Rosh Hashanah 6b ~ How Long is a Year?

There is a special rule about selling a house in a walled city in the Land of Israel. The seller has the right to buy back his property for one year. If he fails to do so in that time it becomes the permanent property of the buyer. This is learned from a verse in Leviticus:

ויקרא 25:30

וְאִ֣ם לֹֽא־יִגָּאֵ֗ל עַד־מְלֹ֣את לוֹ֮ שָׁנָ֣ה תְמִימָה֒ וְ֠קָ֠ם הַבַּ֨יִת אֲשֶׁר־בָּעִ֜יר אֲשֶׁר־[ל֣וֹ] חֹמָ֗ה לַצְּמִיתֻ֛ת לַקֹּנֶ֥ה אֹת֖וֹ לְדֹרֹתָ֑יו לֹ֥א יֵצֵ֖א בַּיֹּבֵֽל׃

If it is not redeemed before a full year has elapsed, the house in the walled city shall pass to the purchaser beyond reclaim throughout the ages; it shall not be released in the jubilee.

The next question is, what is meant by “a full year”? This is the subject of a dispute on today’s page of Talmud:

ראש השנה ו, ב

דְּתַנְיָא: ״שָׁנָה תְּמִימָה״, מוֹנֶה שְׁלֹשׁ מֵאוֹת וְשִׁשִּׁים וַחֲמִשָּׁה יוֹם כְּמִנְיָן יְמוֹת הַחַמָּה, דִּבְרֵי רַבִּי. וַחֲכָמִים אוֹמְרִים: מוֹנֶה שְׁנֵים עָשָׂר חֹדֶשׁ מִיּוֹם לְיוֹם

It was taught in a baraita: With regard to redeeming houses in a walled city the Torah states: “And if it not be redeemed within the space of a full year” (Leviticus 25:30), which indicates that he counts 365 days, in accordance with the number of days in a solar year; this is the statement of Rabbi Yehuda HaNasi. And the Rabbis say: He counts twelve months from day to day.

According to the great Rebbi Yehudah HaNasi (135-217 C.E.) a full year usually means a full solar year of 365 days. A solar year is the time it takes the earth to complete one revolution all the way around the sun. (Or for those of you living in a geocentric universe, that is the time taken for the sun to complete one revolution around the earth). Just like Rebbi, we assume that a solar year is 365 days long, but in fact it is a little less than 365 1/4 days: that is to say, 365 days 5 hours, 48 minutes and 46 seconds. Those extra five and a bit hours are the reason we have a leap year every four years or so, when they add up to almost another 24 hours. And there are other kinds of “solar” years.

The Solar Year

The solar year is the average amount of time it takes the sun to return to the same position as seen from the earth. It is usually measured from one vernal equinox to the next. (The vernal equinox is the day on which the length of day and of the night are equal.) You can see this on the drawing below, taken from H. A. Rey’s wonderful book The Stars (and yes, it is the same H.A. Rey of Curious George fame.) In the drawing, a solar year is the time it takes the sun (as seen from the earth) to orbit from one spring equinox to the next, which is about 365 days 5 hours 48 minutes and 30ish seconds, or 365.242 days.

The solar year is also known as the tropical year and it is also the length of our calendar year. In today’s page of Talmud when Rebbi referred to the “מִנְיָן יְמוֹת הַחַמָּה” or “the the number of days in a solar year” this is what he meant.

The sidereal year

Just like a spinning top the earth wobbles as it orbits the sun.

But there is another way of measuring the year, and it is the amount of time it takes the sun to appear against the same background of fixed stars. It is known as the sidereal year, from the Latin sidus meaning star. And a sidereal year is 20 min 24.5 seconds longer than a tropical year. The sidereal year is longer because of the precession of the equinoxes, the term given to the phenomenon that in addition to orbiting the sun, the earth is wobbling like a spinning top. As the earth returns to the same position it had one full solar orbit ago, the tilt of the earth is not now directly toward the Sun: because of the effects of precession, it is a little way "beyond" this. The sun does not line up against the background of the stars as it did one year ago, and we need to wait an additional few minutes for sun and earth to line up as they did then.

The lunar year

The rabbis were of the opinion that “the year” in question is the period of twelve lunar months that make up a regular (non-leap) Jewish year. That would be 354 days, which is 11 days and some shorter than a solar or sidereal year.

