Gemara

26a - Mishnah: Five things happened on the 17th of Tamuz. The Luchos were broken, Korban ha'Tamid ceased, Yerushalayim was breached, Aposotamus burned the Torah and he set up an idol in the Heichal;

Five things happened on Tish'ah b'Av. It was decreed that our ancestors would not enter Eretz Yisrael, Bayis Rishon and Bayis Sheni were destroyed, Beitar was captured, and Yerushalayim was plowed.

28b - Question: The Mishnah says that the city was breached on the 17th, but it says "Ba'Chodesh ha'Revi'i b'Tish'ah la'Chodesh...; Va'Tiboka ha'Ir"!

Answer (Rava and a Beraisa): In Bayis Rishon it was breached on the ninth. In Bayis Sheni it was breached on the 17th.

Question: "Ume'Es Husar ha'Tamid v'Lases Shikutz Shomem" connotes that one idol was put there. "V'Al Kenaf Shikutzim (plural) Meshomem" connotes that there were two!

Answer (Rava): There were two. One fell on the other and broke its hand. It was found written 'You wanted to destroy the Beis ha'Mikdash (by luring Benei Yisrael to serve you). I punished you properly! (Alternatively - did you succeed?!)'

Rosh Hashanah 18b - Question: "Tzom ha'Revi'i v'Tzom ha'Chamishi v'Tzom ha'Shevi'i v'Tzom ha'Asiri Yihyeh l'Veis Yehudah l'Sason ul'Simchah" - how can they be fasts and also days of Simchah?

Answer (Rav Chana bar Bizna): When there is Shalom, they are for Simchah. When not, they are fasts.

Rav Papa: When there is Shalom, they are for Simchah. When there are royal decrees (against Torah), they are fasts. When there is neither Shalom nor royal decrees, they are optional fasts.

Question: If so, the same should apply to Tish'ah b'Av!

Answer (Rav Papa): Tish'ah b'Av is more stringent, because several tragedies occurred that day. (Five tragedies occurred on the 17th of Tamuz, like on Tish'ah b'Av. However, the fast of Tamuz was fixed on the 17th only after Churban Sheni. Originally it was on the ninth; we do not find other occurrences on the ninth.)

Beraisa - R. Akiva: "Tzom ha'Revi'i" is the ninth of Tamuz, on which Yerushalayim was breached - "Ba'Chodesh ha'Revi'i b'Tish'ah la'Chodesh...; Va'Tiboka ha'Ir". "Tzom ha'Chamishi" is Tish'ah b'Av, on which the Beis ha'Mikdash was burned;

"Tzom ha'Shevi'i" is Tishrei 3, on which Yishmael ben Nesanyah killed Gedalyah ben Achikam. This teaches that the death of Tzadikim is like the burning of the Beis ha'Mikdash. "Tzom ha'Asiri" is the tenth of Teves, on which Melech Bavel began the siege against Yerushalayim - "Ba'Chodesh ha'Asiri be'Asor la'Chodesh...; ...Somach Melech Bavel El Yerushalayim."

R. Shimon: R. Akiva says that the verse lists the fasts in order of the months, even though the events did not occur in this order. I expound that "Tzom ha'Asiri" is the fifth of Teves, on which news of the Churban reached Bavel - "Ba'Asiri ba'Chamishah la'Chodesh l'Galuseinu Vo Elai...Leimor Huksah ha'Ir."

They made the day of the tidings like the day it was burned. This is more reasonable, for (it is in order of the months and) the events occurred in this order.

Beraisa: On Tishrei 3 they abolished writing Hash-m's name in documents; they made the day a Yom Tov.

This occurred during Bayis Sheni, before Megilas Ta'anis was Batel. If it was Batel, we would not add a new Yom Tov!

Question: If so, it was already a Yom Tov, i.e. the day Gedalyah was killed (which is a Yom Tov at the time of Shalom)!

Answer: It was made a Yom Tov to forbid fasting the day before.

Question: It was already forbidden to fast the day before Gedalyah was killed (like it says in Megilas Ta'anis)!

Rishonim

Rambam (Hilchos Ta'anis 5:1,2): There are days on which all of Yisrael fast because of tragedies that occurred on them, to arouse the heart and stimulate Teshuvah. On Tishrei 3 Gedalyah was killed, and the remaining ember of Yisrael extinguished, causing their exile.

Taz (OC 549): The verse lists the fasts in order of the months. What was the source to expound that the death of Tzadikim is like the burning of the Beis ha'Mikdash? We must say that since his death was unrelated to the Churban, it is listed with fasts over the Churban to teach that it is as severe.

Note: According to the Rambam, his death is connected with the Churban. The Rambam must have another answer to the Taz' question.

Rambam (ibid.): On the tenth of Teves, Nebuchadnetzar ha'Rasha laid siege against Yerushalayim and caused great distress. Five things happened on the 17th of Tamuz. The Luchos were broken, Korban ha'Tamid ceased in Bayis Rishon, Yerushalayim was breached before the second Churban, Aposotomus burned the Torah and set up an idol in the Heichal.

Lechem Mishneh: The verse (which was after the first Churban and before the second) alludes to the ninth of Tamuz. We fast on the 17th, when Yerushalayim was breached before the second Churban, for Churban Sheni is more painful to us. We did not similarly change the tenth of Teves to the beginning of the siege before Churban Rishon, for it was not in the same month. Rashi's text says that an idol was set up in the Heichal, i.e. by Menasheh. Chachamim in the Yerushalmi argue like Rashi and the Rambam.

