Balak (5765)

From the book Mida Tova: Articles on the Hermeneutical Principles by Rabbi Michael Avraham. Translated from Hebrew using gpt-5.4 (reasoning_effort=high, batch API).

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With God’s help. Mida Tova — Eve of the holy Sabbath, Parashat Balak, 5766

Questions

1. What is the definition of notrikon (an interpretive expansion of a word into other words) and gematria (numerical interpretation of letters)?
2. Why is Atbash not a notrikon but a gematria?
3. What are the two forms of notrikon interpretation and the two forms of gematria?
4. Why does the baraita of the thirty-two hermeneutical principles bring three examples of notrikon?
5. A passage in Tosafot on Babylonian Talmud, Hullin 28a is extremely difficult.
6. Is there another way to assign numerical values to letters besides gematria?
7. Does the solution of “with the total” indicate that gematria is an allusion rather than a derash?

The principles

Notrikon. Gematria.

“Then the angel of the Lord said to him: Why have you struck your donkey these three times? Behold, I came out as an adversary, because the way has been contrary to me.”

— Numbers 22:32

If one wrote a single letter as a notrikon, Rabbi Yehoshua ben Beteira deems him liable, and the Sages exempt him. Rabbi Yohanan said in the name of Rabbi Yosei ben Zimra: From where in the Torah do we know the language of notrikon? As it is said: “For I have made you the father of a multitude of nations.” I made you a father to the nations, a chosen one among the nations, beloved among the nations, a king among the nations, a veteran among the nations, faithful among the nations. Rabbi Yohanan himself said: Anochi is a notrikon: “I Myself wrote it and gave it.” The Rabbis said: “A pleasant utterance was written and given.” Some say that Anochi, read backward, means: “It was given in writing; its sayings are trustworthy.” The school of Rabbi Natan said: “Because the way was contrary to me” — yarat means: it feared, it saw, it turned aside. The school of Rabbi Ishmael taught: karmel means “a full kernel.” Rav Aha bar Yaakov said: “He cursed me with a grievous curse” — nimretzet is a notrikon: he is an adulterer, he is a Moabite, he is a murderer, he is an adversary, he is an abomination. Rav Nahman bar Yitzhak said: “What shall we say, and how shall we justify ourselves?” — nitsdadak means: we are upright, we are righteous, we are pure, we are crushed, we are holy.

— Babylonian Talmud, Shabbat 105a

A. Summary of Last Year’s Article

In last year’s article we dealt with notrikon. One of the sources cited in the Talmud for notrikon comes from our weekly Torah portion, from the verse “because the way was contrary to me.” On its face, this is not an ordinary hermeneutical principle, even though it appears in the baraita of the thirty-two principles, as principle 30. It seems more like a method that belongs to the realm of allusion than to the realm of derash.

The Talmudic passage above deals with the question of the law governing someone who wrote a single letter that functions as a notrikon on the Sabbath: is he liable as one who wrote two letters, or not? In practical halakha (Jewish law), we follow the Sages, who exempt him. In the same discussion, on Babylonian Talmud, Shabbat 104b, a law is cited that if someone wrote a single letter that completes a scroll, he is liable by Torah law, since in such a case even one letter has significance. We asked: what is the difference between these two cases?

We examined the sources that the Talmud cites there for the principle of notrikon, and saw that most of them are applications of this hermeneutical principle, not its source. The first source cited in the Talmud is Abraham as a notrikon for “father of a multitude of nations.” It is commonly understood as the initials of the words “father of a multitude of nations,” and Rashi indeed explains it this way ad loc., on Genesis 17:5. According to this interpretation, we are not performing a notrikon exegesis on a word in the Torah; rather, the Torah itself is performing the notrikon exegesis — that is, it writes both the word and its full explanation.¹ This reading suggests that the verse is a source for notrikon as a hermeneutical principle.