There is no “correct” way to count a year. It is a matter of convention. We have fiscal years and academic years, draconic years and sothic years. Today’s page of Talmud reminds us that so long as we are clear about what we are saying, a year can be defined in a number of different and interesting ways.

[The same passage appears in Yoma 65b, where this post originally appeared.]

Rosh Hashanah 2 ~ A New Year for Health

The new tractate of Talmud that we begin to study today opens with these famous words:

ראש השנה ב ,א

ארבעה ראשי שנים הם באחד בניסן ר"ה למלכים ולרגלים באחד באלול ראש השנה למעשר בהמה ר' אלעזר ור"ש אומרים באחד בתשרי באחד בתשרי ראש השנה לשנים ולשמיטין וליובלות לנטיעה ולירקות באחד בשבט ראש השנה לאילן כדברי בית שמאי בית הלל אומרים בחמשה עשר בו

There are four days in the year that serve as the New Year, each for a different purpose: On the first of Nisan is the New Year for kings; it is from this date that the years of a king’s rule are counted. And the first of Nisan is also the New Year for the order of the Festivals, as it determines which is considered the first Festival of the year and which the last.

On the first of Elul is the New Year for animal tithes; all the animals born prior to that date belong to the previous tithe year and are tithed as a single unit, whereas those born after that date belong to the next tithe year. Rabbi Elazar and Rabbi Shimon say: The New Year for animal tithes is on the first of Tishrei.

On the first of Tishrei is the New Year for counting years, as will be explained in the Gemara; for calculating Sabbatical Years and Jubilee Years, i.e., from the first of Tishrei there is a biblical prohibition to work the land during these years; for planting, for determining the years of orla, the three-year period from when a tree has been planted during which time its fruit is forbidden; and for tithing vegetables, as vegetables picked prior to that date cannot be tithed together with vegetables picked after that date.

On the first of Shevat is the New Year for the tree; the fruit of a tree that was formed prior to that date belong to the previous tithe year and cannot be tithed together with fruit that was formed after that date; this ruling is in accordance with the statement of Beit Shammai. But Beit Hillel say: The New Year for trees is on the fifteenth of Shevat.

 Declaring different kinds of New Years goes back to the Talmud. But this practice was updated in a remarkable way by a Russian Jewish immigrant to the US in the early twentieth century. Today, Talmudology is proud to tell his remarkable - and overlooked - story.

“Twice a year on the fifteenth day of Shevat and on the eighteenth day of Iyar all the Jewish children from 3 to 13 years of age should undergo a thorough physical examination by the local Jewish physicians free of charge. ”

— Charles Spivak

Charles Spivak and the fight against tuberculosis

Hayyim Haykhl Spivakovski (1861-1927) immigrated to the US from Russia, where he became Charles Spivak. He graduated from Jefferson Medical College in Philadelphia in 1890 (and his thesis, on talmudic theories of menstruation won a prize), and after his wife contracted tuberculosis in 1896 he moved with her to Denver. There she could take advantage of the high altitude which had been shown to help fight the disease. This began his life-long mission to fight the tuberculosis and improve the care of the many Jewish refugees from eastern Europe who contracted it. 

From here.

Spivak founded the Jewish Consumptives’ Relief Society (JCRS), which provided kosher food and a Sabbath atmosphere, but was open to anyone. “We have in our institution chasidim and agnostics,” he wrote in 1914, “Jews and Christians, republicans and progressives, socialists and anarchists, men of all kinds of religious, political and economic options.” Spivak’s personal philosophy was informed by “a unique blend of Yiddishkeit [Jewish values], secularism and socialism” and his approach to the distribution of funds was sometimes at odds with bureaucratic and impersonal ways that some Jewish charities functioned. “We may not be able to return him [the patient] to his family as a useful working unit,” he reminded his benefactors, “we may actually waste money without any hope for any return, nevertheless, we feel that he or she must receive our care and attention, that whole-souled and whole-hearted charity is, after all, the only true, pure and unalloyed charity.” He estimated that of among the 3.3 million Jews then living in the US about 4,600 died each year from the disease, and ten times that number were chronically infected, or as he put it, were “living tuberculous Jews.” It was therefore the duty of the Jewish community to support the fight for to prevent the spread of tuberculosis and search for a cure. 

