This week, we’re going to look at how the effectiveness of arrow fire – especially against armored targets – varies over distance. This is, in a sense, a continuation of my previous post on armor penetration, “Punching Through Some Armor Myths,” so you may want to refer back occasionally.
This is going to be a longer post, with a bunch of math. I think the math is very interesting, but I understand some people don’t – on the flip side, I know if I leave it out, some folks will be very confused as to how I got where I was going. If you just want to contrast how things are shown with how they probably were, read the first two sections, then skip down to ‘Quick Sum Up‘ and pick up from there.
My ‘target’ today (excuse the pun) is less portrayals of pre-gunpowder warfare in film or TV and more how it is portrayed in interactive media like tabletop wargames and video-games. It thus has less to do with how everything looks and more with game mechanics, so let’s start there.
Mechanics and the Problem
Games that attempt instead to simulate the complex effects of wounds (Dwarf Fortress, RimWorld) are very rare. In most games, the effect of weapons is reduced to a mathematical abstraction: ‘damage.’ This weapon does 5 damage, that weapon does 10; this enemy has 25 hit points, so must hit him so many times – that sort of thing. I broadly understand the necessity of that abstraction and I am not going to complain about it here. We can understand the damage statistic as a simplified measure of lethality well enough.
Instead, I want to focus on the way ranged weapons – especially arrows – are handled within these sorts of systems. Let’s take the example of the Total War games – in this case, Total War: Three Kingdoms (because I have been playing a bunch of it).
These basic archers have a range of 200 and an arrow-damage of 53 (31 regular and 22 armor piercing damage). What that means in the game is that these guys can only fire at enemies under 200 units (I think it’s in meters, but it is really hard to tell). When each model (each guy) fires, the game calculates a trajectory for his arrow – if it intercepts another model (friend or foe), it deals the 53 damage listed. It doesn’t matter if the target is 199 meters away, or 2 meters away – it deals 53 damage.
Against armored targets who can resist some of the damage, the arrow does 22 guaranteed damage (the armor piercing value), and the 31 regular damage is reduced by the armor via a complicated formula. Rather than tracking probability of a sudden lethal hit, the target is killed when the cumulative damage from the arrow hits exceeds his ‘hit points.’ Range is still not a factor in damage.
Now, that’s not to say greater ranges do nothing in game. The arrows have ‘scatter’ to them – each shot is a little bit off. The longer the range, the more time for the arrows to spread out on their individual trajectories, and the create chance of missing. Amusingly, a shot that – by accident – goes long still deals full damage (you can see this clearly with artillery), but you can only shoot that far by mistake, never on purpose.
The upshot of this is that the range of the ranged weapons is a hard limit. Inside? Full damage. Outside? No damage. You see this same effect in RPGs (like Divinity: Original Sin or Pillars of Eternity) where ranged weapons either do full damage or no damage. It’s not an uncommon system at all (but we shall not discuss the cringe of systems where the damage stat belongs to the arrow instead of the bow). But I want to stay focused on Total War for a second, because of how it influences the tactics in the game – and thus the common perception of historical tactics.
So what effect does this simplified model of ranged combat have on the tactics of archery?
First: Small differences in range become very important. The difference between a range of, say, 180 and 160 (e.g. the difference between High Elven archers and Empire Crossbowmen in Total War: Warhammer II) seems trivial, but is actually very significant, because at a range of 180 or 170 units, the former group deals full damage and the latter deals none.
Second: tactics that involve skirmishing with fast, long-range units – like horse archers – happen at the very limit of their effective engagement range. That is, horse archers with a range of 200 are going to sit at 200 and fire, moving away so as to maintain that huge range. That makes differences in speed important too – if those horse archers were close, a slower unit (like another kind of cavalry) might be able to surprise them with a sudden dash, but at maximum range, the chance of that happening is basically nil. The only way to bring those horse archers into a fight is to use a faster unit. And in games where there often are no faster units, the only response to horse archers are either foot archers with equal range, or simply enduring the long-range fire. Of course, that further enforces the supremacy of maximum range: bringing inferior-range 150 foot archers does literally nothing to protect an army against range 200 horse archers, since the latter will never let the former in range to even shoot (despite the fact that this very thing was an effective way to at least mitigate the impact of superior enemy archer-cavalry historically).
Why do I care? Because these games are the only experience with tactics – let alone pre-modern tactics – that most people will ever have, and I see these implications seeping into the way my students think about these weapon systems. Students also – without realizing it – bake these assumptions into papers, obsessing about maximum range and movement speed, often over other more important considerations (like morale and battlefield psychology).
So let’s drop the game mechanics and start thinking about real world mechanics.
Range and Power: Thinking in Probabilities
The first thing we need to note is that – at least in Europe, the Middle East and the Mediterranean – by Late Antiquity (and I would argue earlier), most soldiers on a battlefield would wear at least some armor. That might mean only a textile defense, or it might mean more significant metal defenses, but there is typically something an arrow (or crossbow bolt) much defeat in order to score a lethal hit.
Last time, when we discussed penetrating armor, we did so mostly in binaries. Can weapon X – under ideal conditions – defeat armor Y. And that is generally how penetration tests are done: under ideal conditions. The armor being tested is unmoving and secured so that it takes the full force of the hit (the one notable exception was a set of tests Mark Stretton did where the armor was moved towards him at charging speed, making it easier to penetrate to simulate firing into a cavalry charge). It is struck by someone who can concentrate fully on delivering the heaviest blow possible; if a ranged weapon is used, it is fired at point-blank range. The hits that ‘count’ are strong blows, delivered dead on.
But in battle, we’re not dealing with one arrow, but hundreds and thus not with a binary (penetrate or not penetrate), but a range of probabilities (impossible, unlikely, maybe, likely, etc), influenced by a host of factors. The most obvious of these is accuracy – some percentage of shots will miss, and thus fail to inflict casualties regardless of the energy of impact. At the same time, some percentage of shots may strike unarmored or vulnerable areas (face, neck, eye-slits in visors, etc) and kill even at very low energy. Thus at almost any range (within bowshot) there is a least some chance that any shot will be lethal, disabling or miss entirely. The closer the enemy gets, the more accurate the volley becomes and thus the higher lethality goes.
We should understand defeating armor as a ‘probability cloud’ in the same way – and remember, nearly everyone is at least wearing some basic textile armor – most quite a bit more. The main factor here (on the attacker’s side) is the energy of impact and the shape of the arrowhead. High energy of impact, and a properly shaped arrow (read: bodkin) increases the chance to penetrate. At point-blank range, the energy of impact depends on the draw weight and the draw distance (because the longer the ‘power stroke’ of arrow, the more time it has to be accelerated by the string).
But the bows of a group of archers will not be perfectly uniform. The war bows recovered from the wreck of the Mary Rose, when reconstructed, had draw weights running from c. 100lb to c. 170lb, and may have been pulled to slightly different draws (Strickland and Hardy, The Great Warbow (2005)). That in turn means that even within a single volley, the arrows will arrive at a fairly wide range of speeds and impact-energies (further compounded by handcrafted arrows of slightly different weights and dimensions; different types of arrows will have even more variance).
