It seems even if you had bigger numbers, if your army is attacked on multiple sides you lose. Cant soldiers push from a single point strongly to break it? I am thinking of a historic setting, no thecnologically advanced weapons.
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It has to do with the “surface area” of the front lines, compared with their depth.
If an army is surrounded by an army of equal size, the surrounded army is going to start losing space to deploy its people to the front, where melee fighters can actually make a difference. Soldiers standing in the middle of the formation won’t be doing anything, won’t contribute to the battle until they can get to the front, and so may as well not be part of the army. A big blob of soldiers who can’t fight because there isn’t enough space in the lines to deploy them are just a liability.
Meanwhile, the surrounding army has the opposite advantage. All of their ranged attackers will be able to focus their fire in an area with ever more enemies to hit. Lobbing catapult stones at a line a few people deep isn’t going to do much for the defenders, but the attackers get to hit the front lines and also all the people behind them who can’t fit on the front. Meanwhile, at the start of the fight, the surrounding army will be able to deploy all of its melee fighters effectively, only losing area to deploy them after they’ve already pushed the defenders back into a formation easier to pick off from afar.
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