Was it a standard tactic that was accepted as regular practice - i.e. something like the infantry square Napoleonic infantry used to fend off cavalry? Or was it more of a rarity?
Was it only used after the adoption of the rectangular scutum or did it exist before? If so, why did it not survive the fall of the Roman Empire (or is there evidence that it did?)
So, the Testudo. Interesting story. Did you know that, according to Livy, it began life as a circus trick?
Some young Romans turned their training in the Circus games to purposes of war and in this way seized the lowest portion of the wall. Before the extravagant habit came in of filling the Circus with animals from all parts of the world, it was the practice to devise various forms of amusement, as the chariot and horse races were over within the hour. Amongst other exhibitions, bodies of youths, numbering generally about sixty, but larger in the more elaborate games, were introduced fully armed. To some extent they represented the maneuvers of an army, but their movements were more skilful and resembled more nearly the combat of gladiators. After going through various evolutions, they formed a solid square with their shields held over their heads, touching one another; those in the front rank standing erect; those in the second slightly stooping; those in the third and fourth bending lower and lower; whilst those in the rear rank rested on their knees. In this way they formed a testudo, which sloped like the roof of a house. From a distance of fifty feet two fully armed men ran forward and, pretending to threaten one another, went from the lowest to the highest part of the testudo over the closely locked shields; at one moment assuming an attitude of defiance on the very edge, and then rushing at one another in the middle of it just as though they were jumping about on solid ground. — Livy, ab urbe condita book 44
It was quite a useful trick, though:
A testudo formed in this way was brought up against the lowest part of the wall. When the soldiers who were mounted on it came close up to the wall they were at the same height as the defenders, and when these were driven off, the soldiers of two companies climbed over into the city. The only difference was that the front rank and the files did not raise their shields above their heads for fear of exposing themselves; they held them in front as in battle. Thus they were not hit by the missiles from the walls, and those which were hurled on the testudo rolled off harmlessly to the ground like a shower of rain from the roof of a house. — Livy, ab urbe condita book 44
It's unclear whether Livy's explanation, that the testudo was developed as a circus game, is correct. Nor is it clear whether it did indeed first appear in battle during the Third Macedonian War, which is when the events described above took place. What is clear is that the formation was mostly used in siege warfare, and served to protect soldiers from missile fire. It was very effective at this.^1 For example, in a siege during the year of four emperors:
[Vespasian's attacking troops] renewed the "tortoise," and climbing on their comrades' shoulders, they mounted on it and seized their foes' weapons and arms. The unharmed and the wounded, the half-dead and the dying all rolled in one mass; men perished in many ways and death took every form.
[...]When the Vitellian troops could no longer sustain this combined and persistent attack, finding that their shots slipped off the "tortoise" without doing harm, they finally pushed over their ballista itself on the heads of their assailants beneath. This for the moment scattered and crushed those on whom it fell, but in its fall it dragged down the parapet and the upper part of the rampart — Tacitus, Histories book III
Using your catapult as a makeshift missile: rather extreme measures, no?
Here^2 is an illustration of how the testudo formation served to protect the soldiers from missile fire and allowed soldiers to overcome walls by using it as a ramp. However, it was not only used in siege warfare:
[The Romans] use it in two ways: either they approach some fort to assault it, often even enabling men to scale the very walls, or sometimes, when they are surrounded by archers, they all crouch together — even the horses being taught to kneel or lie down — and thereby cause the foe to think that they are exhausted; then, when the enemy draws near, they suddenly rise and throw them into consternation. — Cassius Dio, Roman History book 49
In other words, it was used in open battle against missile fire. This was done particularly against the Parthians and their horse-archers in the eastern wars. In this case, it was deployed in a much larger formation encompassing the entire army rather than a relatively small company attacking in a siege, as described in the earlier quotes and shown in the picture I linked. But the principle remains the same:
"This testudo and the way in which it is formed are as follows. The baggage animals, the light-armed troops, and the cavalry are placed in the centre of the army. The heavy-armed troops who use the oblong, curved, and cylindrical shields are drawn up around the outside, making a rectangular figure; and, facing outward and holding their arms at the ready, they enclose the rest. The others, who have flat shields, form a compact body in the centre and raise their shields over the heads of all the others, so that nothing but shields can be seen in every part of the phalanx alike and all the men by the density of the formation are under shelter from missiles. Indeed, it is so marvellously strong that men can walk upon it, and whenever they come to a narrow ravine, even horses and vehicles can be driven over it. Such is the plan of this formation, and for this reason it has received the name testudo, with reference both to its strength and to the excellent shelter it affords. — Cassius Dio, Roman History book 49
Incidentally, he also answers your question about shield shapes: you need flat ones for making the roof, but other shields can be used to protect the flanks and front.
However, the formation was not without its weaknesses. A formation like that obviously can't move very well, and the men can't very effectively defend themselves either. This could lead to disaster:
When this had taken place, the Roman infantry did not turn back, but valiantly joined battle with the Parthians to avenge his death. Yet they accomplished nothing worthy of themselves because of the enemy's numbers and tactics, and particularly because Abgarus was plotting against them. For if they decided to lock shields for the purpose of avoiding the arrows by the closeness of their array, the pikemen[i.e. heavy cavalry armed with 2-handed lances] were upon them with a rush, striking down some, and at least scattering the others; and if they extended their ranks to avoid this, they would be struck with the arrows. — Cassius Dio, Roman History book 40, describing the battle of Carrhae
In conclusion:
[continued below] [edited because I somehow forgot to paste in half the post]
