Resuming the cliffhanger: the breakdown of classical physics
Shift from the classical to the quantum paradigm
Light is in individual packets of energy whose size is keyed to the frequency of the light.
This is the solution to the blackbody problem: the mathematics of emission of quanta of light energy produces the well-behaved curve with a peak at a given color that we see for hot objects, whether the Sun, iron in a forge, or a light bulb filament
It is also the solution to the photoelectric effect: light of high enough frequency is needed for the individual light quanta to add enough energy to eject electrons. Apparently the atoms can only interact with these photon quanta one at a time; you cannot add multiple red photons to eject an electron, but rather you need a single blue one.
The spectra of the Sun (with absorption lines) and energized gases (like neon or sodium lamps, with emission lines) turned out to be compatible with this quantum theory as well. The structure of the atom itself must be quantized, and electrons must live in well-defined energy shells; when they move from one to another, they emit or absorb photons of a well-defined frequency. Quantum theory began to solve problems chemists did not necessarily even realize they had about, e.g., why chemical bonds tend toward exchanging or sharing electrons so as to reach the number 8 in the outer shell.
Still, even with all this evidence, as the saying went, “Science progresses funeral by funeral.” Many older scientists stayed in the old paradigm to the end of their careers, whether as recalcitrants who refused to even believe in the new paradigm, or perhaps more often as castaways adrift in the new sea, clinging to the old research programs they were comfortable with and hoping, implicitly, that their work would add up to something that would remain untouched in the new world order their students would inhabit.
 
Religion: why do people believe? Is it reasonable or just arbitrary?
Picture a craftsman in Corinth c. 50 AD/CE.
This guy named Paul shows up in the stall next to you and starts making tents. While he’s not making tents, he’s talking about this Jesus guy with these crazy claims that he’s been killed and then “rose from the dead,” whatever that means.
You don’t think too much of it until the day you watch him grab Alexander the cripple by the hand and he suddenly starts walking! You’ve seen this guy for 15 years sitting there begging…
Yet for all the miracles, do you change your beliefs?
It helps that Paul is talking about this Jesus as the Son of, not some Greek god few of you really believe in any more, but some transcendent God that sounds a lot more like Plato’s Form of the Good or Aristotle’s Prime Mover.
It helps a lot more that you see changes in yourself as you listen to Paul: the things you’ve done and the things you’ve suffered make more sense. You want the forgiveness and the joy that Paul says this Jesus brings.
 
All of which is to say, in both scientific paradigm shifts and religious conversions, it takes a convergence of falsified old predictions, verified new predictions, and the ability to fit things that still work from the old paradigm into the new paradigm for the paradigm shift to take place.