What do men in the middle of a phalanx contribute?

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Accepted answer

Most obviously, those in the rear provide reinforcements as those in front are wounded or killed or exhausted. If your phalanx is 10 deep and theirs is 5 deep, odds are you can sustain your line longer than they can.

In general, the immediate front rank form a shield wall and can fight with short swords. The ranks immediately behind can project spears and other long pole arms between and over the shoulders of the front ranks. The result is a very pointy shield wall. The advantage is to allow several ranks of men to participate in the fighting. The front rank can focus on defending, while the rear ranks can focus on attacking.

There is an unconfirmed theory that phalanx fighting was also about pushing, that each formation literally tried to shove its way through the other. Men to the rear would press on those in front to achieve this. There is little contemporary evidence to account for this, but it would explain why some phalanxes are as much as 50 ranks deep.

The people in the rear keep the people in front from running away. Ancient combat as not very lethal so long as the formations held and nobody got flanked. Breaking the enemy's formation, outflanking them, or causing them to flee was how you inflicted great casualties. You could ensure your troops did not run by the simple expedient of giving them nowhere to run to.

Finally, deep ranks can offer opportunity for maneuver. For example, an army on the march might go in 2 deep columns with leaders (dimœrites, decadarchos, and decasterœ) every 8 ranks. Here's an example with 2 columns of 32 ranks on the march.

Di      Di
*       * 
*       * 
*       * 
*       * 
*       * 
*       * 
*       * 
De      De
*       * 
*       * 
*       * 
*       * 
*       * 
*       * 
*       *
De      De
*       * 
*       * 
*       * 
*       * 
*       * 
*       * 
*       * 
De      De
*       * 
*       * 
*       * 
*       * 
*       * 
*       * 
*       *

To deploy for battle, they may split into 4 columns of 16.

Di      Di      De      De
*       *       *       *
*       *       *       *
*       *       *       *
*       *       *       *
*       *       *       *
*       *       *       *
*       *       *       *
De      De      De      De
*       *       *       *
*       *       *       *
*       *       *       *
*       *       *       *
*       *       *       *
*       *       *       *
*       *       *       *

Once engaged, they can use their rear ranks to extend their line. This forces the enemy to either thin their line to match, or allow their vulnerable flank to be enveloped. Even if the enemy matches the extension, the maneuver may disrupt their organization and offer an opening.

Di      Di      De      De      De      De      De      De
*       *       *       *       *       *       *       *
*       *       *       *       *       *       *       *
*       *       *       *       *       *       *       *
*       *       *       *       *       *       *       *
*       *       *       *       *       *       *       *
*       *       *       *       *       *       *       *
*       *       *       *       *       *       *       *

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Question:
I understand how the Phalanx can have an almost impenetrable front, and I understand the rear and sides may have to defend against an attack like the front - but what purpose do men in the middle serve? In other words, why would you ever have a Phalanx deeper than 4 rows?

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Short Answer:
Phalanxes weren't Napoleonic squares, they were vectors. They were designed to fight an enemy to their front. The length of the spear they carried varied and determined how many of the men in the formation contributed to the offensive and defensive capabilities of the phalanx in the moment.

The Persian phalanxes under Darias or Xerces might use just the front two rows of the phalanx. The Greeks Hoplites used the first 3 rows of men. The Macedonians under Philip II and Alexander used the first 5 rows of men.

Men who's spears could not reach beyond the front of the formation stood with their spears in the air, out of the way. They could move forward into one of the fighting positions if a man in their column fell, or if the men in front needed to rest and they were signaled to do so. Rotating men from the back to the front was a tactic to keep men in the front fresh as battles could last a while.

