Earth's Curvature and Battleship Gunnery

Quote of the Day

Eggshells smashing each other with hammers.

— Winston Churchill, describing his feelings on battleship combat.


Figure 1: Factors Affecting Range Ballistics. (Source)

Figure 1: Factors Affecting Range Ballistics. (Source)

I must admit that I am a bit of a battleship junkie. I have been reading some old US Navy manuals on battleship fire control, which discuss the various effects that must be corrected for to ensure accurate fire (Figure 1). In this post, I want to examine how the curvature of the Earth affected the gunnery direction. Curvature corrections are only needed for very long-range artillery.

Figure 2: Range Table for US Navy 16-inch/50 caliber. (Source)

Figure 2: Range Table Excerpt for US Navy 16-inch/50
caliber. (Source)

Gunnery direction calculations usually begin with a range table (Figure 2), which tells the gunner the angle that projectile must be fired at to hit a target at a given range on the same horizontal plane as the gun (i.e. no difference in height between the gun and target). The target height relative to the gun can be either positive or negative, which affects the range that is used to index into the range table . For example, battleships in WW2 doing shore bombardment sometimes needed to attack fortifications on mountains (e.g. Mount Suribachi on Iwo Jima). For sea-level sea battles, the targets are below the horizontal plane of the ship firing the projectile.

Figure 3: Example Where Target is Lower Than the Gun. (Source)

Figure 3: Example Where Target is Lower Than
the Gun. (Source)

Figure 3 shows that firing at a target that is at sea level also involves a difference in heights. The rangefinders on a battleship determined a Line-Of-Sight (LOS) distance, but that distance is not the same as the horizontal distance listed in the table of Figure 2. The LOS distance must be corrected to an effective horizontal distance that can be looked up in the range table. My goal in this post is to show how we can correct the LOS distance to provide the required horizontal distance, which can then be used to read the gun elevation from the  table in Figure 2.

All calculations are performed in Excel – my workbook is here.


Earth Curvature Calculation

I have written about how to compute the curvature of the Earth over a given distance in another post using Equation 1, which relates the deviation from horizontal to the distance from the measurement origin.

Eq. 1 \displaystyle \delta =\sqrt{{{{R}^{2}}+{{R_{LOS}}^{2}}}}-R


  • δ deviation from horizontal, which is called curvature in gunnery.
  • R is the radius of the Earth (3963.2 miles)
  • RLOS is the LOS distance.

These parameters are illustrated in Figure 5.

We can use Equation 1 to compute a curvature versus range table (Figure 4). This table duplicates the results shown in this reference.

To illustrate how to read this table, consider the range of 19,800 yards. We go to the row that corresponds to 19,000 yards and find the column that corresponds to 800 yards. At the intersection of the row and column, we find a curvature of 84 ft.

Figure 4: Table of Curvatures for Different Horizontal Ranges.

Figure 4: Table of Curvatures for Different Horizontal Ranges. This figure shows how to find the curvature for a range of 19,800 yards, which is 84 feet.

Rate of Height Change

The US Navy manuals refer to "Column 19" and the "Change in height of impact for variation of 100 yards in sight bar."  While this sounds like a complex parameter, it is simply the tangent of the projectiles impact angle with respect to horizontal, which is called the angle of fall and is listed in the range table shown in Figure 2. The tangent of the angle of fall tells you how many feet the projectile loses in height for every foot of horizontal distance. We will use this parameter to relate the height difference to the range correction.


Earth Curvature Correction Calculation

Figure 5 defines some variables using the illustration of Figure 3. You can see in Figure 5 hitting target on a requires reducing the range setting of the gun (RH) from the distance measured along the line of the sight (RLOS) by Δ, i.e. {{R}_{H}}={{R}_{{LOS}}}-\Delta.

Figure 5: Illustration of the Range Correction.

Figure 5: Illustration of the Range Correction.

For modeling purposes in Figure 6, we can treat the trajectory of the shell near the target as a straight line. This allows us to use a simple trigonometric function to compute Δ, i.e. \text{tan}\left( {{{\theta }_{{Fall}}}} \right)=\frac{\delta}{\Delta }\Rightarrow \Delta =\frac{\delta}{{\text{tan}\left( {{{\theta }_{{Fall}}}} \right)}}.

 Figure 6: Details on the Correction Term Δ.

Figure 6: Details on the Correction Term Δ.


