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Number

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NUMBER. The Biblical concept of “number,” the nuclear meaning of which is discrete quantity made manifest in series-principle of the numerical time order in the plus and minus directions. The birth and progress of mathematical theoreticization is not apparent in the pre-theoretical revelation of the Biblical sphere.

The background of Biblical numbers

The ultimate origin of the concept of number must on the basis of the Christian worldview be traced to the inherent nature of the creation law-ordinances of God. The concept of number is therefore as old as man, the creature “thinking God’s thoughts after him.” Number is one of the basic modalities of the world-order.

Neolithic evidence of numbers.

The cave paintings and glyptic arts of the paleolithic and mesolithic cultures are evidence of man’s sense of form and relationship. And there is supporting evidence of this sense when an arrangement of multiple simple geometric forms are related to yield a complex design to indicate the aesthetics of geometry. However, in various finds of Neolithic materials around the world sets of holes, posts, stones and massive megalithic boulders have been found all in patterns of regular geometric proportions, often in one to one, 1:1 correspondence. Although a very dangerous methodology must be avoided it is important that number and numeral are two concepts that have been found in every tribe and culture examined since the founding of anthropological science. Undoubtedly the primary Sumer. numbers were one and two. The experience of this is related in the Biblical story of the creation of Eve in which Adam first recognized, in the initial place of all future man, the fact of duality. The evidence from linguistics also indicates that “three” in many languages is equivalent to “many.” And in fact it has been pointed out with a fair degree of evidence that in all three of the most ancient language families of the Near E, Agglutinative, Semitic and Indo-European, the terms for “three” are philologically, if not semantically, related to the terms for “beyond” and “many” e.g. Eng. “three > trans.” The greatest innovation and advance of the Neolithic and protoliterate period was Writing (q.v.); however, in every case it appears to have been preceded by number.

Sumerian numbers.

Aside from the problems of Proto-Elamite and Proto-Danubian, the Sumerians of the alluvial plains of Southern Iraq were the world’s first literate people. Just as lists, poetry, epics, lexica and many other types of writing appear so do various number concepts and operations become manifest in the Sumer. cuneiform tablets. There is no doubt that mental processes which had taken great periods of concentrated effort were suddenly displayed with the Neolithic township establishment of Sumer. culture. Almost all the simple arts of arithmetic operations are found in the Sumer. economic texts as: addition, subtraction, multiplication, division, extraction of simple roots and raising to higher powers as well as the handling of a number of types of fractions. Significantly lacking are place notations and the elusive notion “zero.” The most important feature of Sumer. numbers is their sexagesimal character. That is, the base is not 10, 102=100, 103=1000 and so on; but 60, 602=3600; 603=216,000. The system was adapted to fractions so that individual unit fractions could be expressed in sexagesimal fractions. Thus the numeral 1 can stand for 60, a power of 60, 1/60 or even 1/60n and 2 for 2x60=120, 2/60, etc. The common fractions 1/2, 1/3, 1/4, 1/5 were written accordingly as: 30/60, 20/60, 15/60, 12/60 as in the table of fractions shown below (p. 455). The legacy of this system is interesting because it was admirably superior for weights and measures; in fact, some scholars have surmised that this was its origin. Also take notice because of its importance, almost all subsequent metrology systems in the Near E and Mediterranean were sexagesimal. This system is better adapted to dividing the circle and performing calculations on the circle as astronomical quadrants into degrees, minutes and seconds of arc. Yet, the full development of a true place notation with zero was never fulfilled. In time the Sumer. system was developed to yield cuneiform signs for: 1/2, 1/3, 2/3, 1, 10, 60, 10 x 60 = 600, 602, 10 x 602 = 36,000, and the largest unit, 603. This is vastly beyond the scope of the largest Egyp. unit, 100,000. The sexagesimal system was utilized extensively for the two great proto-sciences of the Sumer. civilization, astrology and the calendrical cult. There is no question but that the sexagesimal system of the Sumerians was known to other peoples of antiquity, Hittites, Akkadians, Greeks and others and that some faint remembrance of it can be detected in the early books of the Heb. Bible.

Egyptian numbers.

The Gr. historians and many authors since them have assumed that mathematics had its origins in Egypt. However, the great antiquity of the cuneiform economic documents with their arithmetical operations which appear in the middle of the third millennium b.c., predate the oldest documents and inscrs. from Egypt. The number system is strictly decimal and unlike the cuneiform yields straightforward 1:1 symbols from 1 to 9, 10 to 90, 100 to 900 and 1,000 to 9,000 (fig. 1). The operations of addition and subtraction were simple enough but multiplication was performed by the process of doubling: thus 14 x 14 could be handled by halving a 14 and used as 7 x 14 plus 7 x 14 or solved by using the 10 as 10 x 14 plus 4 x 14. Division was the inverse of this operation. Complex fractions were reduced to unit fractions; thus: 23/45 is reduced to 1/5 + 1/5 + 1/9. Probably because of their vast experience in manipulating fractions and devising elaborate tables for solutions to problems involving fractions, the Egyptians arrived at a very close approximation of “π, G4073, ”, namely, 3.16. Also, they derived a correct formula for the volume of a pyramid. The fruits of Egyp. mathematics and its practical usage in surveying and construction were passed on to the Semites of Syria-Palestine, but no legacy of theory or the more sophisticated solutions of problems appear among the remains of the cultures to the N. In time Egyp. mathematicians, the scribes entrusted with the royal enumerations, formalized the linear epigraphical script of hieroglyphic into a cursive set of ligatured signs. Numbers and their associated operations were handled in the same manner. The increasing mood of conservatism and intransigence which characterized the last millennium of Pharaonic Egypt, took its toll on the development of numbers and the understanding of the concepts of numbers. To what degree intuitiveness influenced the mechanical notions of this Egyp. culture has not yet been decided, but they utilized their clumsy system to record enumerations as high as 1,422,000. This enumeration was affected by the use of many duplicated signs which had to be totalled to be read. The geometric theory of Egypt, like that of early Ionic Greece, was based largely on constructions. Even the rudiments of algebra were never approached.

