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As for the logic of why it keeps the power on from that point forward is anyone's guess. Maybe it engages another generator system underwater instead of the lighthouse. Solving the puzzle is also delivering power to open the desk with the linking book, so it's more than just turning on the underwater lights. Like I said, it's probably engaging some steady power source.
As a first try, I would press the buttons before the one which you pressed, so the 16th (in case the game decides to floor the angle value, rather than rounding/ceiling it).
Otherwise, I think you did good. A graduation step is 11.25° (8 gradutation steps for 45°). Basically, yours should be half-circle (180°) plus one graduation step, so 17 in total.
Make sure i) that the angle is correct (because I find it weird that it is not a multiple of 4, ii) that it is indeed the lighthouse generator which is at 190°, and not one of the two masts, iii) that you count the buttons clockwise, iv) that the lights are ON when you press the button.
I tried pressing the 16th and got the same result.
Compass be damned ... 'As above so below... ' junk be damned ... Making a connection with the surveyor and the compass be damned
EDIT: There's nothing to suggest that 'assuming' real world knowledge of how magnetic North and a compass work nor the degrees associated. Along side the reading in the game. The 1st island has references to sail boats, but also references celestial navigation. The constellations.
The only thing suggesting this is the solution itself.
In short... The game suggests the characters of the story could measure Linearly. (the journals)
All of a sudden you're presented with a tool that has degree markings and you go assuming things and your mind fills in blanks the game left blank. So you start thinking 'oooo compass, degrees, headings....'
The 13th button didn't work either.
Bro... just count one by one ... which buttons you've pressed and which you haven't.... Not to mention are you on the original solutions or randomized?
If you count North as 1 and move clockwise counting the buttons ... it is button 13
Also, the arrow in the book points exactly to which button.... The right side arrow.
https://steamcommunity.com/sharedfiles/filedetails/?id=2589301373
If you want to argue that the screenshot isn't an 'exact' solution... the game dev's behavior to previous puzzles such as the clock code... they give you the exact code to use in the game, what they don't do is tell you 'exactly' how to use the levers.
Oh... i see now... your previous comment is hidden and i didn't bother to read redacted junk....
You want a solution that appeals to your intellect when the original was playing psychological mind games....
You're part of the crowd that wants the original solution to appeal to their intellects when point in fact the solution is in game and your intellect unnecessary.
Wait, wasn't it what you previously did? South is the 16th button (and North the 32th button). So one step clock-wise from South was the 17th button.
Or was it the 18th button that worked (two steps clock-wise from South)?
So, it was the 18th, right? It is a bit weird because 190° should fall between the 16th (180°) and the 17th button (191.25°).
Or was it that you were counting with the 0° at East rather than North? Here, the puzzle does not use the trigonometric conventions (0° at East, and anti-clockwise direction of angle), but something akin to a clock (0° at 12 hours, so North, and clock-wise direction of angle).
The game in that level never did the best job of having a trail to arrive at that solution. It took me some time to figure out it was like a protractor, long after I had solved it since I just remembered the right button on reflex(years ago I just did it one by one to find the right button and now instinctively know it).