# Did the British navy fail to take into account the ballistics correction due to Coriolis force during WW1 Falkland Islands battle?

In Natural History magazine article, Neil DeGrasse Tyson claims the following:

The military normally knows all about the Coriolis force and thus introduces the appropriate correction to all missile trajectories. But in 1914, from the annals of embarrassing military moments, there was a World War I naval battle between the English and the Germans near the Falklands Islands off Argentina (52° south latitude). The English battle cruisers Invincible and Inflexible engaged the German war ships Gneisenau and Scharnhorst at a range of nearly ten miles. Among other gunnery problems encountered, the English forgot to reverse the direction of their Coriolis correction. Their tables had been calculated for northern hemisphere projectiles, so they missed their targets by even more than if no correction had been applied. They ultimately won the battle against the Germans with about sixty direct hits, but it was not before over a thousand missile shells had fallen in the ocean.

The physics explanation is correct - the correction due to Coriolis force is opposite on Southern hemisphere. This incident is described also in physics textbook Analytical mechanics by Hand and Finch. By direct calculation, Coriolis effect is approximately 100 meters, so this seems significant. However I'm not sure if other factors (such as wind, rocking of the boat and evasive manoeuvres of the target) would be as large.

Did this actually happen during the Battle of Falkland Islands? Was the inaccuracy due to miscalibration of artillery due to wrong Coriolis effect corrections?

EDIT: Some calculational details, as requested by @Querky.

Gravitational acceleration: g = 9.81 m/s2

Angular velocity of Earth: omega = 2*pi/(24*60*60 s) = 1.5*10-4 1/s

Velocity of the artillery shell: v(0) = 500 m/s

Max height:

v(t) = v(0)-gt = 0 -> tmax = v(0)/g = 50.97 s -> h = v(0)/2*tmax = 25.49 km

Coriolis correction (solution of 2. order differential equation):

y = 23/2 omega cos(theta)*h3/2/(2*sqrt(g)) = 0.4515*1.5e-4*cos(52°)*254903/2 = 170 m

• A "missile" is just an object that is thrown or launched. It's only in modern terminology that the "self-propelled" or "guided" modifiers have become implied. Aug 22, 2019 at 6:20
• Ballistics of missiles is what I do. Under about 100 km range the Coriolis effect for a missile at sea level can be neglected (the really big parameter is wind). Also artillery crews watch each missile and adjust accordingly. Sorry no references or I would make this into an answer. Aug 22, 2019 at 7:57
• @Qwerky Sure, I added it above. Unfortunately LaTeX is not supported, so it is ugly but readable. Aug 22, 2019 at 10:28
• The Paris Gun is a better example of the Coriolis effect in WWI. Aug 22, 2019 at 10:59
• @RogerLipscombe given that we live and converse in the modern world, I prefer to try and keep it unambiguous and call them projectiles. While missiles is correct, it does stimulate thinking about modern munitions. Aug 22, 2019 at 15:50

No. It is a complete invention of an urban myth.

Unless anybody produces a Royal Navy document that even does any mention of this influence of Coriolis effect, this seems to be just untrue. A complete fabrication for a good story – good for teaching physics but very bad history.

It seems to be an annoying urban myth. The Coriolis effect is present and significant for shots at long ranges and flight duration. But it is not the only factor involved and naval gunners did not rely on tables alone. The Coriolis effect for any gun on any ship at any latitude in any direction is a constant influence on accuracy if shooting continues into the same direction.

Further, the observations made during peacetime were not made for this range and speed at all. The only time approaching this range was made on the Southern hemisphere en route to the Falklands and the ships involved did a run of gunnery practice the day before the battle. Any systematic error that big as claimed would have surfaced by then and dealt with accordingly.

Such systematic errors should be quickly spotted and corrected for as long as the targets are still visible, that is: above the horizon if spotting is to be done from the same ship that is firing.

This myth is entirely absent from all history books I checked on for this battle. It is also missing from German accounts of the events, as evidenced by how German Wikipedia narrates the analysis:

During the battle, the two British battle cruisers fired the considerable quantity of almost 1200 305 mm grenades at the German armoured cruisers, which allows conclusions to be drawn about the training of the operators and the quality of the grenades. However, the Royal Navy did not draw any conclusions from this circumstance until after the Battle of Skagerrak one and a half years later.

