The Messenger Lecture about The Law of Gravitation:
In the first part and the early second part, he explains why science is superior over social science and why he would focus on Nature rather than Man. Tycho de Brahe is quoted as the originator of science: his idea was to record the observed positions so accurately that different detailed hypotheses could be compared and/or excluded.
The third part is already dedicated to Kepler's laws. He mentions some theories of that time - that angels were pushing the planets. The theories were almost right, RPF says, except that the angels are actually sitting on another side and are pushing the planets radially. :-)
Applause follows. Of course, the discussion turns to Galileo and inertia. Newton added the forces that can change the straight lines to different paths. The universal law of gravity evolves naturally. A lot of dramatic predictions were made - parameters of orbits, tides, etc. High-precision tests followed: the moons of Jupiter. The first discrepancies (moons ahead of the schedule) were explained by the finite speed of light.
In the fourth part, he explains more examples how new laws can be found by taking the old ones serious: the Neptune was predicted and discovered by the 2nd observatory (that didn't share the arrogant and dumb anti-theorist attitude of the 1st one). ;-) Finally, the Mercury anomalies were fixed by Einstein.
The fifth part discusses how gravity holds clusters of stars together: gravity extends very far. For galaxies, he has no way to check but he has no doubt ;-) that the inverse square law will continue to hold (and no dark matter would be needed haha). Gravity acts on clusters of galaxies, too. Still, gravity doesn't have too many applications, except for scholars' jobs and astrologers. :-) Feynman asks whether he attracts a viewer physically. :-)
In the sixth part, Cavendish-like short-distance tests of gravity are being reviewed. Newton's constant was determined for the first time. Equivalently, the masses of celestial objects - in "small" units from everyday life - were measured. The Eötvös experiments to test the equivalence principle are reviewed. He continues with the electrostatic law that has the same inverse square law and argues that the electro-gravitational unification had not worked so far. The huge discrepancy in their strength (between electrons) is picked as an argument against their unification.
Well, this is pretty much what we call the hierarchy problem these days. The last, seventh part continues with this mystery and suggests that the quintessence solutions don't work. More importantly, he mentions quantum gravity. Feynman says that there was no theory of quantum gravity at that time. He promotes the second lecture, on the link between maths and physics (see the URL below), by suggesting that some people may prefer a "mechanism" over maths.
The laws are often known incompletely. However, Feynman concludes by saying that gravity is pretty because its laws are simple in principle (not necessarily in their manifestations) and universal. Nature only uses the longest strings to weave the patterns. So each small piece of the fabric of the cosmos reveals the organization of the entire tapestry.
And that's the memo.
See also: RPF on the arrow of time, the relationships between maths and physics, fourty-six 1979 lectures, Pleasure of finding things out, and Feynman's 90th birthday where additional links appear.