There are a number of recurring themes in debates revolving around battleships, including questions about the ability of quality to triumph over quantity; the relative importance of firepower versus staying power; and the advantages of more guns versus larger guns.  Each great dreadnought power made certain decisions about how to arm their ships and about what kind of ships to build which appear mystifying or ill-conceived to the casual observer.  Foremost among these debates is the value of the dreadnought ("all-big-gun") concept in the first place.  Was it a worthwhile idea to render the entire, overwhelmingly large Royal Navy battle line obsolete at a single stroke for a change in design philosophy?  Perhaps only slightly less contentious are two questions arising from the naval experiences of World War One and a third question from the interwar period.

First, the British Grand Fleet built many battleships with progressively greater firepower but also with indifferent protection.  Their main opponent, the German High Seas Fleet, sailed in ships armed with lighter weapons but whose toughness has become a legendary standard.  The one great contest between the British and the German fleets, the Battle of Jutland, produced an inconclusive tactical result.  From this battle, can it be determined as to which fleet’s design philosophy was correct?

Secondly, across the Atlantic, the United States was in the midst of a run of ships which featured an armament of twelve 14" guns all mounted on the centerline, a veritable thicket of heavy guns compared to the majority of British and German ships with their broadsides of eight guns.  Yet, in the final USN design of this series, a third of these guns were signed away in exchange for eight heavier weapons.  Was this a wise decision?  The same question would emerge again in the later Treaty-era, between the protagonists for designs mounting twelve 14" guns and those for designs mounting nine 16" guns.

The third question, from the far side of the world, arises from upstart Japan’s seeking to buttress imperial ambitions far beyond her modest means.  As her two most likely antagonists began to outstrip any capacity of Japanese shipbuilders to match numbers, the "super-battleship" program emerged, attempting to defeat crushing numerical superiority with individual units of staggering scale and unmatched power.  The resulting Yamato Class battleships were the largest warships of their time and for many years afterwards.  These ships have often been called a waste of steel better used on carriers.  This may be true, but would they have been a similar waste even if battleships were never eclipsed?  Were they, in fact, a bad idea from the very beginning?

The factors involved in the decisions made are manifold and will not all be discussed here.  Similarly, the answers are not always clear-cut or self evident.  The purpose of this article is not to resolve these debates, but to illuminate an often unknown, unappreciated and unperceived influencing factor in these decisions:  Mathematics.

In 1905, USN gunnery theoretician Commander Bradley A. Fiske produced a series of equations which greatly influenced the theory and practice of naval gunfire.  He may have been inspired to do so by the outcome of the Russo-Japanese War and its seminal battle at Tsushima.  That conflict and battle certainly lent themselves well to the cause of designers and theorists in several countries who were calling for a sea-change in the design of battleships.

Amidst the competing thought and design work going on in all the major naval powers, in 1906 Britain’s Admiral Sir John Fisher put the first example of this new thinking into the water, the famous HMS Dreadnought, which gave her name to the "all big gun" type of battleship.  At a single stroke, this vessel seemed to send Britain’s naval supremacy into wholesale obsolescence and initiated a new naval arms race.  Faced with this situation, calculations of force became a matter of serious interest as Europe slid slowly into war.

In Britain, Frederick W. Lanchester produced a series of equations which have retrospectively come to be associated with him, but their basic principle may have been known much more widely and much earlier.  Lanchester’s equations concern the concentration of aimed firepower in ranged combat.  His thesis was that any difference in firepower, assuming no difference between the offensive and defensive character of units on either side, would have an ever increasing detrimental effect on the numerically weaker force, resulting in its annihilation with far fewer casualties to the numerically stronger side.  This was contrary to what the previous linear theory had predicted.  Linear theory held that 100 units would annihilate 75 similar units at a cost of 75 of their own, leaving 25 survivors.  Lanchester’s exponential theory predicted the annihilation of the 75 for a cost of 34, leaving 66 survivors.  This constituted an even great impetus for a nation to posses a numerically stronger fleet as it implied that such a fleet would suffer numerically as well as proportionally fewer casualties than would its opponents.   Lanchester’s equations and theory have become to be known as the "N-Squared Law."

