Michael,
Simply put, the equation is: 1.17 times the square root of your height of eye equals the distance to the horizon (in nautical miles). To figure the max distance you would expect to see an object before it's obscured by the horizon, you have to perform the equation twice (using first the height of your eye and then the height of the object you're looking at) and add the two results together.
As far as the scrap log is considered, I can tell you from personal experience that the scrap log is only used as a rough draft, if you will, until the relevant information is transferred to the ship's log; hence, the name "scrap." Oftentimes, the quartermaster of the watch enters the entries into the scrap log. Since the ship's log is a legal document, it is the responsibility of the Captain, who normally delegates that responsibility to the Chief Officer in his capacity as the Senior Watch Officer, to smooth those entries into an official record of events. That, in itself, explains why the scrap log exists...there is too much error that can be introduced into a log at the time of entry, so the scrap log was adopted to capture the initial impressions, which would later be smoothed for the legal record. After the Chief completes a day's worth of entries, he usually tosses out the scrap log. The reason? There can only be one ship's record.
Now, all I am saying is that there's nothing nefarious about the scrap log being thrown away. There's nothing in this explanation that would prevent a Master or his Chief from excluding or altering entries while smoothing the official log, but then he would also have to bring the watch officers into collusion, for the latter are required by the legal nature of the document to put their signatures to the entries for their respective watches. If you wanted to make the argument that Lord or Stewart altered the log to suit their purposes and then coerced the watch officers to stay silent about the changes, then there's nothing I could say that would disprove that. However, you would have to prove that allegation, and I think that would be hard to do. I am aware of the closed-cabin meetings that Lord had with his officers on the return to Boston, but I'm not aware of any proof as to what they discussed.
Parks