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Numerical Notation.

The letters of the alphabet were used as numerical symbols as early as the Maccabean period (comp. Numismatics). Whether such a usage was known in earlier times also, whether there existed in Israel, as among kindred nations, special signs for figures, or whether numerical notation was entirely unknown, can not be decided by direct proof. That there were no numerical signs at all is hardly possible. The necessities of daily life require such signs, and the example of surrounding nations could not but have suggested their introduction. For an assumption that there were special signs there is no basis. It must, therefore, be assumed that the numerical value of the alphabet was known in earlier times. The fact that figures are not found in the Bible nor in the Siloam inscription, nor on the Moabite Stone, would not militate against such an assumption. In monumental inscriptions the use of figures might have been avoided for various reasons, while the earlier use of figures in the Bible is rather probable, since the discrepancies in numbers which now exist can thus be best explained. Other considerations strengthen such a hypothesis (comp. Gemaá¹­ria).

The use of alphabetical signs was doubtlessly practically the same as in the Talmud, where numbers higher than 400 are formed by composition, as (for 500), (for 900), etc. Such a way of forming higher numbers could not in the end be found other than clumsy, and, therefore, the Masorites introduced the use of the final letters for indicating 500, 600, 700, 800, and 900 respectively; to indicate the thousands the letters representing the corresponding number of units was used. In writing any numerical combination, since the thousands were written before and the units were written after the hundreds and tens (the latter letters of the alphabet), they were easily distinguishable. About 800 C.E. the Jewish scholar Mashallah introduced into the Mohammedan world the use of the so-called Arabic figures (see Harkavy's note to the Hebrew transl. of Grätz's "Gesch." 3:213), which since then have occasionally been used in Hebrew literature also (Oppenheim, in "Monatsschrift," 13:231,462; 15:254,376).

System of Numbers.

The Hebrew system of counting is, like that of all the Semites and like the Egyptian hieroglyphic system, the decimal, which is a later development of a more original quintal system based on the fingers of one hand (L. Reinisch, "Das Zalwort Vier und Neun in den Chamitischen-Semitischen Sprachen"). The blending of the Semitic decimal system with the Sumerian sexagesimal is found in earliest Babylonian times. But in course of time the decimal system prevailed. A trace of the sexagesimal system may still be found in the use of the number sixty (see below). The use of the fingers for numbering occurs in traditional literature (see Yoma 22a, b). In Talmud and Midrash numbers are sometimes formed by subtraction, as in Latin, French, etc.—for example 100 - 2 = 98 (Lam. R. 3:12), 50 - 1 = 49 (Levias, "Aramaic Grammar," § 141)—the reason for which is not clear.

Symbolism of Numbers.

At an early time in the history of man certain numbers were regarded as having a sacred significance or were used with symbolical force, the origin of their symbolism lying in their connection with primitive ideas about nature and God. Such a use of numbers is found also in the Bible, although the Biblical authors were hardly conscious of their origin. In later Jewish literature, however, with Pythagorean doctrines was introduced the use of numbers as symbols, based on their mathematical qualities. The most prominent exponent of the latter custom is R. Abraham ibn Ezra. In cabalistic literature both systems are used. The rhetorical or stylistic use of numbers is largely due to an obsolete symbolism. Even numbers were thought to be unlucky (Pes. 110a). Attempts to find in Biblical numbers references to ideas were made by Aristobulus and Philo, and since their time by many allegorists. Nevertheless, a distinct connection between any given number and a certain idea can not be proved. Among the "thirty-two rules" of the son of R. Jose the Galilean, two refer to numbers—one to gemaṭria, the other, the twenty-seventh, to the symbolism of numbers (see Bacher, "Tannaitische Terminologie," s. ). According to this hermeneutic canon, any number may be explained as corresponding to (, e., "symbolizing") another equal number or sum of numbers. Thus, the "40 days" in Numbers 13:25 correspond to the "40 years" in ib. 14:34; and the number 36 in 2 Chronicles 16 I corresponds to three things in connection with which the same number of years is mentioned (Bacher, c.).

