'Some Terrifying Numbers"

[Anton]

I think that's true and different types of social organisation produce different types of fighting men. 

As fighting was, mainly, very close up and personal in our period I'd wonder about the morale of part time fighters when confronted by professionals who were better equipped and more skilled in weapon handling.  If they were used to combat and had the numbers they might stand until help arrived.  Otherwise I'd imagine they would be strongly tempted to get out the way and leave it to their social superiors.

[Andreas Johansson]

I'm a little confused by Reuter's argument against armies in the thousands on the grounds they'd leave a wake of destruction. In later medieval and early modern times armies in the thousands or tens of thousands were common enough, and their popularity wasn't appreciably diminished by the fact they did leave wakes of destruction.

(He might object that Europe was richer and more populous in the 15th or 17th century than in the eight or ninth, and therefore better able to afford rampaging armies; but the difference was hardly one of orders of magnitude.)

[Erpingham]

I thought it had a number of weaknesses.  For example, just because small forces operated in certain places and times doesn't mean different levels of warfare didn't exist.  And the "small Great Army" arguments alluded to have been rather weakened by recent archaeology of burials and camps.

[RichT]

It's interesting that even for Merovingians and Carolingians, who were pretty well organised, we don't have accurate figures even for their own side.

It's hard for people to really get a good feel for numbers, especially large ones. Inspired by the title of this thread, I've been having a look at the cheerful subject of mortality rates. In England and Wales in 2018 (latest full data), 513,936 people died (that's 1,408 every day). Of these, 15,415 died in accidents - mainly not road accidents (just 1,547), the big ones are 'Accidental falls' (6,103) and 'Accidental poisoning' (3,814). An accident that kills 42 people is big news, but every single day of 2018, 42 people were killed in accidents of one sort or another in England and Wales. There were 638 homicides (just under 2 a day - compare with US figures!). Influenza and pneumonia accounted for 29,516 - about 81 per day, every day, for a year. The big ones of course were cancer (146,357), dementia/Alzheimers (69,478) and heart disease (55,995). Of the big killers of the past, tuberculosis killed 225, 'Vaccine-preventable diseases' killed 160 and 'Vector-borne [ie mosquitos etc] diseases and rabies' killed 8.

The relevance of which to army numbers is... I'm not sure, except that any number that isn't derived from detailed accurate records from their own side (as must have been maintained for logistical purposes, in 'civilised' armies at least) is always going to be suspect. So I wonder which large army numbers we have seem to come from those sorts of sources, and which from guesses, traditions etc. What's the largest Ancient/Medieval army whose size seems likely to come from detailed records?

[Erpingham]

Perception of numbers is an interesting thing.  There's quite a bit on risk perception (stemming often from the insurance industry) which shows that humans are pretty poor at assessing risk based on things like accident statistics.  Your fall statistics are often from home accidents.  But road traffic accidents would probably be seen as much more dangerous.  Anyway, the only relevance there is that perception of numbers, rather than actual numbers, can be important.
To help us on our way, here is an example of the scaling details medieval records can provide .  This is for the Florentine army at Pistoia in 1302.


Among those paid by Buoninsegna, we find 6,292 effective combatants. These can be subdivided into horsemen and footmen (crossbowman, pavesarii , infantryman).

Belonging to the first category, we find 507 cavalrymen. Belonging to the second category, we find 5,785 footmen. There were within the army a series of auxiliary companies, who could be employed for combat, but generally were relegated to a specific strategic function (i.e. the systematic devastation in the countryside through the employment of sappers, in Latin
guastatores). Belonging to this category, we find 977 auxiliary or support troops. Within the host, Buoninsegna also records the employment by the Commune of 221 men assigned exclusively to non-combat related duties. These included carpenters, tailors, administrative officials, messengers and logistics personnel.


I've taken this from this thesis by Drew Calisle.  The full tables are on page 16.  I knew this list existed but have only just found it in this thesis, which may be interesting in itself.

We have various records from the Middle Ages we can draw on - English from mid 13th to mid 15th century, Flemish towns militia lists, Italian records, Swiss musters, Burgundian musters and so on. 

