There isn't much difference in the lateral deviation depending on the plane of measurement, but the deviation of range is quite different from the vertical deviation, equaling the latter multiplied by the cotangent of the angle of fall. While the vertical and horizontal deviations are loosely proportional to range, the deviation of range is not, since the angle of fall also changes. The range deviation remains "nearly the same for widely different ranges" and may indeed decrease slightly. (Alger 253).
At Metz in 1740, a 24-10 on a 78-foot platform was given a 9 pound charge and elevated to 4o. A series of shots were fired (Robins 238) and it's instructive to examine how much their ranges varied. The average was 835 toises (toise=6.39 feet), but the minimum was 715 and the maximum 1010. The standard deviation was 67.9.
In 1833–49 British experiments, with 32-pounders loaded with double shot, at 300 yards, 26/28 shots struck a 10 x 6 foot target, but had 400 yards, only 8/28 did. (Experiment 4).
For an 1850s American naval 32-pounder, "Firing at a vertical screen 40 feet wide by 20 feet high at a distance of 1,300 yards with a 32-pdr. of 57 cwt., only three out of 10 shots hit the target, two direct and one on ricochet. The average range to first splash was 1,324 yards with deviations [spread] from 1,238 yards to 1,383 yards. " (Canfield).
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The vertical and horizontal errors are normally distributed, so if you multiply their means by 2.637, you get the sides of the rectangle (vertical target) that receives 50% of the shots.
Achieving precision is effectively a matter of minimizing the round-to-round variation in muzzle velocity, whereas attaining accuracy requires properly equipping and training gun crews.
There can be significant individual variations in accuracy among different gun crews operating different individual guns of the same type. In the Prize Firing of the British Mediterranean Fleet for 1899, for the nine battleships with 12 inch guns, they scored an average of 33% hits on the standard target, but the best performer was 55% and the worst 11.7%. The same year, for the 4.7" quick fire gun, the best performer was Scylla (80%), followed by Vulcan (51%). (Note that Scylla was captained by gunnery innovator Captain Scott.)(Brassey 30-1).
Range and Accuracy
All else being equal, if you increase the range, you decrease the accuracy. The horizontal extent of an angular error in bearing equals the range times the sine of the angle. The effect of an error in elevation is more complex, thanks to gravity, but the vertical or range error will still increase with range.
Target Size and Accuracy
On the other hand, the larger the target, the easier it is to hit. The principal vertical target (hull side) is defined by the target ship's length on deck and freeboard. The horizontal target (deck) is defined by the target's length and beam.
The angular width of the target, at a particular range, determines what degree of error in traverse can be tolerated. For long-range shooting at ships, the "small angle" approximation works; the angular size is proportional to the range, so, at 1000 yards, a ten foot object is 0.19 degrees.
HMS Victory (1765), a British first-rate, is one of the largest wooden sailing warships ever built, 186 feet long. For a seventeenth-century frigate-equivalent, let's assume a length of 100 feet. So the Victory has an angular width at 1000 yards of 3.55o, and the frigate, of 1.9o.
Danger (Hitting) Space
Because a ship target rises above the water, it is possible for a shot projected for greater than the correct range (at sea level) to still hit the target somewhere on its superstructure. The taller the target, and the flatter the trajectory, the greater the effective "danger space" in which a mis-ranged shot could still strike the side of the target. Since big guns could use lower elevations than small guns for a given range, this gave big guns an advantage. "At 4,500 yards, the 12 in/45 had a danger space of 130 feet... compared to 100 feet for the 6in." (Friedman 18). That assumed a target height of ten feet.
On the Victory, the sides from waterline to bulwarks measured 40 feet (Royal Naval Exhibition 1891), and the hammock rail of a French 82-gun warship was reportedly 26 feet above the water. In contrast, a frigate might have a freeboard of just 8 feet.
