GEMATRIA (from Gr. γεωμετρία), 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 first use of gematria occurs in an inscription of Sargon II (727–707 B.C.E.) which states that the king built the wall of Khorsabad 16,283 cubits long to correspond with the numerical value of his name. The use of gematria (τὸ ὶσόψηφον) 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 (ʾΑβράξας) and Mithras (Μίθρας) 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") occurring 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 τιη, 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. These gematriot are based on the first of four methods of calculating the numeral value of the letters of the Hebrew alpha. Known as Mispar Hekhreḥi, absolute or normative value, each letter is given a specific numerical equivalent. Alef equals 1, bet equals 2, gimmel equals 3, and so on until yod, the tenth letter, which equals 10. The next letter, kaf, equals 20, then lamed, which equals 30, and so on until kuf, which equals 100. The last three letters, resh, shin, taf, equal 200, 300, and 400, respectively. The final forms of the letters, kaf, mem, nun, pei, and ẓadi, used when these letters appear at the very end of a word, are often given the same numerical equivalent as the standard form of the letter. Sometimes, they are given the values 500, 600, 700, 800, and 900, respectively. This brings the numerical equivalencies of the Hebrew alphabet to 1,000, for the alef, the first letter, can also symbolize 1,000. The word alef can also be readas elef, meaning 1,000.
The next two methods of gematria calculation are Mispar Sidduri, ordinal value, where each of the 22 letters of the Hebrew alphabet are given a number between 1 and 22, and Mispar Katan, reduced value, where every letter is equal to a single digit number. This is accomplished by removing the value of 10 or 100. Thus, the alef equals 1, but so do the yod and the kuf, which equal 10 and 100 in the absolute or normative value system. In these last two systems, the five letters that have final forms are usually given the same value as the standard form of the letter. At times, they are assigned special value.
The fourth method, Mispar Katan Mispari, integral reduced value, reduces the total value of the word to a single digit number. If the sum exceeds nine, then the integer values of the total are added together again and again until a single digit number is received. For example, the word, ḥesed (lovingkindness) has an absolute or normative value of 72. The ḥet equals 8, the samakh equals 60, and the dalet equals 4. The numbers of the sum of 72 are then added together (7 plus 2) to equal 9. It should be pointed out that the integral reduced value of the ordinal value and the reduced value of ḥesed also add up to 9.
In Kabbalah, an additional system of gematria is used. The absolute or normative value of a word is calculated by treating each letter as a word and then adding up all of the numerical equivalencies of these letter-words. This system is called milu'i or milu'im. Since some letters can be spelled differently as words, different numerical equivalencies can be achieved for a single word. Thus, the Tetragrammaton, yod, hei, vav, and hei, has the values of 72, 63, 45, or 52, each of which has vital significance in Kabbalah.
The form of gematria which consists of changing the letters of the alphabet according to atbash, i.e., the last letter ת is substituted for the first א, the penultimate ש for the second ב, 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 atbaḥ system, i.e., ט is substituted for א, ח for ב, etc., and is called "the alphabet of Ḥiyya" (Suk. 52b). Rav, the pupil of Ḥiyya, explained that Belshazzar and his men could not read the cryptic writing because it was written in gematria, i.e., according to atbaḥ (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 Ḥaserot 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), including Numbers Rabbah (in Midrash Aggadah, published by S. Buber, 1894) and Bereshit Rabbati (published by Ḥ. 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. Poznański, Mavo al Ḥakhmei Ẓarefat Mefareshei ha-Mikra, 73). A wealth of gematriot occur in Pa'ne'aḥ 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 *Ḥasidei Ashkenaz also caused gematriot to enter the halakhah. In his Ha-Roke'aḥ, *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'aḥ, no 307). Gematriot of the Ḥasidei Ashkenaz occupy a prominent place in their commentaries on the liturgy and on piyyutim. Abraham b. Azriel incorporated the teachings of Judah he-Ḥasid and Eleazar Roke'aḥ in his Arugat ha-Bosem, and followed their lead. These gematriot, which were part of the Kabbalah of the Ḥasidei 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. *Naḥmanides, 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 [see *Hermeneutics] 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).