A more specialised Wiki has its own entry on Coriolis effect and on the page related to the battle puts it this way:

Coriolis Effect An annoying urban legend persists that the Royal Navy's shooting at the Battle of the Falklands was poor due to their equipment applying corrections for Coriolis effect in the wrong direction, as the action was in the southern hemisphere rather than the northern. The truth is, however, that no contemporary aspect of Royal Navy equipment or procedure took Coriolis effect into consideration, an extremely minor deficiency. For, even if the fable were true, if the action took place on a nearly constant bearing, and at a range that changed only slowly, even a blatant mistreatment of Coriolis effect such as its negative consideration would therefore have been a constant error, and one unlikely to be large compared to other factors affecting the proper deflection to use (such as the zig-zagging of a fleeing enemy). This fact implies that the remedy for such a miscue would have been a single spotting correction for deflection which, once made, would counteract the error for the remainder of the action.
While I think it likely that later systems of firing incorporated Coriolis corrections, a system lacking such treatment which is designed primarily to bring fire upon a maneuvering enemy is not a sad system by any means. Taken in context, Coriolis errors are a constant source of deflection error and very small in degree. The need to fire repeated salvoes which for many reasons will require spotting to put them onto the target implies that a failure to handle Coriolis effect, or even handle it entirely backwards, would not prevent a shooter from hitting his target in a prolonged engagement.

And the British Royal Navy summarises, without ever mentioning "Coriolis":

System 1914

1. In the year 1914 a high standard of efficiency in the control of fire was attained; the system used was thoroughly understood and there was no lack of confidence in the ability of the fire control system to compete successfully with the accepted standards of range and general battle conditions.
2. Briefly, the rangefinder equipment was used to feed the fire control gear, and the latter was relied upon to furnish the requisite information for successful attack on a moving target. In conjunction with this, the bracket system of spotting was universally used to correct the best mean range into the actual gun range after opening fire.

The correction of the remaining factors, such as rate and deflection, was primarily dependent upon observation of fire, although great importance was attached to the use of the fire control gear as a guide.

Battle Experience

1. The earliest engagements of the war gave no cause to suspect that the firing rules were inadequate to deal with battle conditions.
2. The Battle of Heligoland Bight in 1914, fought in very low visibility was not of a character to produce any reliable evidence one way or another. It showed, however - The impossibility of taking ranges in low visibility conditions.
3. The action off the Falkland Islands in the same year demonstrated the following:
• The rangefinder equipment failed to provide much information. (This was chiefly due to the range at which the action was fought, which outclassed the 9-ft rangefinders.)
• The use of defensive tactics (zigzagging) rendered the control of fire extremely difficult, and placed a high premium on rapidity of fire as soon as the gun range was found.

USE OF 100-YARD CORRECTIONS

1. Reports were called for at the end of 1932 regarding the desirability of limiting the occasions on which corrections of 100 yards for range are permissible, in view of the frequent occasions on which range spotting corrections are insufficiently bold.
2. As a result of these reports it was not considered that a case had been made out for altering the existing rules, but it was stressed that control officers must constantly bear in mind the necessity for the use of bold spotting corrections, especially when aircraft are not available for spotting.
–– ADM 186/339 C.B. 3001/1914-36: Summary Of Progress In Naval Gunnery, 1914-1936, Training And Staff Duties Division, Naval Staff, Admiralty, S.W., December, 1936.

It is curious for the claim to only focus on the battle at the Falklands. As for the previous battle at Coronel, also on the Southern hemisphere, the outcome was similar. And again at Jutland, this time on the supposedly 'correct' hemisphere for correctly calibrated Coriolis compensation: the British range finding and gun aiming was just inferior. (As the German ships had similar conditions, from perfect manoeuvre-like going over to smoke-induced blindness at high-speed, but remaining at a better hit-rate)

As books focusing just on that specific battle alone never mention Coriolis effect, a work that analysis just the history of naval gunnery and aiming techniques concludes:

