In the 1630s, changing barrel elevation was a little tricky. There was a wedge (quoin) under the breech end. The barrel would be lifted off the quoin with handspikes, and then the quoin would be moved forward or backward to adjust the elevation angle. That was enough for varying the degree of positive (above horizontal) elevation, but depressing the barrel below the horizontal was trickier. A wad had to be rammed down the muzzle so the ball wouldn't roll out, and it might be necessary to insert an additional or thicker quoin under the breech so the barrel would point downward. It should be noted that when the gun recoils, the position of the quoin may be disturbed.
You might logically wonder whether this cumbersome quoin system was adopted because no one had thought of equipping the gun with an elevating screw. The elevating screw per se had already been invented, as is evident from drawings by both Leonardo da Vinci and Albrecht Durer. (Kinard 70). However, it was not used in the field artillery of the Thirty Years' War (Guthrie 15), let alone in the technology-lagging naval artillery. Deane (48) says that a Jesuit, in 1650, was the first to equip a land gun with an elevating screw. As for naval guns, the British introduced elevating screws around 1790, for use on carronades. (Lavery 132).
A screw provides mechanical advantage-it is the equivalent of an inclined plane that has been coiled up. The pitch of the screw determines how much the gun is elevated per turn; the smaller the pitch, the slower the elevation, but the finer the control. Modern tests on nineteenth-century 6-pounders revealed that each turn elevated the piece by 30–60 arc-minutes, and that the obtainable accuracy of elevation was about 2 arc-minutes. (Hughes 19).
Without the elevating screw, it took at least four men to change elevation: at least two with handspikes to lift the breech end, the "first Captain" to sight the gun and judge when it was at the right elevation ("Raise!" "Lower!" "Well!"), and the "second Captain" to adjust the quoins to hold the gun ("Down!) at that elevation. With the screw, one man could sight the gun while turning the screw to suit.
Nonetheless, to make a rapid, albeit crude, change in elevation, quoins were apparently faster, which is why carronades were also given molding under their breeches. (Lavery 132). Quoins were also needed if the elevation change was greater than that permitted by the screw (Douglas 163).
In the case of field guns, "the heavier pieces like the 18-and 24-pounders were still elevated by quoins as late as the early 1800s." (Manuoy 55). Quoins were also still used with siege guns. I suspect that this was because there were technological limitations at the time on the pitch or the compressive strength of the screw, and therefore on how heavy a weight could be lifted. The logical solution was to increase the mechanical advantage by using gears. And from there, the next step was to provide power assistance, e.g., from an auxiliary steam engine or an electric motor, rather than relying on manual operation.
In canon, elevation screws are apparently in use by the Danes in 1634. Offord, "The Bloody Baroness of Bornholm" (Grantville Gazette 18).
Traversal. For a target which is not moving relative to your gun, you traverse the gun so it points horizontally at the target. If the target is moving, you must "lead it" — point to the place it will be when the projectile arrives.
The wheels of the standard naval carriage all rolled forward and backward, and therefore would not have made it any easier to turn the barrel toward the bow or stern. The carriage had to be turned to or fro by brute force.
I own a storage cart with four swivel casters, i.e., wheels with a pivotable connection to the cart. A cannon, of course, is a lot heavier than a storage cart, but internet searching reveals that some caster manufacturers (e.g., Hamilton) claim that their casters can support up to ten tons. Of course, I have no idea whether we have the metallurgical skills to duplicate these casters at a reasonable cost, but it shows that the idea of putting such on a cannon carriage isn't absurd. But perhaps it would make the cannon too easy to move sideways, causing them to shift as the ship pitched.
A pivot mount, of course, would make traversing much easier, and could be equipped with a traversing screw or gear. With a simple slide mount, traversing the gun would be impossible. However, for carronades, the slide bed itself was mounted on a pivot, and on the inboard end there were two small wheels, whose positions established the radius of the traverse. Since the recoil motion was on the slide, and didn't affect these wheels, they could be positioned to roll circumferentially, making the traversal much more efficient. (Blake 140).
There were basically two ways of mounting a turret; it could rotate around a central shaft (Ericsson's USS Monitor) or on a circular track with ball bearings (Eads' USS Winnebago) (cityofart.net).
There were aircraft and tank turrets that were manually rotated, but naval turrets were larger and heavier. While the earliest naval turrets were hand-cranked (Kinard 237), the USS Monitor was equipped with a steam "donkey" engine to turn its turret, and that quickly became the mid-nineteenth-century norm. However, steam engines radiate heat, making conditions in the turret unpleasant, and of course there's the risk of scalding the crew if a leak occurs. There was some experimentation in the late-nineteenth century with compressed air systems, but the necessary high working pressures posed dangers of explosion. By the early-twentieth century, turret power was either hydraulic (British) or electric (American) in character. (Fullam 214ff).
Elevation Measurement
It does you no good to calculate and adjust the elevation of the gun if you can't judge whether you have done so correctly.
Pre-Ring of Fire (RoF), elevation was determined using a gunner's quadrant, first described by Tartaglia (1545). This was an L-shaped instrument with a plumb bob and an arc scale. One arm of the L was placed inside and parallel to the bore; the angle at which the plumb bob intersected the scale was read off.
Elevation may also be read off by a clinometer. A viscous liquid might half-fill a disk, and then the level of the liquid (an artificial horizon) is compared to an angular scale inscribed on the face of a disk. This is analogous to the aircraft inclinometer.
Or an object, a bubble or a bead, moves inside a tube filled with a viscous liquid, as in the spirit level used by carpenters, and the tube is graduated to show the angle of inclination.
Unfortunately, the spirit level doesn't provide much of an angle range. So a military clinometer has the spirit level mounted on a pivotable arm, which points to a scale that specifies the "zero" angle for the spirit level. The arm is attached to a frame whose base is placed on a receiver attached to the gun barrel. Since the outside of the gun barrel is not parallel to the bore, this receiver must be adjusted, just like gun sights, or an offset must be dialed in. To set the gun to a desired elevation, you lay the clinometer on the receiver, adjust the arm to point to the desired value on the main scale, and elevate the gun until the level bubble in the level vial is centered.
Some kind of pocket clinometer, most likely of the kind used by geologists, came through the Ring; see Jones, "Schwarza Falls" (Grantville Gazette 5).
Gun Sights
Open Sights. The simplest method of sighting was to sight along the "line-of-metal," the top of the cannon, directly at the target. However, the cannon was wider at the breech end than the muzzle end, so the line-of-metal was depressed 1–3° (Douglas 293; Beauchant 16) below the line-of-fire, depending on the exact geometry of the cannon. This could be corrected for by adding a "dispart," a vertical sight at the muzzle end, with a height equal to half the difference in diameters. If the bore wasn't quite center, this still wouldn't be quite right, but a gunner could customize the dispart for the peculiarities of a particular gun.