My question is similar, if not identical, to this one, but I don't find the answer satisfying, given the context of experiment.
First, here is an outline my understanding of the motivation behind the experiment:
Galilean invariance states that the laws of motion are identical across inertial frames of reference.
However, it was also known (or suspected), that the speed of light is independent of the speed of its source (as is the case in sound waves).
This presented the apparent opportunity to violate Galilean invariance, and the Michelson Morley experiment is a famous example of an attempt to do just this.
Based on the assumptions and working knowledge of the scientists of the time, if the interferometer was traveling through the ether, then the interference pattern should differ when the contraption was rotated, and this would confirm that the contraption was moving with respect to the ether (i.e. with respect to absolute space), and allow a calculation of the velocity of the Earth (assuming the contraption and the Earth were in the same inertial frame).
The key motif of the Lorentz transformation can be derived from this analysis, which is based upon the Pythagorean theorem:
Here is a schematic of the Michelson-Morley apparatus, borrowed from here
The idea here is that, since we had not yet discovered that the speed of light is invariant across all observers within and across all inertial frames, we would expect two particular round trip times for both beams of light, and these times would depend upon which direction the contraption was moving, and at what speed.
I can fully follow and understand the calculation for the B - E' - B' roundtrip.
I can also fully follow the calculation for the B - C' - B' roundtrip, but I can't understand it. In particular, I don't understand why the first half of this round trip strikes the center of the mirror C'. Remember, those scientists already understood that the speed of light is independent of the speed of its source, so it wouldn't be like bouncing a ball against the roof of a moving vehicle, where the ball inherits the forward speed of the vehicle. Rather, the beam should strike a point behind the center of C' (i.e. to its left). In other words, to an outside observer, the beam would be expected to go straight up and down.
The only way I can reconcile this is that Feynman here is not talking about a laser beam, but rather a point source where the light spreads out in all directions. And in that case, the analysis is done for the particular photon whose angled path through space was such that it struck the center of C'. However, I don't think this is the correct answer to my question, as the same issue crops up later in the discussion of a moving light clock that involves a single photon.
Another way of asking this question is as follows:
According to the way of thinking at the time of Michelson-Morley
If I were to aim a laser beam at a very distant target, and both myself and the target were moving rapidly in a direction perpendicular to the line between myself and the target, then if the distance between myself and the target was great enough, then, by the time the laser beam reaches the target, it will have dodged the laser beam.
Yet, according to my reading of Feynman, their expected calculations imply that the light is carried along by the velocity of the source, which contradicts what they apparently already knew at the time.
What am I missing here?