of the commentator is far from being a neutral undertaking or a question of mere erudition; it is chiefly an opportunity to become more familiar with the text under consideration and to elucidate some intricate passages; hence Simplicius's constant concern to obtain reliable documents and to check the historical value of this information...
Article by: J J O'Connor and E F Robertson