# Earth's Curvature and Battleship Gunnery

Quote of the Day

Eggshells smashing each other with hammers.

— Winston Churchill, describing his feelings on battleship combat.

## Introduction

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 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 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.

## Background

### 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 $\displaystyle \delta =\sqrt{{{{R}^{2}}+{{R_{LOS}}^{2}}}}-R$

where

• 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. 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.

## Analysis

### 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.

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.

### Example

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.

## Conclusion

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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### 12 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.

mark

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.

mark

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.

mark

4. RONAN A MANDRA says:

Mark,
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 😉

Cheers,
Ronan

• mathscinotes says:

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

File updated.

mark

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.

mark

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: https://www.youtube.com/watch?v=W3C_gicuoPw

• 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.

mark