Commentary for Pirkei Avot 3:18

R. ELIEZER CHISMA. This is the correct text, not the one in the printed editions which reads “R. Eliezer son of Chisma”. For he himself was called Chisma, as in the passage in Vayikra Rabbah 23:5 which says that he did not know how to do prisa al shema,¹²³This refers to a public recitation of some part of the daily prayers—possibly the first bracha of birchot kriat shema, possibly also kaddish and barchu. For the range of opinions, see the medieval commentaries on Megillah 23b. and the people dishonored him and said “this is what you call a Rabbi?” He came before R. Akiva and learnt how to do it, at which point the people said “R. Eliezer has grown strong! [Heb. it-chasam]” and called him R. Eliezer Chisma. [*The mishna in Terumot 5:3 also has R. Eliezer Chisma.]

Rabbi Eliezer ben Chisma says: [the laws of] Kinin (bird offerings): The offering of doves by women who give birth.

[That which is cited by Rabbi Bartenura] are the words of Beit Shammai. And according to Beit Hillel, there are only thirty-five [immersions required]. And according to the [actual] law, there is only one immersion required at all. And Rabbi Bartenura is only coming to emphasize the matter here [and not to tell us the law].

"Kinin:" Bird offerings are called Kinin, which is a term related to kan tsippor (bird's nest). And it has strict laws, for example, if obligatory and voluntary offerings get mixed; or an olah which is performed higher [on the altar] and a chatat, which is performed lower.

Introduction Rabbi Eliezer Hisma was a student of Rabban Gamaliel and of Rabbi Joshua. Legend has it that he was a great mathematician.

THE LAWS OF BIRD-OFFERINGS AND THE BEGINNINGS OF MENSTRUAL PERIODS. The laws of bird-offerings are treated in a separate tractate, and the beginnings of menstrual periods are the subject of the mishna in the beginning of the second chapter of Arachin.
[*Rav writes that in some cases the woman requires 95 immersions. This is the opinion of Beit Shammai in a baraita quoted in Niddah 29b, but the opinion of Beit Hillel is that she needs only 35. There might be a scribal error in Rav’s commentary, which should read “35” instead of “95”. But it is possible that Rav simply intends to point out how complicated this can get, for which reason he quotes the opinion of Beit Shammai; rather than issuing a halachic ruling he is merely emphasizing the potential difficulty, even though the halacha does not follow that opinion. The conclusion of Tosafot there (30a, s.v. sh’ma minah), moreover, is that the halacha follows R. Yossi ben Yehuda who says that one immersion at the end suffices. If so, the halacha does not even follow the opinion of Beit Hillel. It is all the more reasonable, then, to suggest that Rav meant only to emphasize the difficulty, for even had he quoted the opinion of Beit Hillel it would not have been the final halacha.]

and the beginnings of Niddah (menstruation): They are three [anatomical] places - the 'house,' the 'attic' and the 'entrance hall'; and also the many sightings of blood that exist with menstruant bloods - these are the body of the laws. And it is like they said (Bava Metzia 84b), "They brought sixty sightings of blood in front of him, and he determined all of them to be pure (non-menstrual) blood."

"these are, etc.": He is not coming to exclude other laws in the Torah, as behold he only excludes "astronomical calculations, etc." But rather he mentioned these because they are similar to astronomical calculations, etc., since they are [also] calculations. And even though these are calculations that come about because of a doubt that arose in them, as opposed to astronomical calculations, etc., [in which] there is no doubt about their calculations, nevertheless, they are preferable - Derekh Chaim. And see Tosafot Yom Tov.

"and the beginnings of Niddah (menstruation)": The laws of a menstruant who has lost her cycle and needs to guard [for the possibility of bleeding] until she goes back to the beginning [of a new cycle]. And sometimes, she is required to immerse ninety-five [times], according to the [opinion] that holds immersion at its proper time is a commandment.

