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In the Mishnah (Rosh ha-Shanah 1.1) we read: “The first of Shvat is the New Year for Trees, according to the House of Shammai; the House of Hillel say the fifteenth.” This difference of opinion is not “one of the leniencies of the House of Shammai” (Mishnah Eduyot chs. 4, 5), but neither it is a question of a lenient ruling by the House of Hillel. What underlies their disagreement in this case? Rashi commented on the text at hand: “This is in regard to tithes,”  i.e. the New Year for Trees marks the beginning of a new year as far as tithes (ma’asrot) were concerned, and on this day a new year also begins with respect to computing the number of years for orlah (whether a tree is too young for its fruit to be permissible to eat).
The Talmud (Rosh ha-Shanah 14a) comments on this mishnah: “What is the reason? Rabbi Eleazar quoted Rabbi Oshaiah as saying:… ‘Even though most of the season is without [or outside], nevertheless most of the rains of that year have already fallen’.” Rabbi Eleazar, quoting Rabbi Oshaiah, was trying to establish the basis for setting the date for the New Year for Trees, but it is unclear whether the reason he gives explains the approach of the House of Shammai or the approach of the House of Hillel, or perhaps both.  In order to understand Rabbi Eleazar in the name of Rabbi Oshaiah, we must understand what is meant by the expression “most of the season is without.”
The Jewish calendar is comprised of lunar months and solar years;  the length of the month is determined by the time it takes for the moon to orbit around the earth, whereas the length of the year is determined by the time it takes for the sun “to go around the earth”.  The average month is 29.5 days long,  and therefore some Jewish months are 29 days long (deficient months) and some are 30 days long (full months). The average year is 365 days long; regular years have 12 lunar months, leap years have 13 lunar months. Simple arithmetic shows that a regular year has about 354 days and a leap year around 383 days. 
The solar year is divided into four equal parts,  called seasons (Heb. tekufot ), each one beginning with yom ha-tekufah , the designated day of the season: the season of Tevet begins at the peak of winter, when the day is shortest and the night longest; the season of Nisan begins in the spring, when day and night are of equal length; the season of Tammuz begins at the peak of summer, when the day is longest and the night shortest; and the season of Tishre begins in the fall, when day and night are of equal length. 
By the end of the period of the Sages, a fixed calendar had been established containing clear and fixed principles regarding deficient and full months, regular and leap years. Leap years were inserted into a cycle of nineteen years, during which there were seven leap years and twelve regular years.  We are also familiar with the general (Gregorian) calendar in use in the world today, which is a fixed solar calendar. In that system, the beginning of winter (the period of Tevet ) always falls on December 21 or 22,  while the beginning of spring (the period of Nisan) falls on March 21 or 22.
Returning to the meaning of R Oshaiah’s words: The Talmud notes that on the New Year for Trees “most of the season is still without,” meaning that on this day the half-way point of the winter season (Tevet ) has not yet been reached. According to the solar calendar, the middle of the season falls on February 4 or 5; thus the New Year for Trees must come before February 4. Here a difficulty arises: examining the Jewish calendar we note that in the nineteen-year cycle of leap years there are seven years in which Tu b-Shvat does not fall when “most of the season is without.”  However, if one follows the system of the school of Shammai, who said that the New Year for Trees is on the first of Shvat , then the principle that “most of the season is without” is always upheld.  One could argue that the explanation that “most of the season is without” pertains only to the approach advocated by the school of Shammai,  nevertheless we should try to understand R. Oshaiah’s reason also according to the approach of the school of Hillel. 
The problem we have raised becomes more severe if we examine the calendar as it was managed in the time of the Sages of the Mishnah. In their day the calendar had not yet been fixed; the new month was declared on the basis of witnesses who sighted the new moon, so that actually the number of days in a regular year fluctuated between 352 and 356,  and in a leap year between 381 and 386.  The solar year, according to the Sages’ knowledge of astronomy,  was 365.25 days long,  and therefore each of the four seasons lasted exactly 91.3125 days.  Leap years were proclaimed at the discretion of a special court established for the purpose. Calculating when it would be a regular year and when a leap year was a complex matter, referred to as “the secret of proclaiming leap years,” and the Rabbis considered that knowledge to be “proof of your wisdom and discernment to other peoples” (Deut. 4:6). 
