TOSFOS DH Kan b'Alin Kan b'Imahos Kan bi'Ymos ha'Chamah...

תוספות ד"ה כאן בעלין כאן באמהות כאן בימות החמה...

(SUMMARY: Tosfos explains unlike Rashi.)

פ''ה דעלין קשין ואמהות יפין אמהות שרשין בימות החמה יפין שמצננין הגוף בימות הגשמים קשין


Explanation #1 (Rashi): The leaves [of radishes] are bad and the Imahos are good. Imahos are the roots. They are good in summer, for they cool the body. They are bad in winter.

משמע לפירושו דעלין לעולם קשין ואפי' אמהות אינן יפין אלא בימות החמה


Inference: The leaves are always bad, and even the roots are good only in summer.

ותימה דבפרק קמא דע''ז (דף יא.) אמר גבי אנטונינוס ורבי שלא פסקו מעל שלחנם לא צנון ולא חזרת לא בימות החמה ולא בימות הגשמים


Question: In Avodah Zarah (11a), it says about Antoninus and Rebbi that radishes and lettuce did not cease from their tables, not in summer and not in winter!

ונראה לפרש דכאן בימות החמה וכאן בימות הגשמים כלומר עלין ואמהות חד מעלי בימות החמה וחד מעלי בימות הגשמים


Explanation #2: This refers to summer, and this refers to winter. I.e. leaves and roots - one is good only in summer, and one is good in winter.

ואית ספרים דגרסי ואיבעית אימא הא והא בעלין אבל באמהות לעולם מעלו:


Alternative text: Some texts say "alternatively, both of these refer to the leaves, but the roots are always good.


TOSFOS DH v'Ein Bein Tekufah li'Tekufah Ela Tzadik Aleph Yom...

תוספות ד"ה ואין בין תקופה לתקופה אלא צ''א יום...

(SUMMARY: Tosfos asks that if so, the Tekufos are not equal.)

וקשה לר''י דאם יש לעולם בין ביום קצר ובין ביום ארוך י''ב שעות ביום וי''ב שעות בלילה אם כן אין התקופות שוות ולעולם האחת ארוכה מהאחרת


Question (Ri): If always, whether a short [winter] day or a long [summer] day, there are 12 hours during the day and 12 hours at night, if so the Tekufos are never equal. One is always longer than the other!

שאם תפול תקופת ניסן בתחלת ליל ד' תפול תקופת תמוז בז' שעות ומחצה בלילה ותקופת תשרי בג' שעות ביום הרי במקום ז' שעות ומחצה הקטנות שבליל תמוז שלא נפלה תקופת תמוז בתחלת לילה אנו משלימין לה צ''א יום וז' שעות ומחצה בלילי תשרי שהן בינוניות


If Tekufas Nisan is at the beginning of Tuesday night, Tekufas Tamuz is at seven and a half hours of the night, and Tekufas Tishrei is at 3 hours of the day. In place of seven and a half small hours of a Tamuz night, for Tekufas Tamuz did not fall at the beginning of the night, we complete for it (until the next to Tekufah) 91 days and seven and a half hours of a Tishrei night, which is average.


Note: Really, there are four and a half hours of a Tishrei night, and 3 hours of the day. Presumably, Tosfos calls it seven and a half hours of the night, for the days and nights are equal then. Why did Tosfos say "Tuesday night"? He does not mention the day of the week in the rest of this Dibur! Perhaps he hints why only Tekufas Nisan can fall at the very beginning of the night. Tosfos (Rosh Hashanah 8a DH li'Tekufos) says that our Sugya is like R. Yehoshua, who says that the world was created in Nisan; the sun was put in place at the beginning of day four, i.e. the start of the night.

וגם בז' ומחצה שבסוף תקופת תשרי העודפות על צ''א היו קצרות ובתקופת טבת יהיו (הגהת הב"ח) בינוניות


Also the seven and a half hours at the end of Tekufas Tishrei above the 91 [days] were short hours, and those [at the end of] Tekufas Tamuz will be average.

ואם נאמר שהשעות לעולם שוות וביום ארוך יש (כן נראה להגיה) ובלילה ארוך י''ח שעות ובלילה קצר ו' או להפך


Suggestion: The hours are always equal (60 minutes), and a long day is 18 hours and a short night is six hours, or vice-versa.

לא משכחת הא דאמרת אין תקופת טבת נופלת אלא כו' אין תקופת תמוז נופלת אלא כו' (כן הוא בדפוס ויניציה)


Rebuttal #1: [If so,] we do not find what [Shmuel] taught "Tekufas Teves falls only... Tekufas Tamuz falls only...