Looking at Curry et al The Soldier in later Medieval England, the largest number put into the field in the 15th century part of the HYW was about 4,000 men-at-arms and 13,000 archers, divided into several forces, in 1436.  The highest numbers in one place never seem to have much exceeded 11,000.

Vaughan shows from records that the Burgundian army of John the Fearless never managed more than 10,000 men and was usually in the 5-7000 range.  He notes that chroniclers accounts of numbers in armies for which we have records are often greatly exaggerated.

[Nick Harbud]

Is this a good point to introduce Lanchester's Laws into the discussion? ^*

These series of differential equations were developed approximately 100 years ago to assess the power relationship between opposing armies.  That is, how much better one side's troops were compared to the other or, looked at another way, how many additional men does the weaker side need to match the stronger side.

Lanchester developed two separate laws for the ancient warfare (Linear Law) where battles are principally decided by hand-to-hand combat and the modern period dominated by long range weapons, such as firearms (Square Law).

One of my Ideas For An Article involves an examination of sundry wargames rules using Lanchester's Linear Law.  Watch out for it in future Slingshots. 



* Nothing to do with the Secretary of the Lance & Longbow Society

[RichT]

I shall look forward to that. I've always struggled to see the point or value of Lanchester's Laws. To quote WP:

"For ancient combat, between phalanxes of soldiers with spears, say, one soldier could only ever fight exactly one other soldier at a time. If each soldier kills, and is killed by, exactly one other, then the number of soldiers remaining at the end of the battle is simply the difference between the larger army and the smaller, assuming identical weapons."

It seems to me that every part of those two sentences is wrong in general and in detail, and I can't imagine what useful results could ever be extracted from such a meaningless 'law'. So I'd be happy to be enlightened...

[Andreas Johansson]

A discretized version of the Square Law is useful for describing many wargames of the chuck dice to score hits variety. A couple of years ago I ran across a guy who'd independently found it as an empirical fact about Axis & Allies.

Some people over at BGG who are more into military modelling than I am tell me that real-world combat, or at least real-world WWII land combat, approximately follows a "Lanchester's logarithmic law" where fighting strength is proportional to the logarithm of numerical strength. Put less mathily, an army twice as numerous is rather less than twice as strong, given that individual prowess is constant.

[RichT]

Some people over at BGG who are more into military modelling than I am tell me that real-world combat, or at least real-world WWII land combat, approximately follows a "Lanchester's logarithmic law" where fighting strength is proportional to the logarithm of numerical strength. Put less mathily, an army twice as numerous is rather less than twice as strong

Forgive my unmathiness, but isn't "an army twice as numerous is rather less than twice as strong" almost the exact oppposite of what Lanchester says, which is (as I understand it) that an army twice as numerous is much more than twice as strong?

given that individual prowess is constant

But that isn't given. That's the trouble with such 'laws' - if you could factor out individual prowess, and differences of equipment, and terrain, and weather, and tactical factors, and human factors, and a whole array of other circumstances, then no doubt what was left might be a simple arithmetic relationship between strength in numbers. But you can't, or if you do you have factored out everything that is important and interesting in the study of combat. What could such a model tell us - that the larger force wins, all else being equal? Well, OK. That seems on one level so obvious as to not be worth saying, and on another level so meaningless as to not be worth saying (because all else isn't equal).

I'd be interested in practical applications or predictions that can be made using Lanchester (or equivalent). I know militaries use similar things in OR and such, but I hope their models are more sophisticated than Lanchester.

[Nick Harbud]

The WP wording is not the best in the world and could probably be improved.  It seems to focus more on highlighting the differences between linear and square model relationships rather than explaining the rationale behind either.

The square law is easiest to understand when considering a group of, say 20 soldiers with muskets shooting at another body containing 40 soldiers.  Assuming the weapons and marksmanship to be the same and that all soldiers can find a target amongst the enemy, if the first unit receives 40 shots (2 shots/soldier in the unit), then the second only receives 20 shots (0.5 shots/soldier).  In other words, half the strength leads to the unit being four times worse off.  Conversely, in order for the lower strength unit to equal the stronger unit, it needs to improve its firing efficiency (ie, shots per firer within a given time) by a factor of 4 rather than 2.

