# Aircraft in Warfare (1916)/Chapter 5

CHAPTER V.
(October 2nd, 1914).
THE PRINCIPLE OF CONCENTRATION. THE N-SQUARE LAW.
§ 19. The Principle of Concentration. It is necessary at the present juncture to make a digression and to treat of certain fundamental considerations which underlie the whole science and practice of warfare in all its branches. One of the great questions at the root of all strategy is that of concentration; the concentration of the whole resources of a belligerent on a single purpose or object, and concurrently the concentration of the main strength of his forces, whether naval or military, at one point in the field of operations. But the principle of concentration is not in itself a strategic principle; it applies with equal effect to purely tactical operations; it is on its material side based upon facts of a purely scientific character. The subject is somewhat befogged by many authors of repute, inasmuch as the two distinct sides—the moral concentration (the narrowing and fixity of purpose) and the material concentration—are both included under one general heading, and one is invited to believe that there is some peculiar virtue in the word concentration, like the "blessed word Mesopotamia," whereas the truth is that the word in its two applications refers to two entirely independent conceptions, whose underlying principles have nothing really in common.
The importance of concentration in the material sense is based on certain elementary principles connected ​with the means of attack and defence, and if we are properly to appreciate the value and importance of concentration in this sense, we must not fix our attention too closely upon the bare fact of concentration, but rather upon the underlying principles, and seek a more solid foundation in the study of the controlling factors.
§ 20. The Conditions of Ancient and Modern Warfare Contrasted. There is an important difference between the methods of defence of primitive times and those of the present day which may be used to illustrate the point at issue. In olden times, when weapon directly answered weapon, the act of defence was positive and direct, the blow of sword or battleaxe was parried by sword and shield; under modern conditions gun answers gun, the defence from rifle-fire is rifle-fire, and the defence from artillery, artillery. But the defence of modern arms is indirect: tersely, the enemy is prevented from killing you by your killing him first, and the fighting is essentially collective. As a consequence of this diff"erence, the importance of concentration in history has been by no means a constant quantity. Under the old conditions it was not possible by any strategic plan or tactical manoeuvre to bring other than approximately equal numbers of men into the actual fighting line; one man would ordinarily find himself opposed to one man. Even were a General to concentrate twice the number of men on any given portion of the field to that of the enemy, the number of men actually wielding their weapons at any given instant (so long as the fighting line was unbroken), was, roughly speaking, the same on both sides. Under present-day conditions all this is changed. With modern long-range weapons—fire-arms, in brief—the concentration of superior numbers gives an immediate superiority in the active combatant ranks, and the numerically inferior force finds itself under a far heavier ​fire, man for man, than it is able to return. The importance of this difference is greater than might casually be supposed, and, since it contains the kernel of the whole question, it will be examined in detail.
In thus contrasting the ancient conditions with the modern, it is not intended to suggest that the advantages of concentration did not, to some extent, exist under the old order of things. For example, when an army broke and fled, undoubtedly any numerical superiority of the victor could be used with telling effect, and, before this, pressure, as distinct from blows, would exercise great influence. Also the bow and arrow and the cross-bow were weapons that possessed in a lesser degree the
(a) Fig. 2. (b)
properties of fire-arms, inasmuch as they enabled numbers (within limits) to concentrate their attack on the few. As here discussed, the conditions are contrasted in their most accentuated form as extremes for the purpose of illustration.
Taking, first, the ancient conditions where man is opposed to man, then, assuming the combatants to be of equal fighting value, and other conditions equal, clearly, on an average, as many of the "duels" that go to make up the whole fight will go one way as the other, and there will be about equal numbers killed of the forces engaged; so that if 1,000 men meet 1,000 men, it is of little or no importance whether a "Blue" force of 1,000 ​men meet a "Red" force of 1,000 men in a single pitched battle, or whether the whole " Blue " force concentrates on 500 of the "Red" force, and, having annihilated them, turns its attention to the other half; there will, presuming the "Reds" stand their ground to the last, be half the "Blue" force wiped out in the annihilation of the "Red" force in the first battle, and the second battle will start on terms of equality—i.e., 500 "Blue" against 500 "Red."