How does range influence those probabilities?
Power over Range
Let’s start by recalling some of our energy thresholds for penetrating armor under ideal conditions. 50J of impact energy behind a spear-point was sufficient to defeat a 16-layer (quite thick) thick linen gambeson, an arrow might need somewhat less. Williams’s mail failed around 100J, but we ought to construct a broad range here: the mail he used was of fairly high quality and he used a lot of padding behind it – we might suppose mail would fail at energies between 60 to 100J (which seems to fit other tests I’ve seen). Plate armors (which we might extrapolate also to lamellar and brigandines) have a range of failure energies based on metal quality and thickness, from 1mm soft iron plates failing at as little as c. 30J to 2mm steel plates withstanding 180J.
We can visualize this in a chart of the energies required to have a chance of defeating armor. Note that to actually kill the person, a hit needs to significantly exceed these energies, perhaps by 10-30J.
Estimates for the firing energy of the most powerful bows vary significantly. Most of the research that I am aware of has been done on English Longbows, but I want to note that the draw weight and draw-length of Steppe composite bows (i.e. Hun, Turk, Mongol bows) are quite similar, giving them similar (as I understand it) range and power characteristics. Many bows were not this strong – these are some of the most powerful bows out there. I’m going to give some of the estimates here in a quote box so that folks who don’t want to stop for them can skip over.
Boring Data Box: E. McEwen, “Experimental Archery” in Antiquity (1988), used an 80lb yew bow to fire a 50g arrow at 53m/s (70J), and a 90g arrow at 43m/s (83J). He also measured a modern 100g crossbow bolt, fired from a crossbow with 90lb at 62m/s (192J). These are the figures Williams (2003) uses.
Mark Stretton in H. Soar et. al, Secrets of the English War Bow (2006) has done testing on Italian yew bows, with a range of energies between 90J and 114J, with a 140lb bow with a 32″ draw (energies depended on arrow type and weight).
Tod’s Workshop tested on of their crossbows, with a 1000lb draw firing a 90g bolt at 47.9m/s (110J).
Magier, Nowak, Tomasz and Zochowski, “Numerical Analysis of English Bows” (2017), available online here, attempted a simulation test, with a maximum energy on the arrow (at 25m) of 130J for a 150lb bow. They used a very heavy (96g) arrow, which launched at a speed of 53m/s (launch energy of 134J). Given the divergence between their results and practical tests suggests something has gone wrong – I suspect the launch energy is too high (cf. Stretton above).
Mike Loades, The Longbow (2013), notes a test fire by Mark Stretton with a launch speed of 170ft/s (51.8m/s), but does not give the arrow-weight.
Very recently, Justin Ma and Blake Cole filmed a test of a 113lb recurve bow, with a 28″ draw, which – depending on arrow weight – had a launch energy of c. 110J.
This is by no means intended as an exhaustive list.
Compared to our chart, those are some formidable numbers indeed! Many of these bows’ launch energy exceeds the ability of mail or even some plate armor by quite a bit. But what does range do to them? A lot, it turns out.
Loades (op. cit.) notes that a test shot from a 150lb bow decelerated from 170ft/s (51.8m/s) to just 137ft/s (41.75m/s) after just 0.8 seconds in flight – though this was on the ascent. Magier et. al.‘s model suggests that, accounting for speed picked up in the descent, a heavy bodkin arrow might lose c. 6% of its speed fired at 100m, and 10% at 200m (though as noted – something seems a bit off about Magier et al.’s simulation).
Loades notes quite correctly that the deceleration for a crossbow is much worse. Crossbow bolts have to be thicker than arrows to withstand the forces they’re subjected to. Crossbows have much shorter ‘draw lengths’ than traditional bows, so the bolt has less time to accelerate, which means you need to put much more force on it (which is why crossbows with massive poundage often fling bolts not so much faster than longbows). But that means, leaving the crossbow, the bolt is thicker, less aerodynamic and moving faster (so more wind resistance), meaning that its velocity drops off even faster. This, if you are curious, is how crossbows manage to be both more powerful and shorter ranged than traditional bows.
Since the kinetic energy (measured in joules) of an object equals (1/2)mv^2, a drop-off in the speed of the arrow actually has a larger effect on kinetic energy. Taking Loades’ example, and assuming a heavy 90g war arrow, the energy at launch was 120J, but after just 0.8 seconds in flight, that would drop to just 78.5J, a one-third drop in kinetic energy for just a 19% decline in speed. Admittedly, this doesn’t account for the arrow accelerating again as it descends on the back of the parabolic arc – following Magier et al.’s experiment suggests an arrow might lose c. 10% of its energy if it impacts at 100m, and c. 20% at 200m. Of course this will depend on all sorts of other variables (wind, arrow shape, weight, etc), but it provides a decent rule of thumb to make the point.
But wait, it gets worse, because:
Angle of Attack
So far we’ve been assuming a dead-on hit from an arrow. But dead-on hits would have been fantastically rare for a variety of reasons. First, rigid armors (plate, brigandine, etc) were shaped to deny flat surfaces to an attack, making dead-on hits improbable even at extreme close range. But range also plays a role in altering the angle of impact even against a flat surface.
As the range of an arrow-shot increases, so does the angle of impact. The reason, of course, is that the arrow has been shot up into the air and is now descending at an angle. Armor provides very few surfaces angled so that a descending object will catch them perpendicularly – if for no other reason than that most humans are taller than they are wide and so most armor surfaces are nearly vertical (the exceptions, like the crowns of helmets, tend to be thickly armored). The angle of descent is essentially a product of how close the shot is to maximum range (generally achieved by a c. 45 degree shot). Because of air resistance, shots always strike with a somewhat more severe angle of impact than they are fired with.
(Note that I am measuring the angles from the perpendicular. So a dead-on perpendicular shot would be 0-degrees, whereas an arrow coming straight down would be 90 degrees. This isn’t the clearest system, but it is how Williams (2003) measures the angles, so it is the easiest for me to use here.)
High angles of impact are very bad for arrow penetration, for two reasons. The first problem is that much of the energy of motion is along the target, rather than into it, meaning the shot might glance off rather than penetrate. Even if the arrow ‘bites’ into the armor, some of the energy will be lost pushing down on the armor rather than pushing through it. But the second problem is that a hit at an angle has to move through more material than a dead-on-strike, making the armor effectively thicker (much like sloped armor on a tank).
The effect of these factors on penetration is severe. Here I am giving only a fairly simplified version of the effects of sloped armor (the question has been examined in far greater detail in reference to high velocity rounds striking tanks). Williams (2003) looks at the problem of glancing. He estimates that the energy to defeat a steel plate at a 30 degree angle is at least 20% higher than a dead-on hit; a 45% angle is 40% higher, due to the arrow not delivering its full energy to the plate. A 60 degree impact requires double the energy to penetrate. This may be less true of armor made from materials the arrow can ‘bite’ into (like mail and textile), but some of the energy of impact will be lost regardless.