Akkadian, Assyrian, Babylonian numbers.

The Sumer. sexagesimal system and the Egyp. decimal system seem to have been known to the Akkado-Babylonians, the E Semite cultures which inherited and refined the ancient non-Sem. culture of Mesopotamia. The Assyrians in the N of the Tigris-Euphrates valley and the Babylonians in the S were dedicated businessmen and traders. Literally hundreds of thousands of economic documents, business ledgers and contracts have been excavated and studied. They also were adept builders and the hard facts of life on the plains of Iraq forced cooperation and authoritative planning for irrigation and defense. The Babylonian mathematical tablets are some of the finest exact scientific treatises still extant from the ancient world. Of special importance is that the late Babylonian scribes were on the verge of discovering two of the chief mathematical tools of later ages, “functions” and algebra. In these matters they were centuries beyond and above any of their contemporaries. The close and often disastrous proximity of Mesopotamia to Pal. made the mathematical insights of Babylon available to Israel, but there is only slight evidence that any of this learning actually found common currency in the twelve tribes. The area of a triangle, quadrangle, trapezoid, and the volumes of many types of figures could be computed by the Babylonians. In the last period of Babylonian culture, the Seleucid, practical knowledge overcame the more difficult solution type of problem, and astronomy dominates the texts. The contents of these texts had been abstracted and refined and were known to the Greeks. Thinkers such as Thales utilized these results. Their greatest insight was in the field of elementary number theory which at this time was unresearched. It is from the E Sem. Akkad. language that the Heb. terms for both cardinal and ordinal numbers were derived. In genera l, the mathematical texts of Mesopotamia may be divided into two classes, the problem texts which offer methods and insights for solving specific problems with examples, and table texts which give tables of successive series of numbers under certain operations. The problem texts appear to have originated in the time of the first dynasty of Babylon and had been recopied with little alteration thereafter. Probably under the era of peace brought about by Hammurabi (1792-1750 b.c.), the great advances in algebra and geometry took place. In Babylon under the Kassite kings, astronomy and astrology were the foremost pursuits. The table texts are prob. a sub-class of the Sumero-Babylonian “listenwissenschaft” or “catalogue-science” by which the vast lexical lists were assembled. Under the Kassites parallel columns of Sumer. terms and phrases with their Babylonian equivalents were executed and long series of these running on to twenty or more tablets have been discovered. The astrological omen series Enūma Anu Enlil was collected at this time. The same method was already in use with tables of numbers. Simple tablets exist in which a column of figures are followed by their reciprocals and other more complex operations as in the tablet, YBC 7354-70g (0. Neugebauer and A. Sachs [1945], p. 17). The text is devised in the sexagesimal system but given below in the modern decimal notation:

Column A lists a set of numbers in the form of sexagesimal fractions, in this case interest charges on loans, column B gives the reciprocals, column C gives B x 2 and the factor of 2 constant is given in column D. With such a table a business clerk or scribe could easily manipulate any set of figures for the appropriate interest rate. Tables such as this one show series of numbers a and a2, a, a2—b2, ə and such similar expressions. Recent investigations have located fermat problems in Babylonian mathematics and even formuli for the length and area of figures such that one of the following expressions hold: ax2+bx=c, ax2-bx=c and the two derived equations bx-ax2=c and bx=ax2. (E. M. Bruins, “Fermat Problems in Babylonian Mathematics,” JANUS LIII, 3 [1966/2], pp. 194-211.) Under the later Babylonian and Assyrian rulers astronomical lists again flourished, and great strides were made in the accuracy with which observations of the heliacal rising of fixed stars, ephemerides of the planets and eclipses of the sun and moon were recorded. When Babylon fell to Cyrus of Persia in 539 b.c. the tradition of Babylonian mathematics passed to Iran. A final flowering of astronomical observation, simple algebra and the tables for lunar, planetary and solar cycles took place after the conquest of Mesopotamia by the Greeks in 333 b.c. The last vestige of this great mathematical tradition was passed on in the Seleucid and Arsacid era and died out in the Medieval period. However, two further aspects of Akkadian, Assyrian and Babylonian numbers were important. The cardinal and ordinal terms for the numbers derived from E. Sem. cuneiform influenced those terms in Ugaritic and Hebrew (p. 455). In addition, the Mesopotamian scribes became so familiar with handling numbers that they often used numerical signs to signify certain common words in cuneiform texts, e.g. 15 = akkad. ẖamiššer “right,” and Sumer. MIN.ES̆ for sexagesimal 2, 30, = decimal 150 Akkad. ẖamšame. (cf. the author’s “An Assyrian Physician’s vade mecum,” Clio Medica, 197ff. [1970]).

Ugaritic and Canaanite numbers.

Posthellenic Semite numbers.

