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The earliest account I know of a scientific experiment is, ironically, the story of Elijah and the priests of Baal.
The people of Israel are wavering between Jehovah and Baal, so Elijah announces that he will conduct an experiment to settle it - quite a novel concept in those days! The priests of Baal will place their bull on an altar, and Elijah will place Jehovah's bull on an altar, but neither will be allowed to start the fire; whichever God is real will call down fire on His sacrifice. The priests of Baal serve as control group for Elijah - the same wooden fuel, the same bull, and the same priests making invocations, but to a false god. Then Elijah pours water on his altar - ruining the experimental symmetry, but this was back in the early days - to signify deliberate acceptance of the burden of proof, like needing a 0.05 significance level. The fire comes down on Elijah's altar, which is the experimental observation. The watching people of Israel shout "The Lord is God!" - peer review.
And then the people haul the 450 priests of Baal down to the river Kishon and slit their throats. This is stern, but necessary. You must firmly discard the falsified hypothesis, and do so swiftly, before it can generate excuses to protect itself. If the priests of Baal are allowed to survive, they will start babbling about how religion is a separate magisterium which can be neither proven nor disproven.
Back in the old days, people actually believed their religions instead of just believing in them. The biblical archaeologists who went in search of Noah's Ark did not think they were wasting their time; they anticipated they might become famous. Only after failing to find confirming evidence - and finding disconfirming evidence in its place - did religionists execute what William Bartley called the retreat to commitment, "I believe because I believe."
Back in the old days, there was no concept of religion being a separate magisterium. The Old Testament is a stream-of-consciousness culture dump: history, law, moral parables, and yes, models of how the universe works. In not one single passage of the Old Testament will you find anyone talking about a transcendent wonder at the complexity of the universe. But you will find plenty of scientific claims, like the universe being created in six days (which is a metaphor for the Big Bang), or rabbits chewing their cud. (Which is a metaphor for...)
Back in the old days, saying the local religion "could not be proven" would have gotten you burned at the stake. One of the core beliefs of Orthodox Judaism is that God appeared at Mount Sinai and said in a thundering voice, "Yeah, it's all true." From a Bayesian perspective that's some darned unambiguous evidence of a superhumanly powerful entity. (Albeit it doesn't prove that the entity is God per se, or that the entity is benevolent - it could be alien teenagers.) The vast majority of religions in human history - excepting only those invented extremely recently - tell stories of events that would constitute completely unmistakable evidence if they'd actually happened. The orthogonality of religion and factual questions is a recent and strictly Western concept. The people who wrote the original scriptures didn't even know the difference.
The Roman Empire inherited philosophy from the ancient Greeks; imposed law and order within its provinces; kept bureaucratic records; and enforced religious tolerance. The New Testament, created during the time of the Roman Empire, bears some traces of modernity as a result. You couldn't invent a story about God completely obliterating the city of Rome (a la Sodom and Gomorrah), because the Roman historians would call you on it, and you couldn't just stone them.
In contrast, the people who invented the Old Testament stories could make up pretty much anything they liked. Early Egyptologists were genuinely shocked to find no trace whatsoever of Hebrew tribes having ever been in Egypt - they weren't expecting to find a record of the Ten Plagues, but they expected to find something. As it turned out, they did find something. They found out that, during the supposed time of the Exodus, Egypt ruled much of Canaan. That's one huge historical error, but if there are no libraries, nobody can call you on it.
The Roman Empire did have libraries. Thus, the New Testament doesn't claim big, showy, large-scale geopolitical miracles as the Old Testament routinely did. Instead the New Testament claims smaller miracles which nonetheless fit into the same framework of evidence. A boy falls down and froths at the mouth; the cause is an unclean spirit; an unclean spirit could reasonably be expected to flee from a true prophet, but not to flee from a charlatan; Jesus casts out the unclean spirit; therefore Jesus is a true prophet and not a charlatan. This is perfectly ordinary Bayesian reasoning, if you grant the basic premise that epilepsy is caused by demons (and that the end of an epileptic fit proves the demon fled).
Not only did religion used to make claims about factual and scientific matters, religion used to make claims about everything. Religion laid down a code of law - before legislative bodies; religion laid down history - before historians and archaeologists; religion laid down the sexual morals - before Women's Lib; religion described the forms of government - before constitutions; and religion answered scientific questions from biological taxonomy to the formation of stars. The Old Testament doesn't talk about a sense of wonder at the complexity of the universe - it was busy laying down the death penalty for women who wore men's clothing, which was solid and satisfying religious content of that era. The modern concept of religion as purely ethical derives from every other area having been taken over by better institutions. Ethics is what's left.
Or rather, people think ethics is what's left. Take a culture dump from 2,500 years ago. Over time, humanity will progress immensely, and pieces of the ancient culture dump will become ever more glaringly obsolete. Ethics has not been immune to human progress - for example, we now frown upon such Bible-approved practices as keeping slaves. Why do people think that ethics is still fair game?
Intrinsically, there's nothing small about the ethical problem with slaughtering thousands of innocent first-born male children to convince an unelected Pharaoh to release slaves who logically could have been teleported out of the country. It should be more glaring than the comparatively trivial scientific error of saying that grasshoppers have four legs. And yet, if you say the Earth is flat, people will look at you like you're crazy. But if you say the Bible is your source of ethics, women will not slap you. Most people's concept of rationality is determined by what they think they can get away with; they think they can get away with endorsing Bible ethics; and so it only requires a manageable effort of self-deception for them to overlook the Bible's moral problems. Everyone has agreed not to notice the elephant in the living room, and this state of affairs can sustain itself for a time.
Maybe someday, humanity will advance further, and anyone who endorses the Bible as a source of ethics will be treated the same way as Trent Lott endorsing Strom Thurmond's presidential campaign. And then it will be said that religion's "true core" has always been genealogy or something.
The idea that religion is a separate magisterium which cannot be proven or disproven is a Big Lie - a lie which is repeated over and over again, so that people will say it without thinking; yet which is, on critical examination, simply false. It is a wild distortion of how religion happened historically, of how all scriptures present their beliefs, of what children are told to persuade them, and of what the majority of religious people on Earth still believe. You have to admire its sheer brazenness, on a par with Oceania has always been at war with Eastasia. The prosecutor whips out the bloody axe, and the defendant, momentarily shocked, thinks quickly and says: "But you can't disprove my innocence by mere evidence - it's a separate magisterium!"
And if that doesn't work, grab a piece of paper and scribble yourself a Get-Out-of-Jail-Free card.