But Williams here seems to only have considered the arrow’s side of the equation – the energy lost to the fact that the arrow isn’t moving perpendicularly to the armor. The arrow also has to move through more material – metal or textile – because the armor is effectively ‘sloped’ relative to the projectile. This increases effective thickness. At 30 degrees, the increase is 16%; at 45 degrees it is 42%; at 50 degrees it is 56%. The benefit to increasing thickness is not linear, but actually exponential (roughly to the power of 1.6), so these increases in effective thickness can have huge impacts. The difference between a 1mm plate and a c. 1.4mm plate (effectively a 45 degree angle of impact) is not 40% more but 75% more.
Determining the impact of these factors on mail and textile – complex materials compared to a steel plate – would probably require a lot of real-world testing. On the one hand, the arrow can bite into the ring or weave of the ‘fabric’ of these armors. On the other hand, these materials have vertical ‘give,’ which may well absorb much of the arrow’s energy as the arrow must push the armor ‘down’ before it can apply full force to splitting the rings or breaking fibers to push through. For mail especially, the arrow’s tip needs to find the center of a ring to split it, which becomes far less likely at impact angles further away from the perpendicular.
Quick Sum Up: Penetration at Range
With all of that information in hand, we can – very roughly – think about armor penetration over range. I want to stress that this is all back-of-the-envelope guesstimation; the studies don’t really exist to be precise yet, since there are so many variables (release energy, release velocity, arrow weight, arrow drag, etc) involves. But we can still get a sense of the issue.
Let’s take three bows: the absolute strongest longbows tested: 140lb draw, 120J release energy (e.g. the Stretton tests). Then we can also consider a more typical longbow with c. 80J release energy, consistent with the 120lb pull ‘average’ Mary Rose bow. Finally, a 60J release energy benchmark, so we can see what happens with a bow that is generally inferior to the English longbow (or perhaps an uncommonly weak English longbow archer). In all cases, we’re assume an appropriate bodkin point. I’m going to use the phrase ‘effective impact energy’ to refer to my guesstimate of the energy delivered after both speed loss and angle of impact are taken into account – this is not a scientific term and should not be treated as such.
When you read this, be thinking that a given mass of archers is likely to have bows with power spread out in this range. So once again, we’re dealing with a probability cloud, not a strict binary.
At short range – 20m or so – all of our bows have nearly their full impact energy and the angle of impact is very low (around 2 degrees; negligible). At this range, the most powerful longbow is very likely to defeat everything short of a 1.5mm steel plate with padding. Thinner plate defenses are likely to be damaged by the impact, even if they are not defeated. The more typical longbow has a good chance of defeating mail with a well-aimed shot that strikes square on and might damage plate defenses. The weakest bow will struggle to defeat (but may damage) mail at this range. Our combined close-range volley is very dangerous to anyone who isn’t wearing well-made and thick plate armor.
At medium range – 100m – things are a bit different. Our arrows have lost some energy (we’ll say c. 10%, but because wind resistance is higher the faster you go, the energy loss will be most extreme for the heaviest bow) and their angle of impact is less favorable (c. 10 degrees or so, assuming our bows could throw an arrow around 250m – so figure 10% more defense material to get through due to effective sloping). For our most powerful bow, (effective impact energy is right around 100J at this range) mail penetration is no longer assured, but still quite possible. For our typical longbow (effective impact energy now c. 65J) defeating mail will require some luck, but simple gambesons should still be defeated frequently. For our weakest bow (effective impact energy now c. 50-55J), defeating a good, thick gambeson is a solid ‘maybe,’ – everything else is solidly out of reach. At this range, well-armored foot soldiers – wearing something like a brigandine over mail, for instance – can advance with some confidence. More poorly armored soldiers, however, are still very much at risk.
At long range – 200m – things get much worse for our archers. Our arrows have lost more energy (we’re up to c. 20% total loss) and the angle of impact is now much less favorable (close to 30 degrees). The natural sloping effect of our angle of impact now also means we have to get through 16% more material. For our most powerful bow (effective impact energy now only around 80J), a mailed enemy is not free from fear, but many of our arrows will strike and fail to penetrate; even a thin 1mm steel plate is sufficient for total safety (if you’re wondering – wait, why isn’t it as good as the typical longbow was at close range? – angle of impact, there’s more mail to get through). For our more typical longbow (effective impact energy now around 55J or so) will really only defeat textile armors, and even then, only some of the time. Our weakest bow (impact energy now c. 40J) will likely fail to penetrate even a good gambeson on a solid hit – if the bow can even put an arrow out this far. At this range, enemies in mail can consider themselves quite safe. Plate armored foes are effectively immune to our fire.
Of course other things are happening as the range extends: accuracy is decreasing as well. At 200m, our best bow’s arrows have been in the air for around 4.5 seconds (of course, at this range, we’re shooting at masses of men, not at individuals). I should also note that, at extreme ranges, the arrows are in a sharp descent, which makes them far more likely to hit the pauldrons (shoulder armor) or the crowns of helmets – some of the thickest and most effectively sloped parts of the armor.
(I should also note that I have, in part, understated the problem of impact angle because of the way armor was shaped. We’ve assumed there’s only one angle in play – the angle off of the vertical – but in fact the armor is shaped in a 3-dimensional space, meaning that in addition to the slope introduced by the arrow’s descent and the vertical slope of the armor, it may also be sloped horizontally (think about striking a globular breastplate an inch to the right of the centerline – the arrow will glance off to the right as well as downward). That means our impact angles are, in most cases, effectively minimums – the actual angle of impact in many cases will be less favorable for penetration.)
The key take-away: the lethality drop-off for bows against armored enemies at range is fairly severe. It is, I should add, even more severe for crossbows, which, as noted, decelerate much faster once fired – alas, more testing is required to figure out exactly how fast.
Thinking about a longbow volley, if we imagine a bell-curve of release energies centered around c. 80-90J against an armored advance, we can figure that at 200m only the strongest bows will have good lethality against armored men (even if just mail or a gambeson). The amount of ‘threat’ that the bows can throw out to their maximum range is actually almost negligible against even moderately armored men. Whereas at 100m all of our bows are potentially lethal and some quite likely to defeat armor. Below that, lethality will rise rapidly, both as arrows arrive with the power to defeat more formidable defenses, but also as archers are close enough to their targets to begin aiming at weak points.
Tactical Implications for Foot Archers
Ok, the math is done, I promise. You can come out now.
The key takeaway here is that against even mildly armored enemies, the lethality of arrows drops off over range. This isn’t linear – the drop off is much more extreme close to the maximum range of the bow or crossbow because of increasingly extreme increases in the angle of impact. But the arrows absolutely do not deal their full 53 damage (remembering our Total War example) out at maximum range.