The following numbers occur in Hebrew literature either as symbols or as round numbers:

• Two: Used in the sense of "a few" in Numbers 9:22; 1 Samuel 11:11; Hosea 6:2; Ned. 66b (comp. the Talmudic rule, ).
• Three: The sacredness of this number is probably due to the fact that primitive man divided the universe into three regions—heaven, earth, and water, respectively represented in Babylonian mythology by the divinities Anu, Bel, and Ea. Its sacred or symbolical use may be illustrated by such passages as 1 Kings 17:21; 1 Chronicles 21:12; Daniel 6:10. Its rhetorical use for a small total is illustrated in Genesis 30:36; 10, 12; 42:17; Exodus 2:2, 3:18, and in Pes. 62b and Yer. Ta'an. 4:8. Multiples of three are similarly used: nine, in Yer. Ta'an. 4:8; twenty-one, in Ethiopic Enoch, 69:2; thirty, in Slavonic Enoch, 36:1; thirty-six, in Ethiopic Enoch, xc. 1; three hundred, in Soá¹­ah 34a; Pes. 62b; Ḥul. 59b, 90b; Yer. Sanh. 7:19; Yer. Ta'an. 4:8; nine hundred, in Yer. Sanh. 7:19.
• Three and one-half: Represents, according to Gunkel ("Schöpfung und Chaos," pp. 309 et seq.), the three and one-half months from the middle of Kislew to the end of Adar—from the winter solstice to the festival of Marduk, the period of the supremacy of Tiamat. The number occurs in Daniel 7:25, 9:27, and 12:7 (Hebr.). In traditional literature three and one-half as a half of seven is frequently used as a round number; see Midrash to Proverbs (ed. Buber, p. 48, note ).
• Four: Sacred as the number of the four cardinal points of the compass; denotes completeness and sufficiency. In cabalistic literature its sacredness isenhanced by the fact that the Tetragrammaton contains four letters. The number is found in Genesis 2:10; Judges 11:40; Jeremiah 15:3; Ezekiel 14:21; Zechariah 1:18; Nehemiah 6:4; etc. The multiples of four used are twenty-eight (in the measurement of the curtains of the Tabernacle) and forty and its multiples.
• Seven: The most sacred number. The origin of its sacredness is found by some in its factors three and four; by others, in its correspondence to the number of the planets; while others assert that it arose from a sacred six by the addition of one. In Judaism its sacredness was enhanced by the institution of the Sabbath. The number occurs in the seven days of Creation, the institution of the seventh year of release, the forty-nine years between the jubilees, the seven altars, the seven lamps, the sprinkling of the blood seven times, etc. (Genesis 7:2 et seq., 21:28-30; 1 Kings 18:43; Deuteronomy 16:9; Ezek. 22, 41:3; et al.). Quite frequently it is met with in Apocryphal literature (Ethiopic Enoch, 21:3-6, 73:5-8; Slavonic Enoch, -, 27:3, 41:1; et al.); in Talmud and Midrash (Pes. 54a; Soá¹­ah 10b; et al.; comp. Lampronti, "Paḥad Yiẓḥaḳ," s. ). The multiple fourteen occurs in Proverbs Rabbah (ed. Buber, p. 92).
• Ten: Had a symbolical character in part because it is the basis of the decimal system, and in part because it is the sum of three and seven. Its simplest use is as a round number (Genesis 24:10,22; Joshua 22:14; Judges 17:10; et al.; comp. Lampronti, c. s.). A more sacred use is found in the ritual (Exodus 26:1,16; Numbers 7, ,; 1 Kings 6,; Ezekiel 45; 2 Chronicles 4). Because of this sacred character "ten" is used in apocalyptic symbolism (Daniel 7:7,20,24).Multiples of ten are used as round numbers: one hundred and two hundred, in Pes. 64b; et al.; one thousand, in Ḥul. 97b; Ned. 50b; Yer. Ta'an. 4:8; ten thousand and two hundred thousand, in Yer. Ta'an. 4:8; one million, in Yoma 33b.
• Twelve: Derived its sacred character from the fact that it is the product of three and four and is the number of the months of the year. There are twelve tribes of Israel and the same number of tribes of Ishmael (Genesis 17:20, 25:16). The number of many representative men and things was made twelve to accord with the number of the tribes (Exodus 24:4; Numbers 17:2,6; Joshua 4; et al.). The number twelve for these reasons entered into Hebrew ritual (comp. Exodus 15:27; Numbers 33:9; Leviticus 24:5; Jeremiah 52:20 et seq.; Ezekiel 43:16). As a round number twelve occurs both in Biblical (2 Samuel 2:15; 1 Kings 10:20) and in post-Biblical literature (see the list of references given by Zunz, "Literaturgesch." p. 601; comp. also Yoma 75b, 77b; Ta'an. 25a; M. Ḳ. 24a; Ḥul. 95).The multiple twenty-four occurs in Lam. R. 1:2; twenty-four millions, in Ned. 50b.
• Twenty-two: Used as a round number in later literature (Gen. R.; Midr. Shemuel ), deriving its significance from the fact that it is the number of the letters in the alphabet (comp. Bacher, "Ag. Pal. Amor." 2:297).
• Forty: Stands in the Bible for a generation (e.g., the forty years of wandering in the desert), hence for any period of time the exact duration of which is unknown (comp. Genesis 7:4,12,17; 8:6; Exodus 24:18, 34:28; Deuteronomy 9:9,11,18; 10:10; 1 Samuel 17:16; 1 Kings 19:8; Jonah 3:4). In later literature forty is commonly used as a round number (comp. Giá¹­. 39b, 40a; Soá¹­ah 34a; Yer. Ta'an. 4:8; et al.). The multiple eighty is found in Yer. Ta'an. 4:8; four hundred, in Ḥul. 59b and Bek. 31a; four hundred and eighty, in 1 Kings 6:1 and Yer. Meg. 3:1; eighty thousand, in Yer. Ta'an. 4:8.
• Sixty: The larger unit of the sexagesimal system; used to express an indefinitely larger number (comp. Song of Solomon 3:7, 6:8). In Talmudic literature it is frequently used as a round number (comp. Ber. 57b; Pes. 94a; B. Ḳ. 92a; B. M. 30b, 107b; Ta'an. 10a; Ned. 39b; Midr. Teh.; Lev. R.; etc.). In the Halakah a thing ritually unfit becomes fit when mixed with something sixty times its own amount.
• Seventy: Has a sacred or symbolical significance because it is made up of the factors seven and ten (comp. Exodus 15:27; 24:1,9; Numbers 11:24 et seq.; Genesis 46:27; Exodus 1:1; Deuteronomy 10:22; Jeremiah 12:11; Daniel 9:24 et seq.). For later Jewish usage compare S. Krauss in Stade's "Zeitschrift," 19:1-14, 20:38-43, and Steinschneider in "Z. D. M. G." 4:145-170; 57:474-507, where he deals also with the number seventy-two.
• Sixty Myriads: Used in later literature to express a very large but indefinite number. It derives its significance from the number of Israelites that went out from Egypt (comp. Lam. R. 2:13; Deut. R. 1:17; etc.). The multiple one hundred and twenty myriads occurs in Lam. R. c.