As the range increases, the elevation must also increase, reducing the danger space. For the 12 in/50 (muzzle velocity 2567 fps), it was 572 yards (for a 30 foot tall target) at 2,000 yards and 33 yards at 12,000 yards. At the latter distance, deck hits were actually more likely than side hits; the greater the target beam, the more likely this was to occur. The beam of the Victory was almost 52 feet; of a typical frigate, 27.
Aiming the Gun
So, what are you trying to hit? A ship is a relatively large target, after all. In the Napoleonic Wars, the British tended to target the enemy hull, and the French, the rigging.
In aiming for the hull, a gunner didn't estimate the range and then look up the proper elevation in a gunnery table. Rather, a rule of thumb was used, for example, at point-blank range, aim at the hull, whereas at half a mile, aim at the fighting top and at one mile, aim for the top of the main mast. (NMRN) (By aiming high, you allowed for the fall of the shot.)
Range Estimation
If the range were great enough that elevating the gun was necessary, then you had to have some way of determining what the range was so you could judge the correct elevation.
The gun captain might, through long experience, be able to estimate visually the range to the target and know the proper elevation to strike it. This depended, of course, in the first instance on the gun captain's visual acuity.
In the late-nineteenth century, American soldiers were required to be able to see a two-foot square black bull's eye on a white background at a distance of 600 yards. (Clowes 385). Training was also important; soldiers would pace off a distance and then study it, or estimate a range and then pace it off. Soldiers were taught that at 600 yards, a man's head was a small round ball, that at 225 yards, his face became distinguishable as a light-colored spot; the eyes can be seen at 80 yards and the proverbial whites of the eyes at 30. (Groome 151; Farrow 697). Presumably, sailors could similarly study the crew of an enemy ship, as well as the visibility of its gun ports, masts, and stays.
In land warfare, visual estimates supposedly had an error of 12–15% at a range of 600–1200 yards. (Hopkins 196). However, at sea, there aren't a succession of fixed reference points, like trees and hills, which you can use to facilitate range estimation. In addition, weather conditions often will degrade visibility. According to Fullam (459), "it is quite impossible to estimate ranges above 2000 yards with anything like sufficient accuracy."
Acoustics: Just as you can estimate how far off a thunderstorm is by timing the interval from lightning flash to thunder rumble, you can count the seconds between the flash and the report of the enemy's guns. This can be made somewhat more precise with an acoustic telemeter; a metal disk is caused to drop through the liquid filling a calibrated tube when the flash is seen, and stopped when the sound is heard. (Cook 593).
Trigonometric Methods: If you know the absolute dimension of any part of the enemy ship, such as the height of its mainmast, you can measure its angular size with the sextant, and calculate the range by trigonometry (or table lookup). Douglas compiled a table of the heights of the parts of French ships of war of various classes. (Douglas 214ff). This works best if the enemy has standardized its warship classes, which unfortunately was not the case in the early-seventeenth century.
Alternatively, as in Buckner's method, you could measure the angle between the enemy's waterline and the horizon; it requires knowledge of the viewer's height above sea level. Use of this method is expedited by what EB11 calls a "depression rangefinder."
These methods were more likely to be used for deliberate shooting by a bow or stern gun during a chase, than for a broadside.
Another trigonometric method is to have observers stationed at the bow and stern of your ship sight the same object and report its bearing. The accuracy of this method depends on the length of your ship, whic
h serves as the baseline (Cook 591). It also required communication between the observers, and wouldn't work if the target were ahead or astern. (Friedman 23).
Consequently, integral rangefinders, with a fixed mirror or prism connected by a rigid base to a rotatable one, were considered. The accuracy of the rangefinder at a given range was proportional to the square root of the base length. (Id.).
In 1891 the Admiralty advertised for a rangefinder which would have an accuracy of 3% at 3000 yards. The winning entry was the Barr-Stroud range finder. As described by EB11/Range-Finder, this used two telescopes, separated by 3–9 feet, to create partial images that were brought into view by reflecting prisms. To overcome the limitation of the short base line, the optical system included a movable deflecting prism. A range could be taken in 8–12 seconds. The 9-foot FQ2 (1906) was theoetically accurate to 1% (150 yards) at 15,000 yards, but in practice, refraction and heating of the tube degraded accuracy, with errors of 1000–1500 yards seen at ranges of 19–21,000 yards. (Friedman 24).