That has major tactical implications. Let’s recap the ‘ideal’ tactics that the Total War model produces: Ranged units begin firing immediately at their absolute maximum range. Moreover, they attempt to keep to this exact range (or just within it). Units that can fire while moving should move backwards, maintaining absolute maximum range at all times. Finally, maximum range is the single most important factor in ranged combat, as the ability to fire at full power while not receiving any return fire is incredibly valuable.
In contrast, actual archers seem to have almost never fired at their maximum range. Mike Loades (cited above) claims that there are no scenes of bowmen firing high arcs outside of siege contexts in the whole of medieval art. I certainly have not seen one. Of course, bowmen fire arrows upwards in siege contexts, but the 45-degree maximum-range arc doesn’t appear in artwork featuring battlefield conditions. Now, at Agincourt (1415), the initial English volleys do seem to have been at very long range, but (following Keegan, The Face of Battle (1976), inter alia) these volleys weren’t intended so much to cause damage as to goad the French into a foolish attack (a psychological impact!). The actual killing the longbows did happened once the French began advancing.
Moreover, we see quite clearly that the existence of the longbow, with its considerable reach, did not drive other missile weapons from the battlefield. Of course the longbow was a socially embedded institution – the princes of Europe could not all just replace their own troops with longbowmen, even had they wanted to (Edward III is said to have quipped, “If you want to train a longbowman, start with his grandfather” – but I do not know the primary source attribution of the quote). But if the longbow had rendered crossbowmen next to useless, one assumes the French would stop using them. They did not, despite the crossbow’s lesser range, rate of fire and only small advantage in penetration ability (we are speaking here specifically of battlefield crossbows – there are siege crossbows with much higher release energies). So clearly the crossbow was not made entirely useless by these factors.
This exercise gives us our answer: maximum range was not only not all important, it was almost unimportant. What mattered was ‘effective range’ – a much fuzzier concept, incorporating not only how far the bow could throw an arrow, but the balance of the average armor of the enemy with the average power of the available bows. While the longbow could out-range a period crossbow, it did so at relatively low lethality, so the crossbowmen (given the protection of some armor, or a shield) might confidently expect to be able to walk the range distance and begin firing. This, of course, happened at Crecy – the French-employed Genoese crossbowmen were able to reach range and begin (and lose) an archery duel with the English (though it was probably quite short – there is some debate on this topic which I will not embark upon here), which the Genoese lost on account of not having been given the time to bring up their pavises (standing shields for cover). In the Total War model, the crossbowmen would never have reached range in the first place.
Compounding this concern is ammunition. In the Total War system, there is little need to save ammunition for the final, high-lethality approach, because shots have the same lethality at all distances. But firing endurance – both of the archer and his ammunition – was a real concern. A longbowman might very well spend all of his ammunition – or the strength of his arms – in just a few minutes. Wasting that window of fire on very ineffective long-range ‘plinking’ would have been extremely foolish.
Instead, the evidence strongly suggests that even English longbowmen held their fire until considerably closer distances. I do not know that anyone has nailed down the exact zone precisely (the evidence seems insufficient to support a whole lot of precision) but it seems to be somewhere around 80-100m (Loades suggests as short as 50m, but this seems excessively short given the primary source descriptions and battlefield topography). This, in turn, neatly fits into the distance where arrow lethality against armor begins increasing rapidly as longbow arrows begin reliably penetrating mail (remember that even for plate-armored knights, some vulnerable areas like the armpits, neck or groin might still only have mail protection).
Tactical Implications for Horse Archers
The mobility of horse archers (by which I mean true Steppe Nomad style horse-archers, not horse-mobile foot archers of the English type) creates an even more complex set of tactical considerations. Moving foot archers around the battlefield can be quite difficult; it is generally agreed, for instance, that the English longbowmen at Agincourt were very vulnerable while they picked up their stakes and moved. The subsequent experiences of English archers being caught out of position in the late phases of the Hundred Years War seems to confirm this (e.g. Formigny, 1450). Such movements are also complicated from a command perspective, if they haven’t been planned.
Horse archers have much more mobility and thus the ability to effectively dictate range. But the evidence suggests they did not ‘plink’ at maximum range either. Timothy May (“The Training of an Inner Asian Nomad Army” JMH 70 (2006) and The Mongol Art of War (2007), 72-4) argues quite persuasively that the Mongols used the ‘caracole‘ to attack with bows. To caracole is to turn a horse about; in battle, this typically means riding up to an enemy at speed and turning around just short of him, while firing some kind of ranged weapon at close range, before galloping away.
What this would look like: an individual Mongol rider would charge in at speed from outside of effective bow range. We’ve already discussed how terrifying an onrushing horse can be and this advance (instead of sitting still at range) gives the onrushing Mongol the full morale advantage of that terror. He can fire as he moves forward into the enemy, putting a storm of arrows in the air which will all seem to arrive almost at once (because of his speed relative to them). But at about 50m of distance, instead of charging home, the rider wheels about and begins riding away at speed, firing backwards (the famous ‘Parthian shot’). In the turn itself, the rider is positioned to fire one, point-blank range, highly accurate and deadly shot.
May argues – on the basis of what he sees as preserved Steppe drills in Mamluk warfare – that the Mongols likely practiced this tactic extensively. The trick – beyond the tremendous skill required to be effective individually as a horse archer – was making it work with a whole large group of horsemen. Doing so requires all of the horses to move in time together and perform the caracole at a predictable distance so that multiple ranks (horses stacked behind each other) and files (horses to the left and right, with a wide distance interval for the turn) could attack at the same time, one rank after another.
The effect for infantry would have been terrifying – instead of a moment of shock at the cavalry charge, the terror of onrushing horses might be maintained for minutes at a time with a continual barrage of arrows (the arrows would also pick up some speed from the forward motion of the horse). And in all of that, the ever-present temptation would exist to either run out after the ‘retreating’ horseman after his turn – with the Mongol riders behind him perfectly positioned to slaughter any man foolish enough to try it – or to run away from danger, to be run down and slaughtered.
For our discussion, what is really crucial here is how this tactic interacts with range and penetration. With steppe composite bows rivaling the power of a longbow, at the apex of his approach, the Steppe warrior is going to be hitting his shots with all of the power of a point-blank longbow shot, defeating mail and even potentially modest plate defenses (Lamellar armor was quite common in China, for instance – a high power close range hit might rupture the connections between plates, e.g. here).
Those shots at the apex are also going to be very accurate. Steppe warriors trained in archery in part by hunting even small game on the Steppe – and an archer who can hit a running rabbit on open ground reliably can also drop his arrow into the unprotected face or neck of a lightly armored infantryman with some frequency. The lethality of this sort of archery would be much higher than the ‘sit and plink’ style of Total War tactics.