There is considerably more evidence of the numerical systems used by the Sem. peoples after the rise of Greece and the establishment of the Gr. colonies in Egypt, Magna Graece, and along the Black Sea coast. Of special importance are the Syrian Palmyrene and S Arabian Nabataean systems. These two numerations developed on the basis of the late Egyp. hieratic script, a separate sign for 5 was introduced. The configuration is similar to certain styles of ’ayin, remotely like Eng. “Y.” The use follows that of the Canaanite and Phoenicians with the exception that the extra ones are set to the left of the five; for example, IY = 6, IIY = 7, IIIY = 8, and IIIIY = 9. The ten is similar to the hieratic Egyp. sign for d ’d(w), a long bar with a sharp down stroke and frequent tight curl. The symbol for one hundred is the Eng. reversed “P.” Symbols for numbers of larger magnitude utilize a hundred determinative with the integer indicator set to the left. The peculiar duplication of the twenty sign is retained up through 70; the largest value below 99 remains to be discovered. None of these possible consonantal signs are remotely similar to the initial consonants of the words for these numbers when spelled out (p. 455). There is little doubt that these symbols are numerical signs. In the same manner as the Phoen. writing system, which was an extensive and very simplified syllabary, the letter system was later modified to serve as an ordered phonetic alphabet under the Punic culture, and was later utilized to indicate numbers similar to the Gr. All evidence points to the Gr. development of this system and its parallel which later was accepted by the Semites. Since this system was illegible to the Greeks and other Indo-Europeans, it was an effective argot among Sem. traders and may be one of the tricks of the Carthaginian merchants spoofed by Plautus in his early Lat. comedies.

OT Biblical numbers

The form of OT numbers.

In the OT MSS now available, all the numbers are spelled out phonetically, but there is no reason to assume that a more direct numeral system was not available. Mason’s marks and what may be simple tallies have been excavated in Israel. The earliest evidence of epigraphic inscrs. yields little in the way of numbers, nothing as general or well distributed as the Aram. and later Sem. inscrs. The few numbers that appear in the earliest Palestinian inscrs. the Gezer Calendar, the Moabite Stone, the Ostraca from Samaria, and the Siloam Inscription of Hezekiah have the numbers either of small magnitude 1 through 3 that they are hardly useful as evidence, or they are written out phonetically. There is no doubt that the modified Egyp. system in use among the Semites of the rest of Asia Minor and the Eastern Mediterranean was also in use among the Jews. The fact that many of the numbers recorded in the earliest autographs of the text were written in this system and later transliterated into phonetic spellings, accounts for many of the primitive textual errors incorporated in the transmission of numbers. The restatement of purely numerical signs in alphabetic numbers, where the consecutive order of the letters of the writing system are not equal to the consecutive order of integers is known. The chief difficulty with such a system is that no associated operations can be defined. Another source of errors is found in transmission of signs of the sexagesimal or vigesimal system into decimal notations. In the extant MSS of the OT and the various VSS, the numbers are spelled out phonetically. The Massoretes pointed such terms as though they were regular nouns and adjectives, and consequently they completely altered any original differentiation of form that may have existed. However, there exist many problems concerning the base and operational procedure utilized for certain notations as in Ug aritic which seem to follow Hieroglyphic Hitt. (Cf. C. H. Gordon, Ugaritic Textbook [1965], par. 7.1, 2.)

Mathematical terms and operations.

The numeral 2 = שְׁנַ֫יִם, H9109, senayim, cardinal, usually dual in all Sem. languages. It is cognate to Ugaritic tnm, and Akkad. šena/šina, and Egyp. śnw(y), and occurs 768 times in the OT. These are related to terms for “repetition,” “succession” and the like. The ordinal, שֵׁנִי, H9108, senīy, occurs 157 times in the OT. Usually it is tr. “second,” “second series” (Gen 22:15, et al.).

The numeral 3 = שָׁלֹשׁ, H8993, sālos, variants שָׁלוֹשׁ, sālōws, שְׁלֹשָׁה, selosāh, cognate to Ugaritic tlt, which occurs only with feminine suffix -t, Akkad. šalāšum, Phoen. as transcribed into Lat. salus. It occurs 430 times in the OT. The ordinal שְׁלִישִׁי, H8958, selīysīy occurs 105 times in the OT. A variation of the term was apparently a military officer of the Hittites (Exod 14:7, et al.).

The numeral 4 = אַרְבַּע, H752, ’arbā'ā, cognate to Ugaritic rb’(t), occurs only in the feminine. However, all Ugaritic numbers from 2 through 10 are used in feminine forms with nouns of both genders. Cognate to Akkad. erbū(m), is Phoen. ’rb’, but not to Egyp. It occurs approximately 250 times in the OT (Gen 2:10, et al.). The ordinal רְבִיעִי, H8055, rebāiyīy, cognate to Akkad. rebû, occurs less than 75 times in the OT (Gen 1:19, et al.).

The numeral 5 = חָמֵשׁ, H2822, ḥāmēs, cognate to Ugaritic ḥms, and Akkad. ḥamšum, occurs 340 times in the OT. The ordinal חֲמִישִׁי, H2797, ḥamīysīy, variant חֲמִשִׁ֔י, ḥamisīy, cognate to Akkad. ḥamšu, occurs 42 times in the OT.

The numeral 6 = שֵׁשׁ, H9252, sēs, cognate to Ugaritic tt, and Akkad. šiššum/šeššum remotely cognate to Ger. sechs, Eng. six, occurs 289 times in the OT. The ordinal שִׁשִּׁי, H9261, sissīy, cognate to Akkad. šeššu, occurs 23 times in the OT (Gen 1:31 et al.).

