כדי שתהא בהמה מצויה לעולי רגלים

The reason is so that animals will be readily available to those ascending to Jerusalem for the pilgrimage Festivals. The animals are tithed shortly before each of the Festivals so that their owners will refrain from selling or killing them until those times, when they gather them together to set aside the animal tithe. As a result, there are plenty of non-sacred animals ready to be sold, for food or offerings, to those on their way to Jerusalem.

ואע"ג דתנן עד שלא הגיע הגורן מותר למכור ולשחוט ניחא ליה דליתעביד מצוה בממוניה והדר לזבון וליכול

And even though we learned in the mishna: Until the time designated for gathering arrives, it is permitted to sell and slaughter the animals, which means that there is no obligation to keep the animals until those dates and there is no concern that the owners will sell the animals before tithing them. This is because a person prefers to perform a mitzva with his money, by separating the animal tithe, and only afterward sell or eat those animals.

ואמאי קרי ליה גורן דטבלה כגורן

The Gemara asks: And why does the tanna call the time designated for gathering animal tithe goren, which literally means threshing floor? The Gemara explains that it is due to the fact that the arrival of this time renders the halakhic status of new animals like that of untithed produce gathered on a threshing floor, as from the time that produce is gathered on the threshing floor one becomes obligated to tithe it. Likewise, from this stage onward it is prohibited to slaughter the new animals until they have been tithed.

וכמה פרס פירש ר' יוסי בר' יהודה אין פרס פחות מט"ו יום

The mishna states that according to Rabbi Akiva the times for tithing animals are adjacent to Passover, Shavuot, and Sukkot. The Gemara asks: And how much is adjacent [peras]? Rabbi Yosei, son of Rabbi Yehuda, explained that the term adjacent means not less than fifteen days.

מאי משמע א"ר אבהו פרוס פלגא פלגא דמאי פלגא דהלכות פסח כי הא דתניא שואלין ודורשין בהלכות פסח קודם לפסח ל' יום רשב"ג אומר שתי שבתות:

The Gemara asks: From where is this inferred? Rabbi Abbahu said that the term peros means half. The Gemara inquires: Half of what? The Gemara answers: Half of the time during which one must learn the halakhot of Passover, and this is like that which is taught in a baraita: One asks about and teaches the halakhot of Passover for thirty days before Passover. Rabban Shimon ben Gamliel says: One begins studying those halakhot two weeks before the Festival.

בן עזאי אומר בכ"ט באדר בא' בסיון: במאי קמיפלגי ר"ע ובן עזאי ר"ע סבר אדר הסמוך לניסן זימנין מלא וזימנין חסר זימנין דמיקלע ביום ל' וזימנין דמיקלע בכ"ט ומשום הכי לא קביע ליה זמן ובן עזאי סבר אדר הסמוך לניסן לעולם חסר הילכך קביע ליה זמן בכ"ט באדר

§ The mishna states that ben Azzai says the times are on the twenty-ninth of Adar, on the first of Sivan, and on the twenty-ninth of Av. The Gemara asks: With regard to what principle do Rabbi Akiva, who says fifteen days before Passover, and ben Azzai, who refers to the twenty-ninth of Adar, disagree? The Gemara answers: Rabbi Akiva holds that the month of Adar that immediately precedes Nisan is sometimes a full month of thirty days and sometimes a defective month of only twenty-nine days. Therefore, sometimes the fifteenth day before Passover occurs on the thirtieth of Adar and sometimes it occurs on the twenty-ninth of Adar. Due to that reason Rabbi Akiva does not fix a time by date, but simply states fifteen days before Passover. And ben Azzai holds that the Adar that immediately precedes Nisan is always defective. Therefore, he fixes a time in the calendar that the animals must be tithed, which is on the twenty-ninth of Adar.