The most important factor in determining whether to proclaim a leap year is the commandment, “observe the month of Abib,” which was a practical matter: the Sages who proclaimed leap years had to make sure that the (solar) season of Nisan, the Spring, always fell before the 16th day of the (lunar) month of Nisan.  In the course of the month of Adar , at the very latest, the Sages who proclaimed the leap years had to calculate when the season of Nisan would begin, and if they foresaw that it would fall on the 16th of Nisan or later, they had to declare that the next month would be Adar II . Precise calculation shows that the season of Nisan can begin between the 19th of Adar and the 15th of Nisan in a regular year, or between the 16th of Adar II and the 29th of Adar II in a leap year.
For the fifteenth of Shvat to fall before the halfway-point of the season of Tevet , to fulfill R. Oshaiah’s reason that “most of the season is without,” the season of Nisan must always begin after the second of Nisan ,  and this, of course is an impossible requirement. On the other hand, for the first of Shvat to fall before the middle of the season of Tevet, following the approach of the school of Shammai, the season of Nisan must begin after the eighteenth of Adar . As we have said, the nineteenth of Adar is precisely the earliest possible date for the new season, according to the rules of determining leap years as well. If so, it appears simply that the talmudic remark, “most of the season is without,” pertains exclusively to the the school of Shammai.
The name Shvat , however, may refer to something else. Generally it means the fifth lunar month (counting from Tishre ), but we find that it is also used as the name for the thirty days that fall in the middle of the season. For example (Rosh Ha-Shanah 15a): “Rabbi Yohanan asked Rabbi Yannai: When is the New Year for citron trees? He said, Shvat . Shvat of the months or Shvat of the season? He said, of the months.”  Rashi comments: “‘Shvat of the season’ means in the solar calendar; after thirty days of the season of Tevet comes the beginning of the solar Shvat.”
In other words, the term Shvat can refer to Shvat of the [lunar] months or to Shvat of the season. The first thirty days of the season of Tevet (Winter) are called Tevet and the next thirty days are called Shvat . Tu b-Shvat (or the fifteenth of Shvat) of the season is the forty-fifth day of the season of Tevet , after which come the remaining forty-six days of the season. If we interpret according to the school of Hillel, that the New Year for Trees is on the fifteenth of “Shvat of the season,” then according to their approach as well “most of the (winter) season is without.” 
Thus one might say that the essence of the dispute between the school of Hillel and the school of Shammai was, ‘Which calendar should be used to determine the New Year for Trees?’ The school of Shammai was of the opinion that one should use the generally accepted lunar calendar, whereas the school of Hillel advocated the solar calendar.  According to the lunar calendar, following the school of Shammai, the New Year for Trees was the first of Shvat , so that “most of the season is without.” In contrast, according to the solar calendar, following the school of Hillel, the New Year for Trees was the fifteenth of Shvat . In actuality, halakhic practice followed both schools: Tu B-Shvat is celebrated on the fifteenth of Shvat like the school of Hillel, but reckoned by the lunar calendar according to the system of the school of Shammai.
Dr. Yoel Shiloh
 The following articles which have appeared in the Bar- Ilan Weekly Parasha Page deal with the Hebrew calendar in general and with calculating the seasons in particular: Y. Loewinger, “This month is the month of Abib”; S. Feigelstock, “The Seasons of the Year”; Y. Loewinger, “Can the founding of the Hebrew calendar be dated by astronomical clarification?”; R. Sar-Shalom, “Establishment of the Calendar by Hillel and its Finalization in the Period of the Geonim”; A. Mirsky, “The Structure of the Hebrew Calendar”; Y. Adler, “On Setting the Years and their Cyclic Nature” (Studies 167, 209, 248, 274, 322, 431).
 Thus Rashi wrote in Rosh Hashana 14a: “One should not give terumah from fruits of trees that began forming their fruits before that date, on behalf of the fruit of trees that began forming after that date.
 Tosefot (s.v. le-tekufot) is of the opinion that the question pertains to both views. The remainder of our paper deals with this issue.
 Maimonides, Kiddush Ha-Hodesh 1.1.
 Of course it is now known that the earth orbits around the sun, but we used the expression that was generally accepted at that time.