But it also posed real risks: the turn of the caracole is executed dangerously close to the enemy, such that a well-timed counter-charge with cavalry might ‘catch’ the caracole as it turned. This might explain how heavier and less maneuverable cavalry sometimes manages to catch and defeat forces of horse archers which should – at least in theory – be more mobile: they don’t need to charge the entire c. 200m distance of bowshot, but merely time a charge over the last 50m or so to hit the steppe nomad as he reaches the apex of his caracole.
This seems to be what happened at Nicaea (1097). Two efforts made by Turkish horse archers under the command of Kilij Arslan to assault the First Crusade in siege around Nicaea were turned back quite roughly with significant losses after getting caught in close-combat with heavier crusader forces. Likewise, at Ain Jalut (1260), the Mamluks were able to pull a Mongol army on to bad ground and ‘catch’ them with the heavier Mamluk cavalry.
Of course, such efforts to ‘catch’ horse archers could be very risky. Publius Licinius Crassus was killed at Carrhae attempting what seems to have been such a maneuver (Plut. Crass. 25) – the Parthians expertly lure him out and then turn and destroy his cavalry and supporting infantry. Interestingly, Plutarch’s description of the Parthian cavalry riding around them “tearing up the ground and bringing up earth from the deep” such that the Romans could neither see nor speak implies the sort of extreme-close-range horse archery consistent with May’s assumptions about Mongols above. The Parthians do this while raining shots into the Roman infantry until it collapses. The Parthians are not showering the Romans with arrows from afar, but riding fast and so close that the Romans are in their dust cloud, even in a dense infantry formation.
It’s also important to remember that in performing this kind of maneuver even steppe horsemen are not the flawless automatons of video games. In video games, players can view the battle from above and instantly give orders which their units again execute nearly instantly. For steppe horseman surprised by a counter-charge or a deluge of unexpected foot-archer fire, an effective and unit-wide reaction would be far more difficult to pull off.
So, to recap: the lethality of archery was highly dependent on distance against even moderately armored enemies. As a result, arrow fire had to be delivered well within the maximum range of bows in order to be reasonably effective. For horse archers in particular, this entailed moving much closer to enemies in order to deliver devastating close-range fire in a caracole maneuver before withdrawing, often repeatedly.
This is, of course, a very different tactical environment than the maximum-range archery exchanges modeled in most computer war games. I’ve picked on the Total War series – which, to be clear, I quite enjoy – but all sorts of titles (Mount & Blade, Chivalry, Age of Empires, etc) indulge in this. Why does it matter? I think there are three main points where having this mistaken mental model of how archery works can lead students of historical battles into error.
The first error is the ‘immortal’ horse archer. In many of these games, it really is nearly impossible to defeat horse archers with anything but other horse archers. Now, I don’t want to overstretch this point: man for man, horse archers were probably the most skilled and fearsome individual warriors on an ancient or medieval battlefield. But they were not immortal. It was possible for light – or even heavy – cavalry to ‘catch’ these horsemen, although doing so required uncommon command skill or at least forced errors by the horse archers. Those steppe horsemen also had to come well within the effective range of other ranged weapons – like crossbows (heavily used in China for this purpose) and even some throwing weapons – to be effective themselves. Han infantry tactics combining pole weapons (the ji) with crossbow troops suddenly make a lot of sense in this context – an infantry formation that can project deadly ranged fire is actually a tough nut to crack for even a Steppe nomad (which is why Steppe nomads often simply avoided large agrarian armies, letting the logistic nightmare of steppe operations do their work for them).
I sometimes wonder if the lionization of the warrior rendered seemingly untouchable by his superior weapon-system in turn influences how we perceive the effectiveness – or lack thereof – of our own highly mobile, seemingly-untouchable missile troops of the sky. If we think that stand-off weapon systems worked this way in the past (but as I hope I’ve shown, they didn’t), we might expect them to work that way in the present. In reality, the morale shock of pounding hoofs did much of the work of the Steppe Horse Archer – a tool that the modern fighter-bomber mostly lacks (cf. sirens on German Stukas).
The second is the unwise assumption of the supremacy of the technological over the psychological. The maximum-range vision of archery encourages students of battle to focus on just that: the maximum range of the bow – making the battle entirely about the weapon and its technological characteristics. But the battles more often turn on what’s in the archer’s head than what is in his hands. To be effective, the English longbowman must keep his wits and his position and continue firing even as the pounding hoofs of the French knight are only 100, then 75, then 50, then only 25 meters away. The striking power of the longbow may be necessary to make this tactic work, but it is not sufficient. The key element is not the bow, but the cohesion that holds the men together.
Likewise, the Steppe composite bow is a fearsome weapon – but what makes it work so well is the psychological strain that the continuous, terrifying assault puts on the (agrarian) enemy. To stand under fire is extremely difficult – to stand in the face of fire and continuing cavalry charge demands extreme discipline and cohesion that fairly few troops could muster. Its important to remember that winning a battle lies in making the enemy run away – in too many games, the power of the steppe horse archer is their ability to kill the enemy, rather than their ability to rout the enemy (and then ride them down).
Finally, this sort of analysis helps to explain why gunpowder weaponry was viewed as desirable, especially against armored infantry – particularly in Europe – even though it was less accurate and slower-firing than bows. This question is a frequent one from students, in light of the low rate of fire and poor accuracy of early arquebuses. By the 15th century, guns were already delivering much higher impact energies (500-1,000J; by the 16th century, this was 1,300-1,700J) than any bow and while a bullet requires (because of its shape) more energy to penetrate, even then, the lethality of firearms quickly eclipsed bows, especially against armored targets. Given a choice between five shots which might wound a target and one shot which would definitely disable him, it’s not hard to see why the preference for the latter developed.
Well, I think it’s time for bows to bow out for now. Next week: agriculture and what the countryside around a pre-modern city should look like. Hint: it won’t be empty.
45 thoughts on “Collections: Archery, Distance and ‘Kiting’”
While I’ve had some criticisms in the past, this is a really excellent post and, besides noting that McEwan used 32″ draw instead of the more likely 28-30″draw, this is a really excellent post. Cheshire has some calculations for angles of deflection for plate armour in his thesis, but you cover this well in any case.
You mentioned Mount and Blade near the end, and I wonder if you’ve heard of the medieval mod 1257ad? It still is not realistic in many ways, but armor is made much more effective. A plate armored enemy is more-or-less immune to arrows at anything but extreme close range, and generally immune to swords as well. The biggest thing it lacks is a better morale system.
I’m amused to think that Heroes of Might and Magic is, in some ways, more accurate than Total War. The former doesn’t even pretend to be at all accurate, and yet projectile weapons, including bows, suffer from range, and ideally you want to move your archers forward on the field so they are at half range and thus get full damage. You can also toggle friendly fire on, so that they need a clear line of sight to not accidentally damage your own infantry and cavalry. And there are morale bonuses and penalties, too!
I looked at nomad archery a while back. Two points struck me then:
– it was not the same throughout. Mongol bows, for instance, are longer (with stiff ears at the end of each limb) than earlier Turkic ones. A Parthian shot is not possible with a Mongol bow (the rider leans out to use it). Turkic bows seems to have been more powerful than Parthian or Scythian bows.