Why did Lanchester postulate exponential vice linear ramifications to a difference in firepower?  He reasoned that the attrition of either side would be proportionate to the remaining firepower of the opposing side.  For the weaker side, that firepower would start out as less, remain less, and become ever smaller in proportion to the opponent as a battle went on.  According to USN Captain Wayne P. Hughes, from whom I draw the above example and background, Lanchester’s equations came into general use for ground warfare theorists but have only infrequently been applied to naval combat.  Hughes makes a very good case in his book, "Fleet Tactics:  Theory and Practice" that Lanchester’s equations are equally or even more applicable to naval combat.  Generally freed of the interfering effects of terrain, particularly in the defender’s favor, naval combat more closely resembles the sterile conditions of mathematics.

Lanchester’s attrition differential equations as given by Hughes are:

dA / dt = -B
dB / dt = -A

A0^2 - At^2 = B0^2 - Bt^2

Case Studies:

Caveats:

For the following explorations, in order to keep the math short and simple, the following, deliberately simplistic, assumptions are made:

1. The basic unit of force is a dreadnought battleship with an eight-gun broadside and the ability to survive three hits, being sunk by the fourth.
2. Guns are assumed to fire one round per minute.
3. 1% of all rounds fired hit their targets.
4. Each hit is assumed to knock out 25% of the guns of the ship struck.

Case One:  Fear God . . . and Dreadnought

In this first case, Fleet A consists of four dreadnought battleships and Fleet B consists of four pre-dreadnought battleships.  For this study, both types are assumed to be identical in every way, including weight of metal in the broadside, except that the pre-dreadnought’s firepower is distributed amongst heavy, medium and light caliber guns with only 50% of the pre-dreadnought’s firepower composed of heavy guns able to hit at the longest ranges.  Theoretically, neither side is advantaged in weight of metal as the broadsides of both sides are the same.  However, because the dreadnoughts have twice as many heavy guns, then during the initial stages of the engagement, the dreadnoughts of Fleet A have a two to one advantage.  If they succeed in holding the range of the engagement to that which favors them, then this advantage would then continue throughout the battle.

Initially, Fleet A has 32 heavy guns firing and Fleet B has 16.
 
 

	Fleet A			Fleet B
Time	Ships	Guns	Shots	Ships	Guns	Shots
00:00	4	32	0	4	16	0
00:04	4	32	128	3.75	15	64
00:07	3.75	30	224	3.5	14	109
00:10	3.75	30	314	3.25	13	151
00:13	3.75	30	404	3	12	190
00:14	3.5	28	434	3	12	202
00:17	3.5	28	518	2.75	11	238
00:20	3.5	28	602	2.5	10	271
00:23	3.25	26	686	2.5	10	301
00:24	3.25	26	712	2.25	9	311
00:28	3.25	26	816	2	8	347
00:32	3.25	26	920	1.75	7	379
00:35	3	24	998	1.75	7	400
00:36	3	24	1022	1.5	6	407
00:40	3	24	1118	1.25	5	431
00:44	3	24	1214	1	4	451
00:48	3	24	1310	.75	3	467
00:52	3	24	1406	.5	2	479
00:56	3	24	1502	.25	1	487
01:01	3	24	1622	0	0	492

From the chart above, Fleet A has sunk its first enemy in 14 minutes and its second in another 14 minutes, while losing one vessel only after 35 minutes have passed.  At this point, the odds are 3:2 in Fleet A’s favor, whether the losses to each side are in actual ships, or in overall firepower.  In fact, henceforth the remaining two virtual ships of Fleet B are destroyed without further loss to Fleet A.

This case clearly illustrates what Lanchester observed.  Fleet B started with less firepower and became weaker in proportion over time, failing to inflict losses in proportion to its initial firepower.  Fleet A has triumphed by a statistically significant margin.  The conclusion must be that the dreadnought constitutes a more efficient combat unit.
 

Case Two:  "Economy" of force?

Alas, dreadnoughts are actually more costly than their ancestors.  For the purposes of this essay, let us say that this is by a factor of 1.5 times the cost of a pre-dreadnought.  For this case study, this means that Fleet C can afford to field two dreadnoughts for every three of Fleet D’s pre-dreadnoughts. Even so, Fleet C possesses greater firepower in terms of number of big guns.

Initially Fleet C has 16 guns firing and Fleet D has 12.
 
