In part 1, I provided an overview of warships and their guns, and in part 2, I discussed how those guns were mounted, laid, sighted and fired, and what happens inside the bore. Here, I talk about what range and accuracy can be expected with seventeenthcentury guns, and what the old time line teaches us about what the uptimers can do to improve on them.
PointBlank Range
Standard practice for seventeenthcentury naval warfare was to engage at pointblank range (or less). Pointblank range (PBR) is the furthest that the gun can be assumed to "shoot straight," that is, the range at which the average gunner will use zero elevation. Strictly speaking, it is the range at which the "drop off" equals the height of the muzzle above the water surface, so the projectile will still hit the target. Yes, that means that PBR should vary depending on which deck the gun is mounted on!
There is considerable disagreement as to the actual value. In 1834, Stevens (25) said from a frigate, PBR is 500 yards, and from a "battleship," 700, assuming that the guns are pointed by the "dispart sight at the hammock rails" of a frigate or larger target. In 1828, Beauchant estimated that the 18, 24, and 32pounders, fired from the main deck of a frigate, had a pointblank range of 400 yards with a onethird charge, 300 with a onequarter, and 250 with onesixth. However, engagements were more typically at 100–200 yards.
Maximum Range
A range table for a gun lists, for each projectile and charge (or muzzle velocity) it typically uses, a series of ranges, and the angle of departure or elevation needed to achieve each range, given standard conditions. Modern range tables also provided the maximum ordinate (height), time of flight, drift, angle of fall, striking velocity, armor penetration, danger space for a target of standard height, and correction factors to use for nonstandard projectile weight, air density, and muzzle velocity, and for wind, gun and target motion.
Simple range tables are available from sixteenth and seventeenthcentury artillerist manuals, but must be taken with great caution. It's quite doubtful that they were based on test firings, rather, just on imperfect recollection. With regard to the maximum ranges given by Luis Collado (1592) and all examined ranges given by Diego Prado y Tovar (1603), ballistics experts at Aberdeen Proving Grounds determined that muzzle velocities of about 6,000 fps would have been required. (Guilmartin 297).
The first reliable range tables were compiled entirely by experimental firings. But these went only so far. Bear in mind that because of all the factors that affect trajectories, it's not enough to fire one shot at each elevation of interest. No, you must fire multiple shots, and average the values, to get reliable predictions. Multiply all this by the number of different guns, projectiles, and standard charges for those projectiles, and you can see that the ammunition expenditure would be quite considerable.
The old military term for a maximum range was a "random", representing apt skepticism as to the chance that such a shot would hit its target. French experiments of 1739–40 revealed that a 2410, elevated to 45^{o}, could range to 2250 toises (4793 yards)(Robins 234). In 1864, a 2.75" Whitworth rifled gun achieved a range of 2,600 yards at 5^{o} and 10,000 yards at 10^{o}. (Bell 445; Newton 87). The effective range was of course much less.
Exterior Ballistics
Modern ballistics can be used to calculate the "first graze" (the initial point of impact of the projectile) and its impact velocity, given the muzzle velocity, the elevation of the barrel and a "drag function" for the projectile. These calculations couldn't be made accurately before the Ring of Fire (RoF) because of certain fundamental misunderstandings and fatal oversimplifications.
The leading downtime work on ballistics was Tartaglia's Nova Scientia (1537). The "physics" underlying Tartaglia's propositions is Aristotelian: a projectile is thought to follow first a straight line in which "impetus" is dominant, then a transitional curve, and then finally fall straight down ("natural motion"): "Wile E. Coyote" physics. Nonetheless, Tartaglia predicted that maximum range would be obtained if the projectile were fired at an elevation angle of 45^{o}—true if the trajectory is in a vacuum.
Galileo has also been studying ballistics; unfortunately, he didn't publish his work (Discourses and Mathematical Demonstrations Relating to Two New Sciences) until 1638. He was the first to point out that gravity wouldn't affect the horizontal motion of the projectile, that a body without an initial upward motion would fall a distance proportional to the square of the time elapsed, and that the combination of these propositions indicated that the path of a projectile would be a parabola. This is all true—in a vacuum.
The range in a vacuum is easily calculated:
R_{vac} = V^{2} sin (2*theta) /g
where V is muzzle velocity, g is gravitation acceleration and theta is the elevation angle. In a vacuum, the maximum range would be at an angle of 45^{o}, and for any lesser range, there would be two equally acceptable elevation angles for achieving it.
Galileo's disciple Evangelista Torricelli published many ballistics theorems in Opera Geometrica (1644). When Giovanni Renieri complained that his experiments did not agree with Torricelli's formulae for the relationship of the pointblank range to the maximum range, Torricelli reminded him that the text was intended for philosophers, not gunners.
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Because of air resistance, the maximum range is lower than R_{vac} and it's achieved at an elevation angle that's less than 45^{o} and may be as low as 30^{o}. (Rinker 332; Douglas 43, 36). The lower the elevation angle (thus, the flatter and shorter the trajectory), the less the effect of air resistance. High angle trajectories nonetheless had their uses, mostly in attacks on fortifications where you needed to put a shell over a wall to hit a higher value target beyond.
In 1668–69, Christaan Huygens demonstrated that water resistance was proportional to the square of the speed of the object moving through it (true for low speeds), and inferred that the same was true of air resistance. Newton later advanced a similar proposition in his Principia.
The air resistance (drag force) is
0.5 * rho * C_{D} * A * V^{2}
where rho is the density of air (which changes with altitude, C_{D} is the dimensionless drag coefficient, A the reference area for which C_{D} is determined (typically the frontal area for a projectile), and V the air speed. C_{D} is a function of projectile shape and, unfortunately, speed.