Dr. Chales Spivak. From here.

As we learn in today’s page of Talmud, we learn that are several different dates that mark the beginning of different new years. The first day of the Spring month of Nissan is the new year for kings, which is used to date legal documents. The new year for trees is marked in the late winter month Shevat, which is used to count tithes, and first day of the late summer month of Tishrei is used to count the number of years since creation. In December 1918, Spivak updated this list and gave it a thoroughly modern twist. Writing in the Journal Jewish Charities, he suggested that the rhythm of the Jewish calendar could be used to improve public health and reduce the toll from tuberculosis.

Twice a year on the fifteenth day of Shvat (New Years for Trees) and on the eighteenth day of Iyar (Lag B'Omar) all the Jewish children from 3 to 13 years of age should undergo a thorough physical examination by the local Jewish physicians free of charge.  

In the evening of the respective days all organized societies in the community should hold Health meetings at which the subject of how to maintain good health and prevent disease should be discussed by health officers and physicians.

 A custom should also be inaugurated that all adults should visit their family physicians during the months of Tishre and Nisson [sic] for the purpose of undergoing a physical examination.

 Spivak’s suggestion was of course dependent on a working knowledge of the Jewish calendar, but the dates he suggested would help. The fifteenth of Shevat was often celebrated in schools, and Lag B’Omer, the thirty-third day of the period leading up to the festival of Shavuot was celebrated as a minor holiday; it marked the end of the pandemic deaths of the students of the talmudic giant Rabbi Akiva. Most Jewish adults, even those who had jettisoned traditional Jewish practice when they arrived in America, would be aware of the timing of the other two months.  The festival of Pesach (Passover) is celebrated in Nissan, and Rosh Hashanah, the start of the Jewish New Year that leads into Yom Kippur (the Day of Atonement) is commemorated in Tishrei.

Helping others, even after his death

Spivak, a member of the Denver Hebrew Speaking Society, developed liver cancer and died in 1927 at the age of 68. His generous spirit is evident in his last will and testament, where he asked that

…my body be embalmed and shipped to the nearest medical college for an equal number of non-Jewish and Jewish students to carefully dissect. After my body has been dissected, the bones should be articulated by an expert and the skeleton shipped to the University of Jerusalem, with the request that the same be used for demonstration purposes in the department of anatomy.

Apparently his request was fulfilled, and somewhere on the campus of the Hebrew University of Jerusalem is his skeleton.

Denver’s National Jewish Hospital

Spivak was not the only Jew who helped Denver’s many “consumptives.” He had traveled to Denver because of its high altitude, and in there in the 1880s a woman by the name of Frances Wisebart Jacobs raised funds to open a new hospital to treat the many “consumptives” who had traveled to the mile high city. She found support from the Jewish community, which agreed to plan, fund and build a nonsectarian hospital for the treatment of respiratory diseases, primarily tuberculosis. That hospital opened in 1899 as The National Jewish Hospital for Consumptives, and after several name changes it is now known as National Jewish Health. Today, it remains a major center for the care of patients with lung and respiratory illnesses.

““…[Pain] knows no creed, so is this building the prototype of the grand idea of Judaism, which casts aside no stranger no matter of what race or blood. We consecrate this structure to humanity, to our suffering fellowman, regardless of creed.” ”

— Rabbi William Friedman at the laying of the cornerstone of the new hospital. From Tom Sherlock. Colorado's Healthcare Heritage: A Chronology of the Nineteenth and Twentieth Century Healthcare, volume 1, p374.

While the Talmud declared four kinds of new year, Spivak declared a fifth. His new year for health was tied to the Jewish calendar, and his memory is a reminder of the importance of getting a routine physical exam from your doctor. It might save your life.

Beitzah 32b ~ Lice

In this page of Talmud there is a pithy comment about three kinds of lives '“that are less than living.”

ביצה לב, ב

תָּנוּ רַבָּנַן: שְׁלֹשָׁה חַיֵּיהֶן אֵינָם חַיִּים, וְאֵלּוּ הֵן: הַמְצַפֶּה לְשֻׁלְחַן חֲבֵירוֹ, וּמִי שֶׁאִשְׁתּוֹ מוֹשֶׁלֶת עָלָיו, וּמִי שֶׁיִּסּוּרִין מוֹשְׁלִין בְּגוּפוֹ

The Sages taught: There are three whose lives are not lives, and they are as follows: One who looks to the table of others for his sustenance; and one whose wife rules over him; and one whose body is ruled by suffering.