 

	Fleet C			Fleet D
Time	Ships	Guns	Shots	Ships	Guns	Shots
00:00	2	16	0	3	12	0
00:07	2	16	112	2.75	11	84
00:09	1.75	14	144	2.75	11	106
00:13	1.75	14	200	2.5	10	150
00:18	1.5	12	270	2.5	10	200
00:21	1.5	12	306	2.25	9	230
00:29	1.25	10	402	2	8	302
00:39	1.25	10	502	1.75	7	382
00:42	1	8	532	1.75	7	403
00:51	1	8	604	1.5	6	466
00:57	.75	6	652	1.5	6	502
01:05	.75	6	700	1.25	5	550
01:15	.5	4	760	1.25	5	600
01:25	.5	4	800	1	4	650
01:38	.25	2	852	1	4	702
02:02	.25	2	900	.75	3	798
02:03	0	0	902	.75	3	801

In this case Fleet C has failed, showing the superiority of investing firepower in a greater number of units, rather than maximizing it in a single unit.  This case reinforces the fact that a dreadnought is a more efficient fighting unit, but it also shows that building dreadnoughts is not going to save any money in procurement costs.  Overall numbers of hulls won’t diminish much and any savings will have to come in maintenance and manning costs.  This case study also helps to show why tanks with a single main barrel became the norm in ground warfare and why multi-turret monstrosities such as HMS Agincourt (ex-Sultan Osman I, ex-Rio De Janiero) were a poor investment.
 

Case Three:  Outranging the enemy

In this case Fleet E has four dreadnought battleships. Fleet F also has four dreadnought battleships. Both fleets’ ships are assumed to be identical in every way, except that Fleet E’s dreadnoughts can open fire at longer ranges, giving them ten minutes during which they can fire and Fleet F cannot.
 
 

	Fleet E			Fleet F
Time	Ships	Guns	Shots	Ships	Guns	Shots
00:00	4	32	0	4	32	0
00:10	4	32	320	3.25	26	0
00:13	4	32	416	3	24	78
00:14	3.75	30	448	3	24	102
00:16	3.75	30	508	2.75	22	150
00:19	3.5	28	598	2.75	22	216
00:20	3.5	28	626	2.5	20	238
00:23	3.5	28	710	2.25	18	298
00:24	3.25	26	738	2.25	18	316
00:27	3.25	26	816	2	16	370
00:29	3	24	868	2	16	402
00:31	3	24	916	1.75	14	434
00:35	3	24	1012	1.5	12	490
00:36	2.75	22	1036	1.5	12	502
00:39	2.75	22	1102	1.25	10	538
00:44	2.75	22	1212	1	8	588
00:46	2.5	20	1256	1	8	604
00:49	2.5	20	1316	.75	6	628
00:54	2.5	20	1416	.5	4	658
00:59	2.5	20	1516	.25	2	678
01:04	2.5	20	1616	0	0	688

Fleet E has triumphed by a significant margin, illustrating the value of being able to hit first and without being harassed by return gunfire, particularly against an equal or possibly superior foe. Thus weapons with greater range would appear to be superior to weapons of lesser range.
 

Case Four:  Does a six-gun really beat four Aces?

In this case Fleet G has four dreadnought battleships.  Fleet H also has four dreadnought battleships.  Both types are assumed to be identical in every way, except that Fleet G’s ships have heavier guns while Fleet H’s ships mount 50% more guns with smaller projectiles.  We will assume that Fleet G's projectiles do 50% more damage per hit than will Fleet H's projectiles.
 
 

	Fleet G			Fleet H
Time	Ships	Guns	Shots	Ships	Guns	Shots
00:00	4	32	0	4	48	0
00:03	3.75	30	96	4	48	144
00:04	3.75	30	126	3.625	44	192
00:05	3.5	28	156	3.625	44	236
00:07	3.25	26	212	3.25	39	324
00:09	3	24	264	3.25	39	402
00:11	3	24	312	2.875	34	480
00:12	2.75	22	336	2.875	34	514
00:15	2.5	20	402	2.5	30	616
00:18	2.25	18	462	2.5	30	706
00:21	2.25	18	516	2.125	26	796
00:22	2	16	534	2.125	26	822
00:25	1.75	14	582	2.125	26	900
00:27	1.75	14	610	1.75	21	952
00:30	1.5	12	652	1.75	21	715
00:34	1.5	12	700	1.375	16	799
00:35	1.25	10	712	1.375	16	815
00:41	1	8	772	1.375	16	911
00:45	1	8	804	1	12	975
00:48	.75	6	828	1	12	1011
00:56	.5	4	876	1	12	1107
01:02	.5	4	900	.625	8	1179
01:05	.25	2	912	.625	8	1203
01:18	0	0	938	.625	8	1307

