Aircraft in Warfare (1916)/Chapter 6

by Frederick William Lanchester

CHAPTER VI.

(October 9th, 1914)

THE N-SQUARE LAW IN ITS APPLICATION.

§ 33. The n-square Law in its Application to a Heterogeneous Force. In the preceding article it was demonstrated that under the conditions of modern warfare the fighting strength of a force, so far as it depends upon its numerical strength, is best represented or measured by the square of the number of units. In land operations these units may be the actual men engaged, or in an artillery duel the gun battery may be the unit; in a naval battle the number of units will be the number of capital ships, or in an action between aeroplanes the number of machines. In all cases where the individual fighting strength of the component units may be different it has been shown that if a numerical fighting value can be assigned to these units, the fighting strength of the whole force is as the square of the number multiplied by their individual strength. Where the component units differ among themselves, as in the case of a fleet that is not homogeneous, the measure of the total of fighting strength of a force will be the square of the sum of the square roots of the strengths of its individual units.

§ 34. Graphic Representation. Before attempting to apply the foregoing, either as touching the conduct of aerial warfare or the equipment of the fighting aeroplane, it is of interest to examine a few special cases and applications in other directions and to discuss certain possible limitations. A convenient graphic form in which the operation of the n-square law can be presented is given in Fig. 7; here the strengths of a number of separate armies or forces successively mobilised and brought into action are represented numerically by the lines a, b, c, d, e, and the aggregate fighting strengths of these armies are given by the lengths of the lines A, B. C, D, E, each being the hypotenuse of a right-angle triangle, as indicated. Thus two forces or armies a and b, if acting separately (in point of time), have only the fighting

Fig. 7.

strength of a single force or army represented numerically by the line B. Again, the three separate forces, a, b, and c, could be met on equal terms in three successive battles by a single army of the numerical strength C, and so on.

§ 35. Special or Extreme Case. From the diagram given in Fig. 7 arises a special case that at first sight may look like a reductio ad absurdum, but which, correctly interpreted, is actually a confirmation of the n-square law. Referring to Fig. 7, let us take it that the initial force (army or fleet), is of some definite finite magnitude, but that the later arrivals b, c, d, etc., be very small and numerous detachments—so small, in fact, as to be reasonably represented to the scale of the diagram as infinitesimal quantities. Then the lines b, c, d, e, f, etc., describe a polygonal figure approximating to a circle, which in the limit becomes a circle, whose radius is represented by the original force a. Fig. 8. Here we have graphically represented the result that the fighting value of the added forces, no matter what their numerical aggregate (represented in Fig. 8 by the circumferential

Fig. 8.

line), is zero. The correct interpretation of this is that in the open a small force attacking, or attacked by one of overwhelming magnitude is wiped out of existence without being able to exact a toll even comparable to its own numerical value; it is necessary to say in the open, since, under other circumstances, the larger force is unable to bring its weapons to bear, and this is an essential portion of the basic hypothesis. In the limiting case when the disparity of force is extreme, the capacity of the lesser force to effect anything at all becomes negligible. There is nothing improbable in this conclusion, but it manifestly does not apply to the case of a small force concealed or "dug in," since the hypothesis is infringed. Put bluntly, the condition represented in Fig. 8 illustrates the complete impotence of small forces in the presence of one of overwhelming power. Once more we are led to contrast the ancient conditions, under which the weapons of a large army could not be brought to bear, with modern conditions, where it is physically possible for the weapons of ten thousand to be concentrated on one. Macaulay's lines

"In yon strait path a thousand
May well be stopped by three,"

belong intrinsically to the methods and conditions of the past.