The numeral 7 = שֶׁ֫בַע, H8679, seba’, cognate to Ugaritic šb’, and Akkad. sebûm, Phoen. šiba’, occurs 390 times in the OT. The ordinal שְׁבִיעִי, H8668, sebīy’īy, occurs 95 times in the OT (Gen 2:2, et al. For some reason it occurs in blocks, and is particularly applied to the seventh day of rest.)

The numeral 8 שְׁמֹנֶה, H9046, semonēh, variant שְׁמוֹנֶה, semōwnēh, cognate to Ugaritic ṯ m n, and Akkad. šamānûm, more commonly samānûm, occurs in the OT 109 times. The ordinal שְׁמִינִי, H9029, semīynīy, cognate to Akkad. samānum, occurs only 31 times; it often is used as the name of a eight-stringed instrument (Ps 61:1, et al.). In spite of numerous exegetical attempts to understand this term in some apocalyptic fashion, its usage in the texts does not warrant such renderings.

The numeral 9 = תֵּ֫שַׁע, H9596, tesa’, cognate to Ugaritic tš', and Akkad. tišûm, occurs less than 30 times in the OT; it appears more often in the form of “nine hundred” (Gen 5:5 et al.). The ordinal תְּשִׁיעִי, H9595, tesīy’īy, occurs only seven times in the OT (Lev 25:22, et al.).

Numbers 11 through 19 are formed by placing the unit number first and the form עֶשְׂרֵה, ’ēsereh, follows. When the form appears by itself with one understood, it means eleven. It occurs some 144 times in the OT (Gen 14:4, et al.). Such forms however are quite rare except for regnal year dates. The Sem. languages have no such forms as Ger. elf, zwölf, Dutch elf, twaalf; Eng. eleven, twelve. This need was supplied in Yiddish by utilizing Ger. elf, tsvelf.

The numbers 20, 30, 40, 50, 60, 70, 80, and 90 are formed by using the dual of 10 for 20; the plurals of 3 through 9 are used for numbers 30 through 90. No separate ordinals of these numbers are extant. The term for hundred, מֵאָה, H4395, me’āh, is cognate to Ugaritic m’t, Akkad. me’u, me’atu; these terms prob. meant simply “crowd,” “large group.” (This frequently has been advanced as an explanation for the long ages assigned various antediluvian figures in Gen 5, et al.) Since the Akkadian-Assyrian-Babylonian system adopted the sexagesimal system of Sumer, the numbers from 30 to 90 and multiples of them are not derived from the same sources as the other Sem. languages. For one hundred, the integer in its positional notation, and on occasion the dual of me’āh is used for two hundred. The term occurs 580 times in the OT, usually in combination with another figure (Gen 5:4, et al.).

Enumerations.

By far the highest frequency of numerical data given in the OT are enumerations either of age or census. These two areas produce some of the most difficult textual problems that arise.

Common enumerations, ages.

The ages assigned to the characters in the OT are all in accord with common experience except those in the antediluvian period (Gen 5-8). The ages given for the line of Adam to Noah are all of great length from Enoch 365 years to Methuselah 969 years. A great deal has been written about this series of ages. The conservative opinion has traditionally been that the conditions of the cosmos before the Flood were such that such longevity was not merely plausible but commonplace. Two factors militate against this simple solution. 1. The longevities play no specific part in the Biblical theology of the scriptural revelation. 2. Other ancient documents which describe vast antediluvian antiquity include such vast ages, e.g. The Sumerian King List. A careful analysis of the ages demonstrates that they are all figures of two sorts: multiples of five (5n) or multiples of five plus seven or two times seven (5n) + (7 x 2). For example, the age of Seth, 912 = 5 x 181 + 7. Since every one of the ten ages quoted in the passage is reducible in this fashion, along with many other ages and chronological totals of the patriarchs, the scheme cannot possibly be accidental. The finality of Lamech’s career is enforced and reenforced by the manner in which his age is stated: “And all the days of Lamech were seven and seventy and seven hundred, and he died.” Unfortunately not one of the major Eng. VSS, KJV, RSV or JPS rightly tr. the MT. The basic structure of the multiples of five in the sexagesimal system plus the perfective seven which is repeated throughout the creation narrative is repeatedly maintained. Furthermore, this pattern persistently appears throughout the Prophets, the Talmud, and the Midrash where the numbers, 600,000; 60,000; 30,000; 12,000; 6,000; 3,000; 1,200; 600; 300 and 120 are commonplace. (U. Cassuto, A Commentary on the Book of Genesis, Eng. tr. [1961], 249-268.) Numerous attempts to reinterpret the term “years” used in the ages of the patriarchs have been unsuccessful; the OT simply does not use the term in any other sense than the solar year. The appearance of the large sexagesimal numbers in the early chapters of Genesis prove beyond a shadow of a doubt the antiquity of the text or literary tradition utilized by Moses.

Large number enumerations, census.

The most consistently confused material in the OT from MS to MS and VS to VS is the record of large census figures. Undoubtedly the source of the difficulty can be traced to several changes in notational systems before and during the transmission of the text. Some passages of noteworthy problems are: 1 Chronicles 19:18 “7,000 chariots,” to 2 Samuel 10:18 “700,” 1 Kings 4:26 “40,000 stalls,” to 2 Chronicles 9:25 “4,000,” 2 Kings 24:8, “Jehoiachin was eighteen years old,” to 2 Chronicles 36:9 “Jehoiachin was eight years old.” Of special importance is the fact that 2 or its multiples often are replaced by 1, 10 etc. or 3 and its multiples. In many cases such problems can be explained by careful analysis without resort to specious and innovative emendations. Some of the difficulties like those already cited can be understood only as primitive textual errors in one or another family of MSS (n.b. the important study: J. W. Wenham, “Large Numbers in the Old Testament,” Tyndale Bulletin 18 [1967], 19-53).