Then the Talmud adds a fourth category. Lice.

וְיֵשׁ אוֹמְרִים: אַף מִי שֶׁאֵין לוֹ אֶלָּא חָלוּק אֶחָד. וְתַנָּא קַמָּא: אֶפְשָׁר דִּמְעַיֵּין בְּמָנֵיהּ

And some say: Even one who has only one robe [since he cannot wash it properly, he suffers from lice and dirt]. And why was this last category not included in the original list? Because it is possible for him to examine his clothes and remove the lice, which would alleviate his suffering.

Today on Talmudology we will focus on the last category: lice

Head lice, Body Lice and pubIC LICE

There are three kinds of lice which infect humans: the tiny head louse Pediculosis humanus capitas, the larger body louse Pediculus humanus humanus, and the crab or pubic louse, Pthirus pubis. There is a discussion of lice in the tractate Nazir, as an aside to a question about how our hair grows:

נזיר לט, א

איבעיא להו האי מזיא מלתחת רבי או מלעיל למאי?... ת"ש מהא אינבא חיה דקאים בעיקבא דבינתא ואי סלקא דעתך מלתחת רבי ברישא דבינתא בעי למיקם. לעולם מלתחת רבי ואגב חיותא נחית ואזיל אינבא

ת"ש אינבא מתה ברישא דבינתא ואי סלקא דעתך מלעיל רבי בעיקבא דבינתא בעי למיקם התם נמי משום דלית בה חילא שרוגי שריגא ואזיל...

A question was asked: Does hair grow from the roots or the tips?...Let us suggest an answer from the live nit [or louse - meaning is not certain] which is found at the root of a strand [of hair]. Now if the hair grew from the root, shouldn't the nit be found at the tip? [The Talmud rejects this suggestion:] The growth may well be from the tip, but the nit, being alive, continually moves down [towards the root].

Let us suggest an answer from the case of a dead nit [or louse, that is found] at the end of a strand [of hair].  If the hair grows from the end, shouldn't the dead nit be found near the root? [The Talmud rejects this suggestion too:]  Perhaps the dead nit has no power [to grasp the hair] and so as the hair grows from the root, the nit slides.

The louse nit (egg) with its adherent cylindrical sheath cemented to the hair shaft.  The free distal end (arrow) would be directed towards the hair tip.  The egg has a domed operculum (arrow) that contains air holes, allowing the maturing larvae to breath.From Burkhat el al.  The adherent cylindrical nit structure and its chemical denaturation in vitro. Arch. Pediatric Adolescent Medicine 1988. 152; 711.

Pediculosis Humanus Capitas

Pediculosis Humanus capitas is the long scientific name of the tiny head louse.  The female, less than 3mm long, lives for about a month, and in that time lays over three hundred eggs.  The eggs are laid on a shaft of hair close to the scalp, where, warmed by the skin of their itchy host, they incubate for two weeks before hatching.  The new lice emerge, grow for about 12 days, mate, and lay their eggs, and the cycle continues. Humans are the only known host of these lice, and somewhere in this cycle you as a parent may get a call to come and take your child out of school because they have been found to have head lice, or nits, the name given to their eggs.  About 15% of school age children in the UK have head lice, while in the US estimates range from 6-12 million infestations per year. In the US, the cost to treat those millions of infestations is more than $350 million.

True story: Many years ago while working in an emergency department in Boston, I received a call from the (warm and loving Jewish) preschool my children then attended.  My daughter had nits, and could not attend class. Despite my explaining that I could not leave my shift in the ED to come and get her for as trivial a reason as head lice, the school was adamant. She remained outside the classroom until arrangements to pick here up were made.  I do hope the psychological damage was minimal.

“No healthy child should be excluded from or allowed to miss school time because of head lice. “No nit” policies for return to school should be discouraged.”

— Barbara L. Frankowski, Leonard B. Weiner, Committee on School Health and Committee on Infectious Diseases. Head Lice. Pediatrics 2002: 110 (3); 642.

