Mathematical discussion of rangekeeping (English Wikipedia)

The plotting was either done manually or by a mechanical plotter in the case of the Dreyer Table.
"Dreyer Fire Control Table". The Dreadnought Project. Retrieved 2006-10-30. The bearing clock was primarily used within the dumaresq, and it could allow a constant bearing rate to be dialed in ... The range clock's constant speed output went into a differential device called the Spotting Corrector, whose gearing multiplexed it out to three further destinations.

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"Chapter 19: Surface Fire Control Problem". Naval Ordnance and Gunnery. Annapolis, MA: United States Naval Academy. 1958 [1950]. NavPers 10798-A. Retrieved 2006-08-26.
"Chapter 20: Gun Directors". Naval Ordnance and Gunnery. Annapolis, MA: United States Naval Academy. 1958 [1950]. NavPers 10798-A. Retrieved 2006-10-15.
"Chapter 19C1". Naval Ordnance and Gunnery: Volume 2 Fire Control. Bureau of Naval Personnel. 1958 [1950]. NavPers 10798-A.
To some extent, the accuracy of this approximation depends on the time of flight of a projectile. At maximum range, the 16-in guns of an Iowa-class battleship had a time of flight of ~95 seconds. Ships engaging at this range would be very unusual. The longest successful between two warships occurred at ~26,500 yards. The projectile in this case would have had a time of flight of ~40 seconds (based on the 16 inch guns of the Iowa class).
A.A. Range Table for 16-inch 50 Caliber Guns. Bureau of Ordnance. 1944-03-11. Ordnance Pamphlet No. 1091. Retrieved 2006-10-29.
"Chapter 17". Volume 2 Fire Control (1958 ed.). Annapolis, MA: United States Naval Academy. 1958 [1950]. NavPers 10798-A. Retrieved 2006-08-26.
"Chapter 19". Surface Fire Control Problem (1958 ed.). Annapolis, MA: United States Naval Academy. 1958 [1950]. NavPers 10798-A. Retrieved 2006-08-26.

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Jackson, USNR, Lt.(jg) J.G. (February 1946). Japanese Torpedo Fire Control (PDF). US Naval Technical Mission to Japan. Fascicle O-1, Target O-32. Archived from the original (PDF) on 2007-07-20.

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"US Army Field Manual 6-40 Chapter 7". US Army. 2003-09-09. Retrieved 2006-10-13.

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The torpedo and long range gun rangekeeping functions are different in terms of scale. For example, a torpedo's time of flight is much longer than that of battleship's main gun projectile. The projectile's advantages in terms of speed and rate of fire make ranging shots possible. A torpedo's time of flight is so long that ranging shots are not feasible. Some papers do address these issues.

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The plotting was either done manually or by a mechanical plotter in the case of the Dreyer Table.

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A. Ben Clymer (1993). "The Mechanical Analog Computers of Hannibal Ford and William Newell" (PDF). IEEE Annals of the History of Computing. 15 (2). Retrieved 2006-08-26.

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Bradley Fischer (2003-09-09). "Overview of USN and IJN Warship Ballistic Computer Design". NavWeaps. Retrieved 2006-08-26.
Jurens, William (2004-12-24). "The Evolution of Battleship Gunnery in the U.S. Navy, 1920-1945". Navweaps.com. Archived from the original on 2006-11-20. Retrieved 2006-10-18. Generally air spot was expected to have little effect at ranges under 20,000 yards, where visual spotting remained supreme. The advantage of air spot increased markedly thereafter. In 1935 the Naval War College estimated that at 29,000 yards air spot would be expected to deliver six times as many hits as observation from spotters aloft.
Norman, Friedman (1981). Naval Radar. Conway Maritime Press. ISBN 0-85177-238-2. Retrieved 2006-10-20. Mk 8 Radar: Range Accuracy: 15 yards (5 m)
Lienau, Peter; Tony DiGiulian (2001-07-16). "German Optical Rangefinders". Navweps.com. Retrieved 2006-10-20. To take another example, the US Battleships of the North Carolina, South Dakota and Iowa classes had main director rangefinders of 25X power with a base length of 26 feet 6 inches (8.0772 m)... For example, to find the error at 20,000 m, simply multiply 0.97 m by 20,000 / 2,000 = 9.7 m.
Bradley Fischer (2003-09-09). "Overview of USN and IJN Warship Ballistic Computer Design". NavWeaps. Retrieved 2006-08-26. If the target's length is known as well as the present range, the operator measures apparent length of the ship in the form of a bearing measurement (using the stern as the reference point). The formula is: inclinometer angle = L × Cos Ø / R, where: L is the length of ship, Ø is target angle and R is present range.
To some extent, the accuracy of this approximation depends on the time of flight of a projectile. At maximum range, the 16-in guns of an Iowa-class battleship had a time of flight of ~95 seconds. Ships engaging at this range would be very unusual. The longest successful between two warships occurred at ~26,500 yards. The projectile in this case would have had a time of flight of ~40 seconds (based on the 16 inch guns of the Iowa class).

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Jackson, USNR, Lt.(jg) J.G. (February 1946). Japanese Torpedo Fire Control (PDF). US Naval Technical Mission to Japan. Fascicle O-1, Target O-32. Archived from the original (PDF) on 2007-07-20.
Jurens, William (2004-12-24). "The Evolution of Battleship Gunnery in the U.S. Navy, 1920-1945". Navweaps.com. Archived from the original on 2006-11-20. Retrieved 2006-10-18. Generally air spot was expected to have little effect at ranges under 20,000 yards, where visual spotting remained supreme. The advantage of air spot increased markedly thereafter. In 1935 the Naval War College estimated that at 29,000 yards air spot would be expected to deliver six times as many hits as observation from spotters aloft.