I copied a section of the range table from the US Navy manual and used it to compute: (1) curvature; (2) change in height of impact for variation of 100 yards in sight bar (i.e. LOS range); (3) danger space (discussed in this blog post). I can verify that (1) and (2) agree with the manual. Item (3) is discussed but not listed in the manual tables.

Figure 7: My Duplication of Curvature Correction Table.

Figure 7: My Duplication of Curvature Correction Table.


I am interested in understanding the gunnery corrections for the Earth's curvature and the Coriolis effect. I believe this post thoroughly covers the curvature correction.  I will put out a post shortly on the correction for the Coriolis effect.

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23 Responses to Earth's Curvature and Battleship Gunnery

  1. matt says:

    earth is not curved. water is always measurably flat. earth is mostly water. therefore earth has to be flat

    • mathscinotes says:

      I am working to understand how the US Navy directed the fire of its now obsolete battleships – correcting for the Earth's curvature was part of their procedure. I do not plan on entering into any flat Earth debate here.


  2. Jim Cole says:

    We had similar problems to consider in field artillery. Our tabular firing tables included corrections for atmospheric density, powder temperature, target above or below gun, effects of cross wind and range wind, direction of fire, weight of projectile, and others. If a target was above the level of the gun, this necessitated a positive correction. Target below gun necessitated a negative correction. These corrections were called site, and could be obtained from the tabular firing tables or from a graphic site stick, somewhat like a slide rule. Site corrections were made for each charge, as most field artillery pieces have multiple charges available affecting velocity and therefor shape of trajectory. Firing due east required a slight negative correction, as the target was moving toward you. Firing due west required a slight positive correction because the target was moving away from you. Any other direction than due east or west the correction was somewhat less, determined by trigonometric calculations (done for us in the tables, as we were all DAGBYs: dumb ass gun bunnies!). The latitude of your position also figured into these calculations. We also had the cotangent of the angle of fall included in table G. Using this function, we could tell how close to the target in defilade a shell would burst if it just cleared the highest point (such as a building). This was useful in determining whether high angle or low angle fire should be used.

    • mathscinotes says:

      Hi Jim,
      Thanks for your comment. It is good to hear from a practitioner. I am always amazed at the level of detail involved in these ballistic calculations.

      My father was in the field artillery (specifically, fire control/communications). As a boy learning to shoot, my father taught me about angular measurement using mils and the effects of weather and gravity. We did not have much time together, and those are times I cherish.


  3. RONAN A MANDRA says:

    Mark, thanks for a fun post. I've also been interested in battleships since I was a child. Looking forward to your Coriolis effect analysis.

    • mathscinotes says:

      I too have always been interested in battleships – they look cool. Also, I am fascinated as to how people can spend so much money on things that end up being of so little practical use. Battleships were produced at great cost, but in the end it was the aircraft carrier that was their downfall. Nuclear weapons are similar in that we have spent an enormous amount on them (US alone spent about 6 trillion dollars on them and is planning to spend more), but you cannot use them.

      I have started on the Coriolis post, but stuck on some nagging details. I will get it done.


  4. RONAN A MANDRA says:

    I downloaded your Excel workbook and found this minor typo in the "RangeCorrection" tab. You state: "Change in height of impact for variation of 100 years in sight bar." Hopefully, no one wants to wait for 100 years for projectile impact 😉


    • mathscinotes says:

      Thanks Ronan! It is tough to proof your own stuff. I always appreciate your help on keeping the content clean.

      File updated.


  5. Jim Cole says:

    Thank you for your comment. All those factors used to have to be computed by hand from tabular and graphic firing tables. Now, as I understand, it is all done by computer. Hopefully though some soldiers still know how to read a TFT and play charts and darts in case a stray bullet or shell fragment goes through the computerTHERE IS NO SUBSTITUTE FOR MANUAL BACK-UP!

    • mathscinotes says:

      Your comment on manual backup reminds me of a great news report I saw a couple of years ago. The universal application of GPS had caused the US Navy to the reduce the amount of celestial navigation taught to sailors. Now with the rise of anti-satellite technology and GPS spoofing, the US Navy has decided that they needed to ensure a manual backup navigation method was available.

      I have always been fascinated by celestial navigation (and all primitive navigation) and it is a hobby interest of mine.

      Thanks for your response.


  6. Ronan Mandra says:

    Mark, FYI Binkov's Battlegrounds, a Youtube guy, has a video on a theoretical engagement between the IJN Yamato and the USS Iowa at:

    • mathscinotes says:

      Hi Ronan,

      That is a good video. I need to try out World of Warships. The only game I play now is Silent Hunter III.

