Ship gun fire-control system
In the director tower, operators trained their telescopes on the target; one telescope measured elevation and the other bearing. Rangefinder telescopes on a separate mounting measured the distance to the target. These measurements were converted by the Fire Control Table into bearings and elevations for the guns to fire on. In the turrets, the gunlayers adjusted the elevation of their guns to match an indicator which was the elevation transmitted from the Fire Control Table—a turret layer did the same for bearing. When the guns were on target they were centrally fired.
The Aichi Clock Company first produced the Type 92 Shagekiban low angle analog computer in 1932. The US Navy Rangekeeper and the Mark 38 GFCS had an edge over Imperial Japanese Navy systems in operability and flexibility. The US system allowing the plotting room team to quickly identify target motion changes and apply appropriate corrections. The newer Japanese systems such as the Type 98 Hoiban and Shagekiban on the Yamato class were more up to date, which eliminated the Sokutekiban, but it still relied on seven operators.
In contrast to US radar aided system, the Japanese relied on averaging optical rangefinders, lacked gyros to sense the horizon, and required manual handling of follow-ups on the Sokutekiban, Shagekiban, Hoiban as well as guns themselves. This could have played a role in Center Force's battleships' dismal performance in the Battle off Samar in October 1944.
In that action, American destroyers pitted against the world's largest armored battleships and cruisers dodged shells for long enough to close to within torpedo firing range, while lobbing hundreds of accurate automatically aimed 5-inch (127 mm) rounds on target. Cruisers did not land hits on splash-chasing escort carriers until after an hour of pursuit had reduced the range to 5 miles (8.0 km). Although the Japanese pursued a doctrine of achieving superiority at long gun ranges, one cruiser fell victim to secondary explosions caused by hits from the carriers' single 5-inch guns. Eventually with the aid of hundreds of carrier based aircraft, a battered Center Force was turned back just before it could have finished off survivors of the lightly armed task force of screening escorts and escort carriers of Taffy 3. The earlier Battle of Surigao Strait had established the clear superiority of US radar-assisted systems at night.
The rangekeeper's target position prediction characteristics could be used to defeat the rangekeeper. For example, many captains under long range gun attack would make violent maneuvers to "chase salvos." A ship that is chasing salvos is maneuvering to the position of the last salvo splashes. Because the rangekeepers are constantly predicting new positions for the target, it is unlikely that subsequent salvos will strike the position of the previous salvo. The direction of the turn is unimportant, as long as it is not predicted by the enemy system. Since the aim of the next salvo depends on observation of the position and speed at the time the previous salvo hits, that is the optimal time to change direction. Practical rangekeepers had to assume that targets were moving in a straight-line path at a constant speed, to keep complexity to acceptable limits. A sonar rangekeeper was built to include a target circling at a constant radius of turn, but that function had been disabled.
Only the RN and USN achieved 'blindfire' radar fire-control, with no need to visually acquire the opposing vessel. The Axis powers all lacked this capability. Classes such as Iowa and South Dakota battleships could lob shells over visual horizon, in darkness, through smoke or weather. American systems, in common with many contemporary major navies, had gyroscopic stable vertical elements, so they could keep a solution on a target even during maneuvers. By the start of World War II British, German and American warships could both shoot and maneuver using sophisticated analog fire-control computers that incorporated gyro compass and gyro Level inputs. In the Battle of Cape Matapan the British Mediterranean Fleet using radar ambushed and mauled an Italian fleet, although actual fire was under optical control using starshell illumination. At the Naval Battle of Guadalcanal USS Washington, in complete darkness, inflicted fatal damage at close range on the battleship Kirishima using a combination of optical and radar fire-control; comparisons between optical and radar tracking, during the battle, showed that radar tracking matched optical tracking in accuracy, while radar ranges were used throughout the battle.
The last combat action for the analog rangekeepers, at least for the US Navy, was in the 1991 Persian Gulf War when the rangekeepers on the Iowa-class battleships directed their last rounds in combat.