– a second point is that steppe archery was optimized for distance shooting, with long draw lengths and light arrows (and the development of the bows emphasized distance more and more). This makes sense since the usual enemy was other nomads. Not only was armour light but the horses were key targets. I suggest the preferred tactic was to disorder an enemy group at long range then make a sudden charge (probably with the heavier horses of the leader and his friends) at the most vulnerable group. This would put a premium on discipline, flexible command and coordination – all areas the Mongols put a heavy emphasis on.
Granted, my knowledge of this comes from secondary and tertiary sources like the videos you sometimes link to, but I thought the historical counter to horse archers was foot archers. Standing on the ground means your bow will be more accurate at a fixed range, which gives foot archers higher effective range than horse archers. Horses also present big, unarmored targets, which increases the effective range of foot archers even more.
In an RTS this can be simulated pretty well even with normal RTS conventions of HP loss and binary hit-or-no-hit range, by giving foot archers more range than horse archers. I don’t know why the Total War series made a game design decision to give horse archers higher range, but game designers who choose the more realistic route of giving foot archers higher range can simulate the way horse archers could wear down immobile melee infantry but lose to higher-range foot archers.
In my experience in Total War games foot archers generally have at least as much range as horse archers, and significantly higher damage output relative to cost, so they are in fact a very good counter. They just need melee infantry to protect them since the horse archers are usually stronger in close combat.
I don’t think this will change the overall conclusion, but I have a bit of a bone to pick with your math. In particular, I think you might be double-counting the effects of impact angle by considering both the arrow’s perspective and the armor’s.
That is, you can say that an arrow hitting 1 mm armor at 45° hits 1 mm with cos(45°) = ~70% of its velocity, or that it hits ~1.4 mm of effective armor with all of its velocity, but you really can’t say it hits 1.4 mm with 70% of its velocity.
(The above of course assumes armor is isotropic)
Two separate effects; the 70% is from deflection due to the angle, whole the increased thickness is due to the angle presenting a thicker cross section.
The arrow loses significant energy right at the point of impact, and then has more material to penetrate.
Overall good discussion though I have to say the repeated emphasis on the ‘bodkin’ type arrows as an armour penetrator is a bit misleading. Perhaps against mail or clothe armour but for plate or lamellar it was most likely a much heavier broadhead type.
There are some reports on the training of English archers that mention 200 yards as the required range for an archer to hit within a less than 1 yard clout target (assuming something above a 30 degree launch angle) but almost all sources report that most archery competitions for English archers were held at butts, shot at wands, or at birds in flight or bird sized objects tied to branches, windmill arms, ship masts, etc at distances from 10 yards to 300 yards with most butt and wand competitions apparently between 50 yards and 100 yards. Militarily in that target ‘window’ an archer can score lethal hits but still leave themselves time to prepare to receive a charge.
A test and some added commentary. From after the above was posted.
Tod’s Workshop — https://www.youtube.com/watch?v=DBxdTkddHaE
Matt Easton’s comments — https://www.youtube.com/watch?v=SWBUFvQvOBQ
Hope you still check these many months later, I don’t use Twitter etc.
Yes! I follow both of them, and have found these additional tests fascinating. I hope they do more!
That’s not exponential! The numbers on the figure you include makes it pretty clear that yes, you do mean it grows as the 1.6 power of the input. That’s a power law, not an exponential; an exponential is when the input is in the exponent, not the base of the exponent. Exponential functions grow much faster than power laws.
Indeed this mistake in terminology is in your previous article on this as well. When I got to that, where you said it’s exponential, I was like, wow, exponential, really? That’s amazing! I could believe quadratic or something but exponential… well, I guess he knows what he’s talking about. Then I get to this one and it becomes pretty clear that the reason what you said sounded so unbelievable is because it isn’t actually what you meant!
Would you mind fixing this error? Thank you!
It’s not either of them! It’s a trigonometric function–specifically, the effective thickness is the perpendicular thickness multiplied by the secant of the impact angle (or, equivalently, divided by the cosine of the impact angle).
That’s a different variable though; we’re not talking about effective thickness as a function of angle, but, um… something… (the text actually isn’t clear on what — is it energy to defeat?) as a function of thickness. The two are discussed near one another, but when he talks about “exponential” and “to the power of 1.6”, you can see that he’s talking about the latter.
That said, this does raise the general point that if one just means “superlinear”, one should perhaps just say “superlinear”. 😛 In this case though he does seem to be specifically claiming it’s (on the order of) a power function.
The first of today’s belated proofreading corrections for your consideration:
(the one notably exception -> (the one notable exception
it is fired a point-blank range -> it is fired at point-blank range
tested on of their crossbows -> tested one of their crossbows
Loades (op. cit.) nots that -> Loades (op. cit.) notes that
Magier et. al.‘s model -> Magier et al.’s model (direction of curl in the apostrophe—what? you thought *you* could be nitpicky!)
bodkin arrow arrow might lose -> bodkin arrow might lose (delete one instance of the word arrow
somewhat more more severe angle -> somewhat more severe angle (delete one instance of the word more)
short), given the primary -> topography). This, (that is, move closing parenthesis/?)
In addition, you end the resources list with this comment: “what is and is not italicized in these quote-boxes.” But ALL the material within this quotebox is italicized, isn’t it? It looks as if you’ve chosen to underline the book titles, rather than showing them in regular type/?
The Last Days of the Third Age mod to Mount & Blade: Warband has both a morale system and much better and more realistic horse archer AI.
A big part of battles is getting the opposing side’s morale to waver (by getting the battle off to a good start) and preventing one’s own side from wavering (by being effective at the beginning of the battle, slaying enemies oneself, and using a “rally” battlecry command that is limited by one’s leadership points). When a side’s morale starts to waver, troops will start to rout from the battle, attempting to run to the edge of the map and disappear from the field of battle, and meanwhile not really defending themselves. If this goes unchecked, you’ll see the message “enemies are fleeing in terror.” This presents a great opportunity to run the fleeing enemies down with cavalry while facing b.
Diplomacy 4.Litdum mod actually handles the AI of this even better than The Last Days mod by having these routing enemies at least put up a token defense if you, for example, slow to a trot alongside them while they are routing, but otherwise not really reacting with much awareness when you swoop in to spear them from behind as they run away.
And then as far as the horse archer AI goes, in The Last Days mod the Rhun, Rohan, Harad, Gundabad, and Rivendell factions all have access to horse (or warg) archers, and when ordered to “charge!”, instead of running head-first into a dense mass of polearms, as the native Mount & Blade AI will usually do…and instead of running around firing ineffective shots at an extreme range (as other games do), the horse archers in The Last Days will swerve to the right of the enemy mass (since all the game’s horse archers are right-handed and fire across their left shoulder) and trot about 10 meters alongside the enemy lines, peppering them with deadly and accurate point-blank fire, gradually moving in a counter-clockwise fashion around the enemy mass (so, not EXACTLY the caracole, but similar in effect). Then, if the enemy force tries to charge to chase them down (either with infantry or cavalry), the horse archers will turn and scoot away while continuing to pepper parthian shots back at the pursuers. Then it generally devolves into a mess of “yackety sax” where the infantry ineffectually chase the horse archers around in a disordered mess and get slaughtered. It is both maddening and genius.