At typical muzzle velocities, the drag force is more than twenty times as strong as the gravitational force.
The equations of motion for a projectile subject to both gravity and air resistance are easy to write, given calculus and knowledge of the resistance as function of speed, but difficult to solve. Some firstclass mathematicians addressed this problem, including Bernoulli and Euler. Their work is touched upon by EB11/Ballistics, which goes on to discuss the SiacciIngalls approximation method. That was still in use in World War I. EB11/Ballistics also describes in some detail how Bashforth constructed his ballistic tables.
It is likely to occur to the mathematicians in Grantville that the equations can be "solved" by numerical integration (calculating position, velocity and acceleration in small time increments, say a millisecond at a time), and indeed that is the only viable approach if the speeds are supersonic and the trajectories are high (so air density varies with altitude). But even with simpler trajectories, it may seem simpler in Grantville's "computer culture" to use numerical integration rather than reconstruct the complex analytic approximations of Bernoulli, Euler and SiacciIngalls. (cp. Cline).
Doing this requires knowing the dependence of air density on air temperature and pressure, and for high trajectories of those on altitude (the aviators in Grantville should have this information). Also, one must determine the drag function (the speed dependence of the drag coefficient) for the projectile of interest, which our characters can do by quantifying (with ballistic pendulum), for each of a variety of charges, its muzzle and downrange velocity from a particular gun.
Just to show that it can be done, I have constructed an Excel spreadsheet that uses the RungeKutta fourth order approximation method (which should be described in a standard textbook on numerical analysis, and thus in the personal library of one of Grantville's mathematicians) to solve the linked differential equations of ballistics. This can be used to construct a range table, if we know the drag coefficient as a function of speed for the projectile, and the muzzle velocity with which the gun projects it. And for that, we need to be able to measure projectile speed.
Projectile Speed Measurement
The real breakthrough in exterior ballistics was the invention (1742) of the ballistic pendulum by Benjamin Robins, who used it to measure the speed of projectiles at the muzzle and at various ranges (the latter permitting the effect of air resistance to be quantified—Douglas 129). This device is described in EB11/Chronograph, but in essence the projectile strikes a pendulum and transfers its momentum to it, causing it to swing.
Robins not only confirmed that the normal drag was proportional to the square of the speed, he detected the sharp (perhaps threefold) increase in resistance at, he reported, 1100–1200 fps, that became known later as the "sound barrier."
Modern shooters have made their own ballistic pendulums with what appears to be reasonable accuracy (main worries are projectile deformation and deflection, friction, gravity and calibration), so this can definitely be done in the new time line. The accuracy of the ballistic pendulum is a respectable 2% (Rinker 148).
The catch is that cannon balls are heavier than bullets, and the ballistic pendulum must be scaled up to match. Hutton used an 1800 pound pendulum for studies on 6 pound balls, and in 1839–40, to measure the speed of 50 pound projectiles, a six ton receiver was employed. (Bashforth 25).
Reverend Bashforth achieved even greater accuracy by timing when the projectile passed through wire screens separated by a known distance; the penetrations interrupted the electric current to a chronograph. (Cantwell 46). Gun chronographs are described in EB11/Chronograph.
Drag Coefficient
The drag coefficient is dependent on the Reynolds number, which is proportional to both the length and the air speed of the projectile. It's also dependent on the Mach number, which is the speed as a fraction of the speed of sound (340.45 meters/second; 1117 feet/second; sea level, 15^{o}C, 59^{o}F). For typical projectile speeds, Mach number is more important.
A projectile traveling at a speed close to the speed of sound (Mach 1.0) exhibits a mixture of subsonic (under speed of sound) and supersonic (over) air flows. This range of speeds at which this mixed flow occurs is called transonic. A projectile is said to be in the subsonic regime if it is traveling at less than Mach 0.8 (some authorities would say 0.6 or 0.7), in the transonic regime at Mach 0.8–1.2, and in the supersonic regime at over Mach 1.2.
At subsonic speeds, drag is primarily frictional drag (air retarded as it passes over the surface) and pressure drag (air pushed out of the way). The lower end of the transonic range (critical Mach number) is where wave drag (air compressed) first becomes significant.
The drag coefficient varies depending on the shape of the projectile. It should be mentioned that since Grantville is in rural West Virginia, it is going to have a higherthannational average of hunters, of firearms (probably more firearms than people), and of books and software relating to firearms, including ballistics. They may thus be able to construct a reasonable drag function without experimentation, at least for shapes similar to the standard G1–G7 shapes.
Finding a dragspeed function for a cannon ball might be tricky. While your characters will have to determine it the hard way, you as an author may look it up. Douglas (132) gives a table of air resistance to a cannon ball at velocities of 100–2000 fps (cp. Guilmartin 296; Allsop 120).
Here are some range data that I calculated with my spreadsheet:
Table 31: Trajectory Computed by Numerical Integration Air Density 1.225545 kg/m^{3}, Gravity 9.80642 m/s^{2} Allsop three tier drag coefficient function for round shot  
elevation  muzzle height m (ft)  muzzle velocity m/s (fps)  shot kg (lbs)  range m (yd)  strike V m/s (fps) 
0^{o}  2 (6.56)  500 (1640)  10.89 (24)  293 (320)  390 (1279) 
"  1.5 (4.92)  "  "  257 (281)  402 (1319) 
"  4.5 (1.76)  "  "  422 (461)  356 (1155) 
2^{o}  2  "  "  1011 (1106)  241 (792) 
5^{o}  "  "  "  1671 (1827)  177 (581) 
"  "  370 (1214)  "  1301 (1423)  178 (583) 
"  "  500  1.36 (3)  1183 (1294)  131 (431) 
"  "  "  21.77 (48)  1856 (2030)  193 (633) 
By way of comparison, in a 1796 Admiralty test, a 24pounder elevated 2^{o} achieved 1274 yards with a onethird charge and 992 yards with a onefourth charge of cylinder (Red LG) powder, and 1020 with onethird charge of the old Blue LG powder. (Gardiner 129).
