Bein Hashmashos-A child who was born during dusk:[1]

If the child was born during Bein Hashmashos[2] then the eight days are to be calculated starting from that evening.[3] [Thus, if the child was born after sunset prior to nightfall on a Wednesday, then the Bris is to be scheduled for the following Thursday which is definitely past the start of the eighth day.] Even if the child stuck out[4] majority[5] of his head[6] during Bein Hashmashos, even though it was only fully removed after nightfall, nevertheless it is considered as if he was born during Bein Hashmashos. [For this reason, special heed must be taken to verify the exact time the baby’s head has protruded in case of a birth taking place near the time of Bein Hashmashos of Erev Shabbos or Shabbos![7] Nevertheless, unless known otherwise[8], we do not assume the head came out much time prior to the body and we thus go in accordance to when the baby was taken out.[9] However other Poskim[10] rule it is always considered a doubt as to whether the head exited during Bein Hashmashos if she was in active labor during Bein Hashmashos and then gave birth at the beginning of the night.]

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[1] Admur 331:5; Michaber Y.D. 262:4-6; 266:8; Shevach Habris 8:2; Pesakim Uteshuvos 262:15-16

Born after Davened Maariv, before sunset: Is considered day. [Michaber 262:7; See Pesakim Uteshuvos 262:17]

[2] When is Bein Hashmashos? Between sunset and nightfall. [Siddur as rules Geonim and so is the custom of all Jewry today; Birkei Yosef, brought in M”B 331:14, writes that the custom has spread like the Geonim in all Eretz Yisrael; This ruling is in contrast to the ruling of Admur in the Shulchan Aruch in 331:5 and 261:5] Practically according to Admur in the Siddur Bein Hashmashos only begins a few minutes [between 3 and 5 minutes-see Kitzur Hilchos Shabbos Miluim p. 99] after the visual sunset. The start of Nightfall varies in each area. An expert Rav is to be contacted in cases of any doubt as to the Halachic definition of the time the child was born.] See Pesakim Uteshuvos 262:15 footnotes 117-123

When is Shekia: See Pesakim Uteshuvos 262:15 footnotes 123-126

Opinion of Rabbeinu Tam-The ruling of the Shulchan Aruch [Admur 331:5 and Michaber Y.D. 266:9: In the Shulchan Aruch Admur [and the Michaber Y.D. 266:9] rules like the opinion of Rabbeinu Tam [unlike the Yereim/Geonim] that Bein Hashmashos begins 15 minutes prior to nightfall, which in turn begins only 72/96 [depending on calculation of Mil in Admur] after sunset. Hence for up to 56.5-78 minutes after sunset it is still considered day for all purposes. Thus, if a child was born during this time on Erev Shabbos he is to be circumcised the next Friday. If he was born at this time on Shabbos he is to be circumcised on Shabbos. The reason we are not stringent like the opinion of the Yireim in this case to suspect that perhaps it is already after Shabbos when he was born is because no one in these provinces are accustomed like the Yireim regarding Melacha on Erev Shabbos, hence there is no need at all to be stringent here and push off the circumcision due to their opinion. [Admur ibid; No mention of the opinion of the Geonim is made by Admur in his Shulchan Aruch. Now although in 251:5 Admur rules regarding one who accepted Shabbos before Plag Hamincha of Rabbeinu Tam, to be stringent like Yireim, perhaps there it is different as by one accepting Shabbos at that time he has shown that he suspects for the opinion of the Yireim. Alternatively, in that case there is nothing to lose by being stringent while here if one is stringent the Bris will be delayed from its proper time according to the main opinion.]

Other Poskim-Opinion of Shach and Maharam Alshiker: The Shach [Yoreh Deah 266:11] brings the Maharahm Alshiker who rules to be stringent like the Geonim and push off the Bris if the child is born on Shabbos anytime between sunset and nightfall [of Rabbeinu Tam]. [This is unlike the ruling of Admur in the Siddur who rules completely like the Geonim that if born after nightfall of Geonim on Erev Shabbos, the Bris is to be done on Shabbos even though it is still Friday according to Rabbeinu Tam.] Birchei Yosef [brought in M”B 331:14] writes that the custom has spread like the Geonim in all Eretz Yisrael.

[3] The reason: As there is doubt as to whether Bein Hashmashos is part of the previous day or the coming day. Hence if born right after sunset of Wednesday, the coming Wednesday is questionable whether it is the child’s 7^th day or 8^th day, and it is forbidden to circumcise a child prior to entering the eighth day. 

[4] See Piskeiy Teshuvos 331:6 and Pesakim Uteshuvos 262:16 footnotes 132-135 for a discussion in Poskim on the exact definition of the head coming out. Is it defined as having come out once it exists the Prozdor to the Beis Hachitzon, or is it only considered to have come out when it enters to the actual air of the world. The custom is like the latter opinion. [Chasam Sofer Yoreh Deah 167; Sidrei Taharah 194:26; Pischeiy Teshuvah 266:8]

[5] It is disputed in Poskim as to how one is to measure the majority of the head. Is it the majority of the forehead, or the entire forehead. [see Piskeiy Teshuvos 331:6 and Pesakim Uteshuvos 262:16 footnotes 128-131]

[6] Regarding a child which was born breached [leg first] then when majority of the body exits it is considered to have been born. Practically once the naval has come out majority of the body is considered to have exited. [see Piskeiy Teshuvos 331:6]

[7] Pesakim Uteshuvos 262:16 footnotes 13138

[8] Such as a case that the mother had difficulty pushing the child out.

[9] See Piskeiy Teshuvos 331:6 in name of Migdal Oz [Yaavetz]; Pesakim Uteshuvos 262:16 footnotes 136-141

[10] Pischeiy Teshuvah 266:8