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Maurikios, book 9 chapter 5, mentions clues including:
1. the frontage and depth of the force, but the figures seem too dense for cavalry, and the total seems too high.
2. the presence of the enemy baggage train.
3. the area trampled by their horses.
4. the size of the camp.
Other possibilities might include:
5. the road space required for the enemy baggage train.
6. the resources available.
I can't recall if any other manuals add other methods, or describe how these work in practice. For example, a method for converting camp size to army size that works for professional armies would require modification for migrating groups.
[Sorry - this belongs in the Roman Army section]
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Maybe it depends on the army you're observing. Romans in organised ranks would seem to be easy to calculate, a swarm of barbarians ( estimate how much they smell if you're downwind? :-)) might be harder. Parhaps the scouts developed some sort of rule of thumb of density and size to make guestimations that might vary with the experience and skill of the scout? I would think it would be a risky business to stay around long enough to get an accurate count and a scout would go for telling the commander the enemy is about a legion and a half with maybe a cohort of cavalry than 7923 foot and 806 cavalry. Not so precise but how precise do you need when a balpark guess on numbers and an idea of composition of auxiliaries, type and approximate numbers would suffice to give the commander a picture with what he has to deal.
Caesar audieritis hoc
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Again rather inexact, but bearing in mind the number of times that commanders throughout history have tried to deceive their enemies by multiplying the number of their campfires, I would assume that fire-counting might have been a common way of assessing the size of a force (it would also enable the scouts to approach in the dark, of course!). Quite how this would have worked in practice I don't know - perhaps if the scout could count the number gathered around a single fire, he could multiply accordingly?
I've also heard of scouts being able to assess the numbers within a small area of ground and multiply the area to cover a larger horde. I can't recall whether this was a genuine practice, or just a theory! As Tom says, it would surely rely to a great deal on the skill and perception of the individual scout...
Nathan Ross
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Even if it's not so important elsewhere, it's important at Hadrianopolis.
The Roman scouts estimated a force of ten thousand Goths, or more likely ten thousand Gothic soldiers and militia along with many more Gothic refugees, camp followers. and so on. Ammianus says they screwed up, but doesn't say how.
The most common interpretation going back, I think, to Delbrück, is that the Roman scouts correctly estimated the Gothic infantry but missed the Gothic cavalry who were away, either foraging or raiding Nice. I for one would assume that these cavalry would leave their families and most of their baggage with the main camp.
So if the Roman scouts are estimating from the number of people, and calculating 1/6 of that, the missing cavalry won't have much effect. If they are estimating from the amount of baggage, the length of the march column, or the size of the camp, they are going end up counting the cavalry's baggage or the cavalry's camp requirements anyway. I don't see any way they can accurately estimate the infantry while missing the cavalry.
There are several other interpretations. I for one suspect there were two Gothic armies converging on Nice, a very risky maneuver since it meant the Romans might defeat them separately.
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1... 2... 3... D'oh counted him twice! 1... 2...
In all seriousness I think the general rules is that in terms of Barbarians with families and baggage, it's about 1/5 the total size is the number of fighting men. Aetius had "60,000 Huns" in 425, but I seriously doubt there were more than around 15000 of them that were actually fighting men. Also, counting campfires sounds about right, if you apply the Roman Contubernium to each campfire that's about the number of men who can huddle around a single fire.
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Civilized armies could be estimated by counting standards, which had to be plainly visible. At least one commander realized that a Roman army had been reinforced overnight when he heard two sets of calls played on the trumpets. I think it was the morning chow call.
Pecunia non olet