Topography of the battlefield
The battlefield is relatively well described: The plains are generally flat. The river crosses on the East, but this simply denies use of this direction in battle. There are fires burning below the plain dating to the first battle of the burning plains though, with the exception of the scene where Galbatorix's messenger gets burned, this does not seem to affect the battle. Of course as an agricultural engineer I cannot but ask how can this fire be possible. Peat bogs are saturated in water and, with the river nearby, the water table must be too high to allow combustion, let alone for over 20 years. This phenomenon serves only one dramatic purpose: deny Eragon the ability to draw power from his surroundings. The forces of Surda and the Varden have constructed earthen defences around their original position which they abandon initially but return to them under the strength of the imperial army's attack
Size of the opposing forces
The size of the force sent by the Empire is well known; 100.000 men. We are not told if this is just the battle soldiers or if it includes the supporting forces. In antiquity armies required at least one support troop for every battle troop, whether that was the helot carrying the supplies of the Spartan, the engineer of the siege machinery supporting the battle or the auxilliary carrying the legionnaire's arms. In the modern era, this ratio has grown in favor of the support troop: every American battle soldier in Iraq requires 5 or 6 people supporting him. We are told that there are siege machinery near the tale of the imperial army and those manning them or carrying them are definitely NOT battle troops. Whether the 100,000 number that Nasuada estimates are opposing her include support troops remains unknown.
Far less is known about the size of the army opposing the Empire. We are only told before the Battle under Farthen Dur by Ajihad that the Varden number 4,000 people. They suffer significant losses in the aformentioned battle but then receive ''a veritable flood of volunteers'' and we can were also able to somewhat replace their losses by troops that were already in Surda before Farthen Dur. Other than that we only learn that the army opposing the empire is "far larger" than those under Farthen Dur, whose number we are also not told.
Let us consider the size of the Varden for a while. We learn at the summary of Eragon at the beginning of Eldest that since the formation of the Varden 20 years have passed. If the Varden received a steady state of volunteers and did not suffer any losses before Farthen Dur (unlikely) they have received 200 people per year. This is our baseline. More likely though they did suffer losses in battle or from natural causes so they probably received more than that. Let us assume it was normally 50% more, that is 300 people per year as a baseline. A 'veritable flood' means several times the baseline. For arguement's sake I assume 1000 people per year. Let us also assume that the losses at Furthen Dur were covered fully by those Varden who were already in Surda or by hereditary succsession (i.e. the son of the one who died in Farthen Dur but did not participate in the battle due to lack of weapons took his father's arms and fights in his place). Since between the two battles passes one year (Eragon was inside Du Weldenvarden during the Dagshelgr Invocation in spring, after Farthen Dur and it is spring when he travels to the Burning plain) the Varden number maximum 5,000 people.
Surdan troops most probably outnumber the Varden, otherwise Orrin would not need to pick affirm military leadership of the Nassuada over the entire troop, it would be rightfully hers. How many are there? Let us assume that the Varden and the dwarves numbered 12,000 under Farthen dur (how I came up with the number I will explain when I analyse that battle). Then if the are twice as many facing Galbatorix than in Farthen Dur the whole army is 24,000 men strong, 5000 of which are Varden thus Surda sent 19,000. At modern draft rates (5% of the population) this implies a total population of almost 400,000 for the kingdom. We have nothing to support or dispel these numbers. For arguements sake though let us take 19,000 as a possibility. Far more than Farthen Dur could mean 20,000 troops as the total number, giving a ratio of 5 to 1 between Galbatorix's troops and those opposing him. This was the ratio between the Athenians and the Persians in the Battle of Marathon. Victory is possible, without Eragon's intervention, if Nasuada is a good general, the Surdans are superior in terms of quality and the other army lacks quality, is overconfident and makes mistakes. Not all of these conditions are met, as we will see later on the analysis.
How many troops accompany Hrothgar? If there were 8,000 dwarves at Farthen Dur and the Dwarves have 5 cities like it, and the entire dwarven nation marches (as Eragon overconfidently tells) then 20,000 dwarves might be accompanying him. But this is only a guess. Next time I will analyse armaments and tactics of the two armies.
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