Note: In the text of Defus Vinitziyah, Shmuel taught Tekufas Tamuz before Tekufas Teves (like in our text), but Tosfos asks first from Tekufas Teves. It is his primary question. How can the Tekufah fall at 10 and a half hours of the day? There are less than 10 and a half hours from sunrise to sunset [even in Eretz Yisrael, and all the more so at higher latitudes, where there is more variation of the length of days]! We can find 10 and a half hours in a day only if we count from dawn to Tzeis ha'Kochavim. This is difficult, for then 1) An average day is more than 12 hours; 2) If we say that the interval between dawn and sunrise is the time to walk four Mil, and likewise from sunset until Tzeis (Zman R. Tam. There are opinions that this is 72, 90 or 96 minutes), in Eretz Yisrael every day exceeds 12 hours! 3) Then, the nights in Tamuz are less than seven and a half hours (according to R. Tam) even in Eretz Yisrael. If Tosfos asked based on what he observed (i.e. in France or Germany), we can say that both are difficult (there are not 10 and a half hours in a day in Teves, and not seven and a half in a night in Tamuz), if Tosfos was further north than Paris, or if four Mil is the interval in Nisan, but it is less in Teves (like Bi'ur Halachah 261:2 DH she'Hu). It seems that Maharshal added the bracketed words in our text, that Tosfos asks from all the Tekufos. I cannot fathom what is difficult about six hours in Nisan (PF).

וגם לא משכחת הא דאמרינן בפרק מפנין בסופו (שבת דף קכט:) דבתלתא בשבתא ובמעלי שבתא קאי מאדים בזוגי ובשאר יומי לא קאי ולפי זה משכחת לה נמי דבשאר יומי קאי ובהאי לא קאי:


Rebuttal #2: We say in Shabbos (129b) that on Tuesday and Erev Shabbos, Ma'adim (Mars) is [dominant] at an even hour [of the day], but not on other days. According to this [suggestion], it [can] dominate also on other days, and sometimes it is not dominant on these days.


Note: Every hour, one of the seven Kochvei Lechet (the five closest planets, the sun and the moon) dominates (has a supernatural influence on earth), according to a fixed rotation that repeats itself every seven hours. Really, Ma'adim dominates at an even hour every day except for Sunday! The Gemara there discusses the best days to let blood. There, the Ritva explains why we need not be concerned for even hours of dominance on Monday, Wednesday and Thursday. Our Tosfos asks that if the number of daytime hours varies, sometimes Ma'adim dominates at an even hour on Sunday, and sometimes it is not dominant at an even hour on Tuesday. This rebuttal does not apply according to R. Chananel, who explained that we discuss days on which Ma'adim dominates twice, whether this is during the day or at night.



TOSFOS DH Palga Havi

תוספות ד"ה פלגא הוי

(SUMMARY: Tosfos explains why he did not include the corners in his question.)

פירוש מגרש דתחומין הוי פלגא דתחומין


Explanation: The Migrash is half the Techum.

ולא רצה לדקדק ולהקשות דמגרש דתחומין והקרנות כך וכך הוי מתחומין וקרנות


Implied question: [The Makshan did not know that the city is 2000 Amos. Based on what he thought,] why wasn't he meticulous to ask that the Migrash is such and such fraction of the Techum and the corners?

משום דפעמים הוי תלתא או רביע או חומש מן התחומין ומן הקרנות לפי מה שהעיר גדולה או קטנה


Answer: Sometimes it is a third, or a quarter, or a fifth of the Techum and the corners, based on whether the city is big or small. (He asked that it is half, for without the corners, it is always half the Techum.)


TOSFOS DH Mishkachas Lah b'Masa

תוספות ד"ה משכחת לה במתא

(SUMMARY: Tosfos discusses why we need to say that the city is round.)

הקשה השר מקוצי דמשכחת לה מגרש רביע דתחומין וקרנות במתא דהוי ח' אלפים על ח' אלפים ומגרש הוי ל''ו ותחומין וקרנות ומתא הוו קמ''ד פעמים אלף על אלף


Question (ha'Sar mi'Kutzi): [Why did the Tartzan need to say that the city and Migrash are circles? Even according to the Havah Amina that they are square,] we find that the Migrash is a quarter of the [city,] Techum and corners in a city that is 8,000 Amos by 8000 Amos. The Migrash is 36,000 [by 1000 Amos. With the city, it is a square of side 10,000 Amos (whose area is 100,000 by 1000 Amos), and 64,000 of it is the city itself]. The city, Techum and corners are 144,000 by 1000 Amos (a square of side 12,000 Amos)!

וי''ל דערי מקלט אינן גדולים כל כך כדאמר בפרק אלו הן הגולין (מכות דף י.) ערים הללו אין עושין אותם לא גדולות ולא קטנות


Answer #1: The refuge cities (which we discuss) were not so big, like it says in Makos (10a) "we do not make these cities large, and not small."