The Barr-Stroud coincidence rangefinder was designed to show the top of the target through one lens and the bottom through the other. The operator looked for a vertical element in the target, the half-images of which would be brought into coincidence by adjusting the angles. To prevent similar rangefinders being used against them, the British "tried to break up the vertical lines of their masts and funnels with spirals around masts and then with triangular inserts (rangefinding baffles)." (Id.)
An alternative design approach was taken by the Germans in 1893. This was the stereoscopic rangefinder, "in which each lens fed its image into one of the operator's eyes." The operator had to have perfect binocular vision, but if so, perceived depth in the image and would move a marker "until it coincided with the target." (25).
If the guns aren't themselves equipped with rangefinders, then the rangefinder information must be communicated by the observer to the gunners.
Active range finders. These "ping" the target with some kind of radiation—radio waves, sound or laser light—and measure the time to receipt of the reflection. The technologies are called RADAR, SONAR and LIDAR, respectively. Military use of RADAR and SONAR began in WW II, and LIDAR is a more recent development. Don't expect any of these to be available in the foreseeable 1632verse future!
All of the above methods determine the geometric range of the observer to the target at the time of observation. Depending on gun and target motion, the gun might have to be set to a different range.
Bracketing: If all else fails, you may "try the range." Observing what proportion of the shot fired fall short of the target can be used to guide how to adjust the aim; if the proportion is much less than one-half, the shots are on average over-shooting.
In 1936, the French began placing marker dyes in shells; each ship would be assigned a particular color so it could identify which splashes were from its guns . . . assuming no other ship was assigned that color. (Friedman 258).
Precision: Smoothbore vs. Rifled Guns
The following table compares the precision of the "best shooting" smoothbore and rifled land artillery circa 1870:
(Owen 334).("Reduced" deflection means relative to the mean point of impact, not the point of aim.)
However, Owen comments that up to 300–400 yards, the smoothbores are just as accurate as the rifled guns, and at very long ranges such that ricocheting is necessary, the smoothbores are superior because round shot ricochets more predictably.
Another source is Abbot, writing about the First Connecticut Artillery. With 32-pounder smoothbore seacoast guns, Fort Barnard achieved 20 feet mean deviation from center for a target 1030 yards away. Fort Richardson didn't fare as well; 28 feet at 950 yards. (51). With the 30-pounder Parrott rifle, Fort Barnard reported 16 feet at 1030 yards (117).
Even at long range, rifling is not a panacea. In 1870, three ironclads tested their big rifled muzzleloaders on a rock 600 feet long, and 60 feet high, 1000 yards away, under favorable conditions, with the following results:
HMS Hercules (1868), 10" guns, 10 hits/17 shots;
HMS Captain (1869), 12" guns, 4/11;
HMS Monarch (1868) 12" guns, 9/12.
(Cooke 182). Note, this was shooting at a stationary target much larger than a ship.
Windage, Balloting and Deviation
So what's the problem with smoothbore precision, and what can be done about it? The principal cause of deviation is balloting, that is, the bouncing of the projectile as it passes down the bore as a result of windage. Without any sabot to center it, round shot must be smaller than and rest on the bottom of the bore. If a projectile is spherical and homogeneous, then the propellant gases will cause it to roll forward (topspin). As the result of the Magnus effect (the effect of the rotation on the airflow around the projectile), the projectile feels an upward force, and bangs against the top. That will reverse the rotation, and the Magnus effect will result in a downward force as it continues down-bore. Now it bangs the bottom, and acquires topspin, sending it up again. Plainly, it's a matter of chance how it emerges.
In 1862, for smoothbores, the angular deviation of the line of departure (how the projectile actually left the bore) from the line of bore was reportedly not more than 5' vertically and 4'30" laterally. (Benton 415).