However, this interpretation is difficult. First, the letter resh is superfluous. Beyond that, the word “nations” does not appear in the notrikon. If so, this is only a rough approximation to a notrikon, at most. Moreover, the Talmud itself proves that this cannot be what Rashi meant. The Talmud gives a list of words that spell out the notrikon, and these are not the words “father of a multitude of nations”: father, chosen, beloved, king, veteran, faithful. These words spell the Hebrew phrase meaning “father of a multitude,” and all the qualities hinted at here are stated with respect to the nations. Thus the initials here are those of the phrase meaning “father of a multitude,” not of the name Abraham. Yet this Talmudic explanation is also difficult, since on this reading there is no source in the Torah for notrikon, but at most a derash, or perhaps only an allusion, on words in the Torah that makes use of the method of notrikon.

All the other sources that the Talmud cites there are not sources at all, but applications. Almost all notrikon expositions look like an arbitrary game. It is unclear by what criterion the words that spell out the notrikon are chosen, and there is a great deal of freedom in arranging them. Sometimes one takes the second letter of the word, as in the example from our Torah portion; sometimes Aramaic words are inserted, and so on.

In the end, we explained that the example of Abraham can indeed be interpreted as a genuine source, whereas the other expositions are applications. Our conclusion was that notrikon is not necessarily a matter of initials. It is a compressed hint to hidden content, related to initials but not bound to be exact initials, as can be seen explicitly in the Mekhilta de-Rabbi Yishmael, Yitro, tractate de-BaHodesh, parashah 8, s.v. “Honor your…”.²

We saw from several sources that the principle of notrikon is not a mere allusion. It is a real hermeneutical principle, but it is not used in halakhic midrashim; see Otzar Midrashim, Eisenstein, “Thirty-two Principles in Aggadah,” p. 269, and Rashi on Genesis 30:8.

We proposed that notrikon expositions are employed in places where a word appears that has no obvious plain-sense meaning, and the assumption is that the Torah is hinting to us to seek a notrikon there. Rashi states this explicitly on our Talmudic passage in explaining the derash from Anochi. According to this, notrikon expositions in a place where there is a straightforward plain sense are not real derivations, but allusions.

We also noted that the principle of gematria appears in the baraita of the thirty-two principles, and the author of Keritut explains that gematria and notrikon do not appear in Rabbi Ishmael’s list because they are principles of aggadic exposition.³ We raised the possibility that the lack of univocity in these hermeneutical principles prevents us from using them in halakhic exegesis. This hypothesis has very broad and important implications: aggadic exegesis, even when it is carried out by more defined and structured tools such as hermeneutical principles, is not univocal, and its conclusions are therefore difficult to rely upon. Aggadic derash is mainly corroborative; halakhic derash, by contrast, can serve as the source of genuinely new laws, a generative derash. See the page on Parashat Vayishlah, 5766, and elsewhere.⁴

In light of all this, we explained the contradiction noted above concerning the prohibition of writing on the Sabbath. The essence of our claim was that notrikon is only a hint to a word, not a representation of the word itself. The reader is the one who constructs the word from the letter; it is not present behind the written letter itself. Therefore the letter has no independent significance. The letter by itself is not a representation of the word but only an allusion to it, and therefore one is not liable to a sin-offering for writing it on the Sabbath. A letter at the end of a scroll, by contrast, has substantial significance, and therefore one is liable to a sin-offering for writing it.⁵

We connected this to Maimonides’ determination, discussed on the page for Parashat Yitro, 5765, that all laws learned from derashot are not embedded in Scripture itself but constitute an expansion of it. The same is true of notrikon.

In the second half of that article we considered several characteristics of the language of Scripture, the sacred language, its difference from other languages — though Maimonides probably would not agree with this description, since he sees the sacred language as an ordinary conventional language — and some implications of those differences. We argued that in the language of Scripture, letters too carry meaning, not only words. We saw that the connection between a word and its meaning is not arbitrary but essential. Toward the end of the article we discussed an implication of these differences for the prohibition of writing on the Sabbath, which according to some decisors applies only to writing in the sacred language. We concluded with an attempt to understand the unique status of the Divine Name in the sacred language.

This year we will expand further on the definitions of notrikon and gematria, and on the difference between these two principles.

B. More on Notrikon and Gematria

Introduction: Notrikon and Gematria in the Baraita of the Thirty-two Principles

In Otzar Midrashim (Eisenstein, “Thirty-two Principles in Aggadah,” p. 266), the principle of notrikon appears immediately after the principle of gematria, in the following language:⁶

“(29) By gematria; (30) by notrikon, which has two aspects: one aspect through transposition of letters, and one through reckoning the letters.”