Fortunately, the American Academy of Pediatrics issued a report on head lice in 2002, and though it came too late for me, their advice supported my decision. Here's what they suggested:

Because a child with an active head lice infestation has likely had the infestation for a month or more by the time it is discovered, poses little risk to others, and does not have a resulting health problem, he or she should remain in class but be discouraged from close direct head contact with others. If a child is assessed as having head lice, confidentiality must be maintained so the child is not embarrassed. The child’s parent or guardian should be notified that day by telephone or a note sent home with the child at the end of the school day stating that prompt, proper treatment of this condition is in the best interest of the child and his or her classmates. 

Head Lice in Antiquity

Head lice have been with us for a long, long time, as evidenced by the Talmud's clear acquaintance with them.  Amazingly though, remains of a head louse have been identified on a louse comb from the Roman period that was discovered near the Dead Sea. (Even older remains have been found on the hair from Egyptian mummies, and nine-thousand year old lice eggs were found on human remains in Nahal Hemar near the Dead Sea.) "The comb was most probably used by inhabitants of the village of En Gedi, who were preparing a place of refuge in the cave, which would have been well equipped with food in baskets, storage jars and a large water pool before the end of the Bar Kokba Revolt in 135 CE." 

Wooden comb found in the Cave of the Pool, near Nahal David in the Dead Sea region. It was discovered in 1961.  From Mumcouglu and Hadas. Head Louse (Pediculus humanus capitis) Remains in a Louse Comb from the Roman Period Excavated in the Dead Sea Region. Israel Exploration Journal, Vol. 61, No. 2 (2011), pp. 223-229 

Head Lice More Recently

The House of Twenty Thousand Books, by Sasha Abramsky is wonderful read about the life, and library, of Chimen Abramsky (1916-2010) who was the son of the great Dayan Yechezkiel Abramsky, (1886-1976) head of the London Bet Din. Chimen (pronounced Shimon) who eventually became a professor of Jewish Studies at University College London, was an expert on Jewish books and built a significant collection of his own, which is detailed in the book. He also served as an advisor to Sotheby's and to Jack Lunzer, who built the greatest privately owned Jewish library in the world. Anyway, I came across this passage in the book, reminding us that presence of head lice was not just an annoyance - it was a way of life: 

Infant and childhood mortality soared in these years [of the First World War] in part because of the prevalence of diseases such as typhus - which  presumably explains why, in early photographs, the heads of Chimen and his brothers are shorn, to county the typhus-carrying lice.  Isaac Bashevis Singer, who was a few years older than Chimen, and like Chimen was brought up in a devout household...recalled having his sidelocks and head hair shaved off for this reason during the First World War...he wrote in his essay "The Book"..."I saw my red sidelocks fall and I knew this was the end of them. I wanted to get rid of them for a long time." (Sasha Abramsky. The House of Twenty Thousand Books. NYRB 2015. 57-58.)

For the Nazir, hair is a central part of his religious identity, and once that identity is no-longer needed, the hair is shaved off.  Which is exactly what Isaac Bashevis Singer felt too. 

The Body Louse and Typhus

Isaac Bashevis Singer (and his younger brother Moishe) caught lice during the occupation of Warsaw by the German army that began in late 1915. The lice certainly made life uncomfortable, as this page of Talmud describes. But the lice brought another more terrible and life-threatening condition. Typhus.

In his classic 1935 book Rats, Lice and History, Karl Zinsser wrote that “swords and lances, arrows, machine guns and even high explosives have had far less power over the fates of the nations than the typhus louse…”It is little wonder therefore that of the many diseases that were associated with Jews and Jewish immigrants to the New World, typhus was among the most feared.  

Typhus (from the Greek word typhos, meaning confused) is caused by a bacterium called Rickettsia prowazeki, but it cannot spread without the body louse. Once inside the louse it causes internal bleeding, and the lice takes on a reddish color as its blood leaks into its tissues. Eventually the carrier louse dies, and if it is on the skin, clothing or bedding of a person the bacterium passes through the skin and into the cells where it reproduces during the incubation period of about two weeks. It then causes a rash, fevers, headaches, confusion, hallucinations, and abdominal pain; it may spread to the lungs where it causes pneumonia. Before the antibiotic era, about 60% of all cases were fatal.

