Maseches Shabbos, chapter 18, mishnah 3. This mishnah deals with assisting at birth, both in animals and in people, on Yom Tov and on Shabbos.
Delivering an animal on Yom Tov:
The mishnah rules that "Ein meyaldin es habehemah" - one does not deliver an animal, and this refers to a large animal. Actual delivery, meaning extracting the newborn from its mother's womb, is forbidden on Yom Tov, and all the more so on Shabbos, because of the exertion and effort involved. And as is known, Chazal (as brought in the Midrash Rabbah) were not pleased with a person burdening himself with heavy exertion on Yom Tov.
The mishnah does, however, permit assisting the animal, and this assistance includes:
Holding the newborn as it emerges and preventing it from falling to the ground.
Pressing on the mother's belly so that the calf, or the like, should come out.
Assisting a woman giving birth on Shabbos:
From here the mishnah moves on to the law regarding a woman in childbirth: a woman may be actually delivered on Shabbos. One may call a midwife for her from place to place, even if this involves going outside the Shabbos boundary, and one may desecrate Shabbos on her behalf, since a woman in childbirth has the status of a dangerously ill patient.
After the birth:
Once the baby has been born, one may tie the umbilical cord and separate the connections between the placenta and the baby. It is also stated that one may even cut the umbilical cord, and this is the halachah.
The mishnah concludes: "V'chol tzorchei milah" - all the needs of the bris milah may be attended to even on Shabbos.
In summary: in this mishnah we learned that delivering an animal is forbidden because of the exertion involved, but assisting - holding the newborn and pressing on the mother's belly - is permitted; a woman, however, may be delivered on Shabbos to whatever extent necessary, including calling a midwife from outside the Shabbos boundary and desecrating Shabbos, because a woman in childbirth is a dangerously ill patient, and likewise tying the umbilical cord and cutting it after the birth.
With this, chapter 18 comes to a close, and the mishnah's words about the needs of the milah serve as an introduction to chapter 19, which is devoted entirely to the laws of bris milah.