Within the definition of notrikon, two sub-principles appear: one mode is transposition of letters; the second is reckoning letters. The meaning apparently is that there are two kinds of notrikon exegesis:
1. replacing one word with another word produced by transposition of the first word’s letters;
2. replacing one word with a group of words hinted at within it by initials.

The second mode is ordinary notrikon exegesis, but the first is a hermeneutical principle of a wholly different kind.

As in Rabbi Ishmael’s baraita, the baraita of the thirty-two principles also gives examples of each hermeneutical principle one after another. In the examples that concern gematria and notrikon, the following נוסח appears there (pp. 269-270):

“(29) Gematria. From where do we know that one expounds gematria in aggadah? If by numerical calculation, Scripture says ‘three hundred and eighteen’ (Genesis 14), which comes out to the numerical value of Eliezer. And if by substitution of letters, it has already said lev kamai (Jeremiah 51), which through letter substitution becomes Kasdim by Atbash. (30) Notrikon. From where do we know that one expounds notrikon in aggadah? Scripture says: ‘Abraham, father of a multitude of nations’ (Genesis 17); karmel (Leviticus 2:14) means ‘softly rubbed,’ something soft that can be crumbled by hand. ‘He cursed me with a grievous curse’ (1 Kings 2) — its initials mean adulterer, Moabite, murderer, adversary, abomination. Yarat (Numbers 22) — its initials mean feared, saw, turned aside.”

The examples of notrikon indeed bring two examples, one from each type. Abraham as “father of a multitude of nations” is the second, ordinary type of notrikon; see last year’s article on this exposition. And karmel, which is expounded through transposition of letters as “softly rubbed,” is the first type. The third example returns to the second type, the type of initials.

Let us note that the gematria examples quoted above also show that gematria has two distinct forms of exposition:
1. calculation — ordinary gematria;
2. substitution of letters — in the Atbash code, which in the fuller description there is also called “transposition of letters.”

Several Difficulties

This version raises a number of difficulties:

1. In what sense is the first type of notrikon similar to the second, ordinary type?
2. Put differently: why is it included under the principle of notrikon at all?
3. Why is the first type of notrikon not included under the principle of gematria? After all, it is a substitution that preserves gematric value.
4. Why is Atbash classified as gematria and not as notrikon, especially since here too it is called “transposition of letters”?
5. Why is there no clear separation between the types in the examples of notrikon that are then cited? Three mixed examples are given there: second type, first type, and again second type. Moreover, the first example cited is specifically a notrikon of the second type, initials, and therefore the order of examples here also seems somewhat problematic.

Explanation: The Numerical and Semantic Value of Letters and Words

To understand all these difficulties, we must note a fundamental distinction. Gematria deals with the numerical value of letters, and therefore of words as well. The basis of the matter is the letters, since the numerical value of a word is the sum of the values of its letters; see below in the remarks of the Rosh. Notrikon, by contrast, as we concluded last year, is not based on numerical value, whether of letters or of words, but on various kinds of hints from letters to words. These are not necessarily initials; see last year’s article.

The Atbash Substitution

It is therefore quite clear that Atbash exposition can indeed fall under the heading of gematria. It does not connect letters to words, but replaces one letter with another letter that is related to it in some way. Here the relation is not the ordinary numerical value of gematria, but rather the letters’ positions in the סדרה of the alphabet — or more precisely, their distance from the edge. Atbash is therefore indeed a substitution of letters, but the rule governing this substitution is not connected to words. It is connected to letters, and to a numerical value attached to them.

As we saw, in our case the numerical value is the distance from the ends of the alphabetic sequence. We in fact find a similar kind of derash regarding blessings over fruits of the Seven Species. Babylonian Talmud, Berakhot 41a states that the order of precedence for blessings, when different fruits of the Seven Species are before us, is determined by the distance between the given fruit and one of the two occurrences of the word “land” in the verse: “a land of wheat and barley, vines, figs, and pomegranates; a land of olive oil and honey.” Here too there are two ends, and the numerical value that determines importance for purposes of the blessing is the distance from those ends.