Having your own horse archers, archers, skirmishers/peltasts, and/or fast light cavalry that can suddenly charge and chase them down, are pretty much the only way to deal with them, aside from maintaining a dense mass of heavily-shielded infantry and just waiting for the enemy horse archers to expend their ~30 arrows while suffering a trickle (or flood) of casualties on your own side. But what’s worse is, the horse archer AI is programmed to then often retreat after they have expended their arrows. So they will run to the edge of the map, and if a large enough group of them get away, you will have to fight them again in a 2nd round of the battle…and maybe a 3rd…and so on. Many forum commentators compare the top-tier Rohan Thengel Guard horse archers to helicopter gunships because of how effective they can be. The best counter to them are Dunnish Wolf Guards who sometimes spawn with wolf companions that run extremely fast, are hard to hit, and can dismount enemy riders.
I’m very surprised Total War doesn’t have a damage falloff mechanic, because that’s a really common thing in other genres – not just for realism reasons, but because it makes the gameplay better – players want close-range engagements to resolve faster than long-range ones, and because getting kited generally isn’t fun for the kite-ee. Maybe they mistakenly thought that falloff was unrealistic?
This is an error in the post. The total war games *DO* take into account range. The formulae (approximations) can be found here https://www.twcenter.net/forums/showthread.php?111344-The-Complete-EDU-Guide#Formulas .
Damage does fall of with range. This is not very noticeable, because archers do way too much damage in unmodded versions of the Total War games.
I play the Europa Barbarorum mod extensively. https://www.europabarbarorum.com/ . This is a mod dedicated to reaching as much historical realism as possible and focuses on the time between 272 BC and 14 AD. It is right up to mr Devereaux alley I would say. I recommend it to everyone who reads the blog and is into the Total War games series.
In the Europa Barbarorum mod, attack values are much lower compared to armor than in the vanilla versions. As a result combat takes longer (though still not too long to be enjoyable as a video game). This also means the effect of range and armour on the effectiveness of your archers is *much* more pronounced. Archery is generally only effective against unarmored or lightly armored opponents. It is much more effective as a morale diminisher than actually killing units.
Since shield values (see guide above) mean that attacking units from the front is basically useless for ranged units, the most useful way to use skrimisher is to fire into the flanks or back of units from close range. Skirmish units in general can be used very effectively this way. Horse archers or horse mounted skirmishers can be even more effective as it is much easier for them to flank.
I have played the Europa Barbarorum mod for quite some time and can confirm it seems much more realistic than vanilla… Archers at long range seem to only disrupt formations slightly and lower morale a bit, but they only manage to cause signing damage on unarmoured/unshielded troops or by flanking them
Likewise one of my biggest surprises was sending one of my elite companion cavalry units to do a frontal charge on a heavy infantry squad that stayed in formation, only to scratch my head after finding out heavy cavalry is not a battering ram and losing half of them on impact with only minor infantry casualties…
WAAAY late to the party, but just to add another voice of support – with due respect to Dr Devereaux’ knowledge and handling to the real life situation (which is considerable and amazing, I truly enjoy this blog and have been binge reading it for last two weeks!), but he sorely underestimates and undersells TW here. Total War games are not perfect and by no means a simulation, but they are much more nuanced than he gives them credit for. Especially the older ones (first Shogun, both Medievals, first Rome), and ESPECIALLY when modded – which, in all honesty, is just something you do if you’re interested in historically semi-accurate games. Vanilla TW is designed for broad(er) appeal and is just slightly more realistic than Age of Empires (…or chess, for that matter) – if you want something that even begins to approach a simulation, you want the mods.
In any mod for an older TW game (like the aforementioned EB, or Rome Total Realism, or Stainless Steel or even fantasy mods such as Third Age), firing arrows at max distance against armored infantry is an exercise in futility. The only reason you’d do this is, guess what, to goad them! Which actually doesn’t even always work – the AI can (sometimes) recognize that your archers are not that much of a thread and will just stay in place, accepting the odd “lucky shot”. If you have your archers shoot at max range, you want to use them for UNARMORED targets. Which I suppose is not quite unrealistic – the blog post doesn’t address it, but I’d expect that the penetration power against skin and flesh would be sufficient even at higher ranges and angles.
Also, the TW games DO feature exhaustion for archers. In the mod I’ve been playing lately, the one type of unit which *always* ends up exhausted are (heavy) bowmen. In fact, the reason why you routinely start shooting against approaching infantry as long as they are in range is NOT because your archers don’t get exhausted, or because their effectiveness at max range is the same – it’s because of a different failing of TW (which *is* IMO a major issue that even the mods don’t properly fix), and that’s the speed of infantry: the buggers are just too damn fast. If you wait to shoot at optimum range (say, half way, which in game would be 100m out of theoretical 200m, or 80 out of 160), your archers will loose one volley, MAYBE two if you’re lucky, before they have to skirmish away. So you don’t order your archers to shot at will at max range because you’re expecting major results, you’re doing that because you have a much smaller shooting window than in real life, and incidental damage is still better than no damage. On the other hand, if you have another use for archers after the melee lines have engaged (say, enemy skirmishers still run around, or you have an opportunity to wheel around the flank and shoot the enemy’s melee infantry in the back!), you’d absolutely do that and NOT waste the archers ammo and stamina on approach.
How much of a difference would being shot downward from a wall or a tower make in the velocity (and thus the impact force) of an arrow?
You can actually calculate this pretty easily with simple physics. But most walls aren’t really tall enough to add more than a small fraction of a second of descending travel to the arrow’s flight, so the effect is probably pretty negligible when we’re talking about real-world walls. Shooting from the top of a tall natural cliff to enemies gathering at its bottom might have more effect, but then why shoot in this situation when dropping objects from the top of the cliff would probably have a more reliable effect? It’s also worth noting that shooting downwards at anything more than a shallow angle with a bow is inherently more difficult than shooting upwards — it’d be harder to use the rear leg to help brace against the “recoil” of the bow, for instance, so archers shooting steeply downwards would either have to use shorter draws (thus reducing the power of the shot right from the start) or specialised techniques like the Middle Eastern “jarmaki” that both need specialised practice and are inherently more exhausting than shooting normally. Or both.
I saw this short movie and just had to share it here: https://i.imgur.com/oFRShKO.gifv
It shows very clearly how armour deflects arrows.