§ 21. Modern Conditions Investigated. Now let us take the modern conditions. If, again, we assume equal individual fighting value, and the combatants otherwise (as to "cover," etc.) on terms of equality, each man will in a given time score, on an average, a certain number of hits that are effective; consequently, the number of men knocked out per unit time will be directly proportional to the numerical strength of the opposing force. Putting this in mathematical language, and employing symbol b to represent the numerical strength of the "Blue" force, and r for the "Red," we have:—
    $^{db}$⁄$_{dt}$ = -r × c (1)
and
    $^{dr}$⁄$_{dt}$ = -b × k (2)
in which t is time and c and k are constants {c = k if the fighting values of the individual units of the force are equal). The reduction of strength of the two forces may be represented by two conjugate curves following the above equations. In Fig. 2 {a) graphs are given representing the case of the "Blue" force 1,000 strong encountering a section of the "Red" force 500 strong, and it will be seen that the "Red" force is wiped out of existence with ​a loss of only about 134 men of the "Blue" force, leaving 866 to meet the remaining 500 of the "Red" force with an easy and decisive victory; this is shown in Fig. 2 (b), the victorious "Blues" having annihilated the whole "Red" force of equal total strength with a loss of only 293 men.
Fig. 3a.
Fig. 3b.
In Fig. 3a a case is given in which the "Red" force is inferior to the "Blue" in the relation 1 : √2 say, a "Red" force 1,000 strong meeting a "Blue" force 1,400 strong. Assuming they meet in a single pitched battle fought to a conclusion, the upper line will ​represent the "Blue" force, and it is seen that the "Reds" will be annihilated, the "Blues" losing only 400 men. If, on the other hand, the "Reds" by superior strategy compel the "Blues" to give battle divided—say into two equal armies—then Fig. 3b, in the first battle the 700 "Blues" will be annihilated with a loss of only 300 to the "Reds" and in the second battle the two armies will meet on an equal numerical footing, and so we may presume the final battle of the campaign as drawn. In this second case the result of the second battle is presumed from the initial equality of the forces; the curves are not given.
Fig 4.
In the case of equal forces the two conjugate curves become coincident; there is a single curve of logarithmic form. Fig. 4; the battle is prolonged indefinitely. Since the forces actually consist of a finite number of finite units (instead of an infinite number of infinitesimal units), the end of the curve must show discontinuity, and break off abruptly when the last man is reached; the law based on averages evidently does not hold rigidly when the numbers become small. Beyond this, the condition of two equal curves is unstable, and any advantage secured by either side will tend to augment.
​§ 23. Graph representing Weakness of a Divided Force. In Fig. 5a, a pair of conjugate curves have been plotted backwards from the vertical datum representing the finish, and an upper graph has been added representing
Fig. 5a.
Fig. 5b.
the total of the "Red" force, which is equal in strength to the "Blue" force for any ordinate, on the basis that the "Red" force is divided into two portions as given by the intersection of the lower graph. In ​Fig. 5b, this diagram has been reduced to give the same information in terms per cent. for a "Blue" force of constant value. Thus in its application Fig. 5b gives the correct percentage increase necessary in the fighting value of, for example, an army or fleet to give equality, on the assumption that political or strategic necessities impose the condition of dividing the said army or fleet into two in the proportions given by the lower graph, the enemy being able to attack either proportion with his full strength. Alternatively, if the constant (= 100) be taken to represent a numerical strength that would be deemed sufficient to ensure victory against the enemy, given that both fleets engage in their full strength, then the upper graph gives the numerical superiority needed to be equally sure of victory, in case, from political or other strategic necessity, the fleet has to be divided in the proportions given. In Fig. 5b abscissæ have no quantitative meaning.
§ 24. Validity of Mathematical Treatment. There are many who will be inclined to cavil at any mathematical or semi-mathematical treatment of the present subject, on the ground that with so many unknown factors, such as the morale or leadership of the men, the unaccounted merits or demerits of the weapons, and the still more unknown "chances of war," it is ridiculous to pretend to calculate anything. The answer to this is simple: the direct numerical comparison of the forces engaging in conflict or available in the event of war is almost universal. It is a factor always carefully reckoned with by the various military authorities; it is discussed ad nauseam in the Press. Yet such direct counting of forces is in itself a tacit acceptance of the applicability of mathematical principles, but confined to a special case. To accept without reserve the mere "counting of the pieces" as of value, and to deny the more extended ​application of mathematical theory, is as illogical and unintelligent as to accept broadly and indiscriminately the balance and the weighing-machine as instruments of precision, but to decline to permit in the latter case any allowance for the known inequality of leverage.