The rangefinder measured the geometric range between shooter and target, which was sometimes called the true range. Given rangefinder errors, this measurement could not be entirely accurate, but it is convenient to identify the rangefinder figure with the actual distance between ship and target. This range was not the same as the gun range, the range to which sights should be set. Gun range took into account the movement of the target while the shell was in the air and even that of the shooter while the shell was in the gun (where it shared the ship’s motion). It thus involved knowledge of how the range was changing: the range rate. The longer the range (ie, the more time the shell spent in the air), the more significant the range rate. At very long range, factors such as the rotation of the earth had to be taken into account. It began to matter that a ship was able to measure her own speed. That was difficult: only in about 1912 did the Royal Navy obtain an electric log (measuring speed). Other navies were probably in about the same position: the Germans license-produced the British log.

The British were fairly sure, moreover, that their understanding of gunnery was far in advance of any other navy: in 1906 DNO’s assistant Captain Harding remarked that foreign navies did not yet understand the difference between geometric and gun range. Only recently had British officers realised how important it was to know the geometric range precisely, rather than depend on spotting beginning with an approximate range. Presumably this referred partly to Captain W C Pakenham’s comments during the Russo-Japanese War (Pakenham was Royal Naval attaché to Japan at the time): ‘Outside the Service the impossibility of continuous use of the rangefinder and therefore the importance of a knowledge of the rate of change [range rate] is not recognised, consequently the means of its determination are unsought for.’ No one had tried to make a rangefinder record its output automatically, and no one (apart from Pollen, see chapter 2) had realised the importance of using a gyro to eliminate yaw from rangefinder bearing readings. The Germans were probably the most advanced foreign navy at this time. Little was known of their thinking, but the evidence of what they were using (sextants with a few unmodified Barr & Stroud rangefinders) and of articles in their main annual publication, Nauticus, suggested that they were not working along British lines.

The range rate
Successful gunnery required that the position of the target be projected ahead, ultimately to the moment at which a shell might be expected to hit. To do that, the shooter had to calculate the rates at which the range and bearing of the target changed; they were usually called the range and bearing rates. Calculation was difficult because neither was constant, and because each depended on the other. Alternatively, one might think in terms of the vector (magnitude and direction) pointing from shooter to target. The change in this vector was another vector which might be called the rate vector. It could be expressed as two components, one along the line of fire and one across it. The rate along was usually called the range rate. The rate across was usually called deflection. Its magnitude was the bearing rate multiplied by the range.

The Falklands
In December 1914 two British battlecruisers fought Admiral Graf von Spee’s Pacific Squadron, which had recently sunk the HMS Good Hope at Coronel. This time visibility was excellent (only in the last hour of the battle did it fall to 15,000 yards), and both squadrons steamed at high speed. Neither British ship had a functioning director or a Dreyer Table. Both ships found their fire control hampered by funnel smoke, so that although her fore conning tower and A turret never lost sight of the enemy, in Invincible the fore top occasionally lost sight, and P, Q, and X turrets were much affected. Rangefinding was very difficult due to the long range, funnel smoke, splashes and spray from the enemy. Rate-keeping was difficult at best, due to the enemy’s zigzagging as well as to the very long range (variations in range were almost undetectable). Gunners found it difficult to stay on a point of aim, sometimes mistaking the target’s bow for her stern. On the other hand, according to prisoners, British shells performed well, penetrating and exploding deep in the ships. Even so, Gneisenau took fifty 12in hits before sinking. It was no great surprise that the British ships used up most of their ammunition: one 12in gun in Inflexible fired 109 rounds (the ship was designed to carry eighty for that gun).

To the surprise of the British, von Spee’s ships zigzagged to avoid being hit, even thought that made hits by their own guns unlikely. As crack gunnery ships, the Germans were expected to fire at maximum range, but the actual figure for their 8.2in guns, 16,000 yards, seems to have surprised the British. The Germans straddled (without hitting) at 15,000 yards. The Germans persistently fired salvoes (the British thought, wrongly, that they were using directors), and their direction and fire discipline were excellent. The British were impressed by the effect of plunging shells at such ranges, and by the blast effect of the German fire. The German survivors stressed, and the British noted, that slow British fire made it easier for their own gunlayers. It also made spotting easier, because the British ships were much less completely enveloped in the smoke of their own guns. This may have been the first of many British observations that their firing techniques were far too deliberate.
-- Norman Friedman: "Naval Firepower. Battleship Guns And Gunnery In The Dreadnought Era", Seaforth Publishing: Barnsley, 2008.