” winning a battle lies in making the enemy run away – in too many games, the power of the steppe horse archer is their ability to kill the enemy, rather than their ability to rout the enemy”
Reminds me of Aral Vorkosigan’s quote from The Vor Game (mil SF by Lois McMaster Bujold): A weapon is device for making your enemy change his mind.
Here’s a related question: how would a densely-packed primarily heavy infantry force defeat a force of skirmishers? For example Hoplites VS javelinists or slingers or something or that nature.
Do they suffer the same problem as heavier cavalry trying to defeat horse archers, and have to rely on how the enemy will end up closing in fairly close to launch their attack, and hopefully not be prepared or agile enough to run away? Does the process of lunging out to catch skirmishers tend to routinely disrupt more organized heavy infantry formations make them constantly have to reform?
Late reply here, but generally you wouldn’t see a single unit of heavy infantry and a single unit of skirmishers have a 1v1; rather you’d see two forces build around a core of heavy infantry with skirmishers scouting ahead and/or guarding the flanks. Combined arms tactics are not new.
The comparison to horse archers and heavy cavalry is superficially persuasive, but it sorta ignores how much easier it is to be a skirmisher than a horse archer, and thus how potentially plentiful skirmishers were in comparison.
Effectively, the unit of heavy infantry should have little trouble maintaining an organic force of skirmishers and quickly regenerating it when necessary. These could be squires and pages in the middle ages, slaves in antiquity, or even just the poorer men in the unit who can’t afford as much armor. It’s not a super difficult job (though, the benefits of skill and practice are noticeable) and I’ve seen brand new reenactors take to it very quickly. It’s a lot easier than learning to keep formation in heavy armour, that’s for sure.
As someone who once worked as a professional (staff designer) in the field of game design, back in the olden days, this all appalls me, the notion that people think they know about military tactics from VIDEO/COMPUTER GAMES! (Which I have not played since the early 1980s.) That drives me up the proverbial wall.
When I was working on game design, or still do as an amateur hobbyist, when I am doing research about weapons systems or tactics or organization, I go to primary and secondary sources, and it would never occur to me to rely on a game for my information. That just seems like relying on movies or TV to learn about ancient history. Yes, you might pick up a few kernels of knowledge, but most of what you see is artifice. If people prefer fantasy over reality, then there’s little hope to reach them.
Isn’t that caracole shot diagram above reversed? Assuming your horse archers are firing right-handed, they will need to wheel their horses about to the right (not left) so they can turn about in the saddle and shoot their bows from the left side of their mounts, with the bow in the left arm, drawing with the right arm. That’s the only effective way to fire to the rear from horseback.
This makes me think about shields’ efficacy at range. That would also be increased. If, for simplicity, we imagine 6-foot infantry with 3-foot-tall shield, as wide as the infantryman, a dead-on shot (0 degrees from horizontal) that was on target (on a path to hit some (random) part of the infantryman would be 50% likely to hit the shield.
But a shot coming down at 45 degrees is going to be “seeing” a target profile roughly of 4-4.5 feet (I am not good at geometry). So an angled shield could go from 50% protection to 75% protection, another hitch in the “fire at extreme range” game plan.
Holding a big shield up over your head, or even just out in front of you, is tiring, though! I’ve heard it suggested that one of the under-rated factors in the Battle of Hastings was that Harold Godwinson’s men, who were already exhausted at the beginning, were being worked over by William’s archers all day and having to constantly keep their shields up to the point where their endurance started to fail.
Unfortunately, what you review is a randomly picked set of combat models, and these properties are result not even of mechanics, but parameters of mechanics. Obviously, with great enough range penalty “hard limit” matters much less.
For a different and old example, take a look at Master of Magic. https://masterofmagic.fandom.com/wiki/Ranged_Attack
No range caps at all. But since to-hit penalties matter, you still don’t want to shoot across the whole tactical map, except with magical attacks and Long Range weapons.
However, inflicted damage is not separate from to-hit, as to-hit chance is checked per Attack die, thus scored hits are reduced proportionally; then Defense rolls negates some.
Crude, but it works.
It’s yet another trap of simple stats. There are two differences that make them less than perfectly comparable with just two parameters.
1. Materials. A laminated board of different materials bends not in the same way a plain wooden stave does. For one, wood has significant “viscosity”, i.e. a branch can un-bend only so fast, even without load. Likewise, plastic bows often give greater velocity than wooden D-bows to the same arrows at the same draw; IIRC, this led to some confusion among the enthusiasts, as the measurements on plastic can deviate far from the “book” ballistics of historically accurate low-draw bows.
2. Shapes. Re-curving is not an aesthetic choice, it’s a game-changer.
While it’s a crude model, physical characteristics of linear tension energy storage (spring) pulled to distance x from its relaxed state are: force of tension F=k*x, energy stored E = k*x²/2.
For a staring bend x₀ and draw length Δ full draw force is F₁=k*(x₀+Δ); energy storage capacity is difference between tensile potential energies of fully drawn E₁ = k*(x₀+Δ)²/2 and strung-but-not-drawn E₀ = k*x₀²/2 states, thus E₁₀ = k*(x₀+Δ)²/2 – k*x₀²/2 = k*(x₀²+2x₀Δ+Δ²-x₀²)/2 = k*(2x₀Δ+Δ²)/2.
The problem: you are comparing F₁ and Δ, but in this context one of your implicit constants becomes a variable: for recurved bows starting x₀ (how far it’s bent from the zero-tension position) is obviously much greater. Or, you are looking at a farther part of a much less steep linear graph intersecting your old graph in one point, but the working cycle is defined by difference between two points. At the same Δ and k*(x₀+Δ), k and E₁₀ will be different.
To skip substitutions, E₁₀ = F₁²*Δ – F₁²*Δ²/2(x₀+Δ). If we compare different values of x₀ at the same F₁ and Δ, can reduce this mess to derived constants and one variable (all positive): E₁₀ = A-A*B/(x₀+B). In this form we can see the energy stored in the draw increases (asymptotically toward A) when x₀ increases.
And for that matter, F₀ = F₁*(1-Δ/(x₀+Δ))
In other words, if a plain and recurved bows have the same full-draw force and the same draw length, the recurved one:
• is noticeably more flexible (in the working direction) when unstrung,
• requires more force even at the start of a draw and
• stores more energy by drawing (takes more work to draw, can transfer more into arrow).
“a tool that the modern fighter-bomber mostly lacks”
I think this was a major argument in favor of retaining the fleet of A-10s: they excelled at CAS in the psychological dimension as well as the kinetic.
This is likely true of any decent combat aircraft, though. Once your unit has been bombed or strafed a few times and taken casualties, just hearing an aircraft engine off in the distance is going to trigger your fight or flight response.
Same with modern artillery, actually; once you’re in range, you could be killed or maimed instantly at any time, with virtually no warning, and there would be almost nothing you could do to fight back. Jean Norton Cru talks about this extensively in his WWI research and it matches accounts of soldiers in WWII who became completely demoralized by air attacks that inflicted seemingly unimpressive loss rates.