Transposition of Letters in Notrikon

By contrast, the transposition of letters that belongs to the first type of notrikon is not connected to numerical value. It is a change in the order of the letters within the word, and therefore its basis is the word, not the letters. When we replace karmel with “softly rubbed,” there is no relation here between first letters, second letters, and so on. Rather, there is a view of the totality of the letters in the word. The relation between the substituted terms is created through their occurrence in the word, and therefore the basic perspective is that of words, not letters. The total number of letters is also preserved, something that is certainly not required in gematria. Presumably for these reasons, this is a method of notrikon and not of gematria.

Let us note that the gematric value of the two substituted words is indeed the same, but this is only an accidental result of the fact that the very same letters appear in both. Clearly, this is not essentially a derash of gematria.

One consequence of this is that such a substitution can be performed even if one changes something in the letters, something that cannot be done in gematria. As long as there is a resemblance between the substituted words, the substitution can still count as notrikon. An example may perhaps be the very example cited here: the phrase “softly rubbed” contains the final form of kaf, whose gematric value differs from that of the ordinary kaf. The final letter forms are counted immediately after tav; the gematric value of final kaf is 500.⁶ Therefore one could infer from this that the substitution is not gematria but notrikon.⁷

The Three Examples of Notrikon

The fact that three examples of notrikon are given can be understood in light of what we argued last year. As noted, our conclusion there was that the exposition from Abraham is a source for performing notrikon, not merely an application of the principle. The Torah itself is doing a notrikon exposition there. The baraita therefore opens with an example that serves as a source for the principle of notrikon. Only afterward does it bring two additional examples of the application of that principle. The first is a notrikon of the first type, transposition of letters, while the second is of the second type, initials.

The order of the examples is now also clear. If the first example is a source, then the examples of application are indeed brought in order: the first application demonstrates transposition of letters, which is the first type, and the second demonstrates the unpacking of initials, which is truly the second type.

We may note that we have here, incidentally, a further proof for what we argued last year: the exposition of notrikon in the case of Abraham is indeed a source, and not merely an example of the application of the principle.

Notrikon and Gematria: Updating the Definitions

Our conclusion is that the definitions of these two hermeneutical principles need slight revision. Notrikon is a hermeneutical principle whose basic unit is the word. Sometimes this takes the form of a hinted connection from letters to words, and sometimes it is a substitution between words composed of the same letters. Gematria, by contrast, refers specifically to letters — more precisely, to some numerical value attached to them. This may be the ordinary gematric value, or the value determined by the distance from the edge of the alphabetic sequence.⁸

This distinction strengthens still further our conclusion from last year’s article: in the sacred language, the basic unit of meaning is the letter, not the word. Gematria, if taken seriously, is a clear expression of this phenomenon.

“With the Total”⁹

A phenomenon that sharply highlights the problematic nature of these principles is the imprecise calculation of some gematrias. At times we find gematric equivalences that do not insist on exact value. Usually the difference is 1, and then the medieval commentators explain that the gematria is correct “with the total.” For example, the Rema, Orah Hayyim 583:2, writes that the custom is not to eat nuts on Rosh Hashanah because egoz has the numerical value of het. But when one actually calculates, egoz has the value 17, whereas heta has the value 18.¹⁰ Here too some commentators explain it as “with the total.”

In Shibbolei ha-Leket, sec. 212, s.v. “Rabbi Yehudah used to say,” it is written: “In gematria we are not exacting about one less or one more.” So too the Rosh writes at the end of Pesahim (sec. 10) to explain the Torah’s own formulation regarding the counting of the Omer: “You shall count fifty days,” whereas the actual counting is only forty-nine days.¹¹ The Rosh explains that this is the way of Scripture, which does not pay attention to the subtraction or addition of one.¹²

In Leket Yosher — by the author of Terumat ha-Deshen — the introduction brings a source for this from Rashi on Parashat Va-Ethanan, Deuteronomy 4:25; see also Babylonian Talmud, Gittin 88a, and Sanhedrin 38a. Rashi explains:

“Venoshantem — He hinted to them that they would be exiled from the land at the end of eight hundred and fifty-two years, the numerical value of venoshantem. But He hastened and exiled them at the end of eight hundred and fifty years, advancing it by two years before venoshantem, so that the end of the verse, ‘you shall surely perish,’ would not be fulfilled in them. This is the meaning of the verse, ‘The Lord hastened the evil and brought it upon us, for the Lord our God is righteous’: He acted righteously with us by bringing it two years before its time.”