 Rosh Ha-Shanah 25a: “Rabban Gamaliel said to them: Thus I have received it from my father’s house: The moon is not renewed in less than twenty-nine and a half days and two thirds of an hour and seventy-three parts of an hour.” The Sages had a mnemonic device for the length of the month – kaf-tet yod-bet taf-shin-tzadeh-gimel, in other words twenty-nine days, another twelve hours, and another seven-hundred ninety-three parts of an hour. A “part of an hour” [heleq, halaqim] refers to the ancient practice of dividing the hour into 1080 equal parts of time, where one “part” equals approximately 3 seconds. It follows that according to the scheme of the Sages, the length of the average month is 29 days, 12 hours, forty-four minutes and three seconds. See Maimonides, Kiddush Ha-Hodesh 6.3. Maimonides, loc. sit. 6.2, explains that the hour was divided into 1080 parts because this number is divisible with no remainder by all the numbers up to ten (excluding 7). See E. Raviv, “Ta’am le-Zayin-Tet Taf-Resh-Mem-Bet,” Sinai 128 (2002), p. 91.
 Arakhin 9b: “Between one Shavuot and the next and between one New Year and the next there are only four days, and if it is a leap year, five.” In other words, if one divides the days of the year into seven (into full weeks), then the remainder is four in a regular year or five in a leap year.
 All the explanations below follow the scheme of Shemuel Yarhinaah, who attested (Berakhot 58b): “The paths of the Heavens are as clear to me as the paths of Nehardea.” At a later period the more complex scheme of Rabbi Adda Bar Ahavah was adopted, but the schools of Hillel and Shammai surely did not know of this scheme, nor is it likely that Rabbi Oshaiah knew of it.
 Astronomists call these days the winter solstice, vernal equinox, summer solstice and autumnal equinox, respectively.
 For example, last year (5764) was the fifth year of the 304th cycle. On the history of setting the calendar, see Z. H. Yafe, Korot Heshbon ha-Ibbur, 1931; H. Y. Bornstein, “Divrei Yemei ha-Ibbur ha-Aharonim,” Ha-Tekufah vol. 14-15, p. 339.
 In the past, the Julian calendar was used, in which the season of Tevet began on December 7 or 8.
 These are the years following a leap year: 1st, 4th, 7th, 9th, 12th, 15th, 18th.
 The ninth year in the cycle is the latest one, and the first of Shvat on such years comes out on January 30 or 31.
 See S. Goren, “Keviat Tu-b’Shvat ke-Rosh ha-Shanah la-Ilan,” Torat ha-Moadim, Jerusalem 1996, p. 234.
 This is the view of Tosefot in note 3 above.
 Mishnah Arakhin 2.2: “There can be no fewer than four full months in a year nor more than eight.”
 Arakhin 9b: “Once Rabbi proclaimed nine deficient months … it was in a leap year.” For an elegant explanation of these calculations see R. Sar-Shalom, “Mispar ha-Yamim be-Luah ha-Shanah,” Shema’tin 147(1999), p. 54.
 See note 8.
 Three hundred and sixty-five days and six hours. Modern astronomical measurements have established that the solar year is 365.2422 days long.
 Maimonides, Kiddush Ha-Hodesh 9.2.
 Shabbat 75a.
 Rosh Ha-Shanah 21a: “Rav Huna bar Avin sent to Rabba: Seeing that the period of Tevet has continued until the twelfth of Nisan, make it a leap year and have no worry. For it is written (Deut. 16): “Observe the month of Abib,” make sure that the season of Spring be in the month of Nisan.” [I.e. that the month of Nisan fall in the springtime.]
 If both Shvat and Adar are full months.
 See Tosefot (s.v. o shvat de-tekufah), who view the question as pertaining not only to citron trees, but to the New Year for Trees in general. According to their approach, however, the question returns, because the Talmudic discussion concludes that it is Shvat of the months; i.e., that the school of Hillel in the Mishnah should not be interpreted as we suggest. The matter requires further study.
 A similar interpretation is given by Ritba on this text.
 The Sages used the solar calendar for other needs, as well, such as: blessing the sun (Berakhot 59b); praying for rain in the Diaspora (Ta’anit 10a; see Resp. Iggerot Moshe, Orah Hayyim 5.7, which uses the secular date); age with respect to a sacrifice (Rosh Ha-Shanah 6b); the laws of the Nazirite (Nazir 8a); mixing the sacred incense (Keritut 6a), and others.