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Historically, experimental firings of a small number of standard projectiles were used to construct retardation (deceleration) functions (retardation=kV^{m}), which in turn allowed approximate calculation of the trajectories for a wide variety of elevations and each service gun's expected muzzle velocities (for each of its standard loads), under "standard conditions". (It was more accurate overall to use the average muzzle velocity for the service life of the gun, rather than the "new gun" design velocity.)
Based on Krupp's experimental firings in 1881, Mayevski and Zaboudski formulated a sevenpiece drag function for a "standard projectile", with e.g. m=2 (corresponding to a constant drag coefficient) for speeds below 790 fps, m=5 for 970–1230 fps, and m=1.55 for 2600–3600 fps. (Ingalls iv). The British developed a somewhat different drag function in which m was as high as 6.45.
Ballistics tables were developed to reduce the work involved in calculating range tables (Hackborn). The earlytwentiethcentury tables precomputed, for various velocities, the space, altitude, inclination and time functions used by Siacci's method for calculating the trajectory. The horizontal and vertical coordinates of the trajectory were dependent on these functions, and on a properly corrected "ballistic coefficient" (unity for standard projectile under standard conditions).
The ballisticians assumed that the retardation functions could be applied to a nonstandard projectile by adjusting a parameter known as the "uncorrected" ballistic coefficient, which in turn considered projectile weight, diameter and "coefficient of form," the last essentially embracing not just form but all other projectile characteristics that would affect its trajectory. The coefficient of form for each nonstandard projectile was itself determined from a more limited set of test firings. The actual drag at a given velocity was assumed to be the standard drag divided by the ballistic coefficient, corrected for nonstandard air density, etc.
The ballisticians erred in assuming that the coefficient of form was a constant, i.e., projectile shape would have the same effect at all velocities. You could, of course, construct a separate drag function for every projectile, and throw the ballistic coefficient "out the window."
In practice, the ballisticians developed a small number of retardation functions, one for each "family" of projectile shapes, and accepted the residual imperfection of the coefficient of form.
Ricochet Fire
This is the reflection of a shot by a surface. EB11/Ricochet says that ricochet fire was first employed by Vauban at the siege of Ath (1697). This is poppycock. Shakespeare refers to the "bullet's crazing" (grazing), causing it to "break out into a second course of mischief, killing in relapse of mortality: Henry V, Act IV, scene iii. And Bourne, The Art of Shooting in Great Ordnaunce (1587), has a section entitled, "How and by what order the shot doth graze or glaunce upon lande or water," which recognizes that the ricochet occurs only if the angle of incidence is shallow. Nonetheless, I suspect that it was not common in the seventeenth century to deliberately elevate the gun so that the shot would strike an enemy ship by ricochet. A particular advantage of ricochet fire was that it tended to strike the target near the waterline.
Ricochet is possible only if the shot strikes water at an angle less than the critical angle; Douglas (108) advises that the angle of incidence shouldn't be greater than 3–4^{o}, while Beauchant (30) favors under 2^{o}. The critical angle depends on the projectile; Birkhoff (1944) proposed that it is 18^{o} divided by the square root of the specific gravity of the projectile (Johnson); for an iron cannonball, that's 6.36^{o} (cp. Adam 111). The angle of incidence depends on the angle of elevation, the height of the muzzle above the water, air resistance, and wave action. Ricochet is more effective when the water is smoother, and firing to windward (lee sides of waves are steeper). There is of course some loss of energy at each bounce, so at least a onethird charge of powder is desirable if you're shooting at a large ship. (Beauchant 31). Ricochet can't be used effectively with rifled projectiles; because of their spin, they are reflected at a high angle. (Scott 25).
It's really amazing how much ricocheting can increase range. By way of an extreme example, the Vesuvius fired (1797) a 43.7 pound shot at a 1^{o} elevation; its first graze was at 358 yards; but it ricocheted a total of 15 times, achieving an extreme range of 1843 yards. (Douglas 209). The following table relates elevation, charge, distance to first graze, and extreme range:
Table 32 Ricochet  
English 24 x 6.5, solid shot  
Charge, lbs  1^{o} elevation  5^{o} elevation  
 first graze  extreme range  first graze  extreme range 
6  560  2027  1565  2230 
4  463  1847  1335  1793 
English 24 x9.5, 5.5 inch shells 

 
3  277  1424  1182  1733 
2  213  1139  836  1670 
(Beauchant 20, 26)
There are empirical formulae of uncertain reliability for estimating ricochet range (Abbot59ff) but I doubt they are in Grantville Literature.
In WW II, the Fifth Air Force experimented with "skip bombing"—essentially, a lowaltitude, highspeed approach so the bombs would ricochet toward the enemy. However, they abandoned this tactic in favor of "masthead height" bombing: "to eliminate the need to calculate the ricochet distance, they timed the release to hit the side of the ships, instead of bouncing short." (Gann 15).
Maximum effective range (MER)
Exterior ballistics can identify the range at which a particular projectile, fired at a particular muzzle velocity and elevation, can strike with a particular impact velocity. And that in turn may be compared with the empirical formulae of terminal ballistics (part 5) to determine how many inches of wood or iron it will penetrate.