      Thanks for the comment.


  7. Al Sidell says:

    I am not a mathematics expert, nor a ballistics expert, but I have a question, kind of back to basics. The longest recorded naval shot was on the Guilio Cesare hit by Warspite at 26,000 yards. From my understanding of WWII naval warfare, LOS or Line Of Sight was a requirement of ballistics, deviation from horizontal, etc. You show this in your drawing. You cannot aim at something that cannot be seen. Your chart clearly shows at 26,000 yards the Cesare would be 154 feet below the event horizon, or the curvature of the earth. The Cesare was approximately 100 feet tall to top of mast, approximately 75 feet to top of funnels. How can one calculate line of sight, or deviation from horizontal to a target that is completely obscured by the curvature of the earth? Magnification is not the answer, you cannot magnify something on the other side of a sphere that is blocked by the body of that same sphere. The best information I can find shows the Musashi, the largest battleship ever built was less than 154 above the waterline, much less a dreadnought class battleship almost 2/3 the size. How can gunnery make adjustments to splashes that are over the horizon of a sphere and not in sight? Take your picture showing curvature of the earth and plot it to scale with Warspite and Cesare. Cesare would be totally beyond the curvature, no Line of Sight is possible. Your Line Of Sight would go through the water. Now do your calculations.

    • mathscinotes says:

      Thank you for your question. I go through the details of the maximum optical range calculation in this blog post. The calculation assumes that the battleship rangefinder can only see the tip of the target mast. This scenario is illustrated as follows.

      The Warspite's rangefinder height is nominally ~28 meters. I say nominally because it varies with the loading of the battleship. I will use your value of 100 ft for the mast height of the Giulio Cesare. The maximum range calculation is performed as shown below.

      The HMS Warspite used 15-inch guns with a nominal range of 33,550 yards. So the max optical range and gun range are consistent.

      My calculations produce similar results to those in this stackexchange post.


  8. Jim Cole says:

    In response to Al Sidell, I wonder if there might not have been aerial spotters radioing the splash locations back to the Warspite. Your evaluation of the distance and visibility seem to ring true, so perhaps further research might show the presence of such aircraft (possibly "String bags", Faiery Swordfish). As i am unfamiliar with this battle, perhaps there were surface ships closer to the Caesare which could have radioed or otherwise signaled necessary corrections. Just a thought.

    • mathscinotes says:

      Hi Jim,

      There were airborne spotters at the Battle of Punta Stilo. See this link for details. My response to Al's question shows that the Warspite and Giulio Cesare were within optical range during their engagement.


    • mathscinotes says:

      Hi Jim,

      One more comment. The splashes can be seen from an amazing distance. Here is a Youtube video showing how high the splashes rise from the USS New Jersey. They are easily seen from the ship. Of course, the New Jersey's gun in this video were at low elevation, but you still can see the size of the splash.


  9. Al Sidell says:

    Perhaps there were spotter aircraft for splash marking, anything is possible. That does not explain how the initial firing solution was derived at without line of sight. You may not be familiar with this engagement, but there are many cases for such long range shooting. The Sharnhorst fired at HMS Glorious at 26000 yards. The Glorious was a converted heavy cruiser to aircraft carrier. Her overall height would be less than a Battleship of the same era, or less than the Cesare. Pictures show what looks to be the same funnel after conversion, the Cruiser super-structure was removed(above the funnel) during conversion. So lesser height above the waterline than the Cesare, built within 6 years of each other. So you have about same range with the target being even smaller and further behind the curve of the earth, according to your chart. There must be an explanation to arrive at the initial firing solutions. An initial firing solution cannot be made with spotter aircraft, or without line of sight, direction and range. Forgive me, but, I cannot fathom a person discussing, in depth, naval targeting, gunnery, and all the intricate details and mathematics of WWII naval gunnery, and not knowing anything about the longest shots taken in battle in Naval history. They would seem to be of the most interest of all. I bow to the mathematician for an explanation.

    • mathscinotes says:

      Hi Al,

      I hope my previous responses provide you sufficient explanation. To summarize:

      • Warspite was led to the Giulio Cesare by airborne spotters
      • At the time of engagement, Warspite and Giulio Cesare were within optical range of each other.
      • Both Warspite and Giulio Cesare fired. Amazingly, Warspite straddled on the first salvo, which is a testament to its rangefinder, director, and crew.
      • You should note that the fire control systems on both the Warspite and Giulio Cesare were able to obtain solutions.