British Royal Navy systems: Dreyer Table
Arthur Pollen's Argo Clock
Admiralty Fire Control Table – from 1920s
HACS – A/A system from 1931
Fuze Keeping Clock – simplified HACS A/A system for destroyers from 1938
Pom-Pom Director – pioneered use of gyroscopic tachymetric fire-control for short range weapons – From 1940
Gyro Rate Unit – pioneered use of gyroscopic Tachymetric fire-control for medium calibre weapons – From 1940
Royal Navy radar – pioneered the use of radar for A/A fire-control and centimetric radar for surface fire-control – from 1939
Ferranti Computer Systems developed the GSA4 digital computerised gunnery fire control system that was deployed on HMS Amazon (Type 21 frigate commissioned in 1974) as part of the WAS4 (Weapon Systems Automation - 4) system.
BAE Systems' Sea Archer – computerised gunnery system. Royal Navy designation GSA.7 from 1980 and GSA.8 from 1985. Production was completed for Royal Navy Type 23 frigates in 1999. Remains in active service as of 2022 on Type 23 (Duke class). Replaced in 2012 on Type 45 destroyers by Ultra Electronics Series 2500 Electro-Optical Gun Control System.

US Navy analogue Gun Fire Control Systems (GFCS): 

Mark 33 GFCS: The Mark 33 GFCS was a power-driven fire control director, less advanced than the Mark 37. The Mark 33 GFCS used a Mark 10 Rangekeeper, analog fire-control computer. The entire rangekeeper was mounted in an open director rather than in a separate plotting room as in the RN HACS, or the later Mark 37 GFCS, and this made it difficult to upgrade the Mark 33 GFCS. It could compute firing solutions for targets moving at up to 320 knots, or 400 knots in a dive. Its installations started in the late 1930s on destroyers, cruisers and aircraft carriers with two Mark 33 directors mounted fore and aft of the island. They had no fire-control radar initially, and were aimed only by sight. After 1942, some of these directors were enclosed and had a Mark 4 fire-control radar added to the roof of the director, while others had a Mark 4 radar added over the open director. With the Mark 4 large aircraft at up to 40,000 yards could be targeted. It had less range against low-flying aircraft, and large surface ships had to be within 30,000 yards. With radar, targets could be seen and hit accurately at night, and through weather. The Mark 33 and 37 systems used tachymetric target motion prediction. The USN never considered the Mark 33 to be a satisfactory system, but wartime production problems, and the added weight and space requirements of the Mark 37 precluded phasing out the Mark 33:

Although superior to older equipment, the computing mechanisms within the range keeper ([Mark 10]) were too slow, both in reaching initial solutions on first picking up a target and in accommodating frequent changes in solution caused by target maneuvers. The [Mark 33] was thus distinctly inadequate, as indicated to some observers in simulated air attack exercises prior to hostilities. However, final recognition of the seriousness of the deficiency and initiation of replacement plans were delayed by the below decks space difficulty, mentioned in connection with the [Mark 28] replacement. Furthermore, priorities of replacements of older and less effective director systems in the crowded wartime production program were responsible for the fact the [Mark 33's] service was lengthened to the cessation of hostilities.
The Mark 33 was used as the main director on some destroyers and as secondary battery / anti-aircraft director on larger ships (i.e. in the same role as the later Mark 37). The guns controlled by it were typically 5 inch weapons: the 5-inch/25 or 5-inch/38.

Deployment: destroyers (1 per vessel, total 48) 
8 Farragut-class(launched ca. 1934)
18 Mahan-class(ca. 1935) (Cassin, Downes later rebuilt with Mk37)
4 Gridley-class(ca. 1937)
8 Bagley-class(ca. 1937)
10 Benham-class(ca. 1938)
heavy cruisers
7 New Orleans-class(launched ca. 1933): for the 5"/25 secondary battery
Wichita(1937): for the 5"/38 secondary battery
light cruisers
9 Brooklyn-class(launched ca. 1937): for the 5"/25 and 5"/38 secondary batteries

Mark 34 GFCS: The Mark 34 was used to control the main batteries of large gun ships.