What it can't do is determine the likelihood that the projectile will strike the target, and is that probability high enough to warrant firing. Is the projectile expensive and available only in small numbers (like a torpedo or missile), so you must make the shot count, or cheap and plentiful?
Accuracy, Precision and Trueness
If there is a single trajectory by which the gun, fired at a particular instant, will hit the point of aim, then it follows that there are essentially three kinds of firing problems that result in a different point of impact: errors in traverse, elevation and muzzle velocity. Errors in traverse result in a lateral error at the range of the target; errors in elevation result in a vertical error (if aiming at a "vertical target" like the side of a ship) or a range error (if aiming at a "horizontal" target like the deck). Errors in muzzle velocity cause vertical and range errors directly, and lateral errors indirectly (by making too much or too little allowance for wind deflection or ship motion during the time of flight).
There are two ways of coping with errors: minimization and compensation. We can minimize variation in powder strength, shot size, etc. There are some errors that we can't avoid, but if they are of predictable magnitude and direction, we can compensate for them. For example, if spin causes a rifled projectile to drift 10 yards left at a range of 1000 yards, we can aim enough to the right of the target so it will drift left onto it. Likewise, we can compensate for known wind, gun platform motion, and target motion.
Precision measures how closely the impacts are grouped together; accuracy, whether they hit the target. It's also helpful to recognize a concept that used to also be called accuracy but which is now (ISO 5725 for the measurement community) called "trueness": the distance of the mean point of impact to the point of aim (the desired point of impact).
It's possible to measure the precision of a gun (although this assumes that you have minimized variation in ammunition, elevation and atmospheric conditions) but a gun doesn't have an inherent accuracy. Any reference here to the "accuracy" of a gun means of that gun operated by a typical trained gun crew.
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"Accuracy" is most often casually stated as the number of hits made on a target at a particular range; hopefully, the records will also state the dimensions of the target.
The most useful method of quantifying either accuracy or precision is in terms of the mean error, laterally and vertically (or in range), of the points of impact from the point of aim (for accuracy) or from the mean point of impact (for precision), for a given range. One may also present the mean absolute error, the mean radial distance of the points of impact from the reference point. Unfortunately, most mean error data is from the midnineteenth century or later.
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 2410 on a 78foot platform was given a 9 pound charge and elevated to 4^{o}. 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 32pounders loaded with double shot, at 300 yards, 26/28 shots struck a 10 x 6 foot target, but at 400 yards, only 8/28 did. (Experiment 4).
For an 1850s American naval 32pounder, "Firing at a vertical screen 40 feet wide by 20 feet high at a distance of 1,300 yards with a 32pdr. 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 roundtoround 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 301).
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 longrange 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 firstrate, is one of the largest wooden sailing warships ever built, 186 feet long. For a seventeenthcentury frigateequivalent, let's assume a length of 100 feet. So the Victory has an angular width at 1000 yards of 3.55^{o}, and the frigate, of 1.9^{o}.
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 misranged 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 82gun 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 pointblank 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 latenineteenth century, American soldiers were required to be able to see a twofoot 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 lightcolored 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 earlyseventeenth 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, which 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 BarrStroud range finder. As described by EB11/RangeFinder, 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 9foot FQ2 (1906) was theoretically 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 BarrStroud 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 halfimages 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 shots fired fall short of the target can be used to guide how to adjust the aim; if the proportion is much less than onehalf, the shots are on average overshooting.
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:
Table 33: Precision, Smoothbore vs. Rifled  
 target range (yards) vs. mean error (yards)  
 1,000 yards  2,000 yards  
 mean range error  mean "reduced" deflection  mean range error  mean "reduced" deflection 
smoothbore  43  4.0  60  10 
rifled  19  0.3  21  0.7 
(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 32pounder 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 30pounder 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 downbore. 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 24pounder 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" 32pounder 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 borewindage 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).
Spinstabilized 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 12pounders) 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 12pounder firing at a continuous screen six feet high, simulating a line of infantry, the Madras Artillery (1810–17) reported that the 12pounder 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 12pounder achieved 45% hits on an embrasure 2.5 feet high and 8 wide 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 74gun 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):
Table 34: Smoothbore Accuracy at Sea, Ideal Conditions  
Shot Wt  Charge  Range (yards); % Hits  

 1500  2000  2500  3000 
32  8  75  45  22  8 
 10  75  45  25  11 
68  1415  "  "  "  " 
(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.
In addition, there would be a continual yawing, pitching and rolling as a result of the action of the sea. If the target were on the beam, these motions would have the following effects. Yawing (left/right) would affect the target bearing and thus the necessary traverse. Pitching would change the height of the gun, if it weren't on the pitch axis, and therefore the proper elevation to account for ballistic drop at the target range, unless the gun happened to be on the pitch axis. And rolling would change the target elevation and thus the required gun elevation. Pitching down when heeled away would create forward traverse and lower effective elevation.
A typical rate of roll would be one degree a second. If the target range were 3,000 yards, the line of sight would sweep across 15 vertical feet in less than onetenth a second. But earlytwentiethcentury German warships were "stiff," rolling at a rate of three degrees a second. (Friedman 163).
Moreover, the target didn't necessarily cooperate by remaining abeam. If it were fleeing, and your ship trying to cut off its escape, chances are that your guns are firing obliquely rather than perpendicularly to your ship's centerline. If so, roll changes not only elevation, but also bearing—the latter was called "crossroll." In the sailing ship era, when ships would be heeled over by the wind, Stevens (26) warned that in a chase, the guns would be inclined to leeward, and the bow or stern guns should therefore be pointed at the "weathermost" part of the enemy's hull.