Rhetorical numbers.

Since numbers were for the most part spelled out and used as words in lit., all Sem. languages developed artistic canons of usage for number terms. The Ugaritic texts, as well as the Akkadian, frequently have a device for building to literary completion with set series of numbers.

Climactic, idiomatic and prose uses.

All ancient Sem. lits. rely upon a series of numbers either syndetically or asyndetically to bring about progression and anticipation in narratives. The standard form is 1, 2 and 3, 4 and 5, 6, then on 7 a change or finale occurs. (Epic of Gilgamesh, XI, 11. 48-76; 140-145; 225-228.) The creation-law order of Genesis is revealed in precisely this fashion: Genesis 2:2; 8:4; Exodus 16:27, et al. frequently. (M. G. Kline, “Because It Had Not Rained,” WTJ XX [May 1958], 146-157.) The idiomatic use of numbers involves the inclusion of a figure in a literary passage where an indefinite quantity and not necessarily the quantity of the figure is meant. For a few indeterminate situations, the OT uses 3 (2 Kings 9:32; Isa 17:6; Amos 4:8, et al.) also for indeterminate large numbers 40 is used, 40 years being the length of a generation (Exod 16:35; Deut 34:7 et al.). In this sense, it must be recognized that ancient societies did not have the passion for objective exact statistics that marks modern man. Often in Persian, Greek and other lits. numbers such as 40 are used simply as synonyms for “many,” “moderate crowd” and other similar expressions.

Poetic series of numbers, X and X + 1.

Symbolic and mystical numbers.

Unfortunately the frequent use of notions of symbolism applied to the Biblical numbers have resulted in little less than soothsaying. The result has been used to reinforce the extreme opposite position, specifically that no mystical use of numbers is anywhere indicated in the text. This is equally false. There is no doubt a proper sequence of numbers representing the creation order 7, the ritual 3, and the unique 1. Larger numbers such as 40, 80, 120, and 1,000 also are used with significance. (For opposing opinions on this difficult question see the two standard works: E. W. Bullinger, Number In Scripture [1913] and O. T. Allis, Bible Numerics [1961].)

Numerological explanations of the OT.

Most of these types of exegetical systems have been based upon the assumption that the later Jewish system of replacing each number 1 - 9, 10 - 90, with the sequential letters of the Heb. alphabet was practiced throughout the Biblical period. Thus, any term in the MT can be deciphered into a code of numbers. For example, the consonantal text of Genesis 1:1 begins with br’syt which is deciphered in terms of numerals as b = 2, r = 200, ’ = 1, s = 300, y = 10 and t = 400, thus the first word of Genesis equals to the total of these numbers of 913 which is then interpreted mystically. This sort of magical nonsense arose during the Hel. age and was applied to many ancient writings under the term “gematria,” a corruption from Semiticized “geometria.” (Cf. summary of this method and its historical development in: J. J. Davis, Biblical Numerology [1968], 125-156.) Such gnostic exegesis contradicts the clear Biblical principle stated in 2 Peter 1:20, “that no prophecy of scripture is a matter of one’s own interpretation.”

NT Biblical numbers

As a whole the NT contains substantially less in the way of numerical material than the OT. In the main they are simple counts of crowds or groups or mercantile figures taken from the world of commerce for purposes of illustration.

The state of Greek numbers and mathematics.

From the early days of the Ionian philosophers the Gr. world considered numbers as worthy of the highest and most sustained study. In the age of Plato and Aristotle (c. 300 b.c.) the great mathematical insights of Gr. civilization were brought forth. The state of this art can be ascertained from the works listed below in the Bibliography.

Hellenistic numerology.

The roots of numerological manipulation of numbers among the Greeks certainly dates from Pythagoras (c. 582- c. 500 b.c.), whose mystic brotherhood of disciples eroded whatever objective scientific value their teacher’s labors may have held and plunged his name and teachings into a veritable swamp of magic and ritual. After Alexander’s conquests (c. 322 b.c.), this residue settled upon the ancient Sem. states of the Near E. Although frequently utilizing the Gr. notational system which still had no operational significance, the Sem. peoples seem to have retained their own simple mercantile art of arithmetic. The impact of Plotinus and Neo-Platonism energized this mystic trend to a point that gematria was practiced widely among various schools of Hel. thought. Not the least important being the Gnostic from which it passed into the post-Nicene church and the Medieval Era.

Form, terms and operations of NT numbers.

Enumerations.

The only difficulty with the NT enumerations arise in contexts where the NT MS differs from the MT as in several cases in Stephen’s defense (Acts 6:8-7:60). The rest of the enumerations of the NT are either direct quotes from the OT (1 Cor 10:8 quoted from Num 25:1-18), or estimates of crowds which can be trusted implicitly as accurate. By and large the numerations of the OT although written out in Gr. follow the Sem. system.

Rhetorical, symbolic and mystical numbers.

