Pesachim, chapter five, mishnah 1. This chapter and those that follow it deal with the offering of the korban Pesach - the offering that was slaughtered and prepared on erev Pesach and eaten at the Seder. Our mishnah discusses the time of its offering.
Before we turn to the time of the korban Pesach, however, we need to establish the time of the afternoon tamid, the daily offering that was brought every morning and every afternoon. The afternoon tamid is especially relevant to our topic, because as a rule all offerings were brought after the morning tamid and before the afternoon tamid. The exception is the korban Pesach, which was specifically brought after the afternoon tamid - and for that reason we begin by clarifying the time of the tamid.
The time of the tamid on an ordinary day:
The mishnah opens: "Tamid nishchat b'shmonah u'mechetzah v'karev b'teisha u'mechetzah" - on an ordinary day the tamid was slaughtered at eight and a half hours of the day, and its offering was completed at nine and a half hours. The hours counted here are seasonal hours: if the sun rose at six in the morning and set at six in the evening, eight and a half hours is two thirty in the afternoon, and nine and a half hours is three thirty. Its offering thus took about an hour.
Strictly by law, one may bring the afternoon tamid already from six and a half hours of the day, since once half an hour has passed after midday the time is already considered "afternoon," and that is a valid time for offering it. Nevertheless, the practice was to delay it another two hours, until eight and a half hours, so that those coming to the Beis HaMikdash would have time during those hours to bring their own offerings.
On the other hand, they did not delay it beyond eight and a half hours, because after the afternoon tamid was completed (or after part of it), there were additional services that had to be performed every day:
The ketores.
The minchah - the meal offering of the libations that accompanies the tamid.
The chavitin of the Kohen Gadol - the daily offering of the Kohen Gadol, which is not an animal but a type of meal offering.
The nesachim - the wine libations.
The kindling of the lamps of the Menorah.
The time of the tamid on erev Pesach:
The mishnah continues: "B'arvei Pesachim nishchat b'sheva u'mechetzah v'karev b'shmonah u'mechetzah" - on erev Pesach the tamid must be brought earlier, because in addition to the services performed every day at the conclusion of the afternoon tamid, the korban Pesach must be slaughtered, its blood sprinkled and the offering made fit, and all of this during those same afternoon hours. Since extra time is needed, on erev Pesach the tamid was slaughtered at seven and a half hours of the day - one thirty in the afternoon - and its offering was completed at eight and a half hours, which is two thirty.
"Bein bachol bein baShabbos":
These times - eight and a half hours on an ordinary day and seven and a half hours on erev Pesach - apply whether the day falls on a weekday or on Shabbos. The mishnah emphasizes this, because on Shabbos only the offerings fixed by the Torah are brought, while vow offerings and voluntary offerings are not brought on it. Seemingly, then, there is no need at all to delay the tamid, and it could have been brought earlier, at six and a half hours of the day. Nevertheless they made no change, and left the time the same on a weekday and on Shabbos.
Erev Pesach that falls on erev Shabbos:
The mishnah also discusses a case in which erev Pesach falls on erev Shabbos. In such a case, in addition to slaughtering the korban Pesach in the afternoon, it must also be roasted that same day. Normally the Pesach was roasted at night, which is Yom Tov, when cooking is permitted; but when the Seder night falls on Shabbos, it cannot be roasted at night, even for the sake of the korban Pesach. They therefore had to roast it while it was still erev Pesach, and additional time was needed.
For this reason we learn: "Chal erev Pesach lihyos b'erev Shabbos, nishchat b'shesh u'mechetzah v'karev b'sheva u'mechetzah" - the tamid is slaughtered at the earliest possible time, six and a half hours of the day, which is twelve thirty, and its offering is completed at seven and a half hours, which is one thirty. The Pesach is brought afterward, and with it too they made every effort to be as early as possible, so that each and every person could roast his Pesach before Shabbos began.
In summary: in this mishnah we learned three sets of times for the slaughter and offering of the afternoon tamid: on an ordinary day - eight and a half and nine and a half; on erev Pesach - seven and a half and eight and a half; and on an erev Pesach that falls on erev Shabbos - six and a half and seven and a half. Everything is determined by how much time is needed for the services performed afterward: the ketores, the meal offerings, the libations and the kindling of the lamps, and on erev Pesach also the slaughtering and roasting of the Pesach.