Timing the Roll
In the 1630s, and indeed for more than two centuries thereafter, gunners only took roll into account. A ship rolled with a pendulum motion; fastest (but at constant speed) at midroll, paused (but accelerating or decelerating) at the top and bottom. Therefore, some gunners favored firing at the top or bottom of a roll. This had several consequences. First, it limited to the rate of fire to the period (or halfperiod) of the roll. Secondly, the gunner had to anticipate when the top or bottom of the roll was approaching, and whether this was possible or not depended on the regularity of the roll.
For this reason, British and French nineteenthcentury naval practice favored firing only on the rising motion, so that a shot intended for the hull would at worst hit the rigging (as opposed to missing altogether). (Douglas 235 ff). I must note that firing at the bottom of the roll was sometimes impractical, as that meant that the ship was in the trough of the waves, so the shot might be delayed until later in the rise. The Americans, in contrast, fired at the top of the sea, or on the falling motion. (245). The rising motion was slowest on the lee side, and the reverse was true on the weather side, but it wasn't always possible to take advantage of this. (Stevens 21).
Firing Interval
Moreover, the ideal would be for the projectile to leave the muzzle at that point, but there were several delays: from observing the roll to the decision to fire; from making the decision to lighting the fuse; from that moment to the ignition of the powder; and finally the time for the projectile to travel down the barrel. Overall, this lag was called the "firing interval," and Alger reported that under the best circumstances it was 0.25–0.30 seconds in the latenineteenth century. Chances are that the gunner wouldn't take all this into account and the projectile would emerge a little late.
We may try to reduce the firing interval as much as humanly possible. Alger conducted an experiment comparing seven different methods of actuating the firing device. The fastest involved biting on (0.198 seconds to the striking of the primer) or puffing air from the cheeks (0.214) into a mouthpiece. These also had the least variation. The classic lanyard pull was slower (0.268 spring lock; 0.354 hammer).
Roll Compensation
The nineteenthcentury British Captain Brooke used a pendulum to correct gun elevation for the normal heel of a ship. One might take this a step further and use it to detect the true angle of elevation during a roll, i.e., the sum of the gun's elevation from the deck and the ship's roll angle. For this to work properly, two criteria must be met.
First, the pendulum must have a long period relative to the roll period of the ship. If it has a short period, it will indicate the apparent vertical, perpendicular to the wave surface, not the true vertical, as discovered by Froude.
Secondly, the pendulum must hang from the center of oscillation (Atwood 252); the pendulum would then indicate the true vertical and the angle between that and the bore would be the true elevation plus 90 degrees. Unfortunately, the guns are distant from the roll axis, which should pass near the center of gravity. Brown (62) says, "one might expect an error of about 20% from a pendulum on the upper deck on an ironclad and of some 50% on a wooden battleship."
There are two choices, then. We can have a master pendulum at the proper location, with a gauge that reads off its angle with the "ship vertical." This angle could be communicated, perhaps electrically, to the gun stations.
Or, we accept the inherent inaccuracy and hang the pendulum near the gun. It's been suggested that the tilt sensor could be as crude as a cannonball hung from a nearby spar. The gunner would ignite the gun when "just before it was parallel to the mast." (NAVORD 15A2). If this was in fact done, it wouldn't have been easy to judge. Sometime before 1855, the French "used a reflector to compare the indication of the pendulum with the real horizon; this combination was called L'Horizon Ballistique." (Friedman 292).
In 1872, Froude designed an automatic roll recorder that featured both short and long period pendula. The long period one was robust, an eccentrically mounted wheel "three feet in diameter and weighing 200 pounds," with a halfperiod of 34 seconds. The roll recorder was used in sea trials of Inflexible (1882), Revenge (1895) and the Vivien gyrostabilizer (1925). It was not used in a firing mechanism. (Brown 62ff).
Bessemer proposed (1873) a firing device that featured a "tumbling bob," a slender triangular element positioned with wide end up, resting against one of two flanking arms. One arm was insulated, the other had an electrical contact, as did the bob. The whole assembly was itself mounted on a graduated quadrant, so it could be inclined at a specified angle to the frame of the quadrant, which corresponded to the desired true elevation at the time of firing. The idea was that with the bob resting against the insulated arm, the gunner would close one switch by pressing the firing button. When the ship rolled enough in the direction of the electrified arm so that the bob would fall over against it, the second and last contact would be closed and a firing signal delivered to the primer. Bessemer recognized that it would require a finite amount of time for the bob to change position, and that the launch of the projectile was also delayed by the "firing interval," so he provided a secondary movement for adjusting the neutral inclination to allow for this. It appears that Bessemer demonstrated a table model at the Royal Naval College, but his offer to fit a British warship with it at his own expense fell on deaf ears (Vincent 507).
Continuous Aim
In the latenineteenth century, if telescopic sights were available, they were used just to make the initial estimate of the range. The gunner dialed in the elevation but still waited for the roll to bring the aiming point into open sights. Accuracy was poor. Firing for five minutes each at a hulk 1600 yards away, five British warships managed to score a grand total of two hits. (Morrison).
In 1899, Percy Scott stunned the Royal Navy when his cruiser Scylla achieved an accuracy of 80% in a prize firing, about six times the normal performance. Rather than set a fixed elevation for the estimated range and try to time the roll, his gunners continuously aimed (i.e., adjusted the elevation) of their guns (Friedman 19).
While this was a procedural rather than a technological change, it was of course made possible by technological improvements, such as breechloading, rifling, elevating gears, and telescopic sights.
Moreover, Scott did some technological finetuning, too. He changed the gear ratio on the elevating gear so that the gunner could follow the target during the roll. And he modified the mounting of the telescope sight so it wouldn't be pushed back (into the gunner's eye!) by the recoil. (Morison).
