Gun sights and other fire control equipment could not be made before the performance of the weapon they were to aim was well understood.  The process for determining this extensive and precise knowledge was based on test firings of the weapon, but there was more to it than that for several reasons.

Format and Content

The most desirable form for such documents was generally a header and a lengthy tabular data set and possibly a few ancillary tables. 

The header indicated the gun, shell and powder charge (what I generally call a "weapon system", though I'm not sure it had a contemporary name) whose joint operation was being recorded, as well as specifics on the muzzle velocity obtained and the atmospheric conditions for which the firing data corresponds. 

The tabular data detailed the performance of the weapon system throughout the entire envelope of ranges it it could attain on mountings it would be fitted to, sorted from minimum to maximum.  The first column indicated ranges (in increments of 100 yards or meters, depending on the nation), and the remaining columns the other parameters such as the angle of elevation required to loft the shell to the range, the time of flight, the terminal velocity at impact, and the angle at which the shell would fall.  Of these, the range/elevation relationship was most important.

Additional tables, when provided, might include other information, such as

• the amount of lateral deviation at each range attributable to ballistic drift (an aerodynamic effect of the spinning shells)
• the "50% errors" along and across the line of fire at each range (a statistical measure of the the weapon system's precision)
• how quickly the gun could be expected to wear out after a certain number of rounds being fired. 

Such data was essential to create equipment and procedures to support the gun's use along its entire service life.

Proof Firing a Weapon System

The Royal Navy had a large range at Shoeburyness where new weapons could be trundled out for firing.  Presumably, no one lived on the reservation (or only those homeowners possessing both a sensitivity to price and a contrarian view on actuarial science).

One might imagine that a sample gun might just be fired so many times at a number of elevations that the range table could simply be the result of those firings where the shell traveled an even multiple of 100 yards (or meters).  But such an ideal method would be both very time consuming and expensive when one considers the cost of shells, powder and that the guns wear out quite quickly and each successive firing occurs at a progressively degraded muzzle velocity.  These factors dictated a different approach be taken.

A ranging and accuracy "practice" was conducted according to a program crafted by the Ordnance Board and resulted in a report that went something like this.  The background conditions were first reported, and contained the following information:

• date of practice
• place and direction of firing ("along line of pegs "G" bearing N. 80 E.")
• gun type and serial number
• complete firing history of the gun ("rounds previously fired, 3 proof and 2 full charges")
• mounting used
• identification of type of projectiles used
• powder charge employed
• temperature of charge
• barometric pressure
• air temperature by dry bulb and wet bulb measures

The test firings would occur in approximately 4 distinct "series" of firings, each series comprised of 5-10 shots all taken at the same quadrant angle of elevation.  For the first series (at the lowest elevation), there were be additional means taken to discover the muzzle velocity and the angle of "jump" of the gun. 

Muzzle velocity was not directly measured, but back-solved for by measuring the shell's velocity a short distance from the gun.  This velocity measurement was obtained by having the shell pass through two skeins of electrified wires around 100 yards downrange.  As it broke these wires, high speed timing devices measured the interval between the breaking of the two separate circuits.  Some mathemagic allowed the small loss in muzzle velocity to that point to be factored out.  "Jump" (a small vertical "flinch" a mounted gun gives as it discharges its shell) was measured by having the projectile pass through a screen some 400 feet from the muzzle, and comparing the height of this hole from a straight line down the bore of the gun before it was fired, possibly correcting also for the short distance the shell would have already fallen at that 400 foot distance.

Additional data recorded for each shot taken were time of flight to impact (in 100ths of seconds), range attained (in yards), lateral deviation (in 10ths of yards), velocity at the skeins (in feet per second) and the distance these were from the muzzle, muzzle velocity calculated from that figure, wind speed in feet per second, wind direction in compass degrees, and apparent "jump" of the gun at firing, in arc minutes, by judging where the shell holed the screen.

Once the data was collected, the individual shots were grouped by series, and mean values for the shots within each series computed.  Additional data in this reduced table also showed that the process was mindful of how the muzzle of the long gun was itself elevated "above the sands" as the gun elevated through its arc, and the wind was helpfully decomposed into components along (head or following?) and across (right to left or left to right?) the line of fire and by some means computed at a height of 101 feet though there is no direct indication of the height at which it was measured.  The last thing in the series table were the measures of the variety in range and lateral deviation observed across the shots within the series.  This last was to serve as the basis for computing 50% errors.

Analysis of Firing

Once the above data was in hand, the considerable task of massaging the numbers began.  Fine points tackled here included:

• turning the global nature of the firing range into a planar case by correcting "muzzle height above sands" to factor in curvature of the earth (the impact point is, in effect, always "downhill")
• calculating the "angle of depression", that extra amount of "looking down" necessary to see the impact point in its "hole" behind the curving earth.
• considering "droop" of the barrel, that small downward arc it has under its own weight and imperfect rigidity

Then, the work must focus on neutralizing the conditions or each firing to a common, standard set of conditions.  Before going further, I'll introduce a few terms here to illustrate the depth at which this was understood and tackled in the era, but will eschew the incomprehensible Greek symbols and inconsistent choice of physical units that seem purposed to ensure that such matters remain "Greek" to the largest audience possible.  Ugh.  I will err on the side of giving them descriptive names that don't invite continual visits back to a legend, and dare to throw in some well chosen computer operators here and there (think C, C++ or Java).

Wind is factored out and is even considered to have a small effect in adding to "jump", according to "Greenhill's treatment".  For each shot, then,

jump += arcsine( sin(angleOfDeparture) * windFollowing / muzzleVelocity)

range += windFollowing * timeOfFlight

velocity -= windFollowing

The interpretation of this last formula is, in my mind, questionable, as I do not see why it is not

velocity -= windFollowing * cosine(angleOfDeparture)

After each shot's various differences are neutralized, an arduous process is undertaken to derive a "coefficient of reduction" for each series. This procedure, as documented in Manual of Gunnery for HM Fleet, Volume II (1917), alludes to the O. B. Ballistic Tables which I do not have, so I'll skip on that part for now.  Suffice it to say that each series has a coefficient of reduction computed for it, and a suitable single value chosen somewhere within the set is then selected to characterize the entire weapon system.

The table was then created, but only around 5 or 6 of the hundreds (200, in the case of a Royal Navy table extending to 20,000 yards) of rows were actually computed.  Rather, 5 or 6 rows scattered along the table (say 1.5, 3, 6, 9, 12, and 15 degrees angle of departure) would be computed from the O. B. using the coefficient of reduction, and all further points then obtained by graphical interpolation of these 6 points, presumably, using a draftsman's spline.

While I am confounded by my lack of insight into the O.B. Ballistic Tables and the computation, meaning, and use of the coefficient of reduction, I have to express considerable surprise that the canonical documents of this time actually contain data so indirectly based upon a tiny number of experimental trials.  While clearly there would be some difficulty in measuring angle of descent and remaining velocity of shells, the other vital relationships of range to elevation (or drift, or time of flight) would seem better sourced from the firing trials themselves, and not to distortions of a single model of projectile flight.

I will perhaps add to this section at a later date.

Simulation

I stripped some core code out of my prototype 3D simulation that creates range tables in formats similar to the historical ones.  I offer some discussion on my method for doing this, and upon the results achieved. Given what I have since learned about the compilation of range tables, I seriously must consider how authoritative the range tables actually are in capturing the performance of the firings they implicitly seem to embody.