Fascinating.

Thanks YOS! I'm a history buff so the information is greatly appreciated.

For all we know it was probably a Roman slave that stumbled upon it!

The kind of cement the Romans stumbled on was a natural material. There is no science in its discovery (not "invention"), just dumb luck. They had no idea how it worked and had not much curiosity on the matter. The Romans were pragmatic to a fault. Nature might be imitated or placated, but probably not understood.

"The Romans had a grave lack of curiosity about the natural world"

. . .when you can make 100 slaves plow the land with rudimentary instruments instead of inquiring on how to improve said rudimentary instruments, yes, the Romans surely lacked curiosity. . . with maybe the exception of the invention of cement. :)

As evolutionary biologists did when they thought the case for eugenics had been well-enough made.

Eugenics was never a scientific theory. It was a proposed application of scientific theory.

It's called metonymy. Sometimes they were trying to avoid being dipped in tar and used to illuminate Nero's banquets. It wasn't always lions, per se.

when the scientific community on the whole thinks the case for the new theory has been well enough made.

Sure, but then they can't speak all virtuous-like over being "empirical" when they are really being Platonic number woo-woos. As Feynmann pointed out, the existence of a term in a mathematical equation does not obligate the physical universe to correspond to it. There's nothing especially wrong with the belief that the order of mathematics is replicated in the order of physical things; but you sorta kinda be either a Neopythagorean woo-woo or you have to believe (as the Chinese did not) in an organizing intellect who "orders all things by number, weight, and measure."

in that case, anyone who contradicted Kepler was contradicting Newton

Not exactly. That's like saying that denying epicycles is the same thing as denying retrograde motion. But to quote our buddy Aquinas:
"The theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained; not, however, as if this proof were sufficient, forasmuch as some other theory might explain them."
-- Summa theologica, I, q.32, a.1, ad. 2
or
The suppositions that these astronomers have invented need not necessarily be true; for perhaps the phenomena of the stars are explicable on some other plan not yet discovered by men
De coelo, II, lect. 17
Which is a nice way of saying that theories about nature are inherently falsifiable; or as the modern analytical philosophers say, scientific theories are "underdetermined." It is always possible to explain the same set of data with more than one theory, even if one theory is pretty and dances well and you take her to the prom.

depends on what you mean by "unproven."

If a theory predicts stellar parallax but not can be observed, then the theory is not yet proven. In fact, it teeters on having been disproven. If to save the theory you then claim that the stars are much farther off but you present no evidence in support, that second hypothesis is unproven. Theories of course can become fashionable and even widely accepted well before they have been proven in an empirical, scientific sense, as Feyerabend noted in his studies of how science is actually done, as opposed to the theory of how science is done.

But neither is there intellectual virtue in parroting "unproven" when the scientific community on the whole thinks the case for the new theory has been well enough made.

As evolutionary biologists did when they thought the case for eugenics had been well-enough made. Or as the Aristotelian physicists did when they thought the geomobile theories had been thoroughly falsified by the evidence.

In the first several centuries, Christians were trying to avoid lions.

What is your observational evidence for that?

It was a masterful feat of mathematical physics, spoiled only by the fact that the diameters of the stars were illusions caused by spherical aberration. But Airy did not establish this until the 1830s, so we can't really fault Tycho.

Airy didn't establish it. The phonemonon had been noticed earlier. What he did was explain why it happened.

We can't blame Tycho for being unaware of the limitations of telescopes, but spherical abberation isn't the only reason stars look bigger than they should. Atmospheric turbulence also increases their apparent size. The first accurate measurement of apparent stellar diameter, compensating for both turbulence and aberration, wasn't made until the 1920s.

Also, you can't save one unproven hypothesis by throwing in another unproven hypothesis

The depends on what you mean by "unproven." Scientific proof doesn't work like mathematical proof. The resolving power of Tycho's instruments, according to the numbers you provided, was roughly 20 arcseconds. A counterargument to heliocentrism based on the assumption that anything he saw had to be at least that size would have been unjustified.

The Keplerian model . . . popped out mathematically from Newton’s theory like Athena from the brow of Zeus.

And in that case, anyone who contradicted Kepler was contradicting Newton. In scientific principle, there was nothing wrong with that, but if you're going to say Newton was wrong, you need an alternative explanation for all the observational data that support Newton, and your alternative needs to be at least as parsimonious as Newton's theory.

What it comes down to is that our ancestors were not stoopid, and they had good empirical reasons for their conclusions.

I'm not arguing for any kind of ancestral stupidity, but very smart people can be very mistaken, even when they have had the best scientific training available in their time.

I agree (if that's a point you're trying to make) that people who accept a new theory before it is sufficiently established by observational evidence are not more intellectually virtuous than those who stick with the old theory for as long as they can. But neither is there intellectual virtue in parroting "unproven" when the scientific community on the whole thinks the case for the new theory has been well enough made.

Procyon and Saturn had about the same brightness and diameter to the eyeball astronomers. There is a simple Euclidean relationship between distance and apparent size. The farther the object, the smaller it appears. So if Procyon were as far as say 100x the distance of Saturn, it would be of humungous proportion, far bigger than the Sun. In fact, all the stars would be enormous, larger than the solar system. The Sun would be the only bee-bee in a universe of cannon balls.

This struck Tycho Brahe, whose instruments could measure diameters as small as a US quarter at the distance of a football field, as violating Ockham's Razor. Therefore, the stars had to be no farther than maybe 100x the distance of Saturn. But if they were that close, then parallax would be easily visible to his instruments if the earth circled the sun. No parallax was seen. Therefore, the earth did not circle the sun.

It was a masterful feat of mathematical physics, spoiled only by the fact that the diameters of the stars were illusions caused by spherical aberration. But Airy did not establish this until the 1830s, so we can't really fault Tycho. His science was superb.

What it comes down to is that our ancestors were not stoopid, and they had good empirical reasons for their conclusions. We look back and say but the Copernicans were correct (sort of). But this was not at all obvious in 1610; and even by 1660 it was not yet empirically verified. A lot had to be taken on faith and was.

As my old history prof liked to say, You must study the Battle of Salamis as if the Persians might still win. That is, any historical event has to be looked at from the POV of that time, and not with hindsight.

if they were too far away for the parallax, their diameters would project them to be larger than the entire solar system!

I'm not getting the logic of this. Could you explain in more detail, please?