We are up to Mishnah 3 in Chapter 7 of Tractate Sanhedrin, and there is some relief in the fact that this is the final Mishnah detailing the methods of these harsh executions. We will deal here with the last two of the four capital punishments of the Beis Din: decapitation and strangulation.
The procedure for decapitation:
How is the death penalty of decapitation performed, when they sever the head of the condemned? "Hayu matizin es rosho besayif" - they sever his head with a sword. The sword is passed through the front of the neck, in the exact place where shechitah is performed: there are the esophagus and trachea, the jugular vein and the rest of the organs, and everything is severed. The condemned stands upright, his head is chopped off, and his body collapses to the floor along with the severed head. Here too, the one wielding the sword is one of the witnesses. It is a gruesome sight, but this is how it was done.
"Kederech shehamalchus osah" - in the same manner that the government, meaning the Roman regime, carried out its executions.
The opinion of Rabbi Yehudah:
Rabbi Yehudah disagrees and says: "Nivvul hu zeh" - this is a disgrace. A death in which the body collapses to the ground like a heap, and the head is cast beside it and blood spills all around - involves disgrace and humiliation, and there is no need for it. The Gemara clarifies the core of his objection: one should not act in this problematic manner, which violates the prohibition of following gentile customs ("bechukoseihem") - this is the gentiles' method of execution, and we must not follow their ways.
Instead, Rabbi Yehudah says: "Manichin es rosho al hasadan vekotzetz bekofitz" - the condemned is laid face down with his head resting on a wooden block, and his head is chopped off with a kofitz. A 'kofitz' is generally a type of butcher's cleaver, and in this case, it is likely referring to something like an axe - a blade with a long handle. The advantage of this according to his opinion: the body does not fall, since it is already lying on the ground, and therefore there is less disgrace.
The response of the Sages:
"Amru lo: ein misah menuveles mizu" - on the contrary, there is no death more degrading and disgraceful than this: striking the back of the neck with an axe, which is not as sharp as a sword. There is no following of gentile customs here, nor is there an arbitrary method dictated by the gentiles, but rather a logical method: decapitation with a sword is the fastest way to kill, and therefore this is how it should be done. And so the halachah was decided.
The procedure for strangulation:
The fourth and final method of execution is chenek, meaning strangulation:
"Hayu meshake'in oso bezavel ad arkuvosav" - they secure the condemned with his legs buried in dung up to his knees. (And we have already discussed the debate over whether this specifically means dung.)
"Venosnin sudar kasheh lesoch harach" - they take a double-layered cloth with two layers: the stiff, coarse cloth, which tightens firmly and kills quickly but might rub and bruise the neck, is placed inside a soft outer cloth, so that it will be gentler on the neck.
"Vechorech al tzavaro" - they wrap it around the neck, even just a single wrap.
"Zeh moshech etzlo vezeh moshech etzlo" - each of the two witnesses pulls one side of the double-layered cloth, and the pressure tightens around the neck.
"Ad shenafsho yotzah" - and they continue doing so until the condemned person's soul departs and he is strangled to death.
In summary: In this section, we learned about the last two death penalties - execution by the sword, including the dispute between Rabbi Yehuda and the Sages whether it is proper to behead with an axe on a chopping block or to strike with a sword in the manner of the government; and strangulation using a hard cloth wrapped inside a soft one, which the two witnesses pull until his soul departs. This completes the four death penalties of the court.
From this point on, the Mishnah will proceed to detail which transgressions are liable for each of these four categories of the death penalty.