      By the way, I am not a mathematician. I am an electrical engineer with experience in designing military hardware. I simply like to use simple math to explore problems that I encounter every day.


    • mathscinotes says:

      As long as we are discussing amazing battleship engagements during WW2, I do want to mention the performance of the USS West Virginia (Wee Vee) at the Battle of Surigao Strait. Here are the points about this engagement I would like to highlight.

      • The Wee Vee had been sunk at Pearl Harbor
      • She was raised and modernized – she had been commissioned in 1914, one year before Warspite.
      • She obtained a firing solution at 30,000 yards.
      • She tracked the IJN force to 22,700 yards, when she opened fire.
      • Five of her first six salvos struck IJN ships.
      • Fischer and Jurens (Fast Battleship Gunnery during World War II) consider Wee Vee's performance as the best example of battleship gunnery during WW2.

      I am taking nothing away from Warspite – it holds the range record. Both WeeVee and Warspite show that the old WW1 battleships could still perform well during WW2.


  10. Al Sidell says:

    Once again, I am no expert, I am just a disabled navy veteran, and became fascinated by WWII naval battles 40 years ago while in service. They had a plethora of naval history in the ships library. I would like to make a couple of observations though. I was wrong when I stated the top of the mast on Cesare was approximately 100 feet. I can't find exact measurements for either the Warspite or Ceasare, what I have found is that most estimates are hugely over concerning a ships height above the water, even my own. The height above the water for a Nimitz Class aircraft carrier is 60 feet, which really surprised me, as I have been on board the USS Enterprise. It is so huge I guess one loses a sense of scale. Which vastly over shadows WWI vintage battleships(both were built around 1917). Therefore I cannot cede you the 28 meter or 90 feet height for the rangefinder. Also, anti-aircraft rangefinders were mounted high on the super-structure, usually above the funnel, gunnery rangefinders were mounted very much lower, almost always on the gun batteries themselves, this is verified by Colin Vass who built an award winning scale model of the HMS Warspite. Each main battery had it's own rangefinder mounted on the battery itself. I cannot measure exactly, but the Warspite draft is listed at 33 feet, that is where the waterline is painted, at nominal draft, draft can vary with loading, but it is a very good baseline. The turrets where the rangefinders were mounted according the model, is almost equal distant from the waterline as is the keel, I would say certainly not more than 40 feet from the waterline. In battle conditions most ships were fully loaded with fuel and armaments, which if anything would lower the rangefinder relative to the water. Naval gunnery control in WWII used almost exclusively stereoscopic rangefinders, which used parallax to determine range, not very complex. What is interesting is these instruments not only provided the range to target, it provided the targets speed and course, all requirements to formulate a firing solution. Here is a quote concerning the use of this instrument. "Measuring the target's speed and course was considerably more tricky than range. It required a combination of looking at the orientation or angle of the target's superstructure and profile along with its bow wave and wake, to determine a base course or angle. Then, how fast it moved across the field of view of the range finder to determine a base speed using the angle of a base course and time mathematically. Also included was how the bearing to it changed as you observed it, taking into account your own ships speed and course. " So, it may or may not be true the top of the mast was visible through magnification alone, a firing solution was not possible using only the mast. According to the documented usage and instructions of the instrument, and appreciable view of the target ship is required for a firing solution. You can verify this information independently. We MUST limit capabilities to what existed at the time for the purposes of this discussion. If the methodology used at the time required viewing a super-structure, profile and even perhaps bow wave and wake for computation, one cannot assume otherwise at this point. Especially in view of the accuracy involved in the engagement. Cesare opened up at a reported range of 29,000 yards and within 3 minutes bracketed Warspite. According to my website, the curvature of the earth is 179.36 feet at 16.4 miles. Given the rangefinder is approximately 40 feet above the water, 139.36 feet of the Warspite would have been obscured by curvature which is more than double that of the USS Enterprise. Seeing as Warspite returned fire at near the same distance, the reverse is true, 139.36 feet of the Cesare was obscured. So the line of sight issue becomes even more pronounced. There are numerous examples of naval gunnery commencing well in excess of 30,000 yards, no hits however, but amazing accuracy never the less.
    Just as a curious side note, the rangefinders in use were the very best available at the time, great effort and expense were expended in the research, development and design by some of the most learned and intelligent people available.
    Curious question, why would they include bow wave and wake in the computation? According to all available information today, the bow wave would have been obscured within approximately 9,000 yards or five miles due to curvature of the earth, even less for smaller ships and wake probably less than a mile. Five miles is a mere chip shot for naval gunnery. One would assume they knew that. Just a curious thing.

