As far as we know (written history only goes back that far) the scientific method was born well before the Christian era. A broad generalization is we had two main centers of thinking from which science irradiated: China and the Greek-speaking world.
The Chinese approach was more “hands-on”, meaning the Chinese concentrated on sciences with an obvious and immediate practical application (printing press, civil engineering, selective breeding etc). The Greek approach was more “theoretical”, meaning they excelled in studies with required a bit of polishing to be used in practice (mathematics, geometry, physics etc).
The Roman civilization represented the natural evolution of the Greek scientific methods: without Greek mathematics and physics the Romans would never have been able to erect their famous acqueducts and fortifications which stand to this day. Roman engineers and physicians went to Citium, Ephesus, Rhodes, Andros etc to study under Greek-speaking teachers and acquire Greek manuscripts. While the pupil sometimes surpassed the master (the Romans invented the bearing arch which, for all their advanced mathematics, the Greeks weren’t able to) it was obviously a symbiotic relationship.
Science started declining in the West with the decline of the Roman Empire. As capital was getting progressively scarcer and economy moved towards substinence, learning started to slip away. The Dark Ages of Science started by convention in 476AD. This year the great library of Constantius II in Byzantium burned to the ground in an accidental fire. By this date the famous library of Alexandria was a shell of its former self, following the destruction of many irreplaceable volumes by Christian extremists at the close of the IV century hence the library in Byzantium was the foremost place of learning in the West.
The revival of learning started in the West in the IX century in Frankish Europe, Greek Byzantium and Muslim Baghdad, for different reasons. In Europe the Carolingian Renaissance laid particular emphasis on the reform and education of the clergy. In Byzantium it was an effect of the much improved conditions of the IX century and, indirectly, Iconoclasm: we are told in 814AD Emperor Leo V ordered a bibliographic search to be made of patristic testimonia favorable to Iconoclasm and teams of clergymen scoured churches, monasteries and other places of learning for every scrap of book they could find to prove or disprove the Emperor’s claims. Also wealthier Byzantines developed an almost insatiable appetite for learning leading to the rediscovery of endless works of science. In Baghdad, the wealthiest city in the world, patronage became common: we are told that caliph al-Mamum offered Leo the Mathematician, a Greek monk of incredible learning, an enormous sum to move to Baghdad and open a scientific school there. He was however outbid by Emperor Theophilos who set up Leo as a paid teacher of secular learning at the Church of the Forty Martyrs. Leo taught mathematics, astronomy, music and geometry. Theophilos’s successor, Caesar Bardas, particularly distressed by the decline of secular science, promoted Leo as the head of a new istitution whose sole scope was teaching secular sciences, the Magnaura. The Magnaura was the first scientific istitution we know about in the West: schools in Frankish Europe mostly concerned themselves with theology and teaching Latin. The Magnaura instead concerned itself only with scientific matters (astronomy, geometry etc) and had a chair of “grammar” whose scope was to teach pupils how to read classic Greek from ancient manuscripts.
After the crisis of the V-VIII century scientific learning stopped declining in the West. For all the highs and lows the financial and social situation kept improving through the centuries. Periods of renovated scientific interest usually coincide with periods during which capital is more abundant and average living conditions improve. For example when Byzantium was reduced to a shell of its former self, the wealthy merchants of Florence stepped in. The same can be said about Industrial England or Merchant Holland. The only exception to this is Spain. Despite being flooded with American silver and having some of the best schools in the World (an heritage of the Moorish occupation), Spain only made marginal contributions to the Western scientific thinking and learning and most of these were relegated to the fields of medical expertise and economic thinking.