באחד בסיון איידי דלא נפישי אי אמרת ליקדים ליעשר עד דמטי רגל שלמי להו:

With regard to the fact that ben Azzai does not require fifteen days before Shavuot but maintains that the animals must be tithed on the first of Sivan, the reason is that since there are not so many animals born between the previous time of tithing and this one, if you say that the owner should tithe earlier, then before the pilgrimage Festival arrives all the animals will already be gone, their sales having been completed. If so, there will be no animals remaining for those coming to Jerusalem for Shavuot.

בכ"ט באב וכו': בן עזאי לטעמיה דאמר האלולין מתעשרין בפני עצמן

The mishna teaches that ben Azzai says the last gathering time is on the twenty-ninth of Av, rather than fifteen days before Sukkot. The Gemara notes that ben Azzai conforms to his standard line of reasoning, as he said: The animals born in Elul are tithed by themselves, because it is uncertain whether the new year for animal tithes begins on the first of Tishrei or the first of Elul. Consequently, he sets the time for animal tithe before the beginning of Elul to avoid the possibility of tithing animals born in one year with animals born in the next year.

וליעשרינהו ביום ל' זימנין דמחסרין ליה לאב ובעינן למיעבד הכירא בחדש וישן:

The Gemara challenges: But if so, let him require that one tithe the animals on the thirtieth day of Av, which is the last day before Elul begins. The Gemara explains: Sometimes the month of Av is defective, i.e., it has only twenty-nine days, and therefore the thirtieth day after the start of Av is the first day of Elul. And we are required to implement a distinction between the new animals born in Elul and the old animals born in Av, which is considered to be the previous year.

ר' אלעזר ור"ש אומרים באחד בניסן בא' בסיון וכו': בא' בניסן כרשב"ג דאמר ב' שבתות בא' בסיון כדאמרן

§ The mishna teaches that Rabbi Elazar and Rabbi Shimon say that the times are on the first of Nisan, on the first of Sivan, and on the twenty-ninth of Elul. The Gemara explains with regard to the first date: On the first of Nisan, this is in accordance with the opinion of Rabban Shimon ben Gamliel, who says one begins studying and teaching the halakhot of Passover two weeks before the Festival. And Rabbi Elazar and Rabbi Shimon hold that the second time for animal tithe is on the first of Sivan, and this is as we said with regard to the opinion of ben Azzai, so that there will be enough animals born since the previous time.

בכ"ט באלול רבי אלעזר ור' שמעון לטעמייהו דאמרי בא' בתשרי ר"ה למעשר בהמה:

Finally, they hold that the third time for animal tithe is on the twenty-ninth of Elul, since Rabbi Elazar and Rabbi Shimon conform to their standard line of reasoning, as they say: On the first of Tishrei is the new year for animal tithe. All animals born prior to that date belong to the previous year for tithe and are tithed as a single unit, whereas those born after that date belong to the next year for tithe, and therefore there is a distinction drawn between the new animals and the old.

ולמה אמרו בכ"ט באלול ולא אמרו באחד בתשרי מפני שהוא י"ט וכו': ותיפוק ליה דבעינן למעבד הכירא בחדש וישן

§ The mishna teaches: And why did Rabbi Elazar and Rabbi Shimon say the twenty-ninth of Elul and why did they not say the first of Tishrei? It is due to the fact that the first of Tishrei is the festival of Rosh HaShana and one cannot tithe on a Festival; consequently, they brought it earlier, to the twenty-ninth of Elul. The Gemara raises a difficulty: But let the mishna derive it from the fact that we require a distinction between the new animals born in Tishrei and the old animals born in Elul, which is considered to be the previous year.

חדא ועוד קאמר חדא דבעינן למיעבד הכירא בחדש וישן ועוד מפני שהוא י"ט ואי אפשר לעשר בי"ט משום סקרתא:

The Gemara answers: The mishna states one reason and adds another. One reason is that we require a distinction between the new animals and the old animals. And another reason is because it is the festival of Rosh HaShana and one cannot tithe on a Festival, due to the sakreta, a lead-based red paint for marking every tenth animal, as explained on 58b. It is prohibited to apply paint on a Festival.