Ascending Enumeration.

The tendency to indicate somewhat more exactly an undetermined number of objects led to the use of two definite numbers instead of one indefinite expression. The smaller numbers are paired in this way in the following passages: one or two: Deuteronomy 32:30; Jeremiah 3:14; Psalms 62:11; Job 33:14, 5; two or three: 2 Kings 9:32; Isaiah 17:6; Amos 4:8; Job 33:29; Ecclus. (Sirach) 23:16, 26:19, 25; three or four: Jeremiah 26:3; Amos 1:3, 2:6; Proverbs 30:15,18,21,29; Ecclus. (Sirach) 26:5; four or five: Isaiah 17:6; five or six: 2 Kings 13:19; six or seven: Proverbs 6:16; Job 5:19; seven or eight: Micah 5:5; Ecclesiastes 11:2. In all these instances the use of a second number calls attention to the fact that the first number is merely approximate; hence such an arrangement of numbers is employed in the so-called "middah" a kind of riddle (Proverbs 6:16-19, 30:15 et seq.; Ecclus. [Sirach] 23:16;; 26:5 et seq., 19; 25 et seq.).

Numerical Grouping.

As an aid to the memory, the ancients frequently grouped themes of traditional law or of haggadah according to numbers; see, for instance, Abot , where various subjects in which the number ten is prominent are grouped together. Such groups are found frequently in Talmud and Midrash. The entire contents of some books were at times arranged in numerical groups, as in the "Pirḳe de Rabbenu ha-Ḳadosh" and, probably, in the "Forty-nine Middot de-R. Nathan," a work now lost.