Further reports include

Excessive smoke was not the only cause of the slow, inaccurate gunfire of the battle cruisers. A British officer in the spotting top of Invincible, Lieutenant Commander Hubert Dannreuther, who happened to be a godson of the composer Richard Wagner, found that his excellent, German-made stereoscopic rangefinder was rendered useless not only by smoke, but also by the vibration caused by the ship’s high speed, and by the violent shaking of the mast whenever A turret fired. In Invincible’s P turret, conditions were impossible. The gun layers could see nothing except enemy gun flashes through enveloping clouds of smoke, and every time Q turret, across the deck, fired over them, everyone in P turret was deafened and dazed by the blast. On Inflexible, Lieutenant Commander Rudolf Verner in the battle cruiser’s foretop was almost the only man aboard his ship who could judge the location of the enemy, and he, handicapped by the smoke from the flagship ahead, had great difficulty observing what damage his gunners were causing. […]

From Invincible’s spotting top, Dannreuther reported, “She was being torn apart and was blazing and it seemed impossible that anyone could still be alive.” On Inflexible, Verner, astounded by the continuing salvos from the German armored cruisers, ordered his crews to fire “rapid independent,” with the result that at one point, P turret had three shells in the air at the same time, all of which were seen to land on or near the target. Yet the German fire continued. “We were most obviously hitting [Scharnhorst,] but I could not stop her firing. […] I remember asking my rate operator, ‘What the devil can we do?’ ”[…]

There were many reasons for what at first sight seemed inefficient ship handling and inept gunnery in the British squadron. Before the war, few British naval officers had appreciated the inherent inaccuracy of naval guns at long range. The only time that Lieutenant Commander Dannreuther, the gunnery officer of Invincible, had been allowed to fire at ranges in excess of 6,000 yards was during the practice authorized by Sturdee on the way south to the Falklands—and he had been gunnery officer of the battle cruiser since 1912. Nor had peacetime practice disclosed the difficulties of shooting accurately from a rapidly moving platform at a rapidly moving target. Further, no one had considered that when ships were traveling at high speed, the intense vibration created by engines and propellers might rattle and blur the gun layers’ and trainers’ telescopes. Nor had prewar maneuvers revealed the obscuring effects of billowing funnel smoke at high speed. As the war went on, the expected rate of shells fired to hits achieved became 5 percent. That was approximately the ratio in the Falklands, but at this early time in the war, everyone expected better and therefore it seemed a failure.
–– Robert K Massie: "Castles Of Steel. Britain, Germany, and the Winning of the Great War at Sea", Ballantine Books: New York, 2003.

To correct a few of the invented history details of the physics teacher, the report of a gunnery officer gives:

Primary Control from Fore Top was used throughout. At times the control was very difficult as we were firing down wind the whole time and the view from aloft was much interfered with by gun smoke and funnel smoke. Range Finders were of little use and any form of range finder plotting was impossible owing to the difficulty of observation and high range.

In fact as far as this particular action was concerned it would have made no difference if the ship had not had a single Range Finder or Dumaresq or any plotting outfit on board.

During the latter part of the action with the Gneisenau (she) continually zig-zagged to try to avoid being hit, altering course every few minutes about two points either side of her normal course. This alteration of course could not be detected by Range Finder or by eye and continual spotting corrections were necessary. The rate being fairly high and changing every few minutes from opening to closing I found the only effective means was to keep the rate at zero and continually spot on the target. By this means we managed to hit her now and again.
-- Quoted from Richard Hough: "Falklands 1914: The Pursuit of Admiral Von Spee", Periscope Publishing, 2003.