§ 36. The N-square Law in Naval Warfare. We have already seen that the n-square law applies broadly, if imperfectly, to military operations; on land however, there sometimes exist special conditions and a multitude of factors extraneous to the hypothesis whereby its operation may be suspended or masked. In the case of naval warfare, however, the conditions more strictly conform to our basic assumptions, and there are comparatively few disturbing factors. Thus, when battle fleet meets battle fleet, there is no advantage to the defender analogous to that secured by the entrenchment of infantry. Again, from the time of opening fire, the individual ship is the mark of the gunner, and there is no phase of the battle or range at which areas are searched in a general way. In a naval battle every shot fired is aimed or directed at some definite one of the enemy's ships; there is no firing on the mass or "into the brown." Under the old conditions of the sailing-ship and cannon of some 1,000 or 1,200 yards maximum effective range, advantage could be taken of concentration within limits; and an examination of the latter 18th century tactics makes it apparent that with any ordinary disparity of numbers (probably in no case exceeding 2 to  1) the effect of concentration must have been not far from that indicated by theory. But to whatever extent thiswas the case, it is certain that with a battle-fleet action at the present day the conditions are still more favourable to the weight of numbers, since with the modern battle range—some 4 to 5 miles—there is virtually no limit to the degree of concentration of fire. Further than this, there is in modern naval warfare practically no chance of coming to close quarters in ship-to-ship combats, as in the old days.

Thus the conditions are to-day almost ideal from the point of view of theoretical treatment. A numerical superiority of ships of individually equal strength will mean definitely that the inferior fleet at the outset has to face the full fire of the superior, and as the battle proceeds and the smaller fleet is knocked to pieces, the initial disparity will become worse and worse, and the fire to which it is subjected more and more concentrated. These are precisely the conditions taken as the basis of the investigation from which the n-square law has been derived. The same observations will probably be found to apply to aerial warfare when air fleets engage in conflict, more especially so in view of the fact that aeroplane can attack aeroplane in three dimensions of space instead of being limited to two, as is the case with the battleship. This will mean that even with weapons of moderate range the degree of fire concentration possible will be very great. By attacking from above and below, as well as from all points of the compass, there is, within reason, no limit to the number of machines which can be brought to bear on a given small force of the enemy, and so a numerically superior fleet will be able to reap every ounce of advantage from its numbers.

§ 37. Individual value of Ships or Units. The factor the most difficult to assess in the evaluation of a fleet as a fighting machine is (apart from the personnel) the individual value of its units, when these vary amongst themselves. There is no possibility of entirely obviating this difficulty, since the fighting value of any given ship depends not only upon its gun armament, but also upon its protective armour. One ship may be stronger than another at some one range, and weaker at some longer or shorter range, so that the question of fleet strength can never be reduced quite to a matter of simple arithmetic, nor the design of the battleship to an exact science. In practice the drawing up of a naval programme resolves itself, in great part at least, into the answering of the prospective enemy's programme type by type and ship by ship. It is, however, generally accepted that so long as we are confining our attention to the main battle fleets, and so are dealing with ships of closely comparable gun calibre and range, and armour of approximately equivalent weight, the fighting value of the individual ship may be gauged by the weight of its "broadside," or more accurately, taking into account the speed with which the different guns can be served, by the weight of shot that can be thrown per minute. Another basis, and one that perhaps affords a fairer comparison, is to give the figure for the energy per minute for broadside fire, which represents, if we like so to express it, the horse-power of the ship as a fighting machine. Similar means of comparison will probably be found applicable to the fighting aeroplane, though it may be that the downward fire capacity will be regarded as of vital importance rather than the broadside fire as pertaining to the battleship.

§ 38. Applications of the n-square Law. The n-square law tells us at once the price or penalty that must be paid if elementary principles are outraged by the division of our battle fleet^[1] into two or more isolated detachments. In this respect our present diposition—a single battle fleet or "Grand" fleet—is far more economical and strategically preferable as a defensive power to the old-time distribution of the Channel Fleet. Mediterranean Fleet, etc. If it had been really necessary, for any political or geographical reason, to maintain two separate battle fleets at such distance asunder as to preclude their immediate concentration in case of attack, the cost to the country would have been enormously

Fig. 9.

increased. In the case, for example, of our total battle fleet being separated into two equal parts, forming separate fleets or squadrons, the increase would require to be fixed at approximately 40 per cent.—that is to say, in the relation of 1 to √2; more generally the solution is given by a right-angled triangle, as in Fig. 9. It must not be forgotten that, even with this enormous increase, the security will not be so great as appears on paper, for the enemy's fleet, having met and defeated one section of our fleet, may succeed in falling back on his base for repair and refit, and emerge later with the advantage of strength in his favour. Also one must not overlook the demoralizing effect on the personnel of the fleet first to go into action, of the knowledge that they are hopelessly outnumbered and already beaten on paper—that they are, in fact, regarded by their King and country as "cannon fodder." Further than this, presuming two successive fleet actions and the enemy finally beaten, the cost of victory in men and matériel will be greater in the case of the divided fleet than in the case of a single fleet of equal total fighting strength, in the proportion of the total numbers engaged—that is to say, in Fig. 9, in the proportion that the two sides of the right-angled triangle are greater than the hypotenuse.

