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.

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]

• 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"

17 October 2000

Updated