The most important fundamental to be addressed by the wargame designer is the resolution of infantry firefights -- that is, the determination of their expected outcomes, prior to the introduction (via dice, cards, etc.) of whatever degree of randomisation about the expected values may be deemed desirable. For this purpose he can choose among three competing theories of fire action:

Targeted or "Concentrated" Fire;
Area Fire; and
Ineffective Fire (performed mainly to entertain the firer).

Each of these theories and its implications shall be examined in turn. It should be noted what the three have in common; they operate independently of measures of time, which is to say that in themselves they offer no guidance as to the actual number of minutes (or hours or days, etc.) required to produce the predicted results. Then again, neither is any of the three incompatible with whatever time scale the designer may otherwise consider appropriate.

The theory of targeted or "concentrated" fire was popularised during the 19th Century by the 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 firepower, 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.

(1) 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 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. (If morale checks are indeed made during the course of the conflict.)

(2) 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).

(3) 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 firepower 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 unfavourable 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 firepower, without adjustment for the varying densities of the opposing forces, are employing the targeted fire theory.

In contrast to targeted fire, area fire requires that fire effect be calculated as the product of:

(1) The volume of fire rained upon the enemy-held area, and

(2) The density of enemy numbers therein, such that in a firefight between numerically unequal forces occupying equal areas, the advantage of the stronger force in volume of fire is exactly offset by the lower density of the weaker, whose own advantage in having a more densely-packed enemy at which to shoot is exactly offset by a lower volume of fire with which to hit it. The loss exchange ratio, both instantaneous and cumulative, begins and remains throughout the battle at 1 to 1, notwithstanding the growing disparity between the opposing forces in gross firepower ratio. As long as the spaces occupied by the two forces remain equal, the loss exchange ratio can be budged from unity only through the introduction of inequality in firepower-per-man, by which one side's volume of fire may be increased without itself presenting any denser a target to its enemy.

As Lanchester puts it:

"The rate of loss is independent of the [ratio of] the numbers engaged, and is directly as the efficiency of the weapons . . . There is no direct value in concentration, qua concentration, and the advantage of rapid fire is relatively great."

Now apply this to the above-mentioned calculations:

(1) 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 only 3/5 of the stronger yet standing - the same number as would survive the total annihilation of the weaker force under the targeted fire theory.

(2) A firefight between two forces of equal number, but with one side enjoying 3/2 the firepower-per-man of the other, continued unto the reduction of the slower-firing force to 1/2 its original number, would see 2/3 of the faster-firing force yet standing.

(3) A firelight 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, continued unto the reduction of the weaker to 1/2 its original strength, sees the stronger reduced to its original number likewise.

To improve the 1 to 1 loss-exchange ratio without introducing an inequality in firepower-per- man requires introducing an inequality between the two forces in the space occupied, such as might be achieved either by:

(1) Spreading the existing number of one's own force over a wider space, thereby reducing its density relative to the enemy's volume of fire without changing one's own volume of fire relative to the enemy's density; or

(2) Directing upon the same enemy the fire of an additional space-full-of-soldiers not otherwise engaged, thereby increasing the volume of friendly fire relative to the enemy's density without changing one's own density relative to the enemy's volume of fire.

A like increase in space occupied using either tactic will produce an equal improvement in the loss-exchange ratio; however, the second tactic will compress the time required to arrive at any particular level of cumulative losses. Against an interval-less or "en muraille" battle line, the additional space required to implement either tactic must be sought on the flanks. To lower the density of one's own force, as under the first tactic, is to increase its vulnerability to enemy cavalry.

An inequality in space-occupied may be used to magnify or to offset an inequality in firepower-per-man. For example, a force equal in numbers but with only 2/3 the firepower-per-man of its enemy might re-establish a loss-exchange ratio of 1:1 [up from 2:3] by spreading itself over a space 3/2 as extensive - provided, again, that the additional space could be found within like range, and that the enemy not be possessed of a force of cavalry able to exact a severe penalty for such dispersion.

Although of no direct relevance to the loss-exchange ratio of the instant firefight, the productivity of one's musketry may also be improved through the generosity of the enemy in deploying his battalions with a non-firing fourth rank, or in stationing his second line-of-battle too close behind his first, or by any other act which puts more of his troops in harm’s way than the number of bullets sent back at your own troops.