    • mathscinotes says:

      Sidell: Therefore I cannot cede you the 28 meter or 90 feet height for the rangefinder.

      I have done some searching for main battery nominal director heights and here is what I found.

      Battleship Rangefinder
      Height (m)
      Iowa 35.3
      Littorio 28.0
      Yamato 39.0
      Nagato 41.0

      If you need see an example of where I found this data, the Iowa data is from the USS Iowa Crew Handbook. As you can see, 28 m is not unreasonable at all. Here is an excellent illustration from the Wikipedia of the battleship Fuso that illustrates just how tall a run-of-the-mill battleship was. The main battery directors are at ~35 m.

      Sidell: Here is a quote concerning the use of this instrument.

      Measuring the target's speed and course was considerably more tricky than range. It required a combination of looking at the orientation or angle of the target's superstructure and profile along with its bow wave and wake, to determine a base course or angle. Then, how fast it moved across the field of view of the range finder to determine a base speed using the angle of a base course and time mathematically. Also included was how the bearing to it changed as you observed it, taking into account your own ships speed and course.

      The function you are describing used to be called rangekeeping. I wrote a Wikipedia article about this subject years ago. A critical part of rangekeeping involves determining the rates of range and angle change. WW2 rangekeepers were sophisticated analog computers that were capable of determining these rates using an iterative process. This process required initial estimates of the angle on the bow and target speed. Poor estimates would work, but it took longer for them to converge to a firing solution. There are great stories of people like Dick O'Kane spending hours practicing determining the angle on the bow by using a Lazy Susan, IJN ship models, and a reversed binocular to shrink the image to the size seen through a periscope. He supposedly got so good he could reliably determine AOB within 1° (reference).

      Target speed was estimated using various methods. Submarine skippers used hydrophones to measure the prop rotation rate and translate that value into speed (reference). Surface ships would use the size of the bow wave to estimate the speed. It was not a great method, but it was close enough for a rough estimate. While not describing surface ship fire control, Dick O'Kane did a wonderful job describing the iteration process with respect to torpedoes in his book Clear the Bridge!.

      I do not have a good reference for how a surface ship director is used operationally, but I do have the following Youtube reference that discusses how a torpedo director worked. Their operation was similar.

      Before I spend too much more time on this, did you review my reference to my post on how refraction affects rangefinding? This article discusses how the line-of-sight (LOS) is not straight.


  11. Al Sidell says:

    I know, how about this. The Warspite used "over the horizon" rangefinding like found on the later Iowa class battleships, and used its 18.1 inch main turret to make the shot, just like the Yamato had. Why not? It seems its other characteristics are totally comparable. Never mind that the Warspite, A Queen Elizabeth class battleship built in 1913 was 244 feet shorter than the Iowa, I am sure its height was equivalent and characteristics were the same. I will check the silhouette cards for WW2, I am sure the Warspite, Iowa and Yamato are virtually identical, its a wonder we didn't sink our own ships. Even though the most casual reference to the Warspite history shows that prior to the engagement in question, its main mast, you know, where the spotting tower is, was removed to make room for a crane and hanger deck for amphibious aircraft. You remember, the Warspite, the ship that made the shot, the topic of discussion. Not the Iowa, not the Yamato, the Warspite. How does that effect your mast to mast calculations? As to that, can you provide ONE documentable instance of any ship in WW2 obtaining a firing solution from a mast alone? The documentation is astounding detailed. Yes, I am aware of refraction, I am aware it MAY increase ones line of sight by refracting along the curve of the earth, also I know it MAY refract away from the surface and reduce ones line of sight also. It also MAY not be there at all. It all depends on a lot of variables. Which you seem to assume were present and in the direction you desire for your math to work. I also tire of the discussion, all you are interested in is the mathematics, just prove your mathematical theory. You have accepted no information I have provided about the configuration of the Warspite, perhaps it doesn't work in your model. Perhaps a wholly different class of ship does fit your model. Actual historical, provable facts seem to have no bearing on your mathematical theory. I think it best if I just go back to my naval history books and you go back to your calculator. I would offer one word of caution, there is no natural law or scientific law that states the universe or even our world must conform to or agree with mathematics. It mostly seems to, but there is no law. Mathematics is a construct devised by humans. Perhaps Tesla was correct.


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