Rabbi Eliezer Hisma said: the laws of mixed bird offerings and the key to the calculations of menstruation days these, these are the body of the halakhah. The calculation of the equinoxes and gematria are the desserts of wisdom. Rabbi Eliezer Hisma states that there are two different types of laws that are “the body of the halakhah”. This means that they are essential halakhot, paradigmatic of the Oral Torah. The “laws of mixed bird offerings” refers to cases where one type of bird offering accidentally became mixed up with a different type. For instance an obligatory offering became mixed up (perhaps in a net or in a dovecote) with a voluntary offering; or a sin offering became mixed up with a whole burnt offering. There is a whole tractate entitled Kinnim that deals with these situations. The second type of halakhah are the calculations of menstruation days. This refers to calculating how many days there are between a woman’s cycle. It is important to figure this out so that women can distinguish between menstrual bleeding and other types of bleeding. Much of tractate Niddah is devoted to this subject. There are two types of laws that are only “desserts of wisdom”. That is they help learning, but they are not essential. One is astronomy, that is the calculation of the seasons. The other is gematria, which refers to the adding up of numbers (obviously this is from the same word as geometry). According to the Meiri, Rabbi Eliezer Hisma wants to guide a person in his course of study. The first thing a person should learn is Torah. The two areas of Torah learning that he mentions are representative of some of the most difficult subjects in the learning of Torah. A person should first learn Torah so well, that he is able to master even tractates Kinnim and Niddah. Afterwards, when he has completely mastered even in these subjects, a person should turn his attention to science and mathematics, represented here by astronomy and geometry.

THEY ARE THE MAIN OF THE LAW. The mishna does not mean to exclude other areas of law, only astronomy and geometry. It picks these areas to contrast them to astronomy and geometry, which also involve calculations [cf. below]—Maharal in Derech Chaim. He also writes that the mishna chooses these areas of law to make an additional point: even though these laws involve calculations that arise from cases of uncertainty, whereas astronomy and geometry involve no uncertainty, the laws are still more valuable.
Kesef Mishne writes on Hilchot Yesodei HaTorah 4:13 that although bird-offerings seem to be a thing of little value and the beginnings of menstrual periods revolve around an unseemly subject, they are the main of the law on account of the great reward one receives for their study. Astronomy and geometry, while their subject is an elevated one [see the explanation of gimatriyot below], are not the most important things but are like side dishes for wisdom. I will reproduce the end of his comment below.

Astronomical calculations and Gematria [numerical calculations]: As they contain many calculations, and the wisdom of mathematics sharpens a man.

"these (literally, these, these)": It means to say, these and these, and as it is explained in the gemara at the end of the first chapter of Chagigah, about "these, these are the body of the Torah." (See there in [the commentary] of Rabbi Bartenura.) And since Rabbi Eliezer Chisma was after the destruction, etc., and the laws of Kinin are only applicable, etc.; therefore he needed to make us understand that even they are the body of the law - even at the time when the Temple does not exist, since their study is in place of the [Temple] service. And that which it did not learn, "body of the Torah," here [is because] here he is speaking about doubts, and there is nothing about that in the written Torah, but rather in the oral halakha.

"these are the body of the laws (halakhot)": The essence of the Oral Law, for which one receives reward.

THEY ARE THE MAIN OF THE LAW [Heb. hen hen gufei halachot]. The mishna at the end of the first chapter of Chagigah uses similar language: hen hen gufei torah, which the Talmud (Chagigah 11b) explains as “these [hen] and these [vehen] are gufei torah”, as Rav writes on the mishna there; the meaning in our mishna is the same. R. Eliezer Chisma was a student of R. Akiva and lived after the destruction of the Temple, as Rambam writes in the fourth of the ten chapters in his Introduction to the Mishna, and the laws of bird-offerings have no practical application when the Temple is not standing. He therefore found it necessary to let us know that these are the main of the law even wen the Temple is not standing, because the study of these laws is a substitute for the sacrificial service, as I write on the mishna in 1:2.

And on account of their circling the world in a circle (hekef), their movement is called tekufah, etc. And his words are only about the knowledge of the movements, and not what follows from them, which is the sanctification of the month, and the intercalation of the years; as concerning that matter, certainly these and these are the body of the laws, etc.

"Astronomical calculations": The matter of the movement of constellations.

THE MAIN OF THE LAW. The reason that the mishna does not say “the main of the Torah” as it does at the end of the first chapter of Chagigah is that these laws deal with cases of doubt, and there is no mention of such things in the written Torah, only in the oral law that the Sages taught.

As in, "go down (redu) to there," which Yaakov said, is a hint to the 214 (the numerical equivalent of the letters in the word, redu) years that Israel stayed in Egypt. And so [too], "With this (Be-zot) Aharon should come," is a hint to the first Temple, which would stand for 410 (the numerical equivalent of the letters in the word, be-zot) years. And so [too] in Arukh: in the place of [the number,] one, we write [the letter], alef, and in the place of [the number,] one hundred, we write [the letter], kof. And in Derekh Chaim, he explained [that is is] the wisdom of measures and mathematics. And it is easy to accept this explanation, since Rabbi Eliezer Chisma knew how to approximate how many drops of water there are in the seas (as is found in Horayot 10a), etc. And therefore it was fit that he should say this, as is exemplified by their statement, may their memory be blessed, in the [Midrash] Rabbah at the beginning of Ecclesiastes.