Rashi explains that the exile was advanced by two years so that the end of the verse would not be fulfilled in them. Leket Yosher notes on this that, apparently, it should have been enough to advance it by one year, since the decree of venoshantem would already then have been broken: the gematria would no longer be identical. From here he proves that if the Holy One, blessed be He, had advanced it by only one year, the gematria of venoshantem would still have counted as 852, because in gematria one does not pay attention to a difference of 1.

At first glance, one could infer from this imprecision that the principle of gematria is not really a hermeneutical principle at all. It would then be a kind of game whose purpose is homiletic, in the negative sense of that word. Of course, this would apply only to expositions that are actually based on such imprecision. Exact expositions could still be assigned to the category of systematic derash.

But it seems that the conclusion that emerges from all this is exactly the opposite. If this were really nothing more than a game, there would be no reason to bring sources and proofs that one need not be precise about a difference of 1. The fact that we found proofs for this rule even in the diction of Scripture, and also among the Sages, means that gematria exposition is a serious form of derash, and that the rule according to which a difference of 1 does not disturb the conclusion is itself one of the rules of gematria, whatever its source and significance may be.

A Common Explanation of the Rule of “With the Total”¹³

A common explanation for the difference in gematria is that the difference of 1 is attributed to the word as a whole. That is, the gematria of the word is the total gematria of its letters, plus the fact that there is here a complete word. This extra unit symbolizes the fact that we are dealing with a word and not merely with a collection of letters. At first glance, that is also what the terminology used by the commentators seems to suggest: “with the total.”

However, this explanation is difficult. On both sides of the gematric comparison we should have to take into account the fact that we are dealing with a word and not with a mere collection of letters. Moreover, why is the difference sometimes in one direction and sometimes in the other?¹⁴

Beyond that, such an explanation somewhat contradicts the whole idea presented above, according to which gematria is founded on letters and not on words. It would seem to create a mixture between notrikon and gematria.

It therefore seems to me that this explanation should be rejected.

The Laws of Slaughtering the Required Organs: The Passage in Hullin 28a

The Talmud in Babylonian Talmud, Hullin 28a expounds the verse “And you shall slaughter… as I have commanded you” (Deuteronomy 12:21), and says:

“As it was taught in a baraita, Rabbi says: ‘And you shall slaughter as I have commanded you’ teaches that Moses was commanded regarding the esophagus and the windpipe, regarding the majority of one in a bird, and the majority of two in an animal.”

The early commentators — see Rashi, s.v. “as,” Rashi on the Torah there, and others — note that the plain sense of the verse proves that Moses received an oral command concerning slaughter, since no such command is stated explicitly in the Written Torah. But the details of the command — the majority of one organ in a bird and the majority of two in an animal — cannot be learned from the wording of the verse through this derash.

And yet, the plain meaning of the Talmud indicates that from here they derived the details of the command as well: namely, that one must slaughter the majority of one organ in a bird — and there the Amoraim dispute whether this means the windpipe or the esophagus, or only the esophagus — and the majority of both organs in an animal. This is very plausible in the context of the passage, because the one who brings this derash is Rabbi, and his dispute with Rabbi Elazar ha-Kappar concerns only the details of the command, not its very existence. This is explicitly stated in the Sifrei there; see the end of paragraph 41 in the Malbim edition.

Nevertheless, Rashi, even in his final view, maintains that the derash proves only that there was an oral command about the laws of slaughter; the laws themselves are not derived from here. See Rashi there, s.v. “majority of one.” Tosafot, however, s.v. “and regarding the majority,” writes as follows:

“Regarding the majority of one in a bird and the majority of two in an animal: some explain that he expounds it by gematria from ka-asher: alef means one in a bird, shin means two in an animal, resh means that the majority of one counts as the whole of it — hence, the majority of one in a bird and the majority of two in an animal. The initials of ka-asher, read backward, are: ‘the majority of one is like the whole of it.'”