However, there's also retained spin to be considered. With ordinary windage for a 24-pounder shell fired with 2.25 pounds of powder, the rotation was 30 fps [2.9 rpm]. (Benton 425). Topspin shortens the range and backspin increases it, again as a result of the Magnus effect.
If the shot's eccentric (the center of gravity doesn't coincide with the geometric center), sidespin is possible, and the Magnus force will then cause a deviation toward the side on which the center of gravity is located. Dahlgren, using "service" 32-pounder shells, determined the location of the center of gravity of each, and positioned them. He found that if the firing were such that a concentric shell would range 1300 yards, with the center of gravity up, the travel was 1415 yards, with it down, 1264 yards, and inwards (toward the breech end), 1360 yards.
Modern smoothbore tank cannon fire projectiles equipped with discarding sabots, that is, sabots that fall away once the projectile leaves the bore (see part 4). A cannonball could be equipped with a discarding sabot, thus reducing bore-windage and consequent balloting, barrel wear, and trajectory errors. R&D is needed to ensure that the sabot separates at the right time.
Cannonballs could also be replaced with elongated projectiles, increasing sectional density and thus reducing retardation by air resistance. The catch is that elongated projectiles must be stabilized. The dominant stabilization method is by spin (imparted by a rifled bore), but that requires replacing smoothbore cannon with rifled ones. But it's also possible to stabilize flight using fins, like the feathers of an arrow (part 4). While the fins would require R&D, finned projectiles might be manufactured faster than rifled cannon (and projectiles to engage the rifling).
Spin-stabilized projectiles fired from rifled cannon also experience the Magnus effect; however, since essentially the same spin (same axis, direction and speed of rotation) is imparted to each projectile by the rifling, the Magnus force exerted on each is the same and the deviation (called "drift") is predictable and can be compensated for. Whereas the rotation of spherical projectiles fired by smoothbore cannon will differ from round to round in an unpredictable way. Projectiles and sabots are discussed in detail in Part 4.
Accuracy: Land versus Sea
The effective range of Napoleonic smoothbore field artillery (4- to 12-pounders) on land was 800–1200 yards. (Nosworthy 359ff). (The guns could probably range farther, but with open sights, aimed fire wouldn't be possible.) For a 12-pounder firing at a continuous screen six feet high, simulating a line of infantry, the Madras Artillery (1810–17) reported that the 12-pounder achieved almost 80% hits at 300 yards, 60% at 900, and perhaps 25% at 1200. (Hughes). Wilhelm Muller (II:195) reported that circa 1811 a 12-pounder achieved 45% hits on an embrasure 2.5 feet high and 8 wi
de at a range of 575 paces, and 18% at 1300.
Why then, was the engagement range of Napoleonic sea ordnance so low? Was it because the close range was needed to ensure penetration of the thick hull of a warship? Were naval guns of inferior precision? Or was marksmanship much worse at sea than on land?
Firing on shipboard presents some difficulties that the land artillery didn't have to consider. Both the firing and the target ship were in motion, perhaps at different speeds on different courses, subject to change at any moment on account of the wind, damage, and tactical decisions, and thus the target range, bearing and aspect were in constant flux. In addition, if the sea wasn't smooth, the firing ship was rolling, pitching and perhaps yawing, too. Even if the two ships were still, estimating range was harder at sea than on land. It's also true that naval gun crews didn't practice firing at long range targets, but that could be because of the other problems set forth above.
There's evidence that ship motion was the principal problem. In 1847, the 74-gun Leviathan was used as a target to test the accuracy of guns firing round shot, with roughly these results under ideal conditions (smooth water, light wind, both ships stationary):
(Experiments, 1).
Those numbers can't be compared directly with those of field artillery, but they show acceptable accuracy at way beyond the normal naval battle range.
Roll, Pitch and Yaw
Let's look at the problem of firing ship motion more closely. If sailing on any course other than directly downwind, a sailing ship would be heeled over, that is, tilted from the vertical. In shooting, its gunners would have to compensate for this constant tilt.