And finally, the first shots being fired by the British did not land sideways to the target. They were just a bit too short at their maximum range of 16,500 yards:

At 1247hrs, Sturdee hoisted the signal ‘Engage the enemy’ and eight minutes later Inflexible opened fire with her forward turret, sending two 12in. shells arcing towards Leipzig at a range of 16,500 yards. The shells fell considerably short, but a few minutes later Invincible began her participation in the battle with a salvo that landed a mere thousand yards short of their target and soon the battle cruisers’ gunnery officers, Hubert Dannreuther and Rudolf Verner were calling out near misses as the German light cruiser was straddled by towering waterspouts.
–– Michael McNally: "Coronel and Falklands 1914, Duel in the South Atlantic", Campaign 248, Osprey: Oxford, New York, 2012.

A similar assessment of the claim is found at The continued badhistory of Neil deGrasse Tyson: This time, it's the slightly esoteric field of WW1 naval fire control.

The source

The origin of this myth can be traced back to admitted hearsay by John Littlewood in 1953 (who is – in non-history science-related papers even said to have served as a gunner in the war):

I heard an account of the battle of the Falkland Islands (early in the 1914 war) from an officer who was there. The German ships were destroyed at extreme range, but it took a long time and salvos were continually falling 100 yards to the left. The effect of the rotation of the earth is similar to 'drift' and was similarly incorporated in the gun-sights. But this involved the tacit assumption that Naval battles take place round about latitude 50 N. The double difference for 50 S. and extreme range is of the order of 100 yards.

Various attempts of mine to set this in examinations failed. ' I had hoped to draw the criticism of unreality', to which there is the following reply. In 1917-18, a range table was called for, for* the first time, and quickly, for a gun in an aeroplane flying at a fixed height, to fire in all directions. A method existed, based on numerical calculation of the vertically upward and vertically downward trajectories. It happened that within the permissible limits of accuracy the values of A and p could be faked to make A--JU, (and downward trajectory could accordingly be read off from a table of sines, and the range table was in fact made in this way (in about two-thirds 1 the time it would otherwise have taken).

I do not deny that the example just given is slightly disreputable…

–– John E Littlewood: "A Mathematicians Miscellany", Methuen: London, 1953 (p51). (archive.org)

During the war

In World War I Littlewood served in the Royal Garrison Artillery. His contributions were highly significant and special allowances were made to keep him happy, such as letting him live with friends in London, and to carry an umbrella when in uniform! Littlewood himself described this war work in 'J E Littlewood, Adventures in ballistics, 1915-1918'. The result was that he improved the accuracy of anti-aircraft range tables and improved the formulae for finding the range, the time of flight and the angle of descent at the end of a trajectory with small elevation.

E A Milne has described how Littlewood was able to discover techniques which greatly reduced the amount of work needed for making these accurate calculation of missile trajectories. Trials were conducted to see if the results of Littlewood's predictions held in practice and, Milne writes:-

... to the astonishment and joy of all concerned the observed positions of the shellbursts fell exactly on Littlewood's trajectories, at the correct time-markings, within very small errors of observation.

This unreliable anecdote seems from 1953 onwards to gain little traction, but slowly found its way from lectures into mathematics and physics textbooks in the following years.

Modern handbooks of artillery simply state to ignore Coriolis for distances of less than 10–20 km

When firing at distances beyond 10 km the Coriolis’ force must be included as its contribution may exceed more than one per cent of the range of the rocket. However, there are no principal difficulties with the inclusion of this force in the equations. The trajectory calculation will consequently depend on the latitude of the firing post and the azimuthal direction of fire. In order to include this force a three-dimensional model must be applied.
–– Ove Dullum: "The Rocket Artillery Reference Book" Norwegian Defence Research Establishment (FFI), 2010.)

Or simply

Projectiles, which travel great distances, are subject to the Coriolis force. This is actually not a force at all, but an apparent acceleration caused the Earth’s rotation. The local frame of reference (north, east, south and west) must rotate as the Earth does. The amount of rotation, also known as the earth rate is dependent on the latitude:

earth rate = (2π radians)/(24 hours) × sin(latitude).

For example, at 30 N, the earth rate is 0.13 radians/hr (3.6 x 10 rad/sec).

As the frame of reference moves under the projectile that is traveling in a straight line, it appears to be deflected in a direction opposite to the rotation of the frame of reference.