The linear model assumes that, more or less, only the front ranks of each body are able to engage or will be killed.  Thus, if one's hoplite in a cuirass is twice as good as a peltast in little more than a t-shirt, a peltast unit will need to be twice a big (ie, deep) as a hoplite unit to achieve a draw.

Note that the above examples bandy about quite large relative effectiveness factors.  In real life, such differences tend to be smaller, in the region of 1.1-1.2.

Of course, the points cost of different troops types is meant to reflect in some way the relative effectiveness with respect to other types.  Therefore, without wishing to give too much of a spoiler on what might appear in Slingshot, one could always compare troop type effectiveness (as determined from the relevant combat mechanism) to the corresponding points costs to see what transpires.

Interesting point on possible logarithmic relationships.  What is the rationale behind it?

[Andreas Johansson]

Forgive my unmathiness, but isn't "an army twice as numerous is rather less than twice as strong" almost the exact oppposite of what Lanchester says, which is (as I understand it) that an army twice as numerous is much more than twice as strong?

The linear law says that a force twice as big is twice as strong.
The square law says that a force twice as big is four times as strong.
The logarithmic law, which wasn't formulated by Lanchester but derives from similar considerations, says that a force twice as big is less than twice as strong.

It doesn't really make sense to say that "Lanchester says" - you have to specify which law is thought to be (more-or-less approximately) applicable.

given that individual prowess is constant

But that isn't given.

It's a pedagogically useful assumption to make when explaining the concept.

I mean, if Bob and Alice have individual fighting powers of eight, it's interesting to learn (assuming it to be true) that their combined fighting power is less than sixteen. It's not appreciably more interesting to learn that if their individual fighting powers are eight and seven their combined is less than fifteen.

I'd be interested in practical applications or predictions that can be made using Lanchester (or equivalent). I know militaries use similar things in OR and such, but I hope their models are more sophisticated than Lanchester.

The only practical application I know are things like the salvo models mentioned in the WP article, where things like morale don't really figure. ETA: Practical application to warfare that is. Wargames is another thing.

[Erpingham]

One thing I can say, based on googling the subject since it came up, is there is a huge debate about validating Lanchester's Laws through the use of historical data (primarily 20th century).  Numerous studies seem to have been done, often contradictory in conclusion.  Very little application seems to have been made to pre-modern warfare.  It will be interesting, therefore, what Nick comes up with.

[Andreas Johansson]

Interesting point on possible logarithmic relationships.  What is the rationale behind it?

Basically that the outnumbered force is in a target rich environment. Increasing your numbers makes it harder for your troops to find a meaningful target and easier for the enemy.

Unlike the two original laws, I've never seen a mathematical derivation of it, and a brief google is not helpful on the subject.

[RichT]

Andreas:

It doesn't really make sense to say that "Lanchester says"

  It makes sense in this sense:

Andreas:

The logarithmic law, which wasn't formulated by Lanchester

IIUC, Lanchester proposes the linear law (pre-gunpowder) and the square law (gunpowder). Some people on BGG propose a logarithmic law for WW2. Therefore the people on BGG are proposing something that contradicts what Lanchester says (in his square law, which is the one he would apply to WW2).

Is that fair?

It's a pedagogically useful assumption to make when explaining the concept.

Yes and I'm not criticising your pedagogy, I'm questioning the value or relevance (to real life) of the concept.

[Andreas Johansson]

IIUC, Lanchester proposes the linear law (pre-gunpowder) and the square law (gunpowder). Some people on BGG propose a logarithmic law for WW2. Therefore the people on BGG are proposing something that contradicts what Lanchester says (in his square law, which is the one he would apply to WW2).

Is that fair?

Lanchester (writing in 1916) would presumably have expected WWII warfare to approximate the square law, yes. But others have applied his laws differently (e.g. the linear law has been used to model indirect artillery fire in WWII), and I'm used to "Lanchester says" meaning "whichever of Lanchester's laws I think is applicable says" rather than "Lanchester the man said".

It's a pedagogically useful assumption to make when explaining the concept.

Yes and I'm not criticising your pedagogy, I'm questioning the value or relevance (to real life) of the concept.

To pre-gunpowder land battles of the kind we typically discuss here, I think the relevance is close to nil. At the most basic level, such battles were not primarily attritional.

I might note that Lanchester's original work is online at Wikisource.