ומהר''י אומר שמתרץ האמת דהשתא קמ''ל דמגרש אינו מרובע כדמסיק


Answer #2 (Mahari): [The Tartzan] answers the truth. Now, he teaches that the Migrash is not square, like he concludes;

ולרבינא קמ''ל דאין מגרש לקרנות ולרב אשי דאין מגרש אלא לקרנות


According to Ravina (57a, who says that the Migrash is square, the Tana) teaches that there is no Migrash in the corners. According to Rav Ashi, [he teaches that] there is Migrash is only in the corners.

אבל בכה''ג לא היה משמיענו שום חידוש


However, in such a case (the city is a square of 8,000 Amos, like ha'Sar mi'Kutzi's question, the Tana) would not teach any Chidush.


TOSFOS DH Kamah Meruba Yeser Al ha'Igul Revi'a

תוספות ד"ה כמה מרובע יתר על העגול רביע

(SUMMARY: Tosfos proves that this is both regarding perimeter and area.)

ואור''י דהא רביע בין בהיקף בין בגוף הקרקע כדאמר הכא נמצא מגרש רביע


Explanation (Ri): [A square] is a quarter bigger [than the circle inside it] both regarding perimeter, and also regarding area, like it says here "it turns out that the Migrash is a quarter" (of the area).

וכן משמע גבי ים שעשה שלמה (לעיל דף יד:)


Support #1: It connotes like this regarding the Yam Shel Shlomo (above, 14b. We deducted a quarter of the area, for it was round, and not square.)

ועל ההיקף יש להוכיח בהדיא כדפ''ה דכל שברחבו טפח יש בהיקפו ג' טפחים ובטפח מרובע יש בהיקפו ד' טפחים


Support #2: Regarding the perimeter, we can explicitly prove [that it is a quarter more] like Rashi explained, since for every Tefach of width [of a circle], there are three Tefachim of circumference, and for every Tefach in [the side of] a square, there are four Tefachim of perimeter;

ועל הקרקע שבפנים יש להוכיח נמי שאם תקיף טפח עגול סביב בחוטין דקין עד הנקודה האמצעי' שבעגול ותחתוך מן הנקודה ולמטה כל החוטין לב' ותפשטם יהיה העליון ארוך ג' טפחים והאחרים מתקצרים והולכים עד הנקודה שהוא חצי טפח מן החוט העליון


Also regarding the area inside, we can prove this. If you surround (i.e. comprise) a circle of a Tefach with thin strips until the point in the center of the circle, and cut from that point downwards (in the picture in Tosfos, "down" is to the left) all the strips into two and spread them out straight, the top strip is three Tefachim long, and the others get shorter until the point, which is a half-Tefach [from the top];

תחתוך שוב כל החוטין לשנים ושים הארוך בצד הקצר נמצא הרבוע אורכו טפח וחצי ורחבו חצי טפח דהיינו שלש חתיכות של חצי טפח על חצי טפח ובטפח מרובע יש ד' חלקים של חצי טפח על חצי טפח


Cut again all the strips into two (in their middle), and put the long [halves] opposite the short (combine the two triangles to make a rectangle, like in the picture in Tosfos), and it turns out that the rectangle is a Tefach and a half long (the longest strip was three, and it was cut in half) and half a Tefach wide, i.e. three pieces (perhaps the text should say "Chalakim" (parts), like below) of a half-Tefach by a half-Tefach, and in a square Tefach there are four parts (quadrants) of a half-Tefach by a half-Tefach.


TOSFOS DH Abaye Amar Mishkachas

תוספות ד"ה אביי אמר משכחת

(SUMMARY: Tosfos resolves the calculation with the conclusion.)

ואם תאמר לפי מה דמוקי לה לקמן במתא עגולה קשה דתחומין הוו טפי כיון (הגהת בארות המים) שאנו מרבעין העיר ולא הוי מגרש רביע מהם


Question: According to how we establish it below (57a) to discuss a round city, this is difficult. The Techumim are more, since we square the city, and the Migrash is less than a quarter of them! (Maharsha - the Migrash (with the city in the middle) is a circle of diameter 3000. Its area is three quarters of such a square, i.e. 6750. (All areas must be multiplied by 1000 to get the number of square Amos.) The city's area is 750, so the Migrash itself is 6000. The Techum is a square of diameter 5000. Its area is 25,000, or 24,250 after we deduct the area of the city. This is slightly more than three times the Migrash!)

ויש לומר דלא קאמר מגרש רביע אלא מתחומין שבחוץ לריבוע העיר:


Answer: [R. Eliezer b'Ribi Yosi] said that the Migrash is a quarter only of the Techumim outside the square of the city. (We deduct not just the area of the city, rather, the area of the square around the city (1000). The Techum outside this is 24,000. Hagahos ha'Gra calls this difficult (there is no reason to do so). Rather, "quarter" is not precise.)