In this case Fleet H has prevailed, but just barely.  While this would appear to vindicate the "more barrels versus heavier barrels" concept, the margin is probably not sufficient to be considered statistically significant.  However, this particular case study ignores synergy.  A heavier gun would also probably enjoy greater range together with a greater ability to penetrate defenses even before its shell employed 50% greater destructive power.  Thus a Fleet H vessel might only have a relative ability to absorb 3.2 vice 4 hits.  Using these considerations would significantly alter the results of this engagement.  A further consideration for this case study would be to determine the effects of using lighter weapons with a more rapid rate of fire.  Simply assume Fleet H to have 50% greater ROF, for example.  Again, the true determinate of which is better will probably lie somewhere in the caveats.
 

Case Five:  Using few to defeat many

Next, let us consider the argument of quality versus quantity.  In this case Fleet J has two dreadnought battleships. Fleet K has four dreadnought battleships. Both types are assumed to be identical in every way, except that Fleet J’s ships have heavier guns which do 50% more damage, and the ships themselves are 50% more rugged, requiring six hits to “sink.”
 
 

	Fleet J			Fleet K
Time	Ships	Guns	Shots	Ships	Guns	Shots
00:00	2	16	0	4	32	0
00:04	1.83	14	64	4	32	128
00:07	1.66	13	206	3.625	29	224
00:10	1.5	12	245	3.625	29	311
00:14	1.33	11	293	3.625	29	427
00:15	1.33	11	304	3.25	26	456
00:17	1.16	9	326	3.25	26	504
00:21	1	8	362	3.25	26	608
00:25	.83	7	394	3.25	26	712
00:26	.83	7	403	2.875	23	738
00:29	.66	5	424	2.875	23	807
00:34	.5	4	449	2.875	23	922
00:38	.33	3	465	2.875	23	1014
00:42	.16	1	477	2.875	23	1106
00:47	0	0	482	2.875	23	1221

In this case, Fleet K has triumphed by a significant margin, illustrating the maxim that “quantity has a quality of its own” past a certain critical threshold.  While, further experimentation might appear to reveal that threshold, it is likely to lie beyond practicable exploitation with respect to many factors subsumed in the caveats.  Thus, attempting to use fewer extremely powerful units to offset a numerical disadvantage does not appear to be wise.
 

Case Six:  The first purpose of a warship is to remain afloat

In this case, Fleet L and Fleet M both have four dreadnought battleships.  Both types are assumed to be identical in every way, except that Fleet L’s ships have lighter guns which do 20% less damage but their ships are 50% more rugged, requiring six hits to “sink.”

Initially Fleet L has 32 guns firing and Fleet M has 32.
 
 

	Fleet L			Fleet M
Time	Ships	Guns	Shots	Ships	Guns	Shots
00:00	4	32	0	4	32	0
00:04	3.83	31	128	3.8	31	128
00:07	3.66	29	221	3.6	29	221
00:10	3.5	28	308	3.4	27	308
00:14	3.33	27	420	3.2	26	416
00:17	3.33	27	501	3.0	24	494
00:18	3.16	25	528	3.0	24	518
00:21	3.16	25	603	2.8	23	590
00:22	3	24	628	2.8	23	613
00:25	3	24	700	2.6	20	682
00:26	2.83	22	724	2.6	20	702
00:30	2.83	22	812	2.4	19	782
00:31	2.66	21	834	2.4	19	801
00:35	2.66	21	918	2.2	17	877
00:37	2.5	20	960	2.2	17	911
00:39	2.5	20	1000	2	16	945
00:43	2.33	18	1080	2	16	1009
00:45	2.33	18	1116	1.8	15	1041
00:49	2.16	17	1188	1.8	15	1101
00:50	2.16	17	1205	1.6	13	1116
00:56	2.16	17	1307	1.4	11	1194
00:57	2	16	1324	1.4	11	1205
01:02	2	16	1404	1.2	10	1260
01:06	1.83	15	1468	1.2	10	1300
01:09	1.83	15	1513	1	8	1330
01:15	1.83	15	1603	.8	7	1378
01:19	1.66	13	1663	.8	7	1406
01:22	1.66	13	1702	.6	5	1427
01:30	1.66	13	1806	.4	3	1468
01:38	1.66	13	1910	.2	2	1492
01:42	1.5	12	1962	.2	2	1500
01:46	1.5	12	2010	0	0	1508

Fleet L has triumphed by a narrow, but significant margin.  It would thus appear that staying power has a value as well as fire power, enough so to offset some degree of disadvantage in fire power past a certain critical threshold.  Further experimentation might appear to reveal that threshold, however, the caveats must be kept clearly in mind.
 