The same sets of figures, 3, 5, 7, 12 which are given symbolic meaning in the OT are used in the NT. The reason for this is the scrupulous attention given in the NT to every aspect of Christ’s Messianic fulfillment, e.g. the twelve apostles as a reinstitution of the sons of Jacob as heads of the twelve tribes of Israel. The only purely symbolic number is the “thousand” applied to lengths of time in the apocalyptic passages. The only purely mystical, in the sense of mysterious, number is the epithet of the antichrist or his agent in Revelation 13:18. The various people identified over the centuries by this number have usually been determined by gematria and the permutation of the resultant numbers. Over the centuries of such speculations Nero has been the most probable choice.

The Biblical theology of numbers

The scriptural revelation is a unified whole, every aspect is concordant in the structure and each word significant. The numbers are no less so. The transcendent monotheism of Jehovah is revealed with 1, the notion of love with 2, the “mystery of the trinity” with 3, and so on. Since they play such a basic role in the enscripturated word, the numbers of the Bible must be taken seriously, and carefully compared from text to text.

Bibliography

L. L. Conant, The Number Concept (1896); H. G. Zeuthen, Geschichte der Mathematik im Altertum und Mittelalter (1896); M. Cantor, Vorlesungen über Geschichte der Mathematik, Vol I (1900); F. X. Kugler, Sternkunst und Sterndienst in Babel, 2 vols. (1907-1935); G. Loria, Le scienze esatte nell’ antica Grecia (1914); K. Sethe, Von Zahlen und Zahlworten bei den alten Ägypten (1916); L. E. Dickson, History of the Theory of Numbers, Vol I. (1919); T. Heath, A History of Greek Mathematics, 2 vols. (1921); F. Cajori, A History of Mathematics (1926); O. Neugebauer, Die Grundlagen der ägyptischen Bruchrechnung (1926); ed. A. B. Chace, L. Bull, H. P. Manning and R. C. Archibald, The Rhind Mathematical Papyrus, 2 vols. (1927-1929); F. Cajori, A History of Mathematical Notations, Vol I (1928); W. W. Struve, “Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau,” Quellen und Studien zur Geschichte der Mathematik (1930); O. Neugebauer, Mathematische Keilschrifttexte, 2 vols. (1935); A. Heller, Biblische Zahlensymbolik (1936); V. Hopper, Medieval Number Symbolism (1938); F. Thoreau-Dangin, Textes Mathématiques Babyloniens (1938); F. Thoreau-Dangin, “Sketch of a History of the Sexagesimal System,” Osiris, vol. 7 (1939), 95-141; C. B. Boyer, “Fundamental Steps in the Development of Numeration,” Isis, vol. 35 (1944), 153-168; E. T. Bell, Development of Mathematics (1945); E. T. Bell, Numerology (1945); S. Gandz, “Complementary Fractions in Bible and Talmud,” Louis Ginsberg Jubilee Volume (1945), 143-157; O. Neugebauer and A. Sachs, Mathematical Cuneiform Texts (1945); E. M. Bruins, Fontes Mathessos (1953); M. Kline, Mathematics in West ern Culture (1953); J. R. Newman, “The Rhind Papyrus,” The World of Mathematics, Vol. I (1956), 169-178; E. J. Dijksterhuis, Archimedes (1957); O. Neugebauer, The Exact Science in Antiquity (1957); T. Dantzig, Number, The Language of Science (1959); K. Vogel, Vorgriechische Mathematik (1959); E. M. Bruins and M. Rutten, Textes mathématiques de Suse (1961); E. J. Dijksterhuis, The Mechanization of the World Picture (1961); B. L. van der Waerden, Science Awakening (1961); M. Lidzbarski, Handbuch der Nordsemitischen Epigraphik, 2 vols. (1962 reprint); F. Lasserre, The Birth of Mathematics in the Age of Plato (1964); ed. O. Becker, Zur Geschichte der Griechischen Mathematik (1965); C. B. Boyer, A History of Mathematics (1968); J. J. Davis, Biblical Numerology (1968). This work contains citations to all the major periodical literature on numerology.; G. de Santillana, Reflections on Men and Ideas (1968), 82-119, 190-201, 219-230.

International Standard Bible Encyclopedia (1915)

num’-ber:

I. NUMBER AND ARITHMETIC

II. NOTATION OF NUMBERS

1. By Words

2. By Signs

3. By Letters

III. NUMBERS IN OLD TESTAMENT HISTORY

IV. ROUND NUMBERS

V. SIGNIFICANT NUMBERS

1. Seven and Its Multiples

(1) Ritual Use of Seven

(2) Historical Use of Seven

(3) Didactic or Literary Use of Seven

(4) Apocalyptic Use of Seven

2. The Number Three

3. The Number Four

4. The Number Ten

5. The Number Twelve

6. Other Significant Numbers

VI. GEMATRIA

LITERATURE

I. Number and Arithmetic.

II. Notation of Numbers.

1. By Words:

No special signs for the expression of numbers in writing can be proved to have been in use among the Hebrews before the exile. The Siloam Inscription, which is probably the oldest specimen of Hebrew writing extant (with the exception of the ostraca of Samaria, and perhaps a seal or two and the obscure Gezer tablet), has the numbers written in full. The words used there for 3,200, 1,000 are written as words without any abbreviation. The earlier text of the Moabite Stone which practically illustrates Hebrew usage has the numbers 30, 40, 50, 100, 200, 7,000 written out in the same way.

