Gematria (one of the aggadic hermeneutical rules for interpreting the Torah (Baraita of 32 Rules, no. 29). It consists of explaining a word or group of words according to the numerical value of the letters, or of substituting other letters of the alphabet for them in accordance with a set system. Whereas the word is normally employed in this sense of manipulating according to the numerical value, it is sometimes found with the meaning of "calculations" (Avot 3:18). Similarly where the reading in present editions of the Talmud is that Johanan b. Zakkai knew "the heavenly revolutions and gematriot," in a parallel source the reading is "the heavenly revolutions and calculations" (Suk. 28a; BB 134a; Ch. Albeck, Shishah Sidrei Mishnah, 4 (1959), 497).

The use of letters to signify numbers was known to the Babylonians and the Greeks. The use of gematria was widespread in the literature of the Magi and among interpreters of dreams in the Hellenistic world. The Gnostics equated the two holy names Abraxas (Abracaj) and Mithras (Miqraj) on the basis of the equivalent numerical value of their letters (365, corresponding to the days of the solar year). Its use was apparently introduced in Israel during the time of the Second Temple, even in the Temple itself, Greek letters being used to indicate numbers (Shek. 3:2). In rabbinic literature numerical gematria first appears in statements by tannaim of the second century. It is used as supporting evidence and as a mnemonic by R. Nathan. He states that the phrase Elleh ha-devarim ("These are the words") occuring in Exodus 35:1 hints at the 39 categories of work forbidden on the Sabbath, since the plural devarim indicates two, the additional article a third, while the numerical equivalent of elleh is 36, making a total of 39 (Shab. 70a). R. Judah inferred from the verse, "From the fowl of the heavens until the beast are fled and gone" (Jer. 9:9), that for 52 years no traveler passed through Judea, since the numerical value of behemah ("beast") is 52. The Baraita of 32 Rules cites as an example of gematria the interpretation that the 318 men referred to in Genesis 14:14 were in fact only Eliezer the servant of Abraham, the numerical value of his name being 318. This interpretation, which occurs elsewhere (Ned. 32a; Gen. R. 43:2) in the name of Bar Kappara, may also be a reply to the Christian interpretation in the Epistle of Barnabas that wishes to find in the Greek letters tih, whose numerical value is 318, a reference to the cross and to the first two letters of Jesus' name, through which Abraham achieved his victory; the Jewish homilist used the same method to refute the Christian interpretation. The form of gematria which consists of changing the letters of the alphabet according to atbash, i.e., the last letter T is substituted for the first a, the penultimate S for the second b, etc., already occurs in Scripture: Sheshach (Jer. 25:26; 51:41) corresponding to Bavel ("Babylon"). The Baraita of 32 Rules draws attention to a second example: lev kamai (Jer. 51:1) being identical, according to this system, with Kasdim. Another alphabet gematria is formed by the atbah system, i.e., t is substituted for a, H for b, etc., and is called "the alphabet of Hiyya" (Suk. 52b). Rav, the pupil of Hiyya, explained that Belshazzar and his men could not read the cryptic writing because it was written in gematria, i.e., according to atbah (Sanh. 22a; cf. Shab. 104a).

Gematria has little significance in halakhah. Where it does occur, it is only as a hint or a mnemonic. The rule that when a man takes a nazirite vow for an unspecified period, it is regarded as being for 30 days, is derived from the word yihyeh ("he shall be") in Numbers 6:5, whose numerical value is 30 (Naz. 5a). Even in the aggadah, at least among the early amoraim, gematria is not used as a source of ideas and homilies but merely to express them in the most concise manner. The statements that Noah was delivered not for his own sake but for the sake of Moses (Gen. R. 26:6), that Rebekah was worthy to have given birth to 12 tribes (ibid. 63:6), and that Jacob's ladder symbolizes the revelation at Sinai (ibid. 68:12), do not depend on the gematriot given there. These homilies are derived from other considerations and it is certain that they preceded the gematriot.

Gematriot, however, do occupy an important place in those Midrashim whose chief purpose is the interpretation of letters, such as the Midrash Haserot vi-Yterot, and also in the late aggadic Midrashim (particularly in those whose authors made use of the work of Moses b. Isaac ha-Darshan) such as Numbers Rabbah (in Midrash Aggadah, published by S. Buber, 1894), and Bereshit Rabbati (published by H. Albeck, 1940; see introduction, 11–20). Rashi also cites gematriot that "were established by Moses ha-Darshan" (Num. 7:18) and some of the gematriot given by him came from this source even if he does not explicitly mention it (Gen. 32:5, e.g., "I have sojourned with Laban"—the gematria value of "I have sojourned" is 613, i.e., "I sojourned with the wicked Laban but observed the 613 precepts," is the interpretation of Moses ha-Darshan, Bereshit Rabbati, 145). Joseph Bekhor Shor, one of the great French exegetes of the Torah, made extensive use of gematriot, and nearly all the tosafists followed him in this respect in their Torah commentaries (S. PoznaGski, Mavo al Hakhmei Zarefat Mefareshei ha-Mikra, 73). A wealth of gematriot occur in Pa'ne'ah Raza, the commentary of Isaac b. Judah ha-Levi (end of 13th century), and in the Ba'al ha-Turim, the biblical commentary of Jacob b. Asher. The Kabbalah of the Hasidei Ashkenaz also caused gematriot to enter the halakhah. In his Ha-Roke'ah, Eleazar of Worms uses gematriot to find many hints and supports for existing laws and customs; with him the gematria at times embraces whole sentences. Thus he establishes by gematria from Exodus 23:15 that work which can be deferred until after the festival may not be performed during the intermediate days (Ha-Roke'ah, no 307). Gematriot of the Hasidei Ashkenaz occupy a prominent place in their commentaries on the liturgy and on piyyutim. Abraham b. Azriel incorporated the teachings of Judah he-Hasid and Eleazar Roke'ah in his Arugat ha-Bosem, and followed their lead. These gematriot, which were part of the Kabbalah of the Hasidei Ashkenaz, established the definitive text of the prayers, which came to be regarded as sacrosanct. Some authorities forbade it to be changed even when the text did not conform with the rules of grammar. Nahmanides, on the other hand, tried to limit the arbitrary use of gematriot and laid down a rule that "no one may calculate a gematria in order to deduce from it something that occurs to him. Our rabbis, the holy sages of the Talmud, had a tradition that definite gematriot were transmitted to Moses to serve as a mnemonic for something that had been handed down orally with the rest of the Oral Law... just as was the case with the gezerah shavah of which they said that no man may establish a gezerah shavah of his own accord" (Sefer ha-Ge'ullah ed. by J. M. Aronson (1959), Sha'ar 4; see his commentary to Deut. 4:25).