Scott's methods revolutionized naval gunnery; in 1905, a gunner "made fifteen hits in one minute at a target 75 by 25 feet at 1600 yards; half of them hit in a bull's eye 50 inches square." (Id.)
However, the bigger the gun, the more difficult it was to move it fast enough to achieve "continuous aim." (Friedman 20). Also, that 1600 yards was about the practical limit without improved range estimation and prediction of target motion (22).
Gun Platform and Ship Stabilization
In 1889, Beauchamp Tower constructed and tested an apparatus for providing a steady naval gun platform. The position of the gyroscope affected the flow of pressurized water into four hydraulic cylinders on which the gun platform rested. The Admiralty tested it on two gunboats; it worked, but the weight was considered excessive. Thornycroft's pendulum, which hyrdraullically shifted a weight within the hold to stabilize the entire ship (1892), had the same problem. (Bennett 97). Still, in 1906 Schlick showed that an 1100 pound steamdriven gyroscope could reduce a torpedo boat's roll from 15^{o} each way to 1.5^{o}. (Airey 49).
There are two modern approaches to ship stabilization. Fin stabilizers have an angle of inclination that is gyroscopically controlled; they are effective only when the ship is traveling. A tank stabilizer operates even when the ship is at rest. One version used a single partially filled tank; others featured two wing tanks connected in some way, but with constricted flow between them. Care must be taken with tank stabilizers to ensure that they decrease rather than increase roll, and of course the tanks take up space and add to the weight of the ship.
Fire Control Systems
The preWW I increase in torpedo range to 1500 yards at high speed and 3500 at reduced speed provided considerable incentive for further increasing effective gun range, as "it was widely understood that a line of battleships would be a virtually unmissable target… [with] little or no underwater protection." (Friedman 22). That meant that further improvements in fire control were necessary for the gun to regain primacy.
Until the gun is fired, and during the "firing interval," the combined motion of the firing ship and the target ship cause the range and bearing of the target ship to be continuously changing. When the projectile leaves the muzzle, its velocity is the vector sum of the velocity imparted by the gun and the velocity of the ship (and the wind, if any). When the projectile is in flight, the firing ship's further motion is irrelevant, but the target's motion during the "time in flight" must have been anticipated, in order for the projectile to strike it.
Friedman (22) says, "Only once a ship's motion had been cancelled out did it really matter whether the range to the target was known." As a result of the relative motion of the ships, and latenineteenthcentury warship speeds, the range could change at a rate of "200 yards or more per minute." (Friedman 23).
The range to look up in the range tables is not the geometric range at the time the decision is made to fire, but rather the range the target is expected to be at when the projectile descends low enough to strike it. If the firing ship and the target ship are moving at constant direction and speed, the range will be changing at a nearly (Friedman 41) constant rate, and the range to set is the sum of the geometric range and the product of the "range rate" and the sum of the firing interval and the time of flight. But the time of flight is itself a function of the range.
The problem of the combined motion of the gun and target could be solved by hand using the same traverse tables that were used for navigation. These were essentially precomputed trig tables for a converting a distance on a course to a latitude and longitude change; you replaced "distance" with "speed," and interpreted "latitude" as rate of change parallel to the line of fire and "longitude" as the perpendicular rate.
During the earlytwentieth century, crude analog mechanical computers were used to take into account the effect of gun and target motion on the proper range and bearing setting. One such device was the "Dumaresq," invented 1902–4 and variations of which were used in WW I.
The Dumaresq subtracted the firing ship's velocity (direction and speed) from the enemy ship's velocity, resolving the difference into components along the line of bearing (the range rate) and perpendicular to it (the deflection rate). First, the inner ring was rotated to the enemy's bearing relative to your bow. Then the outer ring was rotated to your own heading. A slider was mounted on an overhead bar supported by the outer ring, and this slider was moved "aftward" to show your speed forward—thus subtracting it. A ring hung down from the slider and it was rotated to show the enemy heading. And this ring had a slider bar, and the slider was moved to show the enemy speed. A pointer hung down from that slider, marking a point on a graph that indicated the corresponding range and deflection rates (yards/minute).
Of course, to use the Dumaresq, you needed data:
Target Bearing. With experience, target bearing can be estimated by eye within 5–10^{o} (BMR). The downtimers already use an "azimuth compass" to determine the bearing of an object close to the horizon, for navigational purposes, and that's accurate to 0.5–2^{o}.
Own Speed and Course. To determine the "range rate," we must know our own ship's speed and course. Prior to RoF, sailors determined speed by the "common log" (tossing a log attached to a knotted rope behind the ship and timing how fast the knots passed over the rail) . A continuous log using some sort of rotating element was invented by Humfray Cole and published by Bourne in 1578, and another was constructed by Hooke in 1668. However, rotatory logs didn't come into common use until the nineteenth century, under the name "patent log." (EB11/Log; Robinson 53).
Earlytwentiethcentury naval fire control systems used a pitometer log; this measures the total pressure of seawater in the direction of motion and perpendicularly to it, and either measures the pressure difference or generates an equalizing pressure. The side pressure is just the static pressure of the water whereas the forward pressure is augmented by the motion, including a dynamic pressure proportional to the squared velocity. The pitometer log is analogous to a pitot tube airspeed indicator on an aircraft; the tube is used on the Belle built in NTL 1633. (Flint, 1633, Chapter 11).