ר"מ אומר באחד באלול ר"ה למעשר בהמה בן עזאי אומר וכו': תניא אמר בן עזאי הואיל והללו אומרים כך והללו אומרים כך האלולין מתעשרין לעצמן

§ The mishna teaches that Rabbi Meir says: The beginning of the new year for animal tithe is on the first of Elul. Ben Azzai says: The animals born in Elul are tithed by themselves, due to the uncertainty as to whether the halakha is in accordance with the opinion of Rabbi Meir or in accordance with the opinion of Rabbi Elazar and Rabbi Shimon. The Gemara notes that it is taught in a baraita that ben Azzai said: Since these Rabbis, Rabbi Elazar and Rabbi Shimon, say this, that the first of Tishrei is the beginning of the new year for animal tithe, and those, i.e., Rabbi Meir, say that the first of Elul is the beginning of the new year for animal tithe, the animals born in Elul are tithed by themselves.

וליחזי טעמיה דמאן מסתבר וכי תימא לא מצי קאי אטעמייהו והתניא בן עזאי אומר כל חכמי ישראל דומין עלי כקליפת השום חוץ מן הקרח הזה

The Gemara raises a difficulty: But let ben Azzai see whose opinion is more reasonable, and decide accordingly. And if you would say that it is not possible for him to establish their reasoning, that is difficult: But it is taught in a baraita that ben Azzai says: All the Sages of Israel appear to me to be as unimpressive in their intellect as garlic peel, except for this bald one, i.e., Rabbi Akiva.

א"ר יוחנן מפי השמועה אמרוה מפי חגי זכריה ומלאכי

Rabbi Yoḥanan says that ben Azzai was unable to establish their reasoning in this particular case, as they said their opinions by tradition, from a chain of transmission stretching back to Haggai, Zechariah, and Malachi, the last of the prophets. Each Sage claimed his tradition concerning their ruling was the correct one. Since this dispute is based on tradition, it cannot be resolved by logical reasoning.

תניא באיזה צד אמר בן עזאי האלולים מתעשרין לעצמן נולדו לו ה' באב וה' באלול אין מצטרפין ה' באלול וה' בתשרי אין מצטרפין ה' בתשרי וה' באב מצטרפין

It is taught in a baraita: In what manner did ben Azzai say that animals born in Elul are tithed by themselves? For example, if five animals were born in Av and five were born in Elul, they do not combine to make ten animals that require tithing. Likewise, if five were born in Elul and five in Tishrei, they do not combine. But if five were born in Tishrei and five were born in the following Av, they do combine.

פשיטא מהו דתימא כי היכי דמפסקי שנים מפסקי גרנות קמ"ל

The Gemara asks: This is obvious; what is the novelty of the ruling of the baraita? The Gemara answers: This ruling is necessary lest you say that just as years separate between animals that may be tithed together, so too, the designated times for gathering and tithing the animals separate between those animals born before and after each one of them. If so, animals born in Tishrei and those born in the following Av would not combine. Therefore, the baraita teaches us that these designated times do not prevent animals born before and after them from being tithed together.

כדתנן חמשה לפני ראש השנה וחמשה לאחר ראש השנה אין מצטרפין חמשה לפני הגורן וחמשה לאחר הגורן הרי אלו מצטרפין

This is as we learned in the mishna: If five were born before Rosh HaShana and five after Rosh HaShana, those animals do not combine to be tithed; if five were born before the time designated for gathering and five after the time designated for gathering, those animals combine to be tithed.

אמר רבא לדברי בן עזאי נולדו לו חמשה באב וחמשה באלול וחמשה בתשרי כונסן לדיר להתעשר

Rava says: According to the statement of ben Azzai, if five animals were born in Av and five animals were born in Elul and five animals were born in Tishrei, the owner gathers them all in a pen to be tithed,