"and Gematria [numerical calculations]": The calculation of the [numerical value of the] letters.

ASTRONOMY [Heb. tekufot]. Rav: the movement of the constellations. So also Rashi. Because they travel around [Heb. makifim] the world in a circular motion [Heb. hekef] their movement is called tekufah.
A variant text of Rashi’s commentary has: the dimension of the sun’s orbit and when it switches quarters. This is what the word tekufah means in the language of those who calculate the `evron,<sup>124</sup>Lit. “passing”. See notes below. which is the point at which the sun passes from one quarter of its sphere to another,<sup>125</sup>In classical and medieval astronomy, the earth occupies the center of the universe. Surrounding it are several celestial spheres, containing different planets and stars, which revolve around the earth from east to west. The sun, by contrast, has its own celestial sphere which moves from west to east. In addition, while the other celestial spheres complete their orbits around the earth daily, the sun moves incrementally and only completes its orbit every 365 days. The position of the sun in its sphere is divided into four quarters, their starting points corresponding to the solstices and equinoxes, which rabbinic literature calls tekufot. The act of intercalation is called `ibur, a word of unclear etymology. Numerous medieval commentators unterstand it as deriving from the word me`uberet, “pregnant” in the Mishna and Talmud. R. Avraham bar Chiyya, the 11th-century philosopher, astronomer, and mathematician, offers an additional etymology based on `avar, the Biblical and Rabbinic Hebrew “passed”, explaining that the Sages take a month that has already passed and make it pass by again (Sefer Ha`ibur, ed. Filipowski, Ma’amar Sheni, Sha`ar Shelishi). Tosafot Yom Tov here offers a different etymology, making the verb le`aber a denominative deriving from the noun `evron, “a passing”, referring to the point at which the sun passes from one quarter of the sphere into the next. Tosafot Yom Tov seems to be conjecturing that the term was current in the Mishnaic and Talmudic periods, as it does not actually appear in Talmudic or subsequent astronomical literature. I have not made an exhaustive survey, but it appears that this etymology is original to Tosafot Yom Tov. as Rav writes on the mishna in Sahnedrin 1:2 (s.v. bishlosha), that the court must intercalate [Heb. me`abrin] on account of the tekufah.<sup>126</sup>This connects `ibur to the tekufah. Although leap years are now added based on a fixed formula and there is no longer a need for this, in the times of R. Eliezer Chisma they were still calculating the tekufot and detemining when to make a leap year based on their calculations, for this is what they did up until the days of Abaye and Rava, as Rambam writes in Hilchot Kiddush HaChodesh 5:3.
If we accept this explanation of the word tekufot, we must say that R. Eliezer Chisma’s dictum aplies only to the knowledge of the sun’s motion itself and not the legal ramifications thereof in the sanctification of the month and the making of a leap year; these are certainly both the main of the law. R. Eliezer Chisma was addressing himself only to those people that study these things simply to know them and be well-rounded scholars; to them R. Eliezer Chisma says these are merely side dishes.

AND GIMATRIOT. Rav: alphanumeric calculations. Such as the midrash that the word Jacob said to his sons when he told them “go down (Heb. r’du, רדו) there” (Geneis 42:2) is a hint to the 210 years that Israel would spend in Egypt (Bereshit Rabbah 91:2). Or that the verse “with this (Heb. bezot, בזאת) shall Aharon enter the Sanctuary” (Leviticus 15:3) hints to the fact that the first Temple would stand 410 years. Rashi similarly explains gimatriot as alphanumeric calculations and the reading of a single word as an acronym. Aruch likewise writes: gimatriot—writing א instead of 1, writing ק instead of 100.
It seems to me that this is a loan-word from Greek, by which the Greeks call the science of measurement and arithmetic. Our Sages borrowed the word for all matters of calculation and number. So also the anonymous commentary on Hilchot Kiddush HaChodesh 18:13. I also saw that Maharal in Derech Chaim explains the word in our mishna as referring to the science of measurement and arithmetic.
If so, these are truly words that befit the one who said them: in the Talmud, Horayot 10a, R. Yehoshua testifies that R. Eliezer Chisma was able to estimate how many drops of water there are in the sea. This is through the discipline of engineering, for which measurement and arithmetic are prerequisites. It is because he had mastered all of these disciplines that it was proper for him to say this, as it would be improper for some other person who had not mastered these disciplines to speak this way, as the Sages say in Kohellet Rabbah 3:11. Therefore, this explanation of gimatriot as referring to measurement and arithmetic seems most likely. Cf. what I quote below from Maggid Mishne.
Now this discipline must chronologically precede astronomy, for without it a person dare not raise a finger in discussing the paths of the constellations and the hosts of the heavens. Even so, because of the importance of astronomy and its subject, the mishna places it before the chronologically earlier gimatriot.