Typhus should not be confused with typhoid fever, which is caused by the Salmonella bacterium in contaminated food. In both conditions there is confusion, and hence the common root of their names. Modern outbreaks of typhoid fever have been prevented by the chlorination of drinking water, but the World Health Organization estimates that there are at least eleven million cases of typhoid fever worldwide, and about 130,000 deaths.

Typhus was described in eleventh-century Spain, sixteenth-century Italy and nineteenth-century France. In the catastrophic French invasion of Russia in 1812, only 3,000 soldiers of Napoleon’s original army of more than half a million men returned home. Most died of the cold and of typhus. Throughout its history, epidemic typhus was most associated with wars and large population upheavals. Some three million Russians died of typhus between 1917 and 1925. 

Because it was a disease of the those who were malnourished, and who lived in crowded conditions with poor sanitation, typhus was common among the poor Jews of eastern Europe. It is therefore not surprising that it featured some of the stories of Isaac Bashevis Singer (1903-1991), who grew up in the Polish village twenty-five miles north-west of Warsaw. The only child of Reb Mordecai Meir, the central character in his short story Grandather and Grandson, died of typhus. In a poorhouse in Poland we meet Mottke, a failed beadle who had reached America but was deported when he was found to have trachoma in his eye. Singer was not content with giving Mottke just one misfortune; Mottke’s father had died of typhus. And as we have noted, as a young boy Bashevis Singer had himself caught typhus, as had his brother Moishe. Still, Bashevis Singer managed to joke about the awful disease. In the short story Errors one of the characters cleverly notes that “an author doesn’t die of typhus but of typos.” Typhus was also found among the poor who had managed to emigrate, and who faced new living conditions that were not always much better than those they had left behind.  

The Jewish Fight against Typhus

In 1912 a group of Russian physicians and lawyers established the Obshchestvo Zdravookhraneniia Evreev (OSE), The Society for the Protection of the Health of the Jews. Its mission was to battle typhus, cholera and other epidemics among the Jews, and to improve their standards of sanitation. The effort quickly grew and by 1917 there were more than 45 OZE branches that operated in 102 cities in the former Russian Empire; later, offices opened in Berlin, Paris and London. They ran nineteen hospitals and ninety out-patient clinics, two sanatoria for patients with tuberculosis, and one-hundred and twenty-five nurseries that served over twelve-thousand children. A branch opened in Warsaw in 1921, called Towarzystwo Ochrony Zdrowia Ludności Żydowskiej w Polsce (TOZ)– The Society for the Protection of Jewish Health in Poland. The two organizations worked together. They produced a Yiddish health journal for the public called Folks-Gezunt, as well as a Polish-Yiddish scientific journal issued in Warsaw called Medycyna Społeczna (Social Medicine). Before the outbreak of World War II, the TOZ employed more than a thousand doctors, nurses, dentists and teachers.

Among the many public health campaigns that the OSE ran were a series of posters. One of these is especially poignant. It is a poster that uses the motif of the biblical plagues as a warning to prevent the body lice that transmit typhus. “Blood, Frogs, LICE [kinim]: the third plague is the worst. Stop the lice! Lice cause typhus.

Yiddish health poster warning against lice which carried typhus. By Joseph Tchaikov, printed by OSE London/Berlin 1923. Reproduced from Murderous Medicine by Naomi Baumslag, p6

The TOZ was shut down by the Nazis in 1942. Its property and assets were confiscated and most of the staff and patients were murdered. And of course under the deceit of “delousing,” the Nazis killed over a million Jews, along with Gypsies and other “undesirables” using Zyklon B, which had originally been developed as a pesticide and was used to fumigate and delouse clothing, trains and buildings. This page of Talmud is a reminder of the darkest period of Jewish history, when lice really did turn Jewish lives into lives that were not lived, חַיֵּיהֶן אֵינָם חַיִּים.

From the Talmudology Archives: ~ Gauss, Tosafot, and Counting the Sukkot Sacrifices

We are currently celebrating the seven-day festival (or eight outside Israel) of Sukkot (Tabernacles). When the Temple in Jerusalem stood, it was a time when a large number of animals were sacrificed. A very large number. For each of the seven days of the festival, in addition to two rams, fourteen lambs and one goat there were a number of bulls that were sacrificed. Thirteen on the first day, twelve on the second, eleven on the third, ten on the fourth, nine on the fifth, eight on the sixth and finally seven bulls on the last day.