It seems that Tosafot inferred from the wording of the Talmud in our passage that this is a full derash, from which the details of the command are derived by gematria from the word ka-asher in the verse “And you shall slaughter… as I have commanded you.”

Tosafot then details the interpretation. The alef has the numerical value 1, which hints to one organ in a bird. The shin hints to two in an animal. The resh hints to the rule that the majority counts as the whole — both the majority of one organ in a bird and the majority of both organs in an animal. Finally, Tosafot concludes by saying that this word is also expounded as reversed initials: “the majority of one is like the whole of it.”¹⁵

The connection between the end of Tosafot’s remarks and their beginning is not entirely clear. At first glance these seem to be two separate interpretations. The Steipler, in Birkat Peretz there, already noted that this is indeed how it appears in Baalei ha-Tosafot on the Torah. There they added the words “another explanation” and attributed the second explanation to Rabbi Jacob of Corbeil. For that reason he emended the text of Tosafot here as well. But without that emendation, the wording of Tosafot in our edition suggests that they intended a single interpretation. Let us note that Tosafot ha-Rosh here brings only the first part.

It should also be noted that the first exposition is not pure gematria either. Alef is indeed 1, but shin is not 2, and resh is not the numerical equivalent of “majority.” The expositions from the last two letters are clearly notrikon, not gematria. If so, we have here a combination of these two hermeneutical principles.

One could, however, explain that alef too is not expounded by gematria but by notrikon, for the word “one,” which begins with alef. But if so, the entire first explanation is also notrikon, and it is then completely unclear why Tosafot call it a gematria derivation. Indeed, in Tosafot ha-Rosh here and in Baalei ha-Tosafot on the Torah it is not defined as gematria, and there it must certainly be interpreted as notrikon.

In any case, according to Tosafot’s own method, it seems that the alef is indeed expounded by gematria and not by notrikon. But it is not clear why the Sages chose here specifically gematria and not notrikon, especially since the continuation of this very exposition is certainly notrikon.¹⁶

However, when one examines the wording of Baalei ha-Tosafot on the Torah, several interesting discoveries emerge:

“From here our Rabbis said in tractate Hullin: this teaches that Moses was commanded concerning the windpipe and the esophagus, concerning the majority of one in a bird and the majority of two in an animal. Rabbi Jacob of Corbeil said: ka-asher tzivitikha has the same numerical value as ‘the majority of one in a bird and the majority of two in an animal’; calculate and you will find it. Another explanation: asher — alef, one; shin, two; resh, majority. Assign the smaller quantity to the case whose slaughter requirement is smaller, namely the bird. Thus it is one in a bird and two in an animal, while ‘majority’ applies to both.”

Here too it is clear that the approach is like that of Tosafot, against Rashi: the derash teaches the details of the laws and not only the bare fact that there was a command. Rabbi Jacob of Corbeil states that there is an exact gematria between the words ka-asher tzivitikha and the phrase “the majority of one in a bird and the majority of two in an animal.” It appears that the word “two” should be spelled there with two yods; in that case both expressions equal 1057. If so, this is a completely ordinary gematria derivation, and not only from the first letter, as in the version of Tosafot in our editions. Afterward a second interpretation appears, one that is wholly notrikon, including the alef, as in Tosafot ha-Rosh and similarly to the beginning of our Tosafot, aside from the derash on alef. Finally, it explains why the one applies to a bird and the two to an animal, and why “majority” governs both.

If so, there is no mixture here at all. The first explanation is an entirely regular gematria, of the first type we encountered above, and the second is an ordinary notrikon, of the second type discussed above.

Comparison with the wording of Tosafot in our editions shows that there is probably a textual corruption, and once corrected all the above difficulties disappear. The end of Tosafot contains both interpretations found in Baalei ha-Tosafot on the Torah, gematria and notrikon together. The second part then contains an additional explanation of reversed notrikon, which does not appear in Baalei ha-Tosafot on the Torah. As stated, this corrected version is free of all the problems we raised above, and it is fairly clear that this is what Tosafot intended.

Footnotes