In the Northern Hemisphere, the trajectory will be deflected to the right. A projectile traveling 1000 m/s due north at latitude 30 N would be accelerated to the right at 0.07 m/s . For a 30 second time-of-flight, corresponding to about 30-km total distance traveled, the projectile would be deflected by about 60 m. So for long range artillery, the Coriolis correction is quite important. On the other hand, for bullets and water going down the drain, it is insignificant!

–– Craig M. Payne: "Principles of Naval Weapons Systems", US Naval Academy, 2000

If you want to re-calculate the exact amount of deflection that would have been attributable to Coriolis effct at the Falklands:

The problem as presented in courses at Harvard (PDF):

1. The Coriolis Force

Deflection due to Coriolis force is given by ∆y = (∆x)2 Ω sin(λ) / v, where v is the speed, λ is the latitude, ∆x is the distance traveled, and Ω is the angular velocity of the Earth (7.3x10-5 s-1).

(a) Find the displacement of a snowball thrown 10m at 30 km/h in Cambridge (latitude 42°N). (2 points)

∆y = ((10m)2 * 7.3x10-5 s-1 * sin(42°))/(3.0x104/(60*60) m s-1)

∆y = 5.9x10-4 m = 0.59mm

• Nit-pick: "The coriolis effect for any gun on any ship at any latitude in any direction is a constant influence on accuracy if shooting continues into the same direction." For a given point on the Earth, the effect is the same in all directions. At least up to first order. There is a small difference because the strength of the effect changes depending on your latitude, and the projectile does change its latitude during flight depending on the direction it's shot. But that effect is very small for all but the most extreme applications. Aug 22, 2019 at 10:39
• "[...] which allows conclusions to be drawn about the training of the operators and the quality of the grenades." - That's pretty raw for Wikipedia's generally neutral tone. Aug 22, 2019 at 11:18
• Neil DeGrasse Tyson perpetuating myths? tell me it ain't so! Aug 22, 2019 at 19:59
• In other words, the first shot might be calculated using an incorrect formula. But for the remaining salvos the spotter says "a little to the left and little farther" and no more Coriolis influence, if there ever was one. Aug 23, 2019 at 19:07
• "Neither British ship had a functioning director or a Dreyer Table" - so the question is if a human was actually range finding, or if they just used spotting. Can humans even calculate a firing solution including the Coriolis effect in time for it to still be viable?... Could that one guy do it, which is why he got to carry an umbrella? Aug 24, 2019 at 19:21

Rudolph Verner, the gunnery officer of the Inflexible, which took part in the battle, wrote a short set of notes about the successes and difficulties encountered during the engagement. He writes

Deflection caused considerable difficulty, and on two occasions one gun fired some five or six consecutive rounds nearly a ship's length astern of the target.

I traced this to a turret and called for the deflection setting, which proved to be correct.

Since this error was intermittent, and always took effect in the same direction, I am convinced that it was due to the gun-layer mistaking the stem for stern, or foremast for mainmast.

The particular cruisers we were engaging made such an error particularly easy, due to the symmetrical arrangements of their masts and funnels.

Verner gives us three key insights:

• Some deflection was observed, with the worst problems occurring in a single gun.
• This deflection was not regular and continuous, as might be expected if the Coriolis force was responsible.
• The deflection, at worst, was about one ship's length.

The Inflexible measured about 160 meters in length, and the other ships were of similar size. Calculations based on the guns and naval records predict a deflection of only 82 meters, even assuming that corrections were applied for the Coriolis force but in the wrong direction, thereby making the problem worse. It seems much more likely that the bulk of the problem was not due to the Coriolis force but to poor aiming.

Unfortunately, Verner does not specify the "deflection setting" used - or even what sort of deflections it was correcting for (wind, perhaps, or the relative motion of the ships?). When he describes it as "correct", it's not clear whether that means that it is correct if the ship was in the Northern Hemisphere, or correct given that the ship was in the Southern Hemisphere. Therefore, I can't definitively rule out that the setting was incorrect. However, I can confidently say that the major deflection that the British experienced was not due primarily to the Coriolis effect.

• "Deflection" in this setting refers to the combined correction for all the factors that keep the target from being where the gunsight is aimed when the shells arrive: Coriolis, wind, motion, pointing error, etc.
– Mark
Aug 22, 2019 at 21:03