The pre-modern state aims to defend its borders against enemies. The size of its territory, though, has profound implications for how it can do this; which may in turn influence how that state develops--whether, in particular, it seeks aggressive expansion.
Central Active Defense
Assume some state has at least some permanent forces that it deployed about its territory in case of invasion. There are several broad strategies for where these forces are positioned, and thus used. In one strategy, all forces are kept in a centralized location, probably the capital. They can protect the whole realm by mobilizing against attackers.
Such a strategy has many implications. You save overhead costs by building just one fortress instead of many, but must instead build up the infrastructure of travel, namely roads. A large force stands a good chance of repelling an invader, but may not be able to move fast enough to intercept that enemy before he pillages the margins; depending on roads but also advance warning. If multiple attacks occur the single force will be hard pressed to meet them all.
Marching to the Border
The implications above are all inter-related but let's concentrate on the most purely geometric, or geographical: movement of troops to intercept enemies. Ideally this takes place on the periphery of the empire, which we may assume to be circular for simplicity. The larger the state, in terms of area, the greater the radius of the circle becomes; and thus the farther troops must move, which decreases their chance of intercepting enemies. By this reasoning, a smaller state is more defensible than a large one. No state using centralized defenses should try to expand, particularly if it is a very small one, as the increase in travel distance is steepest at the low end.
The size of the defender's forces also depend on territory size though. The more land and thus food a state has, the more soldiers it can choose to support, assuming a constant population density and intensity of agriculture (and neglecting various other factors like economies of scale in maintaining a large army). If enemy armies grow in size, the advantage of a large defensive force may surpass that of response time, pushing a state to grow.
What if a small empire is raided in rapid succession by various enemies in a season? Obviously it becomes more difficult to repel later invasions, if only the points of incursion are harder to reach. Imagine that one attack is repelled on the border, and then a another occurs; how far will the defenders need to travel to get there?
The distance between two points on a circle is a chord, which presumably the defender would traverse. We can also assume the location of each attack to be essentially random. The distance to a second location also depends linearly on territory radius but is greater than it, and thus more problematic. If a state tends to be attacked repeatedly, even by small forces, it will experience an even stronger impetus to remain small and defensible.
The second major strategy for defending a territory is to spread available forces along the border, waiting for incursions; it is essentially the opposite of a centralized defense force. In its favor, this second strategy involves little or no travel on the part of the troops. Instead the deciding factor is how thinly they are spread--that is, the circumference of the circle--relative to their number.
Circumference increases less rapidly than area as a circle is enlarged. Therefore, there are more troops stationed at any point on the border in a larger empire, even though the border is also large. States using peripheral defenses therefore see advantage in growing larger, especially when they are small.
Peripheral defenses are well equipped to handle multiple attacks, simultaneously or in succession, as every position on the border is equally well defended. But conversely, there are very few troops at any position, so a large invading force will breach the defensive line, probably to march unopposed to the capital. If enemies tend to send large forces, even if infrequently, this strategy is problematic--though growing in size always improves it.
Choosing a Strategy
Of course, empires cannot readily choose their own size: they must defend themselves in their current condition. A small empire is relatively better at centralized defense than a large one, which benefits more from the peripheral strategy. But this does not mean an empire should necessarily choose the strategy it is better at; the more important factor is what its enemies do.
An empire that experiences frequent, small attacks can best protect itself with a peripheral defense. Once it has chosen that strategy, it can best ensure its survival by expanding--by whatever means possible. Conversely, an empire that sees infrequent but serious attempts to conquer it must avail itself of a centralized, reactive defense, to oppose each new enemy host. It will then be interested not in expansion but consolidation of its territory: in the building of roads and the elimination of far-flung provinces in favor of a compact shape.
The Evolution of Empire
The above decisions leads to larger consequences for the geopolitics of a region. Consider the scenarios in which each pattern of attack is likely to occur.
Small and frequent attacks are expected when a region is divided into numerous small states: no state can muster a very large attacking force, but there are many of them. Any one of those states is small and therefore good at a centralized defense but should nonetheless choose peripheral defenses because of its circumstances; and it should seek to grow in size by conquering neighbors.
If the geography is changed, however, so that a region is divided into only a few large states, things are different. Any empire can muster a large force, and thus its neighbors must be ready to defend themselves against one; though not frequently. Thus, each empire should choose a centralized defense, even though it is not inherently very adept at such a game. Growing larger makes such a strategy even harder, so it must instead content itself with its current size.
This means that small states will tend to conquer one another (with enough time and effort, some will succeed and absorbing others, even if it is mainly by chance). This will lead to progressively larger states, until they are inclined to switch defensive strategies. If one has, by some circumstance, grown "too" large, a peripheral defense will be inadequate to prevent loss of territory, and a centralized one will be completely ineffective owing to the huge distances its armies must traverse; it will then be nibbled at until it is of a size more in line with its neighbors (or is destroyed completely). There may be, in other worse, a kind of stable equilibrium, which creates an ideal size and number of empires in a region (dependent on technology, culture and innumerable other factors), where each empire is similar to its neighbors. As is the case in our personal lives, it is often inequality that causes disharmony, not the absolute level of wealth.
Forthcoming articles will explore intermediate defensive strategies, the geometry of offense, and more complex geographic scenarios.