In brief, however potent political or geographical influences or reasons may be, it is questionable whether under any circumstances it can be considered sound strategy to divide the main battle fleet on which the defence of a country depends. This is to-day the accepted view of every naval strategist of repute, and is the basis of the present distribution of Great Britain's naval forces.

§ 39. Fire Concentration the Basis of Naval Tactics. The question of fire concentration is again found to be paramount when we turn to the consideration and study of naval tactics. It is worthy of note that the recognition of the value of any definite tactical scheme does not seem to have been universal until quite the latter end of the 18th century. It is even said that the French Admiral Suffren, about the year 1780, went so far as to attribute the reverses suffered by the French at sea to "the introduction of tactics" which he stigmatised as "the veil of timidity;"^[2] the probability is that the then existing standard of seamanship in the French Navy was so low that anything beyond the simplest of manœuvres led to confusion, not unattended by danger. The subject, however, was, about that date, receiving considerable attention. A writer, Clerk, about 1780,^[3] pointed out that in meeting the attack of the English the French had adopted a system of defence consisting of a kind of running fight, in which, initially taking the "lee gage,"^[4] they would await the English attack in line ahead, and having delivered their broadsides on the leading English ships (advancing usually in line abreast), they would bear away to leeward and take up position, once more waiting for the renewal of the attack, when the same process was repeated.^[5] By these tactics the French obtained a concentration of fire on a small portion of the English fleet, and so were able to inflict severe punishment with little injury to themselves.^[6] Here we see the beginnings of sound tactical method adapted to the needs of defence. Up to the date in question there appears to have been no studied attempt to found a scheme of attack on the basis of concentration; the old order was to give battle in parallel columns or lines, ship to ship, the excess of ships, if either force were numerically superior, being doubled on the rear ships of the enemy. It was not till the "Battle of the Saints," in 1782, that a change took place; Rodney (by accident or intention) broke away from tradition, and cutting through the lines of the enemy, was able to concentrate on his centre and rear, achieving thereby a decisive victory.

§ 40. British Naval Tactics in 1805. The Nelson "Touch." The accident or experiment of 1782 had evidently become the established tactics of the British in the course of the twenty years which followed, for not only do we find the method in question carefully laid down in the plan of attack given in the Memorandum issued by Nelson just prior to the Battle of Trafalgar in 1805, but the French Admiral Villeneuve^[7][8] confidently asserted in a note issued to his staff in anticipation of the battle that:—"The British Fleet will not be formed in a line-of-battle parallel to the combined fleet according to the usage of former days. Nelson, assuming him to be, as represented, really in command, will seek to break our line, envelop our rear, and overpower with groups of his ships as many as he can isolate and cut off." Here we have a concise statement of a definite tactical scheme based on a clear understanding of the advantages of fire concentration.

It will be understood by those acquainted with the sailing-ship of the period that the van could only turn to come to the assistance of those in the rear at the cost of a considerable interval of time, especially if the van should happen to be to leeward of the centre and rear. The time taken to "wear ship," or in light winds to "go about" (often only to be effected by manning the boats and rowing to assist the manoeuvre), was by no means an inconsiderable item. Thus it would not uncommonly be a matter of some hours before the leading ships could be brought within decisive range, and take an active part in the fray.

§ 41. Nelson's Memorandum, and Tactical Scheme. In order further to embarass the enemy's van, and more effectively to prevent it from coming into action, it became part of the scheme of attack that a few ships, a comparatively insignificant force, should be told off to intercept and engage as many of the leading ships as possible; in brief, to fight an independent action on a small scale; we may say admittedly a losing action. In this connection Nelson's memorandum of October 9 is illuminating. Nelson assumed for the purpose of framing his plan of attack that his own force would consist of forty sail of the line, against forty-six of the combined (French and Spanish) fleet. These numbers are considerably greater, as things turned out, than those ultimately engaged; but we are here dealing with the memorandum, and not with the actual battle. The British Fleet was to form in two main columns, comprising sixteen sail of the line each, and a smaller column

Fig. 10.

of eight ships only. The plan of attack prescribed in the event of the enemy being found in line ahead was briefly as follows:—One of the main columns was to cut the enemy's line about the centre, the other to break through about twelve ships from the rear, the smaller column being ordered to engage the rear of the enemy's van three or four ships ahead of the centre, and to frustrate, as far as possible, every effort the van might make to come to the succour of the threatened centre or rear. Its object, in short, was to prevent the van of the combined fleet from taking part in the main action. The plan is shown diagrammatically in Fig. 10.