2. By Signs:

After the exile some of the Jews at any rate employed signs such as were current among the Egyptians, the Arameans, and the Phoenicians--an upright line for 1, two such lines for 2, three for 3, and so on, and special signs for 10, 20, 100. It had been conjectured that these or similar signs were known to the Jews, but actual proof was not forthcoming until the discovery of Jewish papyri at Assuan and Elephantine in 1904 and 1907. In these texts, ranging from 494 to circa 400 BC, the dates are stated, not in words, but in figures of the kind described. We have therefore clear evidence that numerical signs were used by members of a Jewish colony in Upper Egypt in the 5th century BC. Now, as the existence of this colony can be traced before 525 BC, it is probable that they used this method of notation also in the preceding century. Conjecture indeed may go as far as its beginning, for it is known that there were Jews in Pathros, that is Upper Egypt, in the last days of Jeremiah (Jer 44:1,15). Some of the first Jewish settlers in Elephantine may have known the prophet and some of them may have come from Jerusalem, bringing these signs with them. At present, however, that is pure hypothesis.

3. By Letters:

In the notation of the chapters and verses of the Hebrew Bible and in the expression of dates in Hebrew books the consonants of the Hebrew alphabet are employed for figures, i.e. the first ten for 1-10, combinations of these for 11-19, the following eight for 20-90, and the remainder for 100, 200, 300, 400. The letters of the Greek alphabet were used in the same way. The antiquity of this kind of numerical notation cannot at present be ascertained. It is found on Jewish coins which have been dated in the reign of the Maccabean Simon (143-135 BC), but some scholars refer them to a much later period. All students of the Talmud are familiar with this way of numbering the pages, or rather the leaves, but its use there is no proof of early date. The numerical use of the Greek letters can be abundantly illustrated. It is met with in many Greek papyri, some of them from the 3rd century BC (Hibeh Papyri, numbers 40-43, etc.); on several coins of Herod the Great, and in some manuscripts of the New Testament, for instance, a papyrus fragment of Mt (Oxyrhynchus Pap., 2) where 14 is three times represented by iota-delta (I-D) with a line above the letters, and some codices of Re 13:18 where 666 is given by the three letters "chi" "xi" "vau" (or digaroma). It is possible that two of these methods may have been employed side by side in some cases, as in the Punic Sacrificial Tablet of Marseilles, where (l. 6) 150 is expressed first in words, and then by figures.

III. Numbers in Old Testament History.

IV. Round Numbers.

Other round numbers are:

(1) some of the higher numbers;

(2) several numerical phrases.

V. Significant Numbers.

Numerical symbolism, that is, the use of numbers not merely, if at all, with their literal numerical value, or as round numbers, but with symbolic significance, sacred or otherwise, was widespread in the ancient East, especially in Babylonia and regions more or less influenced by Babylonian culture which, to a certain extent, included Canaan. It must also be remembered that the ancestors of the Israelites are said to have been of Babylonian origin and may therefore have transmitted to their descendants the germs at least of numerical symbolism as developed in Babylonia in the age of Hammurabi. Be that as it may, the presence of this use of numbers in the Bible, and that on a large scale, cannot reasonably be doubted, although some writers have gone too far in their speculations on the subject. The numbers which are unmistakably used with more or less symbolic meaning are 7 and its multiples, and 3, 4, 10 and 12.

1. Seven and Its Multiples:

By far the most prominent of these is the number 7, which is referred to in one way or another in nearly 600 passages in the Bible, as well as in many passages in the Apocrypha and the Pseudepigrapha, and later Jewish literature. Of course the number has its usual numerical force in many of these places, but even there not seldom with a glance at its symbolic significance. For the determination of the latter we are not assigned to conjecture. There is clear evidence in the cuneiform texts, which are our earliest authorities, that the Babylonians regarded 7 as the number of totality, of completeness. The Sumerians, from whom the Semitic Babylonians seem to have borrowed the idea, equated 7 and "all." The 7-storied towers of Babylonia represented the universe. Seven was the expression of the highest power, the greatest conceivable fullness of force, and therefore was early pressed into the service of religion. It is found in reference to ritual in the age of Gudea, that is perhaps about the middle of the 3rd millennium BC. "Seven gods" at the end of an enumeration meant "all the gods" (for these facts and the cuneiform evidence compare Hehn, Siebenzahl und Sabbath bei den Babyloniern und im Altes Testament, 4 ff). How 7 came to be used in this way can only be glanced at here. The view connecting it with the gods of the 7 planets, which used to be in great favor and still has its advocates, seems to lack ancient proof. Hehn (op. cit., 44 ff) has shown that the number acquired its symbolic meaning long before the earliest time for which that reference can be demonstrated. As this sacred or symbolic use of 7 was not peculiar to the Babylonians and their teachers and neighbors, but was more or less known also in India and China, in classical lands, and among the Celts and the Germans, it probably originated in some fact of common observation, perhaps in the four lunar phases each of which comprises 7 days and a fraction. Conspicuous groups of stars may have helped to deepen the impression, and the fact that 7 is made up of two significant numbers, each, as will be shown, also suggestive of completeness--3 and 4--may have been early noticed and taken into account. The Biblical use of 7 may be conveniently considered under 4 heads:

(1) ritual use;

(2) historical use;

(3) didactic or literary use;

(4) apocalyptic use.

(1) Ritual Use of Seven.

(2) Historical Use of Seven.

(3) Didactic or Literary Use of Seven.

(4) Apocalyptic Use of Seven.

