Quote Originally Posted by article
1. Over-arm: the spear is held in the centre, with the right hand. The hand is held at about head-height, with the elbow of the right arm out to the side. The right thumb of the user is on the head-side of his hand, and his four fingers curl over the top of the spear.
2. Under-arm: the spear is held in the right hand, with the thumb on the top of the spear and the spear held typically at around waist height. The fingers curl under the spear shaft. The spear shaft rests along the underside of the forearm with the butt-spike by the right elbow.
That's not the underarm grip, it's the couched lance, wich was used by medieval knights, but definitely not by hoplites.

Quote Originally Posted by article
This goes dead against the whole idea of a spear. A spear is a device for keeping your enemy at a distance.
Not necesseraly. A spear is also often used because it is much cheaper than a sword and requires less training.

Quote Originally Posted by article
If two formations of spearmen clashed, one using spears underarm, the other over-arm, then the fools using their spears over-arm would face their enemies’ spears before they themselves were in striking range.
Ever heard of shields?

Quote Originally Posted by article
With the over-arm hold, the rear end of the spear acts as a counterweight to the front end. If a foe were to strike the spearhead sideways with a sword, then the counterweight would act against the spear-user. The front end of the spear would act as a lever, twisting the wrist of the spearman, and the swinging rear counter-weight end would act to exaggerate this effect. To close with a spearman, a sword user has to knock the spearhead aside and rush in at his foe. The over-arm grip would make this enormously more easy. With an under-arm grip, the spearman has his spear braced along his forearm, and has much more control of the spearhead. The spearhead may be knocked aside, but it will resume position a great deal more quickly. If a high thrust over a shield is wanted, this can be achieved by bringing the right elbow up to shoulder height. Also, if the swordsman advances, then the under-arm spear user can retreat a great deal faster, to bring his spearhead between them, as he has the ability denied to the over-arm user, of pulling back his spear, and sliding his right hand up the shaft, to shorten the weapon for close use.
Well, what he says here is already wrong because of the fact that he confuses the underhand grip with the couched lance. Because the overhand spear is shorter, the user obviously has more control. He says: "The spearhead may be knocked aside, but it will resume position a great deal more quickly." Well, even idf this were actually true, you'd still be dead before you had your spear back in postition. "the under-arm spear user can retreat a great deal faster" What? Why? How? "the ability denied to the over-arm user, of pulling back his spear, and sliding his right hand up the shaft, to shorten the weapon for close use." And what exactly denies a overhand spearman this ability? Just try it, take a long stick, hold it overhand and try grabbing it closer to the spearpoint, in the same way a fully armed hoplite would have to. Believe me, it works, and probably much faster than an underhand spearman could do it.

He also says this: "If a high thrust over a shield is wanted, this can be achieved by bringing the right elbow up to shoulder height." What the hell is he suggesting here? That the spearman holds his arm completely streched? What the hell? How is he supposed to thrust then? Or even control his weapon at all? And even if he does hit something, the spear will most likely immediately be knocked out of his hand.

Quote Originally Posted by article
With the under-arm grip, a spearman can thrust with his spear downwards at the feet of his foe, or upward at his face. The strongest thrust he can do it at waist height, and he can disguise his intentions easily. He can hold his shield in position during all of this. Using an over-arm grip, the feet of the foe are out of reach. Greaves, protection for the lower leg, were very common in the ancient world, being part of the standard hoplite panoply. This suggests that the lower leg was a common target. Not only are the feet out of reach, but the thighs are difficult targets. A thrust at waist height is difficult, and the spear point will be travelling downwards, and will glance of a shield more easily. The only really strong thrust will be at the face and neck of the enemy. The neck was seldom armoured in ancient times. Greeks and Romans usually had no armour there at all. This thrust will be easy to see coming. Worse still, the spearman thrusting over-arm will of necessity expose himself as he does this, leaning forwards out of formation, and turning his shield to the left to give himself room for the thrust. If an enemy spearman to the right of the over-arm user saw the thrust coming, he would have an easy victim: a man who has stepped with his weight onto his front foot (thus preventing any evasion by footwork) with an exposed shieldless side.
"With the under-arm grip, a spearman can thrust with his spear downwards at the feet of his foe, or upward at his face. The strongest thrust he can do it at waist height" ...wich is where your opponents shield is most likely to be. "The only really strong thrust will be at the face and neck of the enemy. The neck was seldom armoured in ancient times. Greeks and Romans usually had no armour there at all." Greeks often wore helmets that covered most of the face, so that part of the body was well protected. Neck armour was not very practical for the kind of fighting that was done in those days.