Δx = -βy; x > 0, or, if you prefer, death and destruction by numbers!

Capt. Kincaid (see previous post) has put me in military mode.  The mathematical equation in my title is, of course, totally incomprehensible to me, as is its counterpart: Δy = -αx; y > 0.  They are the two parts of F. W. Lanchester's  Square Law of Aimed Fire. 

Lanchester was one of those Victorian/Edwardian geniuses that Britian used to produce so effortlessly.  He was a mathematician, a physicist and an engineer, designing and building the famous Lanchester racing cars in the '20s.  During WWI, he became fascinated with aerial warfare and applied his mathematics to the problem of how to win.  To say that I am not a mathematician would be to understate the case, so I will attempt to translate his equation into plain-ish English.  Basically, he stated that where two forces fired directly at each other, the one with the greater number of firing weapons will win (so far so bleedin' obvious, you might think!) but he went on to point out that the rate of attrition against the losing side would increase exponentially over time.  In other words, as your greater firepower took its greater toll than your enemy's lesser firepower took on you, you would begin to kill more and more of him and faster and faster, too!  So, the essence of a face-to-face battle excluding any other considerations, like terrain, or cover, is to bring to bear more weapons than your opponent.

This mathematical theory of Lanchester came around a hundred years later than Kincaid's battles but then, in the age of the musket, it was experienced rather than theorised!  In the 18th century, Frederich the Great had deployed his musketeers in long lines in order to bring to bear on the enemy as many weapons as possible.  A century later, Bonarparte did the same but added some variations.  First of all he followed the European tradition of forming his infantry in three ranks which, though the rear rank was unable to fire, it was kept busy reloading the muskets of the men in front and passing them forward.  Also, he made use of light infantry as skirmishers to go ahead of his massed ranks in small detachments to sneak up as close as possible to the enemy formations and pepper them with fire to which they could not reply because the innacuracy of the musket made aiming at single men a waste of time and effort.  All you could do was stand there and bear it!  Finally, Bonaparte made another innovation.  By and large he was an aggressive general, always keen to seize the initiative by attacking.  He was a 'gunner by trade' and used his knowledge of artillery to good effect by massing them at critical points and hammering at the enemy formations to knock them about a bit before his assault.  But for the assault, he, unlike previous generals, massed his formations into columns of attack.  So entire divisons consisting of thousands of men would be formed in great oblong phalanxes as they marched towards their, hopefully, already shaken and demoralised foes.  Frequently, the mere appearance of these massed columns of attack advancing inexorably would be enough to break the moral and will-power of the enemy who would simply break and run.

But the British, under Wellington, were different.  For a start, the British infantry formed only two ranks deep and each man was drilled and drilled into reloading his own musket at speed.  They had their Light Infantry as skirmishers, too, which were hugely improved by units like Kincaid's 95th who operated with the Baker rifle which was much more accurate and over longer ranges, too. Also, Wellington, in fighting battles in which he was outnumbered was happy to fight defensively and knowing the damage artillery could cause would try to keep his infantry and cavalry out of sight behind a ridge-line, even allowing the men to lay down under artillery fire.  This would not totally protect them, as Kincaid describes in the fate of the 27th regiment who were all killed whilst standing in square.  However, it did offer some protection and the effect on moral was useful as we will see.  Morale in the age of the musket was absolutely critical.  If even a small fraction of your army turned tail, it would spread at amazing speed like a virulant virus and within minutes your army would be streaming away from the field.  "Morale is to material as three is to one", said Bonaparte and he should know!

Thus, in a battle in the age of the musket, morale was like a see-saw.  The French would open up with long, sustained artillery fire blowing great holes in the ranks of those facing them, let us say, the British, who could do nothing but stand patiently and close up the ranks from the reserves in the rear.  Then the pesky skirmishers would creep up and start to pepper you and again all you could do is stand and wait and wipe off the bits of shattered, bloody flesh from your mate who used to stand next to you.  Then, you would see the French forming their columns of attack, huge masses of men, with bands playing, drummer boys rattling out the advance, as with a steady, measured pace they slowly approached.

Now at this point, the morale of the French would have been high to exultant.  As Bonaparte waved them forward great cheers of "Vive L'Empereur" would roar out and then as they began their trek forward they would begin to sing the martial songs of the revolution. Opposite them the British morale would be, to use a suitably English bit of understatement, anxious!  The NCOs and the junior officers would be up and down holding the men steady and keeping them quiet - no cheering or songs because it was difficult enough to hear the words of command, or the bugle/drum calls, above the sound of the battle without a lot of noise from the ranks.  So they would wait, silent, muskets loaded but arms ordered by their sides as the French drew closer.