במדבר 29:12-34

וּבַחֲמִשָּׁה עָשָׂר יוֹם לַחֹדֶשׁ הַשְּׁבִיעִי מִקְרָא־קֹדֶשׁ יִהְיֶה לָכֶם כָּל־מְלֶאכֶת עֲבֹדָה לֹא תַעֲשׂוּ וְחַגֹּתֶם חַג לַה׳ שִׁבְעַת יָמִים׃ 

וְהִקְרַבְתֶּם עֹלָה אִשֵּׁה רֵיחַ נִיחֹחַ לַה פָּרִים בְּנֵי־בָקָר שְׁלֹשָׁה עָשָׂר אֵילִם שְׁנָיִם כְּבָשִׂים בְּנֵי־שָׁנָה אַרְבָּעָה עָשָׂר תְּמִימִם יִהְיוּ…׃ 

וּבַיּוֹם הַשֵּׁנִי פָּרִים בְּנֵי־בָקָר שְׁנֵים עָשָׂר אֵילִם שְׁנָיִם כְּבָשִׂים בְּנֵי־שָׁנָה אַרְבָּעָה עָשָׂר תְּמִימִם׃… 

וּבַיּוֹם הַשְּׁלִישִׁי פָּרִים עַשְׁתֵּי־עָשָׂר אֵילִם שְׁנָיִם כְּבָשִׂים בְּנֵי־שָׁנָה אַרְבָּעָה עָשָׂר תְּמִימִם׃… 

and so on…

Numbers 29:34

Pablo Picasso: “Bull,” 1945

On the fifteenth day of the seventh month, you shall observe a sacred occasion: you shall not work at your occupations.—Seven days you shall observe a festival of the LORD.— 

You shall present a burnt offering, an offering by fire of pleasing odor to the LORD: Thirteen bulls of the herd, two rams, fourteen yearling lambs; they shall be without blemish. 

The meal offerings with them—of choice flour with oil mixed in—shall be: three-tenths of a measure for each of the thirteen bulls, two-tenths for each of the two rams, 

and one-tenth for each of the fourteen lambs. 

And there shall be one goat for a sin offering—in addition to the regular burnt offering, its meal offering and libation. 

Second day: Twelve bulls of the herd, two rams, fourteen yearling lambs, without blemish… 

Third day: Eleven bulls, two rams, fourteen yearling lambs, without blemish…

and so on… 

So how many bulls would the manager of the Temple inventory have to make ready for the entire festival of Sukkot? Well, you could just add them up, which is not too hard (13+12+11+10+9+8+7=70), but there is another way, which is found in the medieval commentary known as Tosafot, on page 106 of the tractate Menachot. Let’s take a look. 

Another offering, another math problem

We read there that the mincha offering is accompanied by a minimum of a one-issaron measure of flour. But a mincha can also be accompanied by a multiple of that number, up to a maximum of 60 issronot. What happens if a person vows to bring a specific number of issronot of flour to accompany a mincha offering but cannot recall how many he had in mind? What number of issronot of flour should he offer? Well it’s a bit tricky. The sages ruled that a single offering using the full sixty issronot of flour is all that needs to be brought. But the great editor of the Mishnah, Rabbi Yehudah Hanasi disagreed. In a spectacular way. Here is the discussion:

מנחות קו, א

תנו רבנן פירשתי מנחה וקבעתי בכלי אחד של עשרונים ואיני יודע מה פירשתי יביא מנחה של ששים עשרונים דברי חכמים רבי אומר יביא מנחות של עשרונים מאחד ועד ששים שהן אלף ושמונה מאות ושלשים

The Sages taught in a baraita: If someone says: I specified that I would bring a meal offering, and I declared that they must be brought in one vessel of tenths of an ephah, but I do not know what number of tenths I specified, he must bring one meal offering of sixty-tenths of an ephah. This is the statement of the Rabbis. Rabbi Yehuda HaNasi says: He must bring sixty meal offerings of tenths in sixty vessels, each containing an amount from one-tenth until sixty-tenths, which are in total 1,830 tenths of an ephah.