The significance of 7 extends to its multiples. Fourteen, or twice 7, is possibly symbolic in some cases. The stress laid in the Old Testament on the 14th of the month as the day of the Passover (Ex 12:6 and 16 other places), and the regulation that 14 lambs were to be offered on each of the 7 days of the Feast of Tabernacles (Nu 29:13,15) hint at design in the selection of the number, especially in view of the fact that 7 and 7 occur repeatedly in cuneiform literature--in magical and liturgical texts, and in the formula so often used in the Am Tab: "7 and 7 times at the feet of the king my lord .... I prostrate myself." The arrangement of the generations from Abraham to Christ in three groups of 14 each (Mt 1:17) is probably intentional, so far as the number in each group is concerned. It is doubtful whether the number has any symbolic force in Ac 27:27; 2Co 12:2; Ga 2:1. Of course it must be remembered that both the Hebrew and Greek words for 14 (’arba’ah asar; dekatessares) suggest that it is made up of 10 and 4, but constant use of 7 in the sense above defined will have influenced the application of its double, at least in some cases.

Forty-nine, or 7 X 7, occurs in two regulations of the Law. The second of the three great festivals took place on the 50th day after one of the days of unleavened bread (Le 23:15 ), that is, after an interval of 7 X 7 days; and two years of Jubilee were separated by 7 X 7 years (Le 25:8 ). The combination is met with also in one of the so-called Penitential Psalms of Babylonia: "Although my sins are 7 times 7, forgive me my sins."

Seventy and seven, or 77, a combination found in the words of Lamech (Ge 4:24); the number of the princes and elders of Succoth (Jud 8:14); and the number of lambs in a memorable sacrifice (Ezr 8:35), would appeal in the same way to the oriental fancy.

The product of seven and seventy (Greek hebdomekontakis hepta) is met with once in the New Testament (Mt 18:22), and in the Septuagint of the above-quoted Ge 4:24. Moulton, however (Grammar of Greek New Testament Prolegomena, 98), renders in both passages 70 plus 7; contra, Allen, "Mt," ICC, 199. The number is clearly a forceful equivalent of "always."

Seven thousand in 1Ki 19:18 parallel Ro 11:4 may be a round number chosen on account of its embodiment of the number 7. In the Moabite Stone the number of Israelites slain at the capture of the city of Nebo by the Moabites is reckoned at 7,000.

The half of seven seems sometimes to have been regarded as significant. In Da 7:25; 9:27; 12:7; Lu 4:25 parallel 5:17; Re 11:2; 13:5 a period of distress is calculated at 3 1/2 years, that is, half the period of sacred completeness.

2. The Number Three:

The number three seems early to have attracted attention as the number in which beginning, middle and end are most distinctly marked, and to have been therefore regarded as symbolic of a complete and ordered whole. Abundant illustration of its use in this way in Babylonian theology, ritual and magic is given from the cuneiform texts by Hehn (op. cit., 63 ff), and the hundreds of passages in the Bible in which the number occurs include many where this special significance either lies on the surface or not far beneath it. This is owing in some degree perhaps to Babylonian influence, but will have been largely due to independent observation of common phenomena--the arithmetical fact mentioned above and familiar trios, such as heaven, earth, and sea (or "the abyss"); morning, noon and night; right, middle, and left, etc. In other words, 3 readily suggested completeness, and was often used with a glance at that meaning in daily life and daily speech. Only a selection from the great mass of Biblical examples can be given here.

3. The Number Four:

4. The Number Ten:

5. The Number Twelve:

6. Other Significant Numbers:

VI. Gematria.

(GemaTriya’). A peculiar application of numbers which was in great favor with the later Jews and some of the early Christians and is not absolutely unknown to the Bible, is Gematria, that is the use of the letters of a word so as by means of their combined numerical value to express a name, or a witty association of ideas. The term is usually explained as an adaptation of the Greek word geometria, that is, "geometry," but Dalman (Worterbuch, under the word) connects it in this application of it with grammateia. There is only one clear example in Scripture, the number of the beast which is the number of a man, six hundred sixty and six (Re 13:18). If, as most scholars are inclined to believe, a name is intended, the numerical value of the letters composing which adds up to 666, and if it is assumed that the writer thought in Hebrew or Aramaic. Nero Caesar written with the consonants nun (n) equals 50, resh (r) equals 200, waw (w) equals 6, nun (n) equals 50, qoph (q) equals 100, camekh (c) equals 60, resh (r) equals 200: total equals 666, seems to be the best solution. Perhaps the idea suggested by Dr. Milligan that the 3-fold use of 6 which just falls short of 7, the number of sacred completeness, and is therefore a note of imperfection, may have been also in the writer’s mind. Some modern scholars find a second instance in Ge 14:14 and 15:2. As the numerical value of the consonants which compose Eliezer in Hebrew add up to 318, it has been maintained that the number is not historical, but has been fancifully constructed by means of gematria out of the name. This strange idea is not new, for it is found in the Midrash on Ge 43 in the name of a rabbi who lived circa 200 AD, but its antiquity is its greatest merit.

LITERATURE.

In addition to other books referred to in the course of the article: Hehn, Siebenzahl und Sabbath bei den Babyloniern und im Altes Testament; Konig, Stilistik, Rhetorik, Poetik, etc., 51-57, and the same writer’s article "Number" in HDB; Sir J. Hawkins,. Horae Synopticae2, 163-67; Wiener, Essays in Pentateuchal Criticism, 155-69; "Number" in HDB (1-vol); EB; Jewish Encyclopedia;Smith, DB; "Numbers" in DCG; "Zahlen" in the Dicts. of Wiener, Riehm2, Guthe; "Zahlen" and "Sieben" in RE3.

William Taylor Smith