§ 25. Fighting Units not of Equal Strength. In the equations (1) and (2), two constants were given, c and k, which in the plotting of the figures 2 to 5b were taken as equal; the meaning of this is that the fighting strength of the individual units has been assumed equal. This condition is not necessarily fulfilled if the combatants be unequally trained, or of different morale. Neither is it fulfilled if their weapons are of unequal efficiency. The first two of these, together with a host of other factors too numerous to mention, cannot be accounted for in an equation any more than can the quality of wine or steel be estimated from the weight. The question of weapons is, however, eminently suited to theoretical discussion. It is also a matter that (as will be subsequently shown) requires consideration in relation to the main subject of the present articles.
§ 26. Influence of Efficiency of Weapons. Any difference in the efficiency of the weapons — for example, the accuracy or rapidity of rifle-fire — may be represented by a disparity in the constants c and k in equations (1) and (2), The case of the rifle or machine-gun is a simple example to take, inasmuch as comparative figures are easily obtained which may be said fairly to represent the fighting efficiency of the weapon. Now numerically equal forces will no longer be forces of equal strength; they will only be of equal strength if, when in combat, their losses result in no change in their numerical proportion. Thus, if a "Blue" force initially 500 strong, using a magazine rifle, attack a "Red" force of 1,000, armed with a single breech-loader, and after a certain ​time the "Blue" are found to have lost 100 against 200 loss by the "Red," the proportions of the forces will have suffered no change, and they may be regarded (due to the superiority of the "Blue" arms) as being of equal strength.
If the condition of equality is given by writing M as representing the efficiency or value of an individual unit of the "Blue" force, and N the same for the "Red," we have:—
Rate of reduction of "Blue" force:—
    $^{db}$⁄$_{dt}$ = - N r × constant (3)
and "Red,"
    $^{dr}$⁄$_{dt}$ = -&nbspN b × constant (4)
And for the condition of equality,
    $^{db}$⁄$_{bdt}$ = $^{dr}$⁄$_{rdt}$,
or
    $^{-Nr}$⁄$_b$ = $^{-Mb}$⁄$_r$,
or
    N r$^2$ = M b$^2$ (5)
In other words, the fighting strengths of the two forces are equal when the square of the numerical strength multiplied by the fighting value of the individual units are equal.
§ 27. The Outcome of the Investigation. The n-square Law. It is easy to show that this expression (5) may be interpreted more generally; the fighting strength of a force may be broadly defined as proportional to the square of its numerical strength multiplied by the fighting value of its individual units.
Thus, referring to Fig. 5b, the sum of the squares of the two portions of the "Red" force are for all values equal to the square of the " Blue " force (the latter ​plotted as constant); the curve might equally well have been plotted directly to this law as by the process given. A simple proof of the truth of the above law as arising from the differential equations (1) and (2), §21, is as follows:—
In Fig. 6, let the numerical values of the "blue" and "red" forces be represented by lines b and r as shown; then in an infinitesimally small interval of time the change in b and r will be represented respectively by db and dr of such relative magnitude that db / dr = r / b or,
b db = r dr (1)
If (Fig. 6) we draw the squares on b and r and represent the increments db and dr as small finite increments,
Fig. 6.
we see at once that the change of area of b$^2$ is 2b db and the change of area of r$^2$ is 2r dr which according to the foregoing (1), are equal. Therefore the difference between the two squares is constant
b — r$^2$ = constant.
If this constant be represented by a quantity q$^2$ then b$^2$ = r$^2$ + q$^2$ and q represents the numerical value of the remainder of the blue "force" after annihilation ​of the red. Alternatively q represents numerically a second "red" army of the strength necessary in a separate action to place the red forces on terms of equality, as in Fig. 5b.
§ 28. A Numerical Example. As an example of the above, let us assume an army of 50,000 giving battle in turn to two armies of 40,000 and 30,000 respectively, equally well armed; then the strengths are equal, since (50,000)$^2$ = (40,000)$^2$ + (30,000)$^2$. If, on the other hand, the two smaller armies are given time to effect a junction, then the army of 50,000 will be overwhelmed, for the fighting strength of the opposing force, 70,000 is no longer equal, but is in fact nearly twice as great—namely, in the relation ofnb49 to 25. Superior morale or better tactics or a hundred and one other extraneous causes may intervene in practice to modify the issue, but this does not invalidate the mathematical statement.
