But the Byzantines did contribute a lot to science. Here is a link to a thread with more details: http://www.reddit.com/r/AskHistorians/comments/1k1jso/the_byzantine_empire_often_gets_remarkably_little/

I'm afraid your question is based on a number of misconceptions. Firstly, the Greeks were certainly nowhere near an industrial revolution. They never invented the steam engine either. Hero's aeoliphile was a toy that had as much in common with a 17th century steam turbine as a child's spinning top has with a gyroscope.

Secondly, the decline of the Roman Empire didn't result in the loss of the knowledge of the Romans in the West, the writings of the Classical writers were well known in the west all the way through the medieval period, and were discussed and built on by the scientific minds of the time. There was considerable progress made after the Roman period, in both east and west. The medieval period advanced the knowledge of the Romans. The medievals invented cannons, handheld gunpowder weapons and advanced shipbuilding as well, all things the Greeks and Romans had no idea about.

So the answer to your question is that both East and West had access to all the ancient Greek writings and both made considerable contributions to science. See this article for a brief overview.

Part I:

that didn't do more science

"Science". What is science?

According to the Oxford English Dictionary:

The intellectual and practical activity encompassing the systematic study of the structure and behaviour of the physical and natural world through observation and experiment

The key questions for us are "did the Byzantines actually study the structure and behavior of the physical and natural world?" and furthermore, "did they do it through BOTH observation and experiment?". In all cases, the answer is overwhelmingly YES, and I will try to expound upon these as best as I can here. Note: this is a huge subject, and I can't reproduce everything that I have found out in this essay, but I will be gladly willing to answer any specific questions you may have.

Anyways, one must consider Byzantium's unique situation before delving into the question of their apparent "lack of scientific product". For much of the thousand-year history of the Empire, there was a crisis, and one which at some times appeared more dire than at others. That crisis was that the Byzantines, being the last vestige of old Imperial Rome, were constantly at odds with a world which sought to move on. To make matters worse, the Empire, specifically Constantinople, was the prize that everyone wanted to take, because of its high-culture, its smug attitude, and its mountains of stockpiled gold. For much of its history, Byzantium was under attack, and at various times, the light of the Empire was very nearly extinguished. First and foremost, owing to their martial heritage passed down from the old Romans, the Empire first and foremost sought to survive, and the Byzantines achieved this through advancement of their military. The Byzantine army and navy were the life force of the Empire, and were those two to fail, the whole of the Empire would fall into ruin. You will find that many of the innovations of the Byzantines were in the realm of military science and technology: Greek Fire, flamethrowers, trebuchets, terror grenades, standardized military manuals, and klivania to name a few. These are what they are generally remembered for. The Imperial war machine was one based on majesty, adaptability, discipline, and most importantly, intimidation.

When the Empire was in turmoil, whether due to civil war, or external wars, science and literature on the whole seem to diminish drastically, and so we find that in times such as these very little information survives, especially from the dark days of the 7th-9th Centuries, when the Empire was wracked by numerous dire internal and external troubles, and though the Empire would recover afterwards, there were still eras of decline left in store. A notable exception to this decline in learning is the Palaiologan period, but it is the exception because knowledge and education on the whole had increased in Europe and the Near East by the 13th Century. Anyways, knowing this, we can effectively say that even though the Empire lasted for over a millennium, there were periods where scholarship was severely reduced in favor of the very survival of the Byzantine state.

In contrast, we find that the high points of Byzantine history, where the Empire, due to competent leadership and great military success, allowed for amazing and extensive scholarship, and this was furthered by the favored interests of a number of well-read and learnèd philosopher-Emperors such as Constantine VII Porphyrogennetos. Under leaders that understood the benefits brought about by widespread education, the number of scholars within Byzantium exploded, leading to a "golden age" of learning starting in the late 9th Century and lasting until the sack of Constantinople in 1204. This isn't to say that there wasn't any learning before, but during this span of time, learning on the whole was extensively and drastically returned to levels that rivalled that of old Rome.

But you may ask, "what exactly did the Byzantines do to further their interest in education, and, was it science?". You find, upon reading many primary sources, that education during the high point of the Medieval Byzantine Empire was actually quite impressive. According to Lars Brownworth, as well as the Dumbarton Oaks Symposium, literacy (defined as being able to read and write) within the Byzantine Empire around the time of the Macedonian Renaissance might have been as high as 30-40%. There are numerous mentions within both Imperial works, as well as personal letters of contemporary people, of many children attending primary, secondary, and even at times tertiary education. Michael Psellos himself (an upper-middle class scholar) wrote that his daughter Styliane, even at the age of 6 years was "the best and brightest in her class" and he marvelled at the development of her educational ability throughout her teen years, which implies that many other children attended school as well. The Skylitzes manuscript helps to support this by illustrating Byzantine children and their teacher attending school. Additionally, the fact that Byzantine military manuals were intended to be read by both generals and their officers tells us that a good portion of soldiers were also expected to be able to read, and since they were often drawn from all lots of life, this means that a good percentage of regular people must have also been able to read and write. Of course, there is much other evidence to support this, but I shall not dwell on it.

As for science. Well, there IS a lot of scientific thought that was developed in Byzantium. A lot of the work that I am aware of begins with the great Michael Psellos, who I mentioned before. A proverbial "Renaissance Man" who wrote 500 years before the start of the Renaissance, I have had the fortune of reading some of his scientific works, and by God, are they fascinating. Psellos might be thought of as one of the progenitors of the scientific method. One thing that pervades Psellos' works is a keen desire to support rational thinking and support scientific inquiry with evidence, which contrasted heavily with the religious doctrine of the time. Thankfully, it seems that the Orthodox Church became more lenient during this period when it came to this scientific inquiry, and so Psellos was able to make many of his comments without too much trouble. He does, however, seem to thinly veil his true purpose within his works, which is to support education and reason, rather than dogma. To give you an idea of what I am talking about here is an excerpt from the Chronographia:

"At that time I was in my twenty-fifth year and engaged in serious studies. My efforts were concentrated on two objects: to train my tongue by rhetoric, so as to become a fine speaker, and to refine my mind by a course of philosophy. I soon mastered the rhetoric enough to be able to distinguish the central theme of an argument and logically connect it with my main and secondary points. I also learnt not to stand in complete awe of the art, nor to follow its precepts in everything like a child, and I even made certain contributions of a minor character myself. Then I applied myself to the study of philosophy, and having acquainted myself sufficiently with the art of reasoning, both deductive, from cause to immediate effect, and inductive, tracing causes from all manner of effects, I turned to natural science and aspired to a knowledge of the fundamental principles of philosophy through mathematics."

"If the reader does not find me boring in this and will allow me to go on, I will add to what I have already said concerning my own activities the fact to which I am about to refer will undoubtedly win for me high approval among men of learning, quite apart from all other considerations. And you, who read my history today, will bear witness to the truth of my words. Philosophy, when I first studied it, was moribund as far as its professors were concerned, and I alone revived it, untutored by any masters worthy of mention, and despite my thorough research, finding no germ of philosophy either in Greece or in the barbarian world. I had heard that Greece had a great reputation for philosophy, expressed in simple words and simple propositions, and their work in this field set a standard and criterion for the future. There were some who belittled the simplicity of the Greeks, but I sought to learn more, and as I met some of the experts in the art, I was instructed by them how to pursue my studies in a methodical way. One passed me on to another for tuition, the lesser light to the greater, and he again recommended me to a third, and he to Aristotle and Plato. Doubtless my former teachers were well-satisfied to take second place to these two."

Join us over at /r/byzantium and you can read a lot more on the subject...