The theory of targeted or "concentrated" fire was popularized about 100 years ago by a British engineer and mathematician, Frederick William Lanchester. In a firefight between two numerically unequal forces, each able freely to target and re-target its fire upon any and every individual of the opposing force, the instantaneous loss-exchange ratio, being equal to the ratio of the respective firepowers, grows ever more adverse to the weaker force the longer the battle continues. Utilizing the methods of differential calculus, Lanchester showed that the cumulative losses suffered by either side to any point in the battle, even unto the total annihilation of the weaker force, would be such as to cause the difference between the squares of the numbers surviving - multiplied by a coefficient of relative firepower-per-soldier, if unequal - to remain constant.

For example, a firefight between two forces in the initial ratio of 5 to 4, of equal firepower-per-man, continued unto the reduction of the weaker to 2/4 its original number, would see (sq.rt 13)/5 [3] of the stronger yet standing (since 52 - 42 = 9 = (sq.rt 13)2 - 22); continued unto the annihilation of the weaker, would see 3/5 of the stronger yet standing (since 52 -4 2 = 9 = 32 - 02). Of course, the moral effect of the losses sustained would likely be such as to cause the battle to be broken off before this point was reached.

Now consider a firefight between two forces of equal number, but with one side enjoying a 3/2 advantage in firepower-per-man, such as might recreate a Seven Years' War meeting engagement between a Prussian battalion and an Austrian: continued unto the reduction of the slower firing Austrian battalion to 1/2 its original number, the end of the battle would see sq.rt.1/2 (or about 7/10) of the Prussian yet standing (since 3/2(1)2 - (1)2 =1/2 = 3/2 sq.rt 1/2)2 - (1/2)2).

Finally, consider an engagement between two forces, one 3/2 as numerous as the other, but the numerically weaker having 3/2 the firepower-per-man of the stronger: the initial gross firepowers of the two forces are equal, so that the initial instantaneous loss-exchange ratio is 1 to 1; however, each man lost to the numerically weaker force takes with him 3/2 the firepower of each man lost to the stronger, with the result that the struggle becomes increasingly unfavorable to the weaker as it continues. Continued unto the reduction of the weaker to 1/2 its original number, the end of the battle sees (sq.rt 4 1/2)/3 [4] of the stronger yet standing (since (3)2 - 3/2(2)2 = 3 = (sq.rt 4 1/2)2 - 3/2(1)2).

Lanchester himself acknowledges the limited applicability of his formula to land war:

"The basic assumption is that the fire of each force is definitely concentrated on the opposing force ... The conditions may be regarded as corresponding to those prevalent. .. when individual-aimed firing or sniping [is] the order of the day.

"When, on the other hand, the circumstances are such as to preclude the possibility of such concentration, as when searching an area or ridge at long range, or volley firing at a position, or `into the brown,' the basic conditions are violated . . . It is well thus to call attention to the variations in the conditions and the nature of the resulting departure from the conclusions of theory; such variations are far less common in naval than in military warfare; the individual unit - the ship - is always the gunner's mark."

Whether consciously or not, game designers who base their tables of fire effect-per-turn solely upon gross firepowers, without adjustment for the varying densities of the opposing forces, are employing the targeted fire theory.

Footnotes