NavList:

Alex Eremenko, you wrote:
"Kepler's laws are obeyed exactly by a system of two bodies which attract each other by the inverse square law."

Of course, and we owe that understanding to Isaac Newton. Horrocks certainly didn't know that. 

"So all deviations must be explained  by gravitational effect of other bodies."

Yes, that's how we understand it today and ever since Newton. What did Horrocks actually say on this matter? He didn't speak of gravitational influence obviously. Secondary sources imply that he favored an "influence" of other objects on the motions of the planets and Moon and correctly speculated that the long-term "equation" (like the "equation of time", really a correction) in the motions of Jupter and Saturn might be due to their mutual "influence" on each other. Even if there was nothing more to it, conceiving this as a physical influence, and not an astrological influence for which there was centuries of precedent, would be a "modern" insight. But did he have anything more --any mathematical model-- to quantify the influence? Was his attitude towards interplanetary "influences" really any different in practice from the astrologers? The author of the article from 1874 that I included in a post yesterday (Dublin, 1874) attributed his anti-astrological attitude to his Puritan upbringing. Religion makes good science?? That's certainly possible.

"And the first success here is due to Horrocks, who built the best theory of Moon's motion at that time."

Again, just trying to read what secondary sources are saying, it sounds like he had the insight to think of the Moon's motion in a "perturbation theory" fashion. The Moon's motion around the Earth is emphatically not an ellipse. But and ellipse is a good starting point. It sounds as if he suggested that we can think of it as an ellipse with varying parameters driven by the influence of other bodies. Did he? Again, that would be a clever and modern insight for its time, and even if that's all there is, it's significant. Did he have anything more? Again I'm asking whether there is any historical evidence of a mathematical model of some sort? It doesn't have to be perfect (science never is), but it certainly has to be better than his theoretical model (a mere fiction, a bit of nonsense) that inspired his calculation of the distance to Venus.

"He also explained perturbations of Saturn's orbit by Jupiter by gravitation."

He suggested a physical force instead of some more "occult" influence, right? Isn't it putting words in his mouth to say that he understood it as gravitation? Did he suggest that the mutual influence of Saturn and Jupiter on each others' orbits was the same force that caused ground-based gravity (and yes, falling Newtonian apples)? Did he suggest that the influence of the planets on each other was proportional to some innate property of each, comparable to "mass"? Did he build anything like a real mathematical model where the interplanetary influence decreased with distance (even if not inverse-square)? Without some elements of each of those features, he wasn't talking about gravitation. 

"And tides as well"

I have understood from the broad brush accounts of his life that Horrocks was interested in the tides, and certainly the concept of a "magnetic" influence of the Moon on the Earth and its oceans was "popular" among natural philosphers (in that era "magnetic" should probably be understood as some "action at distance" and not literally "magnetic" in the modern sense). But what did Horrocks do?? What did he know about, what did he suggest about the tides, that was not relatively common knowledge? Horrocks gets not even a footnote (that I could find!) in Cartwright's "Tides: A Scientific History" (borrowable at archive.org). Cartwright discusses in some detail the models of Galileo and the ideas of Kepler regarding the tides, as well as lesser known "philosophers". He even quotes the historian known to moderns as the "Venerable Bede" (c.700 AD) who is probably most famous for his accounts of the Viking attacks on England. Until I read it in Cartwright, I had no idea Bede had written about the lunar influence on the tides. Cartwright's treatise borders on being annoying for its focus on minor details and events in the history of the subject --which is why it seems to me that Horrocks should be in there if his work had more than passing significance. That's some very shaky "secondary source logic" that I'm applying, and of course it may well be just plain wrong. Please feel free to knock it down like a house of cards if there's any real evidence. Does anyone have access to what Horrocks actually wrote about the tides? I couldn't find anything, though I admit I didn't spend much time on it (I was focused on the transit originally). Were his thoughts on the tides unusual, worthy of note, beyond enhancing his legend in the years since he exploded back into popular science legend-making in 1874? Better yet, setting aside his "thoughts", did he calculate anything? Did he actually do any math regarding the tides?