And it appears to me that it can be explained that condiments are things that come before the meal to give appetite. And that is [because] one who is in an expert in the tekufot (astronomical calculations) - which means to say the ordering itself, which is the movements of the constellations, etc. - and Gematria - which is the wisdom of [sizing] (so explained Maggid [Mishneh]) - which precedes it;" these are the condiments that [will] give [him] appetite to the wisdom of Divinity. As from the knowledge of the spheres of the heavens, he will know and recognize the Creator, may He be blessed, as He is the Rider of [the] heavens; since from the ridden he will know the Rider. And so [too] according to the explanation of Rashi and Rabbi Bartenura that Gematria are the calculations of the letters, [it is that they are] "condiments to the wisdom," of Divinity that is in Sefer Yetzirah and those similar to it, that greatly use calculations of the letters and acronyms and [other letter] combinations. And it is possible to say that the teacher intended both of them, which are: to the wisdom that is attained by analysis and research - and that is through astrological calculations; and to the wisdom of Divinity, which is passed on to us (Kabbalah). And see Tosafot Yom Tov.

"condiments to wisdom": Just like the condiments, that [people] are accustomed to eat at the end of the meal for dessert in the manner of a treat, honor their owners in the eyes of the creatures; so [too] do these wisdoms.

PARPRA’OT FOR WISDOM [Heb. parpra’ot lechochma]. Rav: like the parpra'ot people eat at the end of a meal for pleasure for dessert, these disciplines give a person honor in peoples’ eyes. If so, the Hebrew lechochma, which usually means “for wisdom”, means “of wisdom”¹²⁷Honor comes from knowledge of these disciplines, and is therefore “of wisdom”. The simple reading would have the word lechochma mean “for wisdom”, implying that knowledge of these peripheral disciplines is somehow beneficial to the main body of wisdom which is Torah. in our mishna. Rashi: because they are only parpra'ot, i.e. matters of wisdom.
The end of the Kesef Mishne I quoted above reads: “for wisdom” means the wisdom of the Talmud which alone is worthy of being called “wisdom” without qualifiers, because it deals with G-d’s commandments and because of the great reward given to those who study it and keep its laws. These disciplines, therefore, are merely parpra'ot for it, for they are not the main thing the way it is. My difficulty with this is that the first section of the mishna should then have said “these are the main of wisdom”.¹²⁸If the “wisdom” referred to at the end of the mishna is the wisdom of the Talmud, the mishna’s contrast should have been between the main of “wisdom” and the side dishes thereof; there would be no reason to call it first law and then wisdom.
I found the Maggid Mishne on Hilchot G’nevah 8:1 giving the following explanation. Tekufot is arithmetic [Heb. cheshbon] per the the Talmud’s language “to be choshev the tekufot” (Shabbat 75a), gimatri’ot is the art of measurement, and “parpra'ot for wisdom” means for the discipline of astronomy, which is called chochma and binah in that same passage in Shabbat. This explanation also encounters difficulties, because tekufot certainly means the path of the stars and their orbit, not arithmetic. His proof from “to be choshev the tekufot” actually proves the exact opposite, that tekufah is not cheshbon!¹²⁹For if it were, the verb and its object would be redundant, along the lines of “calculating calculations”. Also, how could the mishna call these parpra'ot when in fact they come before the meal? He must’ve understood parpra'ot to mean foods that come before the meal to create an appetite, as Rav explains the word in Berachot 6:5.
Now that we’ve mentioned that possibility, I say that tekufot refers to astronomy itself—the paths of the constellations and the whole host of the heavens. It, along with the geometry which precedes it, are appetizers that draw a person to that wisdom beyond which no greater wisdom lies for man: theology. For from a knowledge of the spheres of the heavens he will come to know the Creator, who is the one who “rides the heavens” (Deuteronomy 33:26)—from knowing what is ridden one comes to know the rider. Even according to the explanations of Rashi and Rav that gimatriot refers to alphanumerical calculations, we can say that “for wisdom” means “for theology,” because Sefer Yetzirah and similar works make extensive use of alphanumeric values, words read as acronyms, and permutations of the letters of a word. Based on this, we can also say that the tanna perhaps intended both things at the same time: tekufot are appetizers for human wisdom that can be attained through reasoning and investigation, and gimatriot are appetizers for our received theology.