Rambam (3): Five things happened on Tish'ah b'Av. In the Midbar it was decreed that Benei Yisrael will not enter Eretz Yisrael, Bayis Rishon and Bayis Sheni were destroyed. The great city of Beitar was captured. It had myriads of people, and a great king; Yisrael and the great Chachamim believed that he was Moshi'ach. The Nochrim killed everyone; it was a tragedy like the burning of the Beis ha'Mikdash. On this day prone for punishment Turnusrupus of Edom plowed the Heichal and around it, fulfilling "Tziyon Sadeh Techaresh."

Poskim

Shulchan Aruch (OC 549:1): We are obligated to fast on Tish'ah b'Av, the 17th of Tamuz, the third of Tishrei, and the tenth of Teves because of tragedies which occurred on these days.

Mishnah Berurah (1): Recalling what happened to our ancestors due to their sins inspires us to repent. One is obligated to introspect his deeds on the fast. The fast is a mere preparation for Teshuvah. People who fast and take excursions are missing the point.

Gra (DH uv'Asarah): Regarding the fast of Teves we follow R. Akiva against his Talmid R. Shimon. We follow a Talmid against his Rebbi only after the time of Abaye and Rava.

Beis Yosef (DH Chosav): R. Yerucham says that Gedalyah was killed on Rosh Hashanah. The fast was postponed until a Yom Chol.

Kaf ha'Chayim (5,9): Some say that since Tzom Gedalyah is postponed, a Chasan or Ba'al Bris does not fast the entire day, just like a fast that fell on Shabbos and was postponed. Some do not consider it postponed (unless Tishrei 3 is Shabbos), for it was enacted to be on the third. However, Mateh Yehudah (or Mateh Yosef) proves from Rosh Hashanah (18b-19a) that he really died on the third. Some dispel the proof. Sha'ar ha'Kavanos says that he died on the third, and the Rambam connotes so, and some say that even if he died on Rosh Hashanah the fast is not considered postponed. Therefore, a Ba'al Bris should not be lenient.

Kaf ha'Chayim (11): One should be careful not to get angry on a fast. If one eats more than usual the night before, he defeats the purpose. A weak person may be lenient.

Shulchan Aruch (2): Even though it says "...b'Tish'ah la'Chodesh...; Va'Tiboka ha'Ir", we fast on the 17th. In Bayis Rishon the city was breached on the ninth, but in Bayis Sheni it was breached on the 17th, and Churban Sheni is more painful to us.

Beis Yosef (ibid., and DH u'Mah she'Chosav Af): In Rosh Hashanah it says that the fast of Tamuz is on the ninth. The Tur writes that it is on the 17th because Churban Sheni is more painful to us. Also, the Yerushalmi argues with the Bavli and says that in Bayis Rishon the city really was breached on the 17th. They erred in the calculation of the day of the month due to the great troubles; the verse was left according to their mistake. Even though the verse alludes to a fast on the ninth, after Churban Sheni when it was enacted to fast on the 17th, the fast of the ninth ceased. Chachamim did not want to impose too many fasts on the Tzibur. This fast was enacted for the breaching of the city, therefore the full severity applies even now that we observe it on the 17th.

Note: It is difficult to say that they erred by eight days. The moon was already full two or three days ago! Rav Moshe Shapiro, Shlita, explains that surely, some event occurred in Churban Rishon on the ninth which looked like the breaching of the city. Any news reporter would have said so. Normally, a Navi sees the true significance of events. But at that time Yirmeyahu lacked his usual Ru'ach ha'Kodesh. He saw no deeper than a reporter, and Hash-m allowed his 'mistake' to be written into Tanach.

Kaf ha'Chayim (12): It was fitting that we should fast on the ninth and the 17th, just Chachamim did not want to impose too many fasts on the Tzibur. Therefore, a Ba'al Nefesh should fast on both days. However, according to the Yerushalmi there is no reason to fast on the ninth. Bava Basra 60b connotes that a Ba'al Nefesh need not be stringent about something that the majority cannot endure (e.g. not getting married).

Kaf ha'Chayim (13): Bayis Sheni was consolation for Churban Rishon. We are still in exile from Churban Sheni, therefore it is more painful for us.

Note: The Selichah for Asarah b'Teves 'Ezkerah Matzok...' says that the Ta'anis was fixed for tragedies on three consecutive days. The Torah was translated into Yevanis on the eighth, and on the ninth of Teves there was anger and shame; Ezra died.

Iyun Tefilah (in Otzar ha'Tefilos on this Selichah): The death of Tzadikim is not a reason for shame! Megilas Ta'anis says that a fast was decreed on this day. If it was because this is when Nevu'ah ceased, it should have said so! Rather, it was for some other reason that Chazal did not want to reveal.

Note: I heard from an Adam Gadol that the shame was the birth of a certain Yisrael. January 1, 1996 was Teves 9. Our calendar is based on a 19 year cycle, so 1995 years (105 cycles) earlier Teves 9 was about the same time. The 10 days added to the Gregorian calendar in 5343 compensated for most of the 'extra' leap years until then (see Ta'anis 10 c:2:iii).