However, the closer the French came to this apparently immovable line of silently waiting men, the more the morale see-saw would begin to reverse.  The French would know that as they drew closer and the British refused to break then they would, sooner or later, see their enemy present his arms aimed directly at them and face a dreadful point-blank volley straight into their front ranks.  Here the range of the musket became the critical factor.  Over 100 yards it was useless.  The best range was somewhere between 25 and 75 yards.  At 75 yards, the defenders would inflict considerable damage and would have the chance to reload.  At 50 yards the casualties inflicted would be terrible but at 25 yards it would be mass murder!  Hence the old adage, 'wait 'til you see the whites of their eyes!'  Usually, the British held their fire  to the shorter ranges and then, as the enemy reeled back in shock and confusion from the murderous volleys, they would not bother to fire again but take advantageof the situation and instantly go in with the bayonet.  Now the French would be aware of this and, as their morale sank in knowledge of what they were about to face, they would have to take a decision.  Should they maintain their steady momentum, perhaps even speed it up into a charge, or should they pause and fire - which was difficult to do with thousands of men pressing in behind, and anyway, once stopped would the troops move forward again?  So at this point, British morale would be high and grimly expectant as they waited for the order to present arms and fire, whilst the French would be in a state of dread.

The other factor is that the British being formed in long lines only two ranks deep in which every man could fire against the French who, although outnumbering them, were operating on a much narrower frontage in which only the front two ranks could fire.  Thus, at this crucial point the French were, so to speak, well and truly out-musketeered as the British fired into their flanks as well as their fronts.

All other things being equal, more muskets always beats lesser muskets, as Lanchester formulated in his equation.  Or, as they say in the fight game, a good big'un nearly always beats a good little'un!

 

Comments

[dearieme]

I remember an undergraduate problem I was set to analyse Trafalgar using that "Law of Battle". A bunch of us took it on in the same Mathematics Reading Room where I once saw a job advert on the noticeboard that, it struck me, was recruiting for spies - or, more precisely, probably for counter-espionage, code-breaking and whatnot. Bloody clever, it was - if you couldn't see what the advert must be about, they wouldn't have wanted to hire you anyway. Ever since, I've allowed myself a wee grin whenever I've heard some blowhard assuring the world that all spies and such are stupid. "Aha", I've thought, "you're the sort of tit who wouldn't have understood the job advert."

[dearieme]

By the by, what can French morale have been like when they had to advance on an unseen foe? They are marching up a slope in column; they can see British skirmishers, and officers on horseback with their telescopes out, despatching young messengers at the gallop. A few pieces of light artillery too, perhaps? But just behind the crest lies a British infantry army whose size they don't know, perhaps set out in lines, perhaps to march over the crest at the last moment to deliver volleys. Heart-in-mouth, I'd have thought.

And if the French send their cavalry, the British manoeuvre into defensive array in squares, with their artillerymen tucked up safely inside.

[David Duff]

DM, Lanchester's theory worked beautifully at Trafalgar where, as you know, Nelson cut the Franco-Spanish line one third of the way down thus equalising the two fighting forces. However, his ships, as they approached in two seperate columns in line, split and each ship made its own way through the enemy line passing between a ship to the left and to the right, at which point the gun crew fired off a double-shotted (ie, 2x balls) gun into the rear of one enemy ship, and then, dashing over to the other side, they fired the same way into the front of the other enemy ship. These, of course, were raking shots going the length of the ship rather than across and were thuse many times more deadly.

As for you putative spying career, just think, you might have met John le Carre!

What you describe in your second comment was more or less what happened at Waterloo. Reports from the time indicate that the French, in the absence of any obvious enemy thought they were home and dry - until they crossed the road and the hedge at the top of the ridge. Then, they were truly 'gob-smacked', as they tried desperately to get themselves back in formation, to see lines of British infantry with their muskets at the ready. A few seconds later they were blown away. It was a particular shock to the Imperial Guard who saw nothing to disturb them until the British Guards suddenly stood up in the cornfields where they had been hiding - the corn used to grow very tall in those days!

[Hank]

David

The French at the beginning of the Revolutionary Wars adopted a Column attack formation because of time and cost they did not have to train the conscripts. Anyway Revolutionary elan will always carry the day, right? . It worked well enough for a while, ‘till the allies figured out what to do, of which the British army was unsurpassed.

John Keagan comments that the French columns broke from the rear. In the front with smoke hiding much the Fight side of the “fight or flight” took over. In the rear of the column , where they could see some shat was happening the Flight side would take over. Fight or Flight reaction is some what related to the distance to the danger the brain making subconscious decision on which is the best course to survive. But it could be hellish 60 or 90 seconds until the columan broke.

The British line did not have to fight the whole column, if they defeated the front ranks the rest were in retreat. The math was certainly in their favor and got better once the colun staated to break.

Courage, training, drill, and discipline over elan any time.

http://eclecticmeanderings.blogspot.com/>

[David Duff]

"Courage, training, drill, and discipline over elan any time."

And firepower - remember Lanchester!

By the way, Hank, your link doesn't work.