Conclusions:  Theory and experience

What do these case studies tell us about the great debates revolving around dreadnoughts?  Largely they help explain some of the strategic and design decisions made by the dreadnought building powers.

Case One illustrates the wisdom of fielding dreadnoughts vice pre-dreadnoughts.  On anything approaching a one for one scale, the dreadnought enjoys a clear and decisive advantage in firepower which, in keeping with Lanchester’s observations, would overbear any lesser force to an increasing degree.  However many pre-dreadnought hulls in a given fleet, obtaining an early lead in building the new design constituted a clear naval advantage.  This would validate Admiral Sir John Fisher’s crash program to see that the RN fielded the new design first and in sufficient numbers to prevent any other fleet from catching up.

Case Two, however, illustrates the limitations of the dreadnought design and a secondary observation of Lanchester:  Dreadnoughts could not save their builders much - if any - money over previous designs.  Attempting to rely upon fewer dreadnought hulls to field equivalent or superior firepower, while seemingly attractive in a budgetary sense, constituted a military case of putting too many eggs in too few baskets.  Lanchester foresaw this in advising that firepower be spread amongst as many units as possible.  On land, this yielded tanks with single barrels.  At sea, where such diffusion was impractical, some concentration of firepower into individual platforms was necessary, but the logic of diffusing firepower among many platforms limited the instances of "turret farm" aberrations such as HMS Agincourt (ex-Sultan Osman I; ex-Rio de Janiero).  Case Two also helps show that Britain did not, in fact, throw away any of her naval supremacy by transitioning to the dreadnought, she simply continued it in a different form.

Case Three illustrates the advantage of being able to bring firepower to bear on the enemy while he is unable to respond.  While seemingly obvious, this conclusion is confirmed mathematically by the model.  In a real-world sense, it explains and validates the attempts by various powers to field weapons able to reach progressively farther.  While all major dreadnought fleets sought to exploit such an advantage, the IJN took the matter the farthest, enshrining it as a tactical, strategic and design principle key to their desperate attempts to offset the USN’s crushing numerical superiority.

Case Four, like Case Two, once again illustrates that improved weapons do not necessarily yield a savings proportionate to their superiority.  Nor, on the surface it does it seem to validate the seemingly paradoxical willingness of designers to sacrifice numbers of guns as they increased the size of the weapons.  However, this case is statistically too close to call, and very heavily impacted by the caveats, which may well (and probably do) swing the difference.  This explains the reductions in barrel count as seen in such examples as the USN’s Tennessee to Colorado and the RN’s Iron Duke to Queen Elizabeth.  It also speaks to decisions such as the substitution of a nine-barrel 16” gun armament for a twelve-barrel 14” gun armament in the USN’s North Carolina class.  Actual experience with 8” and 6” gunned 10,000 ton cruisers in combat would also seem to support the apparent parity of a less numerous heavier armament with a more numerous lighter armament.

Case Five closely follows the results of Case Two, once again illustrating the wisdom of not investing too much in any single platform, even if that platform is clearly superior on an individual basis.  To obtain adequate superiority in a single platform, it becomes prohibitively costly in terms of resources on many levels.  Even disregarding the rise of aviation, this appears to validate the futility of Japan’s super-battleship program, which despite massive outlays fielded just two ships of only 15-35% theoretical superiority (considering only tonnage) against a force of ten smaller ships.  These are odds far more extreme than even Case Five considers.

Case Six illustrates the value of staying power, despite the seeming tyranny of fire power brought out in Lanchester’s observations. Lanchester’s theoretical work assumed all platforms to be equal, but clearly some inequalities bring certain advantages.  Admiral Tirpitz’s demand for German capital ships to be survivable, even at the expense of some degree of firepower thus appears valid, but only if one’s fleet comes close to maintaining numerical equality with the enemy.  The fact that the German Navy was out built by the Royal Navy is an often cited cause for Germany’s naval failure in World War I.