Since there is a doubt as to the true intentions of the vow, Rabbi Yehudah HaNasi covers all the bases and requires that every possible combination of a mincha offering be brought. So you start with one mincha offering accompanied with one issaron of flour, then you bring a second mincha offering accompanied with two issronot of flour, then you bring a third mincha together with three issronot, and so on until you reach the maximum number of issronot that can accompany the mincha - that is until you reach sixty. The total number of Rabbi Yehudah HaNasi’s mincha offerings is then calculated: 1,830.

How did the Talmud arrive at that number? We are not told, and presumably you simply add up the series of numbers 1+2+3+4….+59+60, which gives a total of 1,830. That certainly would work. But Tosafot offers a neat mathematical trick to figure out the sum of a mathematical sequence like this:

שהן אלף ושמונה מאות ושלשים. כיצד קח בידך מאחד ועד ששים וצרף תחילתן לסופן עד האמצע כגון אחד וששים הם ס"א שנים ונ"ט הם ס"א ושלש ונ"ח הם ס"א כן תמנה עד שלשים דשלשים ושלשים ואחד נמי הם ס"א ויעלה לך שלשים פעמים ס"א

How did we arrive at 1,830? Take the series from 1 to 60 and add the sum of the first to the last until you get to the middle. Like this: 1+60=61; 2+59=61; 3+58=61. Continue this sequence until you get to 30+31 which is also 61. You will have 30 sets of 61 (ie 1,830).

Tosafot continues with the math lesson, and lets us know the total number of bulls sacrificed over the seven day festival of Sukkot:

וכן נוכל למנות פרים דחג דעולין לשבעים כיצד ז' וי"ג הם עשרים וכן ח' וי"ב הם עשרים וכן ט' וי"א הם כ' וי' הרי שבעים

This method may also be used to count the number of sacrificial bulls on Sukkot, which are a total of 70. How so? [There are thirteen offered on the first day of sukkot, and one fewer bull is subtracted each day until the last day of sukkot, on which seven bulls are offered.] 13+7=20; 12+8=20; 11+9=20… [There are a total of 3 pairs of 20+ an unpairable 10]= 70.

In mathematical terms, the Tosafot formula for the sum (S) of the consecutive numbers in Rebbi’s series, where n is the number of terms in the series and P is the largest value, is S= n(P+1)/2. Which reminds us of…

Carl Friedrich Gauss

Carl Friedrich Gauss (1777-1855) was one of the world’s greatest mathematicians. He invented a way to calculate the date of Easter (which is a lot harder than you’d think), and made major contributions to the fields of number theory and probability theory. He gave us the Gaussian distribution (which you might know as the ”bell curve”) and used his skills as a mathematician to locate the dwarf planet Ceres. The British mathematician Henry John Smith wrote about him that other than Isaac Newton, “no mathematicians of any age or country have ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute rigorousness in demonstration, which the ancient Greeks themselves might have envied.”

There is a delightful (though possibly apocryphal) story about Gauss as a bored ten-year old sitting in the class of Herr Buttner, his mathematics teacher. There are at least 111 slightly different versions of the story, but here is one, as told by Tord Hall in his biography of Gauss:

When Gauss was about ten years old and was attending the arithmetic class, Buttner asked the following twister of his pupils. “Write down all the whole numbers from 1 to 100 and add their sum…The problem is not difficult for a person familiar with arithmetic progressions, but the boys were still at the beginner’s level, and Buttner certainly thought that he would be able to take it easy for a good while. But he thought wrong. In a few seconds, Gauss laid his slate on the table, and at the same time he said in his Braunschweig dialect: “Ligget se” (there it lies). While the other pupils added until their brows began to sweat, Gauss sat calm and still, undisturbed by Buttner’s scornful or suspicious glances.

How had the child prodigy solved the puzzle so quickly? He had added the first number (1) to the last number (100), the second number (2) to the second from last number (99) and so on. Just like Tosafot suggested. The sum of each pair was 101 and there were 50 pairs. And so Gauss wrote the answer on his slate board and handed it to Herr Buttner. It is 5,050.

THE Number of bulls=70

Gauss was raised as a Lutheran in the Protestant Church, and so he did not learn of this method from reading Tosafot. But it is delightful to learn that the same mathematical method that launched Gauss into his career as a mathematician predated him by at least four-hundred years and can be found on page 106a of Menachot, where it also applies to the festival of Sukkot.

Happy Sukkot from Talmudology