Downtimers read their heading (approximately equal to the course) by comparing it to their magnetic compass, which points to magnetic north (or south). The magnetic compass reading requires correction for magnetic variation (caused by changes in the Earth's magnetic field) and deviation (caused by ferromagnetic materials on board). A gyrocompass finds true north; the first practical one was invented in 1908 (Wikipedia). It requires electric power. See "Gyro Sights," Part 2.
The heading differs from the course by the leeway angle. This will be affected by the ship speed and the wind and sea conditions; you can determine a typical leeway angle by maintaining a set heading from one known reference point to another in smooth water with a known current and known steady wind and see how far off course you go.
Target Speed and Course. Friedman (30) states that "at short ranges … it was relatively easy to guess enemy course by how foreshortened the target looked, and speed might be estimated from the appearance of the enemy bow wave." Observers trained themselves to estimate course by studying models of enemy ships, and might be aided by instruments that measured the angular width of the enemy ship and calculated the angle it made to the line of bearing if the range and ship length were known.
In turn, ships were given dazzle camouflage (sometimes including a fake bow wave) to make it difficult to judge course and speed. But even without camouflage, course was estimated by eye only to about a point (11.25^{o}), and speed was typically 15–30% off. (45).
The Dumaresq could also be operated in reverse, inputting the range rate (determined by successive rangefinder measurements) and the deflection rate. (The latter was not directly observable; what you saw was the rate of change of target bearing, which could be divided by the range to get the deflection rate. Some Dumaresqs were inscribed with bearing rate curves to make this easier.) That caused the elements to move to indicate the corresponding target speed and course.
Unfortunately, period rangefinders were not sufficiently accurate to make good estimates of range rates, which after all were the differences between range measurements made at short time intervals. (Id.) As for bearing rates, the trouble was that ships yawed back and forth a great deal, so the bearing rate was very messy. (44).
What you could do, instead, was use the Dumaresq to calculate the range and deflection rates, use them to estimate a later target range and bearing, and see how good the estimate was. If it was off, you adjusted the rates accordingly until you were happy with the prognoses.
Integrator. On the Vickers "range clock," a wheel spun at constant speed, and a spherical roller connected to an output shaft was held against it; the closer to the rim it was, the faster it turned. The roller was set to a position based on the computed range rate, and the outputs were the current true range (black hand) and the adjusted range for targeting (red hand). The latter would be entered into the range tables. I'm not sure how they adjusted the range for targeting without pulling the time of flight from the range tables, but perhaps it was a successive approximation method.
Unfortunately, the range rate wasn't constant even if speeds and courses were maintained. Changing the roller position was a bit awkward, unfortunately, so it was only changed at set intervals and there were "lagging errors." Ideally, you would be able to change the roller position continuously and instantaneously without interfering with the wheel rotation.
Communication. All the number crunching doesn't do you any good if the results aren't timely communicated to the gunners. This could be done by voice, through a sound pipe or phone, or purely electronically, by wire or radio. The concerns were to transmit the data accurately and without disruption by enemy action.
System Evolution. Initially, fire control systems were a "kludge." There would be a large number of operators, some reading off data from gauges and others inputting the data into the Dumaresq or a similar device or transferring the Dumaresq's range rate into a range clock.
By WW II, the data was fed directly into the computer by the various ship sensors and the computer even outputted "gun orders."
Other Sources of Error
Powder Conditions. Muzzle velocity increases with powder temperature. Figure a change of 2 fps per ^{o}F. (NAVORD 17B4 Col. 10). A damp powder burns more slowly, a dry one, more rapidly. Powder that has been in storage a long time may deteriorate, resulting in a lower initial velocity and also in increased pressure within the bore. (Alger 186).
Projectile Variation. Dahlgren, inspecting a heap of 32pounder shot, found diameters of 6.22–6.28 inches, and weights of 32.43–33.00 pounds. With ten such rounds, at 3^{o} elevation, the mean range was 1172 yards and the mean difference 18.4. If he used shot selected so the gauge range was 6.24–6.26 inches (weight 32.43–32.47), the mean range was 1195 yards and the mean difference only 10.3.
Variation in projectile weight has two opposing effects; a heavier projectile is accelerated less, inbore, leaving with a lower muzzle velocity, and decelerated less in flight, giving it greater range for a given muzzle velocity.
Jump and Droop. If the gun, when fired, has a positive elevation, the force of discharge causes the gun to rotate, bringing the muzzle higher ("jump"). On the other hand, the length and weight of the gun may cause the muzzle to droop, so the "line" of bore is not truly a straight line. Jump and droop were not calculated directly, but the coefficient of form calculated by comparing computed ranges with experimental ranges is determined not merely by the projectile per se, but the jump and droop it experienced when fired. (Alger 167ff).
Air density. Earlytwentiethcentury range tables were calculated for a "standard atmosphere" of half saturated air, 15^{o}C, 29.53" pressure. (Alger 189). Atmospheric pressure, temperature and humidity all affect air density and thus air resistance. For example, for Alger's standard problem 12" gun, range 10,000 yards, if the barometer were 29.00" and the thermometer 96^{o}F, the air would be 9% below standard density, and the range achieved would be 194 yards higher. (189).
The effect of pressure and temperature can be determined by the ideal gas law, and that of humidity estimated by comparing the molecular weights of water vapor and dry air.
Pressure, temperature and humidity all vary from place to place and from time to time, and to take them into account, you need to be able to measure them. That measurement, of course, will be for where the gun is located and could be a bit different at the target (or inbetween). The first mercury barometer was invented in the 1640s, whereas the aneroid barometer was introduced in 1843. The first sealed alcoholtype thermometer was made by Grand Duke Ferdinando II de Medici in the 1650s, and the mercury thermometer in 1714. You may measure relative humidity by comparing the readings of wet and dry bulb thermometers after the wet bulb is taken out of water so that evaporation can occur.