Detailed Answer:
The rear and sides were lightly or undefended. If you could flank or out maneuver a phalanx and force it to fight on two sides, that was one way to win and win quickly. The Phalanx was designed to fight with the enemy in the front. If you got around to the left or right that could become the front. The second third and maybe even fourth and fifth rows contributing to the frontal defense depending upon the practitioners of the phalanx. Phalanx were vulnerable to rear or side attacks if they were forced to fight on two sides. This wasn't a major concern for the Greeks however because mostly in Greece, Phalanxes fought other phalanxes and thus neither could exploit this two front weakness. The Macedonian phalanx which was larger and slower than the Hoplite Phalanx employed by Athens and Sparta was also vulnerable to same weakness. But Philip II and Alexander the Great unlike the Greek contemporaries employed a mixed unit army including light infantry, light calvary and heavy calvary which could support the phalanx (heavy infantry) and keep any enemy from exploiting the phalanxes weakness. The Macedonian phalanx employed longer spears and excelled at defending itself from enemies it faced. five layers of spears faced any such enemy. As the enemy was trying to figure out how to mitigate those five spears they had the light infantry's ballista's to deal with or the ballista's from the light calvary, or Alexanders Companion, heavy calvary just running over top of them.

In the phalanx everybody was facing forward and everybody held their spear and shield in the same hand as everybody else in the formation. That means if you could get around on the right of the phalanx, if the phalanx was unable to shift, you would be facing unshielded people, who's shields were all carried in the left hand, if the phalanx was engaged in the front and unable to face to the exposed side. If you got around on the left they could use their shield but you would not face layers of spears as you would if you were coming straight on from the front. The back would be undefended as the formation was designed to fight enemies it faced. Thucydides said this characteristic of the phalanx made undisciplined practitioners naturally drift to the right as every man in the phalanx would naturally try to push his undefended side closer to the shield of the man standing next to him.

Greek Hero's Hoplites and Heros enter image description here

Typically if you weren't contributing to the frontal defense / offense of the phalanx you waiting your turn to do so. You would just be standing their with your spear in the air waiting to replace the man in front of you. If someone in your column fell you would move up, or you could move up if men in the front needed rest. The size of the spear used in the phalanx dictated the minimum size and how many rows of the Phalanx contributed to the offense/defense of the formation. The Greek Hoplite Phalanxes used the spear called the dory about 2 to 3 meters in length, which was longer than the spear used by the Persian phalanx under Darius and Xerxes. The dory allowed the second and third rows of the formation to contribute to the defense of the phalanx.

Dory
The spear used by the Persian army under Darius I and Xerxes in their respective campaigns during the Greco-Persian Wars was shorter than that of their Greek opponents. The dory's length enabled multiple ranks of a formation to engage simultaneously during combat.

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Larger phalanxes were not better, especially if they were using shorter spears which didn't contribute to the offensive defensive capability. Larger phalanxes were less maneuverable, slower, and harder to keep together.

Alexander the Great's Macedonian Phalanxs used the longer Sarissa spear which was 2 to 3 times longer than the dori. It allowed for 5 rows of the formation to contribute to the defense of the unit. but it also required their be five rows of people in the phalanx. It was generally more vulnerable to a two sided attack because the Sarissa speer required two hands to hold. The shield employed by Alexander was smaller and just slung over one arm. If you could flank a Macedonian phalanx they would be especially vulnerable. But Alexander's phalanxes were not called upon to fight independently as were hoplite greek phalanxes. Alexander had complementary units which supported his phalanxes and made it harder for any enemy to exploit it's weeknesses.

Anybody confronting a Macedonian Phalanx head on had to confront 5 rows of spears. This made the Macedonian Phalanx, slower, and harder to maneuver and much harder to keep together especially over rough terrain, but also heavier and more able to hold off enemies in their front. Alexander used them to control the battlefield against smaller units. His tactics were to use his other units to protect the phalanx's vulnerabilities and also to destroy his enemies held at bay by his larger heavier formations.

Sarissa
enter image description here
The sarissa or sarisa (Greek: σάρισα) was a long spear or pike about 4–6 metres (13–20 ft) in length. It was introduced by Philip II of Macedon and was used in his Macedonian phalanxes as a replacement for the earlier dory, which was considerably shorter. These longer spears improved the traditional strength of the phalanx by extending the rows of overlapping weapons projecting towards the enemy, and the word remained in use throughout the Byzantine years to sometimes describe the long spears of their own infantry

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