In regards to the three questions raised at the beginning of this article:

First, it would appear that Britain was wise to build a more efficient battleship in the HMS Dreadnought and - better yet - to get a jump on the competition so as to maintain her naval superiority through a numerical advantage and increasing individual unit firepower.  Essentially, Britain saw the opportunity to stack the deck and took it.  This was the correct decision, considering that in a contest of equal numbers, Tirpitz’s tough German ships might well have outlasted their less rugged British foes.  The High Seas Fleet’s plans wound up fatally compromised by the British numerical advantage, and even Tirpitz eventually came to realize heavier guns were desirable.

Secondly, the change from twelve 14" guns in the USN’s Tennessee Class to eight 16" guns in the otherwise nearly identical Colorado Class has long been regarded as a wash, and illustrates the limitations of simple mathematical models.  A number of factors subsumed in the caveats would certainly make the two armaments equivalent, or perhaps make the 16" armament superior.  Something similar can be said of later debates about 14" and 16" guns in Treaty battleship designs.  With Case Four reconsidered as a rate of fire issue, again the issue appears to be a wash, with the true effects a matter of factors not considered in the mathematical model.  Wartime experience with 8" and 6" gunned 10,000 ton Treaty cruisers would tend to support such a conclusion.

Thirdly, the Japanese efforts to "outrange the enemy" and "use few to defeat many" provide an interesting contrast in the appreciation of mathematic theory.  Lanchester’s equations would certainly lend support to the former idea, but the latter remains questionable.  The Japanese may have recognized this and such recognition may be embodied in the gross excesses of the Yamato Class, a design which almost doubled the size of the previous Nagato Class.  Yet, even assuming the Yamato to have achieved 100% superiority over its potential foes, vice the 50% used in Case Five, the ultimate result still resembles that of Case Two, where the fleet with the smaller number of more powerful warships was defeated by the fleet with the weaker armament but greater numbers.  This lends credence to the idea that super-battleships were a poor investment from the outset.

It must be remembered that the conclusions here are very limited, and for the most part only account for the mathematical model.  The caveats are legion.  Even if such mathematical study directly or indirectly entered into the calculations of the guiding minds of the fleets in question - as it almost certainly did - it was but one of dozens of factors considered in the ultimate decisions.  Some of those factors served to mitigate the ruthless calculus of the “N-Squared Law,” while others would uphold, reinforce and even magnify its ramifications.  “N-Squared Law” and this article do not provide the “last word” on any of the questions examined.  Nor are they intended to.  As previously stated, they merely scratch the surface of one facet of a complex and multi-dimensional process, albeit a facet often overlooked amidst the more glamorous and exciting debates over the relative value of the Japanese Type 93 torpedoes, German optics, a tradition of naval professionalism and radar fire control.

Author’s notes:

For an outstanding theoretical discussion of this and other models of naval warfare, I highly recommend Captain Hughes’ book from which I have drawn some background and equations, and paraphrased an example.  I have no desire to take credit for his fine work.  My work lies in applying his work to the specific questions in this article, something his broader work does not have time to discuss in such detail or specificity.

Lanchester’s differential equations are actually for a continuous process and function in a manner similar to an analog fire control computer.  Fiske’s tables employ discrete time periods to examine the same process, and function in a manner similar to a digital fire control computer.  Lanchester’s equations, like an analog fire control computer are more accurate, but more cumbersome to use.  In keeping with my closest call with academic Darwinization in calculus and the capabilities of my calculator, I chose to employ Fiske’s tables, rather than Lanchester’s equations.  Either is adequate to study Lanchester’s conclusions.  I’ve simply chosen to forego a number of digits to the right of the decimal point.  [Editor's Note:  While the previous statements regarding discrete vs. continuous equations are true, it should be noted that a gun firing a projectile IS a discrete event.  For that reason, processing the data discretely rather than continuously may indeed be a better methodology. - TD]
 

Bibliography:

Dulin and Garzke
    "Battleships:  United States Battleships in World War II"
    "Battleships:  Allied Battleships in World War II"
    "Battleships:  Axis and Neutral Battleships in World War II"

Evans and Peattie, "Kaigun"

Friedman
    "U.S. Battleships:  An Illustrated Design History"
    "U.S. Cruisers:  An Illustrated Design History"

Hughes, "Fleet Tactics:  Theory and Practice"
.

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