The variation of air density with altitude is usually significant only for a high trajectory fire. The US Standard Atmosphere is in CRC and relates density to altitude. So it's easy enough to modify a numerical integration spreadsheet to calculate the density for each altitude rather than treat it as a constant.
A further complication is that temperature and humidity also affect the velocity of sound, and thus the Mach number of the projectile for a given speed. (Ingalls vii).
Per my spreadsheet, if altitude effects are ignored, a 24pound iron ball projected 1600 fps at 45^{o} ascends to 4385 feet, and ranges to 3467 yards. Including the effects of decreased gravity (99.96% at apex), air density ( 87.98%) and speed of sound (98.45%) , it climbs to 4459 feet, and ranges to 3617 yards.
Drift. The spin of the projectile fired from a rifled gun causes it to be deflected to one side; this drift increases with distance flown and is the net result of several forces. (Denny 113ff).
Projectile Angle of Attack. 3DOF methods assume that either the projectile is spherical, or that if elongated, its long axis is always tangential to the trajectory (zero angle of attack). In practice, it isn't. If it's a spinstabilized projectile, the longitudinal axis will precess (rotate around) an equilibrium line, which itself is at an angle (the yaw of repose) to the direction of motion. The line points right for a righthand spin. The yaw of repose thus is the average angle of attack. Air resistance increases as the angle of attack increases. The yaw of repose at apogee is perhaps 2^{o} for an elevation of 50^{o}, but 10^{o} or more for an elevation of 65–70^{o}. (Pope). For a statically stable nonspinning projectile, the nose points slightly above the trajectory. (Carlucci 255).
Wind. If we can measure the wind force and direction, then we can take it into account. Wind direction is easy, that's shown by a wind vane. I am not sure of the practicality of the prenineteenthcentury mechanical anemometers; the cup anemometer was invented in 1846 and I think that it would be fairly easy to duplicate. It may even be possible to measure the speed of rotation by mechanical or electrical means, but one must cope with the way the wind fluctuates. It will be necessary to calibrate the anemometer, i.e., determine the relationship between the speed with which it is rotating and the force of the wind. (EB11/Anemometer notes that some scientific literature falsely assume that the cup speed was onethird the wind speed.) The relation of anemometer rotation to wind speed can be determined by mounting the anemometer on a car and driving the car at a set speed on a day the air is still. That still leaves the problem of relating wind speed to wind force; this may be in a civil engineering handbook since builders of tall buildings must worry about wind loads.
Of course, we will just know the wind felt by the firing ship, and this will be the apparent wind felt by the ship (which is moving), rather than the true wind. So we must be able to convert the apparent wind on the ship to the true wind, and then that to the apparent wind felt by the projectile. That requires knowing the speed and direction of the firing ship.
Wind is less consistent than the other atmospheric conditions; it can change in direction and strength while the projectile is in flight. And there's not much we can do about that other than look for telltales downrange of how the wind is behaving there. Also, in general, ballisticians only consider horizontal winds; updrafts and downdrafts are ignored.
Smoke. The smoke produced by a broadside can be such that it is no longer possible to tell by eye when the cannon is pointing horizontally. One nineteenthcentury British captain used a spirit level while his ship was motionless, in harbor, to make sure that all his guns were pointing horizontally, and then added heel scales to them to indicate the correction for the angle of heel (which could be measured in battle with a ship's pendulum and called off to the gun crews). This became common practice. (EB11/Ordnance; Douglas 218). Fortunately, the enemy's masts would usually be visible above the smoke, even when the hull wasn't, so the horizontal direction of fire could still be determined by sight. (220).
Miscellaneous. At higher ranges, gunners consider the Coriolis force caused by the rotation of the Earth (sixinch deflection at 1000 yards), the decrease of gravitational force with altitude, and the curvature of the Earth.
Farnsworth Gun Error Computer. This was a mechanical device, constructed in 1915, for determining the errors in range and deflection attributable to wind, firing ship motion, target motion, powder temperature, and atmospheric density. It appears to have been a kind of circular slide rule with graphic representations of the values in the ballistics tables. (Alger App. B).
The following table provides a sample of how changing conditions could create range and lateral errors. Do not make the mistake of assuming that the stated errors will hold true for other guns or projectiles, or other ranges; the underlying equations are highly nonlinear.
Table 35: Sample Range Errors and Deflections as a result of Variations in Conditions, per Ballistics Tables, standard atmosphere  
Problem A 3", 13 lb shell, 1150 fps, coefficient of form +1.00 normal range 2000 yards  Problem C 4", 33 lb shell, 2900 fps, coefficient of form 0.67 normal range 3000 yards  Alger page  
condition  range error (yards)  deflection  condition  range chg (yards)  deflection 

drift 
 7.5 

 2.4  152 
+25 fps  +49.8 
 +50 fps  +83.4 
 139 


 +1 lb  33.7 
 140 
Wind 8 kn with  +16.5 
 Wind 11 knots with  +12.7 
 164 
Wind 6 kn right 
 6.3 R  Wind 9 knots right 
 5.7 R  " 
Gun motion 6 kn against  14.7 
 Gun motion 8 kn against  11.5 
 165 
Gun motion 8 kn left 
 23.6 L  Gun motion 9 kn left 
 15.7 L  " 
Target motion 7 kn with  21.4 
 Target motion 9 kn with  17.3 
 " 
Target motion 8 kn R 
 24.4 L  Target motion 10 kn R 
 19.2 L  " 
28.10", 5oF  20.4 
 29.00". 15oF  64.9 
 206 
****
This article continues in Part 4, "Implements of Destruction."