§ 42. Nelson's Tactical Scheme Analysed. An examination of the numerical values resulting from the foregoing disposition is instructive. The force with which Nelson planned to envelop the half—i.e., 23 ships—of the combined fleet amounted to 32 ships in all; this according to the n² law would give him a superiority of fighting strength of almost exactly two to one,^[9] and would mean that if subsequently he had to meet the other half of the combined fleet, without allowing for any injury done by the special eight-ship column, he would have been able to do so on terms of equality. The fact that the van of the combined fleet would most certainly be in some degree crippled by its previous encounter is an indication and measure of the positive advantage of strength provided by the tactical scheme. Dealing with the position arithmetically, we have:—

Strength of British (in arbitrary n² units),
⁠32² + 8² = 1088
And combined fleet,
⁠23² + 23² = 1058
British advantage .... 30

Or, the numerical equivalent of the remains of the British Fleet (assuming the action fought to the last gasp), = √30 or 5½ ships.

If for the purpose of comparison we suppose the total forces had engaged under the conditions described by Villeneuve as "the usage of former days," we have:—

Strength of combined fleet, 46² .... = 2116
Strength of„ British fleet,„ 40² .... = 1600
Balance in favour of enemy .... 516

Or, the equivalent numerical value of the remainder of the combined fleet, assuming complete annihilation of the British, = √516 = 23 ships approximately.

Thus we are led to appreciate the commanding importance of a correct tactical scheme. If in the actual battle the old-time method of attack had been adopted, it is extremely doubtful whether the superior seamanship and gunnery of the British could have averted defeat. The actual forces on the day were 27 British sail of the line against the combined fleet numbering 33, a rather less favourable ratio than assumed in the Memorandum. In the battle, as it took place, the British attacked in two columns instead of three, as laid down in the Memorandum; but the scheme of concentration followed the original idea. The fact that the wind was of the lightest was alone sufficient to determine the exclusion of the enemy's van from the action. However, as a study the Memorandum is far more important than the actual event, and in the foregoing analysis it is truly remarkable to find, firstly, the definite statement of the cutting the enemy into two equal parts—according to the n-square law the exact proportion corresponding to the reduction of his total effective strength to a minimum; and, secondly, the selection of a proportion, the nearest whole-number equivalent to the √2 ratio of theory, required to give a fighting strength equal to tackling the two halves of the enemy on level terms, and the detachment of the remainder, the column of eight sail, to weaken and impede the leading half of the enemy's fleet to guarantee the success of the main idea. If, as might fairly be assumed, the foregoing is more than a coincidence,^[10] it suggests itself that Nelson, if not actually acquainted with the n-square law, must have had some equivalent basis on which to figure his tactical values.

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1. Capital ships:—Dreadnoughts and Super-Dreadnoughts.
2. Mahan, "Sea Power," page 425.
3. John Clerk of Eldin: Essay on Naval Tactics (1779, published 1790) (Wikisource contributor note)
4. See Sailing ship tactics#Importance of the weather gage (Wikisource contributor note)
5. Compare Malian, "Sea Power," page 162.
6. Incidentally, also, the scheme in question had the advantage of subjecting the English to a raking fire from the French broadsides before they were themselves able to bring their own broadside fire to bear.
7. "The Enemy at Trafalgar," Ed. Fraser; Hodder and Stoughton, page 54.
8. Fraser Edward: The Enemy at Trafalgar: the Battle From the Perspective of the French & Spanish Navies and Their Sailors (1906) (Wikisource contributor note)
9. 23 × √2 = 32.5
10. Although we may take it to be a case in which the dictates of experience resulted in a disposition now confirmed by theory, the agreement is remarkable.