§ 29. Example Involving Weapons of Different Effective Value. Let us now take an example in which a difference in the fighting value of the unit is a factor. We will assume that, as a matter of experiment, one man employing a machine-gun can punish a target to the same extent in a given time as sixteen riflemen. What is the number of men armed with the machine gun necessary to replace a battalion a thousand strong in the field? Taking the fighting value of a rifleman as unity, let n = the number required. The fighting strength of the battalion is, (1,000)$^2$ or,
n = √$^{1,000,000}$⁄$_{16}$ = $^{1,000}$⁄$_4$ = 250
or one quarter the number of the opposing force.
This example is instructive; it exhibits at once the utility and weakness of the method. The basic assumption is that the fire of each force is definitely concentrated ​on the opposing force. Thus the enemy will concentrate on the one machine-gun operator the fire that would otherwise be distributed over four riflemen, and so on an average he will only last for one quarter the time, and at sixteen times the efficiency during his short life he will only be able to do the work of four riflemen in lieu of sixteen, as one might easily have supposed. This is in agreement with the equation. The conditions may be regarded as corresponding to those prevalent in the Boer War, when individual-aimed firing or sniping was 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, and the value of the individual machine-gun operator becomes more nearly that of the sixteen riflemen that the power of his weapon represents. The same applies when he is opposed by shrapnel fire or any other weapon which is directed at a position rather than the individual. 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. When we come to deal with aircraft, we shall find the conditions in this respect more closely resemble those that obtain in the Navy than in the Army; the enemy's aircraft individually rather than collectively is the air-gunner's mark, and the law herein laid down will be applicable.
§ 30. The Hypothesis Varied. Apart from its connection with the main subject, the present line of treatment has a certain fascination, and leads to results which, though probably correct, are in some degree ​unexpected. If we modify the initial hypothesis to harmonise with the conditions of long-range fire, and assume the fire concentrated on a certain area known to be held by the enemy, and take this area to be independent of the numerical value of the forces, then, with notation as before, we have—
| - db/dt =b × N r | × constant. |
| - dr/dt =r × M b | × constant. |
or
    M db/dt = N dr/dt
or the rate of loss is independent of the numbers engaged, and is directly as the efficiency of the weapons. Under these conditions the fighting strength of the forces is directly proportional to their numerical strength; there is no direct value in concentration, qua concentration, and the advantage of rapid fire is relatively great. Thus in effect the conditions approximate more closely to those of ancient warfare.
§ 31. An Unexpected Deduction. Evidently it is the business of a numerically superior force to come to close quarters, or, at least, to get within decisive range as rapidly as possible, in order that the concentration may tell to advantage. As an extreme case, let us imagine a "Blue" force of 100 men armed with the machine gun opposed by a Red" 1,200 men armed with the ordinary service rifle. Our first assumption will be that both forces are spread over a front of given length and at long range. Then the "Red" force will lose 16 men to the "Blue" force loss of one, and, if the combat is continued under these conditions, the "Reds" must lose. If, however, the "Reds" advance, and get within short range, where each man and gunner is an ​individual mark, the tables are turned, the previous equation and conditions apply, and, even if "Reds" lose half their effective in gaining the new position, with 600 men remaining they are masters of the situation; their strength is 600$^2$ × 1 against the "Blue" 100$^2$ × 16. It is certainly a not altogether expected result that, in the case of fire so deadly as the modern machine-gun, circumstances may arise that render it imperative, and at all costs, to come to close range.
§ 32. Examples from History. It is at least agreed by all authorities that on the field of battle concentration is a matter of the most vital importance; in fact, it is admitted to be one of the controlling factors both in the strategy and tactics of modern warfare. It is aptly illustrated by the important results that have been obtained in some of the great battles of history by the attacking of opposing forces before concentration has been effected. A classic example is that of the defeat by Napoleon, in his Italian campaign, of the Austrians near Verona, where he dealt with the two Austrian armies in detail before they had been able to effect a junction, or even to act in concert. Again, the same principle is exemplified in the oft-quoted case of the defeat of Jourdan and Moreau on the Danube by the Archduke Charles in 1796. It is evident that the conditions in the broad field of military operations correspond in kind, if not in degree, to the earlier hypothesis, and that the law deduced therefrom, that the fighting strength of a force can be represented by the square of its numerical strength, does, in its essence, represent an important truth.
* ↑ This is not strictly true, since towards the close of the fight the last few men will be attacked by more than their own number. The main principle is, however, untouched.
* ↑ See Archduke Charles, Duke of Teschen#Napoleonic Wars (Wikisource contributor note)