No discussion of infantry fire combat can be complete without some mention of that heretical school of thought which alleges infantry firefights to have been largely sham affairs, infantry fire being next to worthless as a means of working material destruction upon the foe. Interestingly, there exists marked disagreement among subscribing theoreticians as to the cause of this putative phenomenon.

Maurice de Saxe, writing in the 1730's, blames the system of volley fire for the poor results obtained from the fire of line infantry, relative to those obtained from the "voluntary" fire practiced by skirmishers. The physical and nervous strain of awaiting the command to loose a volley, says he, makes accuracy impossible.

The 19th Century analyst Ardant du Picq, on the other hand, claims that volley fire would be devastating if actually used, but that human nature forbids it. The excitement of battle causes its degeneration into fire-at-will, each man firing madly, jostled all the while by the motions of those around him and startled by the discharge of their weapons. Under such conditions neither aiming nor even levelling can be expected. The fire of the third rank is lost entirely: the men fire into the air to avoid hitting their comrades in the front rank, who must stand to reload and cannot be induced to kneel again after the initial volleys.

Regardless of its material ineffectiveness, or the cause thereof, infantry fire is nevertheless of value in occupying the mind of the soldier, to distract him from his fear, and so to hold him in the ranks long enough to see the enemy opposite's morale give way -- an effect produced by agencies other than losses from fire, which are too few to matter. The fire of skirmishers might produce losses enough to matter, were it not directed mainly at other skirmishers.

As for the vaunted superiority in rounds-per-minute of Frederick's Prussians or British infantry, its value lay in the illusion of destructiveness that it imparted to the minds of those delivering and facing it, with concomitant effects upon morale. Usually the high rates of fire could not be sustained as the weapons fouled and/or the smoke obscured the target.

Beguiled though he may be by such theses, the game designer is yet left to wonder how to account for the very heavy casualties, as a percent of total forces engaged - to say nothing of those battalions in the thickest of the fight - characteristic of major battles of the Seven Years War. Perhaps reported losses were exaggerated; or perhaps the principal man-killers were the artillery and the cavalry, hospital statistics on the causes of wounds notwithstanding. It is more likely that the "Ineffective Fire" theory is simply not applicable to this war. Infantry seemed capable of standing “toe-to-toe” with their opponents, probably because many were professional soldiers (at least in the early years before many were killed and their places taken by conscripts).

Which of the remaining theories - Targeted Fire or Area Fire - should be employed? One might say that because the inaccuracy of smoothbore weapons and the blinding effect of black powder smoke made the targeting of individual enemies impossible; the area fire theory must apply by default.

But wait: at the short ranges typical for infantry firefights, bullets do not so much "rain" upon a position as strike it horizontally. If the position in question consists of a unit of multiple ranks, then the chance of scoring successive hits does not diminish as hits accumulate, as long as at least one such rank remains. It pays then to pack into the line as many men as can fire, if what incoming bullets are stopped by the front most rank alone.

The mathematics of the situation revert to those of targeted fire, even though we have not "individual-aimed firing or sniping," but "volley firing at a position!"

Yet, if this model is correct, then how do we account for the anxiety to keep the second line-of-battle far enough back of the first to place it beyond enemy musket range? Why would the deployment of a non-firing fourth rank [to immediately replace losses to the fore] not be considered an unalloyed advantage? Why take any more trouble to shelter a column from fire than a line? No, it cannot be. The infantry "wall" is not solid, but permeable to bullets. The area fire theory applies, although the "area" in question is not upon the earth's surface, but perpendicular, and has depth extending to the firer's effective range.

One last point to consider is that the maximum effective range of the typical smoothbore musket was about 100 yards or metres. So the above calculations are based on firing at much closer distances. This did influence the casualty rate, as the proximity to the target would likely increase the number of casualties as long as the troops fired at the mass of the target rather than just towards the target. The angle at which the ball left the barrel also had an influence as this increased the spread of the balls, many of which would not strike the target area.

So the game designer cannot rely completely on mathematics to develop casualty or “hit” charts. This will also be discussed in the next article concerning artillery performance.