Despite *Naḥmanides' attempt to limit its use, gematria found its way into biblical commentary. The Pane'aḥ Raza by Isaac ben Judah ha-Levi (late 13th century) and Ba'al ha-Turim by Jacob ben Asher (c. 1270 to 1340) both make frequent use of gematria. Indeed, gematria became a staple element in kabbalastic literature. For example, the 17th-century work, Megalleh Amukkot, by Nathan Nata ben Solomon Spira, uses gematria extensively. The followers of *Shabbetai Zevi used gematria as proof of his messianism.
Gematria is still used to this very day. Indeed a search on the "Google" internet search engine reveals over 106,000 references to gematria on the World Wide Web, a great number of these sites deal with Christianity, witchcraft, and general (non-Jewish) mysticism. Numerous contemporary Jewish books have been published about gematria as well as assisting the reader to find his own gematria equivalencies. For instance, one such book, Sefer Gematrikon (Jerusalem, 1990) provides gematria equivalents for the numbers 1 to 1,000.
W. Bacher, Exegetische Terminologie…, 1 (1899), 125–8; 2 (1905), 124; F. Dornseiff, Das Alphabet in Mystik und Magie (19252), 91–118; A. Berliner, Ketavim Nivḥarim, 1 (1945), 34–37; S. Lieberman, Hellenism in Jewish Palestine (1950), 69–74; H. Waton, Key to the Bible (1952); T. Wechsler, Ẓefunot be-Masoret Yisrael (1968); Scholem, Mysticism, index; S.A. Horodetzky, in: EJ, 7 (1931), 170–9. ADD. BIBLIOGRAPHY: H. Gabai, Judaism, Mathematics and the Hebrew Calendar (2002); Y. Ginsburgh, The Hebrew Letters: Channels of Creative Consciousness (1990); Sefer Gematrikon (1990); M. Munk, The Wisdom of the Hebrew Alphabet: The Sacred Letters as a Guide to Jewish Deed and Thought (1983); M. Zuriel, Or ha-Torah: Bi'ur le-Darkhei ha-Gematriot be-Toratenu (1983); S. Sambursky, in: Journalof Jewish Studies, 29:1 (1978), 35–38; G. Locks, The Spice of Torah – Gematria (1985). KABBALAH: D. Abrams, "From Germany to Spain: Numerology as a Mystical Technique," in: JJS, vol. 47 (1996), 85–101; J. Dan, "The Ashkenazi Hasidic Gates of Wisdom," in: G. Nahon and Ch. Touati (eds,), Hommages à Georges Vajda (1980); I.R. Gruenwald, "Uses and Abuses of Gematria," in: M. Bar Asher (ed.), Rabbi Mordechai Breuer Festschrift 2 (1992), 823–32 (Heb.); M. Idel, Absorbing Perfections: Kabbalah and Interpretation (2002); idem, Language, Torah and Hermeneutics in Abraham Abulafia (1989); idem, "Deus sive natura" – the Metamorphosis of a Dictum from Maimonides to Spinoza, in: R. Cohen and H. Levinw (eds.), in: Maimonides and the Sciences (2000), 87–110; D. Abrams and I. Ta-Shma (eds.), Sefer Gematriot of R. Yehudah the Pious (1998); S. Lieberman, "A Mesopotamian Background for the So-Called Aggadic 'Measures' of Biblical Hermeneutics?," in: HUCA vol. 58 (1987), 157–225; D. Segal, Sefer Sodei Razei Simmukhim (2001); A. Wasserstein, in: Tarbiz, vol. 43 (1974), 53–55 (Heb.). WEBSITES: www.inner.org/gematria